温度场中悬索受多频激励组合联合共振响应研究

基于温度变化对拉索张拉力和垂度的影响，利用Hamilton变分原理，引入拟静态假设，推导温度场中受多频激励下悬索的非线性运动微分方程。利用Galerkin法得到离散后的无穷维方程，并考虑一阶正对称模态，利用多尺度法求解系统发生组合联合共振时的幅频响应方程组，并判断稳态解的稳定性。考虑四组垂跨比及四种温度变化工况，通过数值算例探究悬索组合联合共振的响应特性及其受温度变化影响。研究结果表明：多频激励时系统同时展现出组合共振和超谐波共振响应的特性；此时稳态解个数、共振区间、响应幅值及其相位等均会发生改变；温度变化会使得组合共振和超谐波共振发生定性和定量的变化，从而导致联合共振响应亦发生明显的定性和定量的改变；组合联合共振响应受温度变化的影响与悬索的垂跨比和温度变化幅度密切相关；为了更好地区分系统受多频激励下的的稳态解，可以通过研究解的相位来分辨。
Abstract：Here, considering effects of temperature on cable’s tension force and sag, nonlinear equations of motion of a suspended cable under multi-frequency excitation were derived by using Hamilton’s principle and the quasi-static hypothesis.Galerkin procedure was adopted to discretize nonlinear dynamic equations.When the system’s combined joint resonances happened, the amplitude-frequency response equations were solved using the multi-scale method considering the first order symmetric mode, and the stability of their steady-state solutions was judged.Considering 4 sag-to-span ratios and 4 temperature variation cases, through numerical examples the response characteristics of the suspended cable system’s combined joint resonances and the effects of temperature were studied.The results showed that the system under multi-frequency excitation exhibits the characteristics of combined resonances and super-harmonic resonances simultaneously; number of steady-state solutions, interval of resonances, amplitude and phase of responses all vary; temperature variation can make combined resonances and super-harmonic ones have qualitative and quantitative changes; temperature variation affects the system’s combined joint resonance responses, they are closely related to the cable’s sag-to-span ratio and temperature change amplitude; studying phase of steady-state solutions under multi-frequency excitation can distinguish these solutions better.