# Investigation of Vehicle Stability with Consideration of
## 3 Investigation of the Factors Affecting the Vehicle Stability
### Dynamic Quarter-Car Model  
The quarter-car model was used for the tire-road interaction analysis. The model corresponded to the operating conditions of a single vehicle wheel and enabled researchers to simplify the analysis of vehicle dynamics. The quarter-car model is usually used to identify key vehicle suspension parameters and investigate tire performance. Although the quarter-car model only covered the vertical elastic and damping suspension elements, the results enabled analysis of the directional stability of the vehicle at vertical excitation. The data obtained were used as the baseline data to identify the exact spatial position of the wheel (Section 5). A revised quarter-car model (Figure 9) was used for the analysis of vehicle movement on road surfaces of different quality. In contrast to the two-mass model, the revised model also accounted for the tensile properties of the tire tread and the friction in the suspension elements. This model could provide more accurate structural parameters of the vehicle analyzed. Model movement Equations for the vertical direction:  
\[ \begin{split}& m_{1}\ddot{z}_{1}+k_{2}(\dot{z}_{1}-\dot{z}_{2})+k_{ 1}(\dot{z}_{1}-\dot{q})+c_{2}(z_{1}-z_{2})+c_{1}(z_{1}-q)=0,\\ & m_{2}\ddot{z}_{2}+k_{3}(\dot{z}_{2}-\dot{z}_{3})+k_{2}(\dot{z}_{ 2}-\dot{z}_{1})+c_{3}(z_{2}-z_{3})+c_{2}(z_{2}-z_{1})+F_{\mu}\text{sgn}(\dot{z}_ {2}-\dot{z}_{3})=0,\\ & m_{3}\ddot{z}_{3}+k_{3}(\dot{z}_{3}-\dot{z}_{2})+c_{3}(z_{3}-z_{ 2})+F_{\mu}\text{sgn}(\dot{z}_{3}-\dot{z}_{2})=0\end{split}  \tag{13}\]  
where \(m_{1}\), \(m_{2}\), \(m_{3}\) are the tread part of the tire, unsprung, and sprung mass, respectively; \(z_{1}\), \(z_{2}\), \(z_{3}\) are the tire, suspension and body displacements, respectively; \(k_{1}\), \(k_{2}\), \(k_{3}\) are the tread, tire, and suspension damping factors, respectively; \(c_{1}\), \(c_{2}\), \(c_{3}\) are the stiffness of the tread, tire, and suspension, respectively; \(F_{\mu}\) is the friction force at the shock absorber rod and q is the height of road profile unevenness. Figure 8: Tire contact length distribution for different road quality and tire models: 1—smooth asphalt pavement 2D; 2—smooth asphalt pavement 3D; 3—low quality asphalt pavement 2D; 4—low quality asphalt pavement 3D; 5—gravel road 2D; 6—gravel road 3D. When refining the quarter-car model, it is necessary to consider that the elastic elements and dampers of the suspension have distinctly nonlinear characteristics. In the investigated case, the characteristic of the front suspension spring of the modeled VW Golf was linear, as refined by the experiment. However, it must be taken into account that elastic travel stops block with nonlinear characteristics and start to act in extreme positions with respect to the car's suspension recoil and compression travel. Therefore, the overall characteristic of all the elastic elements of the suspension is close to typical (Figure 10a). In the model, the suspension characteristics are described by three lines. For the avoidance of uncertainty, additional refinements were used in the inflection zones of the characteristics. The damping characteristics of the damper were described in a similar way (Figure 10b). The frequency characteristic of the vehicle was determined by numerical modeling using calculation applications, with the road profile being sinusoidal in shape. The developed computer application enabled the analysis of vehicle movement with the set  
Figure 10: Modeling of suspension characteristics: (**a**)—elastic characteristic; (**b**)—damping characteristic of the damper; 1—typical characteristic; 2—simulated characteristic; 3— inflection zone refinement. Figure 9: Quarter-car model. parameters related to the road surface at the respective frequencies. The movement of the model corresponding to the city car parameters was investigated to determine its frequency characteristics (Figure 11). When compared to the theoretical frequency characteristic, low-frequency resonance oscillations of higher amplitude sprung masses of the body were observed.