Non-linear analysis of the hunting phenomenon in a train car using the perturbation method

Dissertation for Master's Degree

Application Design-Dynamic Orientation and Control

Abstract:

Transverse instability is one of the common issues in high-speed rail vehicles that are on rails without arches and due to the frictional-creep nature of the wheel-rail contact forces that They create non-conservatives in the system. This phenomenon is called hunting, which is a type of self-stimulation phenomenon from a dynamic point of view. This phenomenon can cause a lot of wheel and rail wear and tear and reduce travel comfort. The phenomenon of hunting is of two types: primary and secondary hunting.

Secondary hunting is very important from a dynamic point of view, and primary hunting can be easily removed by using appropriate dampers in the suspension system. In this research, an attempt has been made to obtain the speed of the secondary hunting threshold with a close approximation. Wherever hunting is mentioned, it means secondary hunting unless primary hunting is mentioned. Kalker's linear theory has been used in the investigation of the hunting phenomenon so that the creep forces can be calculated easily and with great accuracy in the hunting investigation. Also, the movement of the wheel on a smooth surface without irregularities is considered. Equations related to wheel and axle are derived for moving in a straight path (without arc). A parameter study has been done so that the effect of different factors on hunting can be clearly seen.

Primary hunting is created at low speeds and can be eliminated by using appropriate dampers in the suspension system. Secondary hunting that occurs at high speeds is a self-excitation phenomenon [1] that can never be avoided and can only be brought to speeds higher than the vehicle's operating speed. In the primary hunting analysis, it seems necessary to consider the wagon model, but in the secondary hunting analysis, it is sufficient to model the bogie alone. Linear and non-linear critical velocities have been calculated and bifurcation phenomenon, limit cycle and the range in which the limit cycle may occur have also been determined. The effect of different parameters on linear and non-linear critical speeds is also among the important results of this thesis.

Key words: Verril wheel contact - primary and secondary hunting - linear and non-linear critical speeds - bifurcation - limit cycle.

- Introduction:

Mathematical modeling has been widely used to describe the movement of trains on rails. Considering the variety of parameters in the problem, creating a mathematical model that can cover all aspects of the problem seems very unlikely. It is obvious that using special models of each one with a specific purpose (including limitations) is very helpful.

     Generally, in examining the dynamics of conveyors and especially rail conveyors, it is assumed that the mass elements of each are rigid. Each of these elements has 6 degrees of freedom in longitudinal, transverse, vertical directions (degrees of freedom related to displacement), roll, pitch and yaw (degrees of freedom related to rotation). For the dynamic description of each of these degrees of freedom, we need to solve a 2nd degree differential equation, which is generally coupled with other equations. So, to describe the dynamics of an N degrees of freedom system, it can be said that N×6 coupled differential equations must be solved. Obviously, solving this number of resurrections is very time-consuming and expensive. This is where we have to determine the purpose of the model and its task.

Observations have shown that there is little connection between the movement of the vehicle in the vertical and horizontal directions.  According to this point, it can be said that there is no need to consider the effects of movement in the vertical direction in the transverse inspection of the vehicle. This issue is also true in connection with the investigation of movement in the vertical direction. Here and in this research, considering that the aim was to investigate the movement in the transverse direction, only the yaw and transverse degrees of freedom were considered.

This simplification not only helps us in solving the problem.

This simplification not only helps us in solving the problem, but in the end it also makes it easier to describe the results.

The cross-sectional investigation itself can be divided into two branches:

1- Examining the system's response: Obviously, in this approach, the goal is to investigate the system's response to external stimuli.

2- Stability check in this approach aims to check the transverse stability of the vehicle under different working conditions.

     Just as a system in a static state can be stable or unstable, a dynamic system can also be stable or unstable. As in statics, our criterion for checking stability is the return of the system to its initial state, in dynamic there are similar criteria for checking stability. This criterion is proposed as follows: an unstable dynamic system is a system that diverges the smallest amount of displacement (stimulation) to infinity.

2- 1- Overview of the work done:

    The first research and investigation on the problem of stability of rail transport devices was done by Dipiter [1] [4]. He should investigate the behavior and limit cycle in a two-axle bogie with cylindrical wheels. The research was followed by Matsudaira [2] [5] and Wickens [3] [6] presenting articles. These initial works were the beginning of the investigation of this issue by other scientists. Kuprider [4] [7] was the first person to investigate nonlinear issues in the field of rail systems. He investigated the effects of wheel conicity, flange contact, and bogie bowl friction on bogie dynamics with 6 degrees of freedom. The first bifurcation analysis of the problem of free movement of a wheel and center of air Yalgol [5] [8] was done and led to encountering the limit cycle bifurcation condition. In recent years, many authors have investigated the chaotic oscillatory motion and bifurcation of the limit cycle of wheels and axles and railway vehicles. 

Many theoretical analyzes have been performed to predict critical speeds in the nonlinear dynamics of wheels and axles and rail vehicles. Love and Brand [6] [9] have used the Krylov-Bogolyubov method [7] [20] to extract the wheel and axle vibration amplitudes. In their investigation, they used the perturbation method to analyze the stability of oscillatory motion. They obtained the effect of wheel profile as well as flange contact in this study. They tried to check the behavior of a single axis in the arch by considering the contact of the flange with a similar method. In this model, the flange contact was modeled as a linear spring with a slack equal to the slack of the line gauge. Sheffle [8] [10] theoretically showed the dependence of the hunting stability on the creep coefficients and the weather damping of the suspension system was related to the stability of the vehicle, wheel and axle under study. In the attempt by Hans True [9], Mehdi Ahmad Yan Wayang [10] [11,12,13] were done, they investigated the nonlinear effect of longitudinal yaw damping on the hunting behavior and bifurcation nature of the limit cycle in a wheel and axle as well as in a two-axle bogie. His research indicated that a moderate increase in damping yaw reduces the critical speed of hunting and a large increase in damping yaw improves hunting behavior in terms of better control and hunting range. Horak and Wormley [11] [14] tested the effect of rail surface corrosion on the hunting behavior of a passenger bogie.