## 3 Mathematical Development
### Transient-Dynamic Formulation  
As mentioned, the vehicle mass and inertial properties are approached by a simple rectangular, solid plate element with spring suspension at each corner. Once it was possible to obtain the solution of the modal analysis of the problem from a set of uncoupled equations, the transient analysis would be obtained in a similar practical mode. The dynamic equilibrium equation for the oscillating plate can be re-written as follows:  
\[ \begin{bmatrix}4k&0&0\\ 0&4 kb^{2}/4&0\\ 0&0&4k^{2}/4\end{bmatrix}\begin{Bmatrix}u_{G}\\ \theta_{x}\\ \theta_{y}\end{Bmatrix} + \begin{bmatrix}4c&0&0\\ 0&4 cb^{2}/4&0\\ 0&0&4c^{2}/4\end{bmatrix}\begin{Bmatrix}u_{G}\\ \theta_{x}\\ \theta_{y}\end{Bmatrix} + \begin{bmatrix}4c&0&0\\ 0&4 cb^{2}/4&0\\ 0&0&4c^{2}/4\end{bmatrix} \begin{Bmatrix}u_{G}\\ 0&b^{2}/12&0\\ 0&0&L^{2}/12\end{bmatrix} \begin{Bmatrix}\bar{u_{G}}\\ \bar{\theta_{x}}\\ \bar{\theta_{y}}\end{Bmatrix} = \begin{Bmatrix}F_{u_{G}}(t)\\ M_{\theta_{y}}(t)\\ \end{Bmatrix} . \tag{20}\]  
As previously mentioned, the \(RHS\) of this dynamic equilibrium must be transformed to equivalent moments while dealing with nodal forces. As an illustration, having four independent nodal forces acting transversely on the plate plane, the inverse of Equation (1) produces the expression of the condensed displacement vector (composed of the mass centre displacements and rotations about coordinate axes), therefore:  
\[ \begin{Bmatrix}u_{G}\\ \theta_{x}\\ \theta_{y}\end{Bmatrix} = \begin{bmatrix}1&1&1&1\\ -b/2&-b/2&b/2&b/2\\ -L/2&-L/2&L/2&-L/2\end{bmatrix}\begin{Bmatrix}u_{1}\\ u_{2}\\ u_{3}\\ u_{4}\end{Bmatrix} . \tag{21}\]  
Then, the corresponding force vector, if composed of four time-dependent components, will be:  
\[ \begin{Bmatrix}F_{u_{G}}(t)\\ M_{\theta_{y}}(t)\\ \end{Bmatrix} = \begin{bmatrix}1&1&1&1\\ -b/2&-b/2&b/2&b/2\\ -L/2&-L/2&L/2&-L/2\end{bmatrix}\begin{Bmatrix}F_{u_{1}}(t)\\ F_{u_{2}}(t)\\ F_{u_{3}}(t)\\ F_{u_{4}}(t)\end{Bmatrix} . \tag{22}\]  
Considering that there exists a force \(F_{u_{1}}\) acting at node 1, the equivalent (condensed) force vector is represented in Equation (23). The complete Equation (21) can be extended to dynamic situations of transient forces applied at nodes as:  
\[ \begin{Bmatrix}F_{u_{G}}(t)\\ M_{\theta_{y}}(t)\\ \end{Bmatrix} = \begin{bmatrix}1&1&1&1\\ -b/2&-b/2&b/2&b/2\\ -L/2&-L/2&L/2&-L/2\end{bmatrix}\begin{Bmatrix}F_{u_{1}}(t)\\ 0\\ 0\\ 0\end{Bmatrix} = \begin{Bmatrix}F_{u_{1}}(t)\\ -\frac{b}{2}F_{u_{1}}(t)\\ -\frac{b}{2}F_{u_{1}}(t)\\ \end{Bmatrix} . \tag{23}\]  
A practical and efficient mode to solve this equation for generalized time-dependent loads relies on direct time integration methods. It could be developed from the mode superposition technique, another way to setup the solution for this dynamic problem. However, in the case of complex loads, where their decomposition into elementary modes is necessary, direct time integration is a more practical tool, limited only by the computation time in the case of very large DOF structure modelling. We adopted the [PERSON]'s constant acceleration method [26] for its accuracy, stability, and ease to program for computational purposes. An additional reason for this option is the fact that the solution is structured as an uncoupled system of equations. The dynamic equilibrium of a structure subjected to external time-dependent loads can be analysed by a direct time integration method such as [PERSON]'s algorithm [26].This \begin{table}
\begin{tabular}{c c} \hline
**Natural Frequency, \(f\)(Hz)** & **Motion Description** \\ \hline \(f_{1}^{ANL}=1.06\) & Rotation around \(Y_{G}\) axis; \\ \(f_{2}^{ANL}=1.25\) & Vertical motion, vertical translation of the plate, four springs simultaneously in function; \\ \(f_{3}^{ANL}=2.00\) & Rotation over \(X_{G}\) axis; \\ \hline \end{tabular}
\end{table} 
Table 3: Analytical natural frequency solutions. algorithm calculates the shape of a structure based on the concept of Constant Acceleration Method, where at each time step the structure's acceleration vector is supposed to remain constant.