## 2 Numerical Model
### Vehicle and System Models  
Two types of vehicle model are used in this paper. The first one is a system of a quarter-car, and the second one is a half-car crossing the bridge. A single quarter-car and two discrete quarter-cars are used to assess the feasibility of the acceleration-based BWIM system as these models are not affected by hop mass pitch rotation. The quarter-car model has 2 degrees of freedom (DOFs) which consist of sprung mass bounce translation and axle hop translation [38]. The mechanical properties of the quarter-car [39], other than its sprung mass, velocity and axle-spacing, are shown in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Bridge Property** & **Notation** & **Value** \\ \hline Total no. of degrees of freedom & N & 62 \\ Young’s Modulus & E & \(35\times 10^{9}\) N/m\({}^{2}\) \\
2 nd moment of area & J & 0.114 m\({}^{4}\) \\ Mass per unit length & \(\mu\) & 7000 kg/m \\ Damping & \(\xi\) & 3\% \\
1 st natural frequency & f\({}_{1}\) & 32.2 Hz \\ Approach Length & L\({}_{\text{app}}\) & 100 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Properties of the bridge. A half-car vehicle model, shown in Figure 1, is a more realistic representation of a 2-axle vehicle. This model has four DOF: sprung mass bounce displacement (y\({}_{\text{s,i}}\)), sprung mass pitch rotation (\(\theta\)) and axle hop displacements of the two unsprung axle masses (y\({}_{\text{u,1}}\) and y\({}_{\text{u,2}}\)). The vehicle body is represented by a sprung mass (m\({}_{\text{s}}\)) and the suspension systems by the two unsprung axle masses (m\({}_{\text{u,1}}\) and m\({}_{\text{u,2}}\)). The sprung mass is connected to the axles through a combination of springs with linear stiffness (K\({}_{\text{s,i}}\)) and viscous dampers with damping coefficients (C\({}_{\text{s,i}}\)) and to the road surface via springs with linear stiffnesses, (K\({}_{\text{t,i}}\)). Table 3 shows the mechanical properties of the model [40; 41]. The distance of axles to centre of gravity, i.e., D\({}_{1}\) and D\({}_{2}\), are taken as half of the total axle spacing that is chosen randomly in each simulation run using the typical two-axle traffic data. Equilibrium of forces is used to develop the equation of motion for the vehicle [33]. \[\text{K}_{\text{v}}\dot{\text{y}}_{\text{v}}+\text{C}_{\text{v}}\dot{\text{y}}_ {\text{v}}+\text{M}_{\text{v}}\ddot{\text{y}}_{\text{v}}=\text{f}_{\text{v}} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline
**Quarter-Car Vehicle Property** & **Notation** & **Value** \\ \hline Total degrees of freedom & n\({}_{\text{v}}\) & 2 \\ Each unsprung mass & m\({}_{\text{u}}\) & \(750\) kg \\ Each tyre stiffness & K\({}_{\text{t}}\) & \(3.5\times 10^{6}\) N/m \\ Each suspension stiffness & K\({}_{\text{s}}\) & \(750\times 10^{3}\) N/m \\ Each suspension damping & C\({}_{\text{s}}\) & \(10^{4}\) Ns/m \\ \hline \end{tabular}
\end{table} 
Table 2: Properties of the quarter-car. Figure 1: Section of a half-car vehicle model with general road profile on the bridge. \begin{table}
\begin{tabular}{c c c} \hline
**Half-Car Property** & **Notation** & **Value** \\ \hline Total degrees of freedom & n\({}_{\text{v}}\) & 4 \\ \hline Unsprung masses & m\({}_{\text{u,1}}\) & \(750\) kg \\  & m\({}_{\text{u,2}}\) & \(1100\) kg \\ \hline Tyre stiffness & K\({}_{\text{t,1}}\) & \(1.75\times 10^{6}\) N/m \\  & K\({}_{\text{t,2}}\) & \(3.5\times 10^{6}\) N/m \\ \hline Suspension stiffness & K\({}_{\text{s,1}}\) & \(0.4\times 10^{6}\) N/m \\  & K\({}_{\text{s,2}}\) & \(1\times 10^{6}\) N/m \\ \hline Suspension damping & C\({}_{\text{s,1}}\) & \(10^{4}\) Ns/m \\  & C\({}_{\text{s,2}}\) & \(20\times 10^{3}\) Ns/m \\ \hline \end{tabular}
\end{table} 
Table 3: Properties of the half-car model. where \(K_{v}\), \(C_{v}\) and \(M_{v}\) are the (\(n_{v}\times n_{v}\)) stiffness, damping and mass matrices of the vehicle respectively, and \(y_{v}\), \(\tilde{y}_{v}\) and \(\tilde{y}_{v}\) are the (\(n_{v}\times n_{s}\)) vectors of vehicle displacements, velocities and accelerations, respectively.