An inverse method for structural modification without changing the specified multiple eigen frequencies and their eigen vectors on a whole structure

Abstract

Vibration performance is the one of the important performances in automotive product because of related to not only ride comfort and control performance of vehicle but also durability and noise performance. However, it is difficult to design efficiently because the performance is determined by the influence of all components constituting of a whole structure. In previous paper, we presented an effective way to design vibration performance based on an inverse method for structural modification that keeps one specified eigenfrequency and its eigen vector of the whole structure the same. In this paper, we propose a new method to keep the specified multiple eigenfrequencies and their eigen vectors the same. Firstly, the matrices for structural modification are calculated as dynamic stiffness matrix variation from the zero-divisors of the specified multiple eigen vectors. The design of dynamic stiffness matrix variation is simplified by representing the matrix variation as just stiffness matrix variation which is independence on eigen frequency. Furthermore, it is possible to represent the calculated matrices redundantly by using an arbitrary non-zero weighting matrix. However, the matrices are generally calculated as fully populated and non-symmetric matrices. Therefore, secondly, we transform the matrices calculated as the zero-divisors into the sparse matrix of reduced row echelon form in advance. This transformation simplifies building symmetric and sparse matrices for realizable structural modification. This sparse matrix, redundantly represented by a weighting matrix, allows a value analysis to select among alternative structural changes to simplify structural complexity while keeping several performances originated from vibration performance. Finally, the proposed method was applied to a numerical case study.

Keywords

• structural modification
• modal analysis
• resonance
• modular design
• null space
• zero divisor