Nonlinear dynamics of a Z-shaped structure with validated global analytical mode shapes

Abstract

The nonlinear partial differential governing equations of the planar motion of a Z-shaped structure are derived using Hamilton's principle. The 1:2 internally resonant global analytical mode shapes are validated by figure contrast and the modal assurance criterion (MAC). The partial differential governing equations are truncated into a two-degree-of-freedom ordinary differential system with the validated resonant global analytical mode shapes, and they are further investigated for internal and simultaneous primary resonances. The steady-state responses are studied, and numerical simulations of the system are performed. Complex nonlinear phenomena in the system, such as jumps, bifurcations, quasiperiodic motion and chaos, are observed under specific parameters. The parametric rule of these phenomena obviously depends on the accuracy of mode shapes. This work proposes a nonlinear analysis based on the quantitative validation of mode shapes, which may provide new insights into the optimal design of the parameters and the precise control of the motions of Z-shaped or other multibeam structures in engineering.