# Modelling and Stability Analysis of Articulated Vehicles
## 2 Modelling of Non-Linear Systems for Articulated Vehicles
### Model Simulation  
On the basis of the nonlinear dynamics mathematical model of the previously described articulated vehicle, the articulated vehicle dynamics simulation model was built in the MATLAB/Simulink simulation environment. The simulation model includes the  
Figure 3: Hydraulic steering system model. vehicle dynamics, tire, driver, hydraulic steering system and vertical force calculation models. The parameters in the model are shown in Table 1.  
To verify the snaking, the vehicle travels at an initial speed of 5 m/s (disregarding rolling friction) and exerts a pulse moment interference on the articulated points. The parameters of each vehicle are in a state of oscillatory convergence with decreasing amplitude, as shown in Figure 4. Owing to interference, the ground exerts lateral forces on the tires, thereby generating lateral and angular accelerations about the z-axis in different directions and amplitudes for the front and rear vehicles. The front and rear vehicles produce lateral and angular velocities about the z-axis, and the vehicle's swing angle changes accordingly, thereby resulting in the snaking phenomenon. Owing to vehicle structural parameters, ground and speed, the swing angle is in a state of oscillatory convergence, while the snaking oscillation is in a steady state. Ideally, if the friction between the ground and tires is zero, then the entire snaking oscillation system of the vehicle is oscillating without damping, and the swing angle should be an equal amplitude oscillation. If the snaking is unstable under the influence of vehicle parameters, ground, speed and other factors, then the swing angle and corresponding parameters will oscillate and diverge. Numerous reasons can be cited for snaking, the most important of which is the compressibility of the hydraulic fluid in the hydraulic steering system, thereby results in a certain degree of stiffness of the hydraulic steering system. Other vehicle parameters also affect the stability of vehicles, such as the position of the centre of gravity, tire stiffness, hydraulic system characteristics and other factors. Ideally, when the road is completely flat and in the same condition, the lateral and longitudinal forces on the ground are identical for each tire, thereby substantially producing snaking. However, the fact is that roads will not be in ideal condition, the forces acting on the wheels will be different and non-structural terrain will increase the occurrence of snaking. When drivers operate vehicles under the road surface or other external disturbance, they will have a poor driving experience and misuse the steering system to suppress the effect of the swing or deviation. However, this situation alters the steady state and increases snaking, thereby creating a vicious cycle. To investigate the effects of velocity variation and different loads on the swing angle, several sets of simulations were performed. As shown in Figure 4, the vehicles are all in steady state at three different velocities of 2 m/s, 5 m/s and 10 m/s. As the vehicle velocity increases, the convergence rate of the swing angle gradually decreases which means the vehicle stability tends to decrease as the velocity increases. The vehicles are also in steady state at three different loads of 20%, 60% and 100% load factor of the front vehicle, as shown in Figure 4, the amplitude of the swing angle decreases with the increase of the load factor. The oscillation frequency of the swing angle is lower at high load factor area than that at low load factor area. Due to the influence of vehicle weight and the moment of inertia, the relationship between the swing angle and load factor is not linear. Overall, the increase in vehicle load factor is beneficial for vehicle stability.