# Dynamic Modeling and Analysis of a Driving Passenger Vehicle
## 4 Results and Discussion
### Natural Frequencies and Mode Shapes  
Considering the natural frequencies and mode shapes of the vehicle model developed in Section 2, the natural frequency and mode shape were computed from the eigenvalue problem obtained by ignoring the damping term on the left and the force term on the right  
Figure 11: Comparison of the vertical suspension deformations during the step steer test between the simulation and experiment: (**a**) at the left front wheel; (**b**) at the right front wheel; (**c**) at the left rear wheel; and (**d**) at the right rear wheel. Figure 10: Time variation of the steering wheel angle for the step steer test. in Equation (19). As the damping properties have little influence on the natural frequencies, the damping term is negligible. Assuming \(\mathbf{q}=\mathbf{Q}\mathrm{e}^{\mathrm{i}\omega t}\) where \(\mathrm{j}=\sqrt{-1}\), \(\omega\) is the natural frequency, and \(\mathbf{Q}\) is the modal vector, the eigenvalue problem is given by  
\[(\mathbf{K}-\omega^{2}\mathbf{M})\mathbf{Q}=\mathbf{0} \tag{21}\]  
As the proposed model has seven degrees of freedom, the number of acquired natural frequencies and corresponding mode shapes is seven. The computed natural frequencies are 1.44, 1.62, 1.99, 12.73, 13.78, 15.45, and 16.67 Hz. The obtained mode shapes for these natural frequencies are displayed in Figures 12 and 13, which show the mode shapes for the sprung and unsprung masses, respectively. In these figures, the dashed lines represent the positions where each mass does not move, and the solid lines represent the mode shapes where each mass moves. The mode shapes of the sprung mass are shown in Figure 12. The bounce mode is for a vertical motion of the sprung mass, the pitch is for a rotation about the \(y\) axis, and the roll mode is for a rotation about the \(x\) axis. However, because the two modes shown in Figure 12a,b are modes in which bounce and pitch motions are combined, it is difficult to  
Figure 12: Mode shapes of the sprung mass: (**a**) the bounce–pitch mode with the natural frequency of 1.44 Hz; (**b**) the bounce–pitch mode with the natural frequency of 1.62 Hz; and (**c**) the roll mode with the natural frequency of 1.99 Hz. Figure 13: Mode shapes of the unsprung mass: (**a**) the in-phase front wheel mode (12.73 Hz); (**b**) the out-of-phase front wheel mode (13.78 Hz); (**c**) the in-phase rear wheel mode (15.45 Hz); and (**d**) the out-of-phase rear wheel mode. clearly distinguish these two modes as the bounce and pitch modes. In the mode shown in Figure 12a, bounce motion is more dominant than pitch motion. In contrast, the mode shown in Figure 12b has more pitch motion than bounce motion. In this study, the two modes with combined bounce and pitch motions, shown in Figure 12a,b, are called bounce-pitch modes. Comparing Figure 12a,b, the mode in Figure 12c has a distinct roll motion, indicating that it is a roll mode. The natural frequencies for the bounce-pitch modes are 1.44 and 1.62 Hz, and the natural frequency for the roll mode is 1.99 Hz. The mode shapes of the unsprung masses are displayed in Figure 13, where Figure 13a-d show the in-phase front wheel mode, out-of-phase front wheel mode, in-phase rear wheel mode, and out-of-phase rear wheel mode, respectively. In the in-phase front wheel mode with a natural frequency of 12.73 Hz, the two front wheels vibrate in the same direction, but the rear wheels are motionless (Figure 13a). On the other hand, the out-of-phase front wheel mode with a natural frequency of 13.78 Hz has vibrations of the front wheels in opposite directions and no motion in the rear wheels (Figure 13b). Similarly, in the in-phase rear wheel mode with a natural frequency of 15.42 Hz, the rear wheels vibrate in the same direction (Figure 13c). Additionally, in the out-of-phase rear wheel mode with a natural frequency of 16.67 Hz, the rear wheels vibrate in opposite directions (Figure 13d). Note that the natural frequencies of the vibration modes related to sprung masses are much lower than those of the vibration modes related to unsprung masses.