In this paper the models of the transmission line with one DOF, two DOFs and three DOFs are constructed respectively by using Hamilton principle, where the initial location, the geometric nonlinearity caused by deformation and the aerodynamic nonlinearity caused by flow are considered. These simple models are accurate and can be used for the theoretical analysis. In actual engineering, the critical wind speed of galloping has been paid more and more attention. Therefore, in this paper the critical wind speeds are obtained by Lyapunov stability theory and the effects of structural parameters to the critical wind speed are discussed for different models. And the response curves of the system with one DOF and two DOFs (out-of-plane motion) are obtained by using harmonic balance method and multiple scale method, respectively. The bifurcation of the system moving out-of-plane is analyzed by singularity theory. For the systems whose motion is coupled with torsion, the numerical results are given by using the fourth order Runge–Kutta method. Additionally, the effects of the dynamic vibration absorber and detuning pendulum to the anti-galloping of the transmission line are studied for different cases.

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Bifurcation properties of dynamical systems with two parameters are investigated in this paper. The definition of transition set is proposed, and the approach developed is used to investigate the dynamic characteristic of the nonlinear forced Duffing system with nonlinear feedback controller. The whole parametric plane is divided into several persistent regions by the transition set, and then the bifurcation diagrams in different persistent regions are obtained.

A two-degree-of-freedom bifurcation system for an elastic cable with 1:1 internal resonance is investigated in this paper. The transition set of the system is obtained with the singularity theory for three cases. The whole parametric plane is divided into several different persistent regions by the transition set. The bifurcation diagrams in different persistent regions are obtained.

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