# Investigation of Vehicle Stability with Consideration of
## 2 Vehicle Stability Modeling Methods  
Trend analysis of the traffic and vehicle design patterns suggests that handling, i.e., the ability to move in the direction intended by the driver, is one of the key performance properties of vehicles that influence their safety. The driver sets the intended direction using the steering wheel, and the handling tasks are targeted at the steering mechanism. One of the first tasks addressed was the interrelation between the front wheel angles. The Ackermann-Jeantaud scheme (Figure 1) was used for this purpose. Following the introduction of pneumatic tires into modeling, the dependencies of the tire-road interrelation had to be analyzed in greater detail. The area of contact between the tire and the road is subject to deformation under the action of a lateral force. As a result, the contact patch axis is no longer parallel to the wheel symmetry plane, but forms angle \(\delta\), the former being referred to as the tire slip. In most tasks, the slip angle is determined as the angle between the velocities longitudinally and transversely to the wheel symmetry plane. The lateral force \(F_{y}\) that the wheel has the capacity to withstand depends on the radial load and slip angle. Tire slip angle (Figure 2) causes a considerable distortion of the cornering scheme [48]. There are several theories that can be used to determine the angle of tire-slip. The theory of [PERSON] [48] is most convenient. Under [PERSON]'s theory, lateral tire deformation \(\xi\) and trajectory curvature \(\rho\) are described by the following Equations [47; 48]:  
\[F_{y}=C_{y}\xi,\frac{1}{\rho}=\alpha\xi-\beta\varphi_{t} \tag{1}\]  
where \(C_{y}\) is the lateral stiffness of the tire, \(\rho\) is the trajectory curvature radius, \(\alpha\) and \(\beta\) are the constant coefficients associated with the tire structure, and \(\varphi_{t}\) is the angle between the wheel symmetry plane and wheel trajectory tangent (Figure 2). Figure 1: Ackermann—Jeantaud schema [47]. If \(F_{y}\
eq\)const., then the wheel center movement velocity in the transverse direction equals is given by:  
\[v_{y}=v\varphi_{t}+\dot{\xi} \tag{2}\]  
where \(v_{y}/v=\delta\), then \(\delta=\varphi_{t}+\dot{\xi}/v\), from which it follows:  
\[\varphi_{t}=\delta-\dot{\xi}/v \tag{3}\]  
This would lead to the condition that only the tire slip angle would affect the vehicle movement direction, the prerequisite being \(\dot{\xi}=0\), i.e., \(F_{y}=\) const. For low \(\varphi_{t}\), the wheel trajectory is straight, and the contact patch area is positioned at angle \(\varphi_{t}=\delta\) relative to the symmetry plane of the wheel. With the available inertia forces, the longitudinal and transverse movement of the vehicle can be described. Considering that \(\theta\), \(\delta_{1}\) and \(\delta_{2}\) are low, and their cosines are equal to each other, while sines are equal to the angle values (Figure 3), and forming the expression \(F_{y1}=k_{y1}\delta_{1}\) and \(F_{y2}=k_{y2}\delta_{2}\), the transverse force equilibrium is described by Equation (4) [48]:  
\[F_{iy}=m_{a}(v\omega_{a}+\dot{v}_{y})=k_{y1}\delta_{1}+k_{y2}\delta_{2}-F_{x1}\Theta \tag{4}\]  
where \(m_{a}\) is the vehicle center of mass; \(\omega_{a}\) is the vehicle angular velocity when cornering; \(k_{y1}\) and \(k_{y2}\) are the slip angle drag coefficients; \(F_{x1}\) and \(F_{y1}\), \(F_{x2}\) and \(F_{y2}\) are drag forces for the front and rear wheels respectively and \(\Theta\) is the wheel turning angle.