# Investigation of Vehicle Stability with Consideration of
## 5 The Effect of Vehicle Suspensions on Stability
### Cornering Stability of the Vehicle  
Modeling of the linear motion according to the equations provided in Section 5.1, including additional evaluation of the changes in the spatial position of the wheel, showed the necessity to input the driver's reaction. Without the driver, such modeling is not effective because the car changes direction due to accidental effects and requires frequent correction to simulate direction adjustment using the steering mechanism. However, in this work, the technical specifications for the model with a driver were prepared, while the excitation parameters when driving on roads of different quality could be established using this model. The developed Matlab programs enabled the determination of suspension deformations (and tire-road contact geometry for more accurate modeling). The appropriateness of the solutions was evaluated using a more defined model that examined vehicle behavior under stabilized conditions, i.e., when moving in circles. Vehicle cornering stability was investigated but the studies did not take into account the change in the position of the wheels during vehicle tilt, which may affect vehicle stability. In this work, the vehicle behavior model was refined by evaluating the kinematics of the vehicle's front and rear axles and their influence on the vehicle's behavior. When examining vehicle cornering stability, two criteria are usually evaluated [50]: loss of stability when the car slips and loss of stability when the car rolls over. Our study additionally specified the extent of the change in the vehicle at cornering due to suspension deformations and changes in the position of the wheels at car rollover. The main problem encountered in the evaluation of vehicle stability at corners was that the lateral inertia forces redistributed the vertical wheel loads. The solution to the problem was initiated by investigating the problem of car tilt in cornering. In the programs for the kinematic analysis of the suspension, the position of the center of instantaneous tilting was additionally determined for the front and rear axles. The application with the subroutine provides additional data required to determine the tilt angle, the instantaneous tilt centers, the angular stiffness coefficients of the suspension, and stabilizers. In determining the distribution of vertical reactions, the assumption was made that under the action of a lateral force, the sprung part of the vehicle would rotate about the axis connecting the tilt centers of the two axles. When calculating vehicle tilt, simplified tilt schemes (Figure 27) are usually chosen, which do not take into account body (frame) deformations and changes in the center of position of the tilt center during vehicle tilt. With acceleration in the lateral direction equal to \(\mu\), the damping mass \(m_{a}\) of the vehicle would be subject to force \(m_{a}\mu\). If the designations in Figure 27 are used, the distance from the road surface to the tilt axis at the center of mass is equal to:  
\[h_{p}=h_{1L}+(h_{2L}-h_{1L})/L=(h_{2L}a_{1}+h_{1L}a_{2})/L \tag{20}\]The body tilt is equal to:  
\[\beta=\frac{\Delta R_{1}+\Delta R_{2}}{\frac{(c_{1k}+c_{1 sk})c_{1 pk}}{c_{1k}+c_{1 sk}+c_{kpa}}+\frac{(c_{2k}+c_{2 sk})c_{1 pa}}{c_{2k}+c_{2 sk}+c_{1 qa}}t} \tag{21}\]  
where \(\Delta R_{1}\), \(\Delta R_{2}\) are the changes in radial loads on the front and rear axles, respectively, due to the lateral force, and \(c_{ik}\), \(c_{isk}\), \(c_{kpa}\) are the coefficients of angular stiffness of the suspension, angular stabilizer and the tires, respectively. The tilt of the front and rear wheels is equal to:  
\[\beta_{1}=\beta\frac{c_{kpa}}{c_{1k}+c_{1 sk}+c_{kpa}},\ \beta_{2}=\beta\frac{c_{ kpa}}{c_{2k}+c_{2 sk}+c_{kpa}} \tag{22}\]  
The angle of slipping of the wheel may be calculated after determining the tilt of the wheels.