We present numerical methods for model reduction in the numerical simulation of disk brake squeal. Automotive disk brake squeal is a high frequency noise phenomenon based on self excited vibrations. Our method is based on a variation of the proper orthogonal decomposition method and involves the solution of a large scale, parametric eigenvalue problem. Several important challenges arise, some of which can be traced back to the finite element modeling stage. Compared to the current industrial standard our new approach is more accurate in vibration prediction and achieves a better reduction in model size. This comes at the price of an increased computational cost, but it still gives useful results when the classical modal reduction method fails to do so. We illustrate the results with several numerical experiments, some from real industrial models, some from simpler academic models. These results indicate where improvements of the current black box industrial codes are advisable.
We present a proper orthogonal decomposition (POD) based approach for the accurate solution of parametric eigenvalue problems arising from brake squeal modeling. We compare this approach to a traditional method based on modal decomposition which is very popular in industry. The proposed approach is more accurate than the traditional method especially for parametric studies and for studying non-proportionally damped systems.
We explore dimensionality reduction in the context of model based and model free approaches. In the model based approach there is a governing equation or a set of rules relating quantities of interest, whereas in a model free setting there are no rules or equations, only data is available. As an example consider the problem of squealing noise in a brake. The model based approach relates quantities of interest like mass distribution within the brake, damping, stiffness and other properties of a brake material, speed of rotation etc with a dynamical equation, the steady state behaviour can be obtained by converting it to an eigenvalue problem and finding eigenvalues and
eigenvectors (which are related to squeal frequency and mode shapes of a disc brake). The model reduction problem could be posed as projecting the eigenvalue problem to a lower dimensional space, while preserving important eigenvalues and eigenvectors. In contrast, the model free approach starts with data, i.e., a set of parameter values which correspond to squeal and the values which correspond to no-squeal. If the number of parameters responsible for squeal is very
large, then dimensionality reduction is concerned with reducing the number of these parameters or ranking these parameters in order of importance. We illustrate pros and cons of model based and model free dimensionality reduction with some numerical examples.
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