Dynamic modeling and simulation of trailer with compliance steering system

. Stability of a vehicle and tire wear is affected by the movement inference between the tractor and the trailer when the trailer is turning. Therefore, to solve the problem, Compliant Steering (CS) system is used. The system makes the rear-wheel of the trailer steer. After establishing a dynamic model of the trailer and the CS system, the simulation result shows that the system has a smaller lateral force. This smaller force at the hinge point indicates movement inference is reduced and motion response is far better. Furthermore, this paper contains the analysis of the influence of the key parameters of CS system and the speed on the moving characteristics of the trailer.


Introduction
Rear wheel CS technique is a passive steering technique not requiring steering mechanism that utilizes the lateral force and elasticity of the wheel and the rear suspension when the vehicle turns respectively [1].That is when compared with the active steering technology, which is simple in structure and without energy consumption [2].
Currently, the research is abundant on CS technology used in large-scale vehicle [3].
A full trailer is made up of a tractor and a trailer.It is totally different from a car in terms of handling stability.
On a small curvature road and turning in high speed, the full trailer with over-steer characteristics is easy to lead to folding or swing tail (trailer swing) [4].Although low speed, problem of movement inference between the tractor and trailer during steering exists, thus, aggravating tire wear.In the past, due to small number of trailer, attention was not given.Currently, high rural mechanization and traffic, the number of trailer and accident is increasing gradually.
Recently, many scholars are researching and focusing on the dynamic analysis of trailers.After establishing a dynamic model of the trailer and the compliant steering system, simulation analysis of the trailer result shows the CS technology can reduce movement interference between the tractor and the trailer, improving the stability of the vehicle.

Dynamic Equation of Articulated Full Trailer
The articulated full trailer model is shown in Fig. 1 [5,6].
Here Fyi (i=1~4) represents tire lateral force, δ1 is front  After applying a force-balance analysis [7] to the model in Fig. 1, the dynamic equations can be derived.
Then they are formulated in a standard state space form, thus where the state variable is taken as , βi is the vehicle side slip angle, γ i is yaw rate (i = 1, 2 The tractor is labeled 1 and the trailer is 2), and q is the linear tire [8] lateral force vector

Model of trailer with rear wheel compliant steering
From the dynamic analysis viewpoint, if the rear wheels of the trailer steer a small angle which is opposite to the turning direction [9], the stability of the trailer can be improved, and the lateral interference between the trailer and the tractor can be weakened [10,11].
Therefore, the objective of the study is the full trailer with rear-wheel steering system.The CS system used here is shown in Fig. 2, c2, d2 are its front and rear arm length, 4 is trailer rear wheel CS angle, regardless of the quality of the CS system and the effect of moment of inertia.

Fig. 2 Force diagrams of trailer CS system
FC2 is the elastic and damping force of the CS combined, thus The CS angle meets the following equation: where

Kinematics Simulation of Full Trailer
The main parameters affecting the handling stability of the trailer in the CS system are [12,13],  The stiffness of the CS system ks,   three sets of parameters are selected for comparison in the stable range.In Fig. 6-a, when c2 = 0.1 and d2 = 0.6, it's compliant angle is relatively small, but its response is more stable.In Fig. 6-b and Fig. 6-c, the vehicle side slip angle is more stable, and the side angle of the tractor is relatively larger.When the parameter is chosen as c2 = 0.2 and d2 = 0.5, the system transient response is not good and the steady-state response is the best.Considering the effect of stiffness, we can refer to the dotted line in shows that a group of parameters of CS system is correspond to a certain speed of the vehicle, so that the performances are better relatively, and the increase in speed will reduce the handling of the full trailer stability.
When the speed reaches a certain value, the vehicle becomes instable.When a more stable performance at higher speeds is required, adjust the parameters of the CS system according to the analysis above.

Conclusion
(1) When the rear wheels of trailer steers, the lateral forces of the tires are released, so the lateral force at the hinge point is smaller and the movement interference between the tractor and the trailer is reduced.
(2) Based on the theory of CS technology to study the stability of the full trailer, the dynamic analysis of the MATEC Web of Conferences 139, 00001 (2017) handling stability of the trailer with CS system is conducted.The specific impact of the various parameters and the speed on the performances of trailer are studied.
(3) The simulation results show that the angle of the rear wheel of the trailer can reduce the motion interference at the hinge point, and improve the stability of the vehicle, which is also consistent with the theoretical analysis.The key parameters of the CS system have a comprehensive effect on the vehicle, here the research methods and CS system parameters is provided for the study of the full trailer with compliant steering system.
wheel angle,   is the angle of deviation between the tractor and the trailer, FTx, FTy are the longitudinal and lateral forces of the hinge.Other annotations are shown in Tab 1.

Fig. 1 .
Fig.1.Mechanical analysis of the trailer when steering.

Fig. 5 .
Fig.5.Comparison of traditional trailer and trailer with CS system When the vehicle with the CS turns, the rear wheels of trailer turn an angle opposite to the steering of the vehicle is shown in Fig.3.As shown in Fig.4, the generation of the compliant angle results in a significant decrease in the lateral forces at the hinge points.In Fig.5, it is visible that the yaw rate of the tractor and trailer and the side slip angle of the tractor are significantly improved.The results verifies that when the rear wheel of trailer steers an angle, the lateral force between the tractor and the trailer is released, thereby reducing effectively the motion interference between the tractor and the trailer.In order to analyze the specific characteristics of these parameters, packet simulation is required.Primarily, the effect of the sizes of c2 and d2 are discussed.Secondly, the influence of elastic element stiffness ks and damping ratio ζ is analyzed.Assuming that other parameters are the same, we only adjust the front and rear arm length c2 and d2.The values of the relevant parameters are: u=10 m/s, δ1=0.02rad, ζ=1, ks=45000 n/m.

Fig. 6 -
Fig.6-a and Fig.8-a as shown below.Through the design of specific parameters, these two CS angle response curves are obtained with the same parameters except the stiffness.With the increase of stiffness, the compliant angle response is more stable and the vehicle motion characteristics is improved.Fig.7 shows that when the damping ratio ζ = 1, the system has a long-time oscillation by adjusting the parameters, and then keeping the other parameters unchanged, we only increase the damping ratio, the oscillation time and amplitude of the system are significantly reduced.Although the damping ratio changes, the steady-state response of the rear wheel compliant angle unchanged.The final stability value of the side slip angles of the tractor and the trailer are not changed with the change of the damping ratio.It is clear that the stiffness and damping ratio of the compliant angle system are variable parameters in the moving system structure, and adjusting the values of both are more effective in improving the motion characteristics of the trailer.At a low speed (u=5 m/s), the CS angle of the trailer and the yaw rate of the tractor and the trailer is steady reached a fixed value.At this time tractor and trailer side angle is smaller.In the case of medium speed (u=10 m/s), the side angle of the tractor and trailer increases.Those