Shape optimization for brake squeal
Brake squeal, which usually causes in the frequency range between 1 and 15 kHz, is a phenomenon which has been one of the most difficult concerns in design of brake systems. It causes customer dissatisfaction and increases warranty costs. The present paper proposes a cost function which is expected to be effective to prevent the brake squeal. The cost function is defined with the real part of a complex eigen frequency for a target natural vibration which causes the brake squeal. In the present paper, we assume that a brake system consists of a rotor and a pad between which the Coulomb friction occurs. The Coulomb friction makes an equation of motion asymmetric with respect to displacement and adjoint displacement in the weak form. Then, eigen values for natural vibrations become complex values. On the other hand, it has been known that brake squeal is an unstable vibration induced by friction. From this observation, we considered the real part of a complex eigen frequency for a natural vibration which causes the brake squeal becomes a candidate of a cost function to prevent the brake squeal. A shape optimization problem is formulated using the objective cost function of the real part of a complex eigen frequency and a constraint function of volume for the pad. The main problem consists of a weak form combining the weak forms of the liner elastic continua for the rotor and the pad including the terms of the Coulomb friction and the stiffness in normal direction on the sliding boundary. The shape derivative, which is defined as the Fréchet derivative with respect to domain variation, of the real part of a complex eigen frequency is evaluated with the solutions of the main problem and a adjoint problem which is derived theoretically by the adjoint variable method. To solve the shape optimization problem to minimize the real part of the complex eigen frequency with the volume constraint of the pad, we use an iterative algorithm based on the H1 gradient method. The H1 gradient method is used to keep the smoothness of the boundary. The volume constraint is satisfied using the KKT conditions in the shape optimization problem. We developed a computer program to solve the shape optimization problem based on the iterative algorithm. The boundary value problems are solved by the finite element method using an original code. Numerical example using a simple model is going to be shown in the paper.
Sensitivity analysis of brake squeal tendency to substructures’ modal parameters
Abstract Sensitivity analysis methods are explored to determine the dominant modal parameters of substructures of a brake system for brake squeal suppression analysis. The related formulae of sensitivities of the positive real part of the eigenvalue of squeal mode (RES) to substructures’ modal parameters (SMP) are derived. The sensitivity analysis method can determine the dominant modal parameters influencing the squeal occurrence in a regular way, and the dominant modal parameters will be set to be the target of structural modification attempted to eliminate the squeal mode. Sensitivity analysis of a typical squealing disc brake is performed. The analysis results show that a modified rotor or a modified bracket can be used to eliminate the squealing and that the latter is taken in practice and verified experimentally.
Accurate measurements of output characteristics and detonations of motorbike high-speed racing engine and their optimization at actual atmospheric conditions and combusted mixture composition
Abstract At a motorbike high speed racing engine there is very important an output engine characteristics which depends significantly on a detonation combustion rate, atmospheric conditions and a combusted mixture composition. The combustion process has got a relevant influence on a wear degree of main engine parts or on their damage eventually. The detonation combustion rate is very important because this kind of burning can be not only destructive but a useful phenomenon, as well. If the process of detonation burning is kept on a certain level, it can help significantly to the next increasing of an engine power output. In this article there is described the measurements of output characteristics of a high-speed racing engine which were performed by means of a data-recording system and the detonation combustion was evaluated using a quantitative method. The actual values of atmospheric conditions and the mixture composition were added to the measured values. Then the obtained values are compared visually with a damage degree of an engine piston caused by the detonation burning. The result is a limit value expressed in a number of detonation units related to a run distance interval of the vehicle driven by the high-speed racing engine. In framework of this run distance interval there is the process of detonation burning profitable. As well as it increases an engine power output and together it is not destructive with regard to engine design parts.