A high-performance method of vibration analysis for large-scale systems with local strong nonlinearity (Dimension reduction method using new type of complex modal analysis)

Abstract

A rational dimension reduction method based on a new type of complex modal analysis is developed in order to accurately analyze nonlinear vibrations generated in large-scale structures with local strong nonlinearity, non-proportional damping and asymmetric matrix at low computational cost. In the proposed method, first, the linear state variables are transformed into modal coordinates using complex constrained modes obtained by fixing nonlinear state variables. Next, a reduced model is derived by selecting a small number of modal coordinates that have a significant effect on the computational accuracy of the solution, and coupling them with the nonlinear state variables expressed in physical coordinates. In that process, the remaining modal coordinates that have little effect on the computational accuracy are appropriately approximated and integrated into the equations of motion for nonlinear state variables as correction terms. Furthermore, by using a method of estimating the effect of higher-order modes from lower-order modes, the computation of higher-order eigenpairs becomes unnecessary. From the reduced model constructed by these procedures, periodic solutions and their stability, quasi-periodic solutions and chaos can be computed with a very high accuracy and at a high computational speed.

Keywords

• nonlinear vibration
• stability
• method of vibration analysis
• dimension reduction method
• complex modal analysis
• non-proportionally damped system
• asymmetric matrix system