# Experimental and Mathematical Study of Flexible-Rigid Rail Vehicle Riding Comfort and Safety
## 1 Introduction  
A rigid-flexible analysis is a numerical simulation used to model the dynamic behaviors of complex systems with rigid and flexible components. In the case of rail vehicles, this type of analysis can be used to simulate the vehicle's behavior as it travels along the track. The rigid components of the rail vehicle include the wheels, axles, and main frame. These components are modelled as rigid bodies that move together as a single unit. The flexible members have a suspension system, body, and other elements designed to deform or flex under load [1]. The system is modelled using a combination of mathematical equations and computer simulations to perform a rigid-flexible analysis of a rail vehicle. The simulation considers the forces acting on the vehicle as it travels along the track, including the vehicle's weight, the forces generated by the propulsion system, and the forces generated by the suspension system. The simulation also takes into account the deformation of the flexible components of the vehicle, which can have a significant impact on the vehicle's behaviors. By analyzing the vehicle's behaviors under different conditions, engineers can optimize the vehicle's design to improve performance, safety, and efficiency. A rigid-flexible analysis is a powerful tool for designing and optimizing rail vehicles. By simulating the vehicle's behaviors under various conditions, engineers can identify potential issues and make design changes before the vehicle is built, saving time and money in the long run. In railway vehicles, the vibrations mainly generated due to wheel-rail interaction are isolated by the vehicle suspension system before being transferred to the car body. Thecar body's vertical vibrations arise mainly due to the vertical-lateral irregularities of the track, and also due to the wheel's profile. After hundreds of hours of running on tracks, some wheels lose their correct profile; they start to present wear and generate vibrations. These are incorporated by simple eigenmodes of rigid vehicle masses, i.e., vertical and pitch modes [2]. A continuous effort has been applied by rail dynamics researchers to improve rail vehicle performance indices, i.e., stability, ride, and curving ability [3]. Considering different models, researchers have optimized ride comfort and other performance indices of rail and road vehicles, i.e., vehicle power performance, energy consumption, regenerated power, road holding, etc. [4]. It has been found in the past that the design of railway passenger vehicles has conflicting requirements to obtain maximum ride comfort and minimum track deterioration, as the wheels are affected first, and their profile needs to be corrected on big lathes and maximum payload. Compound structural modes are included due to car body flexibility, i.e., bending and twisting modes. Past research has suggested that a 5 to 10 Hz resonating frequency significantly affects car body vibrations in a symmetrical bending mode [5]. This frequency range is critical to human comfort sensitivity. In this car body, structural vibrations in the bending mode arise in a lightweight structure and are responsible for poor vehicle ride comfort [6]. Control of car body vibrations may be accomplished in several different ways. An essential contrast may be drawn between the approaches concentrating on the flexible car body modes' control and those considering the rigid body modes. This distinction applies to all of these systems. It is possible to lessen the effects of the first flexible car body modes using piezoelectric actuators and technology that provides suspended mass control. Piezo-stack actuators were used to demonstrate the solution's efficacy via experimental testing on a 1:10 scale model and co-simulations for a complete vehicle [7]. Using a 1:6 scaled Shinkansen vehicle, [PERSON] et al. [8] demonstrated that the efficacy of a combination of piezo-stack and linear actuators may be improved upon. As shown by [PERSON] et al.