## 1 Introduction
### Results of the Numerical Simulations
Figure 18 shows five curves corresponding to the maximum of the acceleration frequency response function (Equation (56)), calculated along the experimental model of the carbody, in the frequency range 5-20 Hz, the range in which the vertical bending frequencies of the experimental model of the carbody without/with anti-bending bars were found. Anti-bending bars with the length of 900 mm were considered, mounted at \(h_{b}=40\) mm and with the same four diameter values of the anti-bending bars as in the previous example. For the calculation of the frequency response function of the acceleration of the experimental model of the carbody without anti-bending bars, the relation (56) was used, adapted to this situation:  
\[\overline{H}_{c}(x,\omega)=-\omega^{2}\frac{-\omega^{2}M_{mc}+\overline{k}_{ mc}+4\overline{k}_{zc}\epsilon(\epsilon-\underline{\epsilon})-X_{c}(x)[\omega^{2}M_{c} \underline{\xi}+4\overline{k}_{zc}(\epsilon-\underline{\epsilon})]}{(- \omega^{2}M_{c}+4\overline{k}_{zc})(-\omega^{2}M_{mc}+\overline{k}_{mc}+4 \overline{k}_{zc}\epsilon^{2})-16\overline{k}_{zc}^{2}\epsilon^{2}}-\omega^ {2}\frac{\left(x_{o}-\frac{\underline{\epsilon}}{2}\right)\left(x-\frac{ \underline{\epsilon}}{2}\right)}{-\omega^{2}J_{c}+4\overline{k}_{zc}a_{c}^{2 }}. \tag{58}\]  
All five curves of the maximum of the acceleration frequency response function have the same shape: they recorded a maximum in the middle of the experimental model of the carbody, decreased continuously to the points corresponding to the two nodes of the vertical bending, then increased continuously towards the two ends of the experimental model of the carbody. These results show that the effect of the anti-bending bars in reducing vertical bending vibrations was manifested along the entire length of the experimental model of the carbody, but the greatest effectiveness was recorded in the middle of the experimental model of the carbody, where vertical bending had an important contribution, as shown above (see Figure 17). It was again noted that the effectiveness of the anti-bending bars increased as their diameter increased. For example, in the middle of the experimental model  
Figure 18: Maximum of the acceleration frequency response function along the experimental carbody model. of the carbody without anti-bending bars the acceleration frequency response function was 3.037 (m/s\({}^{2}\))/N and decreased to 1.702 (m/s\({}^{2}\))/N in the case of the experimental model of the carbody with anti-bending bars with the diameter of 5 mm. For anti-bending bars with the diameter of 8 mm, the decrease was more important, reaching up to 0.735 (m/s\({}^{2}\))/N. Diagrams in Figure 19 show the frequency response functions of the acceleration of the experimental model of the carbody without anti-bending bars, respectively, with anti-bending bars of length 900 mm and different diameters, mounted both at the distance \(h_{b}=40\) mm and at the distance \(h_{b}=50\) mm. The presented results show that the increase in resonance frequencies of vertical bending was not only influenced by the diameter of the bar, but also by the distance \(h_{b}\). As can be seen, for the same diameter of the anti-bending bars, the resonance frequency of the vertical bending of the experimental model of the carbody increased as the distance \(h_{b}\) was higher, and this can be explained by the increase in the bending moment of the longitudinal force in the anti-bending bars. Figure 20 shows the acceleration frequency response functions of the experimental model of the carbody without/with anti-bending bars of 6 mm diameter and four length values, respectively, 600 mm, 700 mm, 800 mm and 900 mm. For the distance \(h_{b}\), two values were considered, 40 mm and 50 mm. On the one hand, it was also observed here that increasing \(h_{b}\) had the effect of increasing the vertical bending frequency of the experimental model of the carbody by at least 2 Hz.