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.
Optimal shape design of a brake calliper for squeal noise reduction considering system instability
Abstract Squeal is a noise phenomenon occurring in the last stage of automobile braking with a high-frequency sound. It is very difficult to express the phenomenon using a mathematical model, since the origin of squeal noise is physically complex. However, the possibility of squeal generation can be predicted by solving the vibration equation of the self-excited system using the complex eigenvalue analysis method. The results of the method are expressed as the magnitude of the unstable mode, and the generation of squeal noise can be prevented by reducing the magnitude of the unstable mode of the brake system. The objective of this research is to determine the optimal design process focused on the calliper housing shape to suppress squeal noise generation by reducing the system instability. The objective function is set to minimize the real part of the complex eigenvalue, i.e. the instability index. In the optimization design process, the design variable for topology optimization is established by focusing on the finger part of the calliper housing, which transmits the braking pressure to the pad lining. To supplement the complex shape generated by the topology optimization process, parametric design variables are selected for the subsequent process. Parameters are set to adjust the housing finger stiffness and are defined by considering the topology optimization result. Finally, the asymmetric shape of the calliper housing is obtained to reduce squeal noise generation.
New Sweet-Tasting C21-Pregnane Glycosides from Pericarps of Myriopteron extensum.
Ten novel C21 pregnane glycosides, extensumside C-L (1-10), were isolated as highly sweet-tasting substances from the edible pericarps of Myriopteron extensum (Wight) K. Schum by sensory-guided fractionation and purification. Their structures were determined through 1D and 2D NMR, such as HSQC, HMBC, 1H-1H COSY, HSQC-TOCSY, and ROESY, as well as other spectroscopic analysis combined with chemical evidence. These compounds shared the same aglycone, 3β,16α-dihydroxy-pregn-5-en-20-one, and contained the deoxysugar chain and the glucose chain which were linked to C-3 and C-16 of the aglycone, respectively. The sweetness potency was evaluated by a human sensory panel test and preliminary structure-taste relationship was discussed. The sweetness intensities of these compounds are between 50 and 400 times greater than that of sucrose. Furthermore, quantitation analyses of compounds 1, 3, 4, and 6 in different parts of M. extensum indicated that the concentrations of these sweet components in the pericarps are obviously higher than those in stems and roots.