Research On Bifurcation And Chaos Of Two Degrees Of Freedom Galloping Of Transmission Line

Transmission line galloping is a kind of self-excited vibration with low frequency and large amplitude. Ice covering and wind forcing are the necessary conditions of conductors galloping. The general frequency of galloping is 0.1~3Hz, and the possible amplitude can be as much as 5~300 times of the diameter of the wires. This low frequency, large amplitude vibration can cause flashovers and tremendous damage to support structures and make transmission line system failed to operate. So it is urgent to research the dynamic response during galloping.In the paper, the dynamics including bifurcation and chaos is studied in allusion to the phenomenon of galloping. In this work, analytical methods and numerical methods are combined. The specific work include the following aspects:(1) By using Lagrange equation, an analytical model of two degrees of freedom (vertical movement and torsional movement) is established. By decoupling the differential equations, both the terms of the derivative of the second order and the first power of the two variables are separated at the same time. In this model, the inertial coupling caused by the eccentricity of the ice coated line and the dynamics coupling caused by the changing of the attack angle of the wind force are both considered. From the differential equations obtained, nonlinear factors are very complex.(2) Taking the case of 2:1 internal resonance as an example, the singularity of galloping is analyzed. The method of multiple scales is used to calculate the analytical solutions of the vibration equations and the averaged equations are obtained. By simplifying the averaged equations, the bifurcation equations are acquired. Based on the singularity theory, the bifurcation equations are finally studied to discuss the bifurcation behaviors of the system. In the singularity analysis, a possible idea to calculate the transition set is given. Since the mathematical expression of the transition set is too complicated, the analytical solutions can not be received. Therefore, numerical methods are employed to compute the transition set and the retention bifurcations. Ultimately, the transition sets on three projection planes of the parameter space of three unfolding parameters of the bifurcation equation are obtained respectively, and abundant bifurcation patterns are acquired. (3) Chaos of the wire is studied. The averaged equations of the vibration equations in the cases of 1:1 internal resonance, 2:1 internal resonance and 3:1 internal resonance are obtained respectively through the method of multiple scales. From the averaged equations, numerical methods are used to plot stability boundary curves in various internal resonance cases, and the resonance regions in the parameter plane of frequency ratio and wind speed in the cases of the three internal resonances are received respectively. According to the circumstances of the superposition of all the resonance regions, the parameter plane is divided into six regions. According to Arnold tongue method, the superposition areas of different resonance regions can be determined as the chaotic region. By numerical calculation, various forms of movement in every region are discussed. Complex dynamics, including the times periodic motion, almost periodic motion and chaotic motion, are received in tortional movement. Finally, according to the chaotic time history, the greatest Lyapunov index is calculated. The index in the tortional direction is greater than zero which confirms the existence of chaotic motion.