端部激励相位差下斜拉索的非线性振动

斜拉桥中拉索承受着多种端部激励，可激发大幅空间振动．以斜拉索为对象，探究不同端部激励间相位差对其非线性振动的影响．首先，推导斜拉索无量纲离散控制方程，引入考虑相位的三向端部激励得到一般化模型；然后，针对拉索下端存在的纵桥向、竖向和横桥向激励的两两组合，受大幅或小幅激励，及其在主共振区或主参数共振区几组因素，共计12种工况，采用数值分析法分别研究了各工况下不同激励相位差时的斜拉索稳态响应．研究发现：激励相位差能加剧与激励频率相近的面内、外模态振动；在任意端部激励组合下，激励相位差不仅可使斜拉索非线性振动出现定量变化，还可改变内共振的表现形式．面内、外激励组合下，相位差对拉索响应幅值的影响以π为周期变化，且当相位差趋于π/2 + kπ (k = 0, 1, 2…)时影响最为突出；而面内激励组合下，以2π为变化周期，当相位差为π + 2kπ (k = 0, 1, 2, …)时其对稳态幅值的影响最显著．其原因是：面外激励关于拉索所在的竖直面对称，故其本质上以π为周期；而面内激励无此对称性，仍以2π为周期．因此，有无面外激励参与决定了激励间相位差对斜拉索响应的影响规律．

Nonlinear Vibrations of a Stay Cable under Support Excitation with Phase Differences

Cables of cable-stayed bridges are suffering from various support excitations, which would lead to large amplitude spatial vibrations. The influences of phase differences among support excitations on the nonlinear vibrations of a stay cable have been investigated. Firstly, the discretized nonlinear vibration equations were deduced, and a general mathematical model was obtained by inducing 3-D excitations with consideration of phases. Secondly, totally 12 cases were determined by comprehensively considering the influencing factors such as the in-pair combination of excitations in longitudinal, vertical and horizontal directions, the small and large amplitude excitations, and the primary and sub-harmonic resonance regions. The steady-state responses under those cases in various excitation phase differences were studied by numerical analysis. The study has shown that the excitation phase differences had intensified the in-plane and out-of-plane modal vibrations that had approximately the same frequencies as the excitations. In any support excitation combination, the excitation phase differences had caused quantitative changes of the cable nonlinear dynamic characteristics and even led to the variation of the internal resonance style. Under the combination of in-plane and out-of-plane excitations, the influence of the excitation phase difference on the response amplitude were in periods of π. Moreover, the effect on the vibration responses was most significant when the excitation phase difference tends to π/2 + kπ (k = 0, 1, 2…). On the other hand, the periods were 2π under the combination of in-plane excitations, and the excitation phase difference obtained the most obvious effect on the steady-state amplitudes as the phase differences approached π + 2kπ (k = 0, 1, 2, …). The reason is that the out-of-plane excitation is symmetric on the cable’s vertical plane, so it is essentially in the period of π. While the in-plane excitation is in the period of 2π. Therefore, out-of-plane excitation plays the major role of the effect of excitation phase difference on cable’s responses.