## 2 Dynamic Equations for Vehicle Group-Bridge Coupled Motion  
Before activating the health monitoring system, it is essential to thoroughly investigate and analyze the structural dynamic characteristics, traffic loads, and vehicle-bridge coupling characteristics of the bridge. This ensures a comprehensive understanding of the bridge's behavior before undertaking damage detection. While the theory of vehicle-bridge coupled vibration has been studied extensively in recent years, the development of new dynamic models that consider more influencing factors is still necessary. In this study, the vehicle group is represented as a time-varying dynamic model with body parts and distributed coupling mass. The dynamic equilibrium equation and analytical expressions pertaining to the interaction between the bridge structure and the moving vehicle group are established. Typical bridge structures exhibit bending deformation primarily under the action of vertical vibrations, with the shear deformation and rotary inertia of the central shaft section being negligible. This assumption is illustrated in Figure 1, where the cross-sectional area of the bridge is considered constant along its length, resulting in a constant flexural stiffness (D). Under the assumption that the vehicle group is a uniform moving mass load, the coupling effect and dynamic response of the vehicle group and the bridge can be effectively analyzed directly solving the relevant differential equations. The external excitation acting on the bridge results from a group of moving vehicles. This group is assumed to consist of a main body with uniform mass density, body stiffness, and body damping. If we denote the dynamic deflection of the beam as \(y(x,t)\), the displacement of the main body in the vehicle group as \(z(t)\), the distribution length as \(l_{p}\), and the constant speed of the vehicle group as \(v\), the dynamic equation for the vehicle group is expressed as follows:  
\[\overline{\mu}_{1}l_{p}\bar{z}(t)+\overline{k}_{1}(z(t)-y(x,t))+\overline{c}_{ 1}\big{(}\dot{z}(t)-\partial y(x,t)/\partial t\big{)}=0 \tag{1}\]  
The dynamic equation governing the bridge response under the moving excitation of a vehicle group is formulated as follows:  
\[D\partial^{4}y(x,t)/\partial x^{4}+\overline{m}\partial^{2}y(x,t)/\partial t^ {2}+c\partial y(x,t)/\partial t=p(x,t) \tag{2}\]  
The dynamic loads \(p(x,t)\) exerted on the bridge by the vehicle group involve the gravity load resulting from the mass of the vehicle group, represented as \(p_{G}(x,t)=(\overline{\mu}_{0}+\overline{\mu}_{1})g\); the inertial force, \(p_{I}(x,t)=-\overline{\mu}_{0}\partial^{2}y(x,t)/\partial t^{2}\); the reaction force arising from the stiffness of the vehicle group, \(p_{S}(x,t)=\overline{k}_{1}[z(t)-y(x,t)]\); and the reaction force due to the damping of the vehicle group, \(p_{D}(x,t)=\overline{c}_{1}\big{[}\dot{z}(t)-\partial y(x,t)/\partial t\big{]}\). Hence, \(p(x,t)\) can be expressed as follows:  
\[p(x,t)=[p_{G}(x,t)+p_{I}(x,t)+p_{S}(x,t)+p_{D}(x,t)]\big{\{}\mathrm{H}[x-(Vt- l_{p})]-\mathrm{H}(x-Vt)\big{\}} \tag{3}\]  
where \(H(t)\) is the unit step function. The above equation can be solved using the mode-superposition method. The corresponding transformation expression is shown as:  
\[y(x,t)=\sum_{i=1}^{\infty}q_{i}(t)\cdot\phi_{i}(x) \tag{4}\]  
where \(q_{i}(t)\) is the generalized modal coordinates and \(\phi_{i}(x)\) is the mode function. Substituting Equation (4) into Equation (2) and multiplying each term by the \(n\)th-order mode function, the integral along the beam is calculated, accounting for modal orthogonality. The modal decomposition method is then applied, combining with Equation (1), to derive the coupled dynamic equations for the bridge and vehicle group system. Utilizing  
Figure 1: Illustration of vehicle group with coupling mass and bridge. the stepwise integration method, the \(n\)th-order modal response under the influence of the vehicle group is computed.