考虑温度效应的悬索超谐波共振响应研究

摘 要：引入與张拉力和垂度相关的无量纲参数，可推导出整体温度变化影响下悬索面内非线性运动微分方程.首先利用Galerkin法得到离散后的无穷维方程，然后运用多尺度法求解其二阶和三阶单模态超谐波共振响应的近似解，并得到幅频响应方程，最后通过数值算例探究三种垂跨比的悬索，温度变化对其单模态超谐波共振响应的影响.研究结果表明：当垂跨比较小时，一定程度的温度变化会导致其超谐波共振响应发生定性和定量的改变，改变系统软硬弹簧的性质和程度，即改变其幅频响应曲线的偏转方向和程度；温度变化会改变激励-响应幅值曲线的单值/多解情况.然而当垂跨比逐渐增大后，温度变化对其共振响应仅会产生定量的影响，且温度变化与幅频响应曲线向左偏转程度呈正比，即温度上升，曲线向左偏转程度加剧，软弹簧特性增强，反之则减弱.由于悬索存在初始张拉力，升高和降低相同温度对超谐波共振特性影响呈现出不对称性.

关键词：悬索；温度变化；多尺度法；超谐波共振；幅频响应曲线

中图分类号： 文献标志码：A

Investigation of Super-harmonic Resonances of Suspended Cables Considering Temperature Variation Effects

ZHAO Yaobing 1†， JIN Bo 2， ZHAO Yueyu 2， HUANG Chaohui 1

（1.College of Civil Engineering， Huaqiao University， Xiamen 361021 China；

2. College of Civil Engineering， Hunan University， Changsha 410082 China）

Abstract： By introducing two non-dimensional parameters on the cable tension force and sag， the in-plane nonlinear equations of motion of the suspended cable considering the temperature effects were derived. Firstly， the nonlinear partial differential equation was discretized by the Galerkin method. Then， the approximate solutions of the second and third order single mode super-harmonic resonances of the suspended cable were obtained by the multiple scales method， and the corresponding frequency response equations were also derived. Finally， the effects of temperature variations on the single mode super-harmonic resonances of the suspended cable with three different sag-to-span ratios were illustrated by the numerical calculations. The numerical results show that： in the case of the small sag-to-span ratio， the nonlinear vibration characteristics would be changed by a certain degree of the temperature change quantitatively and qualitatively， and the softening and hardening spring behaviors are changed， and it means that the degree and direction of the frequency response curves are changed. The single/multi-valued of the excitation-response amplitude curves are also varied by the temperature variations. However， with the increasing of the sag-to-span ratio， only some quantitative changes are found under the thermal effects. With the increase of the temperature changes， the frequency response curves are bent to the left much more， and the softening spring characteristic is also increased. Moreover， due to the initial tension force of the suspended cable， the effects of warming and cooling conditions on the vibration characteristics of the suspended cable are not symmetric.