Application of shape optimization to brake squeal phenomena

Abstract

The present paper describes an application of non-parametric shape optimization to disc brake squeal phenomena. 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, a main problem is defined as complex eigen value problem and a real part of the complex eigenvalue causing the brake squeal is chosen as an objective cost function for the shape optimization problem. For the solution to main problem which has a large number of DOF, component mode synthesis method adapting residual stiffness (CMS-R) is used in order to improve the accuracy of eigenpairs by conventional component mode synthesis method (CMS). The Fre´chet derivative of the objective cost function with respect to the domain variation, which we call the shape derivative of the objective cost function, is evaluated using the solution of the main problem and the adjoint problem. A scheme to solve the shape optimization problem is presented using an iterative algorithm based on the H¹ gradient method for reshaping. For an application of the shape optimization method, a numerical example by using a practical disc brake model is presented. From the numerical result, the real part of the target complex eigenvalue monotonously decreases until it reach zero and effectiveness of CMS-R with regard to the shape optimization problem is presented.