Design and implementation of active vibration control in smart structures

Tags: cantilever beam, PPF, Active vibration control, K.V. Gangadharan, piezoelectric actuators, beam, harmonic excitation, stiffness matrices, Euler-Bernoulli, block diagram, Mass matrix, P.K. Tripathi, compensator, vibration suppression, smart beam, SRF, vibration control, beam element, experiments, numerical simulations, pp, vibration, flexible steel cantilever, active damping, damping ratio, J. Fei, space structure, PID controller, beam elements, Smart Structure, Piezoceramic Actuators, feedback control, Smart Materials, CPID, excitation, displacement feedback, flexible beam, feedback control system, M. Umapathy, intelligent systems, Baillargeon, piezoelectric actuator, Aditya Avinash, K. Dhanalakshmi, Journal of Intelligent Material Systems and Structures, International journal
Content: International Journal of Research and Reviews in Mechatronic Design and Simulation (IJRRMDS)
92
Vol. 2, No. 1, March 2012, ISSN: 2046-6234
© Science Academy Publisher, United Kingdom
www.sciacademypublisher.com
Design and Implementation of Active Vibration Control in Smart Structures Prashant Kumar Tripathi and K.V. Gangadharan Department of mechanical engineering, National Institute of Technology, India Email: [email protected], [email protected] Abstract ­ Considerable attention has been devoted recently to active vibration control using intelligent materials as actuators. This paper presents results on active control schemes for vibration suppression of flexible aluminium cantilever beam with bonded piezoelectric actuators. The PZT patches were surface bonded near the fixed end of flexible cantilever beam. The state space model of the flexible cantilever beam with collocated sensor and actuator was derived. The sensor and actuator are collocated to achieve a minimum phase. The aim of this research work was to implement real time control to suppress vibration. To achieve this, a compensated inverse PID controller was developed and tuned to damp first mode using LabVIEW.Positive position feedback control (PPF) and Strain rate feedback control (SRF) were investigated and implemented using cRIO real-time system. Experimental results demonstrate that CPID controller is more stable and effective than PPF and SRF control and achieve effective vibration suppression without more response fluctuations. Keywords ­ Active Vibration Control, Positive position feedback control (PPF), Compensated inverse PID (CPID), Strain rate feedback control (SRF), Smart structures, Smart sensing
1. Introduction The developments in piezoelectric materials have motivated many researchers to work in the field of smart structures. A smart structure can be defined as the structure that can sense external disturbance and respond to that with active control in real time to maintain the mission requirements. Smart structures consist of highly distributed active devices which are primarily sensors and actuators either embedded or attached to an existing passive structure with integrated processor networks. Depending on the characteristics of the smart structures involved and the expected operating conditions, the selection of the sensors and actuators vary considerably. While typical smart structure sensors used in discrete or distributed locations to measure the performance of the system comprise fiber optics, piezoelectric ceramics and polymers, the actuators used in the smart materials technologies include applications of piezoelectric ceramics, piezoelectric polymers (PVDF), electrostrictive (ES) and magnetostrictive (MS) materials, electro-rheological (ER) and magneto-rheological (MR) fluids and piezofibres.PZT reliability, near linear response with applied voltage, exhibiting excellent response to the applied electric field over very large range of frequencies and their low cost make piezoelectric materials (PZT, Lead-Zirconate-Titanate) the most widely preferred one as collocated sensor and actuator pair. Therefore our work mainly considers the application of PZT patches to smart beam-like and smart plate-like structures for the purpose of active vibration control. Finite Element Method was shown to be especially advantageous in handling the multiple design parameters of piezoelectric patches. By enabling the parametric design features of the
technique, the influences of the piezoelectric patch placement and size on the responses of the smart beam are obtained. It is observed that as the patches move closer to clamped-end and increase in the size, the response of the smart beam increases. The technique also allows determination of the maximum admissible actuation value, hence effectively gives the actuator limits. It is also observed that the presence of the patches shifts the natural frequencies of the passive structure to higher frequencies [5]. From finite element model of the smart beam, strain values are obtained by performing modal analysis in order to determine the most suitable location for the strain gauge sensor pair. This corresponds to the location where the strain values attain their highest value for the first two modes of vibrations. K. B. Waghulde (2011) used piezoelectric material as smart material and cantilever beam was considered as a smart structure. Different positions were considered for the model analysis. In this case, the modal analysis was found out by using ANSYS and MATLAB.Gluhihs, S. and Kovalovs,(2006) presented a method to reduce vibration in a helicopter blade under a variable harmonic pressure loading using piezoelectric actuators. The model of a helicopter blade was an equivalent aluminum plate. To decrease the amplitude in the resonant frequency range, piezoelectric actuators are set on the top of the plate. The results were obtained by using the ANSYS finite element code. Choi, Park & Fukuda (1998) investigated the active control of hybrid smart structures under forced vibrations. Two hybrid smart structures considered in the study include PZT film/electro-rheological fluid actuators and piezoceramic/shape memory alloy actuators. J Fei (2008) this paper presents results on active control schemes for vibration suppression of flexible steel cantilever beam with bonded
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piezoelectric actuators. The PZT patches were surface bonded near the fixed end of flexible steel cantilever beam. The dynamic model of the flexible steel cantilever beam is derived. Active vibration control methods, strain rate feedback control (SRF), positive position feedback control (PPF) are investigated. The active vibration control of cantilever beams and plates was studied in the literature by mounting piezoelectric patches as actuators on the beams and plates. In recent years, a special attention is given to the research on the dynamics and control of a flexible satellite. However, dynamic models, especially the control of flexible satellites, still remain as one of the most challenging problems in engineering. This is mostly due to the fact that the types of actuators used to control flexible structures are limited to the control surfaces and engine thrust. In addition to these actuators, piezoelectric actuators can perhaps be used for control purposes. In fact, smart materials are good candidates for the vibration control due to their light weights, deforming controllability and ease in implementation.
modeled using Euler-Bernoulli theory and the Finite Element Method (FEM). Figure 1. Beam with collocated actuator and sensor. The piezoelectric beam element was used to model the regions where the piezoelectric was bonded as sensor/actuator and rest of the structure was modeled by the regular beam elements. 2.1. Mass matrix for Euler-Bernoulli beam element:
2. Modeling of Smart Structure Using EulerBernoulli beam Theory
Theoretical transfer functions were needed for including
different control algorithm, since controller design is based
on the plant transfer function. For this, state space approach
was used to find those transfer functions. The mass and
stiffness matrices are obtained using the Euler-Bernoulli
beam element in conjunction with the principle of virtual
work as given by bandyopadhyay B. (2006). The damping
(1)
matrix is obtained by experimental proportional damping.
Finite element model of the smart structure has four (4) a1 =a3=a4; b1=b3=b4; c1=c3=c4; d1=d3=d4.
equal elements. The sensor and actuator were integrated on
the top and bottom surfaces of the second element of the
beam.
Table 1. Symbolic parameters of Smart Beam
Parameters Length Width Thickness Density Cross ­Section Area Young's Modulus Moment of Inertia Damping Constants used in C* PZT Strain Constant PZT Stress Constant
Symbols
Cantilever Beam Lb b
Piezoelectric(PZT) (Sensor/Actuator) lp b
tb b Ab=b*tb Eb Ib
ta = ts p Ap =b*tb Ep Ip ----------
-----------
d31
-----------
g31
(2) Mass matrix for Euler-Bernoulli beam element with PZT on both sides:
In order to develop the Mathematical Model of the smart beam we first start with the modeling of the regular beam element and the piezoelectric beam element for a two node finite beam element. The flexible beam was divided into a number of finite elements (say, 4) as shown in Figure 1. The piezoelectric element was bonded on one of the section of the host surface, thus give rise to a smart beam. The smart beam model was then developed using a piezoelectric beam element, which includes sensor and actuator dynamics and a regular beam element which was
(3) Global mass matrix of beam with four (4) elements
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and stiffness matrix:
Stiffness matrix for Euler-Bernoulli beam element:
e1 =e3=e4; f1=f3=f4; g1=g3=g4; h1=h3=h4.
Natural frequency is the square root of the Eigen values of M and K matrices and the Mode shapes at each frequency is the row of the Eigen vectors of M and K matrices. The mass and stiffness matrices M and K of the dynamic equation of the smart structure include the sensor/actuator mass and stiffness. The equation of the motion of the smart structure and the sensor output finally given by
(4)
(7)
and
(5) Stiffness matrix for Euler-Bernoulli beam element with PZT on both sides:
(8)
where
are the vector of
displacement, the acceleration vector, the external force
vector, the controlling force vector, the total force vector and
a constant vector of the beam which are of sizes (8 x 1)
respectively.
The generalized coordinated are introduced into Eq. (7)
using a transformation q =Tg in order to reduce it further
such that the resultant equation represents the dynamics of
the first few dominant vibratory modes of the smart flexible
cantilever beam. Here, T is the modal matrix (8x2)
containing the eigenvectors representing the desired number
of modes of vibration.
(9) Pre-multiplying Eq. (9) by TT we get (10) which can be rewritten as (11) Raleigh's proportional damping as
(6) Global stiffness matrix of beam with four (4) elements
Where and are the frictional damping constant and the structural damping constant used in C*. State Space Model of the Smart Structure
Applying Fixed-Free boundary condition to global mass
(12) Using Eq. (12) in Eq. (11) and writing in the state space equation
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(13) for 2 vibratory modes The sensor voltage is taken as the output of the smart beam. The output of the plant is given by
gain positively back to the structure. PPF offers quick damping for a particular mode provided that the modal characteristics are known. The scalar equations governing the vibration of the structure in a single mode and the PPF controller are given as:
(14) The constant h and p in Eq. (13) and Eq. (14) for a 2 node beam element is given by
where is a modal coordinate of structure displacement, is the damping ratio of the structure is the natural frequency of the structure, G is the feedback gain is the compensator coordinate, c is the damping ratio of the compensator, c is the natural frequency of the compensator. The PPF control is illustrated in the Block diagram as shown in Figure. 2.
where, Sc=Gc*g31*zs*b Gc =signal conditioning gain zs=(tb/2)+ta; distance of sensing point from neutral axis of beam Ac=Ep*d31*b*za; za=(ta+tb)/2; distance between neutral axis of the beam and the piezoelectric layer Similarly, the force vector in Eq. (13) is given by force at tip
Figure 2. The block diagram of PPF controller. In PPF control, c should be closely matched to the natural frequency of the structure in order to achieve maximum damping.
The SISO state space model of the smart flexible cantilever beam for the first 2 vibratory modes is thus given by where r(t),u(t),A,B,C,D,E,x(t) and y(t) represent the external force input, the control input, system matrix, input matrix, output matrix, transmission matrix, external load matrix, state vector, system output(sensor output) respectively. 3. Active Vibration Control For this research three vibration suppression methods are used, compensated inverse PID, Positive position feedback and strain rate feedback control. PID controller is the kind of controller of which proportional gain and derivative gain can be determined based on desired specifications and dynamics of a plant. The optimized parameter adjusted CPID controller is widely used in vibration suppression.
3.2. Strain rate feedback (SRF) control Strain rate feedback (SRF) control is used for active damping of a flexible space structure [S. M. Newman, 1992]. Using SRF, the structural velocity coordinate is fed back to the compensator, and the compensator position coordinate multiplied by a negative gain is fed back to the structure. SRF has a wider active damping region and can stabilize more than one mode given a sufficient bandwidth. In this research, the SRF is designed to control the vibration of the first mode. Experimental results demonstrate that the SRF method is effective in actively increasing damping of the flexible beam with the PZT actuator. The SRF model can be presented with the Eq. (15) and Fig.3 block diagram.
3.1. Positive Position Feedback Control Positive Position Feedback (PPF) control was first proposed by Goh and Caughey for he collocated sensors and actuators. Later on Fanson and Caughey demonstrated PPF control in large space structures. The PPF control is applied by feeding the structural position coordinate directly to the compensator and the product of the compensator and a scalar
Figure 3. The block diagram of SRF controller. It is clear from Eq. (15) that the effect of the SRF compensator is a decrease in the damping term, which is referred to as active negative damping. Thus, in implementing SRF, the compensator should be designed so the targeted frequencies are below the compensator
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frequencies. SRF has a much wider active damping frequency region, which gives a designer some flexibility. As long as the compensator frequency is greater than the structural frequency, a certain amount of damping will be provided. 3.3. CPID control algorithm PID algorithm is the most widely used control loop feedback mechanism and has been universally accepted in industrial control. It attempts to minimize the error between the measured process variable and the desired setpoint by performing some calculations and generating a corrective action to adjust the process accordingly. The controller named `Compensated Inverse PID' denoted CIPID was proposed in Gani (2003). The controller is designed by taking the effect of truncated modes which might cause the spillover effects. Hence the controller is designed to attenuate vibration modes, inserting damping to the closed-loop transfer function. where Kp, K I and KD are respectively the parameters of the PID, tuned to damp any of the i-th mode. i is the resonant frequency for the i-th mode. Comparing with a typical second order system, we obtain Using superposition rules, the ClPlD controller can be extended to control first three modes .The controller for each mode will be arranged in parallel. Basically the CIPID is used to damp the resonant peaks by placing poles at each resonant frequency. Changing KP in the above equation will change the damping factor for each mode and can control the resonant peak. However higher damping factor doesn't mean that the peak will be always reduced; hence an optimal value of Kp is required for each mode. A digital lowpass filter is applied to sensor output to remove an interference signal or noise. 4. Experimental implementation Flexible Structure was cantilever beam type structure. The configuration of the structure is shown in Figure 4.
The strain value at the sensor location was taken as the strain feedback and laser pickup was used for tip displacement feedback. The dimensions and the distances for the beams were given in Table 2.
Table 2. Dimension of the beam and a/s locations
S
Dimensions of Dimensions of da:
No. Structurea
Actuatorb
Actuator
ds : Sensor Distance
(mm)
(mm)
Distance (mm)
(mm)
1
300x25x3
25x25x.5
20
30
In active control of forced vibrations, a smart beam having a different configuration was produced bonding two piezoelectric patches to the aluminum beam with conductive epoxy as shown in Figure 5. The piezoelectric actuator was placed near the root of the beam to perform effective control. The piezoelectric shaker placed in the middle of the beam generates harmonic excitation. The ground connections of the two patches are different. The ground connection of the piezoelectric actuator is on aluminum beam while that of the piezoelectric shaker is its back side since it is isolated from the beam. Sparkler ceramics made PZT 5H patches were used for the piezoelectric actuators and exciter. The PZT which was bonded opposite side of the piezoelectric actuator gives charge proportional to strain, which was converted to voltage by charge amplifier. Experimental system consists of a smart beam, NI cRIO with NI C-series input/output modules for real time control implementation, a signal conditioning unit, a non-contact laser displacement sensor, high voltage power amplifiers, multifunction data acquisition cards and a personal computer programming with LabVIEW. A LabVIEW programs were developed to control vibration of the smart beam under harmonic excitation using control algorithms. Two channels of the analog output module NI- 9264 were programmed. One of the channels was used to provide harmonic excitation to the power amplifier. Other channel was used to provide control signal to the piezoelectric actuator over the other power amplifier. Two power amplifiers were utilized for the excitation and control. Charge amplifier was used to convert charge to voltage for PZT sensor and NI-9203 module is used to measure current output of laser pickup (optoNCDT) to get tip displacement. A digital band pass filter of 5 Hz to 50 Hz was used for filtering sensors output. The strain feedback control for gain of 5 provides 40 % reduction in steady-state vibration amplitude. The displacement feedback control for gain of 5 provides 84 % reduction in steady-state vibration amplitude.
4.1. Compensated inverse PID: The beam has been excited at its first resonant, frequency which is 27.05 Hz in order to test the controller ability in damping this mode. The CPID control parameters used in experiment for first mode suppression:
;
;
Figure 4. Configuration for cantilever flexible beam.
The compensated inverse CPID control provides 87 % reduction in steady-state vibration amplitude.
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6. Conclusions The SRF and PPF controller were employed to actively suppress vibration of a flexible aluminium cantilever beam under harmonic excitation. Suppression of the single dominant mode vibration was carried out and the best result was obtained using CPID control. The PPF and SRF control was also effective in suppressing the vibration. The experimental results successfully demonstrated the effectiveness of vibration suppression using the SRF, PPF and CPID controller.
Figure 5. Experimental setup. 5. Experimental Results The experimental results successfully demonstrated the vibration suppression of an aluminium cantilever beam using SRF and PPF control. Moreover, SRF control was better than PPF control under harmonic excitation experiment shown in Figure 6&7. CPID control reduces steady state error and easy to implement for more than one mode in Feedback Control System under harmonic excitation shown in Fig 8. Figure 6. PPF vibration control under harmonic excitation. Figure 7. SRF vibration control under harmonic excitation. Figure 8. CPID vibration control under harmonic excitation.
7. Acknowledgment Our sincere appreciation towards the SOLVE for funding this project. The authors duly acknowledge the generous support provided by the Centre for system design (CSD): A Centre of Excellence at NITK Surathkal. 8. References [1] Gluhihs, S. and Kovalovs, A. Reduction of the vibration in a helicopter blade using piezoelectric actuators, Aviation, 10: 2, 2006, 3-6. [2] M.J. Brennan, J. Garcia-Bonito, S.J. Elliott, A. David, R.J. Pinnington. Active vibration control of beams with optimal placement of piezoelectric sensor/actuator pairs, Journal of Smart Materials and Structures, 1999. [3] S.G. Kelly, Schaum's Outline of Theory and Problems of Mechanical Vibrations. McGraw-Hill, 1993. [4] Kim, J., Varadan, V. V., Varadan, V. K. & Bao, X. Finite element modeling of a smart cantilever plate and comparison with experiments, Smart Materials and Structures, 5, 1996, 165-170. [5] Lim, Y-H., Varadan, V. V., & Varadan, V. K, Closed loop finite element modeling of active structural damping in the frequency domain. Smart Materials and Structures, 6, 1997, 161-168. [6] Celentano, G., & Setola, R., The modelling of a flexible beam with piezoelectric plates for active vibration control. Journal of Sound and Vibration, 223 (3), 1999, 483-492. [7] Manning, W. J., Plummer, A. R., & Levesley, M. C, Vibration control of a flexible beam with integrated actuators and sensors,Smart Materials and Structures, 9, 2000 , 932-939. [8] Gaudenzi, P., Carbonaro, R., & Benzi, Control of beam vibrations by means of piezoelectric devices: theory and experiments. Composite Structures, 50, 2000, 373-379. [9] Singh, S. P., Pruthi, H. S., & Agarwal, V. P., Efficient modal control strategies for Active Control of Vibrations. Journal of Sound and Vibration, 262, 2003, 563- 575. [10] Lin, R. M., & Nyang, K. M., Analytical and experimental investigations on vibrational control mechanisms for flexible active structures. Smart Materials and Structures, 12, 2003, 500-506. [11] Kusculuoglu, Z. K., Fallahi, B., & Roston, T.J.,Finite element model of a beam with a piezoceramic actuator. Journal of Sound and Vibration, 276, 2004, 27- 44. [12] Fei, J., Active vibration control of flexible steel cantilever beam using piezoelectric actuators. IEEE, 2005, 35-39. [13] Vasques, C. M. A., & Rodrigues, J. D., Active vibration control of smart piezoelectric beams: comparison of classical and optimal feedback control strategies. Computers & Structures, 84, 2006, 14021414. [14] Peng, F., Ng A. & Hu, Y.R., Actuator placement optimization and adaptive vibration control of plate smart structures. Journal of Intelligent Material Systems and Structures, 16, 2005, 263-271. [15] Ma, K., & Nejhad, M. N. G., Adaptive simultaneous precision positioning and vibration control of intelligent composite structures. Journal of Intelligent Material Systems and Structures, 16, 2005, 163174. [16] Choi, S. B., Park, S. B., & Fukuda, T. A Proof of Concept investigation on active vibration control of hybrid structures. Mechatronics, 8, 1998, 673-689. [17] Yaman, Y., Ьlker, F. D., Nalbanto lu, V., Зal kan, T., Prasad, E., Waechter, D., & Yan, B., Application of synthesis active vibration control technique to a smart fin.6th Can Smart Workshop, 109-119, 2003, Canada. [18] Kumar, R., & Singh, S. P., Adaptive hybrid control of smart structures subjected to multiple disturbances. Smart Materials and Structures, 15, 2006, 1345- 1357.
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[19] Baillargeon, B. P., & Vel, S. S.,Active vibration suppression of sandwich beams using shear actuators: experiments and numerical simulations. Journal of Intelligent Material Systems and Structures, 16, 2005, 517-530. [20] J. Fei. Active Vibration Control of a Flexible Structure Using Piezoceramic Actuators, Sensors & Transducers Journal, Vol. 89, Issue 3, March 2008, pp. 52-60. [21] K. Dhanalakshmi, Aditya Avinash, M. Umapathy, M. Marimuthu. experimental study on vibration control of shape memory alloy actuated flexible beam, International journal on smart sensing and intelligent systems, VOL. 3, NO. 2, June 2011. [22] K. B. Waghulde, Dr. Bimlesh Kumar. Vibration analysis of cantilever smart structure by using piezoelectric smart material, International journal on smart sensing and intelligent systems, VOL. 4, NO. 3, September 2011. [23] Institution of Mechanical Engineers PArt Journal of Automobile of Engineering vol.215, pp.865-873, 2001.

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