Modal Coordinate Formulation for a Simply Supported Bridge Subjected to a Moving Train Modelled as Two-Stage Suspension Vehicles

Abstract

Modal coordinate formulation for analysing the dynamic interaction between a moving train and a simply supported bridge is presented in this article. The train is composed of a series of identical vehicles, and each vehicle is modelled as a four-wheelset mass-spring-damper multi-rigid body system with two-stage suspension having ten degrees of freedom (DOFs). A simply supported bridge, together with the track, is modelled as a Bernoulli-Euler beam. The deflection of the beam is described by superimposing modes. The train and the beam are regarded as an entire dynamic system, and then the modal coordinate formulation with time-dependent coefficients for this system is directly derived from the principle of virtual work. The formulation is solved by direct time integration method, to obtain the dynamic responses of this system. The correctness of the proposed formulations is illustrated by a comparison with the existing literature. The formulation helps save computer time using a few beam modes for analysing the dynamic responses of an entire train-bridge interaction system. The proposed formulation can also be applied to analyse the dynamic responses of a simply supported bridge subjected to a moving train modelled as two-wheelset four DOFs vehicles. Two numerical examples are given for illustrating the applications of the proposed formulation.