## 1 Introduction
### Dimensioning the Experimental Model of the Carbody
It was noticed that the large static deflection of the elastic supports corresponded to a small stiffness of the elastic supports, which could be obtained by increasing the length and/or reducing their section. In this case, buckling stability problems of the demonstrative experimental system may arise. Considering the above two aspects, the bounce vibration frequency was adopted to a sufficiently high value, to ensure the stability of the experimental model and avoid the interference of the bounce vibration with the vertical bending vibration. By conveniently choosing the frequencies \(f_{(b,xb)1}\) and \(f_{b}\), the vertical bending natural frequency of the experimental carbody model was obtained with the relation  
\[f_{vb}=f_{(b,xb)1}\sqrt{\frac{\kappa f_{b}^{2}-f_{(b,xb)1}^{2}}{f_{b}^{2}-f_{(b,xb)1}^{2}}}. \tag{30}\]  
Knowing the dimensions and material characteristics of the carbody experimental model, the vertical bending natural frequency could be calculated with the relation  
\[f_{vb}=\frac{1}{2\pi}\frac{(\beta l_{c})^{2}}{l_{c}^{2}}\sqrt{\frac{E_{c}l_{c} }{\rho}}=\frac{(\beta l_{c})^{2}}{4\pi\sqrt{3}}\frac{h}{l_{c}^{2}}\sqrt{\frac{ E_{c}}{\rho}}, \tag{31}\]  
where \(\beta l_{c}=4.7300407449\) is the smallest nonzero solution to the characteristic Equation (5). Equation (31) shows that the natural frequency of vertical bending increased directly proportional to the height of the experimental model of the carbody \(h\), decreased with the square of the length \(l_{c}\) and did not depend on the width of the experimental model \(b\). To determine the dimensions of the experimental model of the carbody, it was necessary to know the correlation between the length of the model \(l_{c}\), the natural frequency of the vertical bending \(f_{vb}\) and the height of the model \(h\). It followed from Equation (31)  
\[\omega^{4}-\left(\frac{k_{mc}+8\psi^{2}h_{b}^{2}k_{xb}+4\varepsilon^{2}k_{zc}}{M_ {mc}}+\frac{4k_{zc}}{M_{c}}\right)\omega^{2}+\frac{4k_{zc}}{M_{c}}\frac{k_{mc}+8 \psi^{2}h_{b}^{2}k_{xb}}{M_{mc}}=0 \tag{23}\]0  
The length of the experimental carbody model was obtained by conveniently adopting the scaling factor of the actual passenger coach carbody length. Accepting the scaling factor 1:10, for the carbody of the AVA 200 passenger coach, \(l_{c}=2.6\) m was obtained. By scaling 1:10 of the carbody width of the AVA 200 passenger coach, \(b=0.28\) m results. However, considering that the natural frequency of the vertical bending of the experimental carbody model did not depend on the model width (see. Equation (31)), \(b=0.4\) m was adopted to improve the stability of the carbody model on elastic supports. Further on, considering \(f_{b}=30\) Hz and \(f_{(b,xb)1}=8\) Hz, the frequency \(f_{vb}\) of 10.92 Hz results from Equation (29). According to the relation (32), for \(f_{vb}=10.92\) Hz, \(E_{c}=0.7\cdot 10^{11}\) N/m\({}^{2}\) and \(\rho=2.7\cdot 10^{3}\) kg/m\({}^{3}\) (aluminium model), \(h=0.013\) m was obtained by rounding. With all dimensions set, the mass of the experimental carbody model without anti-bending bars results, \(M_{c}=36.058\) kg. The stiffness of the elastic supports of the demonstrative experimental model was established from the condition of achieving the natural frequency of the bounce vibration \(f_{b}=30\) Hz, with the relation  
\[\omega^{4}-\left(\frac{k_{mc}+8\psi^{2}h_{b}^{2}k_{xb}+4\varepsilon^{2}k_{zc}}{M_ {mc}}+\frac{4k_{zc}}{M_{c}}\right)\omega^{2}+\frac{4k_{zc}}{M_{c}}\frac{k_{mc}+8 \psi^{2}h_{b}^{2}k_{xb}}{M_{mc}}=0 \tag{23}\]1  
Table 2 centralizes the main parameters and material characteristics of the experimental model of the carbody without anti-bending bars established in this section. Table 3 shows the natural frequencies of the experimental carbody model without anti-bending bars.