# Research on the Vehicle Steering and Braking Stability Region
## 5 Verification Based on the Energy Dissipation Method
### Comparison and Verification  
Figure 11 provides both the energy dissipation diagram and projection diagram of the system with the inertia force applied. The initial longitudinal velocities are \(v_{x}\) = 30 m/s and \(v_{x}\) = 50 m/s, respectively. The ranges of \(\delta_{f}\) are \(-\)0.03 rad to 0.03 rad and \(-\)0.02 rad to 0.02 rad, with step sizes of 0.0005 rad. The range of \(T_{b}\) is 0 to 504 N\(\cdot\)m, with step sizes of 10.08 N\(\cdot\)m.  
Figure 11a,c show that the energy dissipation results of the system at two initial velocities have positive and negative values, and the energy dissipation increases with the increase of velocity. The occurrence of negative values is due to the increase of energy caused by the positive work of the inertia force. However, it does not affect the hierarchical distribution trend of energy. The blue part in Figure 11 has a lower energy dissipation, resembling a trapezoid, and is the stable region for the steering and braking condition. The purple part has a high energy dissipation and smooth surface distribution trend, which represents the unstable region of the steering and braking condition. Figure 11b,d compare the boundary lines of the stability region obtained by the energy dissipation method (the boundary lines of the purple and blue parts) with the boundary lines obtained by the bifurcation method (the red circle line). It can be seen that both of them are almost identical. Furthermore, taking Figure 11d as an example, we can observe some quantitative phenomena. When limiting the front wheel steering angle or braking torque, we can easily see the range of stability and instability of the system. For example, when the braking torque is limited to 100 N\(\cdot\)m, the area where the absolute value of the front wheel angle is greater than 0.006 rad is the instability region, and the area less than 0.006 rad is the stability region. When the front wheel steering angle is limited to 0.01 rad and the braking torque is greater than 231.84 N\(\cdot\)m, it is considered a stable region, and, vice versa, it is considered an unstable region. In order to study the stability region under steering and braking conditions using the classical bifurcation analysis method, the virtual force is applied for converting the original model into an equivalent system. However, due to the difference of dynamic characteristics between the equivalent system and the original system after the addition of the virtual force, the energy dissipation method is used to calculate the steering and braking stability region of the original system. Figure 12 provides the schematic diagram and projection of the energy dissipation of the system without applying virtual force, which is also compared with the stability region obtained by the equilibrium point bifurcation method. From Figure 12, it can be seen that there is no external input of virtual force on the system, and the energy dissipation is all positive, showing a clear layered distribution. The blue part has a lower energy dissipation, resembling a trapezoid, which is the stable region for the steering and braking condition. The purple part has a high energy dissipation and smooth surface distribution trend, which represents the unstable region of the steering and braking condition. The boundary of the energy dissipation stable region can be seen in Figure 12 (the boundary lines of the purple energy region and the blue energy region are clearer than those of the system with added inertia force). At the same time, they are compared with the boundary obtained by the bifurcation method. It can be seen that when \(v_{x}=30\) m/s and \(v_{x}=50\) m/s, the results show that the stability regions of the 5 DOF system with and without the application of virtual force almost remain the same. From Figure 12b,d, it can be seen that when the braking torque is limited to 500 N-m and the initial longitudinal velocity is 30 m/s, the absolute value of the front wheel steering angle at the boundary point is greater than 0.02 rad.