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Shock and Vibration
Volume 2018 (2018), Article ID 7208045, 21 pages
Research Article

A Numerical Method Based on the Parametric Variational Principle for Simulating the Dynamic Behavior of the Pantograph-Catenary System

State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, China

Correspondence should be addressed to Qiang Gao; nc.ude.tuld@oagq

Received 25 August 2017; Accepted 12 December 2017; Published 3 January 2018

Academic Editor: Jussi Sopanen

Copyright © 2018 Dongdong He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 time-integration 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.