## 3 Methods
### Road-Vehicle Interaction Model
#### 3.1.2 The Road-Vehicle Interaction Model  
To address the limitations of the bicycle model, it is imperative to enhance its accuracy and effectiveness. By integrating the truck's braking characteristics, the road-vehicle interaction model is development in this study. 1. Truck's braking characteristics  
During braking, there exist three types of braking modes based on the sequential occurrence of wheel lock-up: front-wheel lock-up followed by skidding, rear-wheel lock-up followed by skidding, and simultaneous lock-up and skidding of both front and rear wheels [22; 23]. Clearly, the simultaneous lock-up and skidding of both front and rear wheels demonstrates the most effective utilization of adhesion conditions. The critical conditions for the occurrence of the braking modes are as follows:  
* \(\varphi<\varphi_{0}\): the position of the \(\beta\)_-curve_ lies below the \(I\)_-curve_, implying that during braking, the front wheels consistently experience lock-up before the rear wheels. * \(\varphi>\varphi_{0}\): the position of the \(\beta\)_-curve_ lies above the \(I\)_-curve_, signifying that during braking, the rear wheels consistently experience lock-up before the front wheels. * \(\varphi=\varphi_{0}\): in this scenario, during braking, both the front and rear wheels undergo simultaneous lock-up. The \(\beta\)_-curve_ represents the actual brake force distribution curve, the \(I\)_-curve_ represents the ideal brake force distribution curve, and \(\varphi\) represents the maximum friction coefficient that can be generated between the tire and pavement; namely, \(\varphi_{0}\) represents the synchronizing adhesion coefficient. 2. Model development and analysis  
Considering the brake force distribution under the three types of braking modes, the road-vehicle interaction model can be developed for vehicle dynamics response analysis. 1. Braking mode 1: \(\varphi<\varphi_{0}\)  
The dynamic change process of braking forces on front and rear axles can be divided into three stages during braking (marked in red, green, and blue, respectively), as shown in Figure 3.  
Stage I. This stage is marked in red in Figure 3. Specifically, the braking forces on each axle increase along the \(\beta\)_-curve_. When the braking forces increase to point \(A\), the front wheels lock up while the rear wheels do not, marking the end of stage I. The braking deceleration rate of the truck at point \(A\) is recorded as \(a_{x1}\), and the ratio of the front-axle braking force to the rear-axle braking force at point \(A\) is given by  
\[F_{bf}/F_{br}=\beta(1-\beta) \tag{20}\]  
\[F_{bf}/F_{br}=\frac{\mu_{xmax,supply}\big{(}mgb/L+(m(a_{x1}+gi))h_{g}/L\big{)}}{ m(a_{x1}+gi)-\mu_{xmax,supply}\big{(}mgb/L+(m(a_{x1}+gi))h_{g}/L\big{)}} \tag{21}\]  
where \(\beta\) is the ratio of the front-axle braking force to total braking force and \(a_{x1}\) is the braking deceleration rate of the vehicle at point \(A\), m/s\({}^{2}\). Equations (20) and (21) can be combined to obtain  
\[a_{x1}/g=\Big{(}\mu_{xmax,supply}b\Big{)}/\Big{(}\beta L-\mu_{xmax,supply}h_{ g}\Big{)}-i \tag{22}\]  
The expression of coefficient \(\beta\) is  
\[\beta=\big{(}\varphi_{0}h_{g}+b\big{)}/L \tag{23}\]  
Substituting Equation (23) into Equation (22), one obtains  
\[a_{x1}/g=\Big{(}\mu_{xmax,supply}b\Big{)}/\Big{(}\varphi_{0}h_{g}+b-\mu_{xmax,supply}h_{g}\Big{)}-i \tag{24}\]  
In stage I, \(a_{x}<a_{x1}\), the braking forces on each axle can be expressed as  
\[F_{bf}=\beta F_{b}=\big{(}\varphi_{0}h_{g}+b\big{)}\ (ma_{x}+mgi)/L \tag{25}\]  
\[F_{br}=(1-\beta)F_{b}=\big{(}1-\big{(}\varphi_{0}h_{g}+b\big{)}/L\big{)}(ma_{x }+mgi) \tag{26}\]  
Substituting these expressions into the given motion equations, Equations (14) and (15), and simplifying the results, the longitudinal friction factors on each axle become  
\[\mu_{xf}=F_{bf}/F_{Nf}=\big{(}\varphi_{0}h_{g}+b\big{)}/\big{(}b/(a_{x}/g+i)+ h_{g}\big{)} \tag{27}\]  
Figure 3.0  
Figure 3.1 Variation curve of front- and rear- axle braking (or brake) forces.