## 1 Introduction  
The railway vehicle is a complex oscillating system that has vibration behaviour with specific characteristics, which can have unfavourable effects on ride quality, comfort of passengers, cargo integrity and traffic safety. The vibrations of railway vehicles are mainly caused by the geometric deviations of the track and by the irregularities of the rolling surfaces of the rails [1, 2, 3, 4, 5, 6, 7, 8]. They come from the construction imperfections, track exploitation and change in the infrastructure due to the action of the environment factors or soil movements [9]. These above, along with the constructive discontinuities of the rail-joints, switches and crossings, represent the major causes of railway vehicle vibration. Wheel defects such as eccentricity, ovality, polygonal profile, corrugation, flat wheel, flattening or the defects of rolling surfaces are also causes of railway vehicle vibration [10, 11, 12, 13, 14]. What defines the vibrations in the railway vehicle is that they are low frequency vibrations that occur both vertically and horizontally in the shape of simple vibration modes--rigid modes--and complex vibration modes--structural vibration modes. The construction of the railway vehicle usually complies with the rules of geometric symmetry, inertial and elastic; hence, the vertical vibrations can be regarded as decoupled from the ones in the horizontal vibrations and, for that reason, separately dealt with. The issue of reducing the vibrations of the railway vehicle is open to research, mainly for the high velocity vehicles. Reducing the weight of the vehicle, in particular the carbody, is one of the basic criteria for the design of high-speed rail vehicles. Reducing the weight of the vehicle contributes to increasing speed and reducing energy consumption, ground vibrations and manufacturing costs [15]. The design of the lightweight carbody involves the use of lightweight materials and the modification of mechanical structures. This often leads to a lower structural stiffness of the carbody and therefore to a decrease of the eigenfrequencies. The lighter the vehicle carbody, the greater its flexibility, and this facilitates an easy excitation of the carbody structural vibrations that have a negative effect on the ride comfort [16]. Although the carbody structural vibrations are particularly complex, the first vertical bending mode has the greatest influence on ride comfort. The natural frequency of this mode of vibration is found in the range of 6-12 Hz, where the human body has a deeper sensitivity to the vertical vibrations. The structural vibrations of flexible carbodies have been the object of research in a large number of studies [15; 17; 18; 19; 20; 21], and the reduction or control of these vibration so as to improve the ride comfort have been dealt with in numerous research [22; 23; 24; 25; 26; 27; 28; 29; 30] that can be grouped as follows: approaches concerning the isolation of vibrations, developed on passive, semi-active and active concepts; and approaches regarding the damping of the carbody vibrations, developed in turn on passive or active concepts [39; 40]. Nevertheless, to the best of the authors' knowledge, there is no study that features the contribution of the vibration structural modes upon the vibration behaviour of the railway vehicle carbody. Such study is the subject of this paper, which examines the influence of the first vertical bending mode upon the vibration behaviour of the carbody. To show the impact of the vertical bending upon the carbody vibration behaviour, the results derived from the flexible carbody model and the ones obtained using the rigid carbody model need to be compared. Consequently, the study presented in the paper is based on comparisons between the results from the numerical simulations, developed on two models of a four-axle vehicle with two-stage suspension.