# Dynamic Impact Factor and Resonance Analysis of Curved Intercity Railway Viaduct
## 2. Intercity Vehicle-Curve Bridge System Model  
The vehicle crossing the bridge impacted and produced coupled vibrations in the VTBS. For accurately describing the dynamic performance of the vehicle passing through the curved bridge, the VTBS model is established, as shown in Figure 1. The vehicle follows the mass-spring damping system for simulation. The vehicle body, structure, and wheelset are all assumed to be rigid bodies in the simulation, whereas the damping suspension system is considered a linear spring, as shown in Figure 2. The dynamic vehicle response can be shown as Equation (1). \[\mathbf{M}^{V}\ddot{\mathbf{X}}^{V}+\mathbf{C}^{V}\dot{\mathbf{X}}^{V}+ \mathbf{K}^{V}\mathbf{X}^{V}=\mathbf{F}^{V} \tag{1}\]  
where \(\mathbf{M}^{V}\), \(\mathbf{C}^{V}\), and \(\mathbf{K}^{V}\) are the stiffness, damping, and mass matrix of the vehicle, respectively; \(\dot{\mathbf{X}}^{V},\dot{\mathbf{X}}^{V}\), and \(\mathbf{X}^{V}\) are acceleration, velocity, and displacement vectors of the vehicle, respectively; \(\mathbf{F}^{V}\) is the load vector of the vehicle, and it includes wheel-rail (W-R) contact force, centrifugal force, and self-load. Normally, the bridge model can be established by the finite element method (FEM) or the synthetic modal method, and wave and FEM can also be used straightforwardly [31; 32]. In this model, the bridge model is developed by the FE method. The bridge adopts a spatial Euler-Bernoulli beam element, and the linear and cubic (Hermitian) shape function is used [33]. Each element node contains six degrees of freedom. Through assembly, the mass matrix \(\mathbf{M}^{B}\) and stiffness matrix \(\mathbf{K}^{B}\) of the bridge in the global system can be obtained. The damping \(\mathbf{C}^{B}\) is assumed to be Rayleigh damping, and it is assumed that the damping ratio of each order is 0.02 [28]. The dynamic equation of the bridge is Equation (2). \[\mathbf{M}^{B}\ddot{\mathbf{X}}^{B}+\mathbf{C}^{B}\dot{\mathbf{X}}^{B}+ \mathbf{K}^{B}\mathbf{X}^{B}=\mathbf{F}^{B} \tag{2}\]  
Figure 1. Vehicle bridge system model: (**a**) vertical view; (**b**) side view. The vehicle and bridge are coupled together into a large coupled system by the W-R contact, termed VTBS. The W-R contact adopts the knife-edge model [34; 35; 36; 37], which simplifies the W-R contact into transverse and vertical springs. As shown in Figure 3, the model does not need to search for the contact point and it has high calculation efficiency; moreover, the calculation results can meet the accuracy of bridge dynamic response calculation. The compression amount of Hertz spring can be calculated through the relative vertical displacement of one side wheel rail, so as to calculate the corresponding wheel rail vertical force. The W-R contact lateral force, including creep force and contact force between flange and rail, as well as the creep force, can be calculated by the linear Kalker theory, and the contact force between flange and rail can be calculated by the relative lateral displacement of the wheel rail from one side. The details can be found in [38]. Figure 3: Wheel–rail contact model. Figure 2: Vehicle model: (**a**) side view; (**b**) front view; (**c**) vertical view. The W-R contact was treated in the form of \"weak coupling\". The method is one of the most commonly used methods. Through the simultaneous W-R contact, Equations (1) and (2), the system equation is given in Equation (3). \[\left\{ \begin{array}{l}\mathbf{M}^{V}\ddot{\mathbf{X}}^{V}+\mathbf{C}^{V} \dot{\mathbf{X}}^{V}+\mathbf{K}^{V}\mathbf{X}^{V}=\mathbf{F}^{V}\\ \mathbf{M}^{B}\ddot{\mathbf{X}}^{B}+\mathbf{C}^{B}\dot{\mathbf{X}}^{B}+\mathbf{ K}^{B}\mathbf{X}^{B}=\mathbf{F}^{B}\end{array} \right. \tag{3}\]  
Equation (3) can be solved by the direct integration method introduced by [PERSON] et al. [39]. Then the dynamic response of the bridge can be determined. For detailed information on the methodology and procedure, the work of [PERSON] et al. [40] may be consulted.