## 2 The Bridge-Borne Noise Prediction Model
### The Vehicle-Track-Bridge Coupled Dynamics Model  
The vehicle-track-bridge coupled dynamics model consists of three subsystems: the freight EMU vehicle, the ballastless slab track, and the bridge. In the vehicle model, each car body of the EMU comprises 20 containers with the size of 1.5 m \(\times\) 1.35 m \(\times\) 2 m, and the dead load and carrying capacity of each container are 210 kg and 850 kg. The containers are symmetrically arranged along the centerline of the car body, and containers are mounted to the car body floor via a series of restraining wedges integrating elastomers. The car body, containers, bogie frames, and wheelsets are modeled as rigid bodies, and each of them has six degrees of freedom (DOFs), namely the longitudinal, lateral, vertical, roll, yaw, and pitch motions, which form a 162 DOFs multi-body system. The restraining wedges between containers and the car body are simplified as force elements with three-dimensional stiffness. The primary and secondary suspension systems are modeled as spring-damper elements, in which the air spring provides three-dimensional stiffness and damping, and the anti-hunting damper as well as the lateral and vertical dampers provide appropriate damping.  
For the ballastless slab track, the rails are modeled as simply supported Timoshenko beams considering the vertical, lateral, and torsional motions. The track slabs are described as elastic rectangle plates supported on a viscoelastic foundation, in which only the rigid mode of the slab lateral vibration is considered [25]. The fastenings and rail pads are simplified as discrete spring-damper elements, and the cement asphalt (CA) mortar layer is simulated by uniformly distributed stiffness and damping. The bridge dynamics model is established based on a FEM model, which is introduced in the subsequent chapter in detail. The modal analysis is conducted to obtain the modal frequencies as well as the full mass and stiffness matrices of the bridge, and Rayleigh damping is adopted to describe the damping characteristic of the bridge. The interaction between the track and bridge is discretized as a serial of point-to-point interaction modeled with linear spring-damper elements.  
The dynamic equilibrium equations of the whole system can be expressed as  
\[\left\{\begin{array}{l}M_{\rm c}\ddot{x}_{\rm c}+C_{\rm c}\dot{x}_{\rm c}+K_{ \rm c}x_{\rm c}=F_{\rm c}\\ M_{\rm v}\ddot{x}_{\rm v}+C_{\rm v}\dot{x}_{\rm v}+K_{\rm v}x_{\rm v}=F_{\rm v}- F_{\rm c}\\ M_{\rm t}\ddot{x}_{\rm t}+C_{\rm t}\dot{x}_{\rm t}+K_{\rm t}x_{\rm t}=F_{\rm t}- F_{\rm v}\\ M_{\rm b}\ddot{x}_{\rm b}+C_{\rm b}x_{\rm b}+K_{\rm b}x_{\rm b}=F_{\rm b}\\ \end{array}\right. \tag{1}\]  
where \(M\), \(C\), and \(K\) denote the mass, damping, and stiffness matrixes, respectively; \(\ddot{x}\), \(\dot{x}\), and \(x\) are the acceleration (m/s\({}^{2}\)), velocity (m/s), and displacement (m) vectors, respectively; \(F\) denotes the force vector (N); subscripts c, v, t, and b represent the container, vehicle, track, and bridge, respectively. For the wheel-rail interaction, the normal forces are determined based on the nonlinear Hertz contact theory as  
\[F_{\rm n}(t)=\left\{\begin{array}{l}\left[\frac{1}{G}Z_{\rm wr}(t)\right]^{ 3/2},\ \ Z_{\rm wr}(t)>0\\ 0,\ \In the simulation, the ballastless track spectrum of the Chinese high-speed railway superimposed with a measured shortwave spectrum is adopted for track excitations. A mixed explicit-implicit integration method based on the Zhai method and the Newmark-\(\beta\) method is employed to solve the dynamic response of the train-track-bridge system.