Theoretical Modeling And Analysis On Nonlinear Vibration Of Cable-Beam Structure

Stay cable is the key component of cable-stayed bridge.Its function is to transfer a large part of the load on the main girder to the main tower,thus reducing the mid-span bending moment of the main girder and increasing the span of cable-stayed bridge.However,due to the characteristics of high flexibility,light weight and low damping,cable is prone to large vibration.How to avoid this problem has always been a hot topic in bridge engineering and engineering mechanics.In past,research on cable dynamics mainly focused on the dynamics of the pure cable,but this way isn't accurate enough,because cable is susceptible to the vibration of the main beam.In this study,the cable and beam are chosen as a whole mechanical model.The non-linearity of the cable and the boundary conditions of the structure are carefully studied,so that a refined governing equation is obtained.Based on this model,a set of numerical calculation methods is given.Then,the theoretical solution of the non-linear governing equation is deduced by using the Multi-scale Approach.The main work of this study includes:?1?The dynamic equations of stay cable and beam are obtained based on their dynamic properties,and the quasi-static method of the cable and dimensionless reduction of the motion equations are carried out.The continuity and force equilibrium condition at the junction of the two component are derived.Moreover,a relatively accurate theoretical model is obtained.?2?Based on finite difference method,the numerical solution process of cable-beam model is given.At first,the dynamics model of cable-stayed beam is discretized in the space,and the displacement and velocity of all nodes are combined into a new vector.Then,the displacement and velocity of the intermediate nodes is gotten by Runge-Kutta method based on the dynamic equations,and the edge nodes are included in the calculation process based on boundary conditions.?3?Approximate theoretical solution of motion equations based on direct approach of multi-scale method.After multi-scale expansion of the equations,and the natural frequencies and natural modes of the structure are obtained by solving the O???equation.The O??²?equation reveals the relationship between the amplitude parameter A_m and the time scale T₁,then the O??²?modes is solved.The differential equation between A_m and the time scale T₂ is obtained by solving the O??³?equation under solvability condition.After that,the steady-state solution of A_m can be obtained,then the relationship curve between the single-mode steady-state amplitude and the excitation frequency is determined,which is compared with the numerical method at last.?4?The 2:1 internal resonance of the model is further considered by using the multi-scale method.Then the distinct response results of two excitation of different resonance frequencies are given.Based on that,the property of internal resonance in the structure is analyzed.