# Development of a New Vertical Dynamic Model of a Rail Vehicle for the Analysis of Ride Comfort
## 1 Introduction  
Transportation by railway has seen a continuous increase due to its safe, comfortable and high-speed service. The travel speed of rail vehicles has risen with the improvements in new technologies. The need to remain competitive and be competitive in the future is also a driver for these improvements. However, while traveling at high speeds, a significant issue arises, namely the increase in vibrations that affect the rail car body and passengers due to track irregularities. Therefore, comfortable and safe transportation at higher speeds has become an important engineering problem to be solved [1, 2, 3, 4]. Ride comfort needs to be considered both in the design and evaluation phases of a rail vehicle to overcome the above problems. This is because the vibration behavior of a rail vehicle, and thus the ride comfort, is affected by track irregularities (as excitations) and dynamic characteristics of the rail vehicle. The dynamic characteristics of rail vehicles, at this stage, are significantly important, with the increasing use of advanced materials in the design of light rail vehicles. Structural stiffness--which changes with lightweight construction--accounts for a large part of the vibration level [4]. Most rail vehicles have a body-bending frequency close to 10 Hz, and, unfortunately, the human body is very sensitive to the frequency range of 4 to 10 Hz [5; 6; 7]. This means that, in addition to the rigid car body vibrations, the flexural vibrations of the car body must be also considered in the analysis of ride comfort. There are two methods to evaluate the dynamic behavior and ride comfort of a rail vehicle: measurements and simulations. Measurements are not always possible during the design phase of a new vehicle and are usually very expensive to perform. Therefore, simulations based on dynamic models of rail vehicles provide a good way to predict rail vehicle behavior. Apart from that, dynamic models of rail vehicles form the basis for comfort optimizations and for many other areas, such as train traction systems [8; 9; 10]. Finally, they can reduce the workload and costs of experiments by providing valuable insights into rail vehicle behavior. However, simulations of complex models can take quite a long computation time, depending on the analysis. In some cases, modeling the system in a smarter way for a specific analysis, i.e., not the complex model with full characteristics, is a good approach for computational efficiency. After building an accurate model, it is usually necessary to conduct parametrical and optimization studies targeting the improvements in ride (vertical and/or lateral) dynamics through modifications of suspension parameters using the model. For modern rail vehicles, it is also important to implement the most suitable ride and suspension control algorithms. As an example, in a recent study [9], a dynamic model of a rail vehicle was built to improve the ride comfort level using secondary suspension control algorithms. These algorithms require fast and accurate dynamic models. As mentioned above, the speed of rail vehicles is increasing, leading to more complex ride conditions. Consequently, there is a need to address numerous configurations for examination. In terms of modeling of a car body for vibration analysis, there are three approaches that consider flexural vibrations, as follows:  
1. Modelling the entire car body frame with the finite element method (FEM). 2. Using the free-free beam method (partial differential equation). 3. Assembling rigid bodies with the torsional stiffness elements by distributing the mass of the car body frame (ordinary differential equations--ode). Finite element method has a great accuracy in the calculation, but this method is too complex for computational efficiency. The method becomes especially complex when the full vehicle model with a large number of degrees of freedom (DoF) is included in the analysis [11; 12]. The beam method was first used by Suzuki to analyze the car body flexibility of rail vehicles [13].