Formulation of equations of motion of finite element form for vehicle-track-bridge interaction system with two types of vehicle model

Vehicle, track and bridge are considered as an entire system in this paper. Two types of vertical vehicle model are described. One is a one foot mass–spring–damper system having two-degree-of-freedom, and the other is four-wheelset mass–spring–damper system with two-layer suspension systems possessing 10-degree-of-freedom. For the latter vehicle model, the eccentric load of car body is taken into account. The rails and the bridge deck are modelled as an elastic Bernoulli–Euler upper beam with finite length and a simply supported Bernoulli–Euler lower beam, respectively, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The dynamic contact forces between the moving vehicle and rails are considered as internal forces, so it is not necessary to calculate the internal forces for setting up the equations of motion of the vehicle–track–bridge interaction system. The two types of equations of motion of finite element form for the entire system are derived by means of the principle of a stationary value of total potential energy of dynamic system. The proposed method can set up directly the equations of motion for sophisticated system, and these equations can be solved by step-by-step integration method, to obtain simultaneously the dynamic responses of vehicle, of track and of bridge. Illustration examples are given