## 1 Introduction
### Dimensioning the Experimental Model of the Carbody  
To determine the dimensions of the experimental model of the carbody, the dimensions of the actual carbody of a passenger coach were used. For example, Table 1 shows the main dimensions of the passenger coach series 21-90, also known as AVA 200. The dimensions of the experimental carbody model were determined based on the condition of achieving the natural frequency of the first vertical bending mode of the real carbody. This frequency was determined experimentally and according to the specialized literature, for long and light carbody of passenger coaches, it was located in the range of 8-12 Hz, depending on the vehicle configuration [1; 34; 45; 47; 65]. As shown above, the first vertical bending mode was coupled to the bounce vibration and, to calculate the natural frequencies of the two coupled vibration modes of the experimental carbody model, the following homogeneous equations of motion derived from Equations (15) and (17) were considered  
\[M_{c}\bar{z}_{c}+4k_{zc}(z_{c}+\varepsilon T_{c})=0 \tag{18}\]  
\[M_{mc}\bar{T}_{c}+(k_{mc}+8\psi^{2}h_{b}^{2}k_{xb})T_{c}+4k_{zc}\varepsilon(z_ {c}+\varepsilon T_{c})=0 \tag{19}\]  
Harmonic solutions were imposed  
\[z_{c}=Z_{co}\sin\omega t,\ T_{c}=T_{co}\sin\omega t, \tag{20}\]  
where \(Z_{co}\) and \(T_{co}\) are the amplitudes of the bounce and vertical bending vibrations. Differential equations set (18) and (19) became  
\[(-\omega^{2}M_{c}+4k_{zc})Z_{co}+4k_{zc}\varepsilon T_{co}=0 \tag{21}\]  
\[(-\omega^{2}M_{mc}+k_{mc}+8\psi^{2}h_{b}^{2}k_{xb}+4\varepsilon^{2}k_{zc})T_{ co}+4\varepsilon k_{zc}Z_{co}=0, \tag{22}\] \begin{table}
\begin{tabular}{l l} Length & 26 m \\ Width & 2.825 m \\ Height & 4.050 m \\ Wheelbase & 19 m \\ \end{tabular}
\end{table} 
Table 1: The main dimensions of the carbody of the AVA 200 passenger coach [81]. resulting in the condition  
\[\omega^{4}-\left(\frac{k_{mc}+8\psi^{2}h_{b}^{2}k_{xb}+4\varepsilon^{2}k_{zc}}{M_ {mc}}+\frac{4k_{zc}}{M_{c}}\right)\omega^{2}+\frac{4k_{zc}}{M_{c}}\frac{k_{mc}+8 \psi^{2}h_{b}^{2}k_{xb}}{M_{mc}}=0 \tag{23}\]  
Solving biquadratic equation, the natural frequencies of the bounce and vertical bending coupled vibrations of the experimental model of the carbody with anti-bending bars were obtained,  
\[f_{(b,xb)1,2}^{b}=\sqrt{\frac{1}{2}\left[f_{vb}^{2}+f_{xb}^{2}+\kappa f_{b}^{2} \mp\sqrt{\left(f_{vb}^{2}+f_{xb}^{2}+\kappa f_{b}^{2}\right)^{2}-4f_{b}^{2}(f_{ vb}^{2}+f_{xb}^{2})}\right]}, \tag{24}\]  
where the natural frequencies of the bounce and bending uncoupled vibrations appeared:  
* bounce, \[f_{b}=\frac{1}{2\pi}\sqrt{\frac{4k_{zc}}{M_{c}}};\] (25)
* vertical bending, \[f_{vb}=\frac{1}{2\pi}\sqrt{\frac{k_{mc}}{M_{mc}}},\] (26) and the following notations \[f_{xb}=\frac{1}{2\pi}\sqrt{\frac{8\psi^{2}h_{b}^{2}k_{xb}}{M_{mc}}},\] (27) \[\kappa=1+\varepsilon^{2}\frac{M_{c}}{M_{mc}}=1+\varepsilon^{2}\frac{l_{c}}{ \int\limits_{0}^{l_{c}}X_{c}^{2}(x)\mathrm{d}x},\] (28) where \(\varepsilon\) is calculated with Equation (13) for \(l_{c1}\) and \(l_{c2}\) corresponding to the wheelbase \(2a_{c}\) obtained by scaling the passenger coach wheelbase. Taking \(k_{xb}=0\) (\(f_{xb}=0\)), the natural frequencies of the experimental model of the carbody without anti-bending bars result  
\[f_{(b,xb)1,2}=\sqrt{\frac{1}{2}\left[f_{vb}^{2}+\kappa f_{b}^{2}\mp\sqrt{ \left(f_{vb}^{2}+\kappa f_{b}^{2}\right)^{2}-4f_{b}^{2}f_{vb}^{2}}\right]}. \tag{29}\]  
The natural frequency of the bounce vibration of the railway vehicle was set for comfort reasons around 1 Hz, a value much lower than the natural frequency of the vertical bending vibration, which practically ensured the decoupling of the two types of vibrations. On the other hand, the natural bounce frequency of 1 Hz required the static deflection of the suspension to be approx. 0.25 m [82]. This static deflection value was far too large to be achieved with the demonstrative experimental system.