多频激励下悬索非线性共振特性受温度影响分析

Investigation of temperature effect on nonlinear oscillation characteristics of suspended cable under multi-frequency excitation

推导了考虑温度变化影响的悬索非线性运动微分方程,利用Galerkin法得到离散后的多自由度方程;考虑一阶正对称模态,以悬索同时发生主共振和1/3阶次谐波共振为例,利用多尺度法求解幅频响应方程组,并判断稳态解的稳定性;选取三组垂跨比及两组温度变化,基于幅频响应曲线和调谐相位曲线,探究温度变化影响下的主/次谐波联合共振响应。数值算例结果表明:主/次谐波联合共振时,系统响应变得更加复杂,同时展现出主共振和次谐波共振响应特性;温度变化会定性和定量地改变联合共振特性,改变系统振动的软/硬弹簧特性及程度;联合共振响应的幅值大小、相位和共振区间与温度变化密切相关;相同温度变化对联合共振响应的幅值和相位影响有差异,通过研究联合共振响应的相位,可以区分系统的多个稳态解。

The nonlinear simultaneous resonance responses of the suspended cable subjected to multi-frequency excitation with thermal effect are investigated in this study. Firstly, the nonlinear vibration equations of motion of the suspended cable with thermal effects under multi-frequency excitation are derived. Then, the Galerkin method is introduced to discretize the system into a multi-degree-of-freedom one. After that, considering the first symmetric mode and assuming the primary and the order 1/3 sub-harmonic resonances simultaneously, the method of multiple scales is used to obtain the frequency-response equations, and the stability of the steady-state solutions is also determined. The temperature effect on the simultaneous resonances is studied through investigating frequency-response curves and detuning-phase curves, and three sag-to-span ratios and two temperature variation conditions are considered in the section of numerical examples. The numerical results show that, in the case of simultaneous resonances, the nonlinear vibration characteristics of the system become much more complex, and different vibration behaviors are exhibited simultaneously. The nonlinear vibration behaviors are changed by the temperature variations quantitatively and qualitatively. Especially, the softening/hardening spring behaviors are changed due to thermal effects. The response amplitude, the phase and the resonance range are all closely related to temperature changes. There are some differences between the temperature effects on response amplitudes and phases in the case of simultaneous resonances, and in order to distinguish the entire stable and unstable solutions, the detuning-phase curves should be investigated.