Component mode synthesis with residual stiffness for brake squeal complex eigenvalue problem

Abstract

This paper discusses component mode synthesis (CMS) to solve the complex eigenvalue problem that expresses the disk brake squeal equation. The disk brake squeal is known as self-excited vibration; the real and imaginary parts of the complex eigenvalue indicate the damping coefficient and natural angular frequency, respectively. The modes that have a negative damping coefficient cause disk brake squeal. Therefore, calculating the eigenpairs (eigenvalues and eigenmodes) with accuracy is important. To solve the complex eigenvalue problem, a direct solver, such as the Hessenberg method, and an iterative solver, such as the Lanczos method, are used. By using a direct solver, all eigenpairs can be calculated with high accuracy, but the computational cost is large. Consequently, its application is limited to small DOF problems of approximately 10,000 DOF. On the other hand, calculating the eigenpairs of large DOF problems by using an iterative solver is possible; this also has disadvantages, such as missing actual modes and accepting quasi-modes. To overcome these disadvantages, a CMS that uses both a direct solver and an iterative solver is used. By using CMS, avoiding the disadvantages of missing actual modes and accepting quasi-modes while still using fewer computer resources than the direct solver alone is possible. However, the calculated eigenpairs contain computational error caused by CMS. In this paper, to improve the accuracy of CMS, a novel CMS method adapting residual stiffness is proposed. Here, novel CMS is formulated, and the advantage it offers is shown by applying the direct solver, the previous CMS, and the novel CMS to the finite element models of a disk brake.