This book focuses on the establishment of system dynamics equations and the solutions of a variety of typical problems, each chapter is accompanied by informative examples, detailing the steps and methods of solving the dynamics problems. Chapter 1 introduces the system modelling method. A matrix transformation method is introduced. It is a standardized method and suitable for computer modelling. The creep and gravitational forces induced by the relative movement between the rail and wheel are unique to the railway vehicles, clear interpretations are given graphically. A set of complete expressions of creep and creep force are derived. Three theories of creepage and creep force are introduced. In Chapter 2, four representative static cases (Weight transfer analysis, Adjust wheel loads by spring shimming, Wheel off-loading on twisted track, Linear analysis of bogie steady state curving behaviour) are discussed in detail, their system equations are formed exactly the same way, but the formations of their external force vector are completely different. Eigenvalue and vehicle stability analysis are discussed in Chapter 3. They are illustrated by the examples step by step from simple to complex. Chapter 4 introduces frequency response analysis. It starts with one degree of freedom system, derives a general form of frequency response equation, and then extends to multi-degrees of freedom system. Different forms of interference and time delay expression are introduced. Chapter 5 introduces the system transient response analysis. It gives a detailed introduction to the principles, methods, and calculation steps of the numerical analysis method. Wheel-rail vertical impact analysis deals with the "Hertz" nonlinear contact theory. Responses of wheelsets and bogies disturbed by track lateral irregularities are discussed in great detail by the application of the "square root" creep theory and the "nonlinear" creep theory. Chapter 6 discusses train longitudinal dynamics, the focus of this study is on the buffer-coupler system between vehicles. The analytical path and solving method are different from those mentioned in Chapter 5. After introducing five different buffer characteristics, four different practical cases are discussed in details, of which are the impact, braking, traction, and collision. Chapter 7 discusses the modelling of hydraulic dampers. In engineering practice, hydraulic dampers are considered as a combined component, which consists of a shock absorber (damping) and its end rubber elements (stiffness) in series connection. This chapter is to discuss the calculation, analysis, and application of the impedance of hydraulic damper in engineering practice. Chapter 8 discusses the modelling of shear spring in dynamics analysis. Spring elements that are subjected to a vertical (axial) load, have horizontal degrees of freedom of movement providing lateral suspension function, also known as Shear Springs, which include cylindrical coil springs and air springs. In this chapter, the relationship between lateral forces and displacements of the shear springs are derived; a general lateral mechanical model of the shear spring is proposed. Chapter 9 discusses the modelling of structure flexible vibration. Vehicle vibration caused by track irregularities is transmitted through primary and secondary suspension elements to stimulate the structure vibration of the vehicle. In this chapter, the first-order bending vibration of the vehicle body is taken as an example to discuss the determination of the modal parameters of the single-degree-of-freedom equivalent system, the interaction between the structural vibration and the vehicle suspension elements, and the method of establishing the dynamics equation of the system including structural vibration. This book is suitable for graduate and post-graduate students in universities, researchers in Institutes, software developers and professional designers and dynamics engineers.