## 1 Introduction
### Dimensioning of Anti-Bending Bars  
The main constructive parameters of the anti-bending bar, namely its length and diameter, were established from the condition of achieving a certain vertical bending frequency of the experimental model of the carbody with anti-bending bars,\(f_{vb}^{b}\), convenient from the point of view of ride comfort. Taking in Equation (24)  
\[f_{vb}^{b}=f_{(b\_vb)1}^{b} \tag{34}\]  
results first  
\[f_{xb}=\frac{{(f_{vb}^{b})}^{2}\left[{\int_{vb}^{2}+\kappa f_{b}^{2}-{(f_{vb}^ {b})}^{2}}\right]-f_{vb}^{2}f_{b}^{2}}{{f_{b}^{2}-{(f_{vb}^{b})}^{2}}} \tag{35}\]  
and then the stiffness of the anti-bending bar was obtained from Equation (27)  
\[k_{xb}=\frac{\pi^{2}f_{xb}^{2}M_{mc}}{2\psi^{2}h_{b}^{2}}. \tag{36}\]  
By appropriately choosing the vertical bending frequency of the experimental carbody model with anti-bending bars (\(f_{vb}^{b}\)) and the length of the anti-bending bar (\(l_{b}\)), the cross-section area (\(S_{b}\)) of the anti-bending bar could be calculated using the relation (1)  
\[S_{b}=k_{xb}\frac{l_{b}}{E_{b}}, \tag{37}\]  
where \(E_{b}\) is the longitudinal modulus of the anti-bending bar. To avoid the interference between the structural vibration of the anti-bending bar and that of the experimental model of the carbody, it was necessary to ensure the condition that the vertical bending frequency of the anti-bending bar was sufficiently high in relation to the vertical bending frequency of the experimental model of the carbody. Considering the anti-buckling bar as a Euler-Bernoulli beam embedded in the fasteners of the experimental carbody model, the first vertical bending frequency of the anti-bending bar was given by the relation  
\[f_{bb}=\frac{1}{2\pi}\frac{\alpha^{2}}{l_{b}^{2}}\sqrt{\frac{E_{b}l_{b}}{\rho_ {b}S_{b}}}=\frac{1}{2\pi}\frac{\alpha^{2}}{l_{b}}\sqrt{\frac{k_{xb}}{4\pi\rho_ {b}l_{b}}}, \tag{38}\]  
where \(l_{b}\) is the cross-section inertia moment of the anti-bending bar, \(\rho_{b}\) its density, and \(\alpha\) = 4,7300407449. Requiring that the frequency of the anti-bending bar be at least one octave from the natural frequency of the experimental model of the carbody with anti-bending bars (\(f_{bb}=2f_{vb}^{b}\)), the maximum length of the anti-bending bar was obtained in the form of a nonlinear equation  
\[l_{b\max}=\frac{\alpha}{4\pi}\sqrt[3]{\frac{\alpha k_{xb}}{\rho_{b}{(f_{vb}^ {b})}^{2}}}, \tag{39}\]  
which ca be solved via the simple iteration method. Knowing \(l_{b\max}\), the minimum section of the anti-bending bar was further determined, \begin{table}
\begin{tabular}{l l} \hline \hline Bounce & \(f_{b}\) = 30 Hz \\ \hline Vertical bending & \(f_{vb}\) = 10.06 Hz \\ \hline Natural frequencies of coupled vibration of bounce and vertical bending & \(f_{(b\_vb)1}\) = 7.90 Hz \\ \cline{2-3}  & \(f_{(b\_vb)2}\) = 38.20 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Natural frequencies of the experimental model of the carbody. \[S_{\rm{bmin}}=k_{xb}\frac{l_{\rm{bmax}}}{E_{b}}, \tag{40}\]  
and its diameter \(D_{b\rm{min}}=2(S_{b\rm{min}}/\pi)^{1/2}\). For reasons of effectiveness of the stiffening of the experimental model of the carbody with the help of the anti-bending bars, it was envisaged that they will be made of steel (\(E_{b}=2.1\cdot 10^{11}\) N/m\({}^{2}\) and \(\rho_{b}=7.850\cdot 10^{3}\) kg/m\({}^{3}\)). Figure 15 shows the maximum length of anti-bending bars, and Figure 16 shows their minimum diameter for the vertical bending frequency of the experimental carbody model with anti-bending bars between 12 and 18 Hz. It was specified that the lower limit of the frequency range had been set so that it is outside the range of sensitivity of the human body to vertical vibrations. Calculations were performed for \(l_{b}=40\) mm. It was observed that both the maximum length of the anti-bending bars and their minimum diameter increased approximately linearly as a function of \(f_{\rm{b}p}^{b}\), but the increase was more pronounced in the case of the minimum diameter of the anti-bending bars.