## 3 Numerical Examples  
### Example 1: Suspended Mass Passing Simply Supported Beam Bridge  
A simply supported beam bridge subjected to suspended mass in Ref. [26] is used to demonstrate the effectiveness of the proposed method, as shown in Figure 2. The vehicle is simplified to the 1/4 spring-mass model with only the body's vertical DOF. The vehicle moves from the left end of the beam to the right at a constant speed. The parameters of the model are as follows: [PERSON]'s modulus \(E=2.87\) GPa; section moment of inertia \(I=2.90\) m\({}^{4}\); span \(L=25\) m; mass per unit length \(m=2303\) kg/m; suspended mass \(M_{v}=5757\) kg, suspension stiffness \(k_{1}=1595\) kN/m, driving speed \(v=27.78\) m/s. The fundamental frequency of the bridge is \(\omega_{b}=30.02\) rad/s, and the fundamental frequency of the sprung mass is \(\omega_{v}=16.66\) rad/s. The effect of damping of the bridge is ignored.  
The equation of motion for the mass and the bridge are derived by Biggs [27]. For the dynamic response of the bridge, the analytical solution derived by Biggs only considers the first-order mode of the bridge.  
\[\left\{\begin{matrix}\ddot{y}_{b}\\ \ddot{y}_{v}\end{matrix}\right\}+\left[\begin{matrix}K_{bb}&K_{bv}\\ K_{vb}&K_{vv}\end{matrix}\right]\left\{\begin{matrix}y_{b}\\ y_{v}\end{matrix}\right\}=\left\{\begin{matrix}F_{b}\\ 0\end{matrix}\right\}, \tag{16}\]  
where  
\[\begin{array}{l}K_{bb}=\omega_{b}^{2}+2\omega_{v}^{2}\frac{M_{v}}{mL}\sin^{ 2}\bigl{(}\frac{\pi vt}{L}\bigr{)}\\ K_{bv}=-2\omega_{v}^{2}\frac{M_{v}}{mL}\sin^{2}\bigl{(}\frac{\pi vt}{L}\bigr{)} \\ K_{vb}=-\omega_{v}^{2}\frac{M_{v}}{mL}\sin^{2}\bigl{(}\frac{\pi vt}{L}\bigr{)} \\ K_{vp}=\omega_{v}^{2}\\ F_{b}=-2\omega_{v}^{2}\frac{M_{v}g}{mL}\sin\bigl{(}\frac{\pi vt}{L}\bigr{)} \end{array}\]  
Figure 2: Model of a moving sprung mass through a simply supported beam bridge.  
Figure 3 shows the dynamic responses of the vertical displacement, velocity, and acceleration in the middle of the bridge span compared with the presented model, the analytical model of [PERSON], and the model of [PERSON] nonlinear contact theory [28].  
As shown in Figure 3, the dynamic responses obtained by the presented model using the virtual work principle are in good agreement with the analytical solution. Besides, the solution of the presented model has high-frequency oscillation in the acceleration. Because the analytical model of [PERSON] ignores the excitation effect of vehicle vibration on high-order vibration modes of bridges. However, because the contact stiffness between the mass and the bridge calculated by the [PERSON] nonlinear contact theory is different from the suspension stiffness given in Ref. [26], there is a small phase difference between the presented model and the [PERSON] nonlinear contact model. The maximum displacement,  
Figure 3: Comparative results for the time history curve of the mid-span of the bridge: (**a**) vertical displacement, (**b**) vertical velocity, and (**c**) vertical acceleration.  
velocity, and acceleration in the presented model are 2.395 mm, 0.016 m/s, and 0.336 m/s\({}^{2}\) respectively, while in the nonlinear contact model they are 2.376 mm, 0.012 m/s, and 0.210 m/s\({}^{2}\) respectively. The different contact stiffness between wheel and rail has a limited effect on the displacement but has a slight effect on the velocity and acceleration.  
The calculation times of the presented model, the analytical model, and the nonlinear model are 0.462, 0.030, and 5.069 s, respectively, on a personal computer with Intel(r) (R) CoreTM (TM) i7-10750H CPU, 2.60 GHz. The efficiency of the present method is raised by 10.97 times compared with the nonlinear model but reduced by 15.40 times compared with the analytical model. Therefore, the accuracy of the presented model is higher than that of the analytical model, and the efficiency is higher than that of the nonlinear model.