Mechanical Engineering Journal (Aug 2023)
Substructure elimination method for vibration systems governed by a one-dimensional wave equation
Abstract
This paper presents a vibration analysis using the substructure elimination method for vibration systems governed by a one-dimensional wave equation. The utilization of a one-dimensional continuous body serves as a suitable approach for understanding the fundamental aspects of physical phenomena. Consequently, the significance of one-dimensional computer-aided engineering has witnessed a noticeable upsurge in recent years. The vibration analysis of continuous bodies through modal analysis is effective for reducing the degrees of freedom (DOFs). In addition, modal analysis of continuous bodies using the substructure elimination method can reduce the DOFs further, compared with modal analysis using the substructure synthesis method. However, the substructure elimination method was reported only briefly by the first author, and several problems remained. Focusing on continuous bodies governed by one-dimensional wave equations, this study aimed to address the aforementioned problems by devising solutions and establishing criteria for the effective utilization of the substructure elimination method. As a versatile method for setting arbitrary boundary conditions, a new formulation method based on constraint conditions was proposed. In addition, appropriate material properties of the elimination regions and highest order of the eigenmode were determined through a simulation-based investigation. The effectiveness of the substructure elimination method was verified by comparing the simulation results obtained using the substructure elimination method and exact solutions obtained using the boundary conditions. To investigate the advantage of low DOFs, the simulation results obtained using the substructure elimination method were also compared with those obtained using the substructure synthesis method. As an example, in a simulation of a 0.85 m one-dimensional acoustic field with a non-reflective boundary, highly precise results were obtained below 1200 Hz using 15 DOFs and the substructure elimination method.
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