悬索在外激励作用下的1∶3内共振分析Ⅰ∶离散法

On the onevs.three internal resonances of suspended cables subjected to external excitation Ⅰ∶discretization

研究了悬索在受到外激励作用和考虑1∶3内共振情况下的两模态非线性响应。对于一定范围内的悬索弹性-几何参数而言,悬索第三阶面内对称模态的固有频率接近于第一阶面内对称模态的固有频率的3倍,从而导致1∶3内共振的存在。首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动,可得到两组不同主共振情况下的平均方程。

The two-mode nonlinear response of the suspended cable subjected to the external excitation with onevs.three internal resonance is investigated.Taking into account the geometric non-linearity of the suspended cable and the assumption of quasi-static stretching due to the fact that the transverse wave speed is much lower than the longitudinal one,the non-linear governing in-plane equation of the suspended cable is derived by means of Hamilton principle.And it is shown that onevs.three internal resonances between the first and third symmetric modes may be activated for a range of elasto-geometric parameters.First,the Galerkin method is used to discretize the nonlinear equation of planar motion.Then the method of multiple scales is applied to perturb the discretized equations of motion.And it shows,except the amplitudes of the first and third symmetric mode,other amplitudes will die out due to the presence of damping.Thus,the solvability conditions reduce to a two-dimensional system.At last,the averaged equations under two cases of primary resonances are obtained.