## 2 Fundamentals of Dynamic Flexibility Method
### Vehicle-Rail-Viaduct Coupling Model via Dynamic Flexibility Method  
The vertical wheel rail contact force is assumed to be harmonic, the displacement of the vehicle system follows  
\[Z_{V}(\omega)=\frac{p(\omega)}{-\omega^{2}M_{V}+i\omega C_{V}+K_{V}} \tag{1}\]  
Figure 2: Vertical model of vehicle-track coupling system: (**a**) high-speed railway (HSR) system; (**b**) vehicle-rail-viaduct coupled model. where \(Z_{v}\) means the displacement of the vehicle system, \(M_{v}\), \(K_{v}\) and \(C_{v}\) indicate the mass, stiffness and damping, respectively. \(p(\omega)\) represents the vertical rail-wheel force, and \(\omega\) denotes the excitation circular frequency. This study employs Timoshenko beam to model the rail, according to superposition theory for the wheel rail interaction vibration analysis, the displacement in frequency domain for the rail follows  
\[z_{r}(x)=\sum_{w=1}^{N_{w}}\beta_{r}(x,x_{w})P_{w}-\sum_{n=1}^{N}\beta_{r}(x,x _{n})K_{j}z_{r}(x_{n}) \tag{2}\]  
where \(\beta_{r}(x,x_{w})\) denotes the displacement at \(x\) location when acting unit force at \(x_{w}\) in the rail, \(x_{w}\) indicates the contacting point for the _wh_ wheel rail, \(P_{w}\) means the _wh_ wheel rail contact force for the wheelset, \(N_{w}\) indicates the amount of the wheelsets; \(K_{j}z_{r}(x_{n})\) represents the fastener's force when _nth_ fastener acts on the \(x_{n}\) location of the rail. \(N\) represents the amount of the fasteners. The slab is modeled with free-free Euler-Bernoulli beam, and in the frequency domain, its displacement follows  
\[Z_{s}(x)=\sum_{n=1}^{N}\beta_{s}(x,x_{n})F_{fn}-\sum_{m=1}^{M}\beta_{s}(x,x_{m })F_{jm} \tag{3}\]  
where \(\beta_{s}(x,x_{n})\) indicates the displacement at the \(x\) location of the slab when unit force is loaded on \(x_{n}\) location of the slab, \(F_{fn}\) denotes the fastener's force at location \(x_{n}\) of the slab. \(N\) means the amount of the fasteners. \(F_{jm}\) represents the discrete spring force at \(x_{m}\) in the CA mortar of the slab; \(M\) illustrates the amount of the discrete spring dampers in the CA mortar. The box girder viaduct is considered as simply-supported Euler beam, and by using the modal superposition method, the displacement of the box-girder viaduct follows  
\[Z_{b}(x)=\sum_{n=1}^{N}\beta_{b}(x,x_{m})F_{jm}-\sum_{h=1}^{2}(x,x_{zh})F_{zh} \tag{4}\]  
where \(\beta_{b}(x,x_{m})\) represents the responses at \(x\) location when unit force is loaded at \(x_{m}\) location of the viaduct, \(F_{zh}\) denotes the _hth_ bearing reaction force of the viaduct. When constructing the vehicle-rail-viaduct coupling model, the fastener, the CA mortar, and the bearing are modeled with discrete spring damping elements, and then the fastener's force \(F_{fn}\), the discrete spring force at CA mortar \(F_{jm}\), and the bearing reaction force \(F_{zh}\) can be illustrated as  
\[\left\{ \begin{array}{c}F_{fn}=K_{f}(z_{r}(x_{n})-z_{s}(x_{n}))\\ F_{jm}=K_{j}(z_{s}(x_{m})-z_{b}(x_{m}))\\ F_{zh}=K_{z}z_{b}(x_{h})\end{array} \right. \tag{5}\]  
where \(K_{f}\) denotes the complex stiffness of the fastener, \(K_{j}\) means the complex stiffness of the discrete spring damper element in the CA mortar, \(K_{z}\) indicates the complex stiffness of the viaduct bearing. \[\left\{ \begin{array}{c}K_{f}=k_{f}(1+i\eta_{f})\\ K_{j}=k_{j}(1+i\eta_{j})\\ K_{z}=k_{z}(1+i\eta_{z})\end{array} \right. \tag{6}\]  
where \(k_{f}\) means the stiffness of the fastener, \(\eta_{f}\) denotes the loss factor of fastener, \(k_{j}\) indicates the discrete spring element stiffness of the CA mortar, \(\eta_{j}\) represents the loss factor of the CA mortar discrete spring element, \(k_{z}\) denotes the stiffness of the viaduct bearing, \(\eta_{z}\) represents the loss factor of the viaduct bearing stiffness.