内外共振联合作用下索-梁组合结构非线性振动分析

Nonlinear vibration analysis for a cable-beam coupled system under simultaneous internal and external resonances

研究内共振和外共振联合作用下的索-梁组合结构非线性振动问题。利用Hamilton原理推导索-梁组合结构非线性动力学方程,同时考虑索的垂度以及由梁和索之间模态耦合引起的非线性影响。利用Galerkin方法将索-梁组合结构非线性运动偏微分方程离散为一组常微分方程。最后对数学模型进行数值计算,得到了不同内外共振联合作用下梁和索的模态时程曲线。研究表明,梁的稳态运动呈现周期性振荡,而索在不同的内外共振联合作用下,分别呈现出混沌或周期性振荡,并且索和梁之间持续的模态交替现象只能在特定的内外共振下出现。

Here, the nonlinear vibration of a cable-beam coupled system whose beam was subjected to vertical harmonic force was investigated. The combined effects of nonlinear terms due to the cable＇s geometry and the coupled behavior between the beam and the cable were considered. The nonlinear partial-differential equations were derived with Hamihonian principle. Galerkin method was used to discrete the governing equations into a set of ordinary differential equations and their numerical analysis was conducted by using the fourth order Runge-Kutta method. The results showed that the steady state motion of the beam always is periodic oscillation, the cable reveals chaotic or periodic oscillation under the combination of different internal and/or external resonances; moreover, the phenomenon of mode alternation only occurs in certain case.