# Modelling and Stability Analysis of Articulated Vehicles
## 2 Modelling of Non-Linear Systems for Articulated Vehicles
### Vehicle Dynamics Analysis and Modelling  
The articulated vehicle dynamics model is simplified as follows based on this study's research objectives. 1. The front and rear body centres are located on the longitudinal central axis, and the vehicle is symmetrical with respect to the longitudinal central axis. 2. The influence of the tire camber angle and return torque on wheel dynamics are disregarded. 3. Air resistance is disregarded, and the road surface is flat and two-dimensional. The articulated vehicle dynamics model is shown in Figure 1. The force analysis of the articulated vehicle is performed to obtain the equations of the equilibrium for the lateral, longitudinal and transverse swings of the front and rear vehicles. \[ \begin{cases}m_{1}\big{(}\dot{v}_{x1}-v_{y1}\omega_{1}\big{)}=F_{x1}+F_{x2}+F_{ Ox1}\\ m_{1}\big{(}\dot{v}_{y1}-v_{x1}\omega_{1}\big{)}=F_{y1}+F_{y2}-F_{Oy1}\\ I_{1}\dot{w}_{1}=M_{O1}+(F_{x1}-F_{x2})B+\big{(}F_{y1}+F_{y2}\big{)}L_{1}+F_{Oy1 }L_{2}\end{cases}  \tag{1}\]  
\[ \begin{cases}m_{2}\big{(}\dot{v}_{x2}-v_{y1}\omega_{2}\big{)}=F_{x3}+F_{x4}-F_{ Ox2}\cos\beta-F_{Oy2}\sin\beta\\ m_{2}\big{(}\dot{v}_{y2}-v_{x2}\omega_{2}\big{)}=F_{y3}+F_{y4}-F_{Ox2}\sin\beta+ F_{Oy2}\cos\beta\\ I_{2}\dot{w}_{2}=-M_{O2}+(F_{x3}-F_{x4})B-\big{(}F_{y3}+F_{y4}\big{)}L_{3}+ \big{(}F_{Oy2}\cos\beta-F_{Ox2}\sin\beta\big{)}L_{4}\end{cases}  \tag{2}\]  
Based on the structural and kinematic relationship of the front and rear vehicles, the relationships between the front and rear vehicles are as follows:  
\[ \begin{cases}v_{x2}=v_{x1}\cos\beta-\big{(}v_{y1}-L_{2}\omega_{1}\big{)}\sin \beta\\ v_{y2}=v_{x1}\sin\beta+\big{(}v_{y1}-L_{2}\omega_{1}\big{)}\cos\beta-L_{4} \omega_{2}\end{cases}  \tag{3}\]  
\[ \begin{cases}\dot{v}_{x2}=\dot{v}_{x1}\cos\beta-\big{(}\dot{v}_{y1}-L_{2}\dot {\omega}_{1}\big{)}\sin\beta-(v_{y1}\cos\beta+v_{x1}\sin\beta-L_{2}\omega_{1} \cos\beta)(\omega_{1}-\omega_{2})\\ \dot{v}_{y2}=\dot{v}_{x1}\sin\beta+\big{(}\dot{v}_{y1}-L_{2}\omega_{1}\big{)} \cos\beta-L_{4}\dot{\omega}_{2}+\big{(}v_{x1}\cos\beta-v_{y1}\sin\beta+L_{2} \omega_{1}\sin\beta\big{)}(\omega_{1}-\omega_{2})\end{cases}  \tag{4}\]  
\[\dot{\beta}=\omega_{1}-\omega_{2} \tag{5}\]  
Accordingly, the vehicle dynamics model can be obtained by combining Equations (1) to (5). Figure 1: Articulated vehicle dynamics model. \[ \begin{cases}(m_{1}+m_{2})\hat{v}_{x1}=&(m_{1}+m_{2})\omega_{1}v_{y1}+m_{2}L_{4} \hat{\omega}_{2}\sin\beta-m_{2}L_{2}\omega_{1}^{2}-m_{2}L_{4}\omega_{2}^{2}\cos \beta\\ &+F_{x1}+F_{x2}+(F_{x3}+F_{x4})\cos\beta+\big{(}F_{y3}+F_{y4}\big{)}\sin\beta\\ (m_{1}+m_{2})\hat{v}_{y1}=&m_{2}L_{2}\hat{\omega}_{1}-(m_{1}+m_{2})\omega_{1} v_{x1}+m_{2}L_{4}\hat{\omega}_{2}\cos\beta+m_{2}L_{4}\omega_{2}^{2}\sin\beta\\ &+F_{y1}+F_{y2}+\big{(}F_{y3}+F_{y4}\big{)}\cos\beta-(F_{x3}+F_{x4})\sin\beta \\ I_{1}\hat{\omega}_{1}=M_{\rm O1}+&(F_{x1}-F_{x2})B+\big{(}F_{y1}+F_{y2}\big{)} (L_{1}+L_{2})-m_{1}\big{(}\hat{v}_{y1}+v_{x1}\omega_{1})L_{2}\\ I_{2}\hat{\omega}_{2}=-M_{\rm O2}+&(F_{x3}-F_{x4})B-\big{(}F_{y3}+F_{y4}\big{)} L_{3}+\big{(}F_{y1}+F_{y2}\big{)}L_{4}\cos\beta+(F_{x1}+F_{x2})L_{4}\sin\beta\\ &-m_{1}\big{(}\hat{v}_{y1}+v_{x1}\omega_{1}\big{)}L_{4}\cos\beta-m_{1}\big{(} \hat{v}_{x1}-v_{y1}\omega_{1}\big{)}L_{4}\sin\beta\end{cases}  \tag{6}\]  
An analysis of snaking indicates that articulated vehicles move in a straight line. Given that lateral and longitudinal accelerations cause axial load transfer, the transferred load should be distributed in addition to self-gravity. The articulated vehicle vertical force analysis is shown in Figure 2, and the vertical force of each tire is shown in Equation (7). \[