Using homotopy analysis method to determine profile for disk cam by means of optimization of dissipated energy

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1 International Review of Mechanical Engineering (I.RE.M.E.), Vol. 5, n. 5 Using homotop analsis method to determine profile for dis cam means of optimization of dissipated energ Hamid M. Sedighi 1, Kourosh H. Shirazi Astract Select a particular shape for cam profile can pla an important role in reduction of energ in automoile motors. Engineers tr to reduce this dissipated energ to improve the motor operation. This paper otains nonlinear governing equation of dis cam follower motion and optimizes it with calculus of variation. Because of optimum cam profile, also, maximum acceleration of the follower is decreased. Finall, we solve it analticall means of Homotop analsis method and numerical results have een reported to prove the soundness of the analtical method. Copright 11 Praise Worth Prize S.r.l. - All rights reserved. Kewords: Homotop analsis method; cam profile; optimization of dissipated energ M e M M M I r ω μ t v a l1, l K s Nomenclature Tappet mass Pushrod mass Valve mass Rocer arm mass Rocer arm moment of inertia radius of ase circle of cam Cam angular velocit Friction coefficient Lengths of rocer arm Valve spring coefficient I. Introduction A cam is a mechanical element, which is used to transmit a desired motion to another mechanical element direct surface contact. Generall, a cam is a mechanism, which is composed of three different fundamental parts from a inematic viewpoint [1, ]. A cam, which is a driving element, a follower, which is a driven element and a fixed frame. Cam mechanisms are usuall implemented in most modern applications and in particular in automatic machines and instruments, internal comustion engines and control sstems [3]. This mechanism is an important place in automoile that consume a significant amount of energ and can e effective in wor of motors. Shape optimization ased on genetic algorithm (GA) [4], or ased on evolutionar algorithms (EA) in general, is a relativel oung and potential field of research. The target of the cam shape optimization is to optimize the function (movements) of the operated device without violating geometric and phsical constraints of cam designing. Cam shape optimization is tpicall a heavil constrained multi-ojective optimization prolem in which multiple, design tas dependent, material strength and durailit, cam manufacturing technolog, geometric, inetic and dnamic constraints should e satisfied simultaneousl. For example, the minimum local concave radius of the cam shape ma not e smaller than the radius of the grinding stone used for manufacturing the cam. The minimum local convex radius is constrained the contact pressure (Hertz s pressure) etween the cam and cam follower [5]. Tpical cam design ojectives are maximum rise and return rates for the cam follower movements, minimum instantaneous contact force etween the cam and follower simultaneousl with minimum dnamic force fluctuation. Tpicall also one or more tas dependent measures of operated device performance are among the ojectives. Thus, the ojectives are the target functional properties of the cam mechanism and the device operated it, and the constraints are imposed the restrictive design conditions. In the field of mechanical engineering, in addition to cam shape optimization discussed in this article, evolutionar shape optimization approach have een applied also for shape optimization of: a strain gauge load cell [6], a cantilever eam [7], a torque arm [7], a spherical pressure vessel [7] and a conical pivot earing journal [8]. In this paper, at first, the second order nonlinear differential equation of follower motion is otained with Euler-poison's equation in calculus of variation. For such a sstem, the governing equation is solved analticall in two separate intervals of cam rotation means of homotop analsis method and oundar conditions of each interval are regarded. Finall, the agreement of our results with numerical methods is illustrated. II. The governing equation The main parts of cam mechanism are shown in figure 1. Energ that dissipated in cam- follower for one around of cam rotation is: Manuscript received June 11, revised August 11 Copright 11 Praise Worth Prize S.r.l. - All rights reserved

2 π E = Mdθ (1) to calculus of variation, ( θ ) should satisf Euler- poison's equation [1]: η d η d η + = (9) dθ dθ Where η introduce the term inside integral. Put E into equation (9) instead term η get the second order nonlinear differential equation as follows: ( ) = (1) where 1 1 are defined in Appendix B. Fig. 1, Details of cam mechanism Where, M is the total moment of forces aout cam center, θ is cam rotation that varies from to π in one revolution. The frictional moment M can e written in terms of friction and contact forces as follows: M = F + N () Where is a function of θ and is derivative of with respect to θ. After writing equation of motion of an part of valve mechanism, all equations are merged and otained the general formulation of dissipated energ. Equation (3) represents the energ consumption E of cam-follower sstem, which constants B 1 B 5 in terms of specific characteristics of valve are given elow π ( B1+ B + B3)( + μ ) E = dθ (3) B 4 + B5 where l l B1 = Meg Mvg Ksr l1 l1 l l + Ksd1 Mag + Mtg l l 1 1 B M M M l I = v + + t + e l1 l l = B3 Ks l 1 B4 1 μ ω (4) (5) (6) = (7) μ B5 = (8) ξ where parameters which introduce in the aove equations are defined in Appendix A. Equation (3) is the same target function which we want to optimize it with calculus of variation. So that, we find such a function ( θ ) which minimize this equation. According II.1. Boundar conditions The oundar conditions of equation (1) are ased on availale limitations in follower motion and parameters of cam-follower mechanism. So that, according to, for this oject the following conditions must e satisfied: = r, = (11) ( ) ( ) ( π ) ( π ) = h+ r, = (1) Where r is the radius of ase circle of cam and h is the maximum displacement of follower. Note that, in order to solve differential equation (1) we require two oundar conditions; however equations (11) and (1) tell us that four aove oundar conditions must e satisfied. In according to these conditions, the sign of must e changed in interval[, π ]. Selection of this point is depending on cam length, radius of ase circle and etc. III. Homotop analsis method III.1. Basic ideas Consider the nonlinear differential equation in general form N ( θ ) = (13) where N is a differential operator and ( θ ) is a solution. Appling the HAM to solve it, we first need to construct the following famil of equations: ( 1 q) L φθ ( θ) = hqn φθ, q { } ( ) (14) where L is a properl selected auxiliar linear operator satisfing L ( ) = (15) h is an auxiliar parameter, and ( θ ) is an initial approximation. Oviousl, equation (14) gives φ ( θ,) = ( θ) (16) Copright 11 Praise Worth Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, n. 5

3 when q =. Similarl, when q = 1, equation (13) is the same as equation (15) so that we have φ θ,1 = θ (17) ( ) ( ) Suppose that equation (14) has solution φθ that converges for all q 1, for properl selected h and the auxiliar linear operator L. Suppose further that φ θ, q is infinitel differentiale at q =, that is ( ) φ θ, = 1,,3,... (18) q q= exists. Thus, as q increases from to 1, the solution φ( θ, q) of equation (14) varies continuousl from the initial approximation ( θ ) to the solution ( θ ) of the original equation (13). Clearl, equations (16) and (17) give an indirect relation etween the initial approximation ( θ ) and the general solution ( θ ). A direct relationship etween the two solutions is descried as follows. Consider the Maclaurin s series of φ( θ, q) aout q as φθ, q = φθ, + θ q, (19) where ( ) ( ) ( ) ( θ ) = 1 1 φ θ =! q q=, () Assume that the series (19) converges at q = 1. From equations (16), (17) and (19), we have the relationship ( θ ) ( θ) ( θ) = + (1) = 1 Liao [11] provides a general approach to derive the θ. Sustituting the series (1) governing equation of ( ) into equation (14) and equating the coefficient of the lie power of q, we get the th-order deformation equations ( θ ) χ ( θ) = ( θ) L hr where 1 1 d N φθ R ( θ ) = (3) 1 ( 1! ) dq q= and, 1 χ = (4) 1, It is ver important to emphasize that equation () is linear. If the first (-1)th-order approximations have een otained, then the right-hand side R ( θ ) will e otained. So, using the selected initial θ, θ,..., approximation ( ) 1 () θ, we can otain ( ) ( ) 1 one after the other in order. Therefore, according to equation (1), we convert the original nonlinear prolem into an infinite sequence of linear su-prolems governed equation (). We now consider equation (1) to appl homotop analsis method to otain analtical solution. III.. Application of the HAM According to equation (1), it is straightforward to use the set of ase functions [11] { θ =,1,,3,... }, θ π (5) to represent ( θ ), i.e., ( θ ) = = a θ (6) where a is coefficient. This provides us with a rule, called the rule of solution expression. This rule is important in the frame of the homotop analsis method, as shown later. We introduce 1( θ ), ( θ ) as solutions in first and second intervals of consideration. Based on the initial conditions (11) and (1) and the rule of solution expression descried (6), it is ovious that 1( θ) = r, ( θ) = r+ h, (7) is a good initial guess of 1( θ ), ( θ ). The homotop analsis method is ased on such continuous φθ, q, that, as the emedding parameter q variations ( ) increases from to 1, φθ varies from the initial guess ( θ ) to the exact solution ( θ ), respectivel. To ensure this under the rule of solution expression descried (6), one chooses such an auxiliar linear operator φ( θ, q) L φ( θ, q) = (8) θ that LC [ 1+ Cθ ] = (9) where C 1 and C are coefficients. Note that the rule of solution expression descried (6) plas an important role while determining the initial guess and the auxiliar linear operator L. Then, due to (1), one defines the nonlinear operator N φθ = φθθ + 3 1φθ + φφθ + 3φθ + 4φθ + 5φφθ + 6φθ + (3) 7 + φθ + φ + φ + φ θ, θθ whereφθ = φ θ φ = φ θ. Assume that h is properl chosen, therefore, at q = 1 we have ( θ ) ( θ ) =, (31) = The results at the th-order approximation are given : Copright 11 Praise Worth Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, n. 5

4 m = ( θ ) ( θ ), (3) Notice that at this case Equation L ( θ ) χ 1 ( θ) = hr ( θ) () is sujected to the initial conditions =, =, (33) ( ) ( ) ( π ) ( π ) 1 1 =, =, (34) For the auxiliar function H ( θ ) = 1 and for values r =.156 m and h =.6 m, we successivel otain ( θ) = hθ, 1 (35) π 1( θ) = h θ 1 Similarl, we can get 1 ( θ ),... and ( θ ),... It should e emphasized that our results contain the auxiliar parameter h, which provides us with a simple wa to ensure the convergence of our series solutions. θ is a power series of h. It is found that, at Note that ( ) the 8th-order approximation, ( ) converges to the same value in the region.6 h, as shown in - h curve. This is indeed figure for the so-called ( ) true: ( ) converges to.81 in the region.6 h. Fig. 3, Comparison of the homotop analsis approximation of follower rise when h =.4 with the Numerical solution. Smols: numerical solution; Solid line: 8th-order approximation Velocit and acceleration diagram vs. angle of cam rotation indicated in figures 4,5. As shown in these figures. the 8th-order approximation of (1) agrees well with the numerical results. Fig. 4, Comparison the homotop analsis approximation of follower velocit when h =.4 with the Numerical solution. Smols: numerical solution; Solid line: 8th-order approximation Fig., The 8th-order approximation of ( ) versus h The otained results are nearl identical with the results otained numericall using a fourth order Runge-Kutta method. figure 3 shows, the comparison etween the results otained the present solution and the numerical integration results. Fig. 5, Comparison the homotop analsis approximation of follower acceleration when h =.4 with the Numerical solution. Smols: numerical solution; Solid line: 8th-order approximation The most difficult prolem in shape optimization is to control the contact forces etween cam and follower. In Copright 11 Praise Worth Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, n. 5

5 case of high-speed cam mechanisms, minimizing the contact force to reduce dissipated energ is also one of the most important design targets, whereas, it is necessar to avoid this force to ecomes zero. The value Ks = 5 N / m satisfies two aove conditions and it prevents the jump of dis cam in the motions of cam follower mechanism [9]. IV. Conclusion In this wor, we presented a reliale algorithm ased on the HAM to solve the optimization equation of dis cam profile with initial conditions. Displacement, velocit and acceleration figures of follower are given to illustrate the validit and accurac of this procedure. The series solutions otained HAM contain the auxiliar parameter h. The validit of the method is ased on such an assumption that the series (1) converges at q = 1. It is the auxiliar parameter h which ensures that this assumption can e satisfied. In general, means of the so-called h -curve, it is straightforward to choose a proper value of h which ensures that the series solution is convergent. Thus, the auxiliar parameter h plas an important role within the frame of the homotop analsis method. Unlie all previous analtic techniques, such as HPM, we can adjust and control the convergence region of the solution series assigning h a proper value. Appendix A M e =.183 g; (A.1) M t =.1568 g; (A.) M v =. g; (A.3) M a =.15 g; (A.4) I =.1 g.m ; (A.5) r =.156 m; (A.6) ω = 1 rad/s; (A.7) μ =. ; (A.8) l1 =.1, l =.183 m; (A.9) K = 5 N/m; (A.1) s Appendix B = B B B μ (B.1) = 4B B B μ (B.) μ B3B5 5 4B3B5μ 6 B3B4μ 7 B1B4μ = B B + B B B (B.3) = (B.4) = (B.5) = (B.6) = (B.7) = BBB + BBμ (B.8) = 3B B B μ (B.9) B3B5 μ 1 B B5 = B B B B B μ (B.1) = (B.11) = (B.1) References [1] F. Y. Chen, Mechanics and Design of Cam Mechanisms (New Yor: Pergamon Press, 198). [] J. Angeles, C. S. Lopez-Cajun, Optimization of Cam Mechanisms (Dordrecht: Kluwer Academic Pulishers, 1991). [3] R. Norton, Modern Kinematics: Developments in the Last Fort Years (New Yor: Wile, 1993). [4] D. E. Golderg, Genetic algorithms in search, optimization and machine learning Reading (MA: Addison-Wesle, 1989). [5] J. T. Alander, J. Lampinen, A distriuted implementation of genetic algorithm for cam shape optimization (Civil-Comp Press: Edinurgh, Scotland, 1997). [6] G. M. Roinson, Genetic algorithm optimization of load cell geometr finite element analsis, Ph.D. Thesis, Department of Electrical, Electronic and Information Engineering, Cit Universit, Measurement and Instrumentation Centre,, School of Engineering, London (UK), [7] R. A. Richards, Zeroth-order shape optimization utilizing a learning classifier sstem, PhD Thesis, Mechanical Engineering Department, Stanford Universit, [8] G. Vancsa, T. Bercse, P. Hora, Shape optimization ased on genetic algorithms, Proceedings of International Conference on Engineering Design ICED 97, Finland, Tampere, 1997, pp [9] S. Hasanifard, optimum design of dis cam profile, MS thesis, Department of mechanical engineering, Tariz Universit, Iran,. [1] L. Elsgolts, Differential equation and the calculus of variation (Mir pulishers: Moscow, 1973). [11] S. J. Liao, Beond perturation: introduction to the homotop analsis method (Boca Raton: Chapman & Hall/CRC Press, 3). Authors information 1 Ph.D student, Department of Mechanical Engineering, Shahid Chamran Universit, Ahvaz, Iran. Associate Professor, Department of Mechanical Engineering, Shahid Chamran Universit, Ahvaz, Iran. Hamid M. Sedighi was orn in 1983 in Iran. He is currentl a fourth ear Ph.D student woring under the supervision of Dr. Kourosh H. Shirazi in mechanical engineering at the Shahid Chamran Universit of Ahvaz in Iran. He otained his M.S. degree (7) from the Shahid Chamran Universit of Ahvaz and his undergraduate B.S. degree (5) from the Shiraz Universit. His general academic areas of interest include the Mathematics, Nonlinear Dnamical Sstems, Elasticit and Machine Design. As a teacher and a teaching assistant, he has held various positions at SCU and IAU Ahvaz. The topic of PhD research is Analsis of nonlinear dnamical ehavior of multilaer eams with interlaer slip. Copright 11 Praise Worth Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, n. 5

6 Kourosh H. Shirazi was orn in 1969 in Iran. After studing in Universit of Science and Technolog he received a B.S. degree in mechanical engineering - solid mechanics in 199. He pursued his stud in mechanical engineering in Amirair Universit (Tehran Poltechnic) and received a M.S. degree in 1993 and PH.D. degree in. He started his wor as a full time facult memer in Shahid Chamran Universit since till present. During his wor he offered 7 courses such as Linear Control Theor, Design of Chassis and Bod of Vehicle and Mechanisms Design in undergraduate program and 6 courses such as Advanced Dnamics, Advanced Mathematics and Nonlinear Dnamics in graduate program. His research interest is Kinematic of Mechanisms, Vehicle Dnamics and Chaotic Dnamics. He is a memer of ASME and SAE. Copright 11 Praise Worth Prize S.r.l. - All rights reserved International Review of Mechanical Engineering, Vol. 5, n. 5