## 5 Conclusions  
In this work, a single modular rectangular element was developed to accurately characterise the dynamic behaviour of a bus vehicle model with a passive suspension subjected to generalised transient forces. A mathematical set of equations was developed based on the modal analysis hypothesis to offer an analytical solution for natural frequency calculation. The obtained results infer that the performance of the proposed methodology is accurate in static and steady-state dynamics applications. However, the main conclusions of the present study can be drawn as follows:  
* The most valuable practical implication is a comprehensive vehicle suspension diagnostic system concept, from which a simplified model helps to produce outputs with sufficient accuracy in which an approximate structural behaviour of the physical model can be predicted;
* Only the chassis-induced transverse displacement vibrational modes allow the direct comparison in which the basic model would be cross-checked;
* In this paper, by means of simulation, the analytical control algorithm is used to evaluate the dynamic equilibrium of the structure, subjected to time-dependent domain loads, at any instant. Thus, the obtained results revealed a deviation of less than 8% about the displacement, allowing the model confirmation. This may provide a useful reference for further experimental investigations.
* The analytical solution relies on a single finite element demonstrating a coherent dynamic response under transverse forces. It can practically model a less smooth track in addition to the presence of in-plane transient forces, such as as those due to sharp manoeuvres or the effect of side gusts;
* A quite low number of DOFs and its accuracy make sound arguments for the applicability of this single modular element to automotive engineering in the fields of driving practice and suspension turning for dynamic safety.  
In the future development of this single chassis block-like modulus, the incorporation of rotating inertial elements in the mass matrix and the steady state dynamic analysis integrating damping will result in an improved modal analysis and dynamic vehicle behaviour when subjected to external loads. The presentation of this structural dynamic modelling could be applied to the modal analysis of more elaborate construction vehicles in the future.  
Conceptualization, F.Q.d.M. and R.L.; methodology, F.Q.d.M.; software, B.V.F. and R.L.; validation, B.V.F. and R.L.; formal analysis, R.L.; investigation, B.V.F. and R.L.; writing--original draft preparation, B.V.F. and R.L.; writing--review and editing, B.V.F., R.L., and P.M.G.P.M.; visualization, B.V.F. and R.L.; supervision, P.M.G.P.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Fundacao para a Ciencia e a Tecnologia (FCT), Portugal, under grant number 2022.13216.BD.  
Not applicable.  
The data presented in this study are available on request from the corresponding authors. The data are not publicly available due to their containing information that could compromise the privacy of research participants.  
The authors declare no conflict of interest.