Vertical dynamic responses of a simply supported bridge subjected to a moving train with two-wheelset vehicles using modal analysis method

The vertical dynamic responses of a simply supported bridge subjected to a moving train are investigated by means of the modal analysis method. Each vehicle of train is modelled as a four-degree-offreedom mass–spring–damper multi-rigid body system with a car body and two wheelsets. The bridge, together with track, is modelled as a simply supported 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, in which the contact forces between wheelset and beam are considered as internal forces. The equations of vertical motion in matrix form with time-dependent coefficients for this system are directly derived from the Hamilton’s principle. The equations of motion are solved by Wilson- method to obtain the dynamic responses for both the support beam and the moving train. Compared with the results previous reported, good agreement between the proposed method and the finite element method is obtained. Finally, the effects of beam mode number, vehicle number, beam top surface, and train velocity on the dynamic responses of the entire train and bridge coupling system are studied, and the dynamic responses of beam are given under the train moving with resonant velocity