# Research on the Vehicle Steering and Braking Stability Region
## 3 Solution of Equivalent Equilibrium Point  
### Introducing the D'Alembert Principle into the 5 DOF Model  
The vehicle dynamics system is a typical nonlinear system, and the common classical analysis method is the static bifurcation theory based on the change of the number of equilibrium points of the nonlinear system. Many scholars have studied vehicle steering stability. Under the premise of constant longitudinal velocity, the dynamic essence of vehicle steering instability can be attributed to the bifurcation of the equilibrium point in nonlinear dynamics [2; 3; 5]. However, under the steering and braking conditions, because the longitudinal velocity of the vehicle is constantly changing, there is no equilibrium state according to the strict definition. As a result, the mature research method of steering bifurcation cannot be directly applied to the braking process. Therefore, it is necessary to explore new methods to analyze the bifurcation characteristics of braking. [PERSON] mentioned in [18] that when analyzing the acceleration or braking of a vehicle system, the virtual force in the D'Alembert principle can be introduced so that an accelerated system can be transformed into an equivalent equilibrium system. The conversion method involves applying a virtual external force determined by acceleration to the system. Based on this principle, if a virtual force with the size of \(F=ma\) is applied to the vehicle (where the acceleration is acting on the center of mass of the vehicle), the braking process can be regarded as an equilibrium state (called equivalent equilibrium in this paper) and bifurcation analysis can then be carried out. Figure 3 gives the force analysis diagram of the whole vehicle after adding the virtual force:  
To maintain the longitudinal force equilibrium of the vehicle, the following can be obtained:  
\[ma_{x}=\frac{T_{b}}{R_{e}}+\mathrm{sgn}(v_{x})C_{\mathrm{air\_}x}A_{\mathrm{L \_}x}\varrho_{x}^{2} \tag{11}\]  
Figure 3: Force analysis diagram with virtual force added. The virtual acceleration \(a_{x}\) acting on the center of mass of the vehicle body can be obtained:  
\[a_{x}=\frac{\left(\frac{\frac{T_{b}}{R_{e}}+\text{sgn}(v_{x})C_{\text{air}\_x}A_{ \text{L}\_x}\xi_{\
u}v_{x}^{2}}\right)}{m} \tag{12}\]  
The equilibrium in the plane motion of the vehicle must satisfy the steady state equation, and it is constructed by making the right side of the dynamic equations equal to zero [31]. The accuracy of the equilibrium point solution is evaluated by the value of the fitness function. That is, the smaller the objective function value, the more accurate the solution. The equation for solving the equivalent equilibrium point of the system is as follows:  
\[\left\{ \begin{array}{l}\dot{v_{y}}=-v_{x}\omega+\frac{F_{lf}\sin\delta_{f}+F _{if}\cos\delta_{f}+F_{ur}-\text{sgn}(v_{y})\cdot C_{\text{air}\_y}A_{\text{L} \_y}\xi_{\
u}v_{x}^{2}}{m}=0\\ \dot{\omega}=\frac{(F_{lf}\sin\delta_{f}+F_{if}\cos\delta_{f})\int_{f}-F_{or} I_{r}\int_{z}I_{z}}{I_{z}}=0\\ \dot{v_{x}}=v_{y}\omega+\frac{F_{lf}\cos\delta_{f}-F_{if}\sin\delta_{f}+F_{if} -\text{sgn}(v_{x})\cdot C_{\text{air}\_x}A_{\text{L}\_x}\xi_{\
u}v_{x}^{2}+m \alpha_{x}}{m}=0\\ \dot{\omega_{f}}=\frac{-\text{sgn}(\omega_{f})\int_{f}-R_{e}\cdot F_{lf}}{I_{ r}}=0\\ \dot{\omega_{r}}=\frac{-\text{sgn}(\omega_{r})\cdot\int_{f}-R_{e}\cdot F_{lf}}{I }=0\end{array} \right.