考虑前2阶模态组合的拉索非线性参数共振研究

Nonlinear Parametric Resonance of Cable in Consideration of First Two Modes Combination

针对水平拉索的非线性参数共振问题,首先将桥面简化为集中质量块,建立拉索在桥面谐波位移激励下的非线性微分方程,并考虑拉索前2阶模态组合的影响,进而推导索-桥耦合的无量纲非线性运动方程组,最后以实际斜拉桥工程的拉索为例,对拉索发生参数共振的响应进行数值模拟分析。研究表明:当桥面质量块无量纲固有频率Ω3与拉索1阶无量纲固有频率Ω1M之比为2∶1时,拉索的前2阶模态位移和质量块均出现"拍"振现象,且以第1阶模态为主振动模态;而当桥面质量块无量纲固有频率Ω3与拉索2阶无量纲固有频率Ω2M之比为2∶1时,仅有第2阶模态位移和质量块出现"拍"振现象,拉索以第2阶模态为主模态振动,此时第1阶模态位移未出现"拍"振现象;考虑前2阶模态组合时,当质量块的激励频率与系统的固有频率满足参数共振频比关系2∶1时,系统将发生大幅"拍"振现象,且第1阶主模态的位移响应要远大于第2阶主模态的位移响应。

An approach for investigating the nonlinear parametric resonance of horizontal cable was proposed in this paper. Firstly, the bridge deck was simplified as concentrated mass block, and nonlinear differential equation of cable under harmonic displacement excitation of bridge deck was derived. Meanwhile the influence of first two modes of cable was considered. Then the non- linear vibration differential equations were derived in dimensionless style by considering the cable- bridge coupled effect. At last, a numerical simulation analysis of cable parametric resonance was carried out based on a real stayed-cable bridge. The results show that when the ratio of deck mass block dimensionless frequency Ω3 to first cable dimensionless frequency Ω1M is 2 ： 1, the beat phe- nomenon occurs between the first two cable modal displacements and mass block, and the first mode is the major mode of vibration. However, when the ratio of deck mass block dimensionless frequency Ω3 to second cable dimensionless frequency Ω1M is 2 ： 1, the beat phenomenon occurs on- ly between the second cable modal displacement and mass block, and the second mode is major mode of vibration. This study indicated that for the first two modes of cable, when the paramet- ric resonance ratio relation of mass block excitation frequency and system natural frequency is 2. 1, the large amplitude beat vibration of system occurs, and the displacement amplitude of first mode is much larger than that of the second mode.