STUDY ON NONLINEAR COUPLED VIBRATIONS OF DAMAGED SUSPENDED CABLES WITH SYMMETRY-BREAKING
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摘要: 对称性是振动理论中5大美学特征之一, 然而对称性破缺又难以避免. 本文以工程中常见的易损结构—悬索为例, 探究当该系统遭遇非对称性损伤时, 对称性破缺对其面内耦合振动特性影响. 首先建立受损悬索面内非线性动力学模型, 并采用Galerkin法得到离散的无穷维微分方程. 利用多尺度法计算该非线性系统发生面内耦合共振响应的调谐方程. 截取前9阶模态, 利用数值计算方法得到无损和受损悬索的各类共振曲线及其稳定性, 通过计算最大李雅普诺夫指数来确定系统的混沌运动. 研究结果表明: 已有研究常采用抛物线模拟悬索静态构形, 然而一旦发生不对称损伤, 采用分段函数更能准确描述悬索受损后的静态构形; 对称性破缺会导致悬索固有频率之间的交点变为转向点, 其正、反对称模态均变为非对称模态; 受损后悬索的非线性相互作用系数会发生显著改变, 其内共振响应会产生明显变化; 当激励直接作用在高阶模态时, 无损系统会呈现出单模态解和内共振解, 而受损系统并没有呈现出明显的单模态解; 受损系统的分岔和混沌特性会发生改变, 系统将通过倍周期分岔产生混沌运动.Abstract: Symmetry is one of the five aesthetic characteristics in the vibration theory, but the symmetry-breaking is also inevitable. This paper takes a common vulnerable structure in engineering-the suspended cable-as an example, and the influences of symmetry-breaking on the planar coupled vibrations have been investigated when the asymmetric damage is occurred. Firstly, the in-plane nonlinear dynamical model of damaged suspended cable has been established, and the nonlinear infinite dimensional differential equations have been obtained by using the Galerkin method. The method of multiple scales has been adopted to obtain the modulation equations of the nonlinear systems’ in-plane coupled vibrations. The resonant curves of undamaged and damaged suspended cables including the first nine modes have been obtained by using the numerical methods, and the stabilities of solutions have also been determined. The largest Lyapunov exponent has been calculated to determine the system’s chaotic motions. The numerical results show that the classical parabolic curves have been often adopted to simulate the suspended cables’ static configurations. However, when the asymmetric damage occurs, the piecewise functions should be used to accurately describe the damaged cables’ static configurations. The symmetry-breaking causes crossover points between two natural frequencies of suspended cables to turn into veering points, and the symmetric/anti-symmetric mode shapes before damage are changed into the asymmetric ones after damaged. The nonlinear interaction coefficients are changed significantly, resulting in significant changes in internal resonant responses. When the excitation is directly applied to the higher-order modes, the single-mode solutions and internal resonant ones are obvious in the undamaged system, while the damaged system does not present the obvious single-mode solutions. The bifurcations and chaos of the damaged system are also changed obviously, and some chaotic motions around the period-doubling bifurcation are observed as to the damaged system.
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图 1 受损悬索构形及特性
Figure 1. Configurations and characteristics of the damaged suspended cable
图 2 悬索模态频率和Irvine参数关系曲线及其模态振型
Figure 2. Relationship curves between Irvine parameter and mode frequencies and shapes
图 3 激励响应幅值曲线 (f1 = 0, σ1 = 0.05和σ2 = 0.2)
Figure 3. Excitation response amplitude curves when f1 = 0, σ1 = 0.05 and σ2 = 0.2
图 4 幅频响应曲线 (f1 = 0, f2 = 0.002和σ2 = 0.2)
Figure 4. Frequency-response curves when f1 = 0, f2 = 0.002 and σ2 = 0.2
图 5 受损悬索霍普夫分岔点附近的动态解 (f1=0, f2=0.002和σ2=0.2)
Figure 5. Damaged suspended cable’s dynamic solutions around two Hopf bifurcations when f1=0, f2=0.002 and σ2=0.2
图 6 PD1附近的相位图: 从周期解到混沌 (1→2→4→8→…→混沌)
Figure 6. Phase portraits diagrams around PD1: from periodic motions to chaotic motions (1→2→4→8→…→chaos)
图 7 时程曲线、相位图、频谱以及庞加莱截面(f1=0, f2=0.002, σ1=0.02772, σ2=0.2)
Figure 7. Time history curves, phase portraits, frequency spectrums and Poincare sections when f1=0, f2=0.002, σ1=0.02772, σ2=0.2
表 1 无损和受损悬索的参数与非线性相互作用系数
Table 1. Parameters and nonlinear interaction coefficients of undamaged and damaged suspended cables
Cable types m n λ2 ωm ωn Kmm Knn Kmn/ Knm K1 K2 K3 undamaged 1 2 40.373 6.2640 6.2830 −2650000 1329000 2079000 0 0 1368000 damaged 1 2 41.829 6.1535 6.1538 1091850 902357 −2365780 938598 1024960 −1028280 
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