斜拉桥面内多重内共振下索-梁-索耦合效应的数值研究

NUMERICAL ANALYSIS OF THE CABLE-BEAM-CABLE COUPLED EFFECT UNDER THE IN-PLANE MULTIPLE INTERNAL RESONANCES IN CABLE-STAYED BRIDGES

斜拉桥相邻拉索间局部模态频率非常接近导致结构易发生多重内共振,振动时索-索间耦合效应通过主梁传递或将影响结构动力特性而加剧拉索振动。针对此问题,该文考虑斜拉索的几何非线性,基于多点弹性支承主梁的集中质量参数体系离散方法,建立了八索-变截面梁的斜拉桥面内整体动力学模型。通过有限差分法和Galerkin方法得到了动力学模型的振动方程组,采用特征值法求解其面内自由振动模态参数并对比了有限元计算结果。运用4阶~5阶Runge-Kutta方法对运动方程进行了数值仿真,结果表明:与不同阶竖向整体模态耦合产生的斜拉索振动彼此间相互独立,能量转换仅发生在共振拉索与对应结构整体模态之间。当多根拉索同时与结构某阶整体模态耦合并产生多重“1∶1”内共振时,索-梁-索耦合效应将影响拉索的动力特性。该文结构面内竖向第7阶整体模态下的索-梁-索耦合效应为激励作用,在此效应影响下,拉索的最大振幅接近单索内共振的2倍;此激励效应与拉索间距呈反相关,而与拉索质量、锚固点对应的振型幅值呈正相关。

In cable-stayed bridges,the frequencies of adjacent cables’local modes are close in value,resulting in the high probability of multiple internal resonances in the vertical plane.During the resonance,the coupled effect between cables transmitted by the beam,also named the coupled effect of cable-beam-cable,might intensify the cable’s vibration by changing the dynamic properties of the structure.To investigate the mechanism,considering the geometric non-linearity of the cable,a dynamic model,with eight cables and a variable-section beam,is established.The beam with multi-elastic supports in the model is reduced to a novel integrated dynamic system composed of discrete parametric lumped-mass beam segments.The dynamic equations of the model are obtained by the Galerkin method and amended by the difference methods.The modal properties of the model are solved by the eigenfunctions and verified by the finite element method.Moreover,the dynamic equations were numerically simulated by the 4th~5th-order Runge-Kutta method.The simulation results show that when the local modes of cables are“1∶1”coupled with different global modes,the vibration of cables are independent of each other,and the energy conversion only occurs between the resonant cable and the corresponding global mode.When multiple cables are simultaneously coupled to the global mode of a certain order,the phenomenon of multiple internal resonances can be observed in the model.The coupled effect of cable-beam-cable would affect the characteristics of resonated cables during the multiple resonances.Particularly,the coupled effect is mainly the excitation effect when coupled with the in-plane vertical global mode of the 7th-order.If the coupled effect was considered,the vibration amplitude of the cable would be excited to nearly twice that of the single internal resonance.Additionally,the coupled effect is inversely correlated with the distance between the cables and is positively correlated with the cable mass and the amplitude corresponding to vibration mode at the anchored position.