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Based on the finite element method (FEM), the parametric variational principle (PVP) is combined with a numerical time-domain integral method to simulate the dynamic behavior of the pantograph-catenary system. Based on PVP, formulations for the nonlinear droppers in the catenary and for the contact between the pantograph and the contact wire are proposed. The formulations can accurately determine the tension state or compression state of the nonlinear droppers and the contact state between the pantograph and the contact wire. Based on the periodicity of the catenary and the precise integration method (PIM), a numerical timeintegration method is developed for the dynamic responses of the catenary. For this method, the matrix exponential of only one unit cell of the catenary is computed, which greatly improves the computational efficiency. Moreover, the validation shows that the formulations can compute the contact force accurately and represent the nonlinearity of the droppers, which demonstrates the accuracy and reliability of the proposed method. Finally, the dynamic behaviors of the pantograph-catenary system with different types of catenaries are simulated.

Purpose The purpose of this paper is to propose an accurate and efficient numerical method for determining the dynamic responses of a tensegrity structure consisting of bars, which can work under both compression and tension, and cables, which cannot work under compression. Design/methodology/approach An accurate time-domain solution is obtained by using the precise integration method when there is no cable slackening or tightening, and the Newton–Raphson scheme is used to determine the time at which the cables tighten or slacken. Findings Responses of a tensegrity structure under harmonic excitations are given to demonstrate the efficiency and accuracy of the proposed method. The validation shows that the proposed method has higher accuracy and computational efficiency than the Runge–Kutta method. Because the cables of the tensegrity structure might be tense or slack, its dynamic behaviors will exhibit stable periodicity, multi-periodicity, quasi-periodicity and chaos under different amplitudes and frequencies of excitation. Originality/value The steady state response of a tensegrity structure can be obtained efficiently and accurately by the proposed method. Based on bifurcation theory, the Poincaré section and phase space trajectory, multi-periodic vibration, quasi-periodic vibration and chaotic vibration of the tensegrity structures are predicted accurately.

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