# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 4 Results and Discussion
### Natural Frequencies and Mode Shapes
We analyzed the effects of the vehicle suspension stiffness and anti-roll bar torsional stiffness on the natural frequencies for the sprung and unsprung modes. The natural frequencies for the change in stiffness are shown in Figures 14-19, where the abscissa represents the front suspension stiffness and the ordinate represents the natural frequency. The numbers on the abscissa indicate the multiples of the stiffness given in Section 2. For example, \"0.1\" on the abscissa of Figure 14 means \(0.1\) times \(k_{f}\), which is the front suspension stiffness of \(0.1k_{f}\). In Figures 14-19, the hollow inverted triangles represent the natural frequencies of the two bounce-pitch modes of the sprung mass. These inverted triangles are not connected by lines because the bounce and pitch modes are not clearly distinguished and are mixed, as previously explained. The natural frequencies for the roll mode of the sprung mass are shown as blue triangles connected by solid blue lines. The natural frequencies for the in-phase front wheel mode, out-of-phase front wheel mode, in-phase rear wheel mode, and out-of-phase rear wheel mode are represented by hollow squares with solid lines, hollow circles with red dotted lines, hollow triangles with solid lines, and blue crosses with blue dotted lines, respectively. Figure 14: Natural frequencies for the variation of the front suspension stiffness. Figure 16: Natural frequencies when the front and rear suspension stiffnesses change simultaneously. Figure 17: Natural frequencies for the variation of the torsional stiffness of the front anti-roll bar. Figure 15: Natural frequencies for the variation of the rear suspension stiffness. First, we consider the effect of suspension stiffness variation on the natural frequencies of the modes. Figure 14 shows the natural frequencies for the values of the front suspension stiffness, \(0.01k_{f}\), \(0.1k_{f}\), \(k_{f}\), \(10k_{f}\), and \(100k_{f}\), where \(k_{f}=44,400\) N/m, as given in Section 2. The dimensions and material properties of the vehicle model, excluding the front suspension stiffness, were the same as those presented in Section 2. As shown in the figure, the natural frequencies of the sprung mass modes were much lower than those of the unsprung mass modes. The natural frequencies for the bounce-pitch modes of the sprung mass are 0.16 and 1.61 Hz for \(0.01k_{f}\), 0.49 and 1.61 Hz for \(0.1k_{f}\), 1.44 and 1.62 Hz for \(k_{f}\), 1.61 and 2.95 Hz for \(10k_{f}\), and 1.62 and 3.51 Hz for \(100k_{f}\). Notably, one of the natural frequencies for the bounce-pitch modes does not change significantly and remains close to 1.61 Hz. Other natural frequencies not close to 1.61 Hz increase to 0.16, 0.49, 1.44, 2.95, and 3.51 Hz as the suspension stiffness increases. As observed in this figure, the natural frequencies for the roll mode of the sprung mass, as well as the in-phase and out-of-phase front wheel modes, increase with the front suspension stiffness. However, the natural frequencies of the in-plane and out-of-plane rear-wheel modes remained constant even as the front suspension stiffness increased. In other words, the front suspension stiffness affects the natural frequencies of the front-wheel modes but not the frequencies of the rear-wheel modes. Figure 10:0 Natural frequencies when the torsional stiffnesses of the front and rear anti-roll bars change simultaneously. Figure 10:1 Natural frequencies for the variation of the torsional stiffness of the rear anti-roll bar. The changes in the natural frequencies of the dynamic vehicle model are presented in Figure 15 when the rear suspension stiffness is varied from \(0.01k_{r}\) to \(100k_{r}\), while the front suspension stiffness is fixed at \(k_{f}\). Similar to Figure 14, one of the natural frequencies for the bounce-pitch modes increases and the others remain constant at around 1.45 Hz as the rear suspension stiffness increases. The natural frequency for the roll mode also increased with the rear suspension stiffness.