# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 4 Results and Discussion
### Dynamic Responses  
We analyzed the effects of changes in the design parameters on the dynamic response of the vehicle model. In this study, suspension stiffness, suspension damping, and anti-roll bar torsional stiffness, which are relatively easy to change, were selected as the design parameters related to dynamic responses. The sprung mass, mass moment of inertia for the sprung mass, unsprung mass, and tire stiffness are not considered design parameters because their design changes are difficult. To obtain the dynamic responses of the vehicle model, simulations were performed for the step steer test with the steering wheel angle shown in Figure 10 and a vehicle speed of 100 km/h. The reason for selecting dynamic responses for the step steer test is that the step steer test provides a much harsher driving condition than the slalom and double-lane tests. Therefore, the dynamic responses to the step steer test have been analyzed in many studies [38; 39; 40]. When the step steer test is performed, the dynamic response to the pitch angle is much smaller than the dynamic response to the bounce displacement and roll angle. Therefore, in this study, the dynamic responses for the bounce and roll, excluding the pitch, are analyzed. First, the effects of suspension stiffness on the dynamic response of the sprung mass's bounce and roll were investigated. The dynamic responses of the bounce displacement and roll angle for the step steer test are shown in Figure 20. These responses were computed for the three sets of front and rear suspension stiffnesses: (0.1\(k_{f}\), 0.1\(k_{r}\)), (\(k_{f}\), \(k_{r}\)), and (10\(k_{f}\), 10\(k_{r}\)), where the values of \(k_{f}\) and \(k_{r}\) are the same as those presented in Section 2. As shown in Figure 20a, a positive bounce displacement occurs when the steering wheel angle of Figure 10 is given to the vehicle model. As shown in this figure, the bounce displacement is large when the suspension stiffness is low (in the case of 0.1\(k_{f}\) and 0.1\(k_{r}\)) and is small when the stiffness is high (in the case of 10\(k_{f}\) and 10\(k_{r}\)). However, Figure 20b shows that the roll angles obtained by this step steer test have negative values and become larger as the suspension stiffness decreases. The negative roll angle was caused because a positive steering wheel angle moved the left side of the sprung mass down and the right side up, as shown in Figure 11, resulting in a negative roll angle. Notably, the bounce displacement and roll angle, shown as solid black lines, exhibit high-frequency oscillations when the suspension stiffness is large. The high-frequency oscillation period of the bounce displacement was approximately 0.292 s, and the oscillation period of the roll angle was approximately 0.250 s. These periods correspond to frequencies of 3.42 and 4.00 Hz. The natural frequencies of the bounce-pitch modes are 2.93 and 3.46 Hz, and the natural frequency of the roll mode is 3.96 Hz when the front and rear suspension stiffness are 10\(k_{f}\) and 10\(k_{r}\), respectively. Considering these natural frequencies, the oscillation of 3.42 Hz in the bounce displacement is due to the bounce-pitch  
Figure 20: Dynamic responses of the displacements for three values of the suspension stiffness: (**a**) the bounce and (**b**) the roll. mode with 3.46 Hz, while the oscillation of 4.00 Hz in the roll angle is due to the roll mode with 3.96 Hz. The oscillation amplitudes of the bounce acceleration are shown in Figure 21a. Additionally, the roll angular acceleration shown in Figure 21b increases with suspension stiffness. The oscillation frequencies for these accelerations are the same as those for the displacements in Figure 20. As mentioned above, these oscillations occur due to the natural vibrations of the bounce-pitch and roll modes. The increase in the acceleration values of the bounce and roll may adversely affect the riding comfort and steering stability that passengers can feel.