三阶模态耦合效应下悬索的1:1主共振与稳定性分析

针对悬索的振动,研究了模态耦合效应对悬索振动特征的影响。首先基于哈密顿原理推导了考虑抗弯刚度影响的悬索的偏微分振动方程,采用Galerkin方法得到了悬索的前三阶模态耦合振动常微分方程组。采用多尺度法分析了悬索的一阶、二阶和三阶主共振,得到了一阶、二阶和三阶主共振的幅-频响应方程,接着基于Lyapunov稳定性理论进行了稳定性分析,最后进行了数值算例分析。算例分析表明,当1:1主共振发生时,一阶主共振产生的幅值远大于二阶和三阶主共振产生的幅值,即当悬索振动时,能量主要以一阶模态幅值的形式散发;在同阶次幅值-σ曲线中,随着F的增加,1:1主共振产生的幅值有所增加;在幅值-V曲线中,随着σ的增加,临界跳跃点有向右偏移的趋势,σ增加会导致幅值增加;档距越大,一阶、二阶和三阶1:1主共振产生的幅值越大,但一阶主共振产生的幅值增加最为明显。

1:1 primary resonance and stability analysis about suspension cable based on modal shapes coupling effect

Aiming at the vibration of suspension cable,the influences of mode shapes coupling effect on the vibration characteristics of suspension cables are studied.Based on the variational principle for Hamiltonian,the partial differential equations of suspension cables considering the influences of bending stiffness are derived,and the first three-order mode shapes coupled ordinary differential equations of suspension cables are obtained by Galerkin method.The first-order,second-order and third-order primary resonance of the suspension are analyzed by the multiscale method,and then the amplitude-frequency response equations of the first-order,second-order and third-order primary resonances are obtained.Then the stability analysis is also carried out based on the Lyapunov stability theory and the numerical examples are analyzed.The results of the numerical examples show that when the 1:1 primary resonance occurs,the amplitude of the first-order resonance is much larger than that of the second-order and third-order primary resonances;in the same order amplitude-σ curve,the amplitude of 1:1 primary resonance increases with the increases of F;in the amplitude-V curve,the critical jumping point has a right deviation trend with the increase of σ,and the increases of σ would lead to the increase of amplitude;the larger the span length is,the larger the amplitudes of 1:1 primary resonances of the first-order,second-order and third-order are,but the amplitude of the first-order primary resonance increases most obviously.