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Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations of Multimesh Gears Transmission

Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations... Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 195642, 6 pages doi:10.1155/2011/195642 Research Article Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations of Multimesh Gears Transmission Alexandre Carbonelli, Joel Perret-Liaudet, Emmanuel Rigaud, and Alain Le Bot Laboratoire de Tribologie et Dynamique des Syst`emes, UMR CNRS 5513, Ecole Centrale de Lyon, Universit`ede Lyon, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France Correspondence should be addressed to Alexandre Carbonelli, alexandre.carbonelli@ec-lyon.fr Received 12 January 2011; Accepted 13 April 2011 Academic Editor: Snehashish Chakraverty Copyright © 2011 Alexandre Carbonelli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of this work is to present the great performance of the numerical algorithm of Particle Swarm Optimization applied to find the best teeth modifications for multimesh helical gears, which are crucial for the static transmission error (STE). Indeed, STE fluctuation is the main source of vibrations and noise radiated by the geared transmission system. The microgeometrical parameters studied for each toothed wheel are the crowning, tip reliefs and start diameters for these reliefs. Minimization of added up STE amplitudes on the idler gear of a three-gear cascade is then performed using the Particle Swarm Optimization. Finally, robustness of the solutions towards manufacturing errors and applied torque is analyzed by the Particle Swarm algorithm to access to the deterioration capacity of the tested solution. 1. Introduction (ii) start relief diameter Φ ,that is, the diameter at rel,i which the material starts to be removed until the The STE under load [1] isdefinedasthe difference between tooth tip. Linear or parabolic corrections can be the actual position of the driven gear and its theoretical posi- done, tion for a very slow rotation velocity and for a given applied (iii) added up crowning centered on the active tooth torque. Its characteristics depend on the instantaneous sit- width C . β,i/ j uations of the meshing tooth pairs. Under load at very low Many authors [3–11] worked on the optimization of speed (static transmission error), these situations result from tooth modifications in simple mesh systems. Only few of tooth deflections, tooth surface modifications, and manu- them [12–14] considered multimesh systems as cascade of facturing errors. Under operating conditions, STE generates gears where idler gear modifications affect two meshes. dynamic mesh force transmitted to shafts, bearings, and to In this paper, the application is done on a cascade of three the crankcase. The vibratory state of the crankcase is the helical gears, displayed on Figure 2, for a total of 8 parameters main source of the radiated noise [2]. To reduce the radiated (tip relief and start diameter for the relief for each gear, and noise, the peak-to-peak amplitude of the STE fluctuation added up crowning for a pair of meshing gears). Multipa- needs to be minimized by the mean of tooth modifications. It rameter optimization can easily become a difficult task if consists in micro-geometrical modifications listed below and the algorithm used is not well adapted. We will show that displayed on Figure 1: the Particle Swarm Optimization (PSO) fits efficiently with (i) tip relief magnitude x ,thatis, theamountof ma- that kind of problematic. Indeed, it permits to select a set rel,i terial removed on the tooth tip, of solutions more or less satisfying in the studied torque 2 Advances in Acoustics and Vibration rel,i rel,i β,i Figure 1: Crowning C ,tip relief x , and start relief diameter Φ . β,i/ j rel,i rel,i range. Moreover, the robustness of the optimized solutions is studied regarding large manufacturing errors, lead, and involute alignment deviations. An additional difficulty arises because the modifications performed have to be efficient on a large torque range. The dispersion associated is the source of the strong variability of the dynamic behavior and of the noise radiated from geared systems (sometimes up to 10 dB [15, 16]). 2. Calculation of Static Transmission Error The calculation of STE is relatively classical [17]. For each position θ of the driving gear, a kinematical analysis of the mesh allows determination of the theoretical contact line on the mating surfaces of gearing teeth within the plane of action. Equation system which describes the elastostatic defor- mations of the teeth can be written as follows [17]: Figure 2: Cascade of the 3 helical gears studied: 50 teeth/72 teeth/54 teeth. u,F H (ω = 0) · F = δ(θ) − e − hertz(F), (1) F = F. for tip reliefs and parabolic correction for the crownings. All The following data are needed to perform this interpola- the modifications allow to reduce the STE fluctuation. The tion: most influent parameter is the tip relief magnitude. Indeed, (i) initial gaps e between the teeth: they are function of removing an amount of material on the tooth tip permits to the geometry defects and the tooth modifications, make up for the advance or late position of the tooth induced u,F by elastic deformations. (ii) compliance matrix H , of the teeth coming from in- For the robustness study, the manufacturing errors are terpolation functions calculated by a Finite Element also considered and displayed on Figure 3.The manufactur- model of elastostatic deformations, ing is not directly parameters of the optimization but as they (iii) Hertz deformations hertz, calculated according to have an effect on the STE fluctuation they must be considered Hertz theory. in the robustness study. The calculation of the actual approach of distant teeth δ (i) Lead deviation: f = f + f , Hβ,i/ j Hβ,i Hβ,j on the contact line for each position θ permits to access the (ii) Involute alignment deviation: f and f . gα,i gα,j time variation of STE and its peak-to-peak amplitude E , pp as a function of the applied torque (or the transmitted load Afitness function f to minimize is defined as the in- F) and the teeth modifications. We chose linear correction tegral of STE peak-to-peak amplitude over torque range Advances in Acoustics and Vibration 3 f < 0 gα p(t − 1) Hβ,i V (t) V (t − 1) Figure 4: Particle Swarm algorithm representation. speed to search for the “best location,” according to a defined criterion of optimization. It is commonly called the Theoretical profile fitness function which has to be maximized or minimized Actual profile depending on the problem. For each iteration and each particle, a new speed and so Figure 3: Involute alignment deviation f and lead deviation f . gα Hβ a new position is reevaluated considering: (i) the current particle velocity V (t − 1), [C − C ] approximated by Gaussian quadrature with 3 min max (ii) its best position p , points. (iii) the best position of neighbors p . i=3 max The algorithm can thus be wrapped up to the system of f = E (C)dC −→ a E (C ). (2) i,j pp i pp i C (5)and Figure 4: min i=1 V(t) = ϕ V(t − 1) + ϕ A p − p(t − 1) 0 1 1 i The fitness function of the whole cascade is then f = f + f . (3) + ϕ A p − p(t − 1) , (5) i,j k,j 2 2 g We have thereby 8 parameters for the optimization leading p(t) = p(t − 1) + V(t − 1). to a combinatorial explosion. Meta-heuristic methods allow A and A represent a random vector of number between 1 2 an efficient optimization, and we chose the Particle Swarm 0 and 1 and the parameters of these equations are taken Optimization [18]. Obviously in that kind of problematic, following Trelea and Clerc [19–21]: ϕ = 0.729 and ϕ = 0 1 theaim cannot betoaccess to theoptimum optimorum ϕ = 1.494. but only different local minima whose performances can be quickly estimated over the torque range by a home-built gain function 4. Robustness Study First the tolerance range D of a solution x has been defined, 0 0 G = 10 log ,(4) ref using a vector Δx ={Δx , Δx ,... , Δx },which takes in 1 2 N account the parameters variability. The gears studied have a where f corresponds to the value of the fitness function for ref precision class 7 (ISO 1328). Moreover, the manufacturing a standard nonoptimized gear. errors distribution is considered to be uniform over the range, which is the worst possible case in. Lead and involute alignment deviations and torque variation are associated in a 3. Particle Swarm Algorithm 14-dimensionnal vector as following: The principle of this method is based on the stigmergic behavior of a population, being in constant communication Δx = ΔX , ΔΦ , f , ΔC , f , ΔX , ΔΦ , d´ep,i d´ep,i gα,i β,i/ j Hβ, i/ j d´ep,j d´ep,j and exchanging information about their location in a given space [18]. Typically bees, ants, or termites are animals f ,... , ΔC , f , ΔX , ΔΦ , f , ΔC , gα,j Hβ,l/ j d´ep,l d´ep,l gα,l β,l/ j functioning that way. In our general case, we just consider (6) particles which are located in an initial and random position in a hyperspace built according to the different optimization where i, j,and l correspond to, respectively, the gears with parameters. They will then change their position and their 50, 72, and 54 teeth. 4 Advances in Acoustics and Vibration Table 1: Parameters ranges. Number of teeth Z = 54 Z = 72 Z = 50 Tip relief magnitude and tolerance [μm] [15–150] ± 15 [0–150] ± 15 [15–150] ± 15 Start relief diameter and tolerance [mm] [230–241] ± 0.46 [200–215] ± 0.46 [153–168] ± 0.40 [8–40] ±8— Added up crowning and tolerance [μm] — [8–40] ± 8 0 ± 32 — Lead deviation and tolerance [μm] —0 ± 32 Involution alignment dev. and tolerance [μm] 0 ± 12 0 ± 12 0 ± 12 50 X rel,54 0.8 rel,54 30 ϕ rel,50 0.6 0.4 0.2 0 100 200 300 400 500 600 rel,50 0 C β54/72 Torque (N.m) Reference S1 S3 S2 S5 S4 Figure 5: Optimized and reference solutions versus applied torque rel,72 β,72/50 - - - - torque range boundaries. rel,72 Then, the tolerance range D can be written as Reference S3 S1 S4 D = x : x ∈ R | x − Δx < x < x + Δx . (7) 0 0 0 S2 S5 Figure 6: Optimized parameters of the solutions. Contrary to the case studied by Sundaresan et al. [22], the robustness study concerns micro-geometrical modifications instead of macrogeometrical parameters (i.e., teeth number). The tolerance ranges are moreover noticeably larger than The PSO calculations have been performed using a pop- the ones considered by Bonori et al. [10], especially for ulation of 25 particles and stopped when a precision of the tip relief modifications. The fitness function cannot be −2 10 μrad for peak-to-peak amplitude E is reached. The al- pp assumed monotonic and the study of the extreme boundaries gorithm stops the calculation when no improvement is found of the problem is not sufficient. The PSO is then used to 50 times successively. All the following results have converged locate the maximum of the fitness function in the hyper- after 250 to 400 iterations. That corresponds to 7500 to space D in order to analyze robustness of the solutions. The 14 0, 10000 evaluations of the fitness function (instead of 10 new values for the parameters which maximize the fitness for a Monte-Carlo experiment). Table 1 lists the parameters function define the “degenerated solution,” noted x : ranges. In order to illustrate the optimization process, Figure 5 ( ) ( ) x ∈ D , f x = max f x | x ∈ D . (8) d 0 d 0 displays 5 selected solutions—S1 to S5—corresponding to 5 With this additional criterion, optimal solution corresponds local minima among the computed ones which all obviously to the less deteriorated rather than the minimal E . pp are better than the reference solution in terms of minimal E . Figure 6 displays the optimized parameters of the solu- pp tions rescaled in function of their extremum values. 5. Results According to the gain function (4), we can easily pick up The cascade of three helical gears has to be optimized for the best solutions of the selected ones. Following the results torques from 100 Nm up to 500 Nm. A reference solution, listed in Table 2, solution S5, which provides −4.2 dB of im- with standard and not optimized tooth modifications, is used provement compared to the reference solution, should be se- to emphasize the benefits of the Particle Swarm optimization. lected. STE (μrad) Advances in Acoustics and Vibration 5 Table 2: Gain of the computed optimal solutions compared to the Table 3: Gain of the degenerated solutions compared to optimal reference solution. solutions. Configuration Gain G [dB] Configuration Gain G [dB] 0 1 S1 −1.6 Reference +6.7 S2 −1.9 S1 +11.3 S3 −3.3 S2 +6.0 S4 −3.7 S3 +6.1 S5 −4.2 S4 +2.3 S5 +11.3 200 Table 4: Gain of the degenerated solutions compared to the refer- ence degenerated solution. Configuration Gain G [dB] S1 +2.8 S2 −2.6 0 S3 −4.2 0 100 200 300 400 500 600 S4 −6.2 Torque (N.m) S5 −0.4 Reference S1 S2 S3 S5 S4 6. Conclusion Figure 7: Degenerated solutions versus applied torque - - - - Torque range boundaries. Optimization with an efficient heuristic method (Particle Swarm) has been done to determinate optimized parameters of a multimesh problem. The algorithm permits the gath- ering of many solutions which all lead to really satisfying Figure 7 displays the deteriorated solutions. results over the torque range studied thank to an integration The first analysis of the deteriorating capacity of the of STE peak-to-peak amplitude by Gaussian quadrature. solutions can be done using gain function (9)and listing Finally, a robustness criterion has been defined based on the results in Table 3: deteriorating capacity of the solutions which permits to do a non−deteriorated more accurate choice about the optimal tooth modifications. G = 10 log . (9) deteriorated Indeed, there are many ways of estimating the robustness of the solutions. In some industrial point of view, a solution The deteriorated reference solution has a gain of +6.7 dB which islessefficient than another but much more robust compared with the initial reference solution. The solution should be preferably chosen. S5 is worse considering the gain function (9), but its fitness function value is still less than the deteriorated reference so- lution one. On the other hand, the previous selected solution Acknowledgments S4 appears as the best one with only +2.3 dB of deterioration in the gain function (9)sense. This work has been supported by ANR (National Research The second analysis of the deteriorating capacity of the Agency, contract number: ANR-08-VTT-007-02), ADEME solutions can be done using gain function (10)and listing (French Environment and Energy Management Agency), results in Table 4: and Lyon Urban Trucks&Bus competitiveness cluster. The authors acknowledge gratefully this support and especially S , deteriorated G = 10 log . (10) 10 thank Denis BARDAY from Renault Trucks Company for his ref, deteriorated inestimable help. The solution S1 emphasizes the importance of consider- ing the deteriorating capacity. Indeed, although the optimal References solution brings an improvement compared to the initial reference solution, it is likely to be less efficient taking in [1] L.S. Harris, “Dynamic loads on the teeth of spur gears,” in account the possible manufacturing errors. The previous Proceedings of the Institution of Mechanical Engineers, vol. 172, choice has to be reconsidered. On the other hand, the solu- pp. 87–112, 1958. tion S4 provides a good improvement of −3.7 dB compared [2] D. B. Welbourn, “Fundamental knowledge of gear noise— to the reference solution and is quite robust as a gain of asurvey,” in Proceedings of the Conference on Noise and −6.2 dB is observed if S4 deteriorated solution is compared Vibrations of Engines and Transmissions, vol. C177/79, pp. 9– with the deteriorated reference solution. 29, Cranfield Institute of Technology, July 1979. STE (μrad) 6 Advances in Acoustics and Vibration [3] M.S.Tavakoli and D. R. Houser,“Optimumprofile modifica- [18] R. C. Eberhart and J. Kennedy, “A new optimizer using tions for the minimization of static transmission errors of spur particle swarm theory,” in Proceedings of the 6th International gears,” Journal of Mechanism, Transmissions, and Automation Symposium on Micro Machine and Human Science, pp. 39–43, in Design, vol. 108, no. 1, pp. 86–95, 1986. IEEE Service Center, Nagoya, Japan, October 1995. [4] A. Kahraman and G. W. Blankenship, “Effect of involute tip [19] I. C. Trelea, “The particle swarm optimization algorithm: relief on dynamic response of spur gear pairs,” Journal of convergence analysis and parameter selection,” Information Mechanical Design, vol. 121, no. 2, pp. 313–315, 1999. Processing Letters, vol. 85, no. 6, pp. 317–325, 2003. [5] K.Umezawa,H. Houjoh, S. Matsumura, and S.Wang, [20] M. Clerc, “The swarm and the queen: towards a deterministic “Investigation of the dynamic behavior of a helical gear and adaptive particle swarm optimization,” in Proceedings system dynamics of gear pairs with bias modification,” in of the International Conference on Evolutionary Computation Proceedings of the 4th World Congress and Gearing and Power (ICEC ’99), pp. 1951–1957, IEEE, Washington, DC, USA, Transmissions, vol. 3, pp. 1981–1990, Paris, France, 1999. [6] M. Beghini et al., “A method to define profile modification of [21] M. Clerc and J. Kennedy, “The particle swarm—explosion, spur gear and minimize the transmission error,” in Proceedings stability, and convergence in a multidimensional complex of the Fall Technical Meeting of the American Gear Manufactur- space,” IEEE Transactions on Evolutionary Computation,vol. 6, ers Association (AGMA ’04), Milwaukee, Wis, USA, October no. 1, pp. 58–73, 2002. [22] S. Sundaresan, K. Ishii, and D. R. Houser, “A robust opti- [7] M. Umeyama, “Effects of deviation of tooth surface errors of a mization procedure with variations on design variables and helical gear pair on the transmission error,” Transactions of the constraints,” Engineering Optimization, vol. 24, no. 2, pp. 110– Japan Society of Mechanical Engineers, Part C, vol. 61, no. 587, 118, 1995. pp. 3101–3107, 1995. [8] R. Guilbault, C. Gosselin, and L. Cloutier, “Helical gears, effects of tooth deviations and tooth modifications on load sharing and fillet stresses,” Journal of Mechanical Design, vol. 128, no. 2, pp. 444–456, 2006. [9] A. Kahraman, P. Bajpai, and N. E. Anderson, “Influence of tooth profile deviations on helical gear wear,” Journal of Mechanical Design, vol. 127, no. 4, pp. 656–663, 2005. [10] G. Bonori, M. Barbieri, and F. Pellicano, “Optimum profile modifications of spur gears by means of genetic algorithms,” Journal of Sound and Vibration, vol. 313, no. 3–5, pp. 603–616, [11] S. Kurokawa, Y. Ariura, and M. Ohtahara, “Transmission errors of cylindrical gears under load—influence of tooth profile modification and tooth deflection,” in Proceedings of the 7th International Power Transmission and Gearing Conference, DE 88, pp. 213–217, American Society of Mechanical Engi- neers, Design Engineering Division, San Diego, Calif, USA, October 1996. [12] T. Erltenel and R. G. Parker, “A static and dynamic model for three-dimensional, multi-mesh gear systems,” in Proceed- ings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC ’05), vol. 5, pp. 945–956, Long Beach, Calif, USA, September 2005. [13] H. Vinayak, R. Singh, and C. Padmanabhan, “Linear dynamic analysis of multi-mesh transmissions containing external, rigid gears,” Journal of Sound and Vibration, vol. 185, no. 1, pp. 1–32, 1995. [14] J. Lin and R. G. Parker, “Mesh stiffness variation instabilities in two-stage gear systems,” Journal of Vibration and Acoustics, vol. 124, no. 1, pp. 68–76, 2002. [15] N. Driot, E. Rigaud, J.Sabot,and J. Perret-Liaudet, “Allocation of gear tolerances to minimize gearbox noise variability,” Acta Acustica United with Acustica, vol. 87, no. 1, pp. 67–76, 2001. [16] N. Driot and J. Perret-Liaudet, “Variability of modal behavior in terms of critical speeds of a gear pair due to manufacturing errors and shaft misalignments,” Journal of Sound and Vibra- tion, vol. 292, no. 3–5, pp. 824–843, 2006. [17] E. Rigaud and D. Barday, “Modelling and analysis of static transmission error of gears: effect of wheel body deformation and interactions between adjacent loaded teeth,” M´ecanique Industrielle et Mat´eriaux, vol. 51, no. 2, pp. 58–60, 1998. 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Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations of Multimesh Gears Transmission

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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2011, Article ID 195642, 6 pages doi:10.1155/2011/195642 Research Article Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations of Multimesh Gears Transmission Alexandre Carbonelli, Joel Perret-Liaudet, Emmanuel Rigaud, and Alain Le Bot Laboratoire de Tribologie et Dynamique des Syst`emes, UMR CNRS 5513, Ecole Centrale de Lyon, Universit`ede Lyon, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France Correspondence should be addressed to Alexandre Carbonelli, alexandre.carbonelli@ec-lyon.fr Received 12 January 2011; Accepted 13 April 2011 Academic Editor: Snehashish Chakraverty Copyright © 2011 Alexandre Carbonelli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aim of this work is to present the great performance of the numerical algorithm of Particle Swarm Optimization applied to find the best teeth modifications for multimesh helical gears, which are crucial for the static transmission error (STE). Indeed, STE fluctuation is the main source of vibrations and noise radiated by the geared transmission system. The microgeometrical parameters studied for each toothed wheel are the crowning, tip reliefs and start diameters for these reliefs. Minimization of added up STE amplitudes on the idler gear of a three-gear cascade is then performed using the Particle Swarm Optimization. Finally, robustness of the solutions towards manufacturing errors and applied torque is analyzed by the Particle Swarm algorithm to access to the deterioration capacity of the tested solution. 1. Introduction (ii) start relief diameter Φ ,that is, the diameter at rel,i which the material starts to be removed until the The STE under load [1] isdefinedasthe difference between tooth tip. Linear or parabolic corrections can be the actual position of the driven gear and its theoretical posi- done, tion for a very slow rotation velocity and for a given applied (iii) added up crowning centered on the active tooth torque. Its characteristics depend on the instantaneous sit- width C . β,i/ j uations of the meshing tooth pairs. Under load at very low Many authors [3–11] worked on the optimization of speed (static transmission error), these situations result from tooth modifications in simple mesh systems. Only few of tooth deflections, tooth surface modifications, and manu- them [12–14] considered multimesh systems as cascade of facturing errors. Under operating conditions, STE generates gears where idler gear modifications affect two meshes. dynamic mesh force transmitted to shafts, bearings, and to In this paper, the application is done on a cascade of three the crankcase. The vibratory state of the crankcase is the helical gears, displayed on Figure 2, for a total of 8 parameters main source of the radiated noise [2]. To reduce the radiated (tip relief and start diameter for the relief for each gear, and noise, the peak-to-peak amplitude of the STE fluctuation added up crowning for a pair of meshing gears). Multipa- needs to be minimized by the mean of tooth modifications. It rameter optimization can easily become a difficult task if consists in micro-geometrical modifications listed below and the algorithm used is not well adapted. We will show that displayed on Figure 1: the Particle Swarm Optimization (PSO) fits efficiently with (i) tip relief magnitude x ,thatis, theamountof ma- that kind of problematic. Indeed, it permits to select a set rel,i terial removed on the tooth tip, of solutions more or less satisfying in the studied torque 2 Advances in Acoustics and Vibration rel,i rel,i β,i Figure 1: Crowning C ,tip relief x , and start relief diameter Φ . β,i/ j rel,i rel,i range. Moreover, the robustness of the optimized solutions is studied regarding large manufacturing errors, lead, and involute alignment deviations. An additional difficulty arises because the modifications performed have to be efficient on a large torque range. The dispersion associated is the source of the strong variability of the dynamic behavior and of the noise radiated from geared systems (sometimes up to 10 dB [15, 16]). 2. Calculation of Static Transmission Error The calculation of STE is relatively classical [17]. For each position θ of the driving gear, a kinematical analysis of the mesh allows determination of the theoretical contact line on the mating surfaces of gearing teeth within the plane of action. Equation system which describes the elastostatic defor- mations of the teeth can be written as follows [17]: Figure 2: Cascade of the 3 helical gears studied: 50 teeth/72 teeth/54 teeth. u,F H (ω = 0) · F = δ(θ) − e − hertz(F), (1) F = F. for tip reliefs and parabolic correction for the crownings. All The following data are needed to perform this interpola- the modifications allow to reduce the STE fluctuation. The tion: most influent parameter is the tip relief magnitude. Indeed, (i) initial gaps e between the teeth: they are function of removing an amount of material on the tooth tip permits to the geometry defects and the tooth modifications, make up for the advance or late position of the tooth induced u,F by elastic deformations. (ii) compliance matrix H , of the teeth coming from in- For the robustness study, the manufacturing errors are terpolation functions calculated by a Finite Element also considered and displayed on Figure 3.The manufactur- model of elastostatic deformations, ing is not directly parameters of the optimization but as they (iii) Hertz deformations hertz, calculated according to have an effect on the STE fluctuation they must be considered Hertz theory. in the robustness study. The calculation of the actual approach of distant teeth δ (i) Lead deviation: f = f + f , Hβ,i/ j Hβ,i Hβ,j on the contact line for each position θ permits to access the (ii) Involute alignment deviation: f and f . gα,i gα,j time variation of STE and its peak-to-peak amplitude E , pp as a function of the applied torque (or the transmitted load Afitness function f to minimize is defined as the in- F) and the teeth modifications. We chose linear correction tegral of STE peak-to-peak amplitude over torque range Advances in Acoustics and Vibration 3 f < 0 gα p(t − 1) Hβ,i V (t) V (t − 1) Figure 4: Particle Swarm algorithm representation. speed to search for the “best location,” according to a defined criterion of optimization. It is commonly called the Theoretical profile fitness function which has to be maximized or minimized Actual profile depending on the problem. For each iteration and each particle, a new speed and so Figure 3: Involute alignment deviation f and lead deviation f . gα Hβ a new position is reevaluated considering: (i) the current particle velocity V (t − 1), [C − C ] approximated by Gaussian quadrature with 3 min max (ii) its best position p , points. (iii) the best position of neighbors p . i=3 max The algorithm can thus be wrapped up to the system of f = E (C)dC −→ a E (C ). (2) i,j pp i pp i C (5)and Figure 4: min i=1 V(t) = ϕ V(t − 1) + ϕ A p − p(t − 1) 0 1 1 i The fitness function of the whole cascade is then f = f + f . (3) + ϕ A p − p(t − 1) , (5) i,j k,j 2 2 g We have thereby 8 parameters for the optimization leading p(t) = p(t − 1) + V(t − 1). to a combinatorial explosion. Meta-heuristic methods allow A and A represent a random vector of number between 1 2 an efficient optimization, and we chose the Particle Swarm 0 and 1 and the parameters of these equations are taken Optimization [18]. Obviously in that kind of problematic, following Trelea and Clerc [19–21]: ϕ = 0.729 and ϕ = 0 1 theaim cannot betoaccess to theoptimum optimorum ϕ = 1.494. but only different local minima whose performances can be quickly estimated over the torque range by a home-built gain function 4. Robustness Study First the tolerance range D of a solution x has been defined, 0 0 G = 10 log ,(4) ref using a vector Δx ={Δx , Δx ,... , Δx },which takes in 1 2 N account the parameters variability. The gears studied have a where f corresponds to the value of the fitness function for ref precision class 7 (ISO 1328). Moreover, the manufacturing a standard nonoptimized gear. errors distribution is considered to be uniform over the range, which is the worst possible case in. Lead and involute alignment deviations and torque variation are associated in a 3. Particle Swarm Algorithm 14-dimensionnal vector as following: The principle of this method is based on the stigmergic behavior of a population, being in constant communication Δx = ΔX , ΔΦ , f , ΔC , f , ΔX , ΔΦ , d´ep,i d´ep,i gα,i β,i/ j Hβ, i/ j d´ep,j d´ep,j and exchanging information about their location in a given space [18]. Typically bees, ants, or termites are animals f ,... , ΔC , f , ΔX , ΔΦ , f , ΔC , gα,j Hβ,l/ j d´ep,l d´ep,l gα,l β,l/ j functioning that way. In our general case, we just consider (6) particles which are located in an initial and random position in a hyperspace built according to the different optimization where i, j,and l correspond to, respectively, the gears with parameters. They will then change their position and their 50, 72, and 54 teeth. 4 Advances in Acoustics and Vibration Table 1: Parameters ranges. Number of teeth Z = 54 Z = 72 Z = 50 Tip relief magnitude and tolerance [μm] [15–150] ± 15 [0–150] ± 15 [15–150] ± 15 Start relief diameter and tolerance [mm] [230–241] ± 0.46 [200–215] ± 0.46 [153–168] ± 0.40 [8–40] ±8— Added up crowning and tolerance [μm] — [8–40] ± 8 0 ± 32 — Lead deviation and tolerance [μm] —0 ± 32 Involution alignment dev. and tolerance [μm] 0 ± 12 0 ± 12 0 ± 12 50 X rel,54 0.8 rel,54 30 ϕ rel,50 0.6 0.4 0.2 0 100 200 300 400 500 600 rel,50 0 C β54/72 Torque (N.m) Reference S1 S3 S2 S5 S4 Figure 5: Optimized and reference solutions versus applied torque rel,72 β,72/50 - - - - torque range boundaries. rel,72 Then, the tolerance range D can be written as Reference S3 S1 S4 D = x : x ∈ R | x − Δx < x < x + Δx . (7) 0 0 0 S2 S5 Figure 6: Optimized parameters of the solutions. Contrary to the case studied by Sundaresan et al. [22], the robustness study concerns micro-geometrical modifications instead of macrogeometrical parameters (i.e., teeth number). The tolerance ranges are moreover noticeably larger than The PSO calculations have been performed using a pop- the ones considered by Bonori et al. [10], especially for ulation of 25 particles and stopped when a precision of the tip relief modifications. The fitness function cannot be −2 10 μrad for peak-to-peak amplitude E is reached. The al- pp assumed monotonic and the study of the extreme boundaries gorithm stops the calculation when no improvement is found of the problem is not sufficient. The PSO is then used to 50 times successively. All the following results have converged locate the maximum of the fitness function in the hyper- after 250 to 400 iterations. That corresponds to 7500 to space D in order to analyze robustness of the solutions. The 14 0, 10000 evaluations of the fitness function (instead of 10 new values for the parameters which maximize the fitness for a Monte-Carlo experiment). Table 1 lists the parameters function define the “degenerated solution,” noted x : ranges. In order to illustrate the optimization process, Figure 5 ( ) ( ) x ∈ D , f x = max f x | x ∈ D . (8) d 0 d 0 displays 5 selected solutions—S1 to S5—corresponding to 5 With this additional criterion, optimal solution corresponds local minima among the computed ones which all obviously to the less deteriorated rather than the minimal E . pp are better than the reference solution in terms of minimal E . Figure 6 displays the optimized parameters of the solu- pp tions rescaled in function of their extremum values. 5. Results According to the gain function (4), we can easily pick up The cascade of three helical gears has to be optimized for the best solutions of the selected ones. Following the results torques from 100 Nm up to 500 Nm. A reference solution, listed in Table 2, solution S5, which provides −4.2 dB of im- with standard and not optimized tooth modifications, is used provement compared to the reference solution, should be se- to emphasize the benefits of the Particle Swarm optimization. lected. STE (μrad) Advances in Acoustics and Vibration 5 Table 2: Gain of the computed optimal solutions compared to the Table 3: Gain of the degenerated solutions compared to optimal reference solution. solutions. Configuration Gain G [dB] Configuration Gain G [dB] 0 1 S1 −1.6 Reference +6.7 S2 −1.9 S1 +11.3 S3 −3.3 S2 +6.0 S4 −3.7 S3 +6.1 S5 −4.2 S4 +2.3 S5 +11.3 200 Table 4: Gain of the degenerated solutions compared to the refer- ence degenerated solution. Configuration Gain G [dB] S1 +2.8 S2 −2.6 0 S3 −4.2 0 100 200 300 400 500 600 S4 −6.2 Torque (N.m) S5 −0.4 Reference S1 S2 S3 S5 S4 6. Conclusion Figure 7: Degenerated solutions versus applied torque - - - - Torque range boundaries. Optimization with an efficient heuristic method (Particle Swarm) has been done to determinate optimized parameters of a multimesh problem. The algorithm permits the gath- ering of many solutions which all lead to really satisfying Figure 7 displays the deteriorated solutions. results over the torque range studied thank to an integration The first analysis of the deteriorating capacity of the of STE peak-to-peak amplitude by Gaussian quadrature. solutions can be done using gain function (9)and listing Finally, a robustness criterion has been defined based on the results in Table 3: deteriorating capacity of the solutions which permits to do a non−deteriorated more accurate choice about the optimal tooth modifications. G = 10 log . (9) deteriorated Indeed, there are many ways of estimating the robustness of the solutions. In some industrial point of view, a solution The deteriorated reference solution has a gain of +6.7 dB which islessefficient than another but much more robust compared with the initial reference solution. The solution should be preferably chosen. S5 is worse considering the gain function (9), but its fitness function value is still less than the deteriorated reference so- lution one. On the other hand, the previous selected solution Acknowledgments S4 appears as the best one with only +2.3 dB of deterioration in the gain function (9)sense. This work has been supported by ANR (National Research The second analysis of the deteriorating capacity of the Agency, contract number: ANR-08-VTT-007-02), ADEME solutions can be done using gain function (10)and listing (French Environment and Energy Management Agency), results in Table 4: and Lyon Urban Trucks&Bus competitiveness cluster. The authors acknowledge gratefully this support and especially S , deteriorated G = 10 log . 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