## 1 Introduction  
Ride comfort is a sine qua non condition that must be met by all types of passenger trains, whether for urban, commuter or high-speed traffic. A railway vehicle exhibits vibrations in the three directions, vertical, lateral, and longitudinal, which develop practically independently of each other and have their own characteristics. The vertical vibrations are excited by the irregularity of the track longitudinal level [1, 2], lateral ones are maintained by the alignment deviations and take the form of hunting motion due to the inverted conicity of the running surfaces of the wheels [3, 4], and longitudinal ones are induced by the traction-braking behaviour [5, 6]. As for the vertical vibrations of passenger coaches, they affect the ride comfort especially in long and light vehicles intended for high-speed trains, due to the fact that the first natural bending frequency of the carbody is located in the range of highest sensitivity of human beings to vertical vibrations, between 6 and 12 Hz, as an effect of its slenderness [7; 8]. In these vehicles, the critical point in terms of the ride comfort is in the middle of the carbody because the bending vibration of the first eigenmode has its ventrum (antinode) here. In contrast, the critical points of ordinary vehicles are located at the ends of the carbody [9]. Active, semi-active and passive systems have been proposed to improve the ride comfort to vertical vibration in passenger coaches [10; 11; 12; 13; 14; 15; 16; 17]. They are designed either to isolate the carbody from vibrations using active/semi-active suspension [10; 11; 12], or to reduce the structural vibration of the carbody through the damping provided by the active/passive suspension [13; 14; 15; 16; 17]. In general, a passive system for reducing the vertical vibration of the carbody is simpler to implement, cheaper and requires lower maintenance costs compared to a semi-active or active system. Many passive systems work on the dynamic absorber principle by elastically attaching a mass to the vehicle carbody. In this way, the bending frequency of the carbody can be increased and taken out of the high sensitivity range mentioned above. This solution has been studied in several papers to improve ride comfort in multiple high-speed electric units [18; 19; 20; 21; 22]. The concept of a new passive system for reducing the bending vibration of a railway vehicle carbody was presented in ref. [17]. The functional principle is shown in Figure 1, where the railway vehicle carbody is provided with four rigid supports--two fixed on each side of the carbody chassis, symmetrically in relation to the middle of the carbody, in which two steel bars, called the anti-bending bars, are mounted. When the carbody bends, the cross sections in which the supports work rotate in the opposite direction so that the anti-bending bars are either compressed (Figure 1b) or stretched (Figure 1c). In both situations, the anti-bending bars develop longitudinal forces (red arrows) which, reduced to the neutral axis of the carbody, cause moments that oppose the bending of the carbody (blue curved arrows) and longitudinal forces (blue straight arrows). For clarity, Figure 1a shows the other main components of a four-axle vehicle: wheelset, bogie chassis, primary suspension and secondary suspension. Effect of the anti-bending bar system on the frequency response function of the carbody and on the comfort index in a passenger coach traveling at speeds of up to 300 km/h was investigated in paper [17]. The calculations were performed based on a continuous-discrete  
Figure 1: The functional principle of the passive system with anti-bending bars: (**a**) scheme of vehicle with anti-bending bars: 1. carbody; 2. support; 3. anti-bending bar; 4. wheelset; 5. bogie chassis; 6. primary suspension; 7. secondary suspension; (**b**) carbody when the anti-bending bars are compressed; (**c**) carbody when the anti-bending bars are stretched. analytical model developed for a vehicle like the one in Figure 1a.