# Investigation of Vehicle Stability with Consideration of
## 5 The Effect of Vehicle Suspensions on Stability
### Investigation of the Dynamics of Change in Vehicle Direction  
Following assessment of the kinematic properties of the suspensions, the suspension model was improved by introducing deformable elements at the characteristic locations of the suspension and steering mechanism. The deformability of the arms was evaluated in the suspension, and a nonrigid steering rod was used in the steering mechanism. All deformable elements of the suspension and steering mechanism were reduced to these elements (Figure 20a,b). To determine the spatial position of the wheels during variation of the vertical displacement, \(z\) of the wheel, a coordinate system was used (Figure 23).  
If the vehicle body is subjected to a lateral force, then the reaction resulting from transverse wheel displacements \(\Delta y\) is:  
\[R_{iy}=C_{y}\cdot\Delta y \tag{17}\]  
where \(C_{y}\) is the transverse stiffness of the tire and \(\Delta y\) is the transverse displacement of the wheel. Then:  
\[m_{a}\cdot\ddot{y}_{c}=\Sigma\Delta R_{ih},\ I_{za}\cdot\ddot{\phi}_{c}=\left( \Delta R_{fL}+\Delta R_{fR}\right)\cdot a+\left(\Delta R_{rL}+\Delta R_{rR} \right)\cdot b \tag{18}\]  
When modeling vehicle stability, the effect of nonlinearity of the suspension on the vertical displacements of the body and wheels was investigated in view of the excitation frequency by combining different quality road profiles and vehicle speeds. The quarter-car  
Figure 23: Wheel coordinate system of the 3D model (**a**); lateral forces acting on the wheels (**b**).  
Figure 22: Dependence of the slip angle of the front (**a**) suspension and rear (**b**) suspension wheels on the vertical wheel travel (marking according to Table 2).  
model was used (Figure 9). The driving of the vehicle on three different road profiles (low quality pavement, even asphalt pavement, and gravel road) was simulated during the investigation. The VW Golf vehicle was assumed to travel at a constant speed of 10, 20 and 30 m/s (36, 72, 108 km/h).  
The following suspension performance characteristics were determined: wheel travel in the vertical direction \(\Delta z\); wheel turning angle change \(\Delta\varphi\); wheel displacement in the transverse direction \(\Delta y\); wheel camber \(\xi\) and wheel slip angle \(\delta\) (Figures 24 and 25).  
The Matlab program for the investigation of suspension kinematics was used for description of values \(\Delta\varphi\), \(\Delta y\), \(\xi\). The results obtained that describe the dependence of the parameters of the mentioned parameters on the vertical travel of the wheel are presented in Figure 26.  
Figure 24: Dependence of transverse wheel displacement (**a**) and wheel turning angle (**b**) on vertical wheel travel (marking according to Table 2).  
Figure 25: Dependence of wheel camber (**a**) and slip angle (**b**) on vertical wheel travel (marking according to Table 2).  
The results obtained show that additional investigations of vehicle stability are needed in the dynamic mode, i.e., when the car is moving, as the total slip angles obtained exceed 4 degrees in borderline cases. Previous vehicle stability calculations show that these angles can be critical under transverse stability conditions even at relatively low speeds (up to 100 km/h). To this end, additional investigations on the vertical dynamics of the vehicle were performed to evaluate the influence of nonlinear suspension elements on the tire tread and the dynamics coefficients of unsprung mass and sprung mass:  
\[k_{d}=\frac{z_{i}}{q} \tag{19}\]  
where \(z_{i}\) is the \(i\)-th mass displacement; \(q\) is the height of the unevennesses (\(q\) is assumed to be equal to the average height of the unevenness corresponding to the even asphalt pavement road surface).