端部激励下斜拉索三维振动统一方程的精细推导

为了研究斜拉索在端部激励下的振动机理，考虑了斜拉索垂度、刚度、非线性、空间三维、端部位移等诸多因素，经过一系列精细推导，利用Galerkin多模态截断方法，建立了单自由度斜拉索三维振动的统一方程，并提出了斜拉索三维振动频率分量由各自方向上的固有频率和端部位移激励扰动项组成。斜拉索在考虑三维端部位移理想激励时，采用4~5阶龙格-库塔法编写了端部位移激励与斜拉索固有频率比值以1：1和2：1进行振动的MATLAB数值求解程序。研究表明：当频率比值以1：1振动时，位移响应规律性的拍频消失，转变为拍频叠合的效果，且其振动幅值在时间域上表现出高低起伏的趋势，其位移响应峰值、谷值均呈现出不断增大趋势。此外，斜拉索在端部很小的位移激励下便可产生较大的振动。当频率比值以2：1振动时，其位移响应表现出与一阶强迫共振同样的拍频特征。参数振动位移响应振幅正负值同样出现了偏差值，此条件下，微小的端部位移也能够激起较大的参数振动响应，因此参数振动同强迫振动一样不容忽视。斜拉索在端部位移作用下不仅有频率比为1：1的共振和2：1的参数共振，还发生了1：2的共振，即斜拉索在端部位移激励下具有3个主共振区，分别在频率比值为0.5，1，2周围，1：2共振产生的振幅明显小于1：1共振和2：1参数共振。

Fine Derivation of the Unified Equation for Three-dimensional Vibration of Stayed Cable Under End Excitation

In order to study the vibration mechanism of stay cables under the excitation of cables, this paper considered several factors such as sag, stiffness, nonlinearity, three-dimensional space, and end displacement of cable stays. After a series of fine derivations, the Galerkin multimodal truncation method was used. Based on this method, a unified equation for three-dimensional vibration of a single-degree-of-freedom cable was established, and a three-dimensional vibration frequency component of the cable consisting of the natural frequency in its different directions and the perturbation terms of the end displacement excitation. When considering the ideal excitation of the three-dimensional end displacement using the four-or five-order Runge-Kutta method, the MATLAB program was compiled while the ratios of the end displacement excitation and natural frequency of the stay cable were 1:1 and 2:1. Studies have shown that when the frequency ratio is 1:1, the beat frequency regularity of the displacement response disappears and a beat frequency superposition effect appears. Its vibration amplitude shows a fluctuating trend in the time domain, and the displacement response of the peak value and trough value show an increasing trend. In addition, the cable stays under small displacement excitation of the end can generate large vibrations. When the frequency ratio is vibrated at 2:1, the displacement response exhibits the same beat characteristics as the first-order forced resonance. The amplitude of the displacement response of the parameter vibration also shows a deviation. Similarly, tiny end displacements can excite parametric vibrational responses with large displacements. Therefore, parametric vibration cannot be neglected just like forced vibration. This paper also analyzed the amplitude and frequency of stay cables. Under the action of the end displacement, the stay cables not only had resonance with a frequency ratio of 1:1 and parametric resonance of 2:1, but also had 1:2 resonance. The cable has three main resonance zones under end displacement excitation. The frequency ratio is approximately 0.5, 1, and 2, and the amplitude generated by the 1:2 resonance is obviously less than that by 1:1 vibration and 2:1 parametric vibration.