Nomenclature x: is the dynamic movement of the pusher; x': is the dynamic reduced tappet speed; x'': is the dynamic reduced tappet acceleration; : is the rotation angle of the cam (the position angle); K: is the elastic constant of the system; k: is the elastic constant of the valve spring; x0: is the valve spring preload (pretension); Mc: is the mass of the cam; mT: is the mass of the tappet; m: the angular rotation speed of the cam (camshaft); s: is the movement of the pusher; s': is the first derivative in function of of the tappet movement, s; s'': is the second derivative in raport of angle of the tappet movement, s; s''': is the third derivative of the tappet movement s, in raport of the angle; n: is the motor shaft speed; h: is the follower stroke; r0: is the radius of the base circle; ra: is the radius of the cam; u: is the maximum angle of climb; xc and yc: are the cartesian coordinates of the cam; J*: is the reduced moment of inertia; J*m: is the average of the reduced moment of inertia; J*': is the first derivative of the moment of inertia in raport of the angle; : is the dynamic angular rotation speed of the cam; : is the dynamic angular rotation acceleration of the cam; : is the cam-mechanism efficiency; : is the transmission angle; spp: is the kinematics acceleration of the tappet (valve); xpp: is the dynamic acceleration of the tappet (valve).

I. Introduction The paper presents an original method in determining a general, dynamic and differential equation for the motion of machines and mechanisms, particularized for the mechanisms with rotation cams and followers [1-3]. This equation can be directly integrated by an original method presented in this paper. After integration the resulted mother equation may be solved immediately [3]. One presents an original dynamic model with one degree of freedom, with variable internal amortization. One determines the resistant force reduced at the valve, the motor force reduced at the valve, and the coefficient of variable internal amortization, and the reduced mass [1-3]. We can make the geometrical synthesis of the cam profile with the help of the cinematics of the mechanism. One uses as well the reduced speed, s'. The forces and the velocities at a cam with plate translated tappet can be seen in [18-19]. The driving force Fm, perpendicular on the r in A, is decomposed in two forces: the utile force Fu, which acts the tappet and the lost force Fa, who is a slipping force. The velocities take the same positions. We can make the geometro-kinematics synthesis of the cam profile with the help of the cinematics of the mechanism (see the Figure 1, 8, 9). Now, we can make the geometro-kinematics synthesis of the classic cam profile (see the algorithm). The moments of inertia is determined with THE RELATIONSHIPs from the presented system (algorithm). The angular velocity , and the angular acceleration, are determined with the presented relationships, for a classic cam and tappet mechanism. Where x is the dynamic movement of the pusher, while s is its normal, kinematics movement. K is the spring constant of the system, and k is the spring constant of the tappet spring. It note, with x0 the tappet spring preload, with mT the mass of the tappet, with the angular rotation speed of the cam (or camshaft), where s' is the

Manuscript received and revised September 2013, accepted October 2013

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

Florian Ion T. Petrescu, Relly Victoria V. Petrescu

first derivative in function of of the tappet movement, s [1-36]. New requirements of distribution mechanisms are that they can work at very high speeds without vibration or noise. Because of this, the settings of the valve spring must be pre-calculated rigorously. One sets the spring elastic constant k, and the spring pretension x0. The algorithm used is written in Excel.

II. Presents the Algorithm The paper presents an algorithm for setting the dynamic parameters of the classic distribution mechanism used in the internal combustion engines. This algorithm is written in excel and can be seen below.

A

B

1 h[mm]

0.006

2 [deg]

=B3*180/PI()

3 [rad]

0

4s

=B1/2*(1-COS(PI()*B3/B8))

5 s'

=B1*PI()/2/B8*SIN(PI()*B3/B8)

6 s''

=(PI())^2*B1/2/(B8)^2*COS(PI()*B3/B8)

7 r0 [mm] 0.013

8 u[rad] =PI()/2

9 [deg]

0

10 s [m]

=B4

11 s' [m]

=B5

12 s'' [m]

=B6

13

=-(PI())^3/2*B1/B8^3*

s''' [m]

SIN(PI()*B3/B8)

14 xc

=B16*COS(B3)+B17*SIN(B3)

15 yc

=B17*COS(B3)-B16*SIN(B3)

16 ra [mm] =SQRT(2*B7*B10+B10^2)

17 r0+s [mm] =B7+B10

18

19 Mc [kg] =0.2

20 mT [kg] =0.1

21 n

[rot/min] 22 m [s-1] 23 m2 [s-2] 24 J*m

5000 =PI()*B21/60 =B22^2

[kgm2]

=B19/2*(B7+B1)^2

25

=B19/2*(B7^2+4*B7*B10+2*B10^2)+

J* [kgm2] B20*B11^2

26 J*' [kgm2] =2*B11*(B19*(B7+B10)+B20*B12)

27 2 [s-2]

=B23*B24/B25

28 [s-1]

=SQRT(B27)

29 [s-2]

=-B27/2*B26/B25

30 sin2

=B16^2/(B17^2+B16^2)

31

=SUM(B30:T30)/19

32 rad =ASIN(B16/SQRT(B17^2+B16^2))

33

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

34

35 x0 [m]

=0.09

36 [deg] 37 spp [ms-2] 38 K [N/m]

=B9 =B12*B27+B11*B29 =5000000

39 k [N/m] 40000

40 x'' [m]

see details

41 x' [m]

see details

42 [deg]

=B36

43 xpp [m/s2] =B40*B27+B41*B29

Details 40 x'' [m] 41

=B12-((((B38+B39)*B20*B27*2* (B12^2+B11*B13)+(B39^2+2* B39*B38)*2*(B11^2+B10*B12)+ 2*B39*B35*(B38+B39)*B12)* (B10+B39*B35/(B38+B39))+ ((B38+B39)*B20*B27*2 *B11*B12+(B39^2+2*B39*B38) *2*B10*B11+2*B39*B35* (B38+B39)*B11)*B11-B12*((B38+B39) *B20*B27*B11^2+(B39^2+2*B39 *B38)*B10^2+2*B39* B35*(B38+B39)*B10)-B11* ((B38+B39)*B20*B27*2*B11*B12 +(B39^2+2*B39*B38)*2*B10*B11 +2*B39*B35*(B38+B39)*B11)) *(B10+B39*B35/(B38+B39))^2 -(((B38+B39)*B20*B27*2*B11*B12 +(B39^2+2*B39*B38)*2*B10*B11 +2*B39*B35*(B38+B39)*B11) *(B10+B39*B35/(B38+B39))((B38+B39)*B20*B27*B11^2+ (B39^2+2*B39*B38)*B10^2 +2*B39*B35*(B38+B39)*B10)*B11) *2*(B10+B39*B35/(B38+B39))*B11)/ (2*(B38+B39)^2*(B10+B39*B35/ (B38+B39))^4

=B11-(((B38+B39)*B20*B27*2* B11*B12+(B39^2+2*B39*B38)*2* B10*B11+2*B39*B35*(B38+B39)*B11) *(B10+B39*B35/(B38+B39))((B38+B39) *B20*B27*B11^2+(B39^2+2*B38*B39)* B10^2+2*B39*B35*(B38+B39)*B10) *B11)/(2*(B38+B39)^2*(B10+B39*B35/ (B38+B39))^2)

x' [m] International Review on Modelling and Simulations, Vol. 6, N. 5

Florian Ion T. Petrescu, Relly Victoria V. Petrescu

With this algorithm we can make the dynamic analyze of the tappet or valve movement in function of the cam profile and tappet (valve) dynamic settings. This (presented in the algorithm and in the Fig. 1) profile has the advantage (comparative with a classic profile) that it relaxes the velocities, has lowers noise and vibration, producing a better gas distribution and increases 3-4 times the yield of the mechanism. III. Dynamic Analyze and Settings of the New Presented COS Profile We start with the classical law cos, with u=90 [deg], relaxed, (see the cam profile in the Fig. 1).

Using these modern settings (k and x0), where the valve spring is strong, the valve (tappet) acceleration values are limited. In addition, it may increase the operating speed of the motor shaft. For a motor shaft speed n=10000 [rot/min] we can see the dynamic analysis of the valve acceleration in the below diagram (Fig. 3), for the next settings: the elastic constant of the valve spring, k=40000 [N/m]; the valve spring preload (pretension), x0=0.09 [m].

Fig. 3. The dynamic analysis of the valve acceleration: the law cos; u=90 [deg]; n=10000 [rot/min]; k=40000 [N/m]; x0=0.09 [m]. For a motor shaft speed n=20000 [rot/min] we can see the dynamic analysis of the valve acceleration in the below diagram (Fig. 4), for the next settings: the elastic constant of the valve spring, k=200000 [N/m]; the valve spring preload (pretension), x0=0.2 [m]. Fig. 1. The relaxed cam profile, law cos; u=90 [deg] For a motor shaft speed n=5000 [rot/min] we can see the dynamic analysis of the valve acceleration in the below diagram (Fig. 2), for the next settings: the elastic constant of the valve spring, k=40000 [N/m]; the valve spring preload (pretension), x0=0.09 [m].

Fig. 2. The dynamic analysis of the valve acceleration: the law cos; u=90 [deg]; n=5000 [rot/min]; k=40000 [N/m]; x0=0.09 [m]. Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

Fig. 4. The dynamic analysis of the valve acceleration: the law cos; u=90 [deg]; n=20000 [rot/min]; k=200000 [N/m]; x0=0.2 [m]. For a motor shaft speed n=30000 [rot/min] we can see the dynamic analysis of the valve acceleration in the below diagram (Fig. 5), for the next settings: the elastic constant of the valve spring, k=300000 [N/m]; the valve spring preload (pretension), x0=0.5 [m]. International Review on Modelling and Simulations, Vol. 6, N. 5

Florian Ion T. Petrescu, Relly Victoria V. Petrescu In order to increase the motor shaft speed, the two dynamic settings of valve spring should also increase. IV. The Old COS Profile The old classic cam profile, law COS, is presented in the Figure 8. The dynamic settings are similar with those presented at the new, relaxed, cos, profile. The yield is three to four times lower. Fig. 5. The dynamic analysis of the valve acceleration: the law cos; u=90 [deg]; n=30000 [rot/min]; k=300000 [N/m]; x0=0.5 [m]. For a motor shaft speed n=40000 [rot/min] we can see the dynamic analysis of the valve acceleration in the below diagram (Fig. 6), for the next settings: the elastic constant of the valve spring, k=600000 [N/m]; the valve spring preload (pretension), x0=0.5 [m].

Fig. 6. The dynamic analysis of the valve acceleration: the law cos; u=90 [deg]; n=40000 [rot/min]; k=600000 [N/m]; x0=0.5 [m]. For a motor shaft speed n=50000 [rot/min] we can see the dynamic analysis of the valve acceleration in the below diagram (Fig. 7), for the next settings: the elastic constant of the valve spring, k=900000 [N/m]; the valve spring preload (pretension), x0=0.6 [m].

Fig. 8. The classical cam profile, law cos; u=90 [deg]

Algorithm is modified as follows.

14 xc

=B11*COS(B3)+B17*SIN(B3)

15 yc

=B17*COS(B3)-B11*SIN(B3)

30 sin2

=B11^2/(B17^2+B11^2)

The advantage of the old profile is that it can be modified by shortening the FI angle (see the Fig. 9 where the u=45 [deg]).

Fig. 7. The dynamic analysis of the valve acceleration: the law cos; u=90 [deg]; n=50000 [rot/min]; k=900000 [N/m]; x0=0.6 [m]. Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

Fig. 9. The classical cam profile, law cos; u=45 [deg] International Review on Modelling and Simulations, Vol. 6, N. 5

Florian Ion T. Petrescu, Relly Victoria V. Petrescu

By reducing the angle of lift, push rod stroke occurs quickly, but it and shrinks. V. Conclusion This paper presents an algorithm for setting the dynamic parameters of the classic distribution mechanism used in the internal combustion engines. New requirements of distribution mechanisms are that they can work at very high speeds without vibration or noise. Because of this, the settings of the valve spring must be pre-calculated rigorously. One sets the spring elastic constant k, and the spring pretension x0. In order to increase the motor shaft speed, the two dynamic settings of valve spring should also increase. The algorithm presented was written in Excel. With this algorithm we can make the dynamic analyze of the tappet or valve movement in function of the cam profile and tappet (valve) dynamic settings. The new (presented) cosin profile has the advantage (comparative with a classic profile) that it relaxes the velocities, has lowers noise and vibration, producing a better gas distribution and increases 3-4 times the yield of the mechanism. The advantage of the old profile is that it can be modified by shortening the FI angle. By reducing the angle of lift, push rod stroke occurs quickly, but it and shrinks. References [1] Antonescu, P., Oprean, M., Petrescu, F. Analiza dinamic a mecanismelor de distribuie cu came. Оn al VII-lea Simpozion Naional de Roboi Industriali, MERO'87, Bucureti, 1987, Vol. III, p. 126-133. [2] Petrescu, F., Petrescu, R. Elemente de dinamica mecanismelor cu came. In al VII-lea Simpozion Naional cu Participare Internaional Proiectarea Asistat de Calculator, PRASIC'02, Braov, 2002, Vol. I, p. 327-332. [3] Petrescu, F.I., Petrescu, R.V. Contributions at the dynamics of cams. In the Ninth IFToMM International Symposium on Theory of Machines and Mechanisms, SYROM 2005, Bucharest, Romania, 2005, Vol. I, p. 123-128. [4] Anderson D.G., Cam dynamics. Prod. Engineering, 24(10), 1953, p. 170-176. [5] Barabyi J.S., Cams, dynamics and design. Design News, 1969, 24, p. 108. [6] Dudley W.M., New Methods in Valve Cam Design. Trans. SAE, January 1948, 2, p. 19-33. [7] Hrones J.A., An analysis of Dynamic Forces in a Cam-Driver System, Trans. ASME, 1948, 70, p. 473-482. [8] Cheney R.E., Production of very accurate high-speed master cams. Machinery (London), 1962, 100(2570), p. 380-386. [9] Choi J.K., Kim S.C., Hyundai Motor Co. Korea, An experimental study on the Frictional Characteristics in the Valve Train System. (945046), In FISITA CONGRESS, 17-21 October 1994, Beijing, p. 374-380. [10] Koster M.P., The effects of backlash and shaft flexibility on the dynamic behaviour of a cam mechanism. In, Cams and cam mechanisms, 1974, p. 141-146. [11] Kerle H., How effective is the method of finite differences as regards simple cam mechanisms. Cams and cam mechanisms, 1974, p. 131-135. Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

[12] Mercer S., Dynamic characteristics of cam forms calculated by the digital computer. Trans. ASME, Nov. 1958, 80, p. 16951705. [13] Beard C.A., Problems оn valve gear design and instrumentation. SAE Technical Progress Series, 1963, p. 58-84. [14] Barkan P., s.a., A spring-actuated, cam follower system; Design theory and experimental result. Journal Engineering, Trans. ASME, 1965,(87 B), p. 279-286. [15] Baxter M.L., Qurvature-acceleration relation for plane cams. Trans. ASME 70,1948, p.483-489. [16] Bishop J.L.H., An analytical approach to automobile valve gear design. Inst. of Mech. Engrs. Auto-Division Proc. 4, 1950-51, p. 150-160. [17] Taraza, D., "Accuracy Limits of IMEP Determination from Crankshaft Speed Measurements," SAE Transactions, Journal of Engines 111, p. 689-697, 2002. [18] Petrescu, F.I., Petrescu, R.V. Determining the dynamic efficiency of cams. In the Ninth IFToMM International Symposium on Theory of Machines and Mechanisms, SYROM 2005, Bucharest, Romania, 2005, Vol. I, p. 129-134. [19] Petrescu, F.I., Petrescu, R.V., Popescu N., The efficiency of cams. In the Second InterNational Conference "Mechanics and Machine Elements", Technical University of Sofia, November 4-6, 2005, Sofia, Bulgaria, Vol. II, p. 237-243. [20] Petrescu, F.I., Petrescu, R.V., Contribuii la sinteza mecanismelor de distribuie ale motoarelor cu ardere intern. Оn a V-a Conferin "Economicitatea, Securitatea i Fiabilitatea Autovehiculelor", ESFA'95, Bucureti, 1995, Vol. I, p. 257-264. [21] Petrescu, F.I., Petrescu, R.V., Sinteza mecanismelor de distribuie prin metoda coordonatelor rectangulare (carteziene). In The VIIth Edition of the National Conference With International Participation, GRAFICA-2000, Craiova, Romania, 2000, p. 297302. [22] Petrescu, F.I., Petrescu, R.V., Designul (sinteza) mecanismelor cu came prin metoda coordonatelor polare (metoda triunghiurilor). In The VII-th Edition of the National Conference With International Participation, GRAFICA-2000, Craiova, Romania, 2000, p. 291-296. [23] Petrescu, F.I., Petrescu, R.V., Dinamica mecanismelor de distributie, Create Space publisher, USA, December 2011, ISBN 978-1-4680-5265-7, 188 pages, Romanian version. [24] Petrescu, F.I., Bazele analizei i optimizrii sistemelor cu memorie rigid curs i aplicaii, Create Space publisher, USA, 2012, ISBN 978-1-4700-2436-9, 164 pages, Romanian edition. [25] Petrescu, F.I., Teoria mecanismelor Curs si aplicatii (editia a doua), Create Space publisher, USA, September 2012, ISBN 9781-4792-9362-9, 284 pages, Romanian version. [26] Z. Ge, a.o., Mechanism Design amd Dynamic Analysis of Hybrid Cam-linkage Mechanical Press, Key Engineering Materials Journal, Vol. 474-476 (2011), p. 803-806. [27] Z. Ge, a.o., CAD/CAM/CAE for the Parallel Indexing Cam Mechanisms, Applied Mechanics and Materials Journal, Vol. 4447 (2011), p. 475-479. [28] A. Ghazimirsaied, a.o., Improving Volumetric Efficiency using Intake Valve Lift and Timing Optimization in SI Engine, IREME Journal, March. 2010, Vol. 4, N. 3, p.244-252. [29] M. Hamid, a.o., Using Homotopy Analysis Method to Determine Profile for Disk Cam by Means of Optimization of Dissipated Energy, IREME Journal, July 2011,Vol. 5, N. 5, p. 941-946. [30] M. Liu, Z. Qian, Research on Reverse Design of the Cam Mechanism, Applied Mechanics and Materials Journal, Vol. 43 (2011), p. 551-554. [31] Y. Samim, a.o., Analytical dynamic response of Elastic CamFollower Systems with Distributed Parameter Return Spring, Journal of mechanical design (ASME), Vol. 115, Issue 3, (online June 2008), p. 612-620. [32] R. Shriram, a.o., Design and Development of Camless Valve Train for I.C. Engines, IREME Journal, July 2012, Vol. 6, N. 5, p. 1044-1049. [33] W. Wang, Creation Design of Cam Mechanism Based on Reverse Engineering, Advanced Materials Research Journal, Vol. 230-232 (2011), p. 453-456. International Review on Modelling and Simulations, Vol. 6, N. 5

Florian Ion T. Petrescu, Relly Victoria V. Petrescu

[34] F. Xianying, a.o., Meshing Efficiency of Globoidal Indexing Cam Mechanism with Steel Ball, Advanced Materials Research Journal, Vol. 413 (2012), p. 414-419. [35] H.D. Zhao, a.o., Research on Dynamic Behavior of Disc Indexing Cam Mechanism Based on Virtual Prototype Technology, Key Engineering Materials Journal, Vol. 499 (2012), p. 277-282. [36] G. Zhou, a.o., Seriation Design and Research on Cam Shedding Mechanism of Looms, Advanced Materials Research Journal, Vol. 479-481 (2012), p. 2383-2388.

Authors' information 1Dr. Eng. Florian Ion T. Petrescu, Senior Lecturer at UPB (Bucharest Polytechnic University), TMR (Theory of Mechanisms and Robots) department. 2Dr. Eng. Relly Victoria V. Petrescu, Senior Lecturer at UPB (Bucharest Polytechnic University), TTL (Transport, Traffic and Logistics) department.

1. Ph.D. Eng. Florian Ion T.

PETRESCU

Senior Lecturer at UPB

(Bucharest

Polytechnic

University),

Theory

of

Mechanisms and Robots

department,

with overall average 9.63;

Date of birth: March.28.1958; Higher education: Polytechnic University of Bucharest, Faculty of Transport, Road Vehicles Department, graduated in 1982,

doctoral thesis: "Theoretical and Applied Contributions About the Dynamic of planar mechanisms with Superior Joints".

Expert in: Industrial design, Mechanical Design, Engines Design, Mechanical Transmissions, Dynamics, Vibrations, Mechanisms, Machines, Robots.

Association: Member ARoTMM, IFToMM, SIAR, FISITA, SRR, AGIR. Member of Board of SRRB (Romanian Society of Robotics).

2. Ph.D. Eng. Relly Victoria V. PETRESCU Senior Lecturer at UPB (Bucharest Polytechnic University), Transport, Traffic and Logistics department, Citizenship: Romanian; Date of birth: March.13.1958; Higher education: Polytechnic University of Bucharest, Faculty of Transport, Road Vehicles Department, graduated in 1982, with overall average 9.50; Doctoral Thesis: "Contributions to analysis and synthesis of mechanisms with bars and sprocket". Expert in Industrial Design, Engineering Mechanical Design, Engines Design, Mechanical Transmissions, Projective and descriptive geometry, Technical drawing, CAD, Automotive engineering, Vehicles, Transportations. Association: Member ARoTMM, IFToMM, SIAR, FISITA, SRR, SORGING, AGIR.

Copyright © 2013 Praise Worthy Prize S.r.l. - All rights reserved

International Review on Modelling and Simulations, Vol. 6, N. 5

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