## 3 Analysis of Vertical Vibrations in Track-Bridge Structures Under Short-Wavelength Irregularities  
### Model of the Vehicle-Ballastless Track-Bridge Vertical Coupled System  
In this section, a numerical model of a vehicle with a ballastless track-bridge coupled system is established using the FE-SEA hybrid method. The upper vehicle is represented by a 10-degree-of-freedom vertical model, including the car body (mass \(m_{c}\) and rotational inertia \(J_{c}\)), bogies (mass \(m_{t}\) and inertia \(J_{t}\)), and wheels (mass \(m_{w}\)). Nonlinear Hertzian contact is used to couple the vehicle and track structures. The lower part comprises a CRTS II-type ballastless track structure and a simply supported bridge, modeled as a four-layer Euler Bernoulli beam system with eight nodes per layer: rail, track slab, base layer, and bridge. Each layer shares the same total length. The bridge is composed of five spans supported by piers, as shown in Figure 2.  
The subscripts \(c\), \(r\), \(s\), \(f\), and \(b\) denote vehicle, rail, slab, base, and bridge, respectively. Real nodes are used at boundaries, while internal nodes are treated as virtual. Each layer's degrees of freedom (DOFs) are assigned accordingly. The global displacement vector is  
Figure 2: Vehicle plate track–bridge model. defined \(N_{z}=10+N_{r}+N_{s}+N_{f}+N_{b}\), and interpolation functions are introduced for each element to construct the system matrices. The nodal displacement vector is defined  
\[Q(t)=[q_{c1}\ q_{c2}\cdots q_{c10}\ q_{r1}\ q_{r2}\cdots q_{rN_{r}}\ q_{s1}\ q_{s2} \cdots q_{sN_{s}}\ q_{f1}\ q_{f2}\cdots q_{fN_{f}}\ q_{b1}\ q_{b2}\cdots q_{bN_{b }}]_{(1\times N_{z})} \tag{20}\]  
The shape function corresponding to the element is defined  
\[\Phi(x)=[1\cdots 1\ \ \phi_{r1}(x)\ \ \phi_{r2}(x)\cdots\phi_{rN_{r}}(x)\ \ \phi_{s1}(x)\ \ \phi_{s2}(x)\cdots\phi_{sN_{t}}(x)\ \ \phi_{f1}(x)\]  
\[\phi_{f2}(x)\cdots\phi_{fN_{f}}(x)\ \ \phi_{b1}(x)\ \ \phi_{b2}(x)\cdots\phi_{bN_{b }}(x)]_{(1\times N_{z})} \tag{21}\]  
The vehicle's mass, stiffness, and damping matrices are obtained from the literature [17] and denoted as \(M_{c}\), \(K_{c}\), and \(C_{c}\). According to the FEM, the mass, damping, and stiffness matrices of the track-bridge structure can be systematically derived. The total mass matrix \(M\) of the model is expressed  
\[M=diag[M_{c(10\times 10)}\ \ M_{r(N_{r}\times N_{r})}\ \ M_{s(N_{s}\times N_{s})}\ \ M_{f(N_{f}\times N_{f})}\ \ M_{b(N_{b}\times N_{b})}]_{(N_{z}\times N_{z})} \tag{22}\]  
where \(M_{r}\), \(M_{s}\), \(M_{f}\), and \(M_{b}\) represent the mass matrices of the rail, slab, base, and bridge, respectively, derived from FEM expressions (Equation (12)). The total stiffness matrix \(K\) is defined as  
\[K=K_{1}+K_{1c}+K_{2c}+K_{3c}+K_{4c} \tag{23}\]  
\[K_{1}=diag[K_{c(10\times 10)}\ \ K_{r(N_{r}\times N_{r})}\ \ K_{s(N_{s}\times N_{s})}\ \ K_{f(N_{f}\times N_{f})}\ \ K_{b(N_{b}\times N_{b})}]_{(N_{z}\times N_{z})} \tag{24}\]  
Here, \(K_{r}\), \(K_{s}\), \(K_{f}\), and \(K_{b}\) denote the stiffness matrices corresponding to the bending strain energy of the rail, track slab, base layer, and bridge, respectively, while \(K_{1c}\), \(K_{2c}\), \(K_{3c}\), and \(K_{4c}\) represent the stiffness matrices generated by the discrete elastic support of the rail fasteners (first layer), continuous elastic support of the base CA mortar layer (second layer), continuous elastic support of the bridge (third layer), and discrete elastic support of the bridge piers, respectively. Among them, \(K_{r}\), \(K_{s}\), \(K_{f}\), and \(K_{b}\) can be obtained using Equation (13).