The longitudinal time-delay feedback control strategy is implemented to suppress the subharmonic and superharmonic responses of the suspended cable. Formulated based on the Hamilton variational principle and the longitudinal time-delay feedback strategy, the in-plane nonlinear differential equations for a suspended cable are formulated, and Galerkin method is utilized to transform the equations into delayed differential equations. By using the method of multiple scales, approximate solutions for the 1/2-subharmonic and second-order superharmonic resonance responses of the controlled suspended cable were derived. The effects of the time delay and control gain are obtained through numerical examples. The research results demonstrate that through the adjustment of control gain and time delay, resonance regions can be avoided, effectively suppressing large-amplitude vibrations in the suspended cable, thereby achieving favorable control performance.

中文翻译：

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次谐波和高谐波谐振驱动悬索的时滞反馈控制

采用纵向时滞反馈控制策略来抑制悬索的次谐波和超谐波响应。基于哈密顿变分原理和纵向时滞反馈策略，建立了悬索面内非线性微分方程，并利用伽辽金法将方程转化为时滞微分方程。采用多尺度方法，推导了受控悬索1/2次谐波和二阶超谐波谐振响应的近似解。通过数值算例得到了时滞和控制增益的影响。研究结果表明，通过调节控制增益和延时，可以避开共振区，有效抑制悬索中的大幅振动，从而获得良好的控制性能。