# Bridge Expansion Joint in Road Transition Curve: Effects Assessment on Heavy Vehicles
## 3 Methodology and Case of Study
### Front and Rear Axle Quarter Car Model  
The linear quarter-truck model (Figure 11a) described in [32] was implemented in MATLAB to estimate the dynamic load increment induced by the BEJ at several speeds. The basic equations of motion for the examined model are (1) and (2):  
\[m_{u}\ddot{x}_{u}=c_{s}(\dot{x}_{s}-\dot{x}_{u})-(k_{s}+k_{t})x_{u}+k_{s}x_{s}+k _{t}u, \tag{1}\]  
Figure 10: Pneumatic tire components adopted in (**a**) the vehicle model and (**b**) the tire model front section during a general time step.  
Figure 9: Detail of the rear suspension system.  
\[m_{s}\ddot{x}_{s}=c_{s}(\dot{x}_{u}-\dot{x}_{s})+k_{s}(x_{u}-x_{s}). \tag{2}\]  
where \(m_{u}\) is the unsprung mass, \(m_{s}\) is the sprung mass, \(k_{s}\) is the suspension spring constant, \(k_{t}\) is the tire spring constant, \(c_{s}\) is the suspension viscous damping constant, \(x_{u}\) is the vertical displacement of the unsprung mass, \(x_{s}\) is the vertical displacement of the sprung mass in m, \(\dot{x}_{u}\) is the time derivative of vertical displacement of the unsprung mass, \(\ddot{x}_{s}\) is the acceleration of the sprung mass, \(\ddot{x}_{u}\) is the acceleration of the unsprung mass and \(u\) is the road profile elevations.  
This mathematical model was formulated in the form of state equations. The general form of the state model for a linear dynamic system is reported in Equations (3) and (4):  
\[\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}, \tag{3}\]  
\[\mathbf{y}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u}, \tag{4}\]  
where \(\mathbf{A}\) is the state matrix, \(\mathbf{B}\) the input matrix, \(\mathbf{C}\) the output matrix, \(\mathbf{D}\) the direct transmission matrix, \(\mathbf{x}\) is the state vector and \(\mathbf{u}\) and \(\mathbf{y}\) are, respectively, the input (profile elevations) and the output (reaction forces) vectors (Figure 11b).  
The mechanical parameters of the front and rear axles used in the model are consistent with the data provided in [30] (Table 3).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(m_{s}\) (sprung mass) & 1350 kg (front) \\  & 1470 kg (rear) \\ \hline \(m_{u}\) (unsprung mass) & 335 kg (front) \\  & 450 kg (rear) \\ \hline \(k_{s}\) (suspension spring constant) & 270 kN/m (front) \\  & 660 kN/m (rear, steel) \\ \hline \(k_{t}\) (tire spring constant) & 760 kN/m (standard radial) \\ \hline \(c_{s}\) (suspension viscous damping constant) & 16.8 kN s/m (front) \\  & 89.0 kN s/m (rear) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Mechanical parameters of the quarter-truck model [32].  
Figure 11: Scheme of (**a**) the quarter-truck model and (**b**) the matrices used in the state-space model in MATLAB.  
According to Equation (5), the dynamic load coefficient (DLC) was calculated to characterize the magnitude of dynamic tire forces [9]:  
\[DLC=\frac{RMS\text{ dynamic tire force}}{\text{Static tire force}} \tag{5}\]  
where \(RMS\) is the root mean square of the dynamic tire force signal calculated along the road profile, often indicated using the \(\sigma\) notation. The \(RMS\) tire force (\(\sigma\)) can be calculated by means of Equation (6):  
\[\sigma^{2}=\frac{1}{T}\int\limits_{0}^{T}f(t)^{2}dt=\frac{1}{N}\sum\limits_{k= 1}^{N}f_{k}^{2}, \tag{6}\]  
where \(f_{k}\) is the dynamic tire force \(f(t)\) at time \(t=k\times\Delta t\), \(\Delta t\) is the simulation output time step and \(T\) is the total simulation time (\(T=N\times\Delta t\)).