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

Alexandre Carbonelli, Joël Perret-Liaudet, Emmanuel Rigaud, Alain Le Bot

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Alexandre Carbonelli, Joël Perret-Liaudet, Emmanuel Rigaud, Alain Le Bot. Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations of Multimesh Gears Transmission. Advances in Acoustics and Vibration, 2011, pp.195642. 10.1155/2011/195642. hal-03257198

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Research Article

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

Alexandre Carbonelli, Joël Perret-Liaudet, Emmanuel Rigaud, and Alain Le Bot

Laboratoire de Tribologie et Dynamique des Systèmes, UMR CNRS 5513, Ecole Centrale de Lyon, Université de 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**

The STE under load [1] is defined as the difference between
the actual position of the driven gear and its theoretical posi-
tion for a very slow rotation velocity and for a given applied
torque. Its characteristics depend on the instantaneous sit-
uations of the meshing tooth pairs. Under load at very low
speed (static transmission error), these situations result from
tooth deflections, tooth surface modifications, and manu-
facturing 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 ma-
terial removed on the tooth tip,

(ii) start relief diameter $\Phi_{rel,i}$, 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 $C_{\beta,i/j}$.

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). Multipa-
parameter 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
<figure><img src="image_2.png" /><figcaption>FIGURE 1: Crowning C<sub>β,i/j</sub>, tip relief x<sub>rel,i</sub>, and start relief diameter Φ<sub>rel,i</sub>.</figcaption></figure>

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 deformations of the teeth can be written as follows [17]:

$$ \mathbf{H}^{\text{u,F}}(\omega = 0) \cdot \mathbf{F} = \delta(\theta) - \mathbf{e} - \text{hertz}(\mathbf{F}), \\ \sum \mathbf{F}_i = \mathbf{F}. \qquad (1) $$

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<sup>u,F</sup>, 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<sub>pp</sub>, as a function of the applied torque (or the transmitted load F) and the teeth modifications. We chose linear correction

<figure><img src="image_3.png" /><figcaption>FIGURE 2: Cascade of the 3 helical gears studied: 50 teeth/72 teeth/54 teeth.</figcaption></figure>

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<sub>Hβ,i/j</sub> = f<sub>Hβ,i</sub> + f<sub>Hβ,j</sub>,

(ii) Involute alignment deviation: f<sub>gα,i</sub> and f<sub>gα,j</sub>.

A fitness function f to minimize is defined as the integral of STE peak-to-peak amplitude over torque range
<figure><img src="image_4.png" /><figcaption>FIGURE 3: Involute alignment deviation $f_{g\alpha}$ and lead deviation $f_{H\beta}$.</figcaption></figure>

$[C_{min} - C_{max}]$ approximated by Gaussian quadrature with 3 points.

$$f_{i,j} = \int_{C_{min}}^{C_{max}} E_{pp}(C) dC \rightarrow \sum_{i=1}^{i=3} a_i E_{pp}(C_i). \quad (2)$$

The fitness function of the whole cascade is then

$$f = f_{i,j} + f_{k,j}. \quad (3)$$

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

$$G_0 = 10 \log_{10} \left( \frac{f_{S_i}}{f_{ref}} \right), \quad (4)$$

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

<figure><img src="image_5.png" /><figcaption>FIGURE 4: Particle Swarm algorithm representation.</figcaption></figure>

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:

$$\begin{aligned} \mathbf{V}(t) = & \varphi_0 \mathbf{V}(t-1) + \varphi_1 \mathbf{A}_1 [\mathbf{p}_i - \mathbf{p}(t-1)] \\ & + \varphi_2 \mathbf{A}_2 [\mathbf{p}_g - \mathbf{p}(t-1)], \end{aligned} \quad (5) \\ \mathbf{p}(t) = \mathbf{p}(t-1) + \mathbf{V}(t-1).$$

$\mathbf{A}_1$ and $\mathbf{A}_2$ represent a random vector of number between 0 and 1 and the parameters of these equations are taken following Trelea and Clerc [19–21]: $\varphi_0 = 0.729$ and $\varphi_1 = \varphi_2 = 1.494$.

### 4. Robustness Study

First the tolerance range $D_0$ of a solution $\mathbf{x}_0$ has been defined, using a vector $\Delta\mathbf{x} = \{\Delta x_1, \Delta x_2, \dots, \Delta x_N\}$, 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:

$$\Delta \mathbf{x} = \{ \Delta X_{\text{dép},i}, \Delta \Phi_{\text{dép},i}, f_{g\alpha,i}, \Delta C_{\beta,i/j}, f_{H\beta}, i/j, \Delta X_{\text{dép},j}, \Delta \Phi_{\text{dép},j}, \\ f_{g\alpha,j}, \dots, \Delta C_{\beta,l/j}, f_{H\beta,l/j}, \Delta X_{\text{dép},l}, \Delta \Phi_{\text{dép},l}, f_{g\alpha,l}, \Delta C \}, \quad (6)$$

where $i, j$, and $l$ correspond to, respectively, the gears with 50, 72, and 54 teeth.
<table><caption>TABLE 1: Parameters ranges.</caption><thead><tr><th></th><th>Z = 54</th><th>Z = 72</th><th>Z = 50</th></tr></thead><tbody><tr><td>Number of teeth</td><td>Z = 54</td><td>Z = 72</td><td>Z = 50</td></tr><tr><td>Tip relief magnitude and tolerance [µm]</td><td>[15–150] ± 15</td><td>[0–150] ± 15</td><td>[15–150] ± 15</td></tr><tr><td>Start relief diameter and tolerance [mm]</td><td>[230–241] ± 0.46</td><td>[200–215] ± 0.46</td><td>[153–168] ± 0.40</td></tr><tr><td>Added up crowning and tolerance [µm]</td><td>—</td><td>[8–40] ± 8</td><td>—</td></tr><tr><td>Lead deviation and tolerance [µm]</td><td>—</td><td>0 ± 32</td><td>—</td></tr><tr><td>Involution alignment dev. and tolerance [µm]</td><td>0 ± 12</td><td>0 ± 12</td><td>0 ± 12</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>FIGURE 5: Optimized and reference solutions versus applied torque – – – – torque range boundaries.</figcaption></figure>

Then, the tolerance range $D_0$ can be written as

$$D_0 = \{ \mathbf{x} : \mathbf{x} \in \mathbb{R}^N \mid \mathbf{x}_0 - \Delta\mathbf{x} < \mathbf{x} < \mathbf{x}_0 + \Delta\mathbf{x} \}. \quad (7)$$

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 $\mathbf{x}_d$:

$$\mathbf{x}_d \in D_0, \quad f(\mathbf{x}_d) = \max(f(\mathbf{x}) \mid \mathbf{x} \in D_0). \quad (8)$$

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.

<figure><img src="image_7.png" /><figcaption>FIGURE 6: Optimized parameters of the solutions.</figcaption></figure>

The PSO calculations have been performed using a population of 25 particles and stopped when a precision of $10^{-2} \mu\text{rad}$ 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.
<table><caption>TABLE 2: Gain of the computed optimal solutions compared to the reference solution.</caption><thead><tr><td>Configuration</td><td>Gain G<sub>0</sub> [dB]</td></tr></thead><tbody><tr><td>S1</td><td>-1.6</td></tr><tr><td>S2</td><td>-1.9</td></tr><tr><td>S3</td><td>-3.3</td></tr><tr><td>S4</td><td>-3.7</td></tr><tr><td>S5</td><td>-4.2</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>FIGURE 7: Degenerated solutions versus applied torque - - - Torque range boundaries.</figcaption></figure>

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 Table 3:

$$G_1 = 10 \log_{10} \left( \frac{f_{\text{non-deteriorated}}}{f_{\text{deteriorated}}} \right). \quad (9)$$

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.

The second analysis of the deteriorating capacity of the solutions can be done using gain function (10) and listing results in Table 4:

$$G_2 = 10 \log_{10} \left( \frac{f_{\text{S1, deteriorated}}}{f_{\text{ref, deteriorated}}} \right). \quad (10)$$

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.

<table><caption>TABLE 3: Gain of the degenerated solutions compared to optimal solutions.</caption><thead><tr><td>Configuration</td><td>Gain G<sub>1</sub> [dB]</td></tr></thead><tbody><tr><td>Reference</td><td>+6.7</td></tr><tr><td>S1</td><td>+11.3</td></tr><tr><td>S2</td><td>+6.0</td></tr><tr><td>S3</td><td>+6.1</td></tr><tr><td>S4</td><td>+2.3</td></tr><tr><td>S5</td><td>+11.3</td></tr></tbody></table>

<table><caption>TABLE 4: Gain of the degenerated solutions compared to the reference degenerated solution.</caption><thead><tr><td>Configuration</td><td>Gain G<sub>2</sub> [dB]</td></tr></thead><tbody><tr><td>S1</td><td>+2.8</td></tr><tr><td>S2</td><td>-2.6</td></tr><tr><td>S3</td><td>-4.2</td></tr><tr><td>S4</td><td>-6.2</td></tr><tr><td>S5</td><td>-0.4</td></tr></tbody></table>

## 6. Conclusion

Optimization with an efficient heuristic method (Particle Swarm) has been done to determine 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 thanks 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.

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