The dynamical modeling of a disk brake with respect to squeal and its discretization via the Finite Element Method (FEM), e.g., by using commercial FEM packages, usually results in a large-scale quadratic eigenvalue problem (QEVP) with typically up to a million degrees of freedom. Furthermore, inclusion of squeal relevant physical effects such as gyroscopic and circulatory effects, damping and friction results in a QEVP with parameter dependent, non-symmetric coefficients. To identify the role of different parameters responsible for brake-squeal, a detailed parameter study is necessary, which in-turn requires the solution of many large-scale QEVPs for a variety of choices of the parameter. Thereby complex eigenvalues associated with the audible frequency range should be calculated with high accuracy which is called complex eigenvalue analysis (CEA).
The state of the art modal-transformation approaches used in standard FE software converts the QEVP to a space of modal-coordinates. The modal-transformation matrices are typically constructed by solving a symmetric linear eigenvalue problem, which is obtained by dropping the non-symmetric, parameter dependent and damping terms in the QEVP, i.e., by neglecting all the physical effects essential for self-excited vibrations. This simplistic approach empirically works well for the problems where an approximation of the imaginary part of the eigenvalues are required, but for studying the dynamical stability behavior of a brake with respect to squeal, a good approximation of both the real and imaginary parts of the eigenvalues with a positive real part is of crucial interest.
In this paper, we present a model-order-reduction approach which takes into account the parameter dependent nature of the damping and stiffness matrices. In our approach, we obtain the model-order-reducing subspace by performing a proper orthogonal decomposition (POD) on the matrix of dominant modes of the non-symmetric QEVP for a variety of parameter choices. Numerical experiments suggest that the new POD based approach is more accurate for the brake squeal problem than state of the art algorithms used in FE programs so far.
Nils Gräbner, Sarosh Quraishi, Christian Schröder, Volker Mehrmann, Utz von Wagner - Technische Universität Berlin