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.

本文涉及摩擦系统动力学不稳定性的建模和预测。引入了两种混合方法来进行动态不稳定性分析，并将其应用于简化的盘式制动系统。这些方法基于使用复杂特征值分析技术的随机算法和有限元方法（FEM）。通过将输入参数视为随机变量，通过两种方法进行不确定性分析，以预测制动系统的不稳定频率：（1）使用Mersenne-Twister（MT19937）算法的蒙特卡洛（MC）和（2）周期性采样技术。由于制动尖叫的机制涉及许多设计参数，因此随机有限元方法将与灵敏度算法结合使用，例如 基于方差的敏感性分析和傅里叶敏感性分析测试，以分析每个随机变量对动态不稳定性的贡献。首先，在标准分析模型上对两种随机算法进行了比较。目的是验证准确性并评估FAST对（1）传播模型上游不确定性和（2）计算模型输出的部分方差的数值效率。其次，通过减少的制动系统来执行和测试以前的随机算法与FEM的耦合，该系统包括旋转盘和两个平垫。结果表明，与针对这些类型问题的广泛使用的MC-FE相比，混合方法FAST-FE更加健壮，计算效率更高。FAST-FE求解器收敛，在合理的计算时间内，要么近似随机变量的概率密度函数，要么计算动态不稳定性的部分方差。因此，可以将其视为用于尖叫不稳定性分析的有效数值方法，以减少此类机械系统的尖叫噪声。