## 3 Multi-Scale Coupled Vibration Analysis of the TTB System
### TTB System Analysis Model  
Trains running over a bridge are actually time-variant dynamic loads [21; 22]. Vibrations of the vehicle, track, and bridge are usually coupled and dynamic interactions exist among them [23; 24; 25]. Generally, the TTB system consists of the train subsystem, track subsystem, and bridge subsystem. The train subsystem is modeled as a multiple-degree-of-freedom system composed of car-body, bogie, and wheelset, which can be regarded as rigid bodies. Without considering elastic deformations, both the car-body and bogie have five DOFs associated with rigid body movements, namely, swaying, rolling, yawing, floating, and pitching. Furthermore, the four DOFs of each wheelset are swaying, rolling, yawing, and pitching. The wheelsets are connected with the bogies through the primary suspension, while the bogies are connected to the car-body via the secondary suspension. Both the primary and secondary suspensions are modeled as the springs and dashpots of which the stiffness and damping parameters can be obtained from train manufacturers. Then, the equation of motion of the train subsystem can be established by using the [PERSON]'s equation [15; 26]. The equation of motion of the track subsystem can be expressed as fourth-order partial differential equations [27]. An integral model in which the track is included in the bridge subsystem was presented in [PERSON] et al. [28]. In their study both the track and bridge structure were simulated by finite elements. The equation of motion of the bridge including the track was then formulated through the FE method. Since the present analysis is focused on dynamic stress responses of the bridge structure, the track model is incorporated into the bridge subsystem. Using the multi-scale FE model as discussed in the previous section, the equation of motion of the bridge subsystem can be established. The dynamic interactions between the vehicle and bridge subsystems are associated with the motion of the vehicle and bridge, and track irregularities. In the present study, the linear wheel-rail interaction model proposed by [PERSON] et al. [29] is used to determine these interactions. Then, the equation of motion of the TTB system can be formulated by grouping the equations of motion of the train subsystem and bridge subsystem as:  
\[\left\{ \begin{array}{l}\mathbf{M}_{\mathrm{v}}\ddot{\mathbf{X}}_{\mathrm{v} }+\mathbf{C}_{\mathrm{v}}\dot{\mathbf{X}}_{\mathrm{v}}+\mathbf{K}_{\mathrm{v} }\mathbf{X}_{\mathrm{v}}=\mathbf{F}_{\mathrm{v}}\\ \mathbf{M}_{\mathrm{b}}\ddot{\mathbf{X}}_{\mathrm{b}}+\mathbf{C}_{\mathrm{b} }\dot{\mathbf{X}}_{\mathrm{b}}+\mathbf{K}_{\mathrm{b}}\mathbf{X}_{\mathrm{b} }=\mathbf{F}_{\mathrm{b}}\end{array} \right. \tag{1}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrices; \(\mathbf{X}\), \(\dot{\mathbf{X}}\), and \(\ddot{\mathbf{X}}\) are the time-variant displacement, velocity, and acceleration vectors respectively; subscripts \(\mathrm{v}\) and \(\mathrm{b}\) denote vehicle and bridge; and \(\mathbf{F}_{\mathrm{v}}\) and \(\mathbf{F}_{\mathrm{b}}\) are the vectors of dynamic interactive forces acting on the vehicle and bridge subsystems, respectively. Rayleigh damping is used for the bridge in the present analysis. Consequently, the damping matrix \(\mathbf{C}_{\mathrm{b}}\) can be expressed as the linear combination of mass matrix \(\mathbf{M}_{\mathrm{b}}\) and stiffness matrix \(\mathbf{K}_{\mathrm{b}}\). Equation (1) is subsequently solved by a numerical integration scheme, i.e., the Newmark-\(\beta\) method, in order to acquire nodal displacements \(\mathbf{X}_{\mathrm{b}}\) of the bridge FE model.