Non-linear dynamic model of a cable–mass system with a transverse tuned mass damper is considered. The system is moving in a vertical host structure therefore the cable length varies slowly over time. Under the time-dependent external loads the sway of host structure with low frequencies and high amplitudes can be observed. That yields the base excitation which in turn results in the excitation of a cable system. The original model is governed by a system of non-linear partial differential equations with corresponding boundary conditions defined in a slowly time-variant space domain. To discretise the continuous model the Galerkin method is used. The assumption of the analysis is that the lateral displacements of the cable are coupled with its longitudinal elastic stretching. This brings the quadratic couplings between the longitudinal and transverse modes and cubic nonlinear terms due to the couplings between the transverse modes. To mitigate the dynamic response of the cable in the resonance region the tuned mass damper is applied. The stochastic base excitation, assumed as a narrow-band process mean-square equivalent to the harmonic process, is idealized with the aid of two linear filters: one second-order and one first-order. To determine the stochastic response the equivalent linearization technique is used. Mean values and variances of particular random state variable have been calculated numerically under various operational conditions. The stochastic results have been compared with the deterministic response to a harmonic process base excitation.

中文翻译：

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具有调谐质量阻尼器的电缆系统通过等效线性化技术对随机基础激励的非线性动态响应

考虑了具有横向调谐质量阻尼器的索-质量系统的非线性动力学模型。系统在垂直主机结构中移动，因此电缆长度随时间缓慢变化。在随时间变化的外部载荷下，可以观察到低频和高振幅的主体结构的摇摆。这会产生基础激励，进而导致电缆系统的激励。原始模型由非线性偏微分方程系统控制，在缓慢时变空间域中定义了相应的边界条件。为了离散连续模型，使用伽辽金方法。分析的假设是电缆的横向位移与其纵向弹性拉伸相耦合。由于横向模式之间的耦合，这带来了纵向和横向模式之间的二次耦合和三次非线性项。为了减轻谐振区域中电缆的动态响应，应用了调谐质量阻尼器。随机基激励，假设为窄带过程均方等效于谐波过程，在两个线性滤波器的帮助下被理想化：一个二阶和一个一阶。为了确定随机响应，使用等效线性化技术。特定随机状态变量的平均值和方差已经在各种操作条件下进行了数值计算。随机结果已与对谐波过程基础激励的确定性响应进行了比较。