Vibration Dynamics of an Inclined Cable Excited Near Its Second Natural Frequency
Vibration Dynamics of an Inclined Cable Excited Near Its Second Natural Frequency
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DOI：
10.1142/s0218127414300249
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发表时间：
2014-10
期刊：
影响因子：
--
通讯作者：
V. Tzanov;B. Krauskopf;S. Neild
中科院分区：
文献类型：
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作者：
V. Tzanov;B. Krauskopf;S. Neild
斜拉索是斜拉桥中最常用的重要结构元件。当桥面因外力（例如过往车辆）而振动时，索振动不仅会在激励平面内产生，而且会在垂直平面内产生。这种不良现象可以建模为电缆的面内和面外振动模式之间的自参数共振。在本文中，我们考虑一个三模模型，捕获第二面内模、第一面外模和第二面外模，并用它来研究倾斜电缆的响应，该倾斜电缆在其下（甲板）支撑处以接近电缆第二固有频率的频率垂直激励。对模型应用平均，然后研究所得平均微分方程的解和分岔，并通过数值延拓绘制出来。通过这种方式，我们对电缆的不同可能响应进行了详细的分叉研究。我们首先考虑平均模型的平衡，其中有四种类型，通过电缆响应中是否存在两种面外模式来区分。每种类型的平衡都经过计算并表示为力的幅度和频率平面上的表面。平衡的稳定性发生变化，不同的表面沿着分叉曲线相交，分叉曲线直接延续。总的来说，我们展示了电缆恒幅耦合模式响应的二参数分叉图的全面几何图。然后，我们关注分叉周期轨道，这对应于参与的第二面内和第二面外模式的不同振幅的缆线动力学。激励幅度和频率的范围取决于可能发生这种旋转电缆运动的位置。进一步的分岔——倍周期级联和 Shilnikov 同宿分岔——被发现会导致混乱的电缆响应。完整三模模型的时间步模拟证实了旋转和混沌电缆动力学。不同的电缆响应具有特征，并且可以通过它们在四分之一跨度处的运动和频谱来清楚地区分。
Inclined cables are essential structural elements that are used most prominently in cable-stayed bridges. When the bridge deck oscillates due to an external force, such as passing traffic, cable vibrations can arise not only in the plane of excitation, but also in the perpendicular plane. This undesirable phenomenon can be modeled as an auto-parametric resonance between the in-plane and out-of-plane modes of vibration of the cable. In this paper, we consider a three-mode model, capturing the second in-plane, and first and second out-of-plane modes, and use it to study the response of an inclined cable that is vertically excited at its lower (deck) support at a frequency close to the second natural frequency of the cable. Averaging is applied to the model and then the solutions and bifurcations of the resulting averaged differential equations are investigated and mapped out with numerical continuation. In this way, we present a detailed bifurcation study of the different possible responses of the cable. We first consider the equilibria of the averaged model, of which there are four types that are distinguished by whether each of the two out-of-plane modes is present or not in the cable response. Each type of equilibrium is computed and represented as a surface over the plane of amplitude and frequency of the forcing. The stability of the equilibria changes and different surfaces meet along curves of bifurcations, which are continued directly. Overall, we present a comprehensive geometric picture of the two-parameter bifurcation diagram of the constant-amplitude coupled-mode response of the cable. We then focused on bifurcating periodic orbits, which correspond to cable dynamics with varying amplitudes of the participating second in-plane and second out-of-plane modes. The range of excitation amplitude and frequency is determined where such whirling cable motion can occur. Further bifurcations — period-doubling cascades and a Shilnikov homoclinic bifurcation — are found that lead to a chaotic cable response. Whirling and chaotic cable dynamics are confirmed by time-step simulations of the full three-mode model. The different cable responses are characterized, and can be distinguished clearly, by their motion at the quarter-span and by their frequency spectra.