ZHANG Xintao1, ZHAO Yaobing1,2, CAI Shaohui1, GUO Zhirui1
JOURNAL OF VIBRATION AND SHOCK. 2023, 42(3): 50-56.
As to many kinds of dynamical systems’ resonant responses, the direct and discretized methods could be adopted to obtain the approximate solutions. These solutions’ errors are dependent on two aspects: mode discretization and perturbation analysis. Using finite modes to describe the dynamical behaviors of the continuous systems will induce some errors, and the higher-order mode shapes and natural frequencies are neglected, leading to distortion of nonlinear dynamic phenomena. Therefore, no matter in the practical engineering or theoretical analysis, the errors and convergence of modal truncations in discretized method should be paid much attention. Here, based on the internal resonances between two symmetric modes of horizontal suspended cables, the influences of two different mode truncations on system’s resonant responses are investigated. Firstly, the discretized planar nonlinear vibration equations of motions are obtained by using the Galerkin method. Then, the modulation equations are obtained by using the multiple scales method. By comparing the force/frequency response amplitude curves, time history curves, phase plane diagrams, Poincare sections, and Lyapunov exponents, the fluences of two and nine mode truncations on the system’s dynamical behaviors are illustrated in detail. The numerical results show that: as to the internal resonances, the non-direct excited and non-internal resonant modes would affect the resonant responses definitely, and the cause lies in the resonant terms induced by the quadratic nonlinearity. The external excitation is applied on the lower or higher-order mode, and as to the differences induced by mode truncation, the former is significantly higher than the latter. The influences of mode truncations on the response amplitudes seem more obvious in the large resonant regions. The bifurcations are closely related to the mode truncation, and some saddle-node bifurcations might be missed and some extra Hopf bifurcations may be found when two modes discretization is adopted in the case. In these circumstances, the jump phenomena and the dynamic periodic solutions are changed significantly. Different orders of mode truncation could lead to very different systems’ attractors.