An Efficient Methodology to Predict the Dynamic Instabilities of a Frictional System

Stochastic Finite Elements Method (SFEM) is applied in many fields. For instance, in frictional systems, it helps quantify uncertainties about the parameters controlling the involved process and thus, provides a more reliable prediction of the dynamic instabilities. Usually, SFEM is coupled with sensitivity theory to investigate the effect of a given input on the output. However, the available methods which often couple Monte-Carlo (MC) algorithm with the Finite Element (FE) method have a computational cost that scales linearly as a number of stochastic iteration N and input parameters k (i.e., t ~ N x k). To achieve convergence, the magnitude of N must be on the order of thousands or even millions. Hence, for a frictional system with 5 random variables and requiring 15 min of CPU time per run, the computational cost will exceed 52 days (!). Such a method cannot be applied in an industrial design framework with a high number of random variables since its CPU time becomes prohibitive. In this paper, an efficient SFEM is presented, and its performances demonstrated on a simplified disc brake system. The goal is to predict the most likely dynamic instabilities. The method is formulated to (i) reduce the computational cost while ensuring convergence and (ii) provide a reliable input-output mapping of the model which allows in turn a better prediction and investigation of the friction-induced vibration problem. The approach is based on the Fourier Sensitivity Amplitude Test (FAST) algorithm coupled with FE method through a Complex Eigenvalues Analysis (CEA). First, the uncertainties propagation is carried out using the periodic sampling approach by considering a variety of random variables (e.g., friction coefficient, Young modulus, etc.). Secondly, the random generated data are evaluated in an iterative way by mean of the CEA solver. Lastly, Fourier expansion is introduced to derive the partial variances and the variance of the model output. Based on the last decomposition, a 2D design map is used to visualize the effect of each random variable on the predicted instabilities. The obtained FAST-FE results are systematically compared with the reference approach, namely MC-FE. It is found that the inherent assumption of using a large number of samples is not more needed for reaching the stochastic convergence and thus, the estimation of the moments (i.e., the expected value, the variance and so forth). The periodic properties, carried by FAST research curve function, made it possible to propagate efficiently and within a reasonable computational cost the uncertainties upstream of the FE model. In comparison with the MC-FE, the proposed solver provides good results even when a coarser sample is used. The inefficiency of MC-FE solver is discussed.

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