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

Abstract

SUMMARYVehicle, 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.

Three-Dimensional Rail-Bridge Coupling Element of Unequal Lengths for Analyzing Train-Track-Bridge Interaction System

“…Consequently, each vehicle has 23 independent DOFs. By using the energy principle, such as the principle of the stationary value of total potential energy of a dynamic system (Zeng, 2000;Lou and Zeng, 2005), it is possible to derive the 3D equations of motion written in a sub-matrix for a TTBI system that is shown in Figure 5, as …”

“…Elements in the first row and the first column should be placed in the in the stiffness sub-matrix ss K ; elements in the first row and the last six columns should be placed in the stiffness sub-matrix sb K ; elements in the first column and the last six rows should be placed in the stiffness sub-matrix bs K ; and the remaining elements should be placed in the stiffness sub-matrix bb K . In a similar manner as The displacement sub-vectors, the mass, damping, and stiffness sub-matrices, as well as the force sub-vectors of the train, rail, sleeper, bridge, and pier are explained briefly in the following sections, and a detailed explanation is found in Lou (2005Lou ( , 2007 and Lou and Zeng (2005).…”

“…Thereafter, Wu and Yang (2003) investigated the vertical dynamic responses of a vehicle-rails-bridge interaction system using a condensation technique, which included the steady-state response and riding comfort of the train as well as the impact response of the rails and bridges. Based on the principle of a stationary value of total potential energy of dynamic system, Zeng (2003), Lou (2005), and Lou and Zeng (2005) derived equations of motion in a matrix form for three types of vehicle-track-bridge vertical interaction elements, in which the rails and the bridge deck were represented by an elastic Bernoulli-Euler upper beam with finite length and a simply supported Bernoulli-Euler lower beam, respectively. A later study by Lou (2007) investigated the vertical dynamic responses of a train-track-bridge interaction (TTBI) system using FEM, and by discretizing the slab track subsystem into track elements that flow with the moving vehicle, Lei and Wang (2014) developed a new approach with finite elements in a moving frame of reference to investigate the dynamic behavior of the train and slab track coupling system.…”

An Investigation on the Effect of Rail Corrugation on Track Response

“…The car body has a mass of It is assumed that the mass of each part is concentrated in the centroid of the tack components. the mass, stiffness and damping matrices of train can be expressed as follows: [11], [12] www.ijsea.com 409 …”

“…Therefore, displacements of the beam elements' nodes can be calculated by the following expression. [12], [13]       …”