On Nonlinear Motions of Two-Degree-of-Freedom Nonlinear Systems with Repeated Linearized Natural Frequencies

Author:

An Hua-Zhen1,Yang Xiao-Dong2ORCID,Liang Feng2,Zhang Wei2,Yang Tian-Zhi3,Ren Yuan4

Affiliation:

1. Institute of Advanced Structure Technology, Beijing Institute of Technology, Beijing 100081, P. R. China

2. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, P. R. China

3. Department of Mechanics, Tianjin University, Tianjin 300072, P. R. China

4. Department of Aerospace Science and Technology, Space Engineering University, Beijing, P. R. China

Abstract

In this paper, we investigate systematically the vibration of a typical 2DOF nonlinear system with repeated linearized natural frequencies. By application of Descartes’ rule of signs, we demonstrate that there are 14 types of roots describing different modal motions for varying nonlinear parameters. The method of multiple scales is used to obtain the amplitude-phase portraits by introducing the energy ratios and phase differences. The typical nonlinear in-unison and elliptic out-of-unison modal motions are located for the 14 types of roots and then validated by numerical simulations. It is found that the normal in-unison modal motions, elliptic out-of-unison modal motions are analogous to the polarization of classical optic theory. Further, some kinds of periodic and chaotic motions under out-of-unison and in-unison excitations are investigated numerically. The result of this study offers a detailed discussion of nonlinear modal motions and responses of 2DOF systems with cubic nonlinear terms.

Funder

National Natural Science Foundation of China

Natural Science Foundation of Beijing Municipality

Publisher

World Scientific Pub Co Pte Lt

Subject

Applied Mathematics,Modelling and Simulation,Engineering (miscellaneous)

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