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

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Introduction
The STE under load [1] is defined as the difference between the actual position of the driven gear and its theoretical position for a very slow rotation velocity and for a given applied torque.Its characteristics depend on the instantaneous situations of the meshing tooth pairs.Under load at very low speed (static transmission error), these situations result from tooth deflections, tooth surface modifications, and manufacturing errors.Under operating conditions, STE generates dynamic mesh force transmitted to shafts, bearings, and to the crankcase.The vibratory state of the crankcase is the main source of the radiated noise [2].To reduce the radiated noise, the peak-to-peak amplitude of the STE fluctuation needs to be minimized by the mean of tooth modifications.It consists in micro-geometrical modifications listed below and displayed on Figure 1: (i) tip relief magnitude x rel,i , that is, the amount of material removed on the tooth tip, (ii) start relief diameter Φ rel,i , that is, the diameter at which the material starts to be removed until the tooth tip.Linear or parabolic corrections can be done, (iii) added up crowning centered on the active tooth width C β,i/ j .
In this paper, the application is done on a cascade of three helical gears, displayed on Figure 2, for a total of 8 parameters (tip relief and start diameter for the relief for each gear, and added up crowning for a pair of meshing gears).Multiparameter optimization can easily become a difficult task if the algorithm used is not well adapted.We will show that the Particle Swarm Optimization (PSO) fits efficiently with that kind of problematic.Indeed, it permits to select a set of solutions more or less satisfying in the studied torque 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]).

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 deformations of the teeth can be written as follows [17]: The following data are needed to perform this interpolation: (i) initial gaps e between the teeth: they are function of the geometry defects and the tooth modifications, (ii) compliance matrix H u,F , of the teeth coming from interpolation functions calculated by a Finite Element model of elastostatic deformations, (iii) Hertz deformations hertz, calculated according to Hertz theory.
The calculation of the actual approach of distant teeth δ on the contact line for each position θ permits to access the time variation of STE and its peak-to-peak amplitude E pp , as a function of the applied torque (or the transmitted load F) and the teeth modifications.We chose linear correction for tip reliefs and parabolic correction for the crownings.All the modifications allow to reduce the STE fluctuation.The most influent parameter is the tip relief magnitude.Indeed, removing an amount of material on the tooth tip permits to make up for the advance or late position of the tooth induced by elastic deformations.
For the robustness study, the manufacturing errors are also considered and displayed on Figure 3.The manufacturing is not directly parameters of the optimization but as they have an effect on the STE fluctuation they must be considered in the robustness study.
A fitness function f to minimize is defined as the integral of STE peak-to-peak amplitude over torque range [C min − C max ] approximated by Gaussian quadrature with 3 points. ( The fitness function of the whole cascade is then We have thereby 8 parameters for the optimization leading to a combinatorial explosion.Meta-heuristic methods allow an efficient optimization, and we chose the Particle Swarm Optimization [18].Obviously in that kind of problematic, the aim cannot be to access to the optimum optimorum but only different local minima whose performances can be quickly estimated over the torque range by a home-built gain function where f ref corresponds to the value of the fitness function for a standard nonoptimized gear.

Particle Swarm Algorithm
The principle of this method is based on the stigmergic behavior of a population, being in constant communication and exchanging information about their location in a given space [18].Typically bees, ants, or termites are animals functioning that way.In our general case, we just consider particles which are located in an initial and random position in a hyperspace built according to the different optimization parameters.They will then change their position and their speed to search for the "best location," according to a defined criterion of optimization.It is commonly called the fitness function which has to be maximized or minimized depending on the problem.
For each iteration and each particle, a new speed and so a new position is reevaluated considering: (i) the current particle velocity V (t − 1), (ii) its best position p i , (iii) the best position of neighbors p g .
The algorithm can thus be wrapped up to the system of (5) and Figure 4: A 1 and A 2 represent a random vector of number between 0 and 1 and the parameters of these equations are taken following Trelea and Clerc [19][20][21]: ϕ 0 = 0.729 and ϕ 1 = ϕ 2 = 1.494.

Robustness Study
First the tolerance range D 0 of a solution x 0 has been defined, using a vector Δx = {Δx 1 , Δx 2 , . . ., Δx N }, which takes in account the parameters variability.The gears studied have a precision class 7 (ISO 1328).Moreover, the manufacturing 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 14-dimensionnal vector as following: Δx = ΔX dép,i , ΔΦ dép,i , f gα,i , ΔC β,i/ j , f Hβ, i/ j , ΔX dép, j , ΔΦ dép, j , f gα, j , . . ., ΔC β,l/ j , f Hβ,l/ j , ΔX dép,l , ΔΦ dép,l , f gα,l , ΔC , where i, j, and l correspond to, respectively, the gears with 50, 72, and 54 teeth.Then, the tolerance range D 0 can be written as 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 ones considered by Bonori et al. [10], especially for the tip relief modifications.The fitness function cannot be assumed monotonic and the study of the extreme boundaries of the problem is not sufficient.The PSO is then used to locate the maximum of the fitness function in the hyperspace D 0, in order to analyze robustness of the solutions.The new values for the parameters which maximize the fitness function define the "degenerated solution," noted x d : With this additional criterion, optimal solution corresponds to the less deteriorated rather than the minimal E pp .

Results
The cascade of three helical gears has to be optimized for torques from 100 Nm up to 500 Nm.A reference solution, with standard and not optimized tooth modifications, is used to emphasize the benefits of the Particle Swarm optimization.The PSO calculations have been performed using a population of 25 particles and stopped when a precision of 10 −2 μrad for peak-to-peak amplitude E pp is reached.The algorithm stops the calculation when no improvement is found 50 times successively.All the following results have converged after 250 to 400 iterations.That corresponds to 7500 to 10000 evaluations of the fitness function (instead of 10 14 for a Monte-Carlo experiment).Table 1 lists the parameters ranges.
In order to illustrate the optimization process, Figure 5 displays 5 selected solutions-S1 to S5-corresponding to 5 local minima among the computed ones which all obviously are better than the reference solution in terms of minimal E pp .Figure 6 displays the optimized parameters of the solutions rescaled in function of their extremum values.
According to the gain function (4), we can easily pick up the best solutions of the selected ones.Following the results listed in Table 2, solution S5, which provides −4.2 dB of improvement compared to the reference solution, should be selected.The first analysis of the deteriorating capacity of the solutions can be done using gain function (9) and listing results in Table 3: The deteriorated reference solution has a gain of +6.7 dB compared with the initial reference solution.The solution S5 is worse considering the gain function ( 9), but its fitness function value is still less than the deteriorated reference solution one.On the other hand, the previous selected solution S4 appears as the best one with only +2.3 dB of deterioration in the gain function ( 9) sense.
The second analysis of the deteriorating capacity of the solutions can be done using gain function (10) and listing results in Table 4: The solution S1 emphasizes the importance of considering the deteriorating capacity.Indeed, although the optimal solution brings an improvement compared to the initial reference solution, it is likely to be less efficient taking in account the possible manufacturing errors.The previous choice has to be reconsidered.On the other hand, the solution S4 provides a good improvement of −3.7 dB compared to the reference solution and is quite robust as a gain of −6.2 dB is observed if S4 deteriorated solution is compared with the deteriorated reference solution.

Conclusion
Optimization with an efficient heuristic method (Particle Swarm) has been done to determinate optimized parameters of a multimesh problem.The algorithm permits the gathering of many solutions which all lead to really satisfying results over the torque range studied thank to an integration of STE peak-to-peak amplitude by Gaussian quadrature.Finally, a robustness criterion has been defined based on the deteriorating capacity of the solutions which permits to do a more accurate choice about the optimal tooth modifications.Indeed, there are many ways of estimating the robustness of the solutions.In some industrial point of view, a solution which is less efficient than another but much more robust should be preferably chosen.

Figure 6 :
Figure 6: Optimized parameters of the solutions.

Figure 7
Figure 7 displays the deteriorated solutions.The first analysis of the deteriorating capacity of the solutions can be done using gain function(9) and listing results in Table3:

Table 3 :
Gain of the degenerated solutions compared to optimal solutions.

Table 4 :
Gain of the degenerated solutions compared to the reference degenerated solution.