An efficient numerical strategy to predict the dynamic instabilities of a rubbing system: application to an automobile disc brake system

Abstract

This paper deals with the modelling and the prediction of the dynamic instabilities for a rubbing system. Two hybrid approaches are introduced for dynamic instability analysis and applied to a reduced disc brake system. The methods are based on stochastic algorithms coupled with the finite element method (FEM) using the complex eigenvalue analysis technique. By considering the input parameters as random variables, the uncertainty analysis is performed through two approaches to predict the unstable frequencies of a braking system (1) Monte-Carlo (MC) using Mersenne-Twister (MT19937) algorithm and (2) periodic sampling technique. Since the mechanism of brake squeal involves many design parameters, stochastic finite element approaches will be coupled with sensitivity algorithms, e.g. Variance-Based Sensitivity Analysis and Fourier Sensitivity Analysis Test, to analyze the contribution of each random variable on the dynamic instabilities. First, a comparison between the two stochastic algorithms is performed on standard analytical models. The objective is to validate the accuracy and to assess the numerical efficiency that FAST presents to (1) propagate the uncertainties upstream of the model and (2) to compute the partial variances of the model output. Secondly, the coupling of the previous stochastic algorithms with FEM is carried out and tested through a reduced brake system consisting of a rotating disc with two flat pads. Results show that the hybrid approach FAST-FE is more robust and computationally more efficient compared to the widely used MC-FE for these types of problems. FAST-FE solver converges, within a reasonable computing time, either to approximate the probability density function of the random variables or to compute the partial variances of the dynamic instabilities. Hence, it can be considered as an efficient numerical method for squeal instability analysis in order to reduce squeal noise of such a mechanical system.

Appendix A: Testing the numerical performance of the stochastic algorithms

In this section, the two stochastic algorithms studied in the paper will be tested on standard mathematical models; (1) one is a simple linear model and (2) the other is constructed by the G-functions of Sobol, a classic of sensitivity theory, which is a non-linear and non-monotonic model. The objective of this section is to compare (1) Monte Carlo implementation using Sobol’s approximation, (2) FAST algorithm through the spectrum of the Fourier series decomposition to estimate sensitivity indexes with (3) the analytical solutions inferred through academic models.

Fig. 9

Accuracy of the MC-VBSA algorithm through a linear model in Eq. (33): illustration of the convergence for (a) the first and (b) the second sensitivity index

1.1 A.1 Case 1 : linear model

The simple linear model can be written as follows:

$$\begin{aligned} Y = 3x_1 + 2x_1, \end{aligned}$$

(33)

with $x_i$, $i \in \{1,2\}$ are RVs that follow a standard normal distribution noted by $\mathcal {N}(0,1)$. The variance and the first order sensitivity indexes can be expressed, in this case, by:

• $V(Y) = 13$;

• $S_{x_1} = 0.69$ and $S_{x_2} = 0.31$.

Before presenting the comparative results of the two stochastic approaches (i.e. MC-VBSA and FAST), it can be seen clearly from Fig. 9 that MC-VBSA algorithm converges only with a significant sample size which corresponds to 5000. Despite the simplicity of the linear model in Eq. (33), MC-VBSA has difficulty estimating the analytical solutions on the first attempt. In addition, it is remarkable that the MC-VBSA algorithm converges slowly by increasing the number of iterations towards the analytical solutions. However, the increase in the number of iterations increases the computation time (see Table 6).

Table 6 The computation time of MC-VBSA algorithm for a linear model in Eq. (33)

The response related to sensitivity indexes obtained from FAST algorithm is compared to the reference solution and MC-VBSA approximation in the Fig. 10 using the linear model in Eq. (33). It should be noted that the sample size used to compute the approximations, through MC-VBSA and FAST, respect the “aliasing effect” criteria. It is worth 185 samples generated, as explained in Sect. 2, by the periodic sampling technique and the pseudo-random generator of MT19937. Under the conditions mentioned above, FAST approximation and the reference solutions agree very well over the RVs. In return, MC-VBSA overestimates the first index $S_{x_1}$ and underestimates the second one $S_{x_2}$.

Fig. 10

First order Sensitivity indexes for the linear model in Eq. (33), computed through FAST and MC-VBSA algorithms using $N_s = 185$ samples and compared with the analytic solution

1.2 A.2. Case 2 : G-functions of Sobol

The non-linear and non-monotonic model is based on G-functions of Sobol. It is a classic used to assess sensitivity methods. It is represented through an explicit equations as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} Y = \displaystyle \prod _{i=1}^n g_i(x_i), ~~ \mathbb {K}_x^n = \big \{(x_1,\dots ,x_n) \mid 0 \le x_i \le 1 \big \}\\ g_i(x_i) = \frac{\mid 4 x_i - 2 \mid + a_i}{1 + a_i}, ~~ \forall x_i \in [0,1], ~~ a_i \in \mathbb {R}^+ \end{array}\right. }, \end{aligned}$$

(34)

By exploiting the fact that $x_i$ is in [0, 1] and $a_i \ge 0$, it can be shown that G-Sobol functions vary as follows:

$$\begin{aligned} 1 - \frac{1}{1 + a_i} \le g_i(x_i) \le 1 + \frac{1}{1 + a_i},~~ \forall x_i \in [0,1], \end{aligned}$$

(35)

and the stochastic indicators may be expressed as:

• the variance of the model,

  $$\begin{aligned} V(Y) = \displaystyle \prod _{i=1}^n (V_i + 1) - 1, \end{aligned}$$

  (36)

• the sensitivity indexes,

  $$\begin{aligned} S_{x_i} = \frac{1}{3(1+a_i)^2 \displaystyle \prod _{i=1}^n (V_i + 1) - 1}. \end{aligned}$$

  (37)

Fig. 11

First order Sensitivity indexes for the non-linear model in Eq. (34), computed through FAST and MC-VBSA algorithms using 185 samples and compared with the analytic solution

From the inequations (35) and the sensitivity indexes in Eq. (37), it can be concluded that if $a_i$ is small, the sensitivity index $S_{x_i}$ will be large and therefore, the effect of the random variable $x_i$ on the model Y will be important. The case where $a_i = 0$ represents an extreme case which implies an important first order effect. The testing of stochastic algorithms for sensitivity analysis is performed considering only 3 parameters with factors of the Sobol G-functions corresponding to $a_i = 0$. This is the worst scenario for the Sobol G-functions in which singularities occur in each n dimensions corresponding to the middle point $x_i=\frac{1}{2}$.

Sobol’s G-functions model in Eq. (34) is an interesting model. It is suitable to test the robustness and the accuracy of FAST algorithm since it involves an entirely different behavior compared to the linear model presented above. According the “aliasing effect” criteria, the numerical simulations of the current test model, for FAST and MC-VBSA methods, contains 3 RVs described by 121 samples through an uniform PDF. In analogy with the previous investigation (i.e case 1), both methods are compared in Fig. 11 to the analytical solution given by Eq. (37). As expected, MC-VBSA, which corresponds to the widely used method with FEM [15, 60], under and overestimates sensitivity indexes. However, FAST approximation is in a good agreement with the analytic solutions. Although a difference is observed in the $S_{x_3}$ index between the FAST approximation and the analytical solution due to strong non-linearities in the model, predictions can be improved by increasing the number of samples.