损伤效应对悬索2:1内共振响应影响分析

损伤是结构振动测试和运营维护中不可避免的问题，损伤效应会导致结构振动特性发生改变．本文以受损悬索为例，探究该非线性系统同时发生主共振和2:1内共振时，损伤效应对其面内耦合共振响应影响．首先基于哈密顿变分原理，引入与损伤程度、范围和位置相关的三个无量纲参数，建立受损悬索面内动力学模型，并推导其无穷维非线性运动微分方程．以2:1耦合共振为例，采用Galerkin法和多尺度法得到系统直角坐标形式的调谐方程．数值算例表明：损伤会导致悬索固有频率降低，使得频率间公倍关系发生改变，影响系统耦合共振响应；损伤会引发系统振动特性发生明显定量和定性改变，尤其是共振响应幅值及弹簧特性；损伤对直接激励模态响应幅值的影响比对内共振激发对响应幅值的影响要明显；损伤会导致霍普夫、鞍节点、叉形和倍周期分岔的位置发生偏移，从而影响分岔点附近系统的动力学行为；系统动态解和周期运动与损伤密切相关，损伤会导致系统展现出完全不同类型的吸引子．

Damage Effects on 2:1 Internal Resonant Responses of Suspended Cables

Structural damage is an inevitable problem in vibration testing and operation, and the vibration characteristics of the damaged structures are changed more or less. Therefore, this paper takes the damaged suspended cable as an example to explore the influences of damage effects on the in-plane coupled resonant responses of the nonlinear system when the primary resonance and the two-to-one internal resonance occur simultaneously. Firstly, based on Hamiltonian variational principle, three dimensionless parameters of the damage intensity, extent and position are introduced to establish the in-plane dynamic model of the damaged suspended cable, and then obtain the infinite dimensional nonlinear differential equations. Taking the two-to-one internal resonance as an example, the Galerkin method and multi-scale method are adopted to obtain the harmonious equations of the system in Cartesian coordinate system. Numerical examples show that the damage effetcs can reduce the natural frequencies of the suspended cable, resulting in change of common multiple relations between the modal frequencies, and affecting the coupled resonant responses of the system. The vibration characteristics of the system are also changed quantitatively and qualitatively due to the damage effects. Specifically, the resonant response amplitudes and the spring behaviors are changed. The influences of damage effects on the directly excited amplitudes are much more evident than that on the internally excited ones. The damage effects will cause the shifts of Hopf, saddle-node, pitchfork and period-double bifurcations, as well as the change of the dynamic behaviors around the bifurcation points. The dynamic solutions and the periodic motions of the system are closely related to the damage effects, and some completely different types of attractors are induced by the damage effects.