Nonlinear Vibration Characteristics Of Iced Transmission Lines Based On Perturbation Method

Under the condition of wind load,the iced transmission line is prone to galloping.Long time galloping of the transmission lines may reduce the service life of the transmission line structure,even cause serious accidents such as broken strand or wire and structure collapses,which will cause serious loss of life and property safety of the people.Therefore,the study of the galloping of iced transmission lines has very important theoretical and practical value.Firstly,the paper analyzes the applicability of perturbation method to the governing equation of transmission line galloping,and makes physical modeling of the iced transmission lines under wind load.The nonlinear partial differential motion equation is obtained.The mode-superposition method and the Galerkin method are used to discretize the motion equation,and the nonlinear partial differential motion equation is transformed into the nonlinear ordinary differential galloping governing equation with quadratic and cubic nonlinear restoring force terms.Next,the amplitude equation,phase-shift equation and the approximate analytical solution of the weak nonlinear galloping governing equation are obtained by perturbation method(the averaging method and the multiple scales method).Then,the numerical solution of the galloping governing equation is obtained by using the Runge Kutta function in MATLAB.Finally,the time history displacement and phase diagrams of the analytic solution of perturbation method and the numerical solution are analyzed systematically under the condition of different physical parameters and different wind velocity.The results show that the vibration center of the transmission lines system will drift because of the existence of the quadratic nonlinear restoring force term in the nonlinear galloping governing equation of transmission line.The change in wind velocity,tension and Young’s modulus will change the drift of vibration center.The periodic approximate analytical solution of the averaging method couldn’t reflect the drift of the vibration center of the transmission line system.However,the multiple scales method can accurately solve the weak non-linear equation with quadratic nonlinear restoring force term.In the small range of velocity speed change,the accuracy of periodic approximate analytical solution of the fourth-order multiple scales method is better than that of the low-order multiple scales method.In order to analyze the nonlinear response characteristics of iced transmission lines under the excitation of dynamic wind,a new forced-self-excitation system is established.Firstly,the nonlinear galloping governing equation is solved by the multiple scale method.Based on the averaging equation of response amplitude and phase,the force-amplitude(p-a)curve and amplitude-frequency(σ-a)curve of the forced-self-excited system are obtained by using the mathematics software MAPLE.Then,the applicability of the periodic approximate analytical solution of the multiple scales method to forced-selfexcited system is discussed.Next,the nonlinear galloping characteristics of the principal resonance response under weak excitation and the nonlinear galloping characteristics of the harmonic resonance under strong excitation are analyzed.The results show that with the increasing of excitation amplitude,the self-excited vibration condition of the iced transmission line under wind load is destroyed by forced excitation,and the phenomenon of quenching occurs.In the forced-self-excited system,the condition discrimination equation of self-excited vibration is given.In addition,the change in control parameters(wind velocity,tension,excitation amplitude,Young’s modulus,tuning parameters)has a significant influence on the resonance peak,resonance region and the nonlinear dynamic behavior of the principal resonance and harmonic resonance.By changing these control parameters,the resonance region of the system can be changed,and the response amplitude of the system can be changed from single value to multi-value,so as to suppress or excite the resonance phenomenon of the system.The principal resonance,the harmonic resonance(1/2-order sub-harmonic resonance,the second-order,and third-order superharmonic resonance)affect the galloping characteristics of the iced transmission line.Under the condition of harmonic excitation,the transmission line galloping in a shorter time and at a lower wind velocity,and the peak value of the transmission line response amplitude increases,and the response amplitude also appears jump phenomenon and multi-value phenomenon.The continuous effect of harmonic excitation will reduce the service life of transmission lines,so the influence of principal resonance and harmonic resonance should be considered in the design of structural parameters.In the practical application of engineering,the peak value of principal resonance and harmonic resonance can be reduced by increasing the tension and the Young’s modulus of the transmission line properly.Designers can also improve the galloping conditions and reduce the peak value of galloping by changing the transmission line structure parameters or making measures to prevent the transmission line from galloping.