小参数法在悬垂索共振中的应用

The application of small parameter method in suspended cable resonance

当面外横向振动和面内横向振动频率的比接近1:2时,悬索会出现面内和面外耦合共振现象。为了研究悬索这种复杂独特的非线性特性,利用多尺度法对谐波激励的悬索动力学方程进行求解,得到对应于不同阶小量的偏微分方程组,其中二阶小量偏微分方程中的久期项不为0;采用提出的小参数法可以得到由久期项引起的悬索振动形态,解决久期项频率与系统频率相同但不能直接求解的问题;为了证明小参数法的准确性,采用Galerkin方法离散悬垂索的运动方程,然后利用多尺度法求解离散的运动方程,得到采用基函数描述的由久期项引起的连续系统的振动形态,与小参数法结论一致。

An out-of-plane mode and an in-plane mode are coupled when their natural frequencies are in 1:2 ratio in a suspended cable.In order to investigate the unique and complex nonlinear motion of the suspended cable,equations of the suspended cable under the harmonic excitation are solved by multiscale method.The set of differential equations at different orders of infinitesimal is obtained,in which the second-order secular terms are unequal to zero.Vibration modes of a suspended cable induced by secular terms,which cannot be directly solved due to the same frequency of secular term and suspended cable,are obtained by small parameter method.In order to validate the reliability of the small parameter method,the equations of suspended cable motion are discretized by Galerkin method,and then the discrete equations are solved by multiscale method.The results on the mode shapes of continuous system are in agreement with those of small parameter method.