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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。
In order to solve the nonlinear parametric resonance problem of horizontal cables, the bridge deck is simplified as a concentrated mass, and the nonlinear differential equation of the cable under harmonic excitation of the bridge deck is established. Considering the influence of the second order modal combination , Then the dimensionless nonlinear equations of motion of cable-bridge coupling are deduced. Finally, taking the cable of actual cable-stayed bridge as an example, the numerical simulation of the response of the cables to the parametric resonance is carried out. The results show that when the ratio of the dimensionless natural frequency Ω3 of the bridge deck to the dimensionless natural frequency Ω1M of the first order of the cable is 2:1, the first two modes of displacement and the mass of the cable appear “beat” Vibration phenomenon, and the first-order mode is the main mode of vibration. When the ratio of the dimensionless natural frequency Ω3 of the bridge mass to the second-order dimensionless natural frequency Ω2M of the cable is 2:1, only the second order Mode displacement and mass appear “beat ” vibration phenomenon, the cable is modeled mainly in the second mode vibration, then the first mode shift does not appear “beat ” vibration phenomenon; consider the first 2 Order modal combination, when the excitation frequency of the mass and the natural frequency of the system satisfy the parametric resonance frequency ratio of 2: 1, the system will have a large “beat” vibration phenomenon and the displacement response of the first order main mode Is much larger than the displacement response of the 2nd order master mode.