# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 4 Results and Discussion
### Natural Frequencies and Mode Shapes
However, when the rear suspension stiffness increases, the natural frequencies of the rear wheel modes increase, while the natural frequencies of the front wheel modes remain constant. Based on the analysis of Figures 14 and 15, it can be inferred that an increase in either the front and rear wheel suspension stiffnesses leads to an increase in the natural frequencies associated with the roll mode, wheel modes, and one of the bounce-pitch modes. However, the frequencies of the other modes remain constant. Figure 16 shows the changes in natural frequencies when the stiffnesses of both the front and rear suspensions increase simultaneously from 0.01 times to 100 times the stiffness values given in Section 2. In this case, as expected, the natural frequencies for all modes for the sprung and unsprung masses increased. Next, we analyzed the effects of changes in the torsional stiffness of the front anti-roll bar on the natural frequencies of the vehicle model. Using the torsional stiffness of the front anti-roll bar \(\kappa_{f}=62,085\) N\(\cdot\)m/rad as a reference value, the natural frequencies were computed for five torsional stiffness values of 0.01, 0.1, 1, 10, and 100 times \(\kappa_{f}\). The computed results are shown in Figure 10:2 As shown in this figure, the natural frequency for the roll mode of the sprung mass increases with the torsional stiffness of the front anti-roll bar. However, the natural frequencies for the bounce-pitch modes remain unchanged, regardless of the torsional stiffness. Among the natural frequencies for the modes of unsprung masses, only the natural frequency for the out-of-phase front wheel mode increases with the torsional stiffness, while the natural frequencies of other modes remain unchanged. Figure 18 shows the change in the natural frequencies of the dynamic vehicle model when the torsional stiffness of the rear anti-roll bar is increased. The natural frequencies were computed while increasing the torsional stiffness of the rear anti-roll bar from \(0.01\kappa_{r}\) to \(100\kappa_{r}\) with all other dimensions and material properties fixed. The value of \(\kappa_{r}\) is 58,098 N\(\cdot\)m/rad, as given in Section 2. As shown in Figure 18, the natural frequencies of the sprung mass roll mode and out-of-phase rear wheel mode increased with the torsional stiffness of the rear anti-roll bar. However, this change in torsional stiffness had no effect on the natural frequencies of the other modes. It is observed from Figures 17 and 18 that if only one of the two anti-roll bar 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. Figure 19 shows the changes in natural frequencies of the vehicle model when the torsional stiffnesses of the front and rear anti-roll bars increased simultaneously. The natural frequencies in this figure were computed with values of 0.01, 0.1, 1, 10, and 100 times the values of \(\kappa_{f}\) and \(\kappa_{r}\) presented in Section 2. As shown in this figure, the natural frequencies of the bounce-pitch modes and in-phase wheel modes remain unchanged even when the torsional stiffness increases. However, the natural frequencies of the roll mode and out-of-phase wheel modes increase with torsional stiffness. Comparing Figures 17-19, the result of simultaneously increasing the front and rear torsional stiffnesses (Figure 19) is a combination of the results of increasing the front torsional stiffness (Figure 17) and rear torsional stiffness (Figure 18). Therefore, the natural frequency of the roll mode in Figure 19 is higher than the natural frequencies of the roll modes shown in Figures 17 and 18.