## 1 Introduction
### Theoretical Concept of the Method  
The method of reducing the vertical bending vibration of the carbody of railway vehicles is based on the concept that a passive system can be used to limit the rotation of the cross-sections of the carbody to oppose the bending of the carbody. This system is represented by two anti-bending bars, which are rigidly mounted to the longitudinal beams of the carbody underframe. The effectiveness of using anti-bending bars in terms of reducing carbody bending vibration and improving the ride comfort of the vehicle was investigated using the results obtained through numerical simulations. Numerical simulation applications were developed based on a rigid-flexible-coupled railway vehicle model [29]. The model of the railway vehicle with anti-bending bars, mounted on either  
Figure 8: Overview of the test bench [70]. side of the carbody, is presented in Figure 9. The carbody was modeled by a free-free equivalent beam, of Euler-Bernoulli type, and the bogies and axles were considered as rigid bodies. It was considered that the vehicle is running at constant speed on a perfectly rigid track. The vertical irregularities of the track impose the vertical displacements \(\eta_{1\ldots 4}\) in the axles. The vertical vibration modes of the carbody relevant from the point of view of ride comfort were considered, namely the rigid vibration modes--bounce, \(z_{c}\), and pitch, \(\theta_{c}\), and the first carbody vertical bending, \(w_{c}\). In the case of bogies, the rigid vibration modes were considered, respectively, the bounce \(z_{b1,2}\) and pitch \(\theta_{b1,2}\). The secondary suspension afferent to a bogie was modeled via two Kelvin-Voigt systems, where one was for the representation of the vertical suspension of the vehicle and the other one was used to model the traction longitudinal system between the carbody and the bogie. Each vertical primary suspension corresponding to an axis was modeled by a Kelvin-Voigt system. The anti-bending bars were represented by two elastic elements. The influence of the bars longitudinal and bending vibrations was neglected. The main constructive parameters of the anti-bending bars, namely its length and diameter, were determined by imposing two conditions. The first referred to reaching a certain bending frequency of the carbody equipped with anti-bending bars, convenient from the perspective of the ride comfort. By the second condition, it was necessary that the bending frequency of the anti-bending bars be high enough compared to the bending frequency of the carbody, so as to avoid the interference between the bending vibrations of the anti-bending bars and the ones of the carbody. The influence of the anti-bending bars upon the dynamic response of the carbody was examined based on the results of numerical simulations regarding the frequency response functions of the vehicle. Figure 10 features the power spectral density (PSD) of the carbody vertical acceleration weighted according to UIC 513 and EN 12,299 [78,79] to account for human sensitivity to vertical vibrations. The PSD was calculated in three reference points of the carbody, relevant for ride comfort (see Figure 9)--in the center of the carbody (point C) and above the two bogies (points \(B_{1}\) and \(B_{2}\)). In the case of the carbody without anti-bending bars, the spectrum of the vertical acceleration power density was dominated in the centre of the carbody by the bending vibrations having the frequency of 8 Hz. Due to the anti-bending bars, the peaks of the acceleration power spectral density decreased near the bending frequency of the carbody. As appropriate, these peaks were then found at 10 Hz, 12 Hz and 14 Hz, respectively. The introduction of anti-bending bars had the effect of reducing the level of vibrations due to bending of the carbody and above the bogies. It should be noted, however, that the  
Figure 9: The rigid-flexible coupled model of the railway vehicle [29].