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

Abstract

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

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}.

Figure 1 

Crowning 𝐶𝛽,𝑖/𝑗, tip relief 𝑥rel,𝑖, and start relief diameter Φrel,𝑖.

Many authors [3–11] worked on the optimization of tooth modifications in simple mesh systems. Only few of them [12–14] considered multimesh systems as cascade of gears where idler gear modifications affect two meshes.

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]).

Figure 2 

Cascade of the 3 helical gears studied: 50 teeth/72 teeth/54 teeth.

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 deformations of the teeth can be written as follows [17]: 𝐇^𝐮,𝐅(𝜔=0)⋅𝐅𝐅=𝛿(𝜃)−𝐞−𝐡𝐞𝐫𝐭𝐳(𝐅),_𝐢=𝐅.(1)

The following data are needed to perform this calculation:(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.(i)Lead deviation: 𝑓_{𝐻𝛽,𝑖/𝑗}=𝑓_{𝐻𝛽,𝑖}+𝑓_{𝐻𝛽,𝑗},(ii)Involute alignment deviation: 𝑓_{𝑔𝛼,𝑖} and 𝑓_{𝑔𝛼,𝑗}.

Figure 3 

Involute alignment deviation 𝑓𝑔𝛼 and lead deviation 𝑓𝐻𝛽.

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.𝑓_𝑖,𝑗=^𝐶max_𝐶min𝐸_pp(𝐶)𝑑𝐶⟶^𝑖=3_𝑖=1𝑎_𝑖𝐸_𝑝𝑝𝐶_𝑖.(2) The fitness function of the whole cascade is then 𝑓=𝑓_𝑖,𝑗+𝑓_𝑘,𝑗.(3) 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 𝐺₀=10log₁₀𝑓𝑓^Si_ref,(4) where 𝑓_ref corresponds to the value of the fitness function for a standard nonoptimized gear.

3. 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: 𝐕(𝑡)=+𝜑₀𝜑𝐕₂(𝑡−1)+𝜑₁𝐀_𝟏𝐩_𝑖−𝐩(𝑡−1)𝐩(𝑡)=𝐩(𝑡−𝐀_𝟐1)𝐩+_𝑔𝐕−(𝐩𝑡−(𝑡−1)1.),(5)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 [19–21]: 𝜑₀=0.729 and 𝜑₁=𝜑₂=1.494.

Figure 4 

Particle Swarm algorithm representation.

4. Robustness Study

First the tolerance range 𝐃_𝟎 of a solution 𝐱_𝟎 has been defined, using a vector Δ𝐱={Δ𝑥₁,Δ𝑥₂,…,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=Δ𝑓𝑋_{d́‌ep,𝑖𝑔𝛼,𝑗},ΔΦ_{d́‌ep,𝑖},𝑓_{𝑔𝛼,𝑖},Δ𝐶_{𝛽,𝑖/𝑗},𝑓_{𝐻𝛽,𝑖/𝑗},Δ𝑋_{d́‌ep,𝑗},ΔΦ_{d́‌ep,𝑗},,…,Δ𝐶_{𝛽,𝑙/𝑗},𝑓_{𝐻𝛽,𝑙/𝑗},Δ𝑋_{d́‌ep,𝑙},ΔΦ_{d́‌ep,𝑙},𝑓_{𝑔𝛼,𝑙},Δ𝐶,(6) 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 𝐃_𝟎=𝐱∶𝐱∈𝐑^𝐍∣𝐱_𝟎−𝚫𝐱<𝐱<𝐱_𝟎+𝚫𝐱.(7)

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 hyper-space 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: 𝐱_𝐝∈𝐃_𝟎,𝑓𝐱_𝐝=max𝑓(𝐱)∣𝐱∈𝐃_𝟎.(8) With this additional criterion, optimal solution corresponds to the less deteriorated rather than the minimal 𝐸_pp.

5. 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 10¹⁴ for a Monte-Carlo experiment). Table 1 lists the parameters ranges.

Table 1 

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.

Figure 5 

Optimized and reference solutions versus applied torque - - - - torque range boundaries.

Figure 6 

Optimized parameters of the solutions.

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.

Table 2 

Gain of the computed optimal solutions compared to the reference solution.

Figure 7 displays the deteriorated solutions.

Figure 7 

Degenerated solutions versus applied torque - - - - Torque range boundaries.

The first analysis of the deteriorating capacity of the solutions can be done using gain function (9) and listing results in Table 3: 𝐺₁=10log₁₀𝑓𝑓^{non−deteriorated}_deteriorated.(9) 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.

Table 3 

Gain of the degenerated solutions compared to optimal solutions.

The second analysis of the deteriorating capacity of the solutions can be done using gain function (10) and listing results in Table 4: 𝐺₂=10log₁₀𝑓𝑓^{𝑆𝑖,deteriorated}_{ref,deteriorated}.(10)

Table 4 

Gain of the degenerated solutions compared to the reference degenerated solution.

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.

6. 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.

Acknowledgments

This work has been supported by ANR (National Research Agency, contract number: ANR-08-VTT-007-02), ADEME (French Environment and Energy Management Agency), and Lyon Urban Trucks&Bus competitiveness cluster. The authors acknowledge gratefully this support and especially thank Denis BARDAY from Renault Trucks Company for his inestimable help.