1 Department of Informatics and Mathematical Modeling, Technical University of Denmark2 Department of Applied Mathematics and Computer Science, Technical University of Denmark
The present thesis is concerned with the modelling of the motion of the Three- Piece-Freight-Truck. Although the Three-Piece-Freight-Truck is very simple in its construction, the mathematical model is not simple at all. The model is definitely nonlinear resulting from the nonlinear kinematic and dynamical contact relations between wheels and rails, the nonlinear suspensions and the nonlinear dry friction damping. For low speeds of the truck the kinematic and dynamical nonlinearities might be linearized, but the very strongly nonlinear suspensions and the dry friction damping can not be linearized at all. The motion of the bolsters are at least two dimensional in the lateral and the vertical directions, so the friction on the surfaces of a wedge should be treated as two-dimensional dry friction, and the same is true for the dry friction on the surfaces of an adapter. For the motion with dry friction there exist two motion states: stick motion and slip motion, which leads to a discontinuity in the behaviour of the dynamical system and leads to a collapse of the state space, and consequently, change the degrees of freedom of the system repeatedly. Due to the design clearances between the car body and the side supports on the bolsters the side supports must be modelled as nonlinear dead-band springs. The clearances in the assembly in the wedge damper systems give rise to a relative yaw motion of the bolster with respect to the side frame and a rotation around the truck center line and cause a warping. In addition the assembly clearances between the side frame and the adapter both in longitudinal and lateral directions produce another dead-band spring force. The tractive effort on the car body in the longitudinal direction may be left out of consideration in the modelling of the passenger car, but the normal forces caused by it on the surfaces of the wedges will consequently produce friction forces in the Three-Piece-Freight Truck and should be considered. Therefore, the friction forces on the surfaces of wedges are asymmetrical for one pair of wedges as they should be. The thesis is divided into 10 chapters. In the chapter 1, the research state-of- the art of the dynamics of the Three-Piece-Freight-Truck is reviewed. The framework of the model is introduced. Chapter 2 describes the concept of the friction direction angle with which the stick-slip motion with two-dimensional dry friction can be numerically simulated. Its applications are illustrated in two simple systems. One is an oscillator with a Coulomb dry friction damper in chapter 2 and the other one is the wedge damper in chapter 3. In the mechanical system it is possible that the degrees of freedom will vary with the different friction states. We give a detailed discussion of this type of structure varying systems in chapter 4. For the performances of the vehicle on the track, the contact between a wheel and a rail plays a key role, where there are two types of contacts: One is kinematic and the other is dynamical. For the kinematic contact relation we trace the contact point of the wheel on its possible trajectory and the on-line evaluation of the kinematic contact parameters is introduced. The elastic contact assumption is used to determine the normal loads in the contact patch and then a fully nonlinear contact theory is used to obtain the creep forces. They are discussed in chapter 5. The configuration of the Three-Piece-Freight-Truck and the corresponding positions of the elements, the velocities and some relations among the elements of the system will be described in chapter 6. In chapter 7 the dynamic equations of the system are derived. Chapter 8 provides the numerical methods for the simulation of the system, the discussion focuses on the differential algebraic equations(DAEs) with discontinuous characteristics caused by the two-dimensional friction. In chapter 9 the numerical investigation is provided. The four general irregularities in tangent track are usually described in the form of a power spectrum density(PSD). We transform the PSD into the corresponding series in the time domain and then use the time series as excitations for the dynamical performances of the system. The linear critical speed and nonlinear critical speed and even the chaotic motion of the Three- Piece-Freight-Truck are discussed. Finally in chapter 10 certain conclusions are drawn, and some projects for further research are indicated.