Substructure elimination and binding method for vibration systems governed by a one-dimensional wave equation

Abstract

This paper presents a vibration analysis using the substructure elimination and binding method for vibration systems governed by a one-dimensional wave equation. Coupled vibration analysis has been developed, and the component mode synthesis method is commonly used for dynamic analysis. In the component mode synthesis method, each substructure is formulated, and then coupling between substructures is considered. The component mode synthesis method is a type of modal analysis, and the coupled vibration between vibration systems with different governing equations can be easily formulated. The component mode synthesis method has the problem of increasing the degrees of freedom when the entire structure is complicated and needs to be divided into many substructures. Therefore, the first author proposed methods to analyze the entire vibration system without dividing it into substructures, for example, when a structure is installed inside an acoustic field or when acoustic fields with different media are in contact. These methods have the advantage that only the eigenmodes of the entire acoustic field are used. However, the calculation accuracy has been found to deteriorate because of the discontinuities or non-smooth points in sound pressure and particle displacement at the interface between air and a structure or between two acoustic fields. This study proposed a method to set a virtual elimination region at the interface and then bind the two ends of the virtual elimination region to solve this problem. The analytical model for this method was presented, and a wave equation was derived in this study. Modal analysis was applied to the wave equation. The simulations revealed that the density and bulk modulus of the virtual elimination region should be zero and that its length should be set at 2.5–3.5 times the wavelength of the highest eigenmode of the entire vibration system. To investigate the advantage of low DOFs, the simulation results obtained using the proposed method were compared with those obtained using the component mode synthesis method based on the exact solutions.

Keywords

• wave equation
• modal analysis
• coupled vibration
• continuum vibration
• acoustic field
• simulation