## 3 Mathematical Development
### Matrix Formulation of the Model  
This study focuses on the vehicle's dynamic interaction with the road when the vehicle is moving. Therefore, different states, including driving manoeuvres such as acceleration, braking, and curving, are examined according to the suspension flexibility and damping characteristics. The proposed model is defined as a four-node macro-element with a rectangular shape, dimensioned as wheelbase length \(L\) and bus width \(b\), respectively, in the \(x-\) and \(y-\) coordinate axes, as depicted in Figure 2.  
This rigid plate (the simplified geometry of the bus) materializes the structure of the bus platform. It is assumed to be very rigid compared with the stiffness of the suspension spring/damper assembly at each wheel. With this characteristic, the displacement field bears on a simple algebraic equation defining a plane. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Weight** & **Spring Stiffness** & **Bus Length** & **Bus Width** & **Track Width** & **Wheelbase Length** \\ \(m_{total}\) & \(k\) & \(L_{b}\) & \(b\) & \(b_{W}\) & \(L\) \\ \hline
12 (tonne) & 185 (kN/m) & 12.0 (m) & 2.7 (m) & 2.5 (m) & 5.9 (m) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Characteristics of the studied model. Figure 1: A general perspective of the studied model. To define the spatial position of the plane points by its transverse displacement \(u_{i}\left(x,\ y\right)\) (\(i=1,\ldots 4\)) at each corner node, some parameters must be assigned in the system of equations as: \[\left\{ \begin{aligned} u_{1}&=u_{G}-(b/2)\times\theta_{x }-(L/2)\times\theta_{y}\\ u_{2}&=u_{G}+(b/2)\times\theta_{x}-(L/2)\times\theta_{y}\\ u_{3}&=u_{G}+(b/2)\times\theta_{x}+(L/2)\times\theta_{y} \\ u_{4}&=u_{G}-(b/2)\times\theta_{x}+(L/2)\times\theta_{y} \\ \end{aligned} \right.\] (1) In which: \[\bullet\] \[u_{G}\] is the transverse displacement related to the plate mass centre, \[G\] ; \[\bullet\] \[\theta_{x}\] and \[\theta_{y}\] are the plate rotations about \[x-\] and \[y-\] coordinate axes. If the plate element is supported by 4 springs at corner nodes with stiffness \[k_{1}\], \[k_{2}\], \[u_{G}\]0, and \[u_{G}\]1, then, for a generic position of the rectangular plate defined by the nodal displacements, referring to Equation (1), the elastic deformation energy stored in the plate is obtained by the total contribution of the deformed springs at each node. Indeed, the total amount of the internal energy (elastic energy) must balance the external work performed by the external forces (or moments) applied to the plate as: \[u_{G}\]2 (2) where the nodal displacement \[u_{G}\]3 is a function of parameters presented in Equation (1). In relation to the elastic energy \[u_{G}\]4 of the deformed plate, the nodal equivalent external force system must be obtained to equilibrate internal forces as follows: \[\left\{ \begin{aligned} \frac{\partial U_{elastic}}{\partial U_{G}}& =\sum_{i=1}^{4}k_{i}\times u_{i}\frac{\partial u_{i}}{\partial u_{ G}}=F_{G}\\ \frac{\partial U_{elastic}}{\partial\theta_{x}}&= \sum_{i=1}^{4}k_{i}\times u_{i}\frac{\partial u_{i}}{\partial u_{ G}}=M_{\theta x}.\\ \frac{\partial U_{elastic}}{\partial\theta_{y}}&= \sum_{i=1}^{4}k_{i}\times u_{i}\frac{\partial u_{i}}{\partial\theta_{y}}=M_{ \theta y}\\ \end{aligned} \right.\] (3)  
Taking the derivatives of these equations, the chain rule for the independent variables \(u_{G}\), \(\theta_{x}\), and \(\theta_{y}\) would be acquired, therefore, the linear system of 3 equations can take the following form:  
\[\left\{ \begin{aligned} k_{11}u_{1}+k_{12}\theta_{x}+k_{13}\theta_{y }&=F_{U_{G}}\\ k_{21}u_{1}+k_{22}\theta_{x}+k_{23}\theta_{y}&=M_{ \theta_{x}}\\ k_{31}u_{1}+k_{32}\theta_{x}+k_{33}\theta_{y}&=M_{ \theta_{y}}\\ \end{aligned} \right. \tag{4}\]  
Figure 2: Platform model as a plane resting on 4 springs at corner nodes. Being:  
\[\left\{