equilibrium equation, Magnetostriction, Anders Lundgren, magnetoelastic, mode, displacement, table top, weight functions, gravitational force density, boundary conditions, Kungl Tekniska Hogskolan, basis function, measurement system, laser interferometer, magnetization, mathematical modelling, measurement, frequency direction, mass density, kinetic energy, longitudinal wave, shear modulus, elastic modulus, transmission, magnetostrictive, material properties, energy density, stress tensor, transverse direction, weight function
Content:
On measurement and modelling of 2D magnetization and magnetostriction of SiFe sheets Anders Lundgren Royal Institute of Technology
electric power Engineering Stockholm 1999
Anders Lundgren On measurement and modelling of 2D magnetization and magnetostriction of SiFe sheets
TRITAEEA9901 ISSN 11001593 Department of Electric Power Engineering Royal Institute of Technology SE100 44 Stockholm SWEDEN
On measurement and modelling of 2D magnetization and magnetostriction of SiFe sheets Anders Lundgren Royal Institute of Technology Electric Power Engineering Stockholm 1999
Akademisk avhandling som med tillstand av Kungl Tekniska Hogskolan framlagges till o entlig granskning for avlaggande av teknisk doktorsexamen mandagen den 21 juni 1999 kl 14.00i Kollegiesalen, Administrationsbyggnaden, Kungl Tekniska Hogskolan, Valhallavagen 79, Stockholm. TRITAEEA9901 ISSN 11001593 c Anders Lundgren, 1999 KTH Reprocentral, Stockholm 1999
Abstract The development and technological aspects of a 2D magnetization and magnetostriction measurement setup are documented and described. Local magnetic intensity and ux density are measured with Rogowski and material encircling coils. Inplane strain is measured with a homodyne laser interferometer. Measured and processed timedomain signals, hysteresis plots and signature data such as loss are presented by an e cient and communicative interface. Measurements on quadratic silicon iron sheet samples are included. Material types tested on the setup are with nonoriented and oriented textures. Possible excitations include uniaxial alternating magnetic eld in the rolling and transverse directions between 10 and 300 Hz at least. Rotational excitations are possible at least for the nonoriented and conventional grainoriented types. The value of the setup lies in the possibility of using it for routine measurements on samples. The interplay between mathematical modelling and physical experimenting is described. Investigations by algebraic and numerical methods are done to nd a possible way to parameterize material behaviour and include this behaviour in nite element programs. On the basis of a proposed onedimensional nonlinear model, algorithms are devised to compute magnetostrictive responses to uniaxially alternating magnetic elds. An experimental FEM program to calculate strain elds from inhomogeneous magnetization is developed. Its use for investigation of sample behaviour during the operation of the setup is shown. The value of the proposed modelling methodology lies in the study of possibilities of lowering the production of magnetostrictive vibration in transformer, motor and generator cores. IEEE index terms: Magnetostriction, silicon steel, magnetic cores, strain, inter ferometry, magnetic anisotropy, magnetic elds, magnetic measurements, magnetoelasticity, nonlinear magnetics, power transformers, power distribution acoustic noise, nite element methods. TRITAEEA9901 ISSN 11001593
Acknowledgements I would like to thank the members of the reference committee, Jan Anger (ABB Transformers), Thomas Edstrom (ABB Corporate Research) and Birger Nilsson (ABB Corporate Research) and Elektra programme manager Sten Bergman (Elforsk AB) for their work in supporting this project. On the department side I owe thanks to the
Project Manager Goran Engdahl for energizing the project, applying for funding and proofreading. Head of department Roland Eriksson is thanked for employing me and for administering the nances and agreements. I especially wish to thank former research associate Anders Bergqvist for many stimulating discussions and collaborations. I thank Olle Brannvall, Gote Bergh and Yngve Eriksson for making parts to the experimental setup and transporting it. I send greetings to friendly department colleagues Eckart Nipp, Niklas Magnusson, Fredrik Stillesjo, Mats Kvarngren and Anders Helgesson. I nally express heartily thanks to my girlfriend Cecilia Haggmark for her encouragement, proofreading and general support. Anders Lundgren i
Contents
1 Introduction
1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation and goals . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Credits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4
literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4.1 Magnetic hysteresis models . . . . . . . . . . . . . . . . . . . 5
1.4.2 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.3 Stress dependence . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.4 Measurement methods . . . . . . . . . . . . . . . . . . . . . . 12
1.4.5 Numerics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4.6 Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Measurement system
17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Drawing and design system . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Data acquisition programs . . . . . . . . . . . . . . . . . . . . . . . . 21
ii
2.5 Magnetic circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Excitation frequency limits . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 Voltage or current sti ampli er . . . . . . . . . . . . . . . . . . . . 28 2.8 Bcoils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9 Calibration of the Hcoil . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.10 Measurement table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.10.1 Support placement . . . . . . . . . . . . . . . . . . . . . . . . 33 2.10.2 Optic component placement . . . . . . . . . . . . . . . . . . . 33 2.11 Vibration of material . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.12 Digital control issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.13 Strain measurement by interferometry . . . . . . . . . . . . . . . . . 36 2.14 Stress in uence, frame e ect . . . . . . . . . . . . . . . . . . . . . . . 37 2.15 Yoke design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.16 Magnetic sensor design . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.17 Temperature drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.18 Signal conditioning and Nyquist limit . . . . . . . . . . . . . . . . . 39 2.19 Signal bu ering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.20 Measurement coil misalignments . . . . . . . . . . . . . . . . . . . . 43 2.21 Using the measurement system . . . . . . . . . . . . . . . . . . . . . 43 2.21.1 Magnetic measurements . . . . . . . . . . . . . . . . . . . . . 43 2.21.2 Peak ux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.21.3 Measurement procedure . . . . . . . . . . . . . . . . . . . . . 44
3 Interferometer
46
iii
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Homodyne interferometry . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Heterodyne interferometry . . . . . . . . . . . . . . . . . . . . . . . . 49 3.4 Interferometer alignment . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Doppler e ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.6 Motion of measurement table . . . . . . . . . . . . . . . . . . . . . . 52 3.7 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.8 The acoustooptic modulator . . . . . . . . . . . . . . . . . . . . . . 60 3.9 Beam splitters and prisms . . . . . . . . . . . . . . . . . . . . . . . . 62 3.10 Interference lter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.11 Photodiode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.12 Demodulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.13 Interferometer type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.14 Re ector placements and properties . . . . . . . . . . . . . . . . . . 65
4 Strain analysis
68
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 De nitions of observables . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2.1 2D strain measurement analysis . . . . . . . . . . . . . . . . . 73
4.2.2 Deformation of volume elements . . . . . . . . . . . . . . . . 76
4.3 Stress and 3D elastic material relations . . . . . . . . . . . . . . . . . 77
4.4 2D elastic material modelling . . . . . . . . . . . . . . . . . . . . . . 80
4.4.1 Magnetostriction components and constitutive relations . . . 80
4.4.2 Elasticity and compliance matrices . . . . . . . . . . . . . . . 85
iv
4.5 Equations of equilibrium and motion . . . . . . . . . . . . . . . . . . 87 4.5.1 Force equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5.2 Torque equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 88 4.5.3 Equations of motion, coordinate types . . . . . . . . . . . . . 89 4.5.4 Translatory and rotatory equations of motion . . . . . . . . . 89 4.5.5 Body forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 Magnetic stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Models of magnetostriction
93
5.1 The interplay between mathematical modeling and physical experimenting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Continuum model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 Butter y loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.4 Ratedependency model . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.5 Simple 2D magnetostriction models . . . . . . . . . . . . . . . . . . . 98
5.6 Magnetoviscoelastic models . . . . . . . . . . . . . . . . . . . . . . . 98
5.6.1 Quasistatic linear case . . . . . . . . . . . . . . . . . . . . . . 99
5.6.2 Ratedependent linear case . . . . . . . . . . . . . . . . . . . 99
5.6.3 Ratedependent nonlinear case . . . . . . . . . . . . . . . . . 100
5.7 Model incorporation in plane stress calculations . . . . . . . . . . . . 101
5.7.1 Nonlinear dispersion . . . . . . . . . . . . . . . . . . . . . . . 103
5.8 Macroscopic magnetostrictive response . . . . . . . . . . . . . . . . . 103
5.9 Identi cation of parameters . . . . . . . . . . . . . . . . . . . . . . . 104
5.9.1 Magnetostrictive incompressibility . . . . . . . . . . . . . . . 104
v
5.10 Magnetoelastic shear modulus . . . . . . . . . . . . . . . . . . . . . . 106 5.11 Vector and tensor transformation . . . . . . . . . . . . . . . . . . . . 107 5.12 Magnetic stress alternatives . . . . . . . . . . . . . . . . . . . . . . . 108 5.13 Compliance transformation . . . . . . . . . . . . . . . . . . . . . . . 109 5.14 Piezomagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.15 Physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.16 Material structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.16.1 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 5.16.2 Transformer iron qualities . . . . . . . . . . . . . . . . . . . . 116 5.17 Micromagnetic cause of magnetostriction . . . . . . . . . . . . . . . . 116 5.18 Domains in soft magnetic materials . . . . . . . . . . . . . . . . . . . 117 5.19 Domain walls and magnetostriction . . . . . . . . . . . . . . . . . . . 119 5.20 Domain types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6 Magnetic nite element analysis
123
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.2 Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.3 General motivation and conditions for simulations with computer . . 124
6.4 2D magnetostatic nite element method . . . . . . . . . . . . . . . . 125
6.4.1 A linear isotropic scalar potential problem . . . . . . . . . . . 125
6.4.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4.3 Single triangle element . . . . . . . . . . . . . . . . . . . . . . 127
6.4.4 System of linear equations . . . . . . . . . . . . . . . . . . . . 128
6.4.5 Hollow cylinder test case . . . . . . . . . . . . . . . . . . . . . 129
vi
6.4.6 A nonlinear isotropic formalism . . . . . . . . . . . . . . . . . 130 6.5 3D isotropic formulation . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.6 3D anisotropic formulation . . . . . . . . . . . . . . . . . . . . . . . 134
7 Mechanical nite element analysis
136
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2 E ect of inhomogeneous magnetization . . . . . . . . . . . . . . . . . 136
7.3 Mechanical simulation method . . . . . . . . . . . . . . . . . . . . . 139
7.4 Results and interpretation . . . . . . . . . . . . . . . . . . . . . . . . 139
7.5 Strain eld calculation method . . . . . . . . . . . . . . . . . . . . . 142
7.5.1 Plane stress constitutive relation . . . . . . . . . . . . . . . . 142
7.5.2 Finite element method . . . . . . . . . . . . . . . . . . . . . . 142
7.6 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.6.1 Magnetic eld and force calculation . . . . . . . . . . . . . . 147
7.6.2 Bending formulation . . . . . . . . . . . . . . . . . . . . . . . 147
7.6.3 Extra details . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.6.4 Nonmagnetized case . . . . . . . . . . . . . . . . . . . . . . . 157
7.6.5 Rolling direction magnetization . . . . . . . . . . . . . . . . . 157
7.6.6 Transversal magnetization . . . . . . . . . . . . . . . . . . . . 158
7.6.7 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
8 Measurement and veri cation
163
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
8.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8.3
data processing and nonlinear model . . . . . . . . . . . . . . . . . . 164
vii
8.4 Frequency dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 169 8.5 2D model from measurements . . . . . . . . . . . . . . . . . . . . . . 169 8.6 Magnetization measurements . . . . . . . . . . . . . . . . . . . . . . 172
9 Conclusions and future work
175
9.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.1.1 Setup uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
9.1.2 Sample eld calculation . . . . . . . . . . . . . . . . . . . . . 176
9.1.3 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9.1.4 Magnetostriction harmonics . . . . . . . . . . . . . . . . . . . 176
9.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2.1 SST improvement . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2.2 Magnetoelastic FEM program development . . . . . . . . . . 177
9.2.3 Magnetostriction measurements . . . . . . . . . . . . . . . . . 178
10 List of symbols
179
11 List of units
185
A Design drawings
196
viii
List of Figures 2.1 Sample support table (not hatched) with yokes (hatched). See Fig. 3.1 for its placement in the setup. . . . . . . . . . . . . . . . . . . . . 24 2.2 Magnetic sensors, split sketch. . . . . . . . . . . . . . . . . . . . . . . 25 2.3 Block schematic of electric part of measurement system. . . . . . . . 26 2.4 The magnetic yoke con guration. Dimensions in mm. . . . . . . . . 26 2.5 One Hcoil wound from up to down around a nonmagnetic plate. Hall probe positions for calibration are marked with circles. Dimensions in mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1 Overview of interferometer . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2 Actual IFM setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 The ray in a 90 prism mirrored into a straight ray through a cube. 66 4.1 Relative displacement of length element . . . . . . . . . . . . . . . . 69 4.2 Interpretation of displacement gradient decomposition . . . . . . . . 70 4.3 Interpretation of relative displacement decomposition . . . . . . . . . 71 4.4 Normal strains and shear angle + . . . . . . . . . . . . . . . . . 72 4.5 Polar plot of 011(') and 012(') . . . . . . . . . . . . . . . . . . . . . 74 4.6 90 antisymmetry of shear strains. . . . . . . . . . . . . . . . . . . . 75 ix
4.7 180 symmetry of normal strains . . . . . . . . . . . . . . . . . . . . 75 4.8 Mohr's circle for normal and shear strain in the xy plane. The xy plane is perpendicular to a principal strain direction. 'p is the angle from the xdirection to the direction of the principal strain 1. . . . 76 4.9 Mohr's circles for a complete strain state, three planes perpendicular to each other and to principal directions. . . . . . . . . . . . . . . . . 77 4.10 Moment equilibrium on an area element . . . . . . . . . . . . . . . . 78 4.11 Normal elastic compliance as function of angle of uniaxial stress to rolling direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.12 Orthogonal elastic compliance as function of angle of uniaxial stress to rolling direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.13 Shear elastic compliance coe cients as functions of angle of uniaxial stress to rolling direction. . . . . . . . . . . . . . . . . . . . . . . . . 83 4.14 Uniaxial stress applied obliquely to a texture. Shows rotation of the principal strain system 1 2 compared to the principal stress system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.15 Left: Force on element are from stresses and body force fb. Right: Torque on element are from shear stresses and body torque Tb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.1
Butter vs. By.
y .
loops of .....
negative .....
valued ....
.Mx .
vs. ..
Bx ..
and positive .......
valued ....
. My.
.
95
5.2 Normal magnetoelastic compliance as function of angle of magnetic stress to rolling direction. . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3 Orthogonal (to magnetic stress) magnetoelastic compliance as function of angle of magnetic stress to rolling direction. . . . . . . . . . . 112
5.4 Shear magnetoelastic compliance coe cients as function of angle of magnetic wtress to rolling direction. . . . . . . . . . . . . . . . . . . 113
5.5 (110) 001] crystal orientation. RD is rolling direction and TD is transverse direction of the sheet. . . . . . . . . . . . . . . . . . . . . 115
5.6 Main stripe domains with supplementary lancet domains. . . . . . . 121
x
5.7 Lancet domain viewed from the side. . . . . . . . . . . . . . . . . . . 121 6.1 Equipotential lines for the magnetic scalar potential. Sample magnetized in the rolling (x) direction. Oriented material. . . . . . . . . 135 6.2 Equipotential lines for the magnetic scalar potential. Sample magnetized in the transversal (y) direction. Oriented material. . . . . . . 135 7.1 Magni ed (factor 5000) deformation of sheet from ux density vectors. Nonoriented material. . . . . . . . . . . . . . . . . . . . . . . . 138 7.2 Total strain sx and magnetostrictive strain sMx in the measurement area. Nonoriented material. . . . . . . . . . . . . . . . . . . . . . . . 139 7.3 Magni ed (factor 50000) deformation of sheet at ux peak time when xmagnetized. Flux density vectors drawn. Undeformed boundary dashdotted. Oriented material. . . . . . . . . . . . . . . . . . . . . . 140 7.4 Magni ed (factor 50000) deformation of sheet at ux peak time when ymagnetized. Flux density vectors drawn. Undeformed boundary dashdotted. Oriented material. . . . . . . . . . . . . . . . . . . . . . 141 7.5 Geometry for the cut y = 0 in m with gravity as only load. Deformation of sheet magni ed with factor 50. Undeformed sheet dashdotted.148 7.6 Equilines of de ection (solid) for B 0 case. Outlines of pole surfaces (dashdotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.7 Geometry in m for the cut y = 0 when xmagnetized. Deformation of sheet magni ed with factor 50. Flux density vectors drawn. Undeformed sheet dashdotted. . . . . . . . . . . . . . . . . . . . . . 158 7.8 Equilines of de ection (solid) when xmagnetized. Outlines of pole surfaces (dashdotted). . . . . . . . . . . . . . . . . . . . . . . . . . . 159 7.9 Geometry in m for the cut x = 0 when ymagnetized. De ection of sheet magni ed with factor 50. Flux density vectors drawn. Undeformed sheet dashdotted. . . . . . . . . . . . . . . . . . . . . . . . . 160 7.10 Equilines of de ection (solid) when ymagnetized. Outlines of pole surfaces (dashdotted). . . . . . . . . . . . . . . . . . . . . . . . . . 161 7.11 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 xi
8.1 Measured (solid) and simulated (dashdotted) B2(t). . . . . . . . . . 165
8.2 Measured (solid) and simulated (dashdotted) M(t). . . . . . . . . . 166
8.3
Measured butter y curve, dashdotted.
loops ..
of .
.
My.
vs. By, solid, ........
and singlevalued ..........
tted ...
.
166
8.4 Magnetostriction curves, measured (solid) and simulated with nonlinear model (dashdotted). . . . . . . . . . . . . . . . . . . . . . . . 167
8.5 Magnetostriction curves, measured (solid) and simulated with nonlinear model (dashdotted). . . . . . . . . . . . . . . . . . . . . . . . 168
8.6 Magnetostriciton curves, measured (solid) and simulated with linear model (dashdotted). . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
8.7 Magnetostriction curves, measured (solid) and simulated with linear model (dashdotted). . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8.8 Flux density T] in rolling direction versus eld strength A/m] in rolling direction. Oriented material. . . . . . . . . . . . . . . . . . . 173
8.9 Flux density T] in transverse direction versus eld strength A/m] in transverse direction. Oriented material. . . . . . . . . . . . . . . . 173
8.10 Flux density T] locus. Transverse direction is yaxis and rolling direction is xaxis. Oriented material. . . . . . . . . . . . . . . . . . 174
8.11 Field strength A/m] locus. Transverse direction is yaxis and rolling direction is xaxis. Oriented material. . . . . . . . . . . . . . . . . . 174
A.1 Optic component placement with possible double interferometers . . 197 A.2 Closeup of single interferometer with sample side dimension . . . . . 198 A.3 Side view of interferometer (possibly dual), arm with AOM . . . . . 199 A.4 Side view of interferometer (possibly dual), arm with laser head . . . 199 A.5 Laser mount . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 A.6 Custom tapped rod, for optic rail on diabase spacer fastening . . . . 201 A.7 Acoustooptic modulator, fastening on translation stage . . . . . . . . 201
xii
A.8 Baseplate for AOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 A.9 Diabase spacers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 A.10 Sample support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 A.11 Tall laminated yoke . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A.12 Short laminated yoke . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 A.13 Spacer between yokes . . . . . . . . . . . . . . . . . . . . . . . . . . 206 A.14 Yoke pair assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 A.15 Table top with tapped mount holes . . . . . . . . . . . . . . . . . . . 207 A.16 Experiment table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.17 Table top support . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 xiii
List of Tables 2.1 Calibration factor as function of calibrating Hall probe position. . . 32 7.1 Dynamic normal strains in xdirection when xmagnetized. . . . . . 141 7.2 Dynamic normal strains in ydirection when ymagnetized. . . . . . 142 7.3 Rotations when not magnetized . . . . . . . . . . . . . . . . . . . . . 157 7.4 Rotations when xmagnetized . . . . . . . . . . . . . . . . . . . . . . 159 7.5 Rotations when ymagnetized . . . . . . . . . . . . . . . . . . . . . . 160 7.6 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 xiv
Chapter 1 Introduction 1.1 Overview This book is organized as follows. Chapter 1 contains this overview section, the motivation and goals of the project behind this book and a credits section. It also contains an article literature review on the subject of magnetostriction of siliconiron and related subjects such as magnetization behaviour, ux distribution, hysteresis and magnetic domain processes. Chapter 2 contains a description of the design of a measurement system that can make 2D magnetic measurements and 1D magnetostriction measurements of sheet samples. The setup can be called a rotational single sheet tester (RSST), because the magnetic eld vectors can be made to rotate in the plane of the sheet. This chapter also gives some operation guidelines of the RSST. Chapter 3 contains a description of the interferometer that is used with the RSST to measure strain. The interferometer was built from basic parts to suit the geometry of the RSST. This also meant that the time signal from the photodetector easily could be extracted and correlated with the magnetic time signals. This chapter also contains a section on how to align the reference and measurement beams to get interference and make measurements possible. Furthermore, there are discussions on measurement error causes, particularly the motion of the table top on which the interferometer is mounted and unwanted rotations(tilts) of the re ecting prisms mounted on the sample. 1
Chapter 4 states and illustrates the basic de nitions of displacement, rotation and strain. The strain component dependence of choice of coordinate system is treated. Simple elastic
material models are reviewed and a magnetostrictive constitutive relation is proposed by analogy. The concept of a magnetic stress tensor as a variable for parameterizing measurements of magnetostrictive strain versus magnetic eld is introduced. Chapter 5 describes dispersive lag models of magnetostriction suitable for inclusion in
computer programs. Directional dependence of uniaxial response shows the possibility of including anisotropic materials and gives a guide how to determine parameters from measurements. The chapter also presents an overview of physical descriptions of material microstructure and magnetic domains with focus on the signi cance to magnetostriction. Chapter 6 describes the mathematical details of how to discretize a 2D magnetic scalar potential problem with a nite element method. The procedure of setting up a linear equation system for solution with some acquired solver is shown. The iterated procedure necessary for a nonlinear problem is treated. Scalar potential formulations suitable for 3D problems are stated. The application of a FEM program for solving the magnetic eld distribution inside the sheet sample is shown. Chapter 7 describes the mathematical details and algorithms of how to discretize linear plane stress and thin plate bending problems by nite element methods. Every step before using a commercial or free equation system solver is dealt with. The FEM programs built have been applied to investigate the strain eld in the sample, the percentages of elastic strain and magnetostrictive strain in measured strain and possible rotations of sample re ectors due to sample bending when no sample support table is used. Chapter 8 is devoted to the question of how to represent measured nonlinear magnetostriction data. The representation method is expansion of a single valued magnetostriction in a third order polynomial of the magnetic stress tensor introduced earlier. The double valued magnetostriction as a function of the magnetic stress is represented in the frequency domain by a second order transfer function multiplied by the single valued magnetostriction. Parameter extraction from and comparisons with magnetostriction measured transversely to rolling direction of a sheet excited with a eld strength transversely to rolling direction are shown. Chapter 9 sums up the developments presented and their strengths and weaknesses. The path remaining to go to achieve perfection both with measurement hardware and modelling software is discussed. 2
1.2 Motivation and goals A signi cant di erence in sound level is found between di erent designs of magnetic devices. To understand which parts of the magnetic circuit that are of main importance to sound generation more complete and accurate models are needed than the ones used today. The joints in magnetic cores are of
special interest to study because no explanation of the strong in uence of magnetostriction there has been found. Today no tool exists to model in three dimensions all angle dependent magnetic properties, including magnetostriction and mechanical stress dependence. This has been the driving force behind the project, on which the book is based. The conditions for the models are set by the magnetic design. The project goal was that the models should be possible to apply to a power transformer, where the noise caused by magnetostriction is a technically and commercially important problem. The ultimate goal was that threedimensional magnetomechanical continuummodelling of cores made of oriented silicon iron sheets should be made possible. This model should include nonlinearity, anisotropy and hysteresis, all in a macroscopic sense. The models obtained were to be well adapted to a continuum theory for the vibration of these sheets. Discretization of the continuum problem was then carried out with the nite element method, the algorithms of which was implemented in computer programs. The viewpoint taken in the project was phenomenological. Model development required methods to collect needed
experimental data. Biaxial strain in electric steel could be measured by laser interferometry under rotating magnetic elds. Local magnetic eld strength and ux density was simultaneously measured. The integration of the model in niteelementsoftware was adapted to the simulation of samples and cores. Experimental data covering twodimensional magnetization excitations and magnetostriction responses was collected from a custom designed measurement setup. Finiteelementalgorithms were developed inhouse with MAPLE and MATLAB. MAPLE was also used to parametrize measurement data. The signal generation, data acquisition, presentation and printing programs were written in C with BGI (Borland Graphics Interface) libraries and compiled with the TurboC compiler. This book constitutes the background to, documentation of, and veri cation of the design of the hardware and software that realized the project. No ambitions have been set to contribute to the knowledge about magnetism and magnetostriction on a microscopic level. Information about the physical processes is of course still important, as it poses restrictions on the macroscopic models. Such info has been collected from the literature and is presented in the text for educational purposes. 3
There is quite some material on deductions in relevant areas of mechanics and the nite element method, to give a more complete view to the readers from the electrical engineering departments, and to provide background to the design of the programs and their use. 1.3 Credits Various persons and companies were responsible for various parts of the project implementation. A lot of coding was done by the author and Dr. Anders Berqvist and all the calculations presented in this book and the control of measurements were done with those programs. The hardware was mostly bought or custom built in workshops, and a few major software products was bought as well. Quite many free programs were used, the most important are listed further down. The author coded 2D nonlinear isotropic magnetic scalar potential FE algorithms in Matlab, linear anisotropic plane stress FE algorithms in Matlab, linear thin plate bending FE algorithms in Matlab, basis function and local sti ness matrix derivation algorithms in Maple, nonlinear harmonic interaction relation derivation algorithms in Maple, data acquisition and signal output programs in C, measurement presentation and plotting programs in C and Postscript. Matlab was copied from Mathworks Inc. and Maple from Maplesoft, both under a KTH license to which the department contributed nancially. The C crosscompiler for data acquisition programs was Texas Instruments CL30. The C development environment for PC host programs was Borland TurboC 3.0 bought by the author. Postscript interpreters are built into Apple Laserwriters and some Hewlett Packard Laserjet printers. The workstation used for simulations, drafting and document typesetting was a HewlettPackard 9000/710. Drafting was done in HP ME10. HP equipment was bought by the department. This book was typeset on an Asus/Pentium133 PC bought by the author. The Asus PC operating system was Linux, kernel 2.0.32. Latex by Donald Knuth, Leslie Lamport and Thomas Esser was used for typesetting. Illustrations were done in Tgif, written by William ChiaWei Cheng. Plots were done in Matlab. Postscript plot editing were carried out in Ipe, written by Otfried Chiang. Postscript screen previewing was managed by Ghostview with the Ghostscript interpreter, developed by L. Peter Deutsch. Tgif, Ipe, GNU Ghostscript, Latex and Linux are free. The host for the measurement system was a taiwanese PC motherboard with a 486 and an Ethernet card, all bought from Kallio AB by the department. The host PC operating system was MSDOS 6.20. The data acquisition board was a Data 4
Translation 3818 bought from Acoutronic AB. Communication software between the host PC and the HP workstation was Onnet PC/TCP with ftp and rloginvt. The granite table top under the interferometer was bought from Mikrobas AB. The SpectraPhysics laser head was bought from Permanova AB. The Newport and SpindlerHoyer optic components were bought from Martinsson AB. Department technician Olle Brannvall made the threelegged steel support to the optic table and the PVC sample/Hcoil support. Plasma physics department workshop technician Juhani Hapasaari made the aluminum support to the laser head. The ABB Corporate Research experiment workshop in Vasteras made the two laminated Ccore yokes that magnetically feeds the sample under test. The author selected or designed and drafted the parts, drafted the assembly of parts and put all the pieces together. Former department colleague Dr. Anders Bergqvist developed and coded the 3D linear anisotropic magnetic scalar FE program (in C, compiled by the HP ANSIC compiler) and with it calculated the magnetic eld distribution in and around the sample as fed by the yokes. He also modelled the geometry of the yokesample con guration. Bergqvist calibrated the Hcoils used in the measurement setup with the departemental LDJ electromagnet controlled by his own software running on an HP 9000/300. The section 2.9 is basically a translation of a report he made on the task. He also wound the Hcoils and the coils feeding the yokes. The calculated magnetic eld results were used by the author who calculated the magnetostrictive and total strain elds in the sample as sourced by the magnetic eld. The author also calculated the bending of the sample by gravitational and magnetic load. The author modelled nonlinear, doublevalued transversal magnetostriction and made parameter extractions from measurements. The author made uniaxial and rotating magnetic measurements on nonoriented samples and uniaxial magnetic measurements on oriented samples. 1.4 Literature review 1.4.1 Magnetic hysteresis models Jiles and Atherton 1] renewed the interest in hysteresis models with a mean eld based theory that contained six parameters well interpretable as
physical constants of domain wall translation impediment, initial permeability, saturation magnetization, coercivity, remanence and hysteresis loss. Jiles, Thoelke and Devine 2] clari ed the procedure of how to calculate the Jiles 5
Atherton model parameters from measurements of coercivity, remanence, saturation magnetization, initial anhysteretic susceptibility, initial susceptibility and the maximum di erential permeability. These latter constants are more easily available than the set in which the model was originally formulated. Jiles 3] continued to work on his model to include frequency dependence, something of importance to the operation of ferrites for example. Basically, the idea consists of adding a dynamic part to the static or nearstatic hysteresis loop, where the dynamic part is the solution to a damped harmonic motion equation. The parameters so added are the natural frequency of the material and a second relaxation time for the damping. Mayergoyz, Adly and Bergqvist started the development of Preisach models for magnetostrictive hysteresis 4]. The rst stressdependent Preisach model was presented in 5]. Kvarnsjo 6] applied the stress dependent model to TerfenolD. In 7], Bergqvist continued to write about the di erential based model that he had developed. In 8] he continued over to the magnetomechanical side. The Preisach and laglike models were collected in one work 9]. The models were developed 10] and used for loss determination in a practical example in 11]. Bergqvist 12] 13] went on another trace to treat hysteresis. He started using pseudoparticles, essentially volume fractions of di erent domain types, and included them in a thermodynamical framework. Hysteresis was included by a friction model of pinning 14]. Anisotropy was next to take care of 15] and this model was supported by experiment. Eddy currents and laminates were set in mind by Holmberg 16]. 1.4.2 Magnetostriction Bengtsson 17] reviewed the types of domain structures found on the surface of SiFe sheets with di erent textures. The texture describes the alignment of crystal grains with rolling surface of the material. Three textures are encountered, cubeonedge, cubeonface and nonoriented. The domain types are the main pattern and the supplementary patterns. The main pattern is a band pattern and the supplementaries are spikedomains, facets, and mazepaterns. These structures will be described in more dsetail in chapter 5. Bengtsson also reviewed the rolling direction magnetostriction characteristics found in the di erent materials. In cubeonedge materials, the magnetostriction is negative, quite weak in the rolling direction (about 1 m/m) and reaches a peak at an intermediate eld strength. In cubeonface 6
materials, the negative peak is masqued by a much larger positive contribution. In nonoriented materials, the peak doesn't exist, and the magnetostriction is positive and ten times larger than that for cubeonedge materials. Lee 18] made the rst calculation of Fe 110] magnetostriction in anhysteretic multidomain (i.e unsaturated) single crystals. When comparing to experiments he noted that the demagnetized state is not a proper reference state as the domain magnetization vectors are not equally distributed over material easy axes. Celasco and Mazetti 19] used four parameters to map the saturation magnetostriction behaviour of grainoriented polycrystalline materials with three kinds of texture, Goss (cubeonedge), cubic (cubeonface) and bre. The Goss and cubic textures have three symmetry axes around which the direction of the grains are distributed in a gaussianlike fashion. The bre texture has only one such symmetry axis. Of the four parameters, two of the parameters are composition dependent (single crystal saturation magnetostriction along 100] and 111]). The third parameter is related to the grain dispersion (the average misalignment of grains to the rolling direction). The fourth is the volume fraction of cross domains (domains magnetized in an easy direction transversal to the rolling direction) when the material is in a reference state. The reference state can be any state with the applied eld much lower than the crystal anisotropy equivalent eld. In the formulas and measurements the remanent state is used as the reference state. Experimental results of reference to saturation relative magnetostriction for strip samples have been given. It is worthy of commenting that the widths mostly used for strip tests (often in an Epstein frame) will give di erent results to fullwidth sheet tests as edge in uence will be di erent. Narrow width strip will give a comparably strong demagnetization e ect from the magnetization discontinuity on the edge (with magnetic surface poles as equivalent source). If the rolling direction is parallel to the long strip dimension, the demagnetization in the transversal direction will act perpendicularly aligning to itself. Such a shape e ect will in uence the demagnetized domain distribution and low eld behaviour, both magnetic and magnetostrictive. Allia made a physical model of longitudinal (i.e rolling direction) magnetostriction of high permeability material 20] based on the behaviour of ninety degree spike domains. These spike domains occur when grain lattice planes are misaligned with the lamination surface. Ordinary 180 degree domains would give a strong demagnetization energy contribution in such a case, so spike domains emerge to reduce energy. The spike domain volume is expressed as a function of magnetization and magnetostriction is a negative monotonic function of spike domain volume. Magnetostriction is reported to reach a deep negative peak at 1.75 T, and at a critical applied eld, dependent on the misalignment angle, magnetostriction vanishes. Another condition for strong reduction of this longitudinal magnetostriction is said to be the application of tensile stress in the range of 100 MPA. 7
Allia, Celasco, Ferro, Masoero and Stepanescu 21] calculated the initial magnetization curve of GO sheet with high texture perfections. They stressed the importance of and quanti ed the in uence of ninety degree transverse closure domains present in the bulk of the sample sheets, connecting lancet surface domains with magnetizatons antiparallel to the main stripe domain structure. The collapse of their structure above 500 A/m was also modelled. Bertotti has written lots of papers on the subject of hysteresis and associated power loss in soft magnetic materials. With Mazetti and Soardo 22] he presented a loss model usable for GO SiFe where the traditional anomalous loss was incorporated in the formalism. Bishop 23] 24] simulated the domain wall bowing in materials with di erent crystal orientations between (100) 001] and (110) 001]. This bowing of the eld wall is reversible in itself, but is accompanied by local eddy currents due to the ux density change in space which the wall passes. He found that at intermediate orientations, there would in such a material be an antisymmetry (a shear) in the bending as the wall moves that would cause a reduction of eddycurrent loss. Yamaguchi 25] studied the sheet thickness dependency of magnetostriction in near(110) 001] single crystals and found that a reduction from 0.3 mm to 0.05 mm would lower the magnetostriction peaktopeak value with one fourth. He explained it with annihilation of subdomain structure that occurs due to stronger demagnetization in the thickness direction. Imamura, Sasaki and Yamaguchi 26] explained the increase of eddy loss as the 001] crystal axis is more inclined to the surface. As such an inclination will cause a magnetization component perpendicular to the surface there will follow an inplane circulating eddy current as the magnetization changes. Moses has made a large e ort to practically penetrate the subject of magnetostriction in electrical machinery cores. He measured 27] vibration in transformer cores with accelerometers and noted the importance of harmonics. When it comes to transformer noise, he suggested a method to reduce core vibration by using the stress sensitivity of magnetostriction and applying stress by a bonding technique. Moses 28] continued to perform measurements with high compressive stress applied, a task not easy successfully to complete. The results for SiFe showed that there is a large scattering in the values between di erent samples. Mapps and White 29] explored the transverse magnetostriction with harmonics. They found a twotoone ratio between transversely measured strain and strain in the longitudinal direction, something in accordance with theory. Compressive stress in the range of 5 MPa was reported to cause high harmonics in both directions, and 8
this was coupled to the appearance of ne pattern transverse domain structure. Moses together with Bakopolous 30] tested coatings applied under heat treatment and applied tensile stress. The so lockedin stress for 4 MPa applied stress caused improvements (i.e reduction) in peak magnetostriction and power loss. A higher stress was seen to increase the loss. Allia, Ferro, Soardo and Vinai 31] explained the di erence between magnetostriction behaviour of nonoriented and cubeonface on one hand and cubeonedge material on the other hand. The former materials possesses positive magnetostriction and the latter negative. The negative magnetostriction is connected with transverse spike domains, while the positive magnetostriction is said to appear due to a reorientation of a main structure, that contains domain magnetizations up to ninety degrees from the sheet axis. These initially spread out vectors can be aligned with a tensile stress, and when done so, negative magnetostriction appears even in the materials normally thought to have positive magnetostriction. They conclude that this type of spike domains also appears in nonoriented and cubeonface materials. Pfutzner has been active in the eld for many years. In 32] he brought the subject of domain re nement by scratching under his eye. Scratching of the surface of superoriented (a.k.a HIB) material is done (originally by ballpoint pen, now with laser) to make the domains less wide. Too wide domains lead to higher "anomalous" eddycurrent losses, as does too narrow domains. What Pfutzner here noted was that stacking of the scratched sheet, as is done to form a core, could change the domain width unfavourably (widening), while stacking of unscratched sheet could cause narrowing. Single sheet domain patterns could thus be misleading. Fukawa and Yamamoto 33] calculated the stress distribution from scratched lines on single crystals. They found that stresses are compressive near the surface and tensile in the middle, while being perpendicular to the scratch line. Scratches on sheet are made perpendicular to the rolling direction, so stresses in sheet will appear longitudinally. Pfutzner, Bengtsson and Leeb 34] made investigations on unpolished sheet. Polished sheets are usually prepared to make domain observation possible with scanning electron microscopy. SEM reveals the main domain pattern. A supplementary pattern occurs due to misalignment of 001] to surface. Pfutzner et. al. noted that this pattern is also dependent on the bending of the sample. Together with a magnetic colloid technique (something like a Bitter technique) instead of SEM, they could observe both the treelike supplementary pattern on one side, as well as the main pattern on the other side, without the need for polishing. Eadie 35] checked out the stress and temperature sensitivity of Goss textured SiFe with and without coating. He compared area under the stressmagnetostriction 9
curve and apparent power. Stanbury 36] made an apparatus to measure magnetostriction on strip samples also tting in an Epstein frame. Strips were cut at various distinct material directions to the rolling direction, and values were gotten for strain on each strip. Hribernik 37] measured the in uence of cutting strains on samples. Notably this was really only performed on fully processed nonoriented sheet. Slama and Prejsa 38] observed domain patterns for magnetization processes in di erent directions to rolling direction. Two angle regions were identi ed, separated by di erent dominating domain wall types in motion, types of 180 degree and 90 degree magnetization vector twists. Domain walls through the body of Goss sheet are not straight, but skewed with kinks. On reversing the magnetization, the kinks will change from concave to convex, the so called ruckling process. Morgan and Overshott 39] tested to see if the ruckling process was a fact in electrical sheet steel when reversing the magnetization from saturation. A rmative answer was returned by modelling and image of surface domains. Frequency dependence of domain structure was studied by Ungemach 40] and he showed that there is a critical frequency that marks the onset of dependency of structure on frequency. Bichard 41] observed structures using HVEM (high voltage scanning electron microscopy) and noted that real, rough surfaces have a more complex closure domain structure than polished surfaces. Zhou and Hsieh 42] linearized the electromagnetomechanical interaction in solids listened to by using eddy current transducers and showed that the elastic coupling provides more information than the conventional rigid model. Dynamic behavior of surface closure domains was studied by Nozawa, Mizogami, Mogi and Matsuo 43] through an HVEM. The material they studied was highly advanced GO silicon steel and the material improvements done showed in domain properties and behavior. Masui, Mizokami, Matsuo and Mogi 44] checked out stress dependency of magnetostriction. Deteriorating (i.e. increase of magnitude) in uence of compressive stress was attributed to supplementary domains associated with scratches on surface. The experimentation led to a simple formula for the dependency, usable in the design evaluation of di erent applications. The insight was that a condition wider than previously considered when ful lled leads to the onset of supplementary do 10
main patterns (spike domains) around grooves. The condition was stated with the
strain energy densities This condition in turn
attributable to di erent led to the simple design
deviraelcutaitoinosnafsorem1u00la] .<
e
010]
<
e
001].
Arai and Hubert 45] concentrated on the surface domains, often referred to as supplementary, and wanted to know the depth pro le of those. That goal was achieved by minimization of a wall energy consisting also of directiondependent anisotropy energies and exchangeenergies. Therefore, some inner domain walls do not lie parallel to easy directions, but can also form rounded shapes, as is calculated for the branches of the treelike supplementary pattern.
Nakamura, Okazaki, Harase and Takahashi 46] presented a GO highpurity Fe sheet as an alternative to SiFe. Traditional highpurity Fe has been used for DC applications such as electromagnets, but when applied as sheet in AC eld, its eddy losses become higher than SiFe due to lower resistivity. The material written about is said to be suitable for AC, because its relative permeability is higher at 32000 and that reduces skin depth, compensating losses.
Masui 47] extended the work of previous Japanese researchers and proposed the
condition of total elastic energy form. The appearance of a 180
edt1eo0tg0r]e
weta0ol1tl0]ischange, but 90 degree walls are. The new condition is important, because more
complex stress states can be allowed for adequate analysis.
1.4.3 Stress dependence Stress in uence on magnetic properties has been researched mainly to nd a nondestructive test method for components mostly made of construction steel. Its relevance here is that the authors use a di erent language than people into silicon iron, and the articles may provide a di erent viewpoint on magnetoelasticity. When it comes to silicon iron, stress dependence is considered a means to reduce magnetostriction amplitude. Vasina 48] studied experimentally how a few scalar parameters (coercive force, remanence, saturation ux density, hysteresis loss) depended on stress below a low stress level for low carbon steel. He also measured changes of remanence with coordinate inside the loaded specimen. He writes that the elastic deformation causes a monotonic change of all the above ferromagnetic properties and that the plastic deformation causes nonmonotonic and nonsinglevalued changes of properties as stress is changed. Plastic deformation is connected with motion of dislocations that eventually destroys the magnetic structure.
11
Schneider, Cannell and Watts 49] made a magnetoelastic model based on three material constants for high strength steel, a stress dependent mean magnetic eld, and a constructed saturation anisotropy factor decreasing monotonically up to moderate levels of stress and eld. It ts the experimental data well for four processes with di erent sequences of application and removal of magnetic eld intensity and mechanical stress. The Villari e ect in the form of positive magnetoelastic sensitivity (permeability increase with stress) below the Villari point (at the knee of the BH curve) and negative sensitivity (permeability decrease with stress) above the same point is said to be understood with this model. 1.4.4 Measurement methods Maeda, Harada, Ishihara and Todaka 50] underlined the harmful e ect of a DC excitation on magnetostriction, i.e. the addition of a DC ux component will give an amplitude increase of magnetostriction. Carlsson and Abramson 51] described an alternative to having a continuous wave laser as light source in the interferometer. In their scheme, a pulsed laser was used together with multiple re ections to obtain higher sensitivity than a CW laser with single target re ection or pair of re ections. Mogi, Yabumoto, Mizokami and Okazaki 52] presented an SST (single sheet tester) with nonsinusoidal excitation and harmonic magnetostriction analysis possibility. Lewis, Llewellyn and Sluijs 53] used interferometry to measure piezoelectricity in dielectrics. The basic insight carrying their work was that electromechanical interaction occurs in all dielectrics, and that monitoring of this interaction can be a diagnostic tool to provide information on loss and failure initiation. The same is probably true in ferromagnetomechanical interaction: loss and condition is intimately linked with magnetostriction. Nakata is a living legend in the eld of magnetics. He and Takahashi, Nakano, Muramatsu and Miyake 54] has made magnetostriction measurements with a laser Doppler interferometer. The Doppler principle is used to produce a frequency shift of the measurement beam(s), and the recombined beam will have the frequency shift as main frequency of the intensity. This frequency is proportional to the velocity or velocity di erence of mirrors, and is determined by signal processing circuitry. Positive and negative frequency shifts, corresponding to advancing or retreating mirror, will not be distinguishable due to the squaring of photodetector current for intensity detection. By adding another frequency part, it is possible to lay the shifts around that point on the frequency line, and thus make a distinction between movement direction. 12
Nakata, Takahashi, Fujiwara and Nakano 55] measured ux density in GO SiFe at di erent angles to rolling direction. The equipment used was a crossed yoke SST, making it possible to measure in di erent directions without cutting the sample in di erent directions to the rolling direction. Measurements on such cut samples su er from demagnetization elds not parallel to main material directions. To ease the excitation of the transverse direction, parts of oriented sheet with the rolling direction normal to the edges of the quadratic sample was used to guide the ux in the wanted direction and hinder the ux in the unwanted direction. Another point in the set of measurements was the level of ux density achieved. Sometimes the FE method requires info on higher ux densities than actually possible in the continuum problem, to make a good t of the constitutive relation with parameters. This requirement could here be met by getting rid of constraints made by a waveform shape control device by not using it. The direction of the elds was only determined by the peak values, and it was shown by comparison with a controlled ux density direction technique that the uncontrolled method only deviated within 3 % in measured peak ux densities when plotted against measured peak eld strengths. Ohtsuka and Tsubokawa 56] have made a twofrequency interferometer. This type of interferometer uses an acoustooptic Bragg cell (also known as an acoustooptic modulator, AOM) to produce an oscillating intensity, in this particular setup of both reference and measurement beams. The oscillating intensity can be equivalently described as the e ect of two (or more) superposed waves, slightly separated in frequency (colour). Normally, this frequency split is used to be able to detect movement direction, a method under the name of heterodyne interferometry. The usual demodulation method to get the signal proportional to movement is phase demodulation. The case in the artmcle is that there is a homodyne intensity component and a heterodyne intensity component. The heterodyne component has the movement signal as an amplitude modulation. AM is simpler to demodulate than PM with analogue means, what is used in the article. During the time since the article was written, analogue equipment has to a large extent been replaced by digital and the point may not be crucial any more. Still phase demodulation might su er from phase jump distortions that are di erent in character from amplitude demodulation noise problems. Another aspect is that noise is not frequency independent, there is 1/f ( utter) noise in the photodiode for example. By adding a frequency to the AOM, the electrical signal can be moved upwards in frequency to be better readable. Ohtsuka and Itoh 57] de ned the vibrational modes of the target mirror by its time variation, not spatial (tilting, rotation etc). 13
1.4.5 Numerics Higgs and Moses 58] computed ux distribution with harmonics in transformer cores for three di erent core con gurations. Nakata has also led a number of FE method projects. Him, Takahashi and Kawase 59] analysed singlephase transformers with hysteretic properties. In 60] with Takahashi, he showed to be able to include permanent magnets in a simulation. In 61] he covered ux and loss distributions. Funakoshi and Ito were added 62] to give an early attempt at 3D problems, for the case of axisymmetric and rectangular coupled components. Nakata, Takahashi and Kawase 63] carried on to stacked cores, where the laminations and rst and last sheets make the problem di erent from a twodimensional one but possible to simplify from a full 3D problem. In 64] Kawase and Nakata included anisotropy to model GO cores. Still a limitation to only inplane eld vectors remained. Pavlik, Johnson and Girgis 65] can calculate eddy losses in winding, tank walls, core support frame, lockplates and core laminations. Doong and Mayergoyz 66] implemented the PreisachKrasnoselskii hysteresis model. They used explicit formulas for the Preisach integrals, and the procedure directly involves the experimental data for identi cation of the PK model. Bergqvist has made a large number of papers. Bergqvist has treated vector hysteresis, the case with a rotating exciting eld and a response eld lagging by a (timevarying) angle. One of his models is the di erentialrelationbased model 67]. Magnetomechanical hysteresis was treated by Bergqvist in 68]. Basically he used his di erentialbased model for 2D hysteresis and used it for two other input variables, Hr . A nonlinear anisotropic magnetic model was proposed by Pera et. al. in 69]. It was based on the assumption that the equilines in B~ space of constant magnetic coenergy are ellipses or superellipses for anisotropic materials. While the fundamental postulate is simple and appealing, there enters di cult trigonometric relations when evaluating the permeability for inclusion in a magnetostatic nite element method using the magnetic scalar potential. In 70] a numerical representation for the coenergy material model was presented. Measurement data needed are BH curves for rolling and transverse directions, knowledge of di cult direction (at 54.7 degrees for GO) and the fact that directions are decoupled at low elds. Silvester and Omeragic 71] compared two di erentiation algorithms for nonlinear 14
magnetic material models. Di erentiation has to be used for the Newton iteration, and has to be quite accurate not to set iterates outside convergence range. Gyimesi and Lavers 72] reviewed the scalar potential formulations used for 3D. Kaltenbacher 73] has written a coupled FEMBEM program to calculate the solution to an acoustomagnetomechanical problem. The goal was to simulate an acoustic power source, magnetomechanically driven. In 74] he extended the program to include moving parts in the simulation. Magnetoelasticity as de ned by eddy current forces was written about by 75]. Eddy current forces can occur when there is a ux density component normal to the plate (as viewed by Yoshida et. al.). This component will give a circulating eddy current, that can be acted upon by a plateparallel ux density component, and vibrate the plate in a bending mode for example. Waveform control for the SST with digital feedback was written about in 76]. The estimation of applied voltage is done by a circuit equation, together with a representation of hysteresis from measurement. The hysteresis part greatly reduces the number of digital feedback iterations to be done to achieve stable control. 1.4.6 Technology Nakata and Takahashi have made special studies on transformers. In 77] they studied ux distribution in a velegged transformer. Overlap joint analysis was done in 78]. The straight overlap joint was covered in 79]. The SST:s Hcoil aspects (distance from sample, accuracy) were studied in 80]. Stacking with interleaved rolling direction changes of the sheet have been covered in 81]. Changes between adjacent sheet was 180 degrees, all directions longitudinal. The step lap joint is the unconventional joint type. It has been investigated by 82], for example. Reiplinger has made extensive acoustical investigations of transformers 83]. Together with members of the Study Committee he has made a standard for measurement with the sound intensity method on an array of measuring points 84]. Sievert has led a group researching 2D behavior of electrical steel sheet. Their results and the work regarding standardization of 2D test excitations were summarized in 85]. Nakata and Takahashi and Kawase 86] analyzed proposed transformer core joints 15
with regards to steplap length, length of air gap, number of laminations per one stagger layer and ux density. The nite element method used was able to take care of eddy currents and saturation. Salz, Birkfeld and Hempel 87] have calculated eddy current loss in sheets for a magnetic process with hysteresis in the rotational sense. The calculation was with an elliptical vector tip path, and with a classical description of eddy currents. Apparently their results could be con rmed with experiments. The experiment setup used was a 2D SST. Someone interested in normal to lamination uxes, a few motor people perhaps, can consult 88] for a penetration description. It has been heard that normally the ux will, in the bulk of the stack, be limited by the air gaps between the sheets. These air gaps are present due to the nonmagnetic coatings applied to the sheets when processed. The stacking factor thus produced will be su cient to masque the permeability of the normal magnetic part of the sheet. Kvarnsjo has written a major TerfenolD reference 89] that brings about the subject of giant magnetostriction in rod samples with a single crystal structure, and how to model it for applications such as actuators and transducers. Another paper about rotational magnetization loss treats the phenomenon in induction motors 90]. The authors made measurements of such loss in a 80x80 mm sample of motor steel for ux densities up to 1.1 T. The rotational loss was 7:2 W/kg while the loss from a magnetization process uniaxial in the transverse direction was 5:25 W/kg and the uniaxial loss in the rolling direction was 3:75 W/kg, all at 1.1 T peak ux densities and for a low allow, high loss steel. They simulated the magnetic eld in a stator with the MagNet FEM program and found an elliptical locus of the ux density at the back of a slot and a nearcircular locus at the back of a tooth. They state that rotational losses should occur all along the inner portions of the stator core and to a lesser extent in the rotor due to the slip. They further state that reduction of these losses could signi cantly lower the ac machine operating costs. It can be noted that such rotating processes and loss can be measured with the setup described in later chapters. The author started writing papers about a magnetostrictive generator concept 91]. An RSST (rotational single sheet tester) was shown in 92]. That RSST was built and results were compared to a simple ratedependent model in 93]. A not so simple model was tried to see if it could catch the magnetostrictive response to a transverse ux density excitation in 94]. The knowledge that bending distortions of the sample vibration can be present was taken seriously and analyzed in 95]. 16
Chapter 2 Measurement system 2.1 Introduction This is a presentation of a design of a measurement system for recording local two{dimensional magnetic ux density, eld intensity, and one strain component in silicon{iron sheets. Due to the speci c requirements of the measurement setup, it was designed from scratch. The degrees of freedom needed to recombine light beams, the temporal interference fringes and the current excitation could thus be analyzed and adjusted or processed in detail. In the past, losses and magnetization characteristics of electrotechnically important siliconiron laminations have been measured using single sheet testers providing an alternating applied magnetic ux density or, more recently, a rotating eld vector. The increased interest in the fundamental material responses of the constituents of magnetic devices has encouraged an attempt to bring this area of Measurement Techniques one step further. Creation of the applied waveform has until now largely been realized in the analog domain by frequency generators. With the advent of reasonably lowcost digital signal processors, digital generation of signals can be iibsnetaaebunltseiifttuyollHcy~oalwlnehcditleedsaitcmaieufnrlttolamynedsoeeunvssisloyerdsfelbeoydcianmlglyeaemintsheaeosrfuoCrfintphgreotshgereqaumuasxn.tdiTteihneessibtsyeytBfu~epeadninidngqtuhoeeusttpieouldnt to two voltage ampli ers or two current ampli ers, respectively. The ampli ers in use are connected to separate closed magnetic circuits that will provide the sheet under test with magnetization in two perpendicular directions. Methods for performing planar measurements of H~ and B~ have been subjected 17
to extensive discussions in recent years. Measuring H: For H~ a straightforward method is coils. Hall elements are less suited for this purpose since the measuring elements may well be smaller than the magnetic domains so the measured value depends strongly on the exact positioning of the sensor. B~ is measured by the induced voltage in a coil wound around some appropriate part of the sample. That part might be the center, with holes taking the wire through, or the whole sample. The center is the most interesting region as the eld will be homogeneous there. When the whole width of the sample is used there will be edge e ects di cult to compensate for. The latter alternative is the only choice when holes are regarded to damage the the magnetization process too much. 2.2 Purposes The setup is for the recording of two magnetostrictive strain components in thin siliconiron sheets under arbitrary twodimensional ux density or eld intensity excitations. The excitations of special interest are of course the unidirectionally sinusoidal, in the literature often labelled as alternating, and the vectorially twodimensional sinusoidal, which corresponds to a eld that in some fashion will be rotating. Frequencies are then typically low, at power system rates. Higher frequency tests are of interest to investigate in uence and behaviour of power frequency harmonics and eddy currents. The losses these processes produce in ferromagnetic materials is a classical problem, often hidden in terminology as anomalous or excessive  even though they are perfectly normal, deterministic and calculable, although only calculable by new methods and based upon new characterization measurement procedures. Higher frequencies might also enter when performing transient tests, which are of interest for nonsteady state operation of devices made of this type of magnetic construction materials. Transient tests of a di erent kind, but of no less value, are the quasiDC tests, that are important for investigations of magnetic hysteresis in various forms. In the magnetics group at the department, we feel that the hysteresis models of Bergqvist hold particular strengths, and this setup will enable us to validate that model concept for uniaxially alternating major loops, minor loops and rotational hysteresis. By quasiDC it is meant that the timerate of change of eld is low in the sense that eddy currents are negligible, and of course that case can be extended to nontransient periodic conditions. It is important, though, to recognize the di erence between hysteresis and ratedependent nonsingle valued phenomena hysteresis is the dependence on eld history without regard to the time increment between events (Barkhausen jumps). By doing quasiDC tests we can separate the hysteretic contributions from the ratedependent processes, which we presume are predominant in loss and (rotational) magnetostriction. One must note though, that the pickup coils for magnetic eld entities are relying on speed of ux 18
change to resolve magnetic data, so very low frequencies will give poor accuracy, but in any case there is a possibility scan a frequency range to test ratedependency. 2.3 Drawing and design system Design drawings detailing the assembly of mostly optomechanical components can be seen in Appendix 11. The author used ME10, HewlettPackards program for design and drafting, which uses the internal ME10 le format or HP's interchange le format MI to store geometry. There have been some problems of converting the ME10 format to DXF, which is the popular Autocad format. There have been prospects to convert to and use the IGES format, which seems to evolve as an industry standard, and seems to be more popular with FEM programs. Another standard le format that has emerged lately is the STEP format, which was brought forward on an European initiative to simplify the exchange of production data. ME10 is 2D and working with it is has the great advantage of semiautomatic dimensioning (labelling with lengths) compared to simpler draw programs such as X g, Tgif, PowerPoint or MacDraw, that lack it completely. Other features are (in nite) help lines, various alignment possibilities and methods of length input. Lines can also be drawn in a more sketching style, and then trimmed down or up to other joining reference lines. The basic geometric elements are points, lines and hatch areas. Lines include straight ones, arcs, circles, ellipses, interpolating splines and con ning (control) point splines. An often used operation is to show vertex points of the drawn object and connect other lines to those, or remove unnecessary points. Unnecessary points and duplicate lines (lines on top of each other) can cause problems when selecting a closed curve for hatching its interior area. Another feature is the handling of parts, each of which can be copied multiple times into a larger drawing. What is lacking when it comes to handling of complete objects is the de nition of their topology. When trying to modify a part to create a so called variation, the user has to input constraints between lines. These constraints soon make up a large number for a part with some details. It is di cult to manage all these constraints manually. Just thinking them out is not trivial, let alone change them as they all depend on each other in a way. There is an automatic option so that the program can gure out constraints, but the user is then not really aware of them and cannot make changes, except for redoing the whole procedure. A speci c variation is de ned by values of parameters, outside of constraints. They have given the speci cation method the name parameterized design. The need for creating variations in this project was minimal so the whole constraint business was left, even though it could have been nice to use parameters and a well de ned 19
topology in stead of dimensions and an enormous heap of simple lines and points. The workstation used to run e.g the drafting program was an HP 9000/710. It has an HPPA (Precision Architecture) RISC processor. RISC means Reduced Instruction Set Computer, the CPU type that has a clean instruction set, no so called complex instructions for memory block moves and comparisons. The big thing with RISC is that every instruction is carried out in one clock cycle, at least those that demand integer arithmetic. The HPPA also has oating point capability built in. Complex instructions that can be found on popular CPU's like the Zilog Z80 and Intel 8086/Pentium are really only there for the assembly programmer who wants few code lines per task, speed is not really guaranteed to be optimal for the task. On the HP 9000 series most applications are written in C, because the operating system is UNIX (HPUX) and it is convenient to interface to devices in the native language of the OS. C is a high level language and the compiler will produce and optimize assembly code for iterative tasks, so no complex instructions are needed. HPUX includes a standard windowing system, called X, that was written in C. The combination HPPA, HPUX and C thus ts together and form the platform of the system. For future selection of computer systems to use, it is important to compare the HP with a PC. The PC platform usually consists of a Pentium, DOS (written in 8086 assembly) and Windows (written in C). DOS runs in the real mode of the Pentium processor, with 16bit adressing of memory segments, leading to the infamous 1 MB DOS memory limit. Windows runs in protected mode, but has to switch to real mode to access some drivers, which leads to instabilities. Even though lately produced PC's have a higher clock speed number than the HP 710, the HP is by far more stable than a Windows PC and has a number of other attractive sides. The memory handling is more homogeneous, programming is simple with straight C, there is a vast amount of freeware available, the networked le system is seamless, Internet access is integrated from scratch, and there is good multiuser capacity (just log into the computer on the network which CPU you want to run on). When it comes to hardware the graphics must be mentioned, a nineteen inch screen is very much needed when doing drafting work with multiple part drawings. The author also wants to underline that the HP system is userfriendly. Just log in, start your application by typing its name and then use the mouse interface that most applications have. The VUE desktop can be used to copy les graphically if you really hate typing. When going to 3D, one encounters the problem of generating geometry and elements, setting material parameters as a function of position, setting boundary values and visualizing results in a simple and userfriendly manner. In 3D, there is a version ME30 to produce drawings merely using a set of cuts through the pictured object, and visualizing through interpolation and superimposition of these cuts. 20
A more powerful method of de ning geometry in 3D is to utilize a solid modelling program which bases its actions on a set of primitive solid objects and freeform surface splines. HP has such a product, of course quite expensive, while KTH has a site license for the IDEAS program from Unigraphics, which could be interesting to try out. For data visualization there is also an advanced package called AVS under KTH site license that is probably very nice to work with. There is also a solid modelling program called IRIT, which is freeware, that we have installed on our network. IRIT is text command driven and not particularly userfriendly, but is a candidate to us as a tool for understanding solid modelling and for programming element subdivision routines, for example. 2.4 Data acquisition programs The data acquisition card used to both generate excitation signals to the setup and acquire data from magnetic and interferometric sensors was a Data Translation DT31818 card. The programs for DT3818 card have the following features and limitations. The programs pda.out, pda2.out simply output an in nitely repeated waveform, for making measurement with oscilloscope. In the current setup, it is especially useful to feed the coils while aligning the interferometer. The program su xed with 2 is for DAC channel 2, while the unsu xed is for DAC 1. The main acquisition program dtacq.out outputs a repeated waveform with col lection after a certain number of periods. One to eight channels can be input. The repetition of the waveform stops after one time frame (one repetition) after data have been collected. Both DAC channels are used. If only a single channel is to be used, a dummy signal and a gain of zero can be assigned to the unwanted channel. There is also special demagnetization programs avmag.out, avmag2.out to out put a waveform to demagnetize the sample. It has no collection of data and stops at an exact time. The reason for writing this program was that an earlier version of the main program stopped after the output bu ers had been emptied, which could be in the middle of a time frame. The main program for creating signals to be downloaded to the DT board is called pcgen.exe. It runs on the PC host and can make dual channel twofrequency signals. The addition of a second frequency is aimed at investigating minor hysteresis lops, harmonics and harmonic interaction (nonlinearity). For options and arguments to the program, type pcgen without any option at an MSDOS prompt. 21
gArapprohg.reaxme
for presentation of measurement and contains many features:
results
has
been
written.
It
is
called
Time signal or hysteresis curve plots Data point show Unlimited data scroll and scale in x and y, autozero Signature data: rms, average, max, min, median Systematic naming and processing based on names Processing: Integration, demodulation, scaling Colors from options All options from option les (suitable for batch scripts) Fast printing on Laserwriters by direct Postscript output Scalable 100% vector graphics ASCII column le output FTP transfers of les XMS memory used, high and large room for signals. FFT of signals (experimental) Power time signals (experimental) The number of channels that can be handled by graph is very large, limited only by the available XMS memory. However, only three channels can be viewed simultaneously on screen (but switching to view other channels is quick). The three channels at a time limitation is justi ed by the fact that on paper, more than two curves in the same plot is seldom attractive. If the user wants more curves, he/she just puts them on another plot. The limitation reduces screen clutter and the need for extra symbols or legends telling the curves apart. The sequence of sample alignment, interferometer alignment (screen and oscilloscope veri cations), demagnetization, signal generation, measurement and presentation is automated by an interactive MSDOS batch script. The user should copy a template script le and alter the name and sequence variables in the beginning of the script, to store an exact speci cation of the measurement he/she is doing. In this way, the repetition of an earlier measurement is made easy, and the storage of presentation data is automatic.
22
2.5 Magnetic circuit The magnetic circuit feeding the sample sheet consists of two laminated yokes at right angles to each other. Local ux density is measured with induction coils and eld intensity with so called Rogowski coils. Drawings of yoke geometry and magnetic sensors are depicted in Fig. 2.2. An 8 A current ampli er supplies the coils on the yokes. Sensor bu ers/ampli ers isolate sources of signals and adjust their levels to the AD converter board. A block schematic of the electric part of the measurement system is found in Fig. 2.3. In recent years there have been some groups working on two{dimensional magnetic characterization of electrical steel, see for instance 96]. A few di erent types of magnetic circuit solutions have been proposed. The arrangement with vertical yokes used in the present work , see Fig. 2.5, o ers the possibility of a free line of sight for a laser beam and therefore seems appropriate for the current problem. The sheet sample should be separated from the yoke ends by an airgap of 0.1 mm to improve magnetic eld homogeneity in the sample. To investigate how the yokes would feed the sample with regards to leakage and homogeneity an inhouse 3D brickbased magnetostatic FEM program was used. The basis functions in the formulation were of trilinear type, i.e. the threedimensional extension of the famous bilinear functions often used on rectangular meshes. The results are presented in later chapters. 2.6 Excitation frequency limits In short, conservative lower and upper frequency limits are 10 Hz and 300 Hz due to restrictions imposed by the measurement coil and yoke feeding systems. The lower limit is imposed by the Hmeasurement coils. These produce a fairly weak signal due to the fact that they don't enclose the material, but a small air area right next to the material. The coils depend on the induction of voltage, which decreases with frequency. When going down in frequency, the rst problem encountered is not noise, but drift signals become disturbing. That happens because noise in the induced voltages vanishes when integrating to get the eld strenghts. Slight uncompensated o sets in the voltage signals will be seen as linearly raising or falling drifts in the eld strength signals. Without smart programmed compensation of these drifts, 10 Hz is tested as a safe frequency for measurements that don't have an excessively long duration. For short measurements 5 Hz can work. If the 23
Figure 2.1: Sample support table (not hatched) with yokes (hatched). See Fig. 3.1 for its placement in the setup. 24
By Bx Hy Hx Figure 2.2: Magnetic sensors, split sketch. 25
Laser
HV power supply + tube l. control
Photo detector
PD load/ buffer amp
Bcoils Hcoils Xyoke Yyoke AOM
Buffer amps Current amps RF driver
ADC DAC XTAL osc
DSP
RAM
DMA
ISA
bus
PC Ether net
UNIX system
Figure 2.3: Block schematic of electric part of measurement system.
Test specimen
190 150
y
z x
20
140
280
20
Figure 2.4: The magnetic yoke con guration. Dimensions in mm.
26
experimenter wants even lower frequencies, the measurement of H eld has to be done with Hall probes, that are not currently used in the setup.
The upper limit is due to the impedance of the feeding coils together with the magnetic circuit. The impedance increases with frequency and at high frequencies, the impedance is almost purely inductive. The peak current needed to create the wanted ux density is constant over the frequencies, and within the range of the current ampli er (8 A max). The current ampli er has a limited output voltage (about 60 V max), and this voltage will be reached at the upper frequency limit. The currrent and the upper frequency limit is calculated below.
pharTeieuohrlilxugeegchdartwteepalinn(ud(oscctRifehtt.yaamsn)TBia^cdahrelneieinnRdwaets)ihora=efimsgtsa2hpalp10emle=bmmp(elRam1etwg.4sn0e)Taeenmrhtnediecmlyurbco,ceikt=lryuaecocnu1kptci4eatoe0nsllse.ecemsnetGsgamfoetnthorhdamet(nhsperdaetoemrqlcsyeiapurmligaceranueppcdidltaelmcniesusnahbrtgthreeeteeehrntli0)stautl=ihI^smsicdtl0ok2oe:fn1t=aeeycmrsohm2skmi6eeic0nv,a(eenmyRottmybhhk)eeee,, t = 0:3 mm. The permeabilities of the yoke and the sample can be set to 5000 0 for purposes of estimation. Air permeability is 0. These values give the following reluctances,
Ry
=
2l1 + l2 5000 0wb
= 0:0386=
0 H;1
(2.1)
Ra =
2l0 0wb
= 0:143=
0 H;1
(2.2)
Rs R = Ry + Ra + Rs
= =
l2 5000 8:85=
0wt 0=
= 8:67= 7:0 106
0 H;1 H;1
(2.3) (2.4)
It is seen that the sample reluctance strongly dominates in the example. With 1
mm airgaps and reluctance might in the sample, a
bauexhoifg^ehq=upae1lr:m8mbeatag=bniilt1itu:y8des.a0m:I1f4palep0,e:at0hk0e03uaix=r dg7ea5np:s6itryeWluB^cbt=asnh1co:e8ulTadnidgsowsaraomnutpneldde
the circuit. The required current in the feeding coils becomes
I^
=
P^hi RN
=
1:32
A
(2.5)
as the magnetomotoric force is reluctance times ux. N = 400 is the number of turns of the coils on one yoke in the setup. The current is well below the 8 A max limit of the current ampli er.
The current I^ through the feeding coil inductance L produces a voltage U^ that
27
reaches Umax at the frequency fmax. The inductance L is
L
=
^ I^
=
N^ I^
=
30:2 10;3 1:32
=
22:9
mH
(2.6)
from values in the previous paragraph. is the ux linked with the coils. The coil
reactance is X = !L = 7:2 at 50 Hz. The coil resistance is about 2 , so the
rreeasicsttaannccee,wtihllebveol3t6agteimoevserlatrhgeercothilasnwtihllebreesU^ist=an!ceLaI^t,
500 Hz. By which gives
neglecting the the maximum
frequency
fmax
=
Umax 2 LI^
=
2
22:9
60 10;6
1:32 = 320 Hz
(2.7)
At this frequency, the reactance is 46 .
The maximum frequency can be pushed upwards by two methods. The rst is not
to go to such high peak uxes. This will require a lower current and the frequency
acaltnerbneatiinvcerefaosremdublaeffomreaxhi=ttinUgmtahxe=2vo^lt.ageThmisaxfiomrmumul.a
It can also also shows
be seen that a
by the change
of reluctance (air gap change, material change) won't change the frequency limit.
The only thing that counts is the linked ux, and that normally has to be set to
achieve the wanted ux density. Here one sees the second method to increase the
frequency maximum. By lowering the number of turns of the coils the linked ux
can be decreased and the frequency increased. To keep up the magnetomotoric
force that drives the ux, the current then has to be increased. For the levels in
the previous paragraph, the current can be increased 6 times before reaching the
current maximum of the ampli er. That allows for a reduction to 67 turns on the
coils with the same magnetomotoric force, same ux and six times lower linked ux.
The maximum frequency is then increased to 1.9 kHz.
It should be noted that the maximum frequencies are for the fundamental of the excitation. Smaller harmonics can be added. As an example, for the 320 Hz limit case, a 3.2 kHz current harmonic can be added if it is ten times lower in magnitude than the fundamental.
2.7 Voltage or current sti ampli er Two control methods have been tested: Current sti measurement uses a simple regulator with current sensing of the ampli er that feeds the coils of the yoke. The current on the output can then be made proportional to the voltage on the input.
28
Voltage sti measurement uses the raw ampli er, i.e. the voltage out is proportional to the voltage in. The currentsti ampli er has the bene t of being easy to demagnetize with. Harmonics over the fundamental frequency due to saturation can often be corrected with the regulator's ampli cation. The drawback is that the current is not completely correlated to the magnetic eld intensity in the sheet due to the air gaps, saturation and leakage. The air gap is the main source to this phenomenon. In the extreme case, the coil current will only set up magnetomotoric forces across the air gaps. Due to ux conservation, the ux density in the material will then be controlled by the air gap eld strengths and in turn by the coil current. By lowering the air gap length, the coil current will control a combination of ux density and eld intensity in the material. The uxcurrent correlation is also the reason for the experimental fact that circular (Bx By) is easier to obtain than circular (Hx Hy) when performing measurements with rotating magnetic eld. The voltage sti ampli er has the bene t of having fewer components. The voltage being independent of load also corresponds to a more usual situation in applications, which one might want to simulate in the setup. A problem when measuring uniaxial B ; H curves with the current sti circuit is that due to the rapid change of B with H close to zero, the density of data points can become sparse in that region. With voltage sti measurement it is easier to get equidistant points on the ordinate B. The major drawback with this mode of operation is that it is much harder to demagnetize samples. The voltage on the output corresponds to the ux derivative and there can be a constant component of ux present even as the voltage is made to approach zero. 2.8 Bcoils B coils are used to measure the ux density in the specimen. Either the coils are wound around a central part of the specimen by the use of holes pierced in the sample, or the coils are wound around the whole sample. The ABB program ACE was used to investigate how holes in the sample could in uence the magnetic eld. Such a simulation will take care of magnetization discontinuity on hole edge (equivalent to monopole distribution) and the eld distortion arising to lower energy by avoiding the air. The simulation won't take care of the fact that magnetization will be distributed over domains with continuously di erent size and discretely di erent shape and direction. The holes (as seen by Neel) will on that level act as nucleation centers for needlelike domains (with a transverse to main domain magnetization direction). They will grow under the in 29
crease of eld and can act as secondary source for the creation of main rectangular domains. Thus, the in uence of holes could be larger than seen on simulation. The continuum simulation showed a completely negligible distortion of homogeneity of eld when four 1 mm diameter holes were punched at midpoints of 60 mm sides of a square, both for isotropic and anisotropic material. The author believes the result for isotropic, polycrystalline material, but for modern, intensely anisotropic, textured material the simulations can be doubted. It has been heard that magnetostriction is changed a lot by only pressing sharp tips into the sheets of such material. Such rumours have been taken seriously and the holes have only been used for B pickup coils when measuring on nonoriented (isotropic) motor iron and for conventional types of oriented (anisotropic) transformer iron. The superoriented kinds of SiFe sheets have not been pierced, and coils have been wound around the whole sample. In the case for B coil wound around sample, other factors enter: dead magnetic zone due to cutting of sample and inhomogeneity of eld strength. The dead zone is a fraction of a millimeter in width. Inhomogeneity is due to the pole pieces being 140 mm wide while the sample is recommended to be 305 mm wide. The recommendation is for the sample to span the pole gap of 280 mm, provide some area to rest on table, and avoid edge e ects including dead zone. One might think that measurement with wide coils would give a good result in the low permeability direction ( ux spread out along crossing midline) and a poor result in the high direction ( ux more concentrated from pole to pole). When experimenting, it appears that in the low permeability direction, the ux actually tend to enter the nonactive poles and can cause problems with feeding the magnetic circuit. It is felt that the strong anisotropy in the sample together with the longitudinal yoke forms a magnetic circuit that constitutes a kind of bar to transverse magnetization. Either the ux enters and leaves the same pole without much penetration of the yoke, or the ux actually circulates the yoke. The last hypothesis is likely when there is a misalignment of the sample, and as the yoke is laminated and glued, a transverse enterleave path is quite inhibited due to the reluctance of the glue layers. The exact analysis of the problem has been put on hold as the remedy to be able to make transverse uniaxial measurements is simply to remove the longitudinal yoke, as it is allowed by the setup. For rotational ux density measurements, no 100% solution is given. As rotation in that case basically only will mean a large Barkhausen (discontinuous) jump from the longitudinal to the transverse direction, and it is the nature of that jump that is interesting, the author proposes that the sample be turned so that the hard (to magnetize) direction is aligned straight pole to pole. As the B pickup coils still will be wound around the easy and its transverse direction, information can be collected on how the jump happens by rotating the excitation in the new con guration. Geometry gives minimum sample sides of 300 mm for 45 sample rotation and 385 mm for 30 rotation. 30
20 10
10
30
1,2 2,2
30
1,1 2,1
Plate
Coil 40
Figure 2.5: One Hcoil wound from up to down around a nonmagnetic plate. Hall probe positions for calibration are marked with circles. Dimensions in mm. 2.9 Calibration of the Hcoil
The coils for measurement of H eld in the RSST was calibrated by placing them in the departemental LDJ electromagnet and comparing the results they gave with measurements made by Gaussmeter and Hall probe.
One have
0H =
1Z NA
U dt
where N is number of turns and A is the single loop area. The problem is to determine NA for both directions of the composite Hcoil. For each of the two directions there were four runs with di erent placements of the Hall probe relative to the coil. The placements are shown in Fig. 2.9. The directions that the double Hcoil can measure eld strengths in are called I and II direction I is marked with a little bit brown tape on the connecting wire to be able to relate the correct coil with the calibration data below.
Linear regression gave a standard error for coe cients less than 0.3 %. The results
31
Position 1,1 1,2 2,1 2,2
NNI II AAII I(c(cmm22):):
352.47 370.12
351.55 370.65
349.10 368.12
349.95 366.56
Table 2.1: Calibration factor as function of calibrating Hall probe position.
varied with the Hall probe placement according to Table 2.1. If the average is taken one gets
NIAI = 350:8 2cm2
(2.8)
NIIAII = 368:9 2cm2
(2.9)
2.10 Measurement table The measurement table top that supports the interferometer, the yokes and the sample support is made of black diabase, a kind of granite stone. The table top is depicted in Fig. A.15 and the whole table is in Fig. A.16. The table top was drawn, and a proposition from the French rm Micro Controle for making the granite construction was received. It was found that their facilities for treatment were excellent, with high performance drills for making the optic mount hole picture and proper tools for grinding, why they could obtain a high degree of atness of the stone surface. Granite was of course chosen by us because of the absolute necessity of having a platform made out of a electrically nonconducting, or at least poorly conducting, material to avoid considerable eddy currents, that would inhibit the production of a high ux density in the specimen, and also make the control of the eld in the specimen harder at some of the frequencies we have in mind. Granite also has a high mass density and a fairly high elastic modulus to mass density ratio, which will keep deformation mode amplitudes low and at fairly high frequencies, respectively. While MicroControle had proper tools for machining, it was decided not to order therefrom because of a high price, the long transportation necessary and the cultural barrier to design and fault discussions. Sweden has some granite industries so we consulted the rm Mikrobas instead. They provided a block of black diabase, said to be the granite type of highest quality in terms of internal motion, and made a custom treatment. This treatment consisted of grinding to higher degree of atness, drilling of optic mount holes and cementing of M6 tapped inserts in these holes. Geometry of the block was set to the standard thickness of 75 mm and a square 32
width of 1000 mm. The block weight is at a tolerable level, it arrives at 202 kg. The atness of the stone surface is superb, 55 m tolerance, much higher than needed in the present application. It can be used as a atness reference for various mechanical tools. A drawback with granite is its "ringing" characteristic, high frequencies or impacts seem to give sustained responses with low damping. Keep silence in the lab room. 2.10.1 Support placement The choice of area of the block also has some implications to support placement and method of erection of the setup. It was proven that the weight allowed positioning with inhouse equipment. It would have been better to have tried to reduce the width to 80 cm, to move the setup through door openings in the mounted state. Rigid steel supports are currently used, which do not really damp vibrations entering from the oor. If noise therefrom would be too disturbing (no measurement yet has had any negative e ect of it), one can purchase pneumatic dampers, working with a combination of rubber balloons and mechanical pendulums. There has been a new product entering the market for vibration isolators recently, that is called the subhertz isolator, which uses a support system passively acting as a spring with an e ective negative spring constant, thus counteracting vibrational forces. The product is very expensive though, and is currently only for light single loads such as microscopes, that need vertical vibration reduction. The transmission of vibrations is in this case ten times lower than any other isolator, so performance is supreme in theory. 2.10.2 Optic component placement The arms of the interferometer are quite long, and due to the limited area of the table top, the beam paths have to be folded. The HeNe laser head (from SpectraPhysics) has a diameter of 45 mm and are 400 mm long, and it is quite desirable to mount it such that it doesn't extend beyond the edge of the stone. Guiding equipment is therefore placed before the edge of the siliconiron sample. It is believed that symmetry of the setup must be kept, thus leaving the same space all around the peripheral of the test sheet, in order to be able to t another laser head if wanted. Something to consider when choosing size is also that the sample holder/table should be possible to rotate to be able to measure at about 45 from the longitudinal direction to determine the shear in the unrotated system, and this without tricky beam re ectors. The optic components for guiding reference and 33
measurement beams will be mounted on aluminum rails at some distance to the yokes in order not to in uence ux picture. The standard mount hole picture on commercial metal tables dimensioned in the metric system is a matrix with 25 mm between centers. If the same picture would have been drilled on the granite table, approximately 1000 holes would have been made. This kind of picture is not needed since all optic components are mounted on sledges moveable along the rails. The needed number of holes to x the rails with some options for di erent placements are about twenty.
2.11 Vibration of material
Vibrations due to pulsating magnetic eld are of three kinds.
Rigid or bending vibration due to so called reluctance forces on a magnetic object surrounded by air. The resultant force F R can be calculated with a Maxwell stress equation
~t
=
1 I2
BH
e~n
(2.10)
F~R = ~t dS~
(2.11)
where ~t is a traction (force per unit area) vector, and the integration is taken over a surface enclosing the object, with the surface completely in air, real or in an imagined in nitely thin air gap. Maxwell himself regarded similar traction expressions as valid for the mechanical stress due to magnetization also within bodies. The modern viewpoint is perhaps that only the integrated resultant force is valid, but the expression for the traction (force distribution) on the object surface can also be believed for separate interacting objects.
Forces on eddy currents induced by the the magnetic eld. If an oscillating ux density penetrates a thin sheet obliquely, the normal component can induce a large eddy current circulating in the plane of the sheet. If such an eddy current Je exists, the force volume density will be
f~ = J~e B~t + J~e B~n
(2.12)
where Bt is the ux density component parallel to the sheet and Bn is the normal component. The rst part of the expression will correspond to a normal force that might bend or shake. The circulating current picture suggests that there will be a tilting action when the parallel ux is laminar. The second term can be imagined, from the circulating current picture, to set up
34
a compressive stress towards the center of the circulation. A sheet is much weaker in the lateral direction than in the plane, so a bend is probably more of a worry than strain. As a thin sheet also will be light, the tilting action can be suspected to produce a shaky motion.
Magnetostrictive vibration. Suppose that a core is carrying a ux . The magnetostriction is an even function in the ux density, we can use B2 = P as a model when reasoning, where B2 can be thought of as a magnetic stress, P as a sti ness modulus and as the magnetostrictive strain. If is given in the core, as is often the case for voltage sti excitations, there are a few ways to reduce the vibration amplitude:
{ Increase the area of the core to decrease B. (More core material, cost
increases).
{ Use material with higher P . (Better core material, cost increases).
{ Passively damp the transmission of vibration with sound insulation.
(Polyurethane, maybe not so costly).
{ Actively counteract with actuators out of phase with the vibration. (Dif
cult and risky).
{
Apply tivity
tPenosiflethsetrmesastteoriathl.e
material
when
there
is
a
positive
stress
sensi
Attempts to dampen the vibration by clamping the material perpendicularly to the direction of vibration is not guaranteed to be successful. The magnetic action that shows as magnetostriction is very strong, and can easily make the clamping device vibrate too, perhaps leading to a worse transmission of sound. There might be a positive e ect from a lateral stress dependency if the clamping device suits the nature of the problem.
2.12 Digital control issues If a digital feedback control would be employed, the nature of the sample material would drastically in uence the feedback algorithm. The two industrially used types of siliconiron alloys are being investigated. The nonoriented sheets used in motor applications pose less demand on the control program, since there is no strong macroscopic anisotropy present. In a rotational eld case, the computer only has to store hysteretic lag information along with a direction cosine lookup table for the ampli er outputs to adequately steer the controlled eld vector. Grainoriented sheets (in the case of a circularly rotating ux density) will also require an additional table to additionally enlarge the applied 35
transverse component when the eld direction is moved into the di cult region around the magnetically hard axis, which is at 54:7 from the rolling direction of the sheet. In the case of experiments when only one cycle is measured, past history has to be cleared by saturating the material or magnetically cycle it. Magnetic cycling with the amplitude continuously decreasing from saturation to zero is also called demagnetization. Simple saturation can be used for really large major loop measurements, such that the rst loop guarantees saturation and the second loop is measured. Most measurements are not with that hard saturation, and a demagnetization wave is sent prior to measurement. When conducting basic research, not much is known to the experimenter when a new sample is taken to be measured. The rst measurements become an exploration of the material and are done with no or very primitive control. Based on knowledge gained from the exploration, one can be ready to take control. The simplest is to manually change the input signal to better achieve an intended eld signal. Automatic control using hysteresis models and parameters from initial measurements are possible, but there has not been time to code a suitable algorithm for the acquisition board. If someone would like to try it, he or she should be aware of the existence of delays from board input to memory input and from executed output to board output due to the use of deltasigma analog to digital and digital to analog converters. Deltasigma converters use a bitstream technique with a control/comparison loop to convert analog levels to digital numbers or vice versa. The control loop introduces a time delay between the sampling of the analog level and the output of a number. 2.13 Strain measurement by interferometry The setup is able to retrieve strain information from a laser interferometer. Three pairs of mirrors glued to the surface of the sample are subjected to internal relative translation as the magnetization is altered in the area of measurement. As a result, the laser beam is re ected by the target pair of mirrors whose relative displacement shows as a phase shift between measurement beam and reference beam after those two beams have been recombined. This timevarying phase shift can also be seen as a Dopplershift in frequency due to the di erence velocity of the mirrors. One of the beams can be frequencyshifted (or frequency split which corresponds to intensity modulation) so that a phase carrier (a "running" phase) is superposed on the objectcaused phase shift. This carrier makes the detection of sign of strain change possible. It can also be used to avoid lowfrequency noise. The detector of the phase shift is a photodiode. Its output current (or voltage over a load resistor) is proportional to 36
the light intensity which in turn is proportional to the cosine of the phase shift. The output voltage is sampled by the digital acquisition board. The voltage waveform is then demodulated on the host PC, to arrive at the desired displacement from which the local strain component is calculated. The demodulation code could be moved to the board, but as it depends on the calibration with a known whole wavelength displacement (either from the true object or an arti cial disturbance), it requires some operator validation and iteration that is most conveniently carried out on the host. There is sensitivity to the environment. Tramping on the oor, light knocking on the table and loud voices in the lab room (4x3 m) are clearly seen in the interferometer signal. 2.14 Stress in uence, frame e ect In the above, the handling of the B, H and state variables have been presented. The fourth variable must be adequately analyzed in order to well de ne the measurement conditions. The less magnetized outer regions of the sample will act as a frame around the central region, slightly resisting the strain of the central region. Thus, there will be an elastic component of the strain tending to reduce or smooth out strain arising from magnetostriction. This phenomenon is analyzed in detail elsewhere in this text, with the conclusion that about 10% can di er between true magnetostriction and measured strain. 2.15 Yoke design Field calculations have also been extensively used for the calculation of general setup performance. The yokes constituting the magnetic circuits are laminated to provide eld homogeneity in the sample. Also an airgap between the sheet under test and the end of the lamination is introduced. This increases the homogeneity and makes the magnetic reluctance force more well behaved, compared to letting the sample rest directly on the ends of the yokes. The signi cant reluctance introduced by the air gap makes full saturation harder to achieve. The reluctance force between the sheet and the yokes are carefully balanced out by adjusting the air gap to the upper air of yokes, which are separately xed to the vibration damped measurement table. In fact, to properly balance the force one has to have coils on the upper pair of yokes also. This is due to the fact that a large part of the total ux deviates from the yoke loop in the magnetic circuit into the sheet under test, even though the crossectional 37
area of this sheet is very small. As the force is proportional to the square of the ux, there would be problems to balance this deviation only by adjusting the air gap length. It is also probably only to an advantage to have symmetrical poles with coils wound close to the edges of the yokes from a leakage and drive current point of view. Finite element calculations have also been made to determine the area of homogeneous B, and elds for a variety of excitations. This area is suitable for sensor positioning. The current setup has 24 cm of inner spacing between poles, which are 14 cm wide and 2 cm thick each. This produces an area of homogeneity which is 10x10 cm square. It might be considered a good compromise between easy handling of reasonably light equipment and bene ts of having a large distance between re ectors when it comes to resolving the strain signal for very low magnetostrictive sheet types. On the other hand, for these sheet qualities it is possible to make a pure uniaxial measurement with a mirror spacing of 20 cm, as the homogeneity region is larger for this simple type of excitation. 2.16 Magnetic sensor design The fundamental concept of recording of all state variables re ects itself on the sensor equipment. Induction coils for the measurement of the ux density are made as one loop of 0.1 mm diameter isolated copper conductors thread through drilled holes in the specimen. Again, nite element calculations have been applied to investigate the performance. It is seen that the ux enclosed is very little changed by the introduction of 1 mm holes compared to the ideal homogeneous permeability case. What is worse is that the holes will serve as nuclei for domain growth at magnetization reversal, thus violating the basic assumption of a homogeneous magnetization characteristic of the measurement area. One might therefore consider to use a needle technique, in which two pairs of needles are brought in contact with the sheet to form a closed loop as the induced current will pass through the sheet twice. The low conductivity of the sheet, together with the one loop condition, will make the signal to noise ratio very poor, though. With a good sensor ampli er and DSP noise reduction capabilities, it might be considered as a check of Weiss domain conditions. The induced current in a sensor copper loop will be in the order of microvolts. To be able to feed the signal onto the A/Dboard, it has to be ampli ed 100 dB. This signal level, as the ampli ers are placed close to the sensors on the measurement table, will also protect the signal from environmental noise. The magnetic eld intensity is raised with a factor of approx. two in the holes 38
compared to the homogeneous case. While the ux essentially avoids the holes completely, the H eld will broaden its peaks due to the tangential continuity and a ect the region between the holes slightly. Thus, the Rogowski coils used to measure the H eld has to be geometrically somewhat shorter than the distance between holes. The 1 mm thick Rogowski coils are placed directly tangential to the specimen surface, in order to use the continuity of the eld intensity and the simple relation B = 0H in air to measure the magnetic eld intensity. It has to be realized though, that the H eld is not completely uniform in the direction inwardly normal to the sheet. To get the average eld, one is forced to rely on simulations to calculate a uniformity factor. Of course, this factor is a ected by eddy currents and the so called anomaly occurring when eddy currents are induced in ferromagnetic materials. There is still little quantitative knowledge about this e ect so simulations have to be performed in an ideal vs. worst case manner. 2.17 Temperature drift Another state variable to consider is temperature. During a long measurement, it is possible that a temperature drift in the order of a degree will occur. The best situation is then to record, albeit not very densely in time, the specimen temperature along with the rest of the variables. When interferometric measurements are performed, it is also desirable to measure the temperature of the air, as the index of refraction of air is somewhat dependent upon it. The specimen temperature can be measured by gluing an NTC resistor to its surface. A drift of 1 K will elongate 10 cm of siliconiron with a nanometer. The in uence of such a drift on magnetization is very low. Classical eddy currents at 50 Hz uniaxially alternating magnetization will give rise to approximately 20 mK of heating due to loss. The sources of temperature drift are the power ampli ers in the room, power supplies to control and measurement hardware and the excitation coils. If the excitation coils are overheated in an experiment with high current during a too long time, a slight smell can be felt from the hot insulating lacquer on the coil wires. 2.18 Signal conditioning and Nyquist limit Another source of drift, let alone of a di erent kind, are the analog measurement signal ampli ers. These are realized by the use of low noise TL072 operational IC:s and highquality precision passive components in feedback and signal paths. The main issue regarding these ampli ers are that they add a (constant) o set volt 39
age to the ampli ed signal. These o sets are easiest compensated for by digital
postprocessing of the sampled signals. A routine for such digital compensation is
needed anyway, since the integrated ux signals have to be o set compensated also.
gBroatphhbepfroorgerianmte.grTawtioonoapntidonasftcearninbteegcrhaotisoenn
o set from
compensation when starting
are the
dgornaephbyptrhoe
gram: mean and median. The mean option adds a constant to the signals so that
the time averages are set to zero. This is the usual choice for periodic, symmetric
signals. Also damped oscillatory signals often turns out to be well compensated by
this method. The median option sets the median of the peak and bottom values to
zero. That option can be used for signals that saturate the material both at peak
and bottom values, but are otherwise nonsymmetric. Both options operate at both
the beforeintegration and afterintegration steps.
The A/Dboard comes without so called signal conditioning circuitry and one has to take that into account also. The topics that typically arise are those of antialiasing and sampleandhold during A/Dconversion. When there is a nonnegligible energy in the highfrequency end of the signal spectra, as seen in the magnetostriction measurements, aliasing must be avoided by introducing a passive linearphase Bessel lter of enough order to cancel the frequencies above the Nyquist rate, a lter that can be tricky to build, especially if the order has to exceed two. The Nyquist frequency is 26 kHz for the DT3818 board when the sample frequency is set to the maximum (52 kHz). This range will well cover the magnetic frequencies for normal experiments. The yokes cannot be fed with frequencies much above a kilohertz (see section on frequency limits), and with a kilohertz fundamental the board still allows for twentyfour harmonics. There is a complication here, due to the fact that the ux density time signal is often at for a relatively long time (material is saturated) and then changes rapidly as the ux density passes the steep part of the hysteresis curve. To catch the steep parts of ux changes, which corresponds to spikes in the voltage that is really sampled by the ADC:s, the sample frequency cannot be set too low.
The 26 kHz Nyquist frequency is more of a limitation for interferometric signals. A fty hertz magnetic fundamental yields a hundred hertz magnetostriction fundamental that will be multiplied by the number of temporal bright fringes each strain cycle produces. For a large magnetostriction signal, something like fteen fringes can be produced from bottom to peak strain. That results in a three kilohertz interferometric signal, when the fringes are spaced equally in the temporal dimension. In that case, eight harmonics can be treated, which still is a fair number. The complication with rapid changes in ux density will also be seen in the interferometric signal, the fringes will be crowded around the rises and falls of ux. The sample frequency is most often set to the maximum 52 kHz for these measurements. To spread out the fringes in time, the peak value of the excitation can be lowered (simply by adjusting a gain option to the measurement program) so that
40
the material doesn't go so hard into saturation. If hard saturation is wanted, an exciting waveform can be created that passes the steep part of the hysteresis curve more slowly. The A/Dconverters do have to be fed by a constant signal during the time of the conversion. To provide this piecewise constant signal, a sample and hold circuit, in principle a solid state implementation of a switching transistor and a large capacitor, is provided built into the conversion circuits. The switch trigger signal is available at the A/Dboard. Conversion circuitry uses the now highly popular bitstream technique which, although avoiding signi cant bit errors, could introduce problems connected to jitter in timing signals, as the clock frequency is heavily increased compared to oldfashioned parallel converters. No such problems have been observed. The resolution is 16bits, which means that the data is amplitude resolved in steps of 1/65536. When Frequency analysis then is performed on stored raw data, one has to conceive the semiwhite digitizing noise thus introduced. For ordinary measurements, this does not cause any problem.
2.19 Signal bu ering
There are eight AD converters and two DA converters sampling and working in
parallel and they store and fetch data from two bu er queues (streams). Problems
can easily occur with these streams. If the queue handling is inadequately treated
by the program, bu er ll ups and emptying can happen before new bu ers are
placed in the queue (overruns). This is a bit tricky when the DA stream continu
ously should receive new bu ers to produce signals with very long durations, with
simultaneuous AD input bu er treatment. The library functions supplied with the
board (SPOX functions) that handle bu er gets and bu er puts don't return until
a lled or emptied bu er is available in the queue. The queues normally consist
of three bu ers each of equal length, and when choosing all eight AD inputs the
in queue will ll up much quicker than the out queue is emptied. This causes a
risk of in queue overrun while the emptied to switch in a new bu er
mtoaionutppruogt.raTmhewapirtosgfroarmandtoauctqbucanerhtaonbdleecotmhies
situation by calling the right number of in bu er gets for one or two out bu er
puts, and not initializing the out queue completely to provide room for out bu er
switchin without main program stalling.
The user has to be concerned about how many ADC:s he/she will use. If a DA waveform has 1024 time points, the bu er length might be selected to 1024 points also. If two DA:s are operated in parallel, a bu er will be lled with 512 points from each channel (channel samples are interleaved). That is ne, the program will then only use two bu ers to make a complete signal frame. The bu er length will hold
41
also for the AD side, meaning that the number of AD:s cannot be chosen arbitrarily. Eight AD:s (the most common choice) means that in the current example, 128 samples from each AD is placed in a bu er. That works. If ve AD:s would be chosen, they wouldn't ll up a bu er evenly. The last free space in a bu er would be lled up by three AD:s and two AD:s would have to try to place their samples in the next bu er. It won't work. The board cannot handle this uneven situation, and will lose data. Not just a few samples are lost but complete bu ers. If this happens by mistake, the acquired signals will contain strange jumps or spikes, that are characteristic of this problem. The simplest rule is thus to use a 1024 sample bu er length (for reasons below) and 1, 2, 4 or 8 ADC:s.
The bu er length also has to be set by the user with the bu en option. The natural length for the board is a multiple of 512 samples, with a minimum of 1024. Overly long bu ers will cause board memory ll ups. For example, 10240 samples per bu er gives 61440 samples of total stream bu er memory (2x3x10240). Each sample is a board CPU word which is two octets (PC bytes), which means that 120 kbyte of board memory is then used up. The total board memory is 256 kbyte, so too little room is available for SPOX functions, the main acquisition program and data stored for loading/unloading stream bu ers. The method for long signals is to segment the signal with the bufs option to the program. This option tells the program the ratio of the number of time points of the complete signal to the bu er length. For a single channel signal, it is the number of bu ers it would ll up on its own. The fundamental bu er length can then be kept short at 1024 samples (the usual number) which only gives 12 kbyte of stream bu er space. By creating the output signals with the pcgen program on the PC the lengths are easily adapted to a multiple of 1024. If the use of this program is impossible, the signals can be zeropadded to achieve a suitable length.
Two other arguments have to be given to the dtacq program, but no risks are asso
ciated with these. The rst is the number of signal frame repetitions to be put out.
By using large numbers, very long signals can be generated by the DA converters.
This is good for cases where manual observation, adjustment and experimentation
is necessary. The second option is the number of repetitions before the actual
measurement frame. Using a frame or two in front of the nally stored frame is a
practice to get rid of transients, when these are unwanted. The unmeasured frames
are called delay frames. Larger numbers of delay frames are useful when something
has All
otoptbioenms atonutahlelydvtearciqedproorgrinamuecnancedbeplriesctiesdelbyybienfovroekitnhge
actual measurement. the program without
any argument.
42
2.20 Measurement coil misalignments
Systematic errors in measurements can occur when the magnetic eld sensors are
not lined up strictly in parallel with the preferred and the transverse directions
of the siliconiron sheet. This has been foreseen, and it can be compensated by
two methods. The rst is that the Hsensors are mounted on a detachable board
so that the angle position of these is easy to change. To test that the sensors are
accurately lined up, one can make use of the magnetic
aonf dthH~e sshpeaecte.s
This means, are re ected
that if the trajectories of in the principal directions
eld the
trace re ection magnetic eld
ipnrotpheertB~y
of the sheet, the same trace
shape is obtained. By tracking a quasiDC measurement, one is therefore in the
position to determine the misalignment of the sensors by mathematically adjusting
the deviation angles with coordinate transformation, so that the re ection property
is obtained. Of course, this is simpli ed by very de nitely xing the relative angle
between the Rogowski coils to 90 degrees. This test method might also be put in use
to postprocess already measured, somewhat alignment erroneous, data. To be able
to do this, one must anticipate the possibility and remember to trace out the eld
twice, in 180 degrees opposition, and activating a complete history clearing there
in between. The measurement pair is then stored in parallel and the misalignment
angle is calculated from the angle deviation in expected eld and measured eld.
There is only a need for a single traceout if the intercoil angle is 90 degrees xed.
Misalignment of the Bcoils can only be compensated by the software method when
these are wound through drilled holes in the sample.
2.21 Using the measurement system 2.21.1 Magnetic measurements Hrd, Htd, Brd and Btd can be measured simultaneously during uniaxially alternating H or rotating H  excitations. For a nonoriented material such as DK66 this is easily done. For strongly oriented materials such as ZDKH there might be problems to rotate the magnetization vector out of the rolling direction. When exciting the transversal yoke in such a case, there could be a coupling between the yokes due to the sheet which is harmful to eld homogeneity and possibility of achieving a high peak ux. When only wanting transversal data, it is possible to detach a yoke and make a single yoke measurement. Demagnetization of the RD direction of oriented samples can be hard, due to the relatively high remanent ux density that occurs at zero eld strength. An alter
43
native is to demagnetize the sample in the transversal direction before the rolling direction.
2.21.2 Peak ux How does one get a higher B^ in rolling or transverse direction? It can quite easily be achieved by increasing the gains given to the data acquisition board for output signals. The Techron current ampli er can put 8 amps max into the driving coils, and with 200 turns of each yoke coil, there is plenty of excitation available. limiting factors are leakage (since the sheet under test is very thin), yoke cross coupling (noticeable for superoriented samples), and high reactance of coils especially at higher frequency due to a large number of turns (inductance).
2.21.3 Measurement procedure
The procedure for making a measurement is now described. The operator should
make the interferometer alignment on a dead (nonexcited) object rst, then place
him/herself invocations
abnedhidnadtathaercmhievainsugriesmauentotmsyastteedmbyPaCD. TOhSebwathcohleprsoegqruaemnc,ecaolflepdrmogeraams.
The user can interactively select yoke con guration and sample alignment (direc
tion). Two or three excitations are then on sequence, the rst with a very low
fundamental frequency for the operator to be able to view the interference on the
paper screen with the eye. If approved, the photodetector is inserted in its holder
and the interference can be checked on an oscilloscope, by using the second excita
tion that has a higher frequency. At this stage a ne tuning of the corner mirror
is possible to improve temporal fringe visibility. The nal measurement is done in
the third step of the sequence.
The programs that are called by the master batch program are pcgen for signal generation, exec3801 for downloading the data acquisition program dtacq to the DT3818 board, and a large program graph for post processing, interactive viewing, selecting and scrolling of channels to save or plot. Options to the dtacq program has been covered in section 2.19, and these should be set at the beginning of the meas batch script.
Viewing with graph is basically done in two modes, channel versus time or channel versus channel (hysteresis graph). Two yaxes are present, so a maximum of three channels can be on screen simultaneously. To change channel on an axis and to adjust zero position (scroll), tic mark increment (scale) and dominating scaling exponent, the numeric keypad is used. It is sectioned into rows for operation type
44
and into columns for axis. By pressing the key in the matrix corresponding to the axisoperation wanted and then pressing the + or  key while keeping the other key down, the change is commanded. While it might sound a little tricky when described, it feels very natural when actually using the two nger commands. Most convenient is to place the thumb over the matrix and the index nger over the plus and minus keys. Some clari cation of the matrix is appropriate: 1,2,3 changes channel (1 changes the channel on the left y axis, 2 the channel on the x axis and 3 the channel on the right y axis), 4,5,6 scrolls the channel along corresponding axis, 7,8,9 scales corresponding channel, NumLock,/,* changes exponent. Don't bother to learn it by heart, a help line is always on screen for you to remember, and it will soon stick to your hand.
A little speciality is when no channel is on the x axis, which make the graphs to be drawn against time or point index number. When scrolling the xposition in such a case, scrolling will be faster (coarser) so that the signal can be inspected in detail. Data points can be marked and unmarked by pressing a single key. Single keystroke commands are available for printing the viewed graph directly in Postscript on printer or le, saving the viewed channels in ASCII column format, to FTP a le to a UNIX host, to get signature (RMS, mean value etc) information, etc. The key commands available for the screen present can be read out from a help line.
Postprocessing to be done on signals is commanded by label substrings (extensions). Labelling has to be done to make the channels identi able for the operator. By using these extensions to the labels, extra key commands or options are avoided. The extension "dot" for example, marks a channel as being the time derivative of something, and that something will be formed by integration of the dot signal, and stored as a separate signal. The extension "mod" marks a modulated signal, that is demodulated and stored by the program. The batch program uses dot on pickup coil signals to get the uxes, and mod on the photodetector signal to get the strain as function of time.
A number of options can be given to graph program. In fact the command line will
be long as labels are also given as arguments. DOS has a limit of 127 characters
on the command line so a possibility for using option les has been programmed as
a workaround. The system user won't have to bother about these les since they
are created knowledge.
bOyptthioenms atsotegrrbaapthchcparnogbreamli,stbeudt
the just
existence of them by typing graph
is necessary without any
argument. There we can see how scale factors (multipliers) are given, that two
methods for integration constant determination are present (mean or median to
set the associated property to zero), that a power density time signal (the mean
value of which is loss) can be calculated out of B and H signals, and that eddy
current caused error on the H signal can be compensated from the B signal.
45
Chapter 3 Interferometer 3.1 Introduction The measurement of magnetostrictive strain is a di cult experimental problem. The strain information is retrieved by a noncontact homodyne HeNe laser interferometer. An overview of the interferometer can be seen in Fig.3.1, and a photograph can be seen in Fig. 3.2. The strain information is retrieved by a single noncontact interferometer, which illuminates a pair of sample micro prisms that senses the elongation of a 70 mm element. The MachZender beam path type used simpli es sample re ector placement but makes beam alignment more di cult as there are more degrees of freedom in the setup. That there is a pair of re ectors for all three strain measurement directions makes one assured that displacement recorded is relative. The laser is an intensity stabilized HeNe laser ( =633 nm) that can be switched into frequency stabilized mode if desired. The acoustooptical modulator (AOM) can facilitate intensity level alteration when the interferometer is operated in homodyne mode, and can impose a carrier on the temporal interference pattern to operate the interferometer in heterodyne mode. The sample test bed with feeding yokes is possible to rotate on a Te onglass ber weave, so strain components can be measured in turn while preserving the same excitation. 46
Figure 3.1: Overview of interferometer 47
Figure 3.2: Actual IFM setup 3.2 Homodyne interferometry cETaeh2uldes/eisndttcrbeeoynnss(gvi!tatyh2rtiao)istfiitosEhn1ree/cfeLormceoonbfscit(neh!ee1bdtmeba+eemaa!smu) re.aeldtm!teshntiesrtepotnhphgetotithcpoardaliennptcdaeitcphmtaolaersamisssuoaIrdmeu/mplalee(tnEirote1nb+eoecafEtmof2rr)es2eq(lwumeechintcercrriyoec bcpmoarlxisiccs(krm'=og)srp)oirsumiLlnsa=mod2pv.ehs=pParsaehe'cla2aimsnt=elig2ovdedaweuntmhlodaeotreedeaduicshlianbxtoteietoaihrsnmfeerr're.welnaaS=tcvoieemvalececeonomnsag(tltiIgrch1ier.b2ob=uIrnItp^a1itro2egin)snimvsa(ei'tns_ydd=Iisospf=!cltaa)hIlci1aeen2nmgcd+oegmnIIi0tv0b,eiiwsnsl ehasidstercrb(aoeiiennnIasi1tmt2iaan=/ils)t sensed as proportional current Ipd through the photodetector diode. Fractions of a wavelength are possible to resolve when a calibration measurement is done with at least one guaranteed (and manually eyeproven) complete fringe (i.e. L > ) from a perturbation of the optical path before the actual measurement. Fractions down to 1=100 are expected to be possible with initial phase shift precautions noted below, correct prism mounting insensitive to sample sheet bending at high peak ux densities and avoiding of phase uctuations (due to air ow, subsonic house vibration etc.). 48
3.3 Heterodyne interferometry In the interferometer lab, there is an acousto optic modulator (AOM) that can be used to operate the interferometer in heterodyne mode. The AOM imposes a carrier frequency on the intensity signal (which can also be seen as a frequency shift between the recombining beams) which results in bene ts discussed below. What is lacking to be able to try the mode is a rewrite of the demodulating code to handle the carrier (phase or AM demodulation depending on placement and feeding of the AOM). An alternative is to set an analogue demodulator (built for AM, also in the lab) in operation. With the acousto optic modulator (AOM) operating before the rst beam splitter one can write I / (a(t)E1 + a(t)E2)2 / I12 + I0 where a(t) / cos(!ct) is an oscil lating intensity modulation imposed by the AOM. This will yield a phase carrier 'c = !ct in the interferometric part I12 cos(' + 'c). The phase carrier will basically move the principal signal spectrum up to higher frequencies and it is possible to avoid LF noise (1/fnoise) that can be a problem for low amplitude strain signals. Another advantage is the fact that the phase carrier makes is possible to distinguish between elongation (phase retardation) and contraction (phase advancement) around the carrier, without need to resort to old contact measurements or theoretical results. For small strains L << (and with initial phase shift adjusted to =2 by the use of, e.g, a wave retarding plate in the reference beam path), I12 can be linearized to give an amplitude modulated photodetector signal. This is simpler to demodulate (with analog equipment) than a weak phase modulated signal. In the presence of digital signal processing capabilities, it is probably easier and more accurate to receive the complete HFsignal, reduce noise by ensemble averaging and phase demodulate. The upper limit of high frequency is set by the Nyquist frequency of the data sampling unit (26 kHz). The carrier can be fabricated by using one channel of the DT data acquisition board for example, and will then be highly stable. The phase carrier is sent to the RF driver and serves to modulate the 80 MHz carrier (of the driver) that supplies the AOM crystal with power. An alternative to make the carrier is to use a crystal controlled oscillator, but the frequency is then only changeable with frequency ( ip op) dividers or by manually swapping the crystal for another with a di erent resonance frequency. 3.4 Interferometer alignment As with all interferometers, a big practical issue is how to align the reference and measurement beams so that recombination of these will lead to an interference 49
as visible as possible. The ideal beam path is never attained in practice, since the measurement beam has to be somewhat varied in height and angle to strike the measurement object re ectors correctly. This object adjustment is done rst, followed by a beam parallelity adjustment. To succeed, the use of simple tools has been su cient. The tools needed are a mirror, iron at plates with punched holes, and a glass plate. All have been cut to t between the stabilizing mounting rods in the setup. The rods run parallel to the beam path on top of the rails that carry all optic components, so they provide a reference for alignment. The object adjustment consists of object re ector incident angle correction and spot height correction. Angle can be adjusted by rotating the sample support table (including the yokes) on the te on weave that sticks on to the granite table. Due to the low friction, this rotation is easily managed, even though the yokes weigh about ten kilograms. Height of spots on object re ectors can be corrected by slightly changing the elevation angle of launched beams by rotating the beam splitter. The elevation angle of the reference beam should be coadjusted (by rotating the corner prism) so that on the receiving side, the spot height of the measurement beam on the beam combiner1 is the same as the height of the reference beam on the corner mirror. Control of height at the receiving side can be done with two punched plates, that have the distance from rail or carriage to punched hole equal to the distance to the mechanical center of the beam guiding system. Another method, probably more convenient when the operator adjusts sitting behind the receiving side, is to check height above sample table with a glass plate or a plastic ruler. Position on horizon is adjusted by moving the carriages with the combiner and the mirror as passengers on the rail of the receiving side. Check of this position can be done with a punched plate close after the receiving components. Interference is possible when beams are parallel to each other. Visibility of fringes becomes higher when the spots overlap well, but overlap is not as crucial as parallelity. Laser spots have to be fairly well centered on the photodiode of the photodetector, which is xed at the mechanical center. Parallelizing the beams incident on the diode is done one beam at a time, with the rods as reference. One beam is blocked (with a free mounting plate on the sample bed for example) while the free mirror is put in place of the photodetector. An extra mounting plate is ready there to press on to the mirror and make it perpendicular to the rods. The misalignment of the incoming beam can then be monitored by sticking the glass plate into the paths and watching the di erence of spot position of the incoming beam and its mirrorre ected companion. The misalignment of the measurement beam can then be considerably reduced by rotating the tilt table on which the combiner is mounted. The reference beam is angle adjusted by turning the corner mirror. After 1A combiner is a beam splitter with two incident beams perpendicular to each other. Half of the incident beams will pass straight through the combiner, and half will be de ected by 90 . One straight passing part and one de ected part will make up a recombined beam. 50
removing the free help mirror and inserting a 15 mm focal length lens in its place, it should be possible to see spatial interference fringes on a screen raised somewhere behind the end of the rail. The aim is to get a single spatial fringe on the screen. The best adjustment screws are on the receiving mirror, it is a good choice for the last ne tuning. The nal state of alignment is viewed by the single black or red fringe on the screen. Dynamic interference (oscillation between black and red) can be tested by knocking on the diabase table, or touching/pressing inwardly on the ne adjustment screw of the corner mirror. As stated above, parallelism is most important and the operator should put down e ort on that property. A side e ect occurring when making the beams very parallel to the mechanical center is that re ections from glass surfaces will travel backwards into the aperture of the laser. Such retrore ections will make the laser unstable, a condition that is recognized by an audible signal from a relay switching on and o in the control and power supply unit of the laser. To avoid this condition, it is best to slightly misalign the combining beams with the mechanical center. The re ections from the help mirror in place of the photodetecor should produce spots on the support of the last object re ector. Then one is certain that retrore ections doesn't travel back into the laser. Due to the relative sparsity of degrees of freedom in the setup, the adjustments are slightly dependent on each other. Some iteration of the above procedure might therefore be required to achieve good interference. The steps that are simple to iterate are elevation change, carriage postion change and tilt table rotation change. If needed, the height on the launching side can be changed by inserting or removing thin spacers that are stuck between the laser head in its mount. When no big change is involved, the time to perform alignment is likely to be within half an hour, perhaps ten minutes for a trained operator. 3.5 Doppler e ect The Doppler e ect is the dependency of re ected beam frequency to the velocity of motion of the re ector and the frequency of the incident beam. This e ect is the time derivative view of the phase retardation description mostly used in interferometry. Since the Doppler view is directly connected with the speed of the measured object, it is used in velocimetry of e.g. uid ow. If the re ector speed is v, positive in the ray direction of the incident beam, the frequency at a point on the re ector will be !s = 2 (c ; v)= i, where c is the speed of light and i is incident beam wavelength. The re ected wave will have a frequency on the re ector of 2 (c + v)= r. From the equality of frequency of the two beams as measured on 51
the re ector, one arrives at
r i
=
1 1
+ ;
v=c v=c
1 + 2v=c
(3.1)
where the last approximation holds for small re ector velocities in comparison to the light speed, an assumption that was understood from the beginning. The frequency received by a stationary receiver is given by the reciprocal of the above equation,
!r !i(1 ; 2v=c)
(3.2)
.
3.6 Motion of measurement table
To have a thick table is good, since it reduces amplitude of modes that inject noise in the relative position of sample and reference re ectors. Modes are energized by ambient vibrations of the house and the humans in the house. Particularly, rotating converters in the cellar, walking in the lab room, cooling fans and hard drives to computers and human speech contribute to noise. Direct knocks on the measurement table give a ringing signal characteristic of granite, from which the table is made of. It can be concluded that damping of the material is poor, but it is compensated by a high mass density.
It is in order to go through the possible vibration modes and give some quantitative characteristics of the table in question.
The isotropic Hookes shear modulus and C
law can be written ni = C is the compliance matrix for
ni i normal
s=traG1inis,,
where
G
is
the
C
2 =4
;;Y1YY
;;Y1YY
;;Y1YY
3 5
(3.3)
The elasticity modulus Y is measured at uniaxial stress conditions. The inverse of the compliance matrix is the sti ness (or elasticity) matrix E,
E = (1 +
Y )(1 ; 2
2 )4
1;
1;
3 5 1;
(3.4)
For example, the elasticity coe cient E11 should be used when uniaxial strain is present. By multiplying the scalar factor with the elasticity coe cients the
52
o diagonal entries become the Lame constant = Y =(1 + )(1 ; 2 ). The constant describes the tension needed to counteract transversal contraction from an orthogonal stress and keep a given stress in the former direction. It is also noteworthy that the ondiagonal elasticity entry is not equal to Y , but (1 ; )= , a result from the de nition of the elastic modulus as the sti ness under uniaxial stress conditions. When isotropic, the material should respond the same to every uniaxial tension, regardless of its direction. This property gives a constraint on the shear modulus, it must be equal to Y=2(1 + ) which is in turn equal to =2, with being the second Lame constant.
The equations of motion for a continuum are
@x x + @y yx + @z zx = @t2u
(3.5)
cycl:2
(3.6)
cycl:
(3.7)
(3.8)
By expressing the terms with the isotropic Hookes law one gets
@x x = E0(1 ; )@x x + E0 @x y + E0 @x z
(3.9)
@y yx = @yG yx
(3.10)
@z zx = @zG zx
(3.11)
and by inserting the de nitions of strain one obtain
@x x = E0(1 ; )@x2u + E0 @[email protected] + E0 @[email protected]
(3.12)
@y yx = [email protected] + [email protected]@xv
(3.13)
@z zx = [email protected] + [email protected]@xw
(3.14)
where E0 is Y=(1 + )(1 ; 2 ).
By changing the order of di erentiation, completing the terms to get a derivative of the dilatation r ~u, and using the formula for the isotropic shear modulus, the sum of terms making up the left hand side can be written and equated as
Gr2~u + (E0 + G)@xr ~u = @t2u
(3.15)
cycl:
(3.16)
cycl:
(3.17)
2The cycl. symbol stands for an equation that is gotten by cyclical permutation of the indices
in the equation right above it.
! . yx
zy
Examples of such permutations are @x ! @y,
x!
y and
53
This equation system can be solved with the dilatation as the primary variable. By taking the divergence of both sides of the system, we get
(E0 + 2G)r2r ~u = @t2r ~u
(3.18)
This is a scalar equation in r ~u. The factor E0 +2G is equal to the diagonal entry,
say E11
E11 of the elasticity matrix. (j~k)2 = (;j!)2, which gives
A plane wave solution the wave speed cP = !k
h=asqthEe11d.ispTehresiionndreexlaPtiotno
the speed originates from the fact that dilatation is caused by an hydrolike pressure
acting on elements of the continuum.
At zero dilatation everywhere, there can still be waves of pure shear travelling. Inserting r ~u = 0 in the wave equation Eq. (3.15) one gets
Gr2~u = @t2~u
(3.19)
which yields the wave speed cS = qG. The shear wave speed is lower than the
pressure wave speed.
When examining solid pieces and their vibrational modes, it is interesting to separate longitudinal (horizontal) and transversal (vertical) modes. It is also relevant to take into account possible anisotropic properties. By inserting an orthotropic constitutive relation i = Eij j i0j = Gi0 i0j, into the equations of motion Eqs. (3.5) one gets
[email protected] j + @yG1 xy + @zG3 zx = @t2u
(3.20)
cycl:3
(3.21)
cycl:
(3.22)
where Eij are symmetric elastic constants, Gi are elastic shear constants and unprimed double occuring indices should be summed over.
By inserting displacements and changing order of di erentiation on shear derivatives one gets
[email protected] + [email protected] + [email protected] + (E12 + G1)@xyv + (E13 + G3)@xzw cycl: cycl:
= @t2u (3.23) (3.24) (3.25) (3.26)
3Numeric indices should be cyclically permuted like alphabetic indices, e.g. G1 ! G2.
! E1j
E2j ,
54
Suppose there is only one nonzero displacement component u. The equation system becomes
[email protected] + [email protected] + [email protected] = @2u
(3.27)
(E11 + G1)@[email protected] = @t2v = 0
(3.28)
(E11 + G3)@[email protected] = @t2w = 0
(3.29)
When senses
u propagates in the xdirection, i.e. E11 as resistance and its wave speed
aisloqngEi1t1u.diWnahl evniburaptiroonpaisgaptreesseinntt,hiet
ydirection, i.e. a transversal vibration is present, it senses G1 as resistance, and
the wave speed corresponds to that of a shear wave. If multiple displacement direc
tions are present, there will be coupling between the longitudinal and transversal
vibration types as indicated by Eq. (3.28) and Eq. (3.29) with nonzero right hand
terms.
In granite, the elastic modulus is Y = 65 GPa and Poissons ratio is = 0:125. The stone is probably quite isotropic due to a random distribution of crystallites, so the isGTshhPeeaa=r.nomT2r7hmo0eda0ullolkeunglsga/iscmttaui3ncd.iitbnTyeahdlceiwaaslgahcovuenealaarstlpewcedoaeevdteoicssGpietehn=eetdrYiesifs=oE2tr(1he11ecn+=Lc6=S)5==p=(17p27+92091G:01P192=a052.9)=7(T1207h0;0e=20m=50a::3s13s2:k35dm)ek/n=mss.i/7tsy7.
A standing transversal wave solution of a quadratic plate of side a is
w
=
w0
cos
pix a
cos
piy a
e;j!t
(3.30)
It ful lls free boundary conditions, and a real solution of displacement w is the
superposition of two complex conjugate solutions. The lowest resonance frequency
is gotten from reinsertion of the lowest mode solution in the shear wave equation,
wiosnh3ei:cg3he=tpgsi2vfer=ses;2=:G3 (pk1a2H)cz2aS.
.
2Fwor=a;gra!nr2ietsews.laEbxwpirtehssainsgidtehoef
frequency in the wave speed, 1 m, the resonance frequency
A longitudinal wave in a plate will have a faster wave speed and it will tend to
produce the rst resonance in the slab at a higher frequency than the transversal
waves. On the other hand, a longitudinal wave can only have one direction of
propagation (parallel to the displacement) and there is only two boundaries that
produce resonance, which will lower the frequency. The di erence in resistance
measure is the biggest rc2eLaso=na5n:3c=e2fr=eq2u:7enkcHy zi.s
though, so simply fres
t=hatc2Lae.
ect will dominate somewhat. This rst For the granite one meter slab, fres =
The slab is threedimensional and it is possible to make a wave propagate in the direction normal to the upper surface. A longitudinal wave in that direction will
55
make a resonance at f = cL=2c = 5:3=(2 0:1) = 26:5 kHz, where c is the thickness of the table.
Of course, the mode excited by a particular source is very much dependent on the frequency content and the location of impact/transmission of the source. What is demonstrated above is that the lowest resonances are in the upper end of the spectrum of interest when studying magnetics for electric power applications. Due to the low damping of the granite, it is also possible for a nonresonant disturbance to be harmful the ringing e ect mentioned before. When disturbances travel through the steel support rods to the table, the phase di erence between the legs will play a role for the possibility of exciting an eigenmode of the table top. If legs are vibrating inphase with equal amplitudes, the rst mode won't satisfy bondary conditions and will be suppressed, giving an actual lowest resonance frequency at twice the values calculated above. If legs are outofphase with each other, the lowest table top eigenmode can be excited. In the real setup, circumstances are complicated by the fact that there are three legs, and not four as was pictured when following the above line of thought.
The mass of the table top in uences the amplitude of disturbance waves. Assume
that the surrounding can !, that table top motion
carry out the work W+ on is primarily rigid and that
the table top at the frequency contact between support rods
and table top is always present due to gravity. From the formula W = mw_2=2
adW12umc+roi!ns=2ignwuWt02hssoe(ii!ndnt2ae!lx=ttd.ihs=apD2llfau)cwr;eionmrWgkena(ct0y)hocalf=elf)a.mmwT!pohlr2ikiwtsu02cgd=yie2vc,elwes(,0etqhtwhueiealallmwbtooeprlwkictoounprdnkeerefpcoterermdfoertdmo eotdnhebtywheotrhmkeaWmssas=iss
w0
=
1 !
r 2W+ m
(3.31)
If the mass is increased, the amplitude of the vibration will be decreased by the reciprocal square root of the mass. Compare to the case with a given force applied to the table, then mass in uences sinusoidal amplitude by reciprocal proportion. It is not completely clear how the rigid shaking will couple into an elastic vibration, but the elastic vibration will surely be lower when the rigid amplitude is made lower (for an otherwise unaltered setup). One can say simply that an increased inertia will give a lower sensitivity to external in uences.
The energy of the elastic mode of the table top determines the possible amplitudes of elastic vibration. No damping is assumed, meaning that the energy will oscillate between strain energy and kinetic energy with a constant sum of the two. Kinetic energy density is ukin = ~u_ 2=2. Strain energy density is u = E =2 for a longitudinal wave and u = G =2 for a shear wave. When considering the case of a transversely resonating quadratic table top one can express shear angles in the
56
outofplane displacement w as xz = @xw and yz = @yw, giving an energy density of u = @[email protected]=2 + @[email protected]=2 + w_ 2=2. By inserting the modal solution Eq.
(3.30) into the expression for the energy density, and integrating over the body, one
gets the body energy U of the table and w0 the
d=iscpla2cwe02mGenfotratmhpelitrusdtewohfotleheweadvgeems oofdeth. ec
is the table.
thickness Thus the
amplitude can be written
w0
=
r
c
U 2G
(3.32)
The mass density doesn't enter in the energyamplitude equation, but the shear
modulus does. The ratio of shear modulus and mass density determines the wave
speed and thereby a ects the resonating frequencies of the table top.
The constitution of the legs will a ect the transmission of unwanted vibrations to the table top. An elastic transmission without damping and gravity e ects terminated by the table inertia can be written as
wtabl
=
;2Y A=l + mtrans!2=2 (mtabl=2 + mtrans=2)!2 ; 2Y
A=l
wamb
(3.33)
where Y is crossection
the elastic modulus of area, length and mass.
the transmission (e.g. wamb is the ambient
leg), A,l and displacement
mtrans its (of farther
end of transmission) and wtabl is the displacement of the table top, which has the
mass mtabl. for the steel
Four rods
identical legs have used as legs are Y
been assumed in the analysis. The values = 200 GPa, A = 0:008m2, l = 1:2m and
mtrans = Al = 0:008 1:2 7500 = 71kg. The table top mass is mtabl = 300kg. At
a frequency of 1000 Hz, the ratio of resulting table top motion to driving ambient
motion will be (;2:7 109+1:4 109)=(5:9 109+1:4 109;2:7 109) = ;0:3. We can see
that the inertia of the table begins to dominate over the sti ness of the transmission
rods at the frequency considered, so the elastic properties of the rods have to be
taken into account. One also sees that although there is no material damping, the
displacement is damped thanks to the elasticity of the rods acting as a bu er for
vibrational energy, a bu er which emits its energy back to the surrounding during
the second half of the work cycle. For low frequencies though, the setup will appear
completely rigid and shaking will be fully transmitted (limited only by the available
energy of the source as described earlier). It should also be noted that there can be
horizontal vibrations entering from the oor in addition to the vertical ones treated.
The horizontal vibrations are more harmful as they can excite longitudinal modes
of the tabletop more easily.
It could be possible to include the properties of the underlying oor in the above transmission calculation. Tree has a high material damping, but it will yield to light loads such as humans walking. Probably the high frequencies will be damped out, and lower frequencies will penetrate as vibrations. Concrete seems to conduct
57
audible noise quite well, its higher sti ness will match the support rods to a higher degree and harmful sound picked up by the house can be injected into the setup.
The e ect of mass as inertia was treated above. Mass also has a gravity e ect. Gravity is constant in time and its forces and resulting motions will be superposed on all the timedependent forces and motions considered above. There will be a constant de ection of the tree oor under the setup, a bending of the table top and strained support rods. The importance of gravity on vibrations is that it keeps objects together, more or less well, and provides a path for vibrations to travel or interact. The most severe in uence is between the table top/sample holder and the sample itself. Even a simple rigid motion of the table top can make a light sample shake and cause distortions in measured quantities.
There might be a dynamical component on bending, too. By only looking at the
dynamical parts of the entities, one can write the equation of motion for an element
of a beam as
dT dx
x=
w
x
(3.34)
where T (x) is the lateral force on the left part of element cuts. Instead of a load force term q x as in the static case there is an inertia term ; wbc x. The biharmonic
equation for the dynamic de ection becomes
Y Iw0000 = ; w
(3.35)
where I is the area moment of inertia. A plane wave Ansatz gives a dispersion
relation
Y I(j~k)4w = ; (;j!)2
(3.36)
oqr
k2 YI
= pYI ! or cB
!. =
TqhYe
wave speed becomes dependent on I k. For a parallelepiped the area
frequency, moment of
c2B = !2=k2 inertia is I
= =
bt3=12, where t dicularly toqthe and cB = 65
is the thickness in the lateral direction and b is the
bending plane. The granite block has I = 1 0:13=12
109 8:3 2700
10;5
2
= 280m=s for a one meter wave. The
width = 8:3
1p0e;rp5emn4
resonance fre
quency for a one meter long granite block simply supported as a beam becomes
fres = cB= = 280Hz.
3.7 Laser
The laser is a SpectraPhysics Model 117A stabilized HeliumNeon laser. The speci cation of the laser head is as follows:
58
Dimensions: Cylindrical, 40.1 cm long, 4.5 cm diameter. Weight: 1.0 kg. Frequency stability during 1 minute: 0.5 MHz. Typical value somewhat lower, 0.3 MHz. Frequency drift vs. temperature: < 0:5 MHz/K. Temperature range, in which lock is maintained: 20 10 K. Intensity stability in frequency stable mode: Approx. 1 percent. Intensity stability in special mode during 1 minute: 0.1 percent. Frequency stability in intensity stabilized mode: 3.0 MHz during one minute. Output power at 632.8 nm: >1.0 mW. Typical value 1.4 mW. Frequency : 473.61254 THz, nominal. Beam diameter: 0.5 mm. Beam divergence: 1.6 mrad = 0.0917 deg. , full cone. Transverse resonator mode: TEM00. Polarization: Linear, >1000:1. The laser transition that supplies energy for the gain is very narrow, but is broadened by the Doppler shift caused by motion of the emitting atoms. For HeNe lasers, the width of the gain curve is approximately 1300 MHz. The number of longitudinal modes which might be running in a laser is determined by dividing the width of the gain curve by the mode separation (also called the free spectral range). The mode separation is c=2L, where c is the speed of light and L is the cavity length. For the 117A, there are two modes which operate in the cavity. When temperature of the cavity changes, during warmup or because of ambient changes, the wavelength will shift due to the requirement that 2L= = N, where N is an integer, must hold for the longitudinal modes. The wavelength change results in modes shifting along the gain curve to new positions with di erent amounts of gain. The control circuitry in the 117A monitors the intensity of each of the two modes. A feedback signal is developed to control the tube length. This results in a stable system with a controlled tube temperature. Beam rejection optics are employed to 59
ensure that only one mode is emitted from the laser. The rejection of the second mode is greater than 1000:1. Output instability might occur if retrore ections enter the cavity. When this occurs, the stabilized indicator on the power supply will blink. To correct, attenuate the beam (the re ected beam can be attenuated with a quarterwave plate) or slightly misalign the setup. The beam can also be attenuated with a simple aperture stop lever, which introduces an extended di raction line pattern orthogonally to the lever edge. It is unlikely that contamination will make its way from the outer aperture to the outer surface of the output mirror. The laser has been mounted to the base with a riser block of diabase, an aluminum spacer with tapped holes for rods which run through two Vblocks with 90 degrees veegrooves in which the laser head lies securely. Additional fastening is provided with 5 mm mounting plates pressing on top with nuts on the tapped rods. The diabase riser block is xed to the base by 6 mm rods running through 13 mm holes in the block and through 6 mm holes in the aluminum spacer, on which nuts rest. In the base the rods are fastened in tapped inserts, which have been cemented in an array of holes. The fact that the holes in the block are of a larger diameter than the rods, and also the length of the rods, makes it possible to slightly turn the laser approximately two degrees relative to the base. 3.8 The acoustooptic modulator The acoustooptic modulator is of Bragg cell type, an Isomet 1205C1, with the following speci cations: Spectral range: .442 ! 1. m. Interaction medium: Lead molybdate, PbMoO4. Acoustic velocity: 3.63 mm/ s = 3.63 km/s. Active aperture diam.: 1 mm. Aperture in cover: 2 mm. Center frequency: 80 MHz. RF bandwidth: 30 MHz maximum. 60
Input impedance: 50 nominal. Voltage standing wave ratio (VSWR): <1.5:1 at 80 MHz. DC contrast ratio: >1000:1 RF drive power: <0.6 W (at 633 nm). Bragg angle: 7.0 mrad = 0.401 deg. (at 633 nm). Static insertion loss: <3 percent (at 633 nm). Rise time: 180 ns (at 1.0 mm beam diam.). Modulation bandwidth: 1.9 MHz (at 1.0 mm beam diam., between freqs. with 0.5 in depth of modulation, MTF). De ection e ciency: 85 percent (at 1.0 mm beam diam. and 80 MHz RF frequency). The de ection e ciency increases somewhat with beam diameter, but the bandwidth decreases. At 2.0 mm beam diameter, the bandwidth is reduced to 1 MHz. The de ection e ciency is quite strongly dependent on incident angle, and one should also note that the de ection intensity is nonsymmetrical with respect to the undeviated, transmitted beam. The de ection occurs due to a di raction phenomenon, as a acoustic traveling wave is generated in the medium at the radio frequency provided, a wave which yields a refractive index undulation along the slab. Ideally, the AOM in the case of 633 nm incident light with a beam diameter of 1.0 mm would produce 85 percent intensity in the rst di racted beam on the same side of the normal to the slab surface as the incident beam, and 15 percent in the zero order transmitted beam. The second and minus one orders should be negligible. The AOM has been mounted on a linear stage, allowing one degree of positioning freedom. A spacer has been designed to hold the AOM with apertures coaxial to the optical axis in the mechanical setup. Therefore, to optimize the incident angle ( the optimum should be equal to the Bragg angle ) the incident beam has to be adjusted rather than the rotational position of the AOM. Since the incident beam is more determined by the requirements of no retrore ection into the laser and convenient positioning of light spots on beamsplitters and re ecting prisms, one often has little chance of performing such an optimization by adjusting the laser or rst re ecting prism. In any event, a de ection e ciency of over 50 percent intensity in the plus one order, compared to the intensity of the incident beam is quite possible to achieve. The SpindlerHoyer company o ers a tilt/rotation table which could be used with the AOM, trouble is that the size of the positioning 61
knobs require the table to be mounted from below with a quite thin spacer, a few mm thick. Maybe it could be worth trouble (and money) to gain some positioning freedom and see if the de ection e ciency could be optimized. The AOM is also perfect to obtain a variable intensity level to match the optimum of the photodetector. The intensity level is changed by an RF power potentiometer and a bias potentiometer on its power supply. The RF power should not be set too high, as that will decrease the deviated beam intensity rather than increase it. The bias pot makes it possible ( when talking about level matching ) to reach lower intensities than the power pot will allow, which might be interesting when experimenting with di erent load resistors to the photodiode. 3.9 Beam splitters and prisms The beam splitters used are Spindler & Hoyer cube types, 10 mm edge length. These are broadband antire ection (TBW) coated BK7glass pieces. Transmission percentages for normal incidence on the cube face are 55 for the parallel (p)component and 37 for the scomponent. Absorbance is less than ve percent. The angular beam de ection tolerance built into the splitting layer is only eight angle minutes. As a beam splitter is used to de ect the beam from the test object, the atnesses of the surfaces touched by the beam might be of interest. But as an angular tilt (relative to the normal of the test bed surface) of the object re ectors would cause a lot more optical path di erence in air (the hypotenuse path compared to the horizontal path) than in glass, the atness is of less importance. Beam splitters and prisms are mounted in a metallic insert using plastic screws. The insert (which has a cylindrical shape) is mounted in a cubeshaped holder. The simplest way of positioning the insert, and thereby the optic component, is with three thumbscrews acting normally with 90 degrees spacing around the cylinder. The screw, which acts on top, positions the component in the vertical plane, while the other two screws position the component in the horizontal plane, one counteracting the other and simplifying adjustment. The adjustment range is approximately one degree and is wholly due to play between the insert and the holder. A coarse vertical adjustment of the deviated ray is done with a simple rotation around the cylinder axis. There is a positioning ring to use for this purpose in a ner sense, which might also well replace the top thumbscrew for fastening. To summon it up, the positioning functions required are coarse adjustment, ne adjustment and fastening, and in the present setup, these are often provided all in one knob, which is not an ideal situation. 62
It might be thought that the adjustment procedure using thumbscrews is inadequate, and that one would prefer another type of insert with a tilt type of platform to fasten the raydeviating components on. One type of platform uses a small pressing xture to keep the component in place, which requires a matching dummy prism as a support for the re ecting prism. Another type of platform simply relies on the component being cemented in place. Doubleadhesive tape is probably too loose for that purpose  so in that case some type of glue has to be used, for example of the popular cyanoacrylate kind, which acts fast, is nonremovable and requires little preparation. Platforms with 45 degrees inclination to the incident ray are available, so no dummy support has to be made for the re ecting prisms though. One 90 degree prism has been replaced with a plane mirror mounted on a high resolution angle adjustment stage to facilitate beam recombination. The adjustment of the prisms on the launching side is mainly to get correct elevation angle of the beams and a correct height of the light spots on the receiving components. This adjustment can be adequately done with thumbscrews. Some adjustment of the test beds rotational position also has to be done, but as the yokes are clamped in the pole slots in the test bed, and more or less free hanging from there, this poses no problem. 3.10 Interference lter The interference lter is bought from Spindler and Hoyer. The measured individual characteristic shows a transmittance at 633 nm of 48 percent and a halfvalue width of approximately 10 nm (11.5 nm according to product catalog). The interference lter works by utilizing dielectric lms deposited onto colored glass substrate combinations yielding both re ective and absorptive behaviour of a thin (3.4 mm) plate. The side of the plate with the most re ectance (easily visually identi ed) is to be facing the light source in the mount. 3.11 Photodiode The photodetector used is a SpindlerHoyer model EBAT. This detector consists of a silicon photodiode S23865K, two 6V batteries of type 4 LR 44 (for cameras) and a 10 k load resistor in a cylindrical ( 25 mm) housing. The batteries provide a 12 V reverse bias voltage over the diode PNjunction and enhances sensitivity (it is possible to use the detector with the batteries disconnected, signal strengths will then be in the range of tenths of millivolts). If one is to design a noise resolution 63
limited interferometer, it is satisfying to know that a potential noise source such as a mains connected power supply can be avoided. If it is hard to get hold of camera batteries, one can insert and use one 1.5 V LR 6 / AA type, which is more common. The drawback is then that the usable range of the detector is cut down, since detector saturation occurs when the photocurrent through the load resistor causes a potential drop which is equal to the reverse bias voltage. A drawback with all battery operation is of course that one has to remember to switch o the device to save battery lifetime. The usable spectral range is 3201100 nm, with a spectral responsitivity curve as shown in Appendix C. For 633 nm operation, the spectral responsitivity is given as 4.3 V/mW, provided that the load resistor is the detector internal 1 k . With an external parallel load of 1 k , the responsitivity will be 4:3=10 10=11 = 0:39V=mW. Such a responsitivity is more convenient to use in the current setup, since it has been proven that the use of only the internal resistor will result in detector saturation at reasonable light intensities. This fact remains a little confusing since the laser will typically produce 1 mW of light output, and a nonoptimized AOM will leave approx. 50 percent in the used rst order beam. Using the original gures, one would have 2 volts of detector output at maximum intensity, well below saturation. The usage of the AOM allows one to continuously vary the used intensity level with a bias and a power potentiometer (the bias pot. enables the use of very low intensities) and it is empirically so, that rst order power has to be set to approx. a tenth of the expected value to avoid saturation, still yielding 10 V of detector voltage. With the external resistor in circuit, and the AOM giving full available rst order power, one receives 0.2 V as a max. To analyze that, one sees from the rst empirical case that a detector responsitivity of 50 V/mW seems more correct. The second case yields a responsitivity of 0.2 V/mW in orderofmagnitude agreement with the calculated one. As in the second case the internal resistor is almost completely bypassed by the external, one might wonder if the internal resistor is correctly speci ed. Further tests with di erent external resistances are necessary to check that. The photodetector is connected to an ampli er built around the famous TL072 operational ampli er. The circuit has the external 1 k as input impedance and a simple Feedback Circuit giving an ampli cation of 2.02 according to measurements of the feedback resistors. The output impedance is very low, since the output pin of the IC is connected to the output node of the circuit. The usage of an ampli er is totally necessary, since a direct connection of the detector to a regular ACcoupled oscilloscope input kills the signal (only a few millivolts pp will result instead of several volts). It is possible to view the signal on a DCcoupled scope, but when connecting to impedancewise unknown inputs, the bu er ampli er is always handy. In this case, the viewing input is the scope and the data acquisition board connected in parallel. 64
3.12 Demodulator An analog demodulator of AM detector type for the heterodyne photodetector voltage, which in the twofrequency beam case will be AMmodulated rather than phase modulated, has been built. The circuit is based around the popular MC 1496 IC from Motorola. This circuit will be necessary when the highest heterodyne components exceed the Nyquist frequency of 26 kHz of the data acquisition board. Also, to escape shot noise in the lowfrequency region, the carrier frequency may be deliberately selected higher than that.
3.13 Interferometer type The laser interferometer is of MachZender type. Such an interferometer is recognized by the two arms being parallel, in contrast to the Michelson (or TwymanGreen) interferometer that has the arms at straight angles. The arms of the interferometer are here de ned as the essential paths of the measurement beam and reference beam, i.e. to and from measured object and to and from reference adjustment mirrors. When the MachZender is viewed from the rst beam splitter to the beam combiner, beam paths ideally form a rectangle. This geometry was adequate in the setup because a twomirror relative measurement was wanted on the object, and a "through" beam is simpler to realize than a returning beam. Left to make a complete optic way is folding of the beam for practical purposes. This time a single ninety degree folding was done to the path to make the laser head t on the measurement table. To measure transversal contraction one can rotate the sample bed. By rotation of this bed also shear strain in the rdtdsystem can be measured.
3.14 Re ector placements and properties
The optical path in a prism can be easily determined by using a beam mirroring
technique, see Fig. 3.3. By mirroring the beam path in prism facets, one sees that
the beam length is determined by a straight line through a cube (when the prism
has the
90 and 45 optical path
aLng=leps)s. sInf =thceosbe,amwhleierse
in the plane of the s is the short side
re ecting facet normals, length of the prism, and
is the angle of the incident ray on the inside to the normal of the prism hypotenuse
where the refractive index in n (BK7 glass approx 1.53). is related to the angle
of incidence on the outside i through the refraction law, sin = n0=n sin i, where
65
Яi Я
Figure 3.3: The ray in a 90 prism mirrored into a straight ray through a cube.
n0 is refractive index of air (approx. 1). One also sees that the nally re ected
ray is parallel to the incoming, independent of the inplane rotation of the prism.
Another thing to note is that the path length is not dependent on the spot position
of the incoming a distance h2 =
light. h1 +
pT2hsettahnird
thing to from the
be noted is that the exit spot is 45 degree corner closest to the
located at exit spot,
when h1 is the distance from the other 45 degree corner to the entrance spot.
If the incident ray is oblique with respect to the plane of re ecting facet normals, one can without calculation see from the gure that the angle of the nally re ected ray from the facet normal plane is equal to the angle of the incident ray to that plane, like a plane mirror. The inplane ray projections stay parallel.
Expressing the path length inside the prism in the outside incidence angle, one gets
Lpr
=
p1
p2sn ; (nn0 sin
i)2
p2sn(1
+
1 2
(
n0 n
sin
i)2)
(3.37)
Ap0:rf2Lits1epmxrr2a=pca1hpt0ah2;n5w6g1,ei0lilf;.rebo3ame 0rz:ef2elt1arhoxtio2tvofel1axy0mp;cph6rmaamndgf=aoenrd1go:lw5enxdiet2hinmnmcarina.ddeSa.nomcIeaof udtmnhutieelltisroiadLadepirrao=enfLteophcrfteot=iprltrt(iiwsnlntm0i,lsltiihsngei5viien)ms2oimdnee, and a half nanometer of optical path change inside the prism, which is acceptable in the application currently considered.
66
What might not be acceptable is that, after cementing the prism to the sample,
the pivot axis of the prism is not the midline (parallel to the 90 degree corner) on
the receiving facet. In order to take advantage of the parallelity of the returned
beam, the prism is mounted on its 45 edge (via a small support glued to the
prism base), which could equally well be the true pivot axis. When using a ve
millimeter prism, the air path added when the prism tilts around a nonperfect
pivot axis can be signi cant. If the prism tilts around a 45 degree edge, forming
wanillabneglepo2fsn0(f1ro;mtannorm) sailnincwidheenrcee,
the added (or subtracted) is incidence angle on the
air path length inside, given by
the refraction law. One notes that the additional air path length is independent
oafppraroyxhimeiagthetdpboysiptio2ns.
When which
is
is in 7:1x
the milliradian range, the m when s is 5 mm and
expression is x mrad.
can be So one
can see that a milliradian of tilt can give 7.1 micrometers of added air path, which is
far too much when superposed on a translation signal in the submicrometer range.
It is possible to make the tilt sensitivity less by optimizing the mount angle of the prism. At higher angle of incidence the path length change inside the glass will compensate the air path length change, given a certain pivot axis. At 39.2 degrees from upright mounting (ninety degrees between sample surface and hypotenuse) the small tilt sensitivity is zero, assuming that the pivot axis is a 45 degree edge, and that the glass is BK7. At fourty degrees oblique mounting, the sensitivity is 0.15 m/mrad. While there is no way of knowing the pivot axis, such a compensation method can not be trusted.
Trust can only be gained by verifying that no rotation takes place by measuring the elevation beam angle deviation from a plane mounted prism (i.e mounted on the triangular side), and the azimuthal angle deviation from an edge mounted prism. A method to reduce rotation is to glue sheets together into a packet with a larger bending sti ness than a single sheet, and then measure strain of the packet. Another caution is to purchase prisms with as short side as possible, to minimize air travel when tilted in the edgemounted con guration. In the planemounted con guration, the light beam should be close to the sample surface.
67
Chapter 4 Strain analysis
4.1 Introduction
Strain analysis is important to make correct measurement analyses and to understand the nite element method. Furthermore, it is useful when studying models of magnetostriction, especially of the continuum kind. It is also included as a background for electrical engineers with a weak knowledge of solid mechanics.
4.2 De nitions of observables
The position vector of a particle of the body in the undeformed state is
~x = x1~e1 + x2~e2 + x3~e3 ( = x~ex + y~ey + z~ez )
(4.1)
The position vector of the particle in the deformed state is here denoted by a prime,
~x0 = x01~e1 + x02~e2 + x03~e3
(4.2)
The vectorial distance between particle A and particle B in the undeformed state
is
~x = ~xB ; ~xA
(4.3)
which can be seen as the length element connecting A and B. The length element in the deformed state is denoted
~x0 = ~x0B ; ~x0A
(4.4)
68
B uB du B' dr' A' uA A
B du dr' A
Figure 4.1: (a) Displacement vectors from particles in undeformed state. (b) Distance vectors o set from reference particle.
The displacement of a particle from the position in the undeformed continuum to the position in the deformed continuum is
~u = ~x0 ; ~x = u1~e1 + u2~e2 + u3~e3 ( = u~ex + v~ey + w~ez ) (4.5)
The relative displacement is
~u = ~u(~xB) ; ~u(~xA) = ~x0 ; ~x
(4.6)
The relative displacement expresses the displacement of the particle B relative to the particle A. The second form of Eq. (4.6) is the most useful, since it can be thought of as the vectorial change of the AB length element due to deformation or rotation.
The displacement gradient tensor expresses the limes ratio of components of relative displacement to components of length element in the neighbourhood of a particle given by the position vector,
r~u]ij(~r)
=
@ui @xj
(~r)
(4.7)
This entity is the basis for the analysis of small deformations, which will be assumed in the following. One has to note that the neighbourhood of a particle can undergo local rigid rotation as well as true deformation (strain), both which will be described by the displacement gradient. To separate the strain, an additive decomposition of the displacement gradient can be performed,
@ui @xj
=
ij + !ij
69
=
+
Figure 4.2: Rigid rotation + Strain
ij
d=ef
1 2
(
@ui @xj
+
@uj @xi
)
!ij
d=ef
1 2
(
@ui @xj
;
@uj @xi
)
The The
rsotrtaaitniotnentesnosro$r $i!s
symmetric and thus contains six independent components. is antisymmetric and is without diagonal components in its
matrix representation, leaving three independent components. This kind of tensor
can be represented by a vector ~!, using the following assignment rule,
!ij = ;"ijk!k
(4.8)
where "ijk is the permutation symbol1 and summing over k is implicit. Performing the assignment, one sees that the rotation vector can be written as
!~ = 12r ~u
(4.9)
which is also called the curl of the displacement eld. It can be shown that the absolute value ! is the turning angle of the neigbourhood to the axis of rotation which is parallel to ~!. To interpret the decomposition of the displacement gradient, a picture of the deformation of the neighbourhood of a particle says more than a thousand words, see Fig. 4.2.
We now focus attention on the deformation of single length elements. The linearized relative displacement of a length element is written as
d~u
rot
=
$!d~uds~xtr==~!$
d~u d~x d~x
=
d~u str + d~u rot dui = ijdxj dui = !ijdxj = "ijk!jdxk
(4.10) (4.11) (4.12)
and1T0hfeorpaelrlmelusteatiniodnexsycmomboblin"aitjikoniss.1 for indices 123,231,312, it is 1 for indices 132, 213, 321,
70
dustr
du
durot
l0
xxl0
Figure 4.3: Linearized relative displacement
where the left column is in tensor notation and the right column is in component notation, with implicit summing over indices occurring twice in factors. The interpretation is in Fig. 4.3.
We are now in the position to de ne the normal strain,
~n
d=ef
~
d~u dx
=
lxim!0 ~
~u x
(4.13)
where x = j ~xj is the undeformed lenght element, and ~u is the relative dis
placement as de ned earlier. Normal strain is the fractional length increase in the
direction ~ of a di erential length element originally directed as ~. Expressing this
with the strain tensor we get
~n = ~ $ ~ = i ij j
(4.14)
wherefrom one sees that the normal strain is a quadratic form of the direction cosines i with the strain tensor components as coe cients.
After having de ned the normal strain, one may note that a description of the strain orthogonal to d~x is missing. Rigid rotation contributes to the orthogonal relative displacement, so we have to compare the relative angle change between two line elements to be able to separate orthogonal strain. This strain is called shear strain. The rectangular area element is convenient to use to illustrate shear as well as normal strain, as is done in Fig. 4.4, where a unit area element is pictured (the unit length may of course be arbitrary small). The relations for normal strains
71
y
1 +1 yy
x 1 1+xx
Figure 4.4: Normal strains and shear angle +
in coordinate directions and shear strains between those directions are
xx =
11
=
@ux @x
(4.15)
xy =
12
=
1 2
@ux @y
+
1 2
@uy @x
(4.16)
Expressing the shear strain in the decrease from the straight angle, we get
xy
=
1 2
tan
+
1 2
tan
1 2
+
1 2
(4.17)
where the approximation holds for small strain theory, of main interest here. Note that and cannot be individually determined only from the strain tensor, since rigid rotation might contribute. The sum is not a ected by rigid rotation, though. The shear strain is often represented by the shear angle ,
xy d=ef + = 2 xy
(4.18)
The shear angle is not a tensor component, since it does not obey the tensor component transformation law when changing coordinate system.
Cubical dilatation is a useful entity that describes the fractional volume change of a unit parallelepiped (or brick) volume element,
D0 (1 + xx)(1 + yy)(1 + zz) ; 1 xx + yy + zz = tr( eij]) = kk (4.19)
72
We see that the volume change for small strain theory is adequately described by
the trace of the strain matrix (the sum of the diagonal elements). It is possible
to decompose the strain tensor into deformation without volume change,
aanddevaiastpohrerdiijc,althpaatrtisthraetspisonassisboleciafoterdshwaipthe
uniform volume change,
ij =
dij +
ij
1 3
kk
(4.20)
4.2.1 2D strain measurement analysis
To obtain the complete strain of the surface of a specimen, measurements in three
directions has to be made (three independent nonzero components exist). If one
expands the normal strain quadratic form one gets
~n = 11 12 + 2 12 1 2 + 22 22
(4.21)
so three measurement directions will give a linear system of simultaneous equations
for determining two normal strains in reference strain. As an example, if we measure with 45
coordinate directions angle separation, 1
and the = 2=
1sh=epa2r
and the shear is
12 =
4n5
;
1 2
(
11
+
22)
(4.22)
where 11 and 22 are already given by the orthogonal measurement directions. 60
angle separation is also a possibility, which is called the delta con guration.
If the full set of tensor components are at hand, a description of normal strain and shear strain of every rotated surface area element at the point in question is possible by means of a local coordinate system transformation. Let ' be the angle of the rotated coordinate system (primed) to the reference coordinate system. Use the transformation law
x0ij0i
= =
aij xj aikajl kl
(4.23) (4.24)
where aij is a direction cosine between the i:th primed coordinate direction and the j:th reference coordinate direction,
aij] =
cos ' sin ' ; sin ' cos '
(4.25)
Performing the transformation for the components of interest we get after expressing in the double angle 2',
011 012
= =
1 2
(
12
11 + 22) cos 2' ;
+1112;( 2
11 ; 22) cos 22 sin 2'
2'
+
12 sin 2'
(4.26) (4.27)
73

+
+

+ 
+
Figure 4.5: Polar plot of 011(') and 012(')
whereafter one may instead of the cosinesine sum use a single cosine with an argument shift,
011 012
= = =
1 2
(
11 +
22) +
cos(2' ;
cos(2' ; ( ; =2))
r (
11 ; 2
22 )2 +
212
)
(4.28) (4.29) (4.30)
=
arctan
(
11
12 ; 22)=2
(4.31)
~tercIttohea02feneintshrosseehrnenmeeorca,ewearfleasersrtaeterrnsaiaaysciienenat.loebaTmenedchtdertwenaezetdweee,xrnbotwaryxhseph01miacoeshaalayanrmorpdsfrmptosxrlhdae02oeutitndracioriecorfssenntctartohtahiremoimennanpasxiorsliarimmmpsstruer'aoamdldiinsaus~etsrcra01eeaaladtninneeoodrlrneeimndmba,yaetrslhneseatsehetieerxnFalae01criimglnids.n,etier4nardea.t5cmsiant.itiinonAo'ncinsmli=anoutnehnmddee at 45 to the extremal normal strain directions.
Further study of symmetry can be done as shown in Fig. 4.7
The polar plot is unsuited to simple graphical determination of strains in di erent directions, as a state of strain with both negative and positive values of extrema rBdeeyseclocritobskeiistnsgaelafctiortnchleqeupaiasteirthc(oeem01d1poleue01bx2l)eptoihnlaocruliglnohao,tpioosn,neaaissniasgblseleee2nt'obgyontedhsetthhsahrtoeatuhrgehplloo0ct,u2fsoor]f.etxShaliismgphptalleiyr.
74
y
Negative shear in yx'system x' Positive shear in x'y'system
Positive shear in xysystem x Negative shear in y'xsystem
y' Figure 4.6: 90 antisymmetry of shear strains.
y
Positive x' x' in x'y'system
Positive x in xysystem x
y' Figure 4.7: 180 symmetry of normal strains 75
=2 ( 2 0)
( x xy=2)
2'p x+2 y
( 1 0)
Figure 4.8: Mohr's circle for normal and shear strain in the xy plane. The xy plane is perpendicular to a principal strain direction. 'p is the angle from the xdirection to the direction of the principal strain 1.
rewriting the expression for the shear we de ne what is known as Mohr's circle,
(
1 2
(
11 +
22) +
cos(2' ; ) ;
sin(2' ; ))
(4.32)
and plot it in Fig. 4.8.
4.2.2 Deformation of volume elements
So far, we have mostly studied strain of area elements, and this we have done parallel to a given plane (spanned by reference basis vectors ~e1 and ~e2). The analysis in three dimensions can simply be based on three area elements undeformed being at straight edges to each other at the point of study, which is corresponding to a volume element of brick type.
To get the full picture of the state of strain three Mohr's circles have to be drawn. In doing that, it is convenient to rst nd out the angles of extremal normal strain and zero shear, which are called the directions of principal strain. This is done by noting that for zero shear,
ij j = k i k 2 I II III
(4.33)
which means that the eigenvectors of the strain tensor are principal strain directions with absolute values I II III. By choosing the eigenvector system as the local reference system, we count the tilt angles of areaelements from these basis vectors
76
=2
2
3
1
Figure 4.9: Mohr's circles for a complete strain state, three planes perpendicular to each other and to principal directions. and the double angles from the normal strain axis in the Mohr's circle diagram. Example for a state of plane strain (zero normal strain in the IIIdirection) is shown in Fig. 4.9. Here it is important to note that the maximum shear strain can enter in a plane parallel to the zero normal strain direction.
4.3 Stress and 3D elastic material relations
The deformation of the media of our concern magnetostrictive strain and thermal expansion.
are A
caused by mechanical
tmraecchtiaonnicvaelctsotrre~tss~n,
is de ned as the mechanical force per unit area acting through the area element
with normal ~n at the point in question. The mechanical state of stress at the point
is then de ned by in nitely many traction vectors on the complete set of normals
in the tensor
p$oidnet.
Thus, ned as
the
state
of
stress ij
is =
more attractively ~t ~ei ~ej
described
by
the
stress (4.34)
i.e., component the ~ej direction,
ij is on the where ~ek k
area 2 f1
e2le3mgeanrtewthitehCnaorrtmesaial n~eibaastisravcetciotonrscoamt tphoenpenotinitn.
The traction vector on an arbitrary area element can then be written as
~t ~n = ~n $
tj = ni ij
(4.35)
The mechanical stress physically corresponds to elastic (or possibly plastic) interactions in the lattices of the polycrystalline solid. If we restrict ourselves to the
77
yx
; xy
xy
; yx
Figure 4.10: Moment equilibrium on an area element
elastic case, we de ne a linear elastically isotropic constitutive relation by
x
=
1 Y
x;Y
y;Y
z
y
=
1 Y
y;Y
z;Y
x
z
=
1 Y
z;Y
x;Y
y
xy
=
1 G
xy
yz
=
1 G
yz
zx
=
1 G zx
where i denotes normal stresses, ij denotes shear stresses (stress components
parallel to the area element the action goes through) and the analog notation is
used for strain. Here some points may need to be clari ed. First we note that only
six components of the stress tensor is used, and that is due to the fundamental
assumption of local moment equilibrium, which leads to a symmetric stress tensor,
exlayst=icityyxmeotdcu.luFs iYg.
4.10 best illustrates the derivation of this property. describes normal strain due to parallel uniaxial normal
The stress
(i.e. only one normal stress component nonzero). Poisson's constant describes
the ratio of lateral contraction to longitudinal elongation. The shear modulus G
describes a proportionality of the shear stress to the resulting shear angle. In
this speci c case, the shear modulus can be shown to be expressible in the other
78
constants,
G
=
Y 2(1 +
)
(4.36)
due to the fact that pure shear (no normal stresses) on an element can be expressed as pure normal stresses on a rotated element, the deformation of which is describable in terms only of Y and when isotropy (and linearity ) holds.
An orthotropic material is characterized at a point by three mutually orthogonal directions in which the elastic moduli are extremal2 ,so called principal material directions, and the elasticity for directions re ected in the planes normal to the material directions is symmetric. A shear stress won't a ect the normal strains in the coordinate system of shear stress application, just like the isotropic case. The orthotropic material relation is here given by
8 >>>>>>< >>>>>>:
x y z xy yz zx
92
>>>>>>= >>>>>>
=
6666664
EEE111123 0 0 0
EEE122223 0 0 0
EEE123333 0 0 0
0 0 0 G12 0 0
0 0 0 0 G23 0
0 0 0 0 0 G31
38
7777775
=
>>>>>>< >>>>>>:
9
x y z
>>>>>>=
xy yz
>>>>>>
zx
(4.37)
where ~ei i 2 fx y zg are the material principal directions at the point. We see that nine independent coe cients are used in this case. The most general linear anisotropic material would need twentyone coe cients in the matrix relating local stress to strain, since symmetry of the matrix is sound. A symmetric elasticity matrix is an expression of reciprocity of the material, meaning that an excitation (a given normal strain in a given direction, say) will in uence the measured conjugate quantity (the normal stress in a speci c direction) in the same manner as if the directions of excitation and measurement would be interchanged.
The elasticity matrix is readily inverted to give the compliance matrix. Elasticity coe cients describe the sti ness of the material, which might be slightly unsuitable in certain applications, where the inverse property is more adequate. Looking back to the description of the linear isotropic material, we see that it is stated as a compliance relation, with stress components as input  often regarded as more intuitively attractive. One also has to note that the elasticity modulus introduced in the compliance relation as the reciprocal of a compliance coe cient is not directly identi able with an elasticity coe cient, since lateral contraction ratios (and more generally nondiagonal compliance matrix elements) will in uence the diagonal elasticity matrix elements. 2The directions are mutually orhogonal when the orthotropic elasticity matrix is symmetric.
79
4.4 2D elastic material modelling
Elastic behaviour for a GO material can be written with a matrix,
2 4
xEExEyy
32 5 = 64
;Y01Yxyxx
;Y01Yyxyy
0 0
32 75 4
3 x y5
G1xy
xy
(4.38)
wamlhseoea)rs.eurTeEdheiastdtsihtaeagtoeenlsaaosltfieculnepmiaarextniatolsfsYtthrx;ees1ss.tarnTadihneY(mya;sa1ttrahirxeeraetbhmoevigerhecctoinpbtreaoaicnasmlstahogefnceeotlaomseptlailcisatmniccoepdcauorlti e cients for an orthotropic material. For GO electrical steel, Yy = 200 GP a Ex = 150 GP a xy = 0:4 yx = 0:3 Gxy = 75 GP a is a starting point for numerical experimentation. Data given by Surahammar AB in their product catalogs show the elastic modulus as a function of angle to rolling direction. When applying an uniaxial stress in a direction other than the x or y direction, the compliance matrix can be transformed to the coordinate system with the uniaxial direction as x' direction to get the strain parallel to the stress axis (called normal direction), the strain orthogonal to that axis and the shear in the x'y' system. Such a transformation has been carried out in section 5.13 for the magnetoelastic case, but it holds also for the purely mechanical case. By plotting the so produced compliances over angles, the model can be compared to catalog data. Such a transformation has been carried out for the values given above. The polar plots based on those values are seen in Figs. (4.11), (4.12) and (4.13).
Due to the texture, the principal strain coordinate system won't be codirected with the principal stress system, see Fig. 4.14. There will be an angle between the stress axis and the largest strain axis. The angle is nonzero for stress neither aligned with the preferential direction of the texture nor at right angles to the preferential direction of the texture.
4.4.1 Magnetostriction components and constitutive relations
Magnetostriction lambda is a fractional elongation of a solid ferromagnetic piece
due to homogeneous magnetization with ux density B. For nonsaturating uxes,
a parabolic expression
=
1 0P
B2
(4.39)
seems to be adequate to describe the main quantitative feature of the phenomenon.
Noting that B2 might be perceived as a magnetic stress, the following simple
80
150 180
Normal elastocompliance [1/Pa]
907e12
120
6e12
60
5e12
4e12
3e12
2e12
1e12
30 + 0
210
330
240
300
270
Figure 4.11: Normal elastic compliance as function of angle of uniaxial stress to rolling direction.
81
150 180
Orthogonal elastocompliance [1/Pa]
902e12
120
60
1.6e12
1.2e12
8e13
4e13
30 0
210
330
240
300
270
Figure 4.12: Orthogonal elastic compliance as function of angle of uniaxial stress to rolling direction.
82
Shear elastocompliance [1/Pa] 901.2e12
120
60

+
8e13
150 +
4e13
30 
180
0
210 240
+

270
+ 330 300
Figure 4.13: Shear elastic compliance coe cients as functions of angle of uniaxial stress to rolling direction.
1
2 Figure 4.14: Uniaxial stress applied obliquely to a texture. Shows rotation of the principal strain system 1 2 compared to the principal stress system. 83
isotropic magnetoelastic constitutive relation is proposed
Mx
=
1 BxBx ; 0
0P ByBy ;
BzBz 0
(4.40)
cycl: cycl:
(4.41)
xMy
=
1 Q
1 BxBy 0
(4.42)
cycl: cycl:
(4.43)
By analogy with pure elasticity, P is called the magnetoelastic modulus, Q is called the magnetoelastic shear modulus, and is the magnetoelastic transversal contraction ratio. This constitutive relation is isotropic and linear in a magnetic stress tensor BiBj i j 2 f1 2 3g.
P can be quite widely varying for di erent ferromagnetic metals and alloys, but is
probably often very close to 0:5, since volume magnetostriction (i.e magnetostriction
accompanied by volume change) is seldom seen even for high eld strengths. Volume
magnetostriction is characterized by a nonzero cubical dilatation magnetostriction is then characterized by D0M = 0, which means
D0.
No
volume
D0 = Mx + My + Mz = 0
(4.44)
01P Bx2 +
01P By2 +
01P Bz2 ;
0
P
(By2 )
+ Bz2 + = 0:5
Bz2
+
Bx2
+
Bx2
+
By2)
=
0
8Bi
In solid mechanics, materials with a Poisson ratio of 0:5 are called incompressible, where compression is in the hydrostatic (volume) sense.
The magnetic stress 1 B2 is for 1 T ux density (4 10;7);1 0:75N=mm2. One can compare to st0eel with E = 200 GP a at an elongation of 1 m/m, giving = 0:2 N=mm2. So even though same magnetic stress and elastic stress will produce di erent strain responses due to the di erent nature of the mechanisms in the material, the stresses for strains and uxes in the range expected will not be far apart in order of magnitude.
As previously said, it has to be noted that the constitutive relation 4.40 is suited only for unoriented, isotropic materials, such as those siliconiron alloyed cores used as ux conductors in electrical machines. Oriented siliconiron is used in large generators and transformers, therefore an extension to orthotropic conditions is useful. If ones uses the symmetric magnetic stress tensor BiBj, one can write an
84
magnetostrictively orthotropic relation as
8 1 >>>>>>< 0 >>>>>>:
BBBBxBxyz222 y ByBz BzBx
92
>>>>>>= >>>>>>
=
6666664
PPP111123 0 0 0
PPP122223 0 0 0
PPP123333 0 0 0
0 0 0 Q1 0 0
0 0 0 0 Q2 0
0 0 0 0 0 Q3
38
7777775
>>>>>>< >>>>>>:
MxMyMzxyzMMMxzy
9 >>>>>>= >>>>>>
(4.45)
4.4.2 Elasticity and compliance matrices
The above section stated magnetoelastic relations with a compliance formulation for 2D and an elasticity (sti ness) relation for 3D. It is of importance to have both compliance and sti ness matrices ready for use in various cases of analysis, like with the nite element method, treating cubical dilatation and strain descriptions in di erent coordinate systems. The inverse of the elasticity matric is called the compliance matrix.
When the shear components are simply related as in the preceeding section, a relation which holds for the coordinate axes directed parallel with the structure axes in the material, it is su cient to look at the matrix P relating normal components,
P
=
2 4
PP1112 P13
PP1222 P23
PP1233 P33
3 5
(4.46)
The inverse can be found by writing the cofactor matrix and dividing by the determinant,
cof(P )
=
2 4
;(PPP121222PPP233333;;;PPP121333PPP222233);;P((PP11111P2PP323333;;;PPP11323P3PP113123);PP(P11211P1PP223223;;;PPP21212P2PP113123)
3 5
det(P ) = P11(P22P33 ; P23P23) ; P12(P12P33 ; P23P13) + P13(P12P23 ; P22P13)
P ;1
=
1 det(P
)
cof (P
)
It is seen that both P and its inverse are symmetric. The cubical dilatation can now be written
D0 =
1 0det(P
)
fBx2(P22P33
;
P23P23
;
P12P33
+
P13P23
+
P12
P23
;
P13
P22)
+
By2(;P12P33 + P23P13 + P11P33 ; P13P13 ; P11P23 + P13P12) +
Bz2(P12P23 ; P22P13 ; P11P23 + P12P13 + P11P22 ; P12P12)g
85
;PA2b2=n(o=ate;Pa3bb3)o(=aut+at2hbPe)1.c3aO=senePw2ch3aen=nstePhe1e2trhe=aitsba,isptohotersoittpirvyaenisbsvseinmrsaeplllsaetrciet.nheasAnsftaceorgeisveectsiteianntgnewPgi1al1ltibv=ee (contractive) transverse compliance coe cient. The positive transverse sti ness coe cient describes a sti ening e ect between orthogonal directions, at a given state of strain there will be a transverse contraction tendency requiring a larger orthogonal stress to provide the strain. The determinant of P is (a ; b)2(a + 2b), and zero for b = ;0:5a and a = b, for which the sti ness matrix is not invertible into a compliance matrix. If the sti ness matrix is not invertible, stress cannot appear in any state, there are restrictions on allowable states. As restrictions are more likely to exist on the allowable strain states, like no dilatation or no shear, it is more likely the compliance matrix that is singular. With negative o diagonal entries (necessary in elasticity, not strictly necessary in magnetoelasticity), zero dilatation will happen at o diagonal entry value half of diagonal entry value.
Even though it is the compliance matrix that is more important for analysis, and a bit simpler to understand, the sti ness matrix is needed to form the local sti ness matrix of the FE method. Whatever is given, we need to form the inverse (if possible). A simple approximation can be derived for the 3D case.
P ;1
2 ;; 64
PPP11PP1111111PP232323
;; PPP12PP1212221PP322323
;; PPP12PP1312312PP333333
3 75
(4.47)
From this approximation, one can see that there are six orthogonal contraction
ratios in the orthotropic case, P23=P22. One can also deduct coe cients when there is never
P12=P22, P13=P33, P23=P33, P12=P11, P13=P11 and the approximate relations between the compliance any dilatation: P12=P22 + P31=P33 = 1, cycl., cycl.
Exact inverses are simplest to obtain for 2D cases. Isotropic compliance matrix to elasticity matrix inversion can be written as
;Y1Y
;Y1Y
;1 =
1 1;
2
Y Y
Y Y
(4.48)
with the trivial shear part left out. Orthotropic elasticity matrix to compliance matrix inversion is written as
EE1112
EE1222
;1
=
1 E11E22 ;
E122
;EE2212 ;EE1112
(4.49)
Because the inverse of the symmetric matrix is also symmetric, the relation can be used also for compliance to sti ness inversion.
86
The entries of the inverse of the 3D general matrix P above can be calculated to
P1;11
=
1 P11
(1
;
P223 P22P33
)=N
P1;21
=
;
P12 P11P22
(1
;
P23P31 P12P33
)=N
P2;31
=
;
P23 P22P33
(1
;
P31P12 P23P11
)=N
P3;11
=
;
P31 P33P11
(1
;
P12P23 P31P22
)=N
N
=
1
;
P122 P11P22
;
P223 P22P33
;
P321 P33P11
+
2
P12 P11
P23P31 P22P33
(4.50) (4.51) (4.52) (4.53) (4.54)
where N is a help constant for the denominator of the components of the inverse. In deriving the inverse, it is helpful to note that only two cofactor elements need to be calculated, namely cof(A)11 and cof(A)12, wherefrom the other elements can be gotten from cyclic permutation of indices.
Reciprocity can be investigated more simply with the approximate inverse. If a
traction t direction, direction.
is applied in the zdirection, say, there will be a
an equivalent stress One cannot call the
PPin3313
t in the xdirection and a uence on the orthogonal
dssttirrreaaciinntioPPn111313stPPtr33ie13nststihnbeetchsaaeumxsee
it is natural and stress free. After swapping source (traction) and measurement
(TPs13rt3arnaPPis11n31v)terdisniertechcoetnioztnrdasicrwteiecotngioertna.atniSoosuntdhieaesxcfiaraiclbtsettrhseatsrtsaPiinn13tah=terPxig31hdtimreaacnntgiiolfeensstttshoaittasewlgfiilvilnegnrivesectiraparsiontcr,iatsiyno.
they are less fundamental than the reciprocal moduli.
4.5 Equations of equilibrium and motion
4.5.1 Force equilibrium
The 2D force equilibrium equations are
@x x + @y yx + fbx = 0
(4.55)
@x xy + @y y + fby = 0
(4.56)
The rst equation of these expresses force equilibrium in the x direction, while the second is for the y direction. i i = x y are normal stresses on the side of a cut which has a lower coordinate from the side with higher coordinate. The cut has a
87
;x
; xy
; yx
+y
yy
+ yx
y yx
+ xy
x xy
Tv
fv
+x
xx
; xy
; yx
;y
+ yx
y yx
+ xy
x xy
Figure 4.15: Left: Force on element are from stresses and body force fb. Right: Torque on element are from shear stresses and body torque Tb. normal in the index direction and the coordinate increases in the normal direction. Shear stresses ij have rst index as surface normal and second index as positive component direction. fb is the body force density that isn't an action from the surrounding but from an externally applied eld on an internal property (e.g gravity on mass or magnetic eld on magnetic poles / magnetization inhomogeneities xed to matter). With some risk for ambiguity, it can be called an internal force.
4.5.2 Torque equilibrium
Torque equilibrium around the zaxis is written
xy ; yx + Tbz = 0
(4.57)
Tbz and
is an internal torque, e.g. magnetic eld on torques can be seen in Fig. 4.15, where the
dipoles xed to short operators
matter. The i = [email protected] i
forces =x y
have been used, and i is element side in i direction. In comparing the force
equilibrium equations with the one for torque, it is seen that only stress gradients
matter for the local resulting force, while shear stresses enter directly into the local
torque expression.
Usually, solid mechanics calculations are performed on very passive materials that don't have any internal torques, and one arrives at xy = yx in torque equilibrium. That is why one most often sees the 2D stress tensor expressed in three components
88
instead of four (six instead of nine in 3D). The addition of an internal torque will create a di erence between shear stresses, a di erence which will balance the internal torque in equilibrium. Since the unsymmetrical shear stresses also will appear in the force equilibrium equation (where the torque doesn't enter), the di erence will give rise to an additional rotation of the element. In the constitutive relation, the shear stress asymmetry will give a shear strain that is a ected only by sthyyxme)m=sy2em,tarminzedattrtiiohzneendoftshhteheaenroshsrtmeraearslsss,thrseeoassrthcmaeonsdtburealuiwns rctitaetnnesnboreasiusasessdtiimltlopslyreemlaamvteeertatrogicet. hAxe0 ystt=eran(itnax.tyiv+e
4.5.3 Equations of motion, coordinate types
If the acceleration of elements is nonnegligible, the equilibrium equation has to be modi ed into the equation of motion,
@x x + @y yx + fx = @t2u
(4.58)
cycl:
(4.59)
cycl:
(4.60)
(4.61)
Another the eld
way of writing it is as r $+ f~ = is really initial/material coordinates
[email protected]~ue..
The coordinates used to write It matches with the de nitions
of strain used. If the space coordinates were used, the acceleration of the single
particle or element at (x t) would be the sum of the eld speed change at a space
point (with di erent particles passing) and the eld gradient times the distance
the actual particle travels per time unit. That case would give a second term to
the right hand side, which would not dominate for small strains ( rst term linear
in strain and second quadratic from a plane wave ansatz). In fact the rst term
becomes equal to the right hand side in the material coordinate case. Thus there is
no need to make a distinction between the coordinate types from a strain viewpont,
but when visualizing magni ed computed solutions in terms of displacements, it is
clearest to use grid coordinates as o sets for displacement (i.e grid coordinates are
material coordinates) and not as results of displacement.
4.5.4 Translatory and rotatory equations of motion
Basic postulates governing the motion of the continuum are
F~ = p~_
(4.62)
T~ = L~_
(4.63)
89
where F is the force acting on the continuum enclosed by a volume V , T is the torque on that volume, p is the momentum and L is the angular momentum. The equations should hold for any part V of the continuum. From this statement the equation of translatory motion and the equation of rotation can be derived.
The equations of motion are simple to derive in di erential form by componentwise
equating the force on an element with inertia. Force comes from stress and so
called body forces, typically gravity (see section below). In the xdirection for a
2Delement one gets
@x x + @y yx + fbx = u
(4.64)
Cyclic permutation of di erentiation variable and component index give the equation for the ydirection. A 3D object can be treated with the two equations if there is no traction on the znormal surface, a case called plane stress. When counting dimensions as the number of independent variables, it is a 2D problem.
The equations of equilibrium are achieved when the continuum is at rest, simply leaving the right hand side in the equations of motion identical to zero.
4.5.5 Body forces
The body force is typically a volume force from gravity, f~b = ge~g.
Magnetic body forces, i.e. magnetic force density distributed over the volume,
can be present when the magnetic material is inhomogeneous. Stratton gives the
formula
fv
=
;
1 2
H
2r
(4.65)
for this body force, saying that the forces are directed from high permeability spots
to spots of low permeability. As the materials de nitely are nonlinear, there will
be di erences in permeability when the specimen is inhomogeneously magnetized.
Another formula given by Stratton states the surface force density fs,
f~s =
(H~
~n)
;
1 2
H2~n
(4.66)
which can be simpli ed to
f~s
=
1 2
H2~n
(4.67)
when the magnetic eld strength H~ is normal to the surface, that could be a good
approximation for real high permeability materials.
90
Cheng 97] presents a derivation of attraction of two perfect permeable materials where he nds the e ective surface force density from the magnetic energy derivative with respect to position change of the yoke. The result is
f~s = 12BH~n
(4.68)
where the value of H is taken on the air side of the airyoke interface.
Binns and Lawrenson 98] make an equivalent con guration with magnetic surface
poles at the airyoke interface with replacement of the magnetic material with air.
The force equation is
f~s = mH~
(4.69)
where m is the magnetic surface pole density.
Becker 99] states the body force by using the magnetization M~ directly,
fv = (M~ r)B~
(4.70)
In applying this equation, the user will still get problems due to kinks in the approximated ux lines.
4.6 Magnetic stress
Based on the constitutive relations 4.40 and 4.42 , a tensor was identi ed that can be regarded as the driving magnetic stress,
xM = yM = xMy = yMx =
1 0
Bx2
1 0
By2
1 BxBy
0
1 0
ByBx
Maxwell stated an asymmetric stress by using the externally applied eld strength H in addition to B,
xM
=
BxHx
;
1 2
0H2
yM
=
ByHy
;
1 2
0H2
91
xMy = BxHy
(4.71)
yMx = ByHx
He called the tensor P and used another unit system 100] but the above is the BtsB~haxmeHreHyHw;ixteBhl;diynHs12hadxdox.sucBla0dyl(aHbrue2sfai)dnciigvtnoettrroh.geefHnbfoemcrexdcleee=srsai.vndedTdxdhtBioetxrfbqHoyurxem;durselta21nsrfieo0tsrHytat2thto]ienusgttnordtreheqsrseueetehqxseuptaarctetoeisondsnidoasint,siofTnvbmmtxzha==t
fbmx
=
d dx
xM
+
d dy
yMx
(4.72)
Tbmz = xMy ; yMx
(4.73)
he could identify the expressions 4.71. His equations for stress are the general ones and covers the case with nonaligned magnetization to magnetic eld intensity. Field intensity doesn't have to be strictly externally applied, it can also come from magnetization discontinuities or inhomogeneities as discussed earlier.
92
Chapter 5 Models of magnetostriction 5.1 The interplay between mathematical modeling and physical experimenting Measurements are needed both to validate made models and to inspire the making of models. This interplay can occur on di erent levels of physical scale and on di erent types of problems. Three kinds of problems are separable to the engineer: material, component and system problems. Measurements can be done on all types and levels of problems, but in the present instance, measurements are only done on a macroscopic material problem. The measurement provides parametric input to a model of the material properties. The model is made to t into a nite element program, that is in turn able to model a component such as a core. The core model might be studied to provide a simpli ed, single element, model to be included in a greater system model. Ideally, the component and the system should be veri ed with measurements of the corresponding type. All these steps are timeconsuming but not impossible to be carried out by people. The most di cult task is to bridge the gap between the microscopic and the macroscopic levels. A microscopic model often consists of an ideal part and part originating from a small number of defects or only one defect. The real situation di ers in that the defects are neither of a small number nor approaching an in nite number. In the in nite limit the model might be possible to average, but real components well represented by it could be bad. In the intermediate range automated mathematical tools together with elaborate physical equipment are needed to map the defects and their e ects. Parameter selection from a standpoint on the grounds of 93
thermodynamics and causality could lead to simpli ed macroscopic models. Other divisions of models into classes can also be done. One such division is with continuum, parameterized and physical models. Again, a complete description also suitable for use in engineering analysis would need to bridge the model types. There is seldom the case that this can be achieved. In his speci c instance, the author wanted to make measurements and use a parameterization that was sound and could t the nite element method of solving engineering problems.
5.2 Continuum model
The activity to include continuum magnetostriction phenomena in computational
tools for magnetomechanics 92] has spawned an interest in the exploration of possi
ble mathematical ways to represent the material response in a su ciently accurate
manner. Here, we take into account the dynamic behaviour of magnetostriction as
described by what is commonly known as butter y loops. When a harmonic ux
density is present, the measured magnetostriction loop is a curve with two branches
when plotted against the ux density. Although often thought of as related to hys
teresis, we believe that the lag of magnetostrictive strain can be e ectively modelled
by a ratedependency model, which properly assigns the phase shifts to the mag
netostriction harmonics. We deal at length with the case of linear orthotropic
elasticity ticity it is
amnedanlitnetharatatnhiseomtraogpnicetmosatgrnicettioveelsatsrtaicinb$eMhavisioliunre.aBr yinltinheeamr amgangentiectosterleasss
tensor BiBj. By knowing that there is a weak coupling on permeability from stress,
we are able to separate the simulation into one magnetic part and one mechanical
part, where the magnetostrictive strains appear as sources to the total strain. The
main interest in the results lies in an evaluation of the in uence of elastic properties
on measurable total strain in a sheet excited by the yoke pair described elsewhere
in this text.
By using the ratedependent, dispersive, model in stead of a true hysteresis model we get mathematical simplicity and easier veri ed thermodynamic compatibility. Simplicity gives speed of computation and a more de nitive ability to consider rotational hysteretic phenomena. As mentioned before, we foresee that there will also be considerable ratedependency in magnetostriction, for which we will use a nonlinear dispersion law.
94
Magnetostrictive strain [microm/m]
6
5
4
3
2
1
0
1
2
3
1.5
1
0.5
0
0.5
1
1.5
Flux density [T]
Figure 5.1: Butter y loops of negative valued vs. By.
Mx vs.
Bx and positive valued
My
95
5.3 Butter y loops
Are the M B] loops hysteretic in its true sense or are they a re ection of a timeratedependent, dispersive, phenomenon? The simplest proposition of a dispersive governing di erential equation is
_M B] = k( Mntr(B) ; M B])
(5.1)
wThheisreideMnatlr
is the strain
strain at an ideal process with can be investigated by exciting
no time rate the material
of change of with a ux
strain. of very
low frequency. The bracketing of the arguments to the strains in the formula shows
the dependence of history of the argument. ( ) means that there is no dependence
on past history and ] means that there is a dependence of the time history of the
argument.
The time domain equation can be Fourier transformed into a frequency domain equation for a speci c ux density process (time behaviour),
j!~M(!) = k(~Mntr(!) ; ~M(!))
(5.2)
from which an equation between the notimeratestrain and the actual strain is
obtained,
~M
(!)
=
k j! +
k
~Mntr(!)
(5.3)
The frequencies present in the spectrum of the magnetostrictive strain is di erent from those present in the ux density, since the no time rate dependent strain is eopwvfrioellndMbuinecwehieulilgvxhebdneeerhn2pasfior.tmwy.AeornIssficBtisnheiinsthmheaMargnmnocenootnlmoiinscpetaararitrcettdhiMnoettnorf(rctBeuhq)reuvfeBeunnschhcyatofirwom,nsto.haneTiscfah.utonusdreaamteivoeennntaeploferwceteq,rustehwnecirlyel
If B is anharmonic with a fundamental of f, the spectrum of M becomes more complicated. There will be harmonic interaction through the nonlinearity, and the crossproducts will give addition and subtraction of frequencies in B to form mirrored frequencies in . These mirrored frequencies will ll out gaps in the spectrum of the strain compared to when having a purely harmonic ux density.
The equation 5.1 was a dispersive relation for a simple case of ratedependency. A general dispersive relation can be written
u(t) = K(t) ? i(t)
(5.4)
u~(!) = K~ (!) ~i(!)
(5.5)
where i is the input variable, u is the output variable, ? is the convolution operator and K is the kernel describing the properties of the medium without reference to
96
nKa~osnp=lienckeia=cr(ji!tnim+theket)raaucnxeddoeif(ntts)hite=y,initpMntuirst(Btvea(mtr)pi)at.binleAg.stoItnhceatlhtleitmhreeatrea;tdeeBpinerndedelpaetneincoydnecnaatsnesotnralabiinonevaeisr, dispersive one. It is not a good term though, as a di erential equation governing the nonlinear dispersive process would contain a timederivative term with a higher power than one, rather than the driving term containing higher powers as the ratedependency case above.
5.4 Ratedependency model
We introduce a scalar magnetostriction ratedependency model in the time and frequency domains as
_
=
k(
B2 0P
;
)
(5.6)
~
=
k= jn!1
0P +k
F
fB2g
(5.7)
where k is a lag parameter and P is a magnetoelastic modulus. The following relations apply when there are only discrete harmonics,
g(t) = F;1fg~g = X 1 Refg~(n)ejn!1tg
(5.8)
n=0
g~(n) = Ffgg
(5.9)
where in Eq.
n is the (5.8) is
number of the harmonic the Fourier series of the
ttoimtheesfiugnnadlamg(etn),taalnfdreEquqe. n(c5y.9!)1s.yTmhbeolsiuzems
the Fourier decomposition of g(t) into the Fourier coe cients g~(n) that are the
discrete spectrum of g(t). Restricting what follows to the case of a harmonic ux
density of angular frequency !1, one has
F
fBiBj
g
=
Bd iBj
1 2
n0
+
Bd iBj
1 2
n2
(5.10)
where BiBj i j 2 fx yg is a magnetic stress tensor and nm is the Kronecker hcdoaemlrtmap.oonnWiecnhtcesanscteah,newbeueaxrwedreaintbtselietnytBoisd iiuBdnejina=txifiyaB^lltiyhB^eajl.tkerApnasartabimnugtet,teetrhr eiynpleaoaokspimvsapalulreeesmgoiavfnetnnheerfot. ernOtshnoeer can show that an approximation of the vertical width b of the loop is
b=
1 0P
4!1 k
q B^2
;
Bm2 Bm
(5.11)
97
from which k can be determined, as Bm is the ux density where the width is measured. This holds when the damping introduced by the model is negligible, i.e when 2!1 k. Phase shift can in that case still be considerable, allowing loops with fair width to be represented.
5.5 Simple 2D magnetostriction models
A simple isotropic constitutive relation for nonoriented silicon iron in thin (0.5 mm) sheets (motor steel) can be written as
8 < :
MxMyxMy
92 = =4
;P1P 0
;P1P 0
0 0 2(1P+
)
3 5
8 1< 0:
BBBxBxy22 y
9 =
(5.12)
A parameterization are the components
like of a
this is tensor
ianttlheeasxtymsaytshteemma, ti.iec.aliltyosboeuynsdstarsaiBnx2l,iBkey2caonmdpBonxeBnyt
transformation to rotated coordinate systems. Statements taken by analogy from
elasticity say that = 0:5 holds due to magnetoelastic isotropy and that the shear
modulus can be written as P=2(1 + ) due to isotropy and linearity.
Oriented silicon iron in 0.3 or 0.23 mm thin sheets (transformer steel) is tried to be included in the simple scheme by the relation
8 < :
MxMyxMy
92 = = 64
DP12x1 0
D12 P01y
0 0 G1xy
3 75
8 1< 0:
BBBxBxy22 y
9 =
(5.13)
where the Dij:s and reciprocals of moduli can be called magnetocompliance coe cients.
5.6 Magnetoviscoelastic models Magnetostriction models can be classi ed into two main groups: magnetoelastoplastic and magnetoviscoelastic. Plastic models are timerate independent of the excitation, while viscous models are ratedependent. The model development for the magnetoviscoelastic case is reported in the following subsections.
98
5.6.1 Quasistatic linear case
This case can stress 1 B2.
be written on the form M = The proportionality constant P
Ph1a0sBb2e. enIt
is linear in the magnetic called the magnetoelastic
modulus0 101]. No time derivative is present in this constitutive relation, so it can
be useful when timerate of the magnetic stress is low, which is the quasistatic case.
For a twodimensional continuum one might apply an isotropic version as
Mx =
1 0P
;Bx2
;
By2
(5.14)
My =
1 0P
;By2
;
Bx2
(5.15)
xMy
=
2(1 + 0P
) BxBy
(5.16)
where the sumption
oefxbporetshsiloinnefaorritythaensdheiasortmroapgyn. etTohsetriccotniosntrucxMtyedis
obtained magnetic
from the asstress tensor
l$$a10EssBtaiacitnBisPdfjyotiihnsisegojmnt2harefagxteniqeoyut.ogilOeioblanbrseietuyihmcsappsaarntorodtpne$broMottuee.nntTdshaoharretyiaiclneoqtmcruaopanntletsioiftnoenurscmuowmanitlsilptoicrntooubnltaliswevismest.sro,efltaiahstneioatenmolatfaosatgrlincslietnprteaoaaiernrt
isotropy is then
x=
1 Y
(
x;
y) +
1 0P
;Bx2
;
By2
(5.17)
y=
1 Y
(
y;
x) +
1 0P
;By2
;
Bx2
(5.18)
xy
=
2(1 + Y
)
xy
+
2(1 + 0P
) BxBy
(5.19)
where elastic
tshterasitnre$sEs
$ is related only.
through
the
elastic
modulus
Y
and
Poisson
ratio
to
5.6.2 Ratedependent linear case
One simple type of ratedependency can be written in the timedomain as _M =
kre(pBr0e2Pse;nt
M). It is response
of rst speed.
order in time derivatives and uses a single The reciprocal of k is a lag time constant.
parameter k to The simple lag
behaviour might be found for excitations that do not signi cantly enter frequency
regions with material resonances. By plotting M versus B one gets the butter y
99
curve. The curve for such a simple case is with only one crossing at B = 0, an example of which is shown in Fig. 5.1 at 50 Hz and for two directions of ux and strain.
Continuing in the frequency domain, the rst order relation is written as ~M =
sPo10i1jrPbjjn!t!kkh+1e+k kBff.r2eF,qouwrehnaecnryead!neispiosetnraodnpegniuctlamfraacftrteeorqriauilennotcnoye.mcaIanngnateht2eonDcowcmraipstelei,anitceis
convenient coe cients
to abDij =
eMx
=
D11 0
Bfx2
+
D12 0
Bfy2
eMy
=
D21 0
Bfx2
+
D22 0
Bfy2
exMy
=
D33 0
Bg xBy
(5.20)
in a coordinate system positioned relative to the material texture such that shear strain is independent of x; and y; magnetic stress components. In fact magnetostrictive shear strain seems to be negligible when x; and y; directions are
chosen to be coincident with rolling and transverse directions of highly anisotropic
siliconiron 102]. In those cases D33 can be set to zero. This does not mean that shear magnetostriction is zero in all coordinate systems. The linear case is easy to
implement in computation programs.
5.6.3 Ratedependent nonlinear case
Of interest is the plot M to B2, which can be seen as a strainstress diagram, yielding magnetoelastic potential energy from an averaged singlevalued curve and a loss proportional to the area of the loop. This case uses an arti cial nonlossy, i.e singlevalued, butter y curve MA(B2). The singlevalued constitutive relation is found from least squares polynomial tting to the vertical mean curve of the two branches. One can use scaled Legendre polynomials translated to the argument interval 0 1] to form an orthogonal function sequence ffig. This sequence makes it easy to alter and evaluate the polynomial order of the approximation. For a third order model, able to cover moderately wavy butter y loops, one can write
MA(B2)=^MA = d0 + d1f1(B2=Bs2) + d2f2(B2=Bs2) + d3f3(B2=Bs2) (5.21)
where Bs is netostriction
thMe
ux density at the striction peaks. Lossy (doublevalued) is dependent on frequency and nonlossy magnetostriction,
~mMag=
H(f)~MA. A three parameter resonant transfer function H(f) can be formulated as
H(f
)
=
;(f=fd2)2 + 1 ;(f=fr2)2 + jf=fr1
+
1
(5.22)
100
This function was used in 94] for transversal strain and ux density. Resonance can be seen as additional crossings at nonzero ux densities in the butter y curve or larger than 90 phase shifts between nonlossy and lossy magnetostriction at the resonance frequency fr2. The zero response at fd2 is needed to restore amplitude and phase for higher measured harmonics. Additional measurements can be carried out to investigate if this zero is physical or if an additional parameter is needed to move the zero out into the complex plane. To simulate the magnetostriction response for cyclic processes it is convenient to use frequency domain techniques which allows the application of H(f) directly instead of solving the corresponding ordinary di erential equation with numerical timestepping. The di culty arising is the amount of algebra that has to be done to sort out the harmonic interaction of Fourier components that occurs due to nonlinearity. As an example for a case without excessive waveform distortion, three odd harmonics of the ux density signal might be enough to represent it. The quadratic relation between ux density and magnetic stress then gives ve even harmonics in the magnetic stress for the example and then the third order nonlinearity in Eq. (5.21) gives 15 even harmonics in the magnetostriction. The algebra will be presented in detail in chapter 8.
5.7 Model incorporation in plane stress calculations
We assume that plane stress prevails in the sheet whose strain eld is to be computed. It is also assumed that inertia e ects can be neglected, which is the case if we consider the sheet being massless or if the time derivative of excitation is low. Performing the decomposition as shown in Eqs. (5.8), equilibrium equations and straindisplacement equations are written in the frequency domain as
@x~x + @y~xy = 0
~x = @xu~
@x~xy + @y ~y = 0
~y = @yv~
(5.23)
~xy = @xv~ + @yu~
We have to use a constitutive relation suitable for representing both elastic and magnetostrictive strain. When assuming elastic orthotropy and magnetoelastic anisotropy in a somewhat restricted sense one can write
~x
=
C11 ~x
+
C12 ~y
+
D11 0
F
fBx2g
+
D12 0
F
fBy2g
~y
=
C12 ~x
+
C22 ~y
+
D21 0
F
fBx2g
+
D22 0
F
fBy2g
(5.24)
101
~xy
=
C33 ~xy
+
D33 FfBxByg 0
CxT2ha2en=edla1ys=tdYiciyrce, ocCmt3io3pnl=isanr1ec=seGpcexocyet.ivYecxliyea.nntdsxCYyyijaanardreetChyex11eal=ares1tti=chYemxo,orCdtuh12loi g=foorn;aulnxciyao=xnYitayrla=scttri;eosnsyirxna=ttYihoxes, of the strain in the rst index direction to the strain in the second index direction under uniaxial stress in the second index direction. Gxy is the shear modulus in the xycoordinate system, which is directed with basis vectors parallel to the material principal axes. The xdirection is the rolling direction of the sheet and the ydirection is the transversal direction. Reciprocity holds because of elastic energy conservation and orthotropy then holds as material principal axes are at right angles to each other. In the simulations we have used typical values of Yy = 200 GP a, Yx = 150 GP a, Gxy = 74 GP a, xy = 0:4 and yx = 0:3.
g0DTr:h010a1e0in1m=3oarGig;ePnn0eta:te0;od01e1lma3asantGtdiecPrDicaao30l;3mi1n,p=lDi1a00n20.22c]e.T=Bhcoey0ese:c0o2caoirereGndtiPtnsaaakD;teen1ij,tfrrwDaonem10s2fhoda=rvametaa;uti0pso:ern0ed3si7etanritGesePdsbeaaefs;oner1dt,ahoDahnt20ig1nahonl=rdy
mal magnetoelastic compliance is by far greatest in the ydirection and even greater
is the negative compliance to an orthogonal direction from applied normal magnetic
stress in the ydirection, see Figs. 5.2, 5.3 and 5.4. . The experimental data in
102] show no shear strain in the x,ysystem for a variety of angles of ux density
to rolling frequency
direction, dependent
scooeweciseenttsthDeijmaargenwetriicttsehneaursincogmthpeliamnocedeDl i3n3
to zero. The Eqs. (5.7),
Dij
=
Di0j
j
k n!1
+
k
(5.25)
By using the same kparameter for all directions, it is seen from the simple formula
Eq. (5.11) that an inherent assumption is that the relative butter y loop width is
constant over all directions when directions are magnetized with the same ampli
tsuetdet,oi.1e6.0i0ncsr;ea1sainngdloboupttweridythlsowopitshfoinrctrheaissicnagsenoarrme adlrmawagnnientoFciogm. p5l.i1a,ncwei.thk
was ux
density amplitudes of 0.63 T in the transverse direction and 1.2 T in the rolling
direction at frequency 50 Hz. Large relative butter y loop widths are mostly con
nected with low magnetocompliance, so here are space for improvements in the
description. The restriction on magnetoelastic anisotropy is here that magnetic
shear stress does not in uence normal x,ystrains, something that might be loos
ened in the future to obtain a better t with experiments.
102
5.7.1 Nonlinear dispersion The notion of nonlinearity needs here to be clari ed. In the beginning of this section it was said that magnetostrictive strain was modelled being linear in the magnetic stress tensor BiBj i j 2 fx y zg. Converting that to the customary butter y curve relation between and B for the magnetostriction of a homogeneously magnetized sample at ux density B, it corresponds to a parabolic expression of the "anhysteretic", or rather , the singlevalued approximation, since loop behaviour probably is an e ect of ratedependency rather than hysteresis in a more strict sense. So, magnetoelastic linearity is the same as a parabolic ;B relation. By nonlinear dispersion we mean the nonlinearity of the singlevalued approximation with respect to ux density. This singlevalued approximation is in fact of more value than just an estimate, it is the equilibrium points obtained at quasistatic conditions. The other aspect of nonlinearity is when magnetoelastic nonlinearity is present, which surely has been seen in data from specimens experiencing saturation, leading to higher order terms in the ; B approximation. With a proper rate dependency law, which is conceptually free from bindings to a particular form of a singlevalued representation of magnetostriction with respect to ux density, there should be no problem to include the magnetoelastic nonlinearity at saturation.
5.8 Macroscopic magnetostrictive response
The response is often seen as graphs known as butter y curves. There the magnetostriction is read out on the vertical axis and the ux density on the horizontal axis. The magnetostriction is an even, doublevalued "function" of the ux density. The two branches of the curve enclose two wing like areas, therefore its name. The branches are rounded and fairly smooth, and if one compares to a hysteresis curve, the latter has sharp tips where the eld is reversed and is essentially independent of the frequency of the eld. Hysteresis is the phenomena of eventlag rather than timelag between the cause and its e ect. Due to the shape of the butter y curve one can believe that it depicts a phenomenon related to the time change of the driving entity (B), i.e. it is ratedependent, frequency dependent.
This frequency dependence can be incorporated in the material model through
realizable phase shift factors (causal and real signal in time), the simplest case
being
8 < :
MxMyxMy
9 = =
2
k j! +
k
4
DD1211 0
DD1222 0
0 0 D33
3 5
1 0
8 < :
BBBxBxy22 y
9 =
(5.26)
where the eld entities are in complex representation. In the time domain, the
103
model can be symbolically written _ = k( e ; ). e is a singlevalued function
and represents the equilibrium for no change in driving eld, or when changing the
eld very slowly, material relation,
aqnudasbisetcaatuicsaellye.
The di erential is nonlinear in B
equation describes the resulting ; B
a dispersive dependency
is nonlinearly dispersive.
5.9 Identi cation of parameters
If the parameters are D11 used to obtain the values
D22 D12 of them:
Dk 2i1s
and k, the following information can taken from the butter y wing width
be or
area. D11 is found the GO materials.
from D22 is
determined from the ratio
e ; B in the RD direction. It might be seen from e ; B in the TD direction. of RD contraction to TD expansion. D21
negative for D12 can be is evaluated
by transverse expansion to rolling direction length change. If there is zero volume
magnetostriction, not all of the parameters will be independent. The relations
between them in that case is derived below.
5.9.1 Magnetostrictive incompressibility
The restrictions on magnetoelastic coe cients when no volume change is assumed are now derived. 2D forms are uninteresting, area conservation is not an issue, so only 3D is dealt with. It is simple to look at the expression of the dilatation using magnetocompliance coe cients,
Mx +
My +
Mz =
1 0
Bx2fD11
+
D12
+
D31g+
1 0
By2fD22
+
D23
+
D12g
+
1 0
Bz2fD33
+
D31
+
D23g
=0
8Bi2
(5.27)
For the equation to hold always, the expressions between braces must be zero. There will be a simple equation system of three equations and six unknowns when the magnetocompliance matrix is symmetric as above,
D11 + D12 + D31 = 0 D12 + D22 + D23 = 0 D31 + D23 + D33 = 0
(5.28) (5.29) (5.30)
yielding three dependent parameters and three independent ones. The solution in terms of the diagonal compliances is
D31
=
1 2
(;D11
+
D22
;
D33)
(5.31)
104
cycl:
(5.32)
cycl:
(5.33)
which is helpful when lateral contraction ratios are wanted from measured data of compliances in main directions 1.
In situations where only D11, D12 and D22 are given from measurements, like when only measuring in the plane of the specimen, the other coe cients are from
D13 = ;D11 ; D12 D23 = ;D12 ; D22 D33 = D11 + 2D12 + D22
(5.36) (5.37) (5.38)
In some cases one might be given values of sti ness coe cients rather than compliance coe cients. But getting a complete set of sti ness coe cients from the incompressibility condition is impossible. The columns of the compliance matrix in such a case are linear combinations of each other, which means that the compliance matrix is singular, noninvertible. It also means that all stress states are permitted, but not all strain states. Thus the sti ness matrix in the ordinary sense doesn't exist. However, we can write a lower dimensional description of sti ness, and use one stress component as a parameter to determine the stress state. One splitting of the compliance relation is
MxMy
= D0 1 0
BBxy22
+ d03
1 0
Bz2
Mz
= d03T
1 0
BBxy22
+
D33
1 0
Bz2
(5.39) (5.40)
Inversion of the D0 matrix and using the incompressibility constraints, one gets
fact
that
D0;1d03
=
;1
;1]T
under
the
1 0
BBxy22
= D0;1
MxMy
+
1 1
1 0
Bz2
Mz = ; Mx ; My
(5.41) (5.42)
1It might be clarifying to write out the de nitions of lateral contraction and tension ratios.
They are here denoted by ij and ij respectively, and are expressed by
ij
def =
; j ; M = i M B~ =Bj~ej j
PPji;;jj 11
(5.34)
ij
def =
j Bi2 Bj2
kl=
kl
kl
lj =
Pij Pjj
(5.35)
The lateral tension ratio is de ned with the denominator being the major applied normal stress and the numerator being the minor normal stress that has to be applied orthogonally to the major stress to obtain an uniaxial strain state, due to the Poisson transversal contraction e ect.
105
It is seen that knowledge of two strain components and one stress component gives kshneoewtlesadmgepolefsthcoensstirdeesrsesdtaintet.hIifssbtroeosks,inontehecaznduirseectthioen2iDs zsetrio,naesssitmoaftterinxiDs i0n;1t,he
D0;1
=
1 D11D22 ;
D122
;DD2212
;D12 D11
(5.43)
5.10 Magnetoelastic shear modulus
To get some grip on the magnetoelastic shear modulus, below called Q, one can study the magnetoelastically isotropic case, which might hold as an approximation for nonoriented materials. Isotropy means that the magnetocompliances are independent of coordinate system. By transforming a sheared state in the xysystem to the unsheared principal system called the ntsystem, one can relate the shear modulus Q to the normal magnetocompliance D11 and the o diagonal (orthogonal) magnetocompliance D12.
The general strain component transformation from nt to xy can be written
Mx = Mn cos2 ' + 2 Mnt cos ' sin ' + Mt sin2 '
(5.44)
My = Mt cos2 ' ; 2 Mnt cos ' sin ' + Mn sin2 '
(5.45)
xMy = 2(;2 cos ' sin ' Mnt + cos ' sin ' Mn ; cos ' sin ' Mt ) (5.46)
The ntsystem is a zero shear strain system,
2 Mnt = nMt = 0
(5.47)
which yields the principal strain to xy strain transformation,
MxMy
= =
MnMt
cos2 cos2
' '
+ +
MtMn
sin2 sin2
' '
(5.48) (5.49)
xMy = sin 2'(; Mt + Mn )
(5.50)
For an isotropic condition, the principal strain system is also a principal stress
system. The vector nature of magnetic ux density leads to the fact that the ptghirveinecsatipreanslosrmmaaxagilsnsedtturicaeinsttordecusosemitspoluicanonimacxepialDilatna=cnedDDg1inv2.e=nTbDhye11m1,0aaBgnn2nd.etaTochsotisrmaupinnliiaaonxrctihaeslogsintornetashlselywntitlol system (Dn Dt) are equal to the ones in the xysystem (D11 D12), due to material isotropy. The biaxial principal strain will be
Mn
=
D11
1 0
Bn2
Mt
=
D12
1 0
Bn2
(5.51) (5.52)
106
By inserting into Eq. (5.50) one transforms back to the xy system to get the shear
in that system expressed in the uniaxial stress and the normal and orthogonal
compliances,
xMy
=
sin
2'(;D12
+
D11)
1 0
Bn2
(5.53)
By comparing this expression of the shear with an expression that uses the shear
modulus, one can identify the relation between shear modulus and normal and
orthogonal compliances. It is simple to transform the ux density components to
the xy system and therefrom write the magnetic stress in the xy system,
1 0
Bn
cos 'Bn
sin
'.
This
gives
the
shear
using
the
shear
modulus,
1 0
Bx
By
=
xMy
=
1 Q
1 0
Bn2
cos
'
sin
':
(5.54)
By comparing Eqs. (5.53) and (5.54) one gets the expression for the magnetic shear
modulus at isotropy,
Q
=
2(D11
1 ;
D12)
(5.55)
The shear modulus Q is the reciprocal gives D33 = 2(D11 ; D12), the relation
of the shear compliance (here D33) which between isotropic magnetocompliances.
5.11 Vector and tensor transformation
It is of interest to study the transformation properties of the magnetic stress and the magnetocompliance. Explicit formulas will be given for the plane case, but general ideas hold for three dimensions also. Wuppose there is an nt coordinate system rotated in the xy plane, with an angle ' between the x and n axes. The transformation of ux density components from the nt system to the xy system can be written
Bx By
=
cos ' ; sin ' sin ' cos '
Bn Bt
= A;1
Bn Bt
(5.56)
The transformation matrix A has orthonormal columns which means that the inverse is the transposition of A. The strain tensor is transformed as
0 = A AT
(5.57)
where the shear component has to be half the shear angle, v x), in order to follow tensorial transformation. The outer
xpyro=du12ct
xBy i=Bj21f(ourmye+d
from the ux density vector is a tensor. The transformation of the product is
deduced as
Bi0 = AikBk ) Bi0Bj0 = AikAjlBkBl
(5.58)
107
swyhseteremB. i0Oanree
the sees
components in the tosystem and Bi are components in the fromthat the product formed in the tosystem is formulawise invariant
compared to the product formed in the fromsystem for any pair of systems, which
is the property de ning a construction as a tensor. This property makes it a good
candidate as an entity to parameterize magnetoelastic strain against, as the strain is
also this
a tensor. The outer product book, and it has been called
1t0hBe imBaj ghnaestibceestnreusssedwiftohr
this purpose a dimension
throughout of Pa.
The strain transformation Eq. (5.57) can be worked out to
2 4
MnMtnMt
32 5=4 
cos2 ' sin2 ' ;2 cos '
sin2 ' cos2 ' 2 cos ' sin ' {z
cos ' sin ' 3 2 ; cos ' sin ' 5 4 cos2 ' ; sin2 ' }
MxMyxMy
3 5
T
(5.59)
where the shear angle has been duly taken care of. The stress transformation is
2 4
BBBnBnt22 t
3 5
2 =4
cos2 ' sin2 ' ; cos ' sin '
sin2 ' cos2 ' cos ' sin '
2 cos ' sin ' ;2 cos ' sin ' cos2 ' ; sin2 '
32 54
BBBxBxy22 y
3 5
(5.60)

{z
}
T
The inverses of the transformations are gotten by substituting ' ! ;' in the above formulas.
A note about superposition of ux densities and magnetic stresses. Since the mag
netic stress is the outer product of the ux density, superposition of the ux densities
doesn't yield stress contributions in an additive manner. If the ux density consists
of two parts, seen by
Bi
=
Bi1
+
Bi2,
there
will
be
crossproducts
in
the
magnetic
stress
as
BiBj = (Bi1 + Bi2)(Bj1 + Bj2) = Bi1Bj1 + Bi2Bj2 + Bi1Bj2 + Bi2Bj1
6 Bi1Bj1 + Bi2Bj2
(5.61)
and the stresses associated with each ux density part cannot be added to get the resulting stress.
5.12 Magnetic stress alternatives
One can think of other possibilities of constructing a magnetic stress tensor than
1 0
BiBj
discussed
in
the
previous
section.
The
alternatives
BiHj
and
BiMi
are
the
108
ones most obvious. They have a magnetic material in uence
I(tlairsgepopsesrimbleeatboilsitiympwliitfhyoButiMcojutpolin1g0
BiBj when rjk 1 : between directions) as
built j=k seen
in, like Hk rjk = 0 :
jik6=Hjk.
from the formula
Mj to
= the
1a0pBpjl;ieHd j
.
It is unknown how eld from external
much H really in uences. H is more coils and edges of the specimen.
connected Inside the
material, B and M have very equal directions, due to the ferromagnetic material
property, and strain is believed to depend only on these internal entities. When
the external entity H is not codirected with M, it will probably only tend to rotate
the state of strain (when the sample is xed) as it will rotate M. If this rotation
of strain will also be accompanied with a rotation of matter probably depends on
the type of magnetic material. For soft2materials the atomic moments perhaps
can rotate on the lattice sites and stay internally parallel, without rotation of lines
connecting lattice points. For hard materials, there are certain easy directions
in which the atomic magnetization vectors probably lie. Switching between these
directions occur when the applied eld rotates. One can imagine that there can be
some rotation of lattice lines as the cells try to keep the magnetization in the initial
cell easy direction before switching occurs. The phenomenon might be consistent
with a torque action description. As the applied eld rotates, the torque on cells
increases until the atomic moments turn relative the cells and the torque becomes
zero, or less than critical. To conclude, B and M might be adequate to describe
strain, while H has to be used when describing rotation. In an inhomogeneously
magnetized material, there will be local distortions of H, so the local H has to be
distinguished from the applied H.
When there is hysteresis between the magnetic stress and the resulting strain, it would be interesting to keep track of both B and H during the hysteresis cycle. There is a possibility that the hysteresis is purely magnetic, and that the construction of a more proper magnetic stress tensor (perhaps BiHj) would lead to a hysteresisfree magnetic stressmagnetoelastic strain relation.
5.13 Compliance transformation It is appropriate to write out how to transform the magnetocompliance. By transforming the stress in the xysystem to the ntsystem, multiplying with the ntsystem compliance matrix Dnt to get the strain, and transforming the strain back to the 2Soft and hard in this context states the ease of rotation of magnetization. Normally, soft and hard states the ease of changing the sign of magnetization along an axis. These two qualities might di er.
109
xysystem, one can identify the xysystem compliance matrix Dxy,
Mxy = T ;1DntT Mxy  {z } Dxy
(5.62)
wo1Mn0hxetByrhex2e=BaMny2gxMxlyBe ix'sBMytbhye]eTtxMwc, oeyalen]uTndm.tnhMFeooxfrnymtaisncaotgahonnereditcsiiooncltausrttomerpensiscysosmfcteomammtaepgarnoinanedlte,onttshthtsereiixcnctyoitvmhseypxsslttyireaamsnyicn.setcewommil,lpdoMenpexney nt=sd,
In the magnetoorthotropic case with the x,ycoordinate axes parallel to the characteristic material axes, the compliance matrix is written
Dxy
=
2 4
DDxxxy
DDxyyy
03 05
0 0 Dxyxy
(5.63)
with Dxx 6= Dyy. The orthotropic model doesn't always give a principal strain system coincident with a principal stress system. The principal strain tends to be rotated towards the easy material characteristic axis. Moreover, symmetry in the compliance matrix means something for the contraction ratios between di erent directions. The application of stress in a hard characteristic direction will give a higher strain ratio between easy and hard direction than the strain ratio between hard and easy direction when the same magnitude of stress is applied to the easy direction.
By transforming with Dnt = T DxyT ;1 one gets a full matrix for the ntsystem,
Dnt
=
2 4
Dnn DDnnntt
Dnt DDttntt
Dnnt 3 DDnttnntt 5
(5.64)
where the entries are
Dnn = 1=8 cos4'Dxx + 1=2 cos2'Dxx + 3=8Dxx + 1=8Dxyxy
;1=8 cos4'Dxyxy ; 1=2 cos2'Dyy + 1=8 cos4'Dyy + 3=8Dyy
;1=4 cos4'Dxy + 1=4Dxy
(5.65)
Dnt = 1=8Dxx ; 1=8 cos4'Dxx + 3=4Dxy + 1=4 cos4'Dxy
+1=8Dyy ; 1=8 cos4'Dyy ; 1=8Dxyxy + 1=8 cos 4'Dxyxy (5.66)
Dnnt = ;1=4Dxx sin 4' ; 1=2Dxx sin 2' + 1=2Dxy sin 4' + 1=2Dyy sin 2'
;1=4Dyy sin 4' + 1=4Dxyxy sin 4'
(5.67)
Dtt = 3=8Dxx ; 1=2 cos2'Dxx + 1=8 cos4'Dxx + 1=4Dxy
;1=4 cos4'Dxy + 1=8 cos 4'Dyy + 1=2 cos 2'Dyy + 3=8Dyy
110
+1=8Dxyxy ; 1=8 cos4'Dxyxy
(5.68)
Dtnt = ;1=2Dxx sin 2' + 1=4Dxx sin 4' ; 1=2Dxy sin 4' + 1=4Dyy sin 4'
+1=2Dyy sin 2' ; 1=4Dxyxy sin 4'
(5.69)
Dntnt = 1=2Dxx ; 1=2 cos4'Dxx ; Dxy + cos 4'Dxy
+1=2Dyy ; 1=2 cos4'Dyy + 1=2 cos4'Dxyxy + 1=2Dxyxy (5.70)
This compliance transformation can be used to nd the axes of extremal compliances to uniaxially applied stresses. A uniaxial stress can be written as
2 M3 Mnt = 4 0 5 0
(5.71)
where M is a scalar, with the the ntsystem chosen with the naxis parallel to the stress application axis. The strain response is then
2 4
MnMtnMt
3
2
5 = Dnt 4
M 0
32 5=4
Dnn(') Dnt(')
3 5
M
0
Dnnt(')
(5.72)
Dacaopxnmipsnl,p(ic'laaina)tndiioscnDethi'nnen,ctttohhimese fpteohlxlielatornwsechmieenaapglra.srcDoaoflnmlteth(lp'etlio)caotnishmcetephalieinpaopntrlhcitceheasotngcitoaonnnsyaabslxetciesosm,emai.tprcwBlihaiyelnldvcb.aeerTtycohainetllhgeeedxttahtrpheepemalnniacoglarsltemaioroanefl easily found for a model by plotting the functions 5.65, 5.66 and 5.67. Such plots can be used to compare a model with strain measurements where uniaxial stresses have been applied in di erent directions. The plots for the values given in section 5.7 is given in Figs. 5.2, 5.3 and 5.4.
5.14 Piezomagnetism
The total dipole moment of a crystal may be changed by the movement of the walls between domains or by the nucleation of new domains. Only walls separating domains with 90 domain magnetization direction di erence will contribute to magnetostriction with its motion. The response of piezomagnetic crystals in transducer applications is characterized by the magnetomechanical coupling factor k, de ned by
k2
=
energy
convertible to mech: mag: energy stored
work
(5.73)
111
Normal magnetocompliance [1/Pa]
902e11
120
60
+ 1.6e11
1.2e11 150 8e12
4e12
180 
30  0
210 240
+ 270
330 300
Figure 5.2: Normal magnetoelastic compliance as function of angle of magnetic stress to rolling direction.
Orthogonal magnetocompliance [1/Pa]
904e11
120
60
 3e11
150
2e11
30
1e11
180 +
+ 0
210 240
270
330 300
Figure 5.3: Orthogonal (to magnetic stress) magnetoelastic compliance as function of angle of magnetic stress to rolling direction. 112
Shear magnetocompliance [1/Pa]
904e11
120
60
3e11
150
+
2e11
30
1e11
180
0
210 + 240 270
330  300
Figure 5.4: Shear magnetoelastic compliance coe cients as function of angle of magnetic wtress to rolling direction. The constitutive relation between smallsignal magnetic ux density B, stress , magnetic eld strength H and strain
B = H+d
(5.74)
= dH + C
(5.75)
The piezomagnetic constant d is the same in both equations due to the idea of reversibility, energy can ow equally well in both directions between electrical and mechanical terminals. is the permeability at constant stress and C is the elastic compliance at constant magnetic eld strength. For the coupling coe cient to be nonzero, the smallsignal quantities have to be imposed on bias quantities. The way they are written, Eqs. (5.74) and (5.75) are suitable for and H as independent variables. Another choice of independent variables gives another (equivalent) material relation. When analyzing the conversion of energy through the material, B and H can be chosen as independent variables. By exciting the sample with a smallsignal B at constant stress (zero smallsignal stress), the magnetic energy density stored will be B2=2 . By mechanically loading the sample at constant B, a decrease of magnetic energy follows. The magnetic energy di erence
113
is converted to stored mechanical energy. By closing the smallsignal B ; H loop with a decrease of B at constant strain, we know from the symmetry of the transduction matrix that the enclosed B ; H loop area has been actually transferred to mechanical work. The loop area divided by the rst phase magnetic storage gives the expression for k2 = ( ; )= , or k2 = d2= C by using the transduction coe cents in Eqs. (5.74) and 5.75. and are the permeabilities at constant stress and constant strain, respectively. 5.15 Physical models The phenomenon of magnetostriction is the ability of pieces of ferromagnetic materials to elongate or contract by the presence of a magnetic eld. The spontaneous magnetostriction of a Weiss domain is obtained as the material becomes ferromagnetic by cooling below the Curie temperature. Curie temperatures for the three principal ferromagnetic elements are several hundred degrees centigrade, which means that spontaneous magnetostriction in these elements and their common alloys is present at room temperature. What we will mean by magnetostriction in the following is the observed elongation of an initially unmagnetized piece as a result of an applied magnetic eld. This occurs as magnetic domain magnetization vectors are oriented from a pseudorandom con guration at zero applied eld strength to a con guration with a resultant macroscopic magnetization by the external application of a magnetic eld strength. In the following, by magnetization we mean, if not otherwise stated, the macroscopic quantity observed as a spatial average of domain magnetizations. Of course, for this to be a relevant description of the magnetic response, the object under consideration has to contain a large number of domains initially (the perfectly saturated magnetic state is a single domain) . If the specimen is a singlecrystal, this would probably mean that a large number of impurities or lattice defects has to be present, otherwise domain wall motion would be uninhibited and single domain behaviour would easily be achieved. Strict single domain behavior is associated with domain rotation between magnetically easy directions in the lattice and the in uence of the so called forme ect from the discontinuous change of magnetization at the edges of the specimen. The principal elements are Cobalt, Nickel and Iron, and alloys of interest to us are SiliconIron (SiFe), CobaltIron (CoFe), NickelIron (NiFe) and TerfenolD (TbFeDy). Applications include power transformer cores and ux conductors in large electric power generators (oriented SiFe), electric motors (unoriented SiFe), relays (NiFe), ultrasonic transducers (CoFe), actuators for prospecting, shaking and vibration control (TbFeDy). 114
100 001
TD
RD
010 Figure 5.5: (110) 001] crystal orientation. RD is rolling direction and TD is transverse direction of the sheet. 5.16 Material structure 5.16.1 Texture Texture is the important structure property here. Since all the rolled materials are polycrystalline, there can be a structure of the alignment of the crystallites that make up the body. That structure is called the texture of the material. In NO (nonoriented) materials, there is no preferred direction and the material will supposedly be isotropic. In GO (grain oriented) materials there will be a distinct preferential direction close to the rolling direction. Furthermore, the crystal unit cube is characteristically rotated around the preferential direction. Two rotational positions are encountered, the cubeonface variant and the cubeonedge position. The latter is the common texture for GO SiFe. The three polycrystalline textures are (110) 001], (100) 001] and nonoriented (with grains randomly oriented). The (110) 001] texture has the unit cell cube of the grain crystals oriented with the cube diagonal plane (110) parallel to the rolling plane and the cube edge 001] parallel to the rolling direction, see Fig. 5.5 The production of such textured SiFe material was invented by Goss in 1933 and the texture is frequently named after him. The name is "cubeonedge" which is short, but imprecise since the rotation of the cube around the edge is unspeci ed. The (100) 001]texture is called "cubeonface" since the cube face is parallel to the rolling plane, but to be exact one has to add that the cube edge is parallel to the rolling direction.
115
5.16.2 Transformer iron qualities The commercial grades of grain oriented silicon iron sheet are
CGO, conventional grain oriented, (Mx, e.g. M5, are American AISI standard names). HIB, "high B", superoriented material, (no independent standard). Material improved during last couple of years.
The di erences between the classes of materials lie in the mean deviation of the misalignment angle between the grains in the sheet to the rolling direction of the sheet. The grain direction is taken as the direction of the cube edge of the crystal, which is also a direction of easy magnetization.
The directional magnetic properties of a sheet with Goss texture comes from the
fact that the cube edges are directions of easy magnetization of the SiFe crystal. The
rolling direction (RD) is very close to an easily magnetized direction of the crystals
and is therefore used in the longitudinal direction of limbs and yokes in a core. The
transverse direction (TD), where transverse is with respect to rolling direction, is
on a cube face diagonal, which is not an easy direction of the crystal, resulting in a
much lower the hardest
directional permeability than RD. In direction (HD) on the cube diagonal,
between which is
RD at
aatnadn(TpD2)o=ne
nds 54:7
from RD.
5.17 Micromagnetic cause of magnetostriction The magnetostrictive strain is a relative displacement of the lattice planes, a change of the lattice parameter, due to ux density change. The ux density changes the equilibrium con guration of lattice planes in the quantummechanical system of the crystals. It might be possible to solve for quantummechanical equilibrium by stating a cell problem which is de ned on the basis unit of the crystal. The Schrodinger equation would then be solved numerically with the appropriate periodic boundary conditions. That kind of simulations are performed in the area of materials science. To get the e ect of the complete medium, a homogenization could follow such a calculation to get the macroscopic magnetostriction to ux density relation.
116
Another possibility is to solve with so called micromagnetic simulation: A set of interaction relations between the lattice points is stated and is simulated in time with an externally applied eld. Such interaction relations for purely magnetic response have been formulated and go under the names of Ising (nearestneighbour interaction) and mean eld interaction. A di culty with both the cell problem and the micromagnetic formulation is how to model grain boundaries. Those surfaces are sources of disturbances as well as sites of impediment. Disturbances in this case are the nucleation of domains, and impediment is pinning and release of domain walls. 5.18 Domains in soft magnetic materials Power losses and acoustic noise are due to domain wall pinning domain nucleation/formation domain annihilation domain magnetization rotation Domains occur as an answer to the global energy minimization principle. Phenomena on many levels contribute to the energy: stray eld energy anisotropy energy exchange energy magnetoelastic energy external eld energy The stray eld energy is most important, since large amounts of energy can be saved by keeping the ux inside the material. This phenomena is a balance between the possibility of getting a lower H eld inside the material than in the air, and the possibility of getting a lower B eld by spreading out the ux in the air. Minimizing stray eld only in an isotropic material would not give rise to domains but ux 117
lines would be smooth and only re ect the specimen shape in order to achieve ux closure with optimal spreading out of ux. In an anisotropic material there could be something resembling domains, since the ux lines would have kinks in going from one preferred direction to another. Anisotropy energy is linked with the magnetization in a single crystal, where there will be directions di cult to magnetize and others easy to magnetize, dependent on the distribution of atomic sites and the interaction between atomic magnetizations. Exchange is the underlying quantummechanical phenomenon of ferromagnetism. There will be a nonclassical contribution to the magnetic energy from interchange of spin between atomic sites in a pair of atoms. This contribution can lead to a favourable energy situation when spins are parallel, which occurs in ferromagnetic substances. Magnetoelastic energy enters as there is probably always at least a weak coupling between magnetic eld and strain eld. The phenomenon can be analyzed on two di erent levels: lattice level and macroscopic level. On the lattice level, one could make a cell problem and solve the Schrodinger equation with variable lattice parameter and magnetic ux density, and see how the lattice would expand or contract as magnetic eld was changed. On a macroscopic level, it is possible to introduce coupling parameters between the pure mechanical entities and the pure magnetic entities. Stratton made suggestions for such parameterizations regarding the electromechanical case for dielectrics. Linearized material relations are used for biased signals in piezoelectric and magnetostrictive (a.k.a piezomagnetic) transducers. The grain size is important in a magnetic context, since very small grains might become single domain particles as demagnization e ects from the boundary of the grain will take over, a phenomenon used to make permanent magnets. In soft magnetic materials, the grain boundaries will act as pinning sites to the moving domain walls during dynamic excitation, and the grain size and shape will a ect power lost to the lattice through these sites. Misorientation of an easy axis to a specimen surface will a ect the domain pattern viewed on the surface. In SiFe with a perfect orientation of a (100) surface, the pattern viewed in the middle of the plane should be broad stripe domains at low Helds and narrow stripe domains at high H elds. Close to the edges of the surface there will be triangular domains providing ux closure between the stripes, as neighbouring stripes have opposite magnetization directions. On a slightly misoriented surface, s.c. supplementary domains will occur, forming a tree pattern with spiky branches extending from the wall separating the main stripe domains. There could also appear a lancetshaped supplementary pattern, with spikes oriented along the stripes and scattered over the stripes. Loss and noise doesn't simply depend on domain wall movement, the reorganization of domain structure (including appearance 118
and disappearance of domains) will also come into the picture. 5.19 Domain walls and magnetostriction Domain walls are named after the di erence in magnetization direction between the domains divided by the wall. Between stripe domains there are 180 walls and between closure domains and stripe domains there are 90 walls. In every domain M is constant, equal to the bulk saturation magnetization achievable with ordinary equipment ("technical saturation"). If the walls present in the specimen were only of 180 type and the motion of the walls was perpendicular to a xed direction (possibly the excitation direction) there wouldn't be any noticeable magnetostriction, because domain magnetostrictive strain would be of same magnitude and state regardless of the motion of the domain walls. In a real sample there will be closure domains at the edges and surfaces (at least for a soft magnetic material) and there might be wall irregularities, domain nucleation processes or closure domains at grain boundaries. For samples with poor grain alignment, there will be deviation of domain magnetization direction between neighbouring grains when domains try to span multiple grains. The microscopic strain is therefore positionand applied eld strengthdependent even during uniaxial excitation, leading to a changing macroscopic strain during the excitation cycle. For wellaligned cubeonedge materials, the negative magnetostriction in the rolling direction is attributable to spike domains (also called lancets) observable on the sample surface, as understood by Shur (1947). The lancets occur due to misalignment of a grain easy axis with sheet surface, and provide ux closure for the stray eld caused by the misalignment. This closure is achieved by a volume domain directed from one surface of the sheet to the other, parallel to the normal of the sheet and at ninety degrees to the main domains. The dynamics of the associated ninety degree walls will lead to an observable magnetostriction. Due to the unknown details and quantities of the processes leading to a nonconstant strain, it is hard to nd a mathematically accurate eld strength to strain expression directly from physical reasoning. What one can say is that the strain is equal for opposite signs of applied eld strength at opposite signs of eld strength time derivative (strain at equal amount of reversal from saturation is independent of sign of bulk magnetization). An even form of the anhysteretic magnetostriction curve can then be postulated. A higher density change of 180 domain walls during the cycle will lead to a higher degree of non180 wall activity, leading to a higher magnetostriction valleytopeak value. Therefore the maximum wall density comes in when predicting the strain magnitude, together with kind of material (saturation magnetostriction value and degree of grain alignment) and individual sample dependency (spread due to manufactur 119
ing process or handling). 5.20 Domain types The domain types are band(or stripe)patterns, spike(or lancet)domains, and mazepatterns. On Goss textured SiFesheet surfaces with a grain easy axis nearly parallel to the surface the primary domain structure seen is a stripe pattern (the primary structure) and the secondary, smaller, structure is a spikepattern, see Fig. 5.6. The spikedomains occur at grain interfaces and at grain surfaces. Maze patterns occur on unpolished surfaces. At grain interfaces spikedomains result from the misalignment of neighbouring grain easy axes, by the fact that the magnetization component normal to the interface is discontinuous and sources an increased magnetic stray eld, a eld that is decreased by the introduction of spike domains. The domains provide a path for some ux to close within the material, which is energetically favourable compared to closing the path through air. The same reason holds for spike domains that can occur along domain walls separating stripe domains for (100) 001]textured sheet. In this case, the magnetization discontinuity occurs due to the grain easy axis misalignment with the sheet surface, and the observable pattern is a treelike array of spikes, each of which is like a small magnet needle, bent at the middle. The grain misalignment with sheet surface sources spike domains scattered over the stripe domains for (110) 001]textured sheet as stated above. There is a correlation between grain length and domain width. In grainoriented (commonly Goss textured) siliconiron, the grains are about 25 mm long and the domains are roughly 0.5 mm wide. If the grains are made longer, the domains will be wider (perpendicularly to the grain length dimension). This is due to the fact that the angle between the grain boundary and the transverse direction will be lower after the grain size has been increased, which reportedly lowers magnetization discontinuity between grains. The domains can then a ord to get wide and escape the energy needed to create domain walls. The wider domains will increase stray eld at interfaces because the equivalent N and S poles will in the mean be farther away from each other, but walls occur across grains and not only at interfaces. The resulting domain width balances the two energy contributions. There is also a correlation between losses and domain width. The local eddy current losses will decrease with lower domain width, indicating that a material with a ne domain structure should be chosen for transformer and machine applications. But a low domain width is also a sign of high interfacial discontinuity, that will be accompanied by spike domains. Spike domains will source much larger losses than 120
z Figure 5.6: Main stripe domains with supplementary lancet domains. Figure 5.7: Lancet domain viewed from the side. 121
the primary domain structure, leading to a minimum loss at a grain length of about 0.5 mm, which balances domain neness and lack of spike domains. There are simple models that predict positive magnetostriction in the rolling direction of SiFe sheets, as an e ect of domain magnetization directional changes from other easy axes to the easy axis parallel to the rolling direction. In (110) 100]textured sheet, most domain magnetizations are already 100] or 100] in the demagnetized state, so the e ect is very small. Empirically, negative magnetostriction up to ;2 m=m is found. Allia explained the unexpected behaviour by formation of volume domains in the body of the sheet having 90 walls, occuring due to grain misalignment with sheet surface. These volume domains are connected to surface spike or lancet domains described above. The spike domains vanish at a critical eld and the magnetostriction becomes less negative as the eld is increased. Nonoriented sheet has a large (up to 40 m/m) positive magnetostriction due to alignment to 100] magnetizations from a wider distribution of 010] and 001] magnetizations. By applying a tension one can introduce an anisotropy in this sense and make the demagnetized state contain more 100] domains. Then the negative magnetostriction contribution from spike domains can be seen again. 122
Chapter 6 Magnetic nite element analysis 6.1 Introduction Simulations are interesting in two respects. Firstly, the measurement setup (yokes, sample and sample table) can be analyzed. Such analysis was carried out to evaluate the concept before the setup was built, later to investigate the magnetic and magnetoelastic elds in the sample with a material model hypothesis. Also an error source, bending of the sample, was analyzed by simulation. The second respect is that to be of greater engineering use, material models obtained from hypothesis and experimentation should be stated in such a form to allow them be included in simulation programs. That is why the study of the simulation method, at least to some level, is important to the experimenting researcher. The project which made this book as an o spring even had as an ultimate research goal to parameterize the phenomenons encountered, so greater weight has been put on the aspect here. When studying di erent alternatives of software to buy, it was soon clear that none of them really allowed the user to experiment with unconventional, nonlinear and/or frequency dependent material models. At the time of evaluation only linear or splined magnetic material functions could be entered, and magneto elastic formulations, if at all present, was only for smallsignal linearized behaviour. It was decided to write the programs by own hand, and the MATLAB language and interpreter was chosen to get full control over formulation and material models, while still providing a decent solver to the nal equation system, so hand coding of 123
or library search for such a solver was avoided. MATLAB's pretty plotting facilities also charmed the author. The simulation technique for the cases presented in this book is the nite element method, and the variants used will be presented in some detail. First a program to calculate the magnetostatic two dimensional magnetic excitation of the sample was written. Of special interest are the results concerning the area of uniform magnetic and mechanical elds where sensors are placed. An inhouse 3D program has also been used. One can de ne geometry and carry out calculations on rectangular parallelepiped elements (also called brick elements). The program was written in C and is for magnetostatic approximations. Trilinear basis functions to the magnetic scalar potential are used and a linear anisotropic magnetic material relation ( a constant tensor) is used as material model. The crossed yokes of Ccore shape with the by them fed sample has been geometrically described. Coils on the yokes have been modelled by equivalent surface poles. 6.2 Coupling niTsohvteadrleiidriesocvatleymraaacgchnoieentvtoeinmduefurcomhmawnBi~ictahdlubpeoruotobnldeelamarsytbicecocinanduteistreiaotcnhtseioacnno.dnTsB~thitenucototivuhepolrmienlogagtcieoannneobu(esw, Bir.eiitBtejins) as B ! $ . There is a weak coupling between and , and the additional small deformation approximation make it possible to solve the magnetic problem rst, followed by a mechanical simulation. The simulations are thus decoupled. Eddy currents are neglected due to the sheets used being thin and with a relatively high resistivity. A magnetostatic analysis will therefore do. 6.3 General motivation and conditions for simulations with computer Problem: How predict a vibration level/noise level from a conceived design change ? Experiments and small scale prototype manufacture can be expensive or misleading, and full scale experiments are impossible in many cases. Solution: A good characterization, magnetic and magnetostrictive, of the core material, with tting software for computer simulations. The software should thus be 124
able to represent the material characteristics in a proper way. The program must also be fast to allow a human to make lots of changes and trials. It should also be easy to make these changes in an orderly fashion. The nal computed result must be accurate so that guarantees safely can be given.
6.4 2D magnetostatic nite element method This is a presentation of a nite element method for the computation of the magnetic eld inside a magnetic material. First a method for a linear material is presented, then a method for a nonlinear material.
6.4.1 A linear isotropic scalar potential problem
In this section a scalar potential problem is presented for linear media. The problem is twodimensional, either in Cartesian or axisymmetric coordinates. The di erential formulation of the problem can be stated as
;r r = 0 r 2
(6.1)
= g r 2 ;D
(6.2)
@n = 0 r 2 ;HN
(6.3)
Eq. (6.2) can be called the quasiLaplace equation. In the magnetic case, is the magnetic scalar potential, and the problem is that of magnetostatics, where is a currentfree domain without equivalent magnetic charges. The boundary ;D is a nonhomogeneous Dirichlet boundary. The Dirichlet condition is used when a given magnetomotoric force (g above), possibly a function of spatial coordinates, is prescribed on the boundary. The word nonhomogeneous means nonzero, or not everywhere zero to be more precise. ;HN is a homogeneous Neumann boundary. The hom. Neumann condition is used where there is no magnetic eld normal to the boundary. In the language of computational magnetics, the nonhom. Dirichlet condition can be termed the normal ux condition and the hom. Neumann condition can be termed the tangential ux condition. Nonhom. Neumann (given eld strenght) boundaries can occur when treating equivalent pole distributions or given ux problems, but these are not treated in the below. The material modelling is carried out by using a scalar coe cient , which might be a function of position. This is the permeability to use for isotropic and linear magnetic problems. When the permeability is dependent upon coordinates, the media is nonhomogeneous.
125
6.4.2 Discretization
The nite element method provides a scheme to obtain a discretized, approximate, version of the space continuous problem. The approximate problem has a solution that is also de ned at every point in space, but is only determined by a nite number of (discrete) values. The global domain is subdivided into elements, and over each local element, the approximate solution is chosen with a simple (polynomial) form. The values of the approximant (or derivatives of it) at the vertices of the elements (or some other discrete nodes) will determine the approximate solution. Those values are called degrees of freedom (dofs). The dofs are determined by minimization of the energy of the approximant. To conclude, the local approximant form will together with energy integrals tting the di erential equation give the best approximate solution possible for the form choice.
In the case of a linear local approximation (piecewise linear globally), the solution has to be sought of the potentials in the nodes where no Dirichlet boundary condition is imposed. These nodes will be called active nodes in the following, and consist of inner nodes as well as Neumann boundary nodes. Furthermore, if the equation is linear, i.e. if the permeability does not have any dependency upon the potential (or any derivative thereof), the discretization leads to a system of linear equations, which will be described in the following.
A good approximate solution will satisfy the integrated weighted di erential equation for many weighting functions. By using integration by parts, one can transfer one di erentiation of the solution to the weight function, and allow approximants
with less regular behavior to be solutions.
Z
Z
Z
; wr rphid = ; w r ~nd; + rw r d = 0 (6.4)
;
w is the weight function. By writing the solution as a linear combination of simple basis functions Ni (Ni e.g. piecewise linear in x y) and choosing the weight functions as the basis functions, one gets the Galerkin formulation of the problem. It is
a well known fact 103] that the Galerkin solution function is orthogonal to the true
solution function and that the The Galerkin solution u(x y) and solving
e=rroPr ifuNnic(txioyn)tuhiecreabnybies
minimized in energy norm. determined by integrating
Z ;
Nj
X rNiui~nd; + Z
rNj
X rNiuid = 0
(6.5)
;
i
i
for all combinations of basis function indices i j. If the elements are triangular and the basis functions are piecewise linear, continuous, and nonzero in only one node
with index equal to basis function index, the element parts of the integration can
be carried out with help from the next section.
126
The basis functions to the global problem only have a small localized support, and the support of a gradient of a given basis function will only overlap with a small number of other basis function gradient supports, so the resulting equation system matrix is sparse.
6.4.3 Single triangle element
When using a linear approximant over a triangular element that has its vertices as nodes, it can be observed that the values of the linear approximant that is unity in one node and zero in the other two, will constitute a kind of coordinate of the distance normally to the baseline between the zero nodes. By permuting the unity node and the baseline, another coordinate is gotten, and the two coordinates can be used to specify a point location. By permuting once more, a redundant third coordinate is gotten. These three coordinates are called the area coordinates N1 N2 N3 and are equal to three simple linear approximants (shape functions) that make up the total element approximant by superposition.
The area coordinates Ni, i = 1 2 3, are
2 4
NNN123
3 5
=
1 2A
2 4
xxx231yyy312
; ; ;
xxx312yyy221
yyy231
; ; ;
yyy312
xxx312
; ; ;
xxx231
32 54
1 x y
3 5
(6.6)
where xi yi i = 1 2 3 are the coordinates of the vertices of the triangle. To remember the structure of the formula one can notice how the indices permute. One should also note that the area coordinates are linearly dependent. The use of this choice is evident if one explores the property
Ni(rj) = ij i j = 1 2 3
(6.7)
i.e the i:th area coordinate is equal to unity in the i:th vertex of the triangle and is equal to zero in the two other vertices, and according to the transformation above the coordinate varies linearly in between the vertices. Thus, an arbitrary linear function over the triangle can be decomposed into a superposition of area coordinates. In the above equations, A is the area of the triangle,
A
=
1 2
f(x2
;
x1)(y3
;
y1)
;
(y2
;
y1)(x3
;
x1)g
(6.8)
The gradients of the area coordinates will be of further use,
rNi
=
1 2A
(yj
;
yk
xk ; xj)
i=1 2 3 j=i
1 k=i
2
(6.9)
127
where is the modulo 3 addition operator1 . Note that the gradients of Ni are constant vectors. The direction of the i:th gradient is normal to the line connecting the vertices number i 1 and i 2. One is now able to construct a symmetric matrix s of scalar products between the area coordinate gradients de ned by a triangle,
sij = 4A2rNi rNj
(6.10)
s11 = (y2 ; y3)2 + (x3 ; x2)2
(6.11)
s21 = (y3 ; y1)(y2 ; y3) + (x1 ; x3)(x3 ; x2)
(6.12)
s22 = (y3 ; y1)2 + (x1 ; x3)2
(6.13)
s31 = (y1 ; y2)(y2 ; y3) + (x2 ; x1)(x3 ; x2)
(6.14)
s32 = (y1 ; y2)(y3 ; y1) + (x2 ; x1)(x1 ; x3)
(6.15)
s33 = (y1 ; y2)2 + (x2 ; x1)2
(6.16)
This matrix contains biquadratic terms of the coordinates of the vertices of the triangle and will be used in the FEM algorithm. By multiplying s with the permeability and a coordinate system scale factor h (h = r for axisymmetric coordinates r,z) and then integrating over the triangle, one gets the so called local sti ness (or system) matrix for the triangle.
6.4.4 System of linear equations
Proceeding with the practical handling of the discretized problem, one can study the case of what can be called an undetermined system  the matrix problem corresponding to a di erential equation with homogeneous Neumann conditions on all boundaries. As will be seen later on, the undetermined system is the discretized problem without boundary conditions imposed. Algorithmically it is simpler to set up the undetermined system rst and then impose constraints from the Dirichlet nodes. By writing the undetermined problem
u = X Njuj
(6.17)
jZ2A D
S^ij = rNi rNjhd i j 2 A D
(6.18)
S^u = 0
(6.19)
where u = u(x y) is the approximate solution eld, u is the column of nodal values of u, A are active nodes (inner nodes + Neumann boundary nodes), D are Dirichlet
1The modulo 3 addition counts with wraparound. If counting would begin with 0, wraparound to 0 would occur at 3, normally counted. In this text, counting begins with 1 and wraparound occurs at 4, e.g 3 1 = 1, 2 3 = 2.
128
nodes and S^ is the undetermined system matrix. It is clear that S^ is singular and that there are in nitely many solutions u  a not properly posed problem, linked to the notion of a oating potential. To make it properly posed, constraints from the known Dirichlet nodes uj j 2 D are imposed by
fi =
; ui
P j2D
S^ij
uj
i2A i2D
(6.20)
Sij =
S^ij ij
i j2A otherwise
(6.21)
Su = f
(6.22)
where f is the excitation column, surrounding node i. The Dirichlet
where nodes
an can
entry fi comes from Dirichlet be stored in the same column
nodes as the
active nodes, and the construction of the system matrix S and the contribution from
the boundary to the excitation column f can be carried out in a single process, the
so called assembly. This process will be described below.
The assembly is regarded as building up the undetermined system matrix S^ by
ssyusmtemminmg actornixtris^b(umt)i,oniss
from each formed by
triangle Km. A single integrating the gradient
contribution, the local scalar product matrix
s(m), where m is the number of the triangle in question. Concisely,
S^ij = X s^(ijm)
s^(ijm)
=
Zm Km rNi
rNjhd =
1 4A2
s(ijm)
(m)
A
1 3
(x1
+
x2
+
x3
)2
h=1 h=x
(6.23) (6.24)
where d = dxdy and h = 1 when x y are Cartesian coordianates and h = x when x y are axisymmetric coordinates (=r z). Note that on the right side of Eq. (6.24), mu has been taken out of the integration and the formula is therefore strictly valid only for a piecewise constant permeability.
6.4.5 Hollow cylinder test case This test case was made to show the order of accuracy to expect from the method for a certain mesh density. The case is obtained by solving Eq. (6.2) in axisymmetric coordinates with z = 1 and z = 13 as homogeneous Neumann boundaries and r = 1 and r = 13 as Dirichlet boundaries, with a potential of 1 At assigned to the inner side and 0 At assigned to the outer side. The mesh used was a triangulation with a 13 13 grid of nodes. 129
When comparing the numerical results to the analytical solution (a decaying logarithmic potential), one could see that in actual nodes the error is typically 1:3 % and in between nodes the error can reach 1:9 %. One should note that the error considered is the potential error. Often in these kinds of eld problems one is more interested in the negative gradient of the potential (the magnetic eld intensity H in this case). By using a piecewise linear approximation of the potential , the H eld is piecewise constant and therefore probably more prone to errors. One must also consider the e ect of such an approximation when it comes to the ful llment of the interface conditions (continuity of normal ux density and continuity of tangential magnetic eld intensity), which is by no means guaranteed.
6.4.6 A nonlinear isotropic formalism
The nonlinear isotropic magnetic scalar potential problem consists of a scalar permeability dependent on the negative potential gradient, i.e. the magnetic eld intensity. The formal discretization scheme is the same as for the linear case, but the end product is a set of simultaneous nonlinear equations. These equations can be solved by iterative methods such as successive approximation (Chord method) or successive linearization (NewtonRaphson method) or optimization methods such as conjugate gradient methods (especially the incomplete Cholesky preconditioned conjugate gradient method, the ICCG) or simplex methods. In the following, the NewtonRaphson scheme will be adopted.
The NR technique is outlined as follows.
nd starting approximation u(0)
n=0
while stop criteria not ful lled
form residual form jacobian solve for Newton correction form approximation
Pruini(uj+u(n1un=)n=)=Pu=Sn;(@[email protected]+ururi(jnu)inuj)nuj ; fi
(6.25)
n=n+1
In this scheme there are a number of things to clarify. First of all, the solution vector contains all node variables regardless of type, so one will have to extend the jacobian with trivial entries so that the Dirichlet node values will not be altered during the iteration. The jacobian depends on all node values so that matrix will have to be reassembled in each iteration. The fact that we are using piecewise linear approximation here, will lead to a fairly simple expression of the jacobian as an outer product of a single vector. Lastly, the stop criteria for the iteration has to be stated. It is important here to remember to check not only the solution, but
130
also the residual, so the residual itself also has to be assembled in each iteration. First, though, we have to express the residual and the jacobian. The residual is
ri ji2A
=
X Sijuj ; fi =
X
Z rNi
rNjhd uj = X S^iju(j6.26)
j2A
j2A D
j2A D
ri ji2D = 0
(6.27)
Note that the residual is de ned as zero for the Dirichlet nodes, while a multiplication of the undetermined system matrix with the solution vector not necessarily will produce zeroes in the Dirichlet node positions. The reason for introducing the undetermined system matrix is once again because of its suitability for assembly.
The jacobian is a bit more elaborative to express. The de nition is easily expanded as
Pik
=
@ri @uk
=
X fS^ij j2A D
@uj @uk
+
(
@ @uk
S^ij
)ujg
=
S^ik
+
X( j2A D
@ @uk
S^ij
)uj
8i k 2 A
(6.28)
[email protected]@tuukjis =
ik.
The second term is now examined. The derivative of a matrix
@ @uk
S^ij
=
Z
rNi
@ @uk
rNj
hd
(6.29)
and it is clear that it is only a ected by a nonlinear . The desired dependency to express the permeability in is customarily, and probably the most convenient, the magnetic eld intensity squared. This is useful when isotropic materials are present, since the directional properties of the eld is not of interest. One should also remember that a change in a single nodal variable changes the global eld, let alone with a small local support as the basis functions are constructed as such. Writing the eld and the nodal variable derivative thereof,
H~ = ; X rNlul
(6.30)
l2A D
@H~ @uk
=
;rNk
(6.31)
one obtains a mathematical statement of that. Now, it is possible to write down the nodal variable derivative of the permeability,
@ @uk
=
@(H~ H~ ) @uk
@ @H2
=
2H~
@H~ @uk
@ @H2
=
2 X rNk l2A D
rNlul
@ @H
2
(6.32)
Thus, for linear triangular FEM it is seen that the derivative depends on the nodal values in the vertices of adjacent triangles to the node in question, as well as
131
the permeability vs H squared, which is a function of the triangle number as the eld approximation is piecewise constant. Rewriting the second term suitable for assembly one obtains
P m
2
@@H2
jH2
(Km)
(RKm
hPd j2) APlDj([email protected]@ukDS^iujl)urjN=l
rNk rNi
rNj]Kmuj (6.33)
where Km denotes that correspond to
the triangle vertices on
with number m. For a given m only those i Km will contribute to the second term. By
klj intro
ducing the vector
bk = H~
@H~ @uk
= X ulrNl l2A D
rNk
(6.34)
the assembly is somewhat simpli ed, when evaluating this vector on the triangle
Km as
b(km) = X ul rNl l:~rl2Km
rNk]Km
=
X l:~rl2Km
1 4A2
s(kml )ul
(6.35)
and writing a factor of the second term of the jacobian as an outer product of the evaluated bvector,
b(km)b(im) = X ul rNl rNk rNi rNj]Kmuj l j2A D
(6.36)
Recall that local vectors and matrices superscripted by m only have nonzero entries
for indices corresponding to restricted to these. One can
vertices on triangle also notice that the
HKmealdndsqtuhaeresdtocraangebies
therefore expressed
in terms of the bvector,
H2(~r 2 Km) = ( X rNjuj)2 = X uib(im)
j:~rj 2Km
i:~ri2Km
(6.37)
which is obtained after rewriting the square of the sum as a quadratic form of the s(m)matrix.
Another thing to sort out is how to de ne the jacobian for entries that correspond to Dirichlet nodes, when the equation
r(un) = P un
(6.38)
is solved for the NewtonRaphson correction un. Prior to solving this equation,
the residual elements to zero. Now setting
corresponding Pik i 2 D to
to ik
Dirichlet nodes, i.e. ri i will produce zeroes in the
2 D, are put Dirichlet node
elements of the Newton correction. This can be seen as extending the equation set
for the active nodes with trivial equations for the Dirichlet nodes. The equations
132
for the active nodes should remain unchanged, so Pik i 2 A k 2 D have to be set to zero.
The stop criterion also have to be stated. It is immediately clear that the Euclidean
norm of the Newton correction should be close to zero, and this is often expressed
in a sense that
k unk kunk
<
tol
(6.39)
The residual also has to be close to zero, inferring that
k rnk krnk
<
tol
(6.40)
In the case that un = 0 is a possible iterate, the norm of the iterate may be safely
swapped to unity in the test. The tolerance tol is chosen according to the sought
accuracy, after the machine double precision, something
lipkreec1i0si;o5n
has been taken into might be considered.
account.
When
using
6.5 3D isotropic formulation
To estimate the magnitude and homogeneity of the produced magnetic eld in the measurement area, three{dimensional magnetostatic nite element simulations have been performed using the following formulation. The relevant equations are
r H~ = J~ r B~ = 0 B~ = H~
(6.41)
The problem can be expressed on the form
H~ = H~ p ; r r ( r ) = r ( H~ p)
(6.42)
where region
is and
Ha~ pcoinstainnuyoaursbsitinragrley{vveaclutoerd
solution that is particularly convenient
sheceladrleasraistpiHos~ftypein=ntgiaRrl(Ju~seHd~~epxin)=dtxhJ~,e
entire solution 72]. A general where ~ex is the
cuhnoiticveeoctfoH~r pinisthtehaxt{dfoirretchteiopnr.eAsepnatrgteforommetbrye,initg
simple, an attractive feature of this becomes zero everywhere except in
etheecrteginionthsecotnesttaisnheedetwwithhiinchthoecccuorisls.wAhesnajHr~epsuj lt, wjH~e ja.voTidhethteercmanrcella(tiHo~np)erisroar
surface density at the coil ends.
Fig. 7.1 shows results when a current was applied to one coil pair only. The sheet
had The
a size of 140 linear B ; H
140 0:5 relation
mm3 and a constant isotropic permeability = 1000 adopted is appropriate for lower eld intensities and
0. is
133
su cient to investigate homogeneity close to saturation. The system was solved using 7168 trilinear block elements. The inhomogeneity of the magnetic eld in the central 60 60 mm2 area was found to be approximately 10 %, while the leakage ux was about 25 %.
6.6 3D anisotropic formulation
In the calculation of the ux density distribution, eddy currents are neglected and it is thus su cient to perform a single magnetostatic run for a ux peak time instant. The solution achieved can be used to nd the magnetic stress components according to Eq. (5.10). The magnetostatic equations are
r H~ = J~ r B~ = 0 B~ = H~
(6.43)
In the sheet, the permeability is a tensor with x = 52000 in the rolling direction and y = 3200 in the transversal direction. These are typical values for a highly grainoriented material in the linear region 55]. In the lateral direction z = 3200 was used, and all o {diagonal entries were set to zero. The system can be solved using a single continuous scalar potential by writing it on the form
H~ = H~ p ; r r ( r ) = r ( H~ p)
(6.44)
where H~ p is any vector function satisfying r H~ p = J~. We here set
Z
H~p = (J~ ~1x) dx
(6.45)
where ~1x is the unity vector along the x axis. This choice of H~ p is for the current geometry akin to replacing the coils by permanent magnets or, equivalently, using magnetic charge surface densities at the coil ends. The system was solved using 8092 trilinear block elements. Some results are shown for the cases when the sample was magnetized in the rolling direction (Fig. 6.1) and the transversal direction (Fig. 6.2) respectively. The currents were adjusted so that the maximum ux density in the xdirection of the sheet was 1.2 T in both cases, corresponding to maxima in jByj of 0.22 T and 0.63 T, respectively.
134
y
z
x
x
Figure 6.1: Equipotential lines for the magnetic scalar potential. Sample magnetized in the rolling (x) direction. Oriented material.
y
z
x
y
Figure 6.2: Equipotential lines for the magnetic scalar potential. Sample magnetized in the transversal (y) direction. Oriented material.
135
Chapter 7 Mechanical nite element analysis 7.1 Introduction Mechanical FEA has been carried out to investigate the strain eld in the sample and possible bending of the sample. The rst problem required a nite element program for plane stress that could use magnetostrictive strains as a source. The second problem required a plate bending program with gravitational and reluctance force loads, as well as a possibility to experiment with inplane loads from strain to examine buckling. There was no program at the department that could calculate these cases so they were written from scratch. This also gave the opportunities to experiment with nonstandard loads and to import data freely from other calculations. An inhomogeneous source strain will likely set up stresses in the body, unless the source strain ful lls the equilibrium conditions with zero stresses. 7.2 E ect of inhomogeneous magnetization When the magnetization is inhomogeneous, it is coupled into mechanical stresses that tend to smooth out the measurable total strain. We have attempted to estimate this e ect by using the magnetic solution to nd the una ected magnetostrictive 136
strains, followed by a nite element analysis to approximately solve the elastic boundary value problem. In this analysis, we assume that a state of plane stress prevails in the sheet and formulate the numerical procedure in terms of material point displacement (u v) parallel to the sheet. The equations of equilibrium and the straindisplacement relations can then be expressed as 104]
@x x + @y xy = 0 x = @xu
(7.1)
@x xy + @y y = 0 y = @yv
(7.2)
xy = @xv + @yu
(7.3)
where xy is
ii the
= x y are the shear stress,
mechanical i i=xy
normal stress components parallel to are normal strain components and
the xy
sheet, is the
shear angle (or engineering shear). We have excluded the in uence of body forces,
typically gravity, since the sheet is light and rests horizontally. Next, a stressstrain
relation is needed. Here we have to use a simple representation from data available
in the literature, while still retaining mathematical soundness to obtain a properly
fpuonscetdiopnroobfleB~m..
It A
is well known that the anhysteretic magnetostriction fair approximation for a nonsaturated material is
is to
an even assume
a quadratic dependence. Moreover, it is reasonable to expect that the strain is
a ected by the tensor BiBj in a qualitatively similar manner as it is by the stress
tensor. In this way we can get an expression for the shear in uence of the magnetic
eld. If we assume the material to be linear and isotropic, we can express the total
strain as
x=
1 Y
(
x;
y) +
1 0P
;Bx2
;
By2
(7.4)
y=
1 Y
(
y;
x) +
1 0P
;By2
;
Bx2
(7.5)
xy
=
2(1 + Y
)
xy
+
2(1 + 0P
) BxBy
(7.6)
$whoernetEheisritghhet{mhaonddulsuisdeofeexlparsetsicsietsytahnedconivsePntoiiosnsoanl 'Hs oraotkieo'.s
The law.
terms involving The remaining
terms are the magnetostrictive strains. P is a magnetoelastic modulus, while
is a magnetoelastic Poisson ratio. If we assume that, as is often the case, there
is no volume magnetostriction, we get = 0:5. The condition of linear elastic
isotropy re ects itself on the expression E=2(1 + ) for the shear modulus, and
in analogy with that, one is able to write the magnetoelastic shear modulus as
0P=2(1+ ) when the magnetostrictive strain is linear in BiBj and magnetoelastic
material isotropy is present. P can be found from experiments by noting that
for are
a homogeneous, onedimensional magnetic eld, = magnetostriction and ux density respectively. Here
we01PhBav2e
where used a
and value
B of
P = 32 109 N/m2 based on experiments reported in 105] . The other constants
137
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Figure 7.1: Magni ed (factor 5000) deformation of sheet from ux density vectors. Nonoriented material.
used are E = 200 109 N/m2 and = 0:30, which are typical for a nonoriented sheet grade.
A displacementbased nite element algorithm with bilinear rectangular elements has been developed and used. The results for one quadrant of the sheet are shown in Fig. 7.1. Symmetry boundary conditions are used on the left and lower sides, and the upper and right sides are free. The ux density from the magnetic calculation is approx. 0.6 T in the measurement area and 1.1 T at its highest, close to the feeding pole.
It is seen that the di erence between magnetostrictive strain and total strain in the
measurement area is 16 % at its highest. If strain measurements are performed by
interferometry, a mirror pair spacing of 6 cm will result in a relative displacement
of 0.65 m. The change in permeability due to the induced mechanical stress can
be found using the Maxwell relation 0 @[email protected] the relative change in was estimated to be
ij in
[email protected] the
ij [email protected] Hk . order of
I1n0;th3eacnudrrceannt
case, thus
safely be neglected.
138
y(mm) 28 20 12 4
sx (um/m) at x,y
9.24 9.61 9.85 9.96
9.36 9.72 9.98 10.10
9.64 9.99 10.23 10.38
10.05 10.42 10.66 10.77
y(mm) 28 20 12 4
sMx (um/m) at x,y
7.58 7.97 8.19 8.31
7.74 8.08 8.36 8.48
8.06 8.41 8.65 8.83
8.56 8.93 9.18 9.31
x (mm) > 4
12 20 28
Figure 7.2: Total strain sx and magnetostrictive strain sMx in the measurement area. Nonoriented material. 7.3 Mechanical simulation method By the term mechanical we here denote the force interactions in the material which give rise to strain of both elastic and magnetoelastic nature. By magnetoelasticity we call the process of pure magnetostrictive strain occurring as a response to the magnetic stress tensor, a process that is measured as total strain in homogeneously magnetized samples. On each harmonic component of the magnetic stress tensor, a magnetomechanical simulation is performed using proper nite element software as developed earlier 92]. In this case, one static simulation with real nodal displacements and one harmonic simulation with complex displacements su ces. Assembly of the sti ness matrix is carried out concurrently with the incorporation of the magnetostrictive strains in the load column 104]. The problem was solved on 225 bilinear rectangular elements. All cases had boundary conditions of upper and right edges free, whilst left edge had u~ zero and lower edge had v~ zero due to symmetry. Results at a ux peak time are shown in Fig. 7.3 for magnetization by the vertical yokes and in Fig. 7.4 for magnetization by the horizontal yokes.
7.4 Results and interpretation Investigations of the strain in the central region of the sheet (lower left corner in simulations) are presented in Tables 7.1 and 7.2. It is seen that for the sheet being magnetized by the horizontal yokes, the dynamic normal strain with frequency
139
0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Figure 7.3: Magni ed (factor 50000) deformation of sheet at ux peak time when xmagnetized. Flux density vectors drawn. Undeformed boundary dashdotted. Oriented material. 100 Hz in the horizontal direction is 20 % lower than the pure magnetostrictive strain. When magnetizing by the vertical yokes, it is seen that the amplitude of the ux density in the ydirection is as weak as 0.33 T in the central region together with maxima of 0.62 T at coordinates (0,0.1) m, closer to the feeding pole. This gives an in uence of the surrounding to the central part. That local minima in magnetostrictive strain then will show as higher local total strain is re ected in Table 7.2. When magnetizing and allowing for saturation, we will get a di erent picture at ux peak time as the quadratic dependence of magnetostrictive strain to ux density will have a major e ect. Inhomogeneity of ux density is fairly low in the central region in both cases, 5 and 8 percent respectively. In total strain, the inhomogeneity is 16 and 9 percent respectively. The maxima at the origin is 1.0 m/m for negative x in the rst case and 2.1 m/m for y in the second case.
140
y mm] 28.00 20.00 12.00 4.00 x mm]
j~x(2)j m=m] at x,y 0.4432 0.4473 0.4531 0.4638 0.4649 0.4687 0.4755 0.4867 0.4795 0.4839 0.4910 0.5020 0.4873 0.4908 0.4989 0.5101 4.0 12.0 20.0 28.0
6 ~x(2) ] 158.6 158.6 158.6 158.6
y mm] 28.00 20.00 12.00 4.00 x mm]
0.5685j~Mx 0(2.5)7j 24m=0m.5]7a2t6
x,y 0.5804
0.5953 0.5953 0.5992 0.6032
0.6069 0.6109 0.6109 0.6148
0.6148 0.6148 0.6187 0.6227
4.0 12.0 20.0 28.0
6
~Mx
(2) ] 158.6
158.6
158.6
158.6
Table 7.1: Dynamic normal strains in xdirection when xmagnetized.
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Figure 7.4: Magni ed (factor 50000) deformation of sheet at ux peak time when ymagnetized. Flux density vectors drawn. Undeformed boundary dashdotted. Oriented material. 141
y mm] 28.00 20.00 12.00 4.00 x mm]
j~y(2)j m=m] at x,y 1.1036 1.1023 1.0974 1.0908 1.0690 1.0666 1.0627 1.0564 1.0461 1.0436 1.0390 1.0308 1.0349 1.0324 1.0270 1.0176 4.0 12.0 20.0 28.0
6 ~y(2) ] 21.4 21.4 21.4 21.4
y mm] 28.00 20.00 12.00 4.00 x mm]
j~My (2)j m=m] at x,y 0.8979 0.8981 0.8985 0.8878 0.8314 0.8315 0.8317 0.8320 0.7885 0.7885 0.7886 0.7887 0.7675 0.7675 0.7675 0.7675 4.0 12.0 20.0 28.0
6 ~My (2) ] 21.4 21.4 21.4 21.4
Table 7.2: Dynamic normal strains in ydirection when ymagnetized.
7.5 Strain eld calculation method
7.5.1 Plane stress constitutive relation
The constitutive relation between B has been written
8 < Cij] :
x y xy
9 = +
Dij ]
1 0
8 < :
BBBxBxy22 y
98 =< = :
9 x= y xy
(7.7)
for an elastic and magnetoelastic material. For a magnetized material, without
externally applied stress, there will be an elastic reaction from the surrounding to a
magnetic spot. The reaction is modelled with the conventional stress, and its e ect
on strain through the compliance coe rise to a strain through the magneto
cients Cij compliance
. The local magnetization give coe cients Dij. The D matrix
is proper when no internal torque is present, which is the case when no externally
applied eld is present, or when the material has aligned its magnetization with
such a eld.
7.5.2 Finite element method In a numerical approximation, the strain eld is the derivative of the approximated displacement eld. The displacement is described by basis functions of coordinates 142
with nodal displacements as parameters. The derivatives of the basis functions that
describe the strain eld are collected in a matrix B, that depends on coordinates.
For a local rectangular nite element with corner nodes, the strain approximation
is
8 < :
xy((xx
y) y)
xy(x y)
9 = = B(x
8 >>>>>>>>>>< y) >>>>>>>>>>:
u1 uvv122 uuvv3434
9 >>>>>>>>>>= >>>>>>>>>>
(7.8)
This equation is called the straindisplacement relation. u v are displacement components indexed by node number. To get an equation system for the eight displacement values (degrees of freedom) there has to be eight simultaneous equations. The straindisplacement relation above only has three equations, so if the state of strain is known at one point in the element (e.g. at center of mass), one must reduce the number of equations to make the system determinate. The choice of way to do it is guided by physics, one wants to minimize the energy di erence between strain eld approximation/knowledge and displacement eld approximation. An equation with energy densities is gotten from multiplying with UT BT E from the left on the strain expressions. U is the nodal displacements column, B is the shape of the strain eld and, E is the elasticity matrix. The multiplication with E gives stress, and UB is strain, so the result is energy density. To get energy, one simply integrates over nite element area. When equating, we get
8
9
fUigT
Z Sl
BT
EBdS
fUig
=
fUi
gT
Z Sl
BT
E
< :
x y
= dS
xy
(7.9)
for the energy equation. To solve for U, one can identify the parts to the left
of of
BUTT
, which is a statement of an eight equation system. (x y)EB(x y) the local sti ness matrix k, which is a
We call the integral characteristic of the
element shape, material properties and strain approximant shape. Thus,
Z
k = BT EBdS
Sl
8
9
k fUig
=
Z BTE <
Sl
:
x= y dS xy
(7.10) (7.11)
To solve a real problem, nite elements have to be connected (reducing number of degrees of freedom) and boundary conditions have to be imposed. When there is a part of the strain prescribed from strong magnetoelastic interaction, a sound
143
method to evaluate the right hand side has to be adopted. For a ne subdivision into elements, a simple method is to set that strain constant over elements. Elements with constant strain and linear displacement approximations (CST, constant strain triangles) are suitable for the prescribed strain contribution chosen piecewise constant. The CST has been used in the history of the nite element method, but su er from not being capable of representing certain modes of motion of the element. For rectangles, at least a bilinear displacement approximation must be used to have four parameters per displacement component. Bilinear rectangles are probably less compatible with a constant strain part, and they also su er from some mode restrictions. Modern formulations use cubic interpolants to cover in plane bending. Still, with a ne element subdivision constant strain prescriptions will be practical, and will probably work. The alternative is to give strain data in the points used for numerical integration with the interpolant in question (Gauss quadrature). Another thing is that the driving strain in the magnetoelastic case considered will come from a decoupled magnetic eld simulation, where a linear or bilinear potential is probably used. So strain data must be compatible with both the magnetic and the elastic elements. In reality one cannot really hope to get everything one wants, so there will be compromises and room for improvement.
The above energy approach is equivalent to a Galerkin method. It is here symbolically gone through, because it shows more clearly the connection to the equilibrium equation that is solved and the approximations made. Also, the global viewpoint is simpler to take. The di erential equation problem is, with primes for spatial derivatives,
0 = 0 equilibrium eq:
(7.12)
E = constitutive eq:
(7.13)
= u0 strain ; displacement eq:
(7.14)
u j;= g boundary cond:
(7.15)
= E + M strain contrib:
(7.16)
u = uE + uM displacement contrib:
(7.17)
It is possible to solve directly for stresses, but here the solution is sought in terms
ohfasdiEspulaEc00em=en;tE.
BMy0
inserting the other relations , where the magnetoelastic
into the equilibrium equation, one strain on the right hand side acts
as a source to the elastic displacement on the left hand side. If the sought elastic
dbiespnldaocfemnuenmtbiesrinotfesripmoulalttaendewouitshenqudoaftionnusm. bTerheoyf
degrees of freedom, there has to are made from the second order
di erential equation by forming the scalar product (multiplication and integration)
with a set of weight functions fwjg,
Z S
wj E uE00
=
Z ; S
E
M0
j = 1::ndof
(7.18)
144
where the integration di erential dS is suppressed. A wider class of solution functions is allowed by integrating the left hand side by parts to get a less singular factor from uE (weaker restrictions on u) in the integrand,
wj E uE0
j;
Z ; S
wj0 EuE0
=
Z ; S
E
M0
j = 1::ndof
(7.19)
The interpolation of uE is index i is understood, and
NwiriattreencaulEled=baNsii(sxfuyn)cUtiio, nwshoerreshsaupmemfuatnicotnioonvs.erTdhoef
question is now how to choose basis functions and weight functions. The Galerkin
method uses the basis functions as weight functions also,
Nj E Ni0 UiE
j;
Z ; S
Nj0ENi0UiE
=
Z ; S
E
M0
j = 1::ndof i = 1::ndof
(7.20)
It can be proven that the solution from this weight function choice minimizes the en
ergy di erence between left and right hand side of the original di erential equation
(the approximate solution is orthogonal to the true solution). The basis functions
remain to set. They are not orthogonal to each other, but are constructed so that
each function has a local support (one element wide) around a certain node to make
it associated with (scaled by) one dof only. In this way, dofs are made independent
of each other and the solution at a point is mostly dependent of the closest sur
rounding which is physical. The resulting equation system is fairly simple to state
(assemble) and requires little memory to store as it is sparse. Boundary conditions
enter as ;. That
atesromurwceillinvathneishintfoergrhaotmedotgeernmeo,udsep(zeenrdoe)nDt iornichploestit(iuoEn)aolornNgetuhme abnonun(duaEr0y)
boundary parts and nodes with such Dirichlet conditions won't need associated
shape functions. Nonhomogeneous boundaries will require special treatment, as
the dofs there will be constants (Dirichlet) or unknowns (Neumann). Simplest is
to keep the shape functions for all boundary dofs and replace the associated rows
(equations) with identities and move the associated column entries (terms) to the
right hand side by subtracting. The equation system is then solved for all dofs with
a slight overhead for known dofs, but the assembly process is kept the same for all
boundaries and conditions on the boundaries.
sAptlranacoientmeeoennltdshhaoanwpdetehfnuetnescrhtaiionpnetdhfueernidcvotaiftoitnvosessatNrreai0iunasremedtahitseritbxhueBilsd,uiwnbghjeibcclthoocikfsstmhoeuf ltbthiepelloaiwepd.prwToixhtiehmdtahitseetransposition of itself and the (constant) elasticity matrix and followed by integration over element coordinates to form the local sti ness matrix k. We look now on the displacement shapes themselves: A dof is a node scalar that scale an associated basis function, with the association made so that the only nonzero nodal value of the basis function is at the node of the dof. This scheme is called Legendre interpolation, and has been used in this work for plane stress problems. A nodal basis
145
function can be used for all displacement directions if there are multiple dimensions. Hermite interpolation is used when the dof is associated with the basis function to scale through a single nonzero nodal zeroth (as in Legendre) or rst derivative. In this work, Hermite interpolation has been used for bending problems. To be very clear in the Legendre case, the interpolation and derivative of interpolation matrixes can be written out,
u(x y) v(x y)
=
N1 0
0 N1
N2 0
0 N2
N3 0
0 N3
N4 0
0 N4
fUig
(7.21)
2 4
3 x
2 N1 x
y xy
5=4
0 N1 y
0
NN11
y x
N2 x 0 N2 y
0
NN22
y x
N3 x 0 N3 y
0
NN33
y x
N4 x 0 N4 y
03
NN44
y x
5 fUig
(7.22)
fUig = u1 v1 u2 v2 u3 v3 u4 v4 T
(7.23)
Earlier paragraphs have dealt with forces, stresses and energy, and it was understood that the equilibrium equations were not ful lled at every point for the approximate solution. In elasticity, there is another requirement to ful ll called the compatibility condition. It states that mass can neither penetrate itself (implode) nor leave holes in itself (crack) under the circumstances present. This condition can be cast in a di erential equation form. Solutions that are approximate with respect to the equilibrium equation will probably be worse with respect to the compatibility equation, deformed elementtoelement continuity is not exact, for example. The fact that the interpolants on elements are separate for the displacement components complicates the evaluation of intraelement compatibility.
7.6 Bending A problem with high B elds is that the sheet has a tendency to bend. In the speci c measurement setup an oscillating bending is due to reluctance forces from the yoke that supports the sheet. When the reluctance force is present, the sheet will be sucked to the yoke pole surfaces, and be clamped. When absent, the sheet will be simply supported and only acted upon by gravity. This oscillation will be an error source to magnetostriction measurements with optical means. One can also suspect that magnetostrictive strain energy easily can make a transition into bending energy due to the low bending to tensile sti ness ratio. So a program to investigate bending has been written. Comparisons with experimental light beam deviations from re ectors on the sample have also been done. A numerical scheme is presented below to solve the plate bending problem for a thin conventional grain oriented siliconiron sheet in a heterogeneous magnetic force 146
eld produced by a yoke con guration asymmetric with respect to the sheet plane 7.7. De ections of the sheet midsurface are small.
7.6.1 Magnetic eld and force calculation
The magnetic problem was solved with the method presented in section 6.6. Eddy
currents were neglected. The setup is typically operated at 50 Hz, so there will
be some eddies in the sheet where the ux enters from the (laminated) yoke pole
pieces. The sheet is 0.23 mm thin so the assumption is fair. In the sheet, the
permeability is a tensor with x = 52000 in
x) direction and (H~zp)=diRre(Jc~tion~1xw)adsx
y = 3200 in used z =
the transversal 3200 and all o
the rolling (or longitudinal, LD, or (TD or y) direction. In the lateral {diagonal entries were set to zero.
To get the magnetic force distribution from the FEsolution, one can use the expression for the Maxwell stress
f~M = (2 0);1Bz2e~z
(7.24)
where f~M is the surface force density acting on the sheet in the lateral direction and Bz is the ux density on the air side of the airsheet interface. As it is the scalar potential that is continuous over the interface and not necessarily the computed normal ux density, the force density is prone to errors. An alternative used here is to integrate iron element uxes and determine the air ux for each element from Gauss' theorem. Flux density vectors can be seen in Fmg. 7.7 for LDexcitation. Maximum ux densities were 1.2 T in the sheets for both LD and TD excitation cases.
7.6.2 Bending formulation
The governing equilibrium equation is
Mx xx + My yy + 2Mxy xy + fM + fg = 0
(7.25)
where Mi is bending moment per The gravitational force density is
unit length associated fg = tg, where mass
wdeitnhsibtyendi=ng7:s8trkesgs=edsm3i,.
free acceleration g = 9:8 m=s2 and t is sheet thickness. We make the following
assumption of the strain distribution from the bending stresses over the crossection
of the sheet,
xyxy(x((xxyyyzzz0)00))===;;;zzz0 00
xy((xx xy(x
y) y) y)
z yz
(x (x
zx(x
y y y
zzz000)))
= = =
0 0 0
(7.26)
147
z 0.2
0.15
0.1
0.05
0 x
0.15
0.1
0.05
0
0.05
0.1
0.15
Figure 7.5: Geometry for the cut y = 0 in m with gravity as only load. Deformation of sheet magni ed with factor 50. Undeformed sheet dashdotted.
where z0 is the lateral coordinate from the sheet midsurface and i are curvature components. This is the approximation of Kircho plate theory. Most notable is the assumption of zero lateral shear forces, which is only valid for thin sheets. The curvature components are
x = w xx y = w yy xy = 2w xy
(7.27)
where w is the de ection of the midsurface in the zdirection. By using a moments to curvature relation one can symbolically see (7.25) as 0 = M xx / xx / w xxxx which means that the di erential equation expressed in w is the biharmonic equation 103]. The approximate solution is here approached using energy arguments. We use rectangular elements with twelve degrees of freedom of a Hermite de ection eld approximation w(x y) = Ni(x y)di. di is a degree of freedom (dof) that scales a single nonzero nodal de ection or derivative of de ection (which is midsurface rotation), via its associated shape function Ni. A full cubic has ten parameters so two fourth order terms are also needed, here chosen as xy3 and x3y in local
148
coordinates with origin in element center of mass. One can show that N1= 2 a3b3;3 xa2b3;3 ya3b2+4 x8yaa32bb32+x3b3+y3a3;x3yb2;xy3a2 N2= a3b;xa2b;a3y;x2a8b+a2xbya2+x3b+x2ya;x3y N3=; ;ab3+xb3+yab2;xy8b2a+b2y2ab;xy2b;ay3+xy3 N4 = N1(;x y) N5 = ;N2(;x y) N6 = N3(;x y) N7 = N4(x ;y) N8 = N5(x ;y) N9 = ;N6(x ;y) N10 = N7(;x y) N11 = ;N8(;x y) N12 = N9(;x y)
wdihreecretioNn1+w3ix
is connected and N3+3i y
with with
nodal nodal
de w
ye.ctaioann, dNb2+a3riex
with nodal rotation element halfwidths.
in xOne
is now in the position to write out the double strain energy due to bending in an
element as
Z
Z
2ub = Ve iEij jdV =
iDij jd e
(7.28)
where integration over sheet thickness using (7.26) has been done over the right equal sign, and double index summation is understood. Eij are planestress elas [email protected][email protected]=icihij=3y=ai;enlDtdd3si=jk12ijdE.o=ifjTnoBaurimgekedbtkeetrxwhsu.ehreacIrlunerrsvieignaritdtdiuinicrtgeieessthietilhsrduaintonngoteivhveseerettlcsohoceumaplmpeoBonnmieekrengnt=yst
expression (7.28) dofs can be factored out,
Z 2ub = dl BilDijBjkd dk d=ef dlklkdk e
(7.29)
giving a de nition of the local bending sti ness matrix klk. Equalling the strain energy with work done from a force free state to loaded equilibrium one can write
Z
dlklkdk = dl Nl(fM + fg)d
(7.30)
e
This relation holds strictly only for the whole body, so assembly summation over all elements has to be done, giving the global sti ness matrix
Kig
jg
=
X e
kie
j=f
(ig
e)
f (jg
e)
(7.31)
where f is a global to local dof renumbering function and e is element index. The
same holds for the right and Ri assembled shape
hand side giving DiKijDj = DiRi function weighted loads. Boundary
with Di global dofs conditions (BC) are
149
0.15
0.1
0.05
0
0.05
0.1
0.15
0.15
0.1
0.05
0
0.05
0.1
0.15
Figure 7.6: Equilines of de ection (solid) for B 0 case. Outlines of pole surfaces (dashdotted).
then imposed. For the present problem, two types of BC occur simply supported (w = 0) and clamped (w = w x = w y = 0), which set dofs connected with the conditions and their energy contributions to zero. It is well known that the solution to the linear equation system KijDj = Ri will minimize the residual of the energy equation. The BC:s are incorporated in the system by zeroing out associated terms in the equations and replacing associated equations with the BC:s themselves. The linear system is solved with a sparse Gaussian elimination method.
7.6.3 Extra details
The Kircho bending assumption of zi = 08z i 2 x y means that there is no shear strain on a 2D element cut perpendicular to midsurface of sheet so there is only a pure rotation of this element. If no additional inplane strain is present (u(z=0)=v(z=0)=0), the in(undeformed)plane displacements over the thickness of the sheet are due to this rotation and can be written
u = ;zw x
(7.32)
v = ;zw y
(7.33)
where w that the
i are slopes de ection
of the midsurface, eld w(x y) is su
i.e. a cient
measure of the rotation. One now sees as the unknown. When di erentiating
Eq. (7.33) to get strains, the derivative of midsurface rotation enters, which is the
150
curvature (x y) = w xx w yy 2w xy]T . The straincurvature relation becomes
2
32
x
4 y 5=4
u v
x y
3
2
5 = ;z 4
w w
xx yy
3 5 = ;z
xy
uy+vx
2w xy
(7.34)
From this relation it is trivial to use the plane stress elasticity coe cients to get the stress. The relation also states that the strain and stress pro les over the thickness of the sheet are linear in the Kircho approximation. The bending stress is
i=
i(z = t=2) t=2
z
(7.35)
This distribution of stress is antisymmetrical w.r.t. the sheet midsurface and can be quanti ed as the bending moment inside the material (on the matter above a coordinate from the matter below the coordinate). The quanti cation is done by integration of the product of stress with torque lever distance z,
Z t=2 Mi = ;t=2
i(z)zdz =
i(t=2)
t3=12 t=2
=
;Eij
jt3=12 d=ef ;Dij
j
(7.36)
where the bending sti ness D = Et3=12 has been introduced, and the index is i 2 x y xy. The obtained moment curvature relation M = ;D simpli es the description of bending, and will be used onwards. The torque component labels correspond to stress directions, e.g. Mx is from stresses x(z) and tries to turn matter around the yaxis. In an another indexing system the torque components would be labeled after the associated rotation axes.
The energy in a bent con guration comes from the applied bending moment having produced a curvature of the surface so that each element has a zero resultant force in equilibrium. The opposing quality of the surface is the bending sti ness, between element and surrounding as well as between surface and external load/applied bending moment. The element energy can be deduced from strain energy using the approximation above,
U
=
1 2
Z Ve
TE
dV
=
1 2
Z Ae
Z t=2 ;t=2
z2
TE
dzdA
=
1 2
Z Ae
T D dA
(7.37)
NrTohwde)wapitphr=o1x2iBmedna,ttriwioenhse,[email protected]@[email protected][email protected]]eT
ection interpolation, w = . For a rectangle, N is a curvature approximation
in the energy expression gives
U
=
1 2
Z
TD
dA =
1 2
dT
Z
BT DBdAd d=ef
1 2
dT
kd
(7.38)
151
where the element bending sti k is 12x12 if the element is a
ness matrix k quadrilateral
w=itRhAnBoTdDesBodnAithsavs ebretiecnesi.ntrBodius ctehde.
derivative of the de ection interpolant or the scaled curvature, so to write out the
sti ness matrix, one needs an appropriate de ection interpolant w = N(x y)d,
where d is the unknown degrees of freedom (dof) or scale factors. The curvature
is from second order di erentiation of the de ection, so the interpolant must be
of Hermite type, i.e. with scalable rst derivatives at nodes. The scalability of
derivatives and magnitudes are decoupled, so that a change of the slope at a node
doesn't change its height, providing a simple association scheme between node,
order of derivative, dof and basis function (interpolant term to be scaled by the
dof). Basis functions associated with zeroth derivatives have vanishing derivatives
at all nodes of the element and only one nonzero nodal height value, which is at
the associated node. A basis function for a rst derivative has zero nodal heights
and a single associated nonzero partial derivative. The association scheme for the
xed node number n can be written as
d = 3 (n ; 1) + 1
(7.39)
j = 1234
(7.40)
k = xy
(7.41)
Nd(j) = nj
(7.42)
Nd k(j) = 0
(7.43)
Nd+1(j) = 0
(7.44)
Nd+1 k(j) = nj xk
(7.45)
Nd+2(j) = 0
(7.46)
Nd+2 k(j) = nj yk
(7.47)
where d is the de ection dof number and j is a number running over the element nodes. The rst index to N is the basis function number which is equal to the associated dof number. For a 2D element, there are three dofs per node, one for de ection and two for partial derivatives (slopes). The nonzero nodal values are often put to unity, so the scale factors will directly give the eld values at the nodes, even though the factors themselves are dimensionless. The dof column for the rectangle can be written
d = w(1) w x(1) w y(1) w(2) w x(2) w y(2) w(3) w x(3) w y(3) w(4) w x(4) w y(4)]T
where the node number is between brackets. It is also understood that entities with dimension are from multiplication with an appropriate unit.
The basis functions must be composed from monomials of orders covering the possible bending modes of the surface. Since the curvature is a second derivative of
152
the de ection, the lowest thinkable order would be two. When looking at analytical solutions to bending of onedimensional beams with simple load distributions, third order polynomials describe the midlines. To t a Hermite interpolant over a rectangular element, there are four de ection magnitudes and eight partial derivative values to match, so there should be twelve independent coe cients in the polynomial. A full cubic of two variables has only ten coe cients, so two higher order terms must be added. In view of the twodimensionality, the "bicubics" x3y, xy3 could be appropriate to take care of transversal changes of a onedimensional cubic shape. Storing the constituent monomials in X one can write a basis function and its derivatives as
Ni = XT i = 1 x y x2 iy y2 x3 x2y xy2 y3 x3y xy3] i(7.48) Ni x = X x i = 0 1 0 2x y 0 3x2 2xy y2 y3 3x2y y3] i (7.49) Ni y = X y i = 0 0 1 0 x 2y 0 x2 x2y 3y2 x3 x3y2] i (7.50)
where i is a column of coe cients for the i:th basis function. These columns are determined by inserting node coordinates and equating to nodal interpolant values. Using the node indexes 1 $ (a b) 2 $ (;a b) 3 $ (;a ;b) 4 $ (a ;b) one can
write an equation system for the rst basis function, which is associated with node
1 height, as
2 1 3 2 X(1) 3
6666666666666666664
0 0 0 0 0 0 0 0 0 0
7777777777777777775
= 6666666666666666664
X X
x(1) y(1)
X(2)
XXXxy(((322)))
XXXxy(((433)))
X x(4)
7777777777777777775
1 = XDB
1
0
X y(4)
(7.51)
where XDB is the matrix of the X row and its derivatives determined in the nodes.
By going through the nodal properties of all the basis functions one gets a multiple
unknown column equation system
I = XDB ]
(7.52)
where I is the 12x12 identity matrix and is the matrix of coe cient columns to solve for. The solution is found by fully inverting the XDB matrix. Because the XDB matrix holds integer entries, the inverse should contain simple fractions. A MAPLE1 program to set up XDB and calculate the basis functions from an arbitrary choice of X is given below.
1MAPLE is a system for doing algebraic/symbolic and numerical (in almost in nite or 
153
# this script calculates # shape functions N for deflection field # second derivatives of N in B for curvature field # local stiffness matrix k # nodal loads factors re_f # for the plate bending problem # with rectangular twelve dof Kirchhoff elements # # remove previous session assignments restart: # e:=array(1..3 , 1..3 ): e 1,3]:=0: e 2,3]:=0: e 3,1]:=0: e 3,2]:=0: e 1,2]:=E12: e 2,1]:=E12: e 2,2]:=E22: e 1,1]:=E11: e 3,3]:=E33: #print(e) # # flexural rigidity d:=t^3/12*evalm(e): # # deflection shape function polynomial terms in local coordinates X:=array( 1, x, y, x^2, x*y, y^2, x^3, x^2*y, x*y^2, y^3, x^3*y, x*y^3]): # # dofs contains rotations which are associated with derivatives of shapes # differentiation of array has to be done elementwise # map function helps to remove one explicit iteration Xx:=map(diff,X,x): Xy:=map(diff,X,y): # # now evaluate terms and differentiated terms at nodes XDB:=array(1..12, 1..12): # this matrix will be filled columnwise for j from 1 by 1 to 12 do: # first node lower left nite precision) calculations. The system includes an interpreter for user programs, called scripts. MAPLE scripting is well suited for manipulation of large collections of equations or parts of equations. The user can concentrate on developing and keeping an ordered scheme of data and operations, without wasting e ort on checking factors and trying to t calculations to A4 paper. Particularly, symbolic data includes expressions (intended for numerical evaluation or not) and matrices of expressions, while operators contain matrix composition, inversion and inde nite integration. 154
XDB 1,j]:=subs(x=a, y=b, X j]): XDB 2,j]:=subs(x=a, y=b, Xx j]): XDB 3,j]:=subs(x=a, y=b, Xy j]): # second node lower right XDB 4,j]:=subs(x=a, y=b, X j]): XDB 5,j]:=subs(x=a, y=b, Xx j]): XDB 6,j]:=subs(x=a, y=b, Xy j]): # third node upper right XDB 7,j]:=subs(x=a, y=b, X j]): XDB 8,j]:=subs(x=a, y=b, Xx j]): XDB 9,j]:=subs(x=a, y=b, Xy j]): # fourth node upper left XDB 10,j]:=subs(x=a, y=b, X j]): XDB 11,j]:=subs(x=a, y=b, Xx j]): XDB 12,j]:=subs(x=a, y=b, Xy j]): od: # # invert to get matrix of shape function coefficients alphas:=linalg inverse](XDB): # # fix the shape functions as elements of a single row N:=linalg innerprod]( X, alphas): # note that evalm doesnt work here as it doesnt count a single row as # a matrix # # B with double derivatives of shape fcns # as bending uses curvature which is from twice diffs of deflection # construct BCOM for B at element center of mass (local x,y=0,0 ) # B needed for stiff matrix and BCOM for fast eval of curvature B:=array( 1..3, 1..12): BCOM:=array( 1..3, 1..12): for j from 1 by 1 to 12 do: B 1,j]:=diff(N j],x,x): B 2,j]:=diff(N j],y,y): B 3,j]:=diff(N j],x,y)+diff(N j],y,x): BCOM 1,j]:=subs(x=0,y=0,B 1,j]): BCOM 2,j]:=subs(x=0,y=0,B 2,j]): BCOM 3,j]:=subs(x=0,y=0,B 3,j]): od: # # then set the integrand to the stiffness matrix F:=evalm(transpose(B)&*d&*B): # 155
# integrate # here one can use the map function since all elements will be # equally operated upon k:=map(int,F,x=a..a): k:=map(int,k,y=b..b): k:=map(simplify,k): #print(k) # # construct nodal loads factor # assume transversal force surface density constant over element re_f:=map(int,N,x=a..a): re_f:=map(int,re_f,y=b..b): re_f:=map(simplify,re_f): #print(re_f) # # the below code fragment is suitable for matlab readable output # note that if the save command would be used instead, matrix elements wont be # stored in order writeto(`bendstiff.sol`) for i from 1 by 1 to 12 do: for j from 1 by 1 to 12 do: lprint(cat(`k(`,i,`,`,j,`)=`),k i,j],` `): od: od: for j from 1 by 1 to 12 do: lprint(cat( `re_f(`,j,`)=` ),re_f j],` `): od: for i from 1 by 1 to 3 do: for j from 1 by 1 to 12 do: lprint(cat( `BCOM(`,i,`,`,j,`)=` ),BCOM i,j],` `): od: od: writeto(terminal)
Loading forces that are applied laterally to the sheet will give rise to bending. The forces must be properly integrated to be used in the source column of the discretized bending equation. If the load is q N/m2, the element source column is
Z
re =
NT qdA A
(7.53)
where N is the matrix with basis functions.
156
y(mm) 40 32 24 16 8 0
w x(mm=m) 0 0.930 1.855 2.767 3.660 4.525 0 0.987 1.967 2.935 3.882 4.799 0 1.031 2.057 3.068 4.057 5.015 0 1.064 2.121 3.164 4.184 5.171 0 1.084 2.1608 3.222 4.261 5.265 0 1.090 2.173 3.242 4.286 5.297 0 8 16 24 32 40 x(mm)
y(mm) 40 5.309 5.278 32 4.305 4.280 24 3.262 3.242 16 2.189 2.176 8 1.099 1.092 000 08
w y(mm=m) 5.185 5.031 4.816 4.543 4.204 4.078 3.903 3.680 3.184 3.088 2.955 2.786 2.137 2.073 1.983 1.869 1.073 1.040 0.995 0.938 0000 16 24 32 40 x(mm)
Table 7.3: Rotations when not magnetized
7.6.4 Nonmagnetized case
A simulation with only gravity as lateral force was carried out to give a reference
shape at zero ux density. This case occurs repetitively when ux density is sinu
soidal and will give the largest de ection, as the pole pieces won't be magnetized
and only GPa, E22
act as simple =227.3 GPa,
supports. E12=68.2
Elastic sti ness coe cients used were GPa and E33=74 GPa. Maximum de
E11=170.5 ection was
calculated to 0.8 mm/m on a space discretization with 225 elements for a quadrant
of the sample. The midline de ection curve can be seen in Fig. 7.5 and equilines
of de ection in Fig. 7.6.
7.6.5 Rolling direction magnetization
Active poles feeding the sample are on the xdirected yoke. Nodes on top of those poles are set as clamped. Nodes on top of the poles of the yyoke are set as free, except on the inner edges where nodes are simply supported. De ection of the y = 0 midline can be seen in Fig. 7.7. Numerical results are stated in Table 7.4, where also curvatures and ux densities of the middle element row are printed. An overview of the de ection eld can be seen in Fig. 7.8. It is seen that clamping due to reluctance force gives a low de ection gradient, which is midsurface rotation,
157
z 0.2
0.15
0.1
0.05
0 x
0.15
0.1
0.05
0
0.05
0.1
0.15
Figure 7.7: Geometry in m for the cut y = 0 when xmagnetized. Deformation of sheet magni ed with factor 50. Flux density vectors drawn. Undeformed sheet dashdotted.
close to the active poles and leakage seems not to be large enough to counteract this behaviour. Leakage reluctance force density is 16 N=m2 localized to an element column 12 mm wide around poles. Compare to gravity force density 17.6 N=m2 over the whole surface. 7.6.6 Transversal magnetization
Simulations with ydirected yoke exciting the sheet transversely have also been done. Boundary conditions are as in the xexcitation case, but rotated 90 degrees in the sheet plane. De ections for the midline connecting the poles are seen in Fig. 7.7 and numerically stated in Table 7.5. Equilines of de ection can be seen in Fig. 7.8. Leakage is larger when trying to excite the anisotropic sheet transversely, but it shows no greater e ect as it is still quite localized around the pole pieces.
7.6.7 Experiments Experiments with measuring sample re ector tilts have been performed. The setup is schematically drawn in Fig. 7.11. The screen where the spot position was measured was located 3.1 m from the re ecting micro prism. The light beam deviation change from the xyplane with parallel incidence on prisms placed at (x y) with
158
y(mm) 40 32 24 16 8 0
w x(mm=m) 0 0.848 1.682 2.485 3.242 3.932 0 0.898 1.781 2.632 3.433 4.164 0 0.938 1.860 2.748 3.584 4.347 0 0.967 1.916 2.832 3.693 4.478 0 0.984 1.951 2.882 3.758 4.557 0 0.990 1.962 2.899 3.780 4.584 0 8 16 24 32 40 x(mm)
y(mm)
w y(mm=m)
40 3.358 3.330 3.248 3.112 2.925 2.691
32 2.716 2.693 2.626 2.516 2.364 2.174
24 2.053 2.036 1.985 1.901 1.786 1.642
16 1.376 1.364 1.330 1.274 1.196 1.100
8 0.690 0.684 0.667 0.638 0.600 0.551
0000000
0 8 16 24 32 40 x(mm)
Table 7.4: Rotations when xmagnetized
0.15
0.1
0.05
0
0.05
0.1
0.15
0.15
0.1
0.05
0
0.05
0.1
0.15
Figure 7.8: Equilines of de ection (solid) when xmagnetized. Outlines of pole surfaces (dashdotted).
159
z 0.2
0.15
0.1
0.05
0 y
0.15
0.1
0.05
0
0.05
0.1
0.15
Figure 7.9: Geometry in m for the cut x = 0 when ymagnetized. De ection of sheet magni ed with factor 50. Flux density vectors drawn. Undeformed sheet dashdotted.
y(mm) 40 32 24 16 8 0
w x(mm=m) 0 0.486 0.9699 1.448 1.919 2.378 0 0.528 1.054 1.575 2.086 2.584 0 0.562 1.122 1.676 2.220 2.748 0 0.587 1.172 1.750 2.317 2.868 0 0.602 1.202 1.795 2.376 2.941 0 0.607 1.212 1.810 2.396 2.965 0 8 16 24 32 40 x(mm)
y(mm) 40 4.022 3.999 32 3.329 3.310 24 2.560 2.546 16 1.737 1.727 8 0.877 0.872 000 08
w y(mm=m) 3.930 3.814 3.654 3.449 3.252 3.157 3.024 2.856 2.501 2.428 2.326 2.197 1.697 1.647 1.578 1.490 0.857 0.832 0.797 0.753 0000 16 24 32 40 x(mm)
Table 7.5: Rotations when ymagnetized
160
0.15
0.1
0.05
0
0.05
0.1
0.15
0.15
0.1
0.05
0
0.05
0.1
0.15
Figure 7.10: Equilines of de ection (solid) when ymagnetized. Outlines of pole surfaces (dashdotted).
front surface normals (cos , sin ) is
'de = ;2 (w x(x y) cos + w y(x y) sin )
(7.54)
The change can be computed from Table 7.4 or Table 7.5 subtracted with the reference values in Table 7.3. Measured values are found in Table 7.6. They agree very well for the TDexcited case, while the LDexcitation presents 40% lower xrotation change and 25% higher yrotation change in computations. This might be due to elastic and magnetic anisotropy data of the actual sample di ering from the typical data used in simulations.
B~ dir. LD
1 2
j 'de 1.1
j (mrad) 1.6
TD
2.3
1.3
(40,0) (0,40) x y(mm)
0
90 (deg)
Table 7.6: Experimental results
161
Figure 7.11: Experimental setup 162
Chapter 8 Measurement and veri cation
8.1 Introduction
Measurements with focus on magnetostriction are reported. Magnetization measurements, uniaxial and rotating, are presented brie y at the end.
tBheecatruasnesvtheresaclodnvireencttiioonna(lmsheaeseut riesd cmonasgindeetroacbolympmlioarnecem4a:g1ne1t0o;s1t1rPicati;v1e
with > 10
B~ in times
having B~ in the rolling direction) there has been a focus on measuring with this
direction of excitation.
Measurements of the transverse magnetostriction from a likewise oriented magnetic ux density in a conventional grain oriented siliconiron sheet are presented. A data processing scheme to extract nonlinearity and frequency dependency parameters from such measurements is shown. A good t is obtained with six reals representing ux density excitation, four for material nonlinearity and three for time rate dependency.
The By.
magnetostrictive strain component is The rolling and transverse directions of
tMyhe
and the sheet are
ux density here x and
component is y respectively.
The butter y curve of doublevalued strain vs. ux density is made singlevalued
by a least squares procedure. A t to a nonlinear function of the magnetic stress
1 0
By2
is performed.
The time lag of strain to magnetic stress is modelled by a
163
ratedependent equation. The equation is solved in the frequency domain with a magnetic stress from a ltered ux density. At present it is unclear if ratedependency is dominating over hysteresis in the 50 to 250 Hz ux frequency region considered. The model is nevertheless useful as a parameterization in simulation programs and as a wellde ned hypothesis to further test with experiment. Model use in a nite element surrounding is indicated. 8.2 Experiments The magnetic setup has been described in 101], and it is here used to feed the sample with an alternating ux density in the ydirection, which is known to cause the greatest magnetostrictive response of these materials. The optical setup has been presented in earlier chapters. The strain information is retrieved by a single noncontact interferometer, which illuminates a pair of sample re ectors M that senses the elongation of a 70 mm element. L is a stabilized HeNe laser, P are prisms, AOM is an acoustooptic modulator, BS are beamsplitters, M1 is a mirror and PD is a photodiode. The AOM facilitates intensity alteration when the interferometer is operated in homodyne mode, and can impose a carrier on the temporal interference pattern to operate the interferometer in heterodyne mode. The sample test bed with feeding yokes is possible to rotate, so strain components can be measured in turn while preserving the same excitation. In this paper results are restricted to the transverse strain component. The ux density waveform can be seen as squared in Fig. (8.2). The peak ux density is 0:7 T and the rst harmonic is at 50 Hz. Two odd harmonics are signi cant, the third and fth at 150 and 250 Hz respectively. imTrseMyhastpeghvonetenehrtspsioeruldeosltaMyhBsoatyfshicaoasMynpmeoatatgepogenenttaBsiitkayt2ulh,dveeaewnlbueohurefigct1hoty9efrfc1%rao5ynmtcobmuaetrn=hvsmeeea,evasnerhnrsoadatwsghiensaadrsimsnhstioornFwangiiinlgnce.isavn8tnar.ldF4eus.ietsgdhO.defc8iau.f2gufrt.rvrhateBhm1aey0,rndpy%inlioe.atltedtTrliioennhssggest proportional to the area of the loop. 8.3 Data processing and nonlinear model The creation of a singlevalued magnetostriction curve is done by interpolation, averaging and least squares polynomial tting. The butter y curve is interpolated in equally spaced strain points, to catch the M peak in the lossless curve rather than the B peak. The averaged curve is then least squares tted to a polynomial. The last action can be shown as equal to nding the polynomial that minimizes the 164
Flux density squared [T^2]
0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Time [s] Figure 8.1: Measured (solid) and simulated (dashdotted) B2(t).
distance to both ux density branches in a least squares sense. The lossless strain curve is here expressed as a third order polynomial in powers of the magnetic stress. By introducing the scaling values ^M = M(tp), Bs = B(tp) where tp is strain peak time, one can write the polynomial using an orthogonal function sequence as
MA(B2)=^ = d0 + d1f1(B2=Bs2) + d2f2(B2=Bs2) + d3f3(B2=Bs2) (8.1)
f1 = x ; 1=2
(8.2)
f2 = x2 ; x + 1=6
(8.3)
f3 = x3 ; 3=2x2 + 3=5x ; 1=20
(8.4)
Fitting to singlevalued data gave d0 =
= 1.3266. It Esliigh=t.d2i kfik2
is=sede2i nR01thfai (txe)n2 derxg.y
quotas So the
0.5378, E3=E1
d1
1=%0a.8n4d85E,2d=2E=1
0.00:50934%awndhedr3e
need for nonlinear terms in this region is
To get a short description of the exciting ux density, a Fourier expansion is done with following ideal ltering out of the signi cant harmonics. One can write the cut o Fourier expansion and the identi cation of the coe cients Bi as
Bsim(t) = X 3 (B2i;1ej(2i;1)!1t + B2i;1e;j(2i;1)!1t)
(8.5)
i=1
165
Magnetostriction [mum/m]
16 14 12 10 8 6 4 2 0 2 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 Time [s] Figure 8.2: Measured (solid) and simulated (dashdotted) M(t). 16
14
Magnetostrictive strain [microm/m]
12
10 8
6
4
2
0
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
Flux density [T]
Figure 8.3: Measured butter y loops of curve, dashdotted.
My vs. By, solid, and singlevalued
tted
166
16
14
12
Magnetostriction [mum/m]
10
8
6
4
2
0
2
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
Flux density [T]
Figure 8.4: Magnetostriction curves, measured (solid) and simulated with nonlinear model (dashdotted).
Bi = X N B(kTs)e;ji2 k=N k=1
Ts
=
2 N !1
(8.6)
where the last expression is seen to be equal to 1=N times the fast Fourier transform. The number of sample points N in the time trace is 256 and the rst harmonic angular frequency !1 is 2 50 rad/s. The squared ltered ux density is then
Bs2im(t) = C0 + X5 (C2iej2i!1t + C2ie;j2i!1t)
(8.7)
i=1
C2i = g(Bl) i = 0 1:::5 l = 1 3 5
(8.8)
where g is given by the relations
C0 = 2jB1j2 + 2jB5j2 + 2jB3j2
(8.9)
C2 = 2B1B3 + 2B3B5 + B12
(8.10)
C4 = 2B1B3 + 2B1B5
(8.11)
C6 = 2B1B5 + B32
(8.12)
C8 = 2B3B5
(8.13)
C10 = B52
(8.14)
167
Magnetostriction [mum/m]
16 14 12 10 8 6 4 2 0 2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Flux density squared [T^2] Figure 8.5: Magnetostriction curves, measured (solid) and simulated with nonlinear model (dashdotted). 168
It is necessary to use g since direct ltering of B2 gives a signal with negative values which is unphysical. Carrying on to the expression for the lossless magnetostriction, with the knowledge of there being a third order relation to this entity, Eq. (8.1), one can write
MsimA(t) = F0 + X 15 (F2iej2i!1t + F2ie;j2i!1t)
(8.15)
i=1
F2i = e(C2m dn) i = 0 1:::15 m = 0 1:::5 n = 0 1:::3 (8.16)
The relation (8.16) is analogous to the previous case of squaring of the ux density, but this time for a third order combination of ve harmonics and thus too lengthy to write out here. The coe cients Fi are determined via e g from the ux density coe cients Bj and the model parameters dk. The signals simulated in this way are seen in Fig. (8.2).
8.4 Frequency dependence
Loop ts are carried out by a frequency domain method. It is assumed that the magnetostriction obeys a linear di erential equation with the lossless magnetostriction as the primary driving entity. The greatest loop width comes from a lag of the 100 Hz component of M to MA. There is an approximate 90 degree phase shift of the 200 Hz component in the presented measurement, so a resonant model has to be used. The resonance can also be seen in the butter y curve as a crossing of the branches at a nonzero ux density. The 300 Hz component has a low leading phase, so to catch that, there has to be a zero in the transfer function ~M=~MA at some complex frequency. For simplicity that frequency is chosen as real here. The above yields
~M
(f
)
=
;(f
;(f =fd2)2 =fr2)2 + jf
+1 =fr1
+
1
~MA
d=ef
H(f
)~MA(f
)
(8.17)
where ~M(f) is the Fourier coe cient at frequency f. The parameters for the curves shown are fr1 = 900 Hz, fr2 = 193:5 Hz, and fd2 = 215:5 Hz.
8.5 2D model from measurements
The use of a linear model in simulation programs is now illustrated. The magnetostriction to be incorporated as a strain source in a nite element scheme 93]
169
16
14
12
Magnetostriction [mum/m]
10
8
6
4
2
0
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
Flux density [T]
Figure 8.6: Magnetostriciton curves, measured (solid) and simulated with linear model (dashdotted).
170
Magnetostriction [mum/m]
16 14 12 10 8 6 4 2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Flux density squared [T^2] Figure 8.7: Magnetostriction curves, measured (solid) and simulated with linear model (dashdotted). 171
is
2 4
~~~MxMxMyy
3
2
5 = H(f) 4
DD012ii 11
DD012ii 22
0 0 D3i 3
32 54
Bx~BBB~~yxy22===BBBxxy22sssBys
3 5
(8.18)
awnhderBe yDskial raerescsaclainlegd
magnetocompliances at frequency f = levels. The scaling levels are adjusted
2if1 i to the
= 0 1::5. Bxs measurement
ranges which should ideally cover the region up to saturation. The factoring out of
a single transfer function H(f) is strictly not possible, but merely indicates the rst
step of a tensorial extension of the frequency dependency. The construction of the
magnetic shear stress Bx~By is straightforward from harmonic interaction of Fourier
coe cients Bxi and Byi. For the present single direction low peak measurements,
the linear model yields fair results as seen in Fig. (8.7).
8.6 Magnetization measurements The setup can make nice ux density versus eld strength measurements. This capability is shown for three examples of excitation of an oriented sheet. In Fig. 8.8, the sample was subjected to a eld strength uniaxially alternating in the rolling direction. In Fig. 8.9, the sample was subjected to a eld strength uniaxially alternating in the transverse direction. The characteristic initial bend of the magnetization curve in the transverse direction is seen. The third case was with a rotating eld excitation applied to the specimen. The rst gure Fig. 8.10 shows the ux density locus of the rotational process. One can see that the natural locus of the material resembles a rhombus (also known as the lozenge or diamond shape). The hardest directions are at right angles to the side of the rhombus, 56 from the rolling direction when estimated by eye from the graph. A slight misalignment of 12 of the measurement coils can also be seen. The second gure Fig. 8.11 shows the eld strength locus for a cycle. The angle to the maxima of the eld strength is roughly 53 , also indicating the direction of hard magnetization. For more results on magnetization the reader is referred to the articles cowritten by the author 15] 106].
172
BH_rd
2
1.5
1
0.5
0
0.5
1
1.5
2
1000 800
600
400
200
0
200
400
600
800
1000
H_rd
Figure 8.8: Flux density T] in rolling direction versus eld strength A/m] in rolling direction. Oriented material.
1.5
1
0.5
B_td
0
0.5
1
1.5
2500 2000 1500 1000 500
0
500
1000 1500 2000 2500
H_td
Figure 8.9: Flux density T] in transverse direction versus eld strength A/m] in transverse direction. Oriented material.
173
1.5
1
0.5
B_td
0
0.5
1
1.5
2
1.5
1
0.5
0
0.5
1
1.5
2
B_rd
Figure 8.10: Flux density T] locus. Transverse direction is yaxis and rolling direction is xaxis. Oriented material.
2000
1500
1000
500
H_td
0
500
1000
1500
2000
1500
1000
500
0
500
1000
1500
H_rd
Figure 8.11: Field strength A/m] locus. Transverse direction is yaxis and rolling direction is xaxis. Oriented material.
174
Chapter 9 Conclusions and future work 9.1 Conclusions 9.1.1 Setup uses The described measurement setup is a good supplier of magnetostriction and magnetization material characteristics. Together with nite element friendly parametrization techniques, like the one shown in the last chapter, it can provide support for the analysis and optimization of magnetically excited siliconiron structures. Because of the e ciency of the user interface, magnetization measurements can be done quickly. It can therefore be used for routine hysteresis, loss, permeability and saturation measurements. Magnetostriction measurements take longer time because of the need of interferometer alignment. Magnetization of nonoriented and conventional grainoriented samples can easily be measured in di erent directions or with a rotating vector because of the double yoke system. Superoriented samples are more di cult to saturate in the transversal direction if double yokes are used, and the operator might have to only use one yoke in that situation. The measurement of magnetostriction in di erent directions is simpli ed by the possibility of rotating the sample holder with the yokes. 175
9.1.2 Sample eld calculation Field calculations have resulted in an estimation of ux density and strain eld size and homogeneity in the central part of a nonoriented siliconiron sample sheet subjected to inhomogeneous magnetostrictively induced strain without applied external loads. The ux density homogeneity is fair and magnetic eld sensors can be made from coils that have signi cant length. The local strain corresponds fairly well to the local magnetostrictive strain. A constitutive relation for ratedependent magnetostrictive strain suitable for continuum magnetomechanical simulations of oriented siliconiron sheets has been proposed. The shape of the magnetostriction loop when excited by a sinusoidal magnetic ux is fairly well represented in the rolling and transverse directions. Computations of total strain magnitude and phase elds are feasible. In a typical grainoriented material one is able to calculate the relative in uence of elasticity on magnetostrictive strain to total strain. 9.1.3 Bending The bending rotation of a siliconiron sheet in a magnetic and gravitational eld was studied. It was found that rotation changes during the magnetization cycle could give laser beam deviation changes up to 4.6 mrad when re ectors were mounted 40 mm o from the sheet center. This beam vibration can have serious e ects on recombination in an interferometer for displacement or magnetostriction measurements if not considered and componentwise properly compensated for. Use the sample holder table, and ll out the air gaps from the yoke to the sample to make the surface under the sample plane. Use fairly light loads such as steel bars to atten any nonplanar imperfection of the sample. Check with measurement of the spot vibration of a beam re ected by a prism mounted at on the triangular side. 9.1.4 Magnetostriction harmonics In the single sheet tester that is more current than voltage controlled, the ux density waveform will be distorted due to saturation e ects. When the ux density contains three frequencies, the magnetic stress and the singlevalued magnetostriction can be well represented with harmonic interaction formulas from nonlinearity. The lagging magnetostriction is gotten by a transfer function operating on the singlevalued magnetostriction. 176
9.2 Future work 9.2.1 SST improvement With perfect sinusoidal waveform control of the SST it would be simple to pick out single ux frequency magnetostriction responses to compare models with. That requires nonlinear hysteretic magnetization models that are available for 1D but have not been dealt with in this book. It also requires a quick digital feedback algorithm, available in the literature. It would simplify the investigation of the proper magnetic stress to use as independent variable in magnetostriction response modelling. It would also simplify the investigation of the nonlinear magnetostrictive response to this independent variable. For a transformer application, the ux signal is spectrally pure, but ux density signals in di erent parts of the core might not be pure due to varying amounts of saturation reached. How measure the material and model the magnetostrictive response of the core? Using uncontrolled or only slightly controlled SST measurements, the magnetization could be modelled by available models. These models would enable better control of the SST, and enable ux distribution and ux density waveform distortion calculation in a transformer core. The predicted ux density waveforms could be reproduced in the controlled SST and the magnetostriction measured for those cases. This would enable a more real insight into the performance of a design. In motor applications, the motor can be fed by frequency converters that are very spectrally unpure, and high frequency fundamentals can be present. Low frequency fundamentals (major hysteresis loops) and higher frequency harmonics (minor loops) have been tested in the SST, but are not the focus of this book. It would be interesting to test the limits of the SST with high frequency fundamentals or switched excitation (rectangular waves). Such experiments are quite simple to make. Due to increased frequency, the Bcoil sensor voltages might saturate the sensor ampli ers, and the voltages have to be divided with resistors. 9.2.2 Magnetoelastic FEM program development Magnetoelastic programs with rectangular elements have been developed. It would be good to have a version for triangular elements. Such a formulation is in the authors archive but no coding of numerics has been done yet. The formulation worked on was for CST:s (constant strain triangles) that can model the sheet if they are not too few. To be able to couple triangle elements to rectangular elements would be nice, because one could take a mesh generated somewhere else and use 177
it for magnetoelastic calculations with the same element numbering. Making an extension to 3D would be interesting because one could try to model stacks of sheets with 3D elements, where each element corresponds to a large number of sheets within its thickness. This would allow the study the magnetoelastic e ects of clamping devices applied normally to the sheet, for example. The ux could be allowed to be unevenly distributed over the normal direction to the stack. The weakening introduced by the lamination compared to a solid block would have to be considered. The shear moduli in the zx and zy planes, where z is normal to stack, would be lowered. 9.2.3 Magnetostriction measurements More magnetostriction measurements need to be done. Complete three axis strain measurements should be done for di erent directions of magnetic excitation. Higher ux densities should be tested. The onset of frequency dependency should be determined with di erent frequency tests. The use of the acoustooptic modulator to determine the sign of strain should be put into practice. 178
179
Chapter 10 List of symbols
Symbol styles
a scalar
~a vector
$aai
Cartesian component of vector tensor
aij Cartesian component of tensor
C matrix
d column (single column matrix)
dxy column for xy coordinate system
u column
uuin
column entry iteration numbered column
s(m) element numbered matrix or column
CCixjy
matrix entry matrix used in xy coordinate system
CT transposed matrix
CCC? iC;;j11
complex conjugate inverted matrix component of inverted matrix convolution operator
g~ Fourier decomposed function
I^ peak value of function
S^ indetermined form of S matrix
@@@xxy
partial derivative w.r.t. xcoordinate (@i w.r.t. i:th coord.) second partial derivative w.r.t. x and y coordinates domain boundary
180
Uppercase Latin symbols
A~ (magnetic) vector potential
A area
A transformation matrix
B~ (magnetic) ux density
B shape function derivative matrix
C (elastic) compliance matrix
D magnetocompliance matrix
D bending sti ness (can be a matrix)
E elasticity matrix
EF~ij
elasticity coe cient force
F Fourier decomposition
G shear modulus (isotropic case)
GI i
shear modulus (orthotropic case) (electric) current
J~ (electric) current density
K global sti ness matrix
K dispersion kernel
L inductance
M~ magnetization
M bending moment per unit length
NOi
shape (or basis) function major ordo
P magnetoelastic modulus (uniaxial stress)
Q magnetoelastic shear modulus (isotropic case)
QRi
magnetoelastic shear modulus (orthotropic case) resistance
R reluctance
S global magnetic sti ness matrix
S^ indeterminate global magnetic sti ness matrix
T temperature
T transformation matrix for stress column
T transformation matrix for strain column
U column of unknowns
U voltage, electromotive force
V (electric scalar) potential
V (magnetic) scalar potential
X reactance
Y Young's modulus (uniaxial stress elastic modulus)
Z impedance
181
Lowercase Latin symbols
aij direction cosine between i:th primed coordinate direction and j:th unprimed dir.
a half of rectangular element length
a table side length
b half of rectangular element width
bi derivative of approximation of H2=2 w.r.t. nodal value ui
c table thickness
c speed of light or of sound
cL longitudinal wave speed
cP pressure wave speed
ccSB
shear wave speed bending wave speed
f excitation column
f~ body force, force per volume unit
f frequency
fc carrier frequency
fi m
modulating frequency input variable
i index (integer)
j index
j imaginary unit
k index
k local sti ness matrix
l index
m index
n number of nodes
nx number of nodes in x direction
ny number of nodes in y dir
nz number of nodes in z dir
q load, force/area
q (equivalent) magnetic charge
~r position vector
r residual column
s^ local magnetic sti ness matrix
s shape function gradient scalar product matrix
t element thickness
~t traction vector
u column of nodal values
u exact or approximate solution
u output variable
u energy density
182
u(i) displacement at node i ui displacement at node i ~u displacement vector u displacement in x direction v displacement in y direction w displacement in z direction w weight function x horizontal coordinate y vertical coordinate z lateral coordinate
Uppercase Greek symbols ; boundary di erence, increase x coordinate di erence x linearized change operator for x direction (= [email protected]) product sum ux linked ux domain
Lowercase Greek symbols
thermal expansion coe cient
i thermal expansion coe cient in i:th direction
angle of material direction change
angle of material direction change
i direction cosine, relative to i:th direction
xy shear angle (= 2 xy)
di erence, increase
$ij $M
Kronecker delta, ( ij = 1 if i = j, else 0) (total) strain tensor magnetostrictive strain tensor
ij strain tensor component
x normal strain in xdirection (= xx)
y xy $M
normal strain in ydirection (= shear strain (= xy=2)
yy)
engineering strain column = x y
xy]T
magnetostrictive strain
angle from speci ed direction or plane
angle from zdirection
curvature
183
magnetostriction
s magnetostriction of magnetically saturated material
wavelength
rst Lame constant
second Lame constant
permeability
i permeability in i:th direction
0 vacuum permeability
r relative permeability
Poisson ratio (lateral contraction ratio at isotropy)
~i
directional cosines direction, unity vector
magnetoelastic lateral contraction ratio
acos(1)
mass density
charge density
m magnetic pole density
s surface charge density
$ $M
(electric) conductivity stress tensor magnetic stress tensor
ij
stress tensor component stress column = x y xy]T
normal stress
shear stress
'ij
shear stress component, i 6= j polar angle
(magnetic) scalar potential
diameter
(magnetic) susceptibility
! angular frequency
184
Chapter 11 List of units
A A/m A=m2 H mH;H1 H/m Hz kHz J J=m3 N N/m N=m2 Nm Pa S T V V/m W W/kg W=m2
ampere ampere per meter ampere per square meter henry millihenry per henry henry per meter hertz kilohertz joule joule per cubic meter newton newton per meter newton per square meter newtonmeter ohm pascal siemens tesla volt volt per meter watt watt per kilogram watt per square meter
185
Wb mWb Wb Wb/m deg g kg m mm m nm m2 mm3;1 m=m m=m m/s m=s2 mrad rad rad/s s
weber milliweber microweber weber per meter degree gram kilogram meter millimeter micrometer nanometer square meter cubic meter per meter meter per meter micrometer per meter meter per second meter per second per second milliradian radian radian per second second
186
Bibliography 1] D. Jiles and D. Ahterton, \Theory of ferromagnetic hysteresis," Journal of Magnetism and Magnetic Materials, no. 61, pp. 48{60, 1986. 2] D. C. Jiles, J. B. Thoelke, and M. K. Devine, \Numerical determination of hysteresis parameters for the modeling of magnetic properties using the theory of ferromagnetic hysteresis," IEEE Transactions on Magnetics, vol. 28, pp. 27{35, Jan. 1992. 3] D. C. Jiles, \Frequency dependence of hysteresis curves in ferromagnetic materials," in IEEE Transactions on Magnetics, INTERMAG, 1993. 4] A. Adly, I. Mayergoyz, and A. Bergqvist, \Preisach modeling of magnetostrictive hysteresis," J. Appl. Phys., vol. 69, p. 5777, 1991. 5] A. Bergqvist and G. Engdahl, \A stressdependent magnetic Preisach hysteresis model," IEEE Trans. Mag., vol. 27, p. 4796, 1991. 6] L. Kvarnsjo, A. Bergqvist, and G. Engdahl, \Application of a stressdependent magnetic Preisach hysteresis model on a simulation model for TerfenolD," IEEE Trans. Mag., vol. 28, p. 2623, 1992. 7] A. Bergqvist and G. Engdahl, \A phenomenological di erential{relation{ based vector hysteresis model," J. Appl. Phys., vol. 75, p. 5484, 1994. 8] A. Bergqvist and G. Engdahl, \A phenomenological magnetomechanical hysteresis model," J. Appl. Phys., vol. 75, p. 5896, 1994. 9] A. Bergqvist, On magnetic hysteresis modeling. PhD thesis, KTH, 1994. 10] A. Bergqvist and G. Engdahl, \A model for magnetomechanical hysteresis and losses in magnetostrictive materials," J. Appl. Phys., vol. 79, p. 6476, 1996. 187
11] G. Engdahl, F. Stillesjo, and A. Bergqvist, \Performance simulations of magnetostrictive actuators with resonant mechanical loads." presented at ACTUATOR 96, Bremen, Germany. 12] A. Bergqvist and G. Engdahl, \A thermodynamic representation of pseudoparticles with hysteresis," IEEE Trans. Mag., vol. 31, p. 3539, 1995. 13] A. J. Bergqvist, \A thermodynamic vector hysteresis model for ferromagnetic materials," in Nonlinear electromagnetic systems (A. J. Moses and A. Basak, eds.), p. 528, Amsterdam: IOS Press, 1996. 14] A. Bergqvist, \Magnetic vector hysteresis model with dry friction{like pinning," Physica B., vol. 233, p. 342, 1997. 15] A. Bergqvist, A. Lundgren, and G. Engdahl, \Experimental testing of an anisotropic vector hysteresis model," IEEE Trans. Mag., vol. 233, p. 4152, 1997. 16] P. Holmberg, A. Bergqvist, and G. Engdahl, \Modelling eddy currents and hysteresis in a transformer laminate," IEEE Trans. Mag., March 1997. in press. 17] C. Bengtsson, \Domanstrukturer i sife plat," Postgraduate course UPTEC 82131 R, ISSN 03468887, Institute of Technology, Uppsala University, Avd. for fasta tillstandets fysik, Inst. for teknologi, Uppsala Universitet, Box 534, 751 21 Uppsala, 1982. Bengtsson is now (1994) manager of ABB Transformers development laboratory in Ludvika, Sweden. 18] E. W. Lee, \The 110] magnetostriction of some single crystals of iron and siliconiron," Proc. Phys. Soc., vol. LXVIII, no. 2A, pp. 65{71, 1954. 19] M. Celasco and P. Mazetti, \Magnetostriction of grainoriented ferromagnetic cubic materials," J. Phys. D: Appl. Phys., vol. 5, pp. 1604{1613, 1972. 20] P. Allia, A. FerroMilone, G. Montalenti, G. P. Soardo, and F. Vinai, \Theory of negative magnetostriction in grain oriented 3and applied stresses," IEEE Transactions on Magnetics, vol. MAG14, pp. 362{364, Sept. 1978. 21] P. Allia, M. Celasco, A. Ferro, A. Masoero, and A. Stepanescu, \Transverse closure domains and the behavior of the magnetization in grainoriented polycrystalline magnetic sheets," J. Appl. Phys., vol. 52, pp. 1439{1447, Mar. 1981. 22] G. Bertotti, P. Mazetti, and G. P. . Soardo, \A general model of losses in soft magnetic materials," Journal of Magnetism and Magnetic Materials, no. 26, pp. 225{233, 1982. 188
23] J. Bishop, \Asymmetric domain wall bowing and tilting motins in materials with orientations between (100) 001] and (110) 001]," Journal of Magnetism and Magnetic Materials, no. 26, pp. 247{251, 1982. 24] J. E. L. Bishop, \Simulation of skew domain wall bowing in sife laminations with asymmetric roll orientation," IEEE Transactions on Magnetics, vol. MAG18, no. 4, pp. 970{981, 1982. 25] T. Yamaguchi, \E ect of sheet thickness on magnetostriction characteristics of 3 percent sife single crystals with slightly inclined (110) 001] orientation," IEEE Transactions on Magnetics, vol. MAG20, pp. 2033{2036, Sept. 1984. 26] M. Imamura, T. Sasaki, and T. Yamaguchi, \Domainwall eddycurrent loss in a stripe domain structure of sife crystals inclined slightly from the perfect (110) 001] orientation," IEEE Transactions on Magnetics, vol. MAG20, pp. 2120{2129, Nov. 1984. 27] A. J. Moses, \Measurement of magetostriction and vibration with regard to transformer noise," IEEE Transactions on Magnetics, vol. MAG10, pp. 154{ 156, June 1974. 28] A. J. Moses, \Magnetostriction measurements at high stress levels in sife and cofe alloys," Journal of Magnetism and Magnetic Materials, no. 26, pp. 185{ 186, 1982. 29] D. J. Mapps and C. E. White, \Longitudinal and transverse magnetostriction harmonics in (110) 001] siliconiron," IEEE Transactions on Magnetics, vol. MAG20, pp. 1566{1568, Sept. 1984. 30] A. J. Moses and C. P. Bakopolous, \E ect of stress annealing on loss and magnetostriction in grainoriented siliconiron," manuscript copy. 31] P. Allia, A. Ferro, G. P. Soardo, and F. Vinai, \Magnetostriction behavior in isotropic and cubeonface 3vol. 50, no. 11, pp. 7716{7718, 1979. 32] H. Pfutzner, \Domain interactions between stacked hib sife sheets," IEEE Transactions on Magnetics, vol. MAG18, pp. 961{962, July 1982. 33] K. Fukawa and T. Yamamoto, \Domain structures and stress distributions due to ballpoint scratching in 3(110) 001]," IEEE Transactions on Magnetics, vol. MAG18, pp. 963{969, July 1982. 34] H. Pfutzner, C. Bengtsson, and A. Leeb, \Domain investigation on coated unpolished sife sheets," IEEE Transactions on Magnetics, vol. MAG21, pp. 2620{2625, Nov. 1985. 189
35] G. C. Eadie, \The e ects of stress and temperature on the magnetostriction of commercial 3.25Journal of Magnetism and Magnetic Materials, no. 26, pp. 43{46, 1982. 36] H. J. Stanbury, \Magnetostriction e ects at angles to the rolling direction in grainoriented 3.25% siliconsteel," Journal of Magnetism and Magnetic Materials, no. 26, pp. 47{49, 1982. 37] B. Hribernik, \In uence of cutting strains on the magnetic anisotropy of fully processed silicon steel," Journal of Magnetism and Magnetic Materials, no. 26, pp. 72{74, 1982. 38] J. Slama and M. Prejsa, \Investigation of magnetization mechanisms in oriented sife sheets," Journal of Magnetism and Magnetic Materials, no. 26, pp. 239{242, 1982. 39] J. V. S. Morgan and K. J. Overshott, \An experimental veri cation of the ruckling process in electrical sheet steel," Journal of Magnetism and Magnetic Materials, no. 26, pp. 243{246, 1982. 40] V. Ungemach, \Dependence of the domain structure of sife single crystals on the frequency of demagnetizing elds," Journal of Magnetism and Magnetic Materials, no. 26, pp. 252{254, 1982. 41] V. M. Bichard, \The observation of domain structures in grain oriented sife using a high voltage scanning electron microscope," Journal of Magnetism and Magnetic Materials, no. 26, pp. 255{257, 1982. 42] S. A. Zhou and R. K. T. Hsieh, \A theoretical model of eddy current nondestructive test for electromagnetoelastic materials," Int. J. Engng Sci., vol. 26, no. 1, pp. 12{26, 1988. 43] T. Nozawa, M. Mizogami, H. Mogi, and Y. Matsuo, \Domain structures and magnetic properties of advanced grainoriented silicon steel," Journal of Magnetism and Magnetic Materials, vol. 133, pp. 115{122, 1994. 44] H. Masui, M. Mizokami, Y. Matsuo, and H. Mogi, \A proposal of predicting formulae for in uence of stress on magnetostriction in grain oriented silicon steel," ISIJ International, vol. 35, no. 4, pp. 409{418, 1995. 45] S. Arai and A. Hubert, \The pro les of lancetshaped surface domains in iron," Phys. stat. sol. (a), vol. 147, pp. 563{568, 1995. 46] Y. Nakamura, Y. Okazaki, J. Harase, and N. Takahashi, \The dc and ac magnetic properties of a highpurity grainoriented iron sheet," Journal of Magnetism and Magnetic Materials, vol. 133, pp. 153{155, 1994. 190
47] H. Masui, \In uence of stress condition on initiation of magnetostriction in grain oriented silicon steel," IEEE Transactions on Magnetics, vol. 31, no. 2, pp. 930{937, 1995. 48] R. Vasina, \Changes of ferromagnetic parameters during cyclic stressing," phys stat sol (a), no. 137, pp. 189{196, 1993. 49] C. S. Schneider, P. Y. Cannell, and K. T. Watts, \Magnetoelasticity for large stresses," IEEE Transactions on Magnetics, vol. 28, pp. 2626{2631, Sept. 1992. 50] H. Maeda, K. Harada, Y. Ishihara, and T. Todaka, \Performance of magnetostriction of silicon steel sheet with bias eld," in Proc. of the 12th Soft Magnetic Materials Conference, 1995. PB26. 51] T. E. Carlsson and N. Abramson, \Shortcoherencelength multipass interferometer," Optics letters, vol. 20, no. 10, pp. 1187{1188, 1995. 52] H. Mogi, M. Yabumoto, M. Mizokami, and Y. Okazaki, \Harmonic analysis of ac magnetostriction measurements under nonsinusoidal excitation," in Proc. of the Intermag'96 Conference, 1996. 53] T. Lewis, J. Llewellyn, and M. van der Sluijs, \Electrokinetic properties of metaldielectric interfaces," IEE ProceedingsA, vol. 140, no. 5, pp. 385{392, 1993. 54] T. Nakata, N. Takahashi, M. Nakano, K. Muramatsu, and M. Miyake, \Magnetostriction measurements with a laser doppler velocimeter," Joint INTERMAG MMM, 1994. Paper No. AQ04. 55] T. Nakata, N. Takahashi, K. Fujiwara, and M. Nakano, \Measurements of magnetic characteristics along arbitrary directions of grainoriented silicon steel up to high ux densities," IEEE Transactions on Magnetics, vol. 29, pp. 3544{3546, November 1993. 56] Y. Ohtsuka and M. Tsubokawa, \Dynamic twofrequency interferometry for small displacement measurements," Optics and Laser Technology, pp. 25{29, February 1984. 57] Y. Ohtsuka and K. Itoh, \Twofrequency laser interferometer for small displacement measurements in a low frequency range," Applied Optics, vol. 18, pp. 219{224, January 1979. 58] C. R. G. Higgs and A. J. Moses, \Computation of ux distribution and harmonics in various transformer cores," Journal of Magnetism and Magnetic Materials, no. 26, pp. 349{350, 1982. 191
59] T. Nakata, N. Takahashi, and Y. Kawase, \Finite element analysis of magnetic elds taking into account hysteresis characteristics," IEEE Transactions on Magnetics, vol. MAG21, pp. 1856{1858, Sept. 1985. 60] T. Nakata and N. Takahashi, \Application of the nite element method to the design of permanent magnets," IEEE transactions on Magnetics, vol. MAG18, no. 6, pp. 1049{1051, 1982. 61] T. Nakata, \Numerical analysis of ux and loss distributions in electrical machinery," IEEE Transactions on Magnetics, vol. MAG20, pp. 1750{1755, Sept. 1984. Invited paper. 62] T. Nakata, N. Takahashi, Y. Kawase, H. Funakoshi, and S. Ito, \Finite element analysis of magnetic circuits composed of axisymmetric and rectangular regions," IEEE Transactions on Magnetics, vol. MAG21, pp. 2199{2202, Nov. 1985. 63] T. Nakata, N. takahashi, and Y. Kawase, \New approximate method for calculating threedimensional magnetic elds in laminated cores," IEEE Transactions on Magnetics, vol. MAG21, pp. 2374{2377, Nov. 1985. 64] \Approximation analysis of threedimensional magnetic elds in iron cores laminated with grainoriented silicon steel," Electrical Engineering in Japan, vol. 1987, no. 6, pp. 90{97, 1987. 65] D. Pavlik, D. C. Johnson, and R. S. Girgis, \Calculation and reduction of stray and eddy losses in coreform transformers using a highly accurate nite element modelling technique," in IEEE/PES 92 Winter Meeting, IEEE Power Engineering Society, 1992. Auhors 1 & 2 are with Westinghouse, 3 is with ABB T&D Indiana. 66] T. Doong and I. D. Mayergoyz, \On numerical implementation of hysteresis models," IEEE Transactions on Magnetics, vol. MAG21, pp. 1853{1855, Sept. 67] A. Bergqvist and G. Engdahl, \A phenomenological di erentialrelationbased vector hysteresis model," J. Appl. Phys., vol. 75, no. 10, pp. 5484{5486, 1994. 68] A. Bergqvist and G. Engdahl, \A phenomenological magnetomechanical hysteresis model," J. Appl. Phys., vol. 75, no. 10, pp. 5496{5498, 1994. 69] T. Pera, F. Ossart, and T. Waeckerle, \Field computation in non linear anisotropic sheets using the coenergy model," IEEE Transactions on Magnetics, Sept. 1993. 70] T. Pera, F. Ossart, and T. Waeckerle, \Numerical representation for anisotropic materials based on coenergy modelling," J. Appl. Phys., vol. 73, pp. 6784{6786, May 1993. 192
71] P. P. Silvester and D. Omeragic, \Di erentiation algorithms for soft magnetic material models," in IEEE Transactions on Magnetics, INTERMAG, 1993. Paper no. EQ05. 72] M. Gyimesi and J. Lavers, \Generalized potential formulation for 3{D magnetostatic problems," IEEE Transactions on Magnetics, vol. MAG28, pp. 1924{ 1929, July 1992. 73] M. Kaltenbacher, H. Landes, and R. Lerch, \An e cient calculation scheme for the numerical simulation of coupled magnetomechanical systems," in Proc. of CEFC'96, March 1996. 74] M. Kaltenbacher, H. Landes, and R. Lerch, \Simulation tool for magnetomechanical systems with moving parts," in Proc. of the ISEM Conf., 1995. 75] Y. Yoshida, K. Demachi, and K. Miya, \Scaling law for magnetoelastically coupled vibration of thin conducting components in a fusion reactor," in Proc. of the ISEM Conf., 1995. 76] K. Matsubara, N. Takahashi, K. Fujiwara, and T. Nakata, \Acceleration technique of waveform control for single sheet tester," in Proc. of the Intermag Conf., 1995. 77] T. Nakata and N. Takahashi, \Analysis of ux distributions in threephase velegged recti er transformer cores," Electrical Engineering in Japan, vol. 100, no. 6, pp. 42{50, 1980. 78] T. Nakata, N. Takahashi, and Y. Kawase, \Flux and loss distribution in the overlap joints of laminated cores," Journal of Magnetism and Magnetic Materials, no. 26, pp. 343{344, 1982. 79] T. Nakata and Y. Kawase, \Analysis of the magnetic characteristics in the straight overlap joint of laminated cores," 1982. 80] T. Nakata, Y. Ishihara, N. Takahashi, and Y. Kawase, \Analysis of magnetic elds in a single sheet tester using an h coil.," Journal of Magnetism and Magnetic Materials, no. 26, pp. 179{180, 1982. 81] T. Nakata, N. Takahashi, Y. Kawase, and M. Nakano, \In uence of lamination orientation and stacking on magnetic characteristics of grainoriented silicon steel laminations," IEEE Transactions on Magnetics, pp. 1774{1776, Sept. 1984. 82] T. Erlandsson and L. Martinsson, \Investigation of noload losses and distribution of magnetic ux in transformer cores with steplap joints," Examensarbete 1993 03 04, ABB Transformers, 1993. 193
83] E. Reiplinger, Beitrag zur Berechnung der Lautstarke von Transformatoren. PhD thesis, Technischen Universitat Hannover, 1972. 84] E. Reiplinger, J. P. Fanton, A. Adobes, A. W. Darwin, P. van Leemput, F. Claeys, H. Huttner, J. G. Paulick, A. J. M. Verhoeven, and H. F. Reijnders, \Transformer noise: Determination of sound power level using the sound intensity method," Electra, pp. 79{95, Oct. 1992. 85] J. Sievert, \Recent advances in the one and twodimensional magnetic measurement technique for electrical sheet steel," IEEE Transaction on Magnetics, vol. 26, pp. 2553{2558, Sept. 1990. 86] T. Nakata, N. Takahashi, and Y. Kawase, \Magnetic performance of steplap joints in distribution transformer cores," IEEE Transactions on Magnetics, vol. MAG18, pp. 1055{1057, Nov. 1982. 87] W. Salz, M. Birkfeld, and K. A. Hempel, \Calculation of eddy current loss in electrical steel sheet considering rotational hysteresis," in IEEE Transactions on Magnetics, INTERMAG, 1993. 88] T. Yagisawa, Y. Takekoshi, and S. Wada, \Magnetic properties of laminated steel sheets for normal uxes," Journal of Magnetism and Magnetic Materials, no. 26, pp. 340{342, 1982. 89] L. Kvarnsjo, On Characterization, Modelling and Application of Highly Magnetostrictive Materials. PhD thesis, Royal Institute of Technology, 100 44 Stockholm, Sweden, 1993. TRITAEEA9301. 90] R. D. Findlay, N. Stranges, and D. K. MacKay, \Losses due to rotational ux in three phase induction motors," IEEE Transactions on Energy Conversion, no. 3, pp. 543{548, 1994. 91] A. Lundgren, H. Tiberg, L. Kvarnsjo, and A. Bergqvist, \A magnetostrictive electric generator," IEEE Trans. Mag., pp. 3150 { 3152, November 1993. 92] S. Lundgren, A. Bergqvist, and S. Engdahl, \A system for dynamic measurements of magnetoMechanical Properties of arbitrarily excited siliconiron sheets," in Nonlinear electromagnetic systems (A. J. Moses and A. Basak, eds.), (Amsterdam), pp. 528 { 531, IOS Press, 1996. 93] A. Lundgren, A. Bergqvist, and G. Engdahl, \A ratedependent magnetostrictive simulation method for siliconiron sheets under harmonic excitation," IEEE Trans. Mag., March 1997. 94] A. Lundgren and G. Engdahl, \Measurements and modeling of magnetostrictive strain of a transversely excited silicon iron sheet," Journal de Physique IV, vol. 8, pp. 587{590, June 1998. Pr2. 194
95] A. Lundgren, A. Bergqvist, and G. Engdahl, \Analysis and veri cation of anisotropic magnetic sheet bending in an inhomogeneous lateral eld," in XI Conference on the Computation of Electromagnetic Fields (J. Cardoso, J. Bastos, and N. Ida, eds.), (Belo Horizonte, Brazil), pp. 287 { 288, Sociedade Brasileira de Eletromagnetismo, November 1997. 96] T. Nakata, N. Takahashi, and M. Nakano, \Improvments of measuring equipments for rotational power loss," in 1st International Workshop on Magnetic Properties of electrical sheet steel under two{dimensional excitation, (Braunschweig), Physikalisch{Technische Budesanstalt, 1992. 97] D. K. Cheng, Field and wave electromagnetics. AddisonWesley, 2 ed., 1990. 98] K. Binns, P. Lawrenson, and C. Trowbridge, The analytical and numerical solution of electric and magnetic elds. Wiley, 1992. 99] R. Becker, Electromagnetic elds and interactions. Dover, 1982. Originally published by Blaisdell, 1964. 100] J. C. Maxwell, A Treatise on Electricity and Magnetism, vol. 2. Oxford University Press, 3 ed., 1998. First published by Clarendon Press, 1891. 101] A. Lundgren, A. Bergqvist, and G. Engdahl, \A system for dynamic mea surements of magnetomechanical properties of arbitrarily excited siliconiron sheets," in Nonlinear electromagnetic systems (A. J. Moses and A. Basak, eds.), (Amsterdam), pp. 528 { 531, IOS Press, 1996. 102] H. Pfutzner and A. Hasenzagl, \Fundamental aspects of rotational magnetostriction," in Nonlinear electromagnetic systems (A. J. Moses and A. Basak, eds.), (Amsterdam), pp. 374{379, IOS Press, 1996. 103] C. Johnson, Numerical solution of partial di erential equations by the nite element method. Lund, Sweden: Studentlitteratur, 1987. 104] R. D. Cook, D. S. Malkus, and M. E. Plesha, Concepts and applications of nite element analysis. New York: John Wiley & Sons, 1989. 105] A. Schulze, \Magnetostriction i.," Z. Physik, vol. 50, p. 448, 1928. 106] A. Bergqvist, A. Lundgren, and G. Engdahl, \Computationally e cient vector hysteresis model with ux density as known variable," in Proceedings of the ISEM 97 conf., 1998. 195
Appendix A Design drawings Dimensions are in mm. 196
Figure A.1: Optic component placement with possible double interferometers 197
Figure A.2: Closeup of single interferometer with sample side dimension 198
Figure A.3: Side view of interferometer (possibly dual), arm with AOM Figure A.4: Side view of interferometer (possibly dual), arm with laser head 199
Figure A.5: Laser mount 200
Figure A.6: Custom tapped rod, for optic rail on diabase spacer fastening Figure A.7: Acoustooptic modulator, fastening on translation stage 201
Figure A.8: Baseplate for AOM 202
Figure A.9: Diabase spacers 203
Figure A.10: Sample support 204
Figure A.11: Tall laminated yoke Figure A.12: Short laminated yoke 205
Figure A.13: Spacer between yokes Figure A.14: Yoke pair assembly 206
Figure A.15: Table top with tapped mount holes 207
Figure A.16: Experiment table 208
Figure A.17: Table top support 209
A Lundgren