# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 5 Summary and Conclusions  
This study presents a dynamic vehicle model that can analyze the dynamic behavior of a vehicle without an actual driving test. Considering the centrifugal force acting on the vehicle during turning and the torsional stiffness of the anti-roll bar, a vehicle model with seven degrees of freedom was constructed to investigate the dynamic characteristics and responses to driving conditions. The equations of motion for this vehicle model were derived using Lagrange's equation. The constructed vehicle model was verified by comparing the suspension deformations computed by the simulations with the deformations measured through actual driving tests. Using this model, the natural frequencies and mode shapes of the vehicles were analyzed. Moreover, the effects of the suspension stiffness and anti-roll bar torsional stiffness on the natural frequencies were also analyzed. In addition, the effects of suspension stiffnesses, suspension damping coefficients, and anti-roll bar torsional stiffnesses on the dynamic responses of the vehicle were also investigated when the vehicle was under the step steer test. The results obtained in this study can be applied not only to internal combustion engine vehicles but also to electric and hybrid vehicles. The results of analyzing the dynamic characteristics of the driving vehicle using the proposed dynamic model are summarized as follows:  
* The natural frequencies of the vibration modes associated with the sprung mass were much lower than those of the modes associated with the unsprung masses. * When one of the front and rear wheel suspension stiffnesses increases, the natural frequencies for the roll mode, the associated wheel modes, and one of the bounce-pitch modes increase, whereas the frequencies of the other modes remain constant. * When both the front- and rear-wheel suspension stiffnesses increase simultaneously, the natural frequencies of all modes for the sprung and unsprung masses increase. * When only one of the two anti-roll bars' torsional stiffnesses increases, the natural frequencies of the roll mode and the corresponding out-of-phase wheel mode increase, while the natural frequencies of the other modes remain unchanged. * When the two torsional stiffnesses of the front and rear anti-roll bars increase simultaneously, the natural frequencies of the roll mode and out-of-phase wheel modes increase, while those of the bounce-pitch modes and the in-phase wheel modes remain unchanged. When performing the step steer test, the analysis results of the effects of design parameters such as suspension stiffness, suspension damping, and anti-roll bar torsional stiffness on the dynamic responses of the vehicle model are as follows:  
* The bounce displacement and roll angle decrease with the suspension stiffness. * When the suspension stiffness is large, the bounce displacement and roll angle have high-frequency oscillations because of the bounce-pitch mode and the roll mode, respectively. * The oscillation amplitudes in the bounce acceleration and roll angular acceleration increase with the suspension stiffness. * Suspension damping contributes to reducing the dynamic responses of the vehicle, including both the displacements and accelerations for bounce and roll motions. * The torsional stiffness of the anti-roll bar reduced both the displacements and accelerations of the bounce and roll motions. Conceptualization, J.C. and M.C. ; methodology, J.C. and S.Y. ; software, S.Y., J.L. and D.K. ; validation, W.J. and M.C. ; formal analysis, S.Y. and J.L. ; investigation, S.Y., J.L., W.J. and D.K. ; resources, D.K. and M.C. ; data curation, S.Y. and W.J. ; writing--original draft preparation, S.Y. ; writing--review and editing, M.C. and J.C.; visualization, S.Y. ; supervision, J.C.; project administration, J.C.; funding acquisition, J.C. All authors have read and agreed to the published version of the manuscript.