## 2 Methodology
### Finite Element Modeling  
For conciseness, only the interaction system of the tractor (2-DOF vehicle) and the beam in Figure 1b will be presented here. The 2-DOF vehicle motion is described by its vertical translational displacement (\(u_{\mathrm{c}}\)) and angular displacement (\(\theta_{\mathrm{c}}\)). \[ \begin{array}{l}\left[\begin{array}{cc}M_{\mathrm{c}}&0\\ 0&k_{\mathrm{c}}\end{array}\right]\left[\begin{array}{c}\ddot{u}_{\mathrm{c}}(t )\\ \ddot{\theta}_{\mathrm{c}}(t)\end{array}\right]+\left[\begin{array}{cc}c_{F }+c_{\mathrm{R}}&(c_{\mathrm{F}\alpha}t_{\mathrm{F}}-c_{R\alpha}g_{\mathrm{L} })L_{\mathrm{c}}\\ (c_{\mathrm{F}\alpha}t-c_{R\alpha}g_{\mathrm{L}})L_{\mathrm{c}}&(c_{\mathrm{F} \alpha}t_{\mathrm{F}}^{2}+c_{R\alpha}^{2}t_{\mathrm{L}}^{2})L_{\mathrm{c}}^{2} \end{array}\right]\left[\begin{array}{c}\dot{u}_{\mathrm{c}}(t)\\ \dot{\theta}_{\mathrm{c}}(t)\end{array}\right]+\left[\begin{array}{cc}k_{F }+k_{R}&(k_{F}\alpha_{F}-k_{R\alpha}g_{\mathrm{L}})L_{\mathrm{c}}\\ (k_{F\alpha}t-k_{R\alpha}g_{\mathrm{L}})L_{\mathrm{c}}^{2}\end{array}\right] \left[\begin{array}{c}\dot{u}_{\mathrm{c}}(t)\\ \theta_{\mathrm{c}}(t)\end{array}\right]\\ =\left[\begin{array}{cc}c_{F}&c_{R}\\ c_{F\alpha}t_{\mathrm{F}}&-c_{R\alpha}g_{\mathrm{L}}\end{array}\right]\left( \left[\begin{array}{cc}\Gamma_{F}(t)&0\\ 0&\Gamma_{R}(t)\end{array}\right]\left[\begin{array}{cc}\dot{u}_{\mathrm{c}}(V _{\mathrm{c}}(t-t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t-t_{ \mathrm{T}1}))\\ \dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t-t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_ {\mathrm{c}}(t-t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t-t_{ \mathrm{T}1}))\\ \dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t-t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_ {\mathrm{c}}(t-t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t-t_{ \mathrm{T}1}))\end{array}\right]\right)\\ +\left[\begin{array}{cc}k_{F}&k_{R}\\ k_{F\alpha}t_{\mathrm{F}}&-k_{R\alpha}g_{\mathrm{L}}\end{array}\right]\left( \left[\begin{array}{cc}\Gamma_{F}(t)&0\\ 0&\Gamma_{R}(t)\end{array}\right]\left[\begin{array}{cc}\dot{u}_{\mathrm{c}}(V _{\mathrm{c}}(t-t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t-t_{ \mathrm{T}1}))\\ \dot{u}(V_{\mathrm{c}}(t-t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t- t_{\mathrm{T}1}))&\dot{u}_{\mathrm{c}}(V_{\mathrm{c}}(t-t_{\mathrm{T}1}))\end{array} \right]\right)\end{array}  \tag{18}\]  
The vehicle suspension system is simulated by the translational springs and dashpots, including \(k_{F}\) and \(c_{F}\) for the front axle and \(k_{R}\) and \(c_{R}\) for the rear axle. \(a_{F}\) and \(a_{R}\) are the ratios of the distance of the front axle and the rear axle from the vehicle's center of gravity to the wheelbase (\(L_{\mathrm{c}}\)). In addition, \(u(x,t)\) is the beam displacement response, which will be presented subsequently. Similarly, \(\Gamma_{F}(t)\) and \(\Gamma_{R}(t)\) are used to indicate the statuses of the front and the rear axles on this beam or the neighboring regular rigid road. \[ \begin{array}{l}\Gamma_{F}(t)=H_{\mathrm{e}}(t-t_{\mathrm{F}1})-H_{ \mathrm{e}}(t-t_{\mathrm{F}2}),\Gamma_{R}(t)=H_{\mathrm{e}}(t-t_{\mathrm{R}1} )-H_{\mathrm{e}}(t-t_{\mathrm{R}2})\\ t_{\mathrm{F}2}=t_{\mathrm{F}1}+t_{\mathrm{L}},t_{\mathrm{R}2}=t_{\mathrm{F} 2}+t_{\mathrm{c}},t_{\mathrm{R}2}=t_{\mathrm{F}1}+t_{\mathrm{L}}+t_{\mathrm{c }},t_{\mathrm{L}}=\frac{L}{V_{\mathrm{c}}},t_{\mathrm{c}}=\frac{L}{V_{\mathrm{c }}}\end{array}  \tag{19}\]  
where \(t_{\mathrm{F}1}\), \(t_{\mathrm{F}2}\), \(t_{\mathrm{R}1}\), and \(t_{\mathrm{F}1}\) are the time instants when the front and rear axles enter and leave the beam, respectively. When this uniform Euler-Bernoulli beam is discretized by two DOFs per node element, the beam responses \(u(V_{\mathrm{e}}(t-t_{\mathrm{f}1}),t)\) and \(u(V_{\mathrm{e}}(t-t_{\mathrm{r}1}),t)\) and their derivates can be presented through Hermitian cubic interpolation as the shape function. \[\begin{array}{l}\left[ \begin{array}{c}u(V_{\mathrm{e}}(t-t_{\mathrm{f}1}),t)\\ u(V_{\mathrm{e}}(t-t_{\mathrm{r}1}),t)\end{array}