# Investigation of Vehicle Stability with Consideration of
## 5 The Effect of Vehicle Suspensions on Stability  
### Kinematic Model of Vehicle Suspensions  
Three variants of vehicle suspension were modeled in the work: 1--front suspension: double arm Mercedes Benz type suspension (Figure 20a); 2--front suspension: double arm Honda type suspension (Figure 20a); 3--front suspension: MacPherson VW Golf type suspension (Figure 20b); 4--rear suspension: single-arm BMW type suspension (Figure 20c); 5--rear suspension: semi-dependent single-arm VW Golf type suspension (Figure 20c); 6--rear suspension--MacPherson Audi type suspension (Figure 20b). Figure 19: Comparison of vertical displacements of the vehicle body: 1—real vehicle; 2—suspension test bench. Driving of the vehicle with each of the suspension combinations was simulated on three different road profiles (low quality pavement, even pavement, and gravel road) during the investigation. For this purpose, programs developed in Matlab were used. The main version was designed to study the kinematics of suspensions. The programs were developed on the basis of kinematic analysis of multiassembly mechanisms. In preparation for the use of the provided data for stability calculations, a calculation of the instantaneous centers of suspension on the transverse and longitudinal planes of the vehicle was introduced, and auxiliary values for the evaluation of the angular stiffness of the suspensions were determined. The parameters required for the calculation of the angular suspension stiffness values and the calculation results were transferred to the files and used directly for stability assessment programs. It was assumed that the vehicle was traveling at a constant speed of 90 km/h. The following characteristics of suspension performance were determined: wheel travel in the vertical direction \(\Delta z\); wheel turning angle \(\Delta\varphi\); wheel displacement in the transverse direction \(\Delta y\) and wheel camber \(\xi\). The effect of the variation of these parameters on the stability was evaluated by using the generalized slip angle \(\delta\):  
\[\delta=\Delta\varphi+\delta_{\xi}+\delta_{y} \tag{14}\]  
where \(\Delta\varphi\) is the change in wheel turning angle; \(\delta_{\xi}\) is the tire slip angle due to wheel camber and \(\delta_{y}\) is the tire tread slip angle due to the transverse displacement of the wheel. Tire slip angle \(\delta_{\xi}\) was determined as follows:  
\[\delta_{\xi}=\frac{\xi}{k_{\xi}} \tag{15}\]  
where \(\xi\) is the wheel camber and \(k_{\xi}\) = 4-6. Additional tire slip angles \(\delta_{y}\) were calculated using the Equations:  
\[\delta_{y}=\frac{F_{y}}{k_{y}}=\frac{\Delta yC_{y}}{k_{y}} \tag{16}\]  
where \(F_{y}\) is the load acting on the wheel in the transverse direction; \(\Delta y\) is the transverse displacement of the wheel, \(C_{y}\) is the transverse stiffness of the tire and \(k_{y}\) is the slip angle drag coefficient. Table 2 shows the vertical travel of the wheel \(\Delta z\) with the car traveling on different road profiles. Based on these results, the suspension travel limits were set to calculate the changes in wheel position due to vertical suspension excitation. Investigation of the kinematic characteristics of the suspensions showed that different road profiles had significant effects on suspension performance. When driving on low quality asphalt, the total suspension angles of the front suspension for the Mercedes Benz suspension ranged from \(-0.5\) to \(0.4^{\circ}\), for Honda--from \(-0.5\) to \(0.8^{\circ}\), and for VW for front suspension from \(-0.7\) to \(1.3^{\circ}\). For the rear suspensions, the slip angles were much smaller. For example, in the case of the BMW they varied from \(-0.005\) to \(0.03^{\circ}\), and for VW from