## 4 Experimental and Theoretical Results
### Accelerance of the Scale Model of the Carbody--Theoretical Results  
For reasons of double reflective symmetry and symmetrical loads, numerical simulations of the accelerance of the scale model of the carbody are performed on a quarter of the finite element model of the experimental scale demonstrator of the carbody in order to reduce the simulation time (see Figure 10). Harmonic analysis was performed with the modal superposition method [33; 34]. Figure 15: Accelerance of rail coupon above the rubber supports. Figure 16 shows accelerance in the middle of the scale model of the carbody without the anti-bending bar system calculated using the finite element model considering the harmonic excitation force located in the centre of symmetry of the model. The material constants are: aluminium density \(\rho_{\rm AI}\) = 2700 kg/m\({}^{3}\), Young's modulus 70 GPa and Poisson coefficient \(\
u_{\rm AI}\) = 0.33. The dimensions of the CAD model are as shown in Section 2. The accelerance peaks are located at 9.019 Hz, 45.32 Hz, 78.24 Hz and 170.7 Hz with the following values: 14.85 m/Ns\({}^{2}\), 0.7975 m/Ns\({}^{2}\), 1.713 m/Ns\({}^{2}\) and 1.961 m/Ns\({}^{2}\). Figure 17 shows accelerance in the middle of the scale model of the carbody with anti-bending bar system. Numerical simulation using the finite element model shows that due to the use of the anti-bending bar system, the accelerance peak corresponding to the first bending mode of the scale model of the carbody increases to 13.81 Hz, and its amplitude decreases to 6.494 m/Ns\({}^{2}\). In both simulations, modal damping was determined by explorations based on experimental results on acceleration peaks. Tables 1 and 2 summarise the comparison between the experimental results and those obtained with the finite element model in terms of vibration mode frequencies of the scale model of the carbody without/with anti-bending bar system. Errors were calculated against experimental values. Figure 16: Acceleration in the middle of the scale model of the carbody without anti-bending bar system (FEM model). Figure 17: Accelerance in the middle of the scale model of the carbody with anti-bending bar system (FEM model). In general, the errors are of several percentage points, which is acceptable. The errors are greater in the case of the scale model of the carbody with anti-bending bar system because its structure is more complicated, and in modelling, these intervene with a series of simplifying assumptions regarding the contacts between components, the constraints generated by the own weight and the tightening of the anti-bending bars, etc. Regarding Table 2, it should be noted that mode 2 is of the carbody, as shown in Figure 18a, where this mode is shown for the carbody without an anti-bending bar system. When the carbody is fitted with the anti-bending bar system, it also drives the anti-bending bars in an anti-phase motion (Figure 18b). In contrast, mode 2* corresponds only to the vibration of the anti-bending bars--the vibration of the carbody is insignificant (Figure 19). Table 3 shows the comparison in terms of damping in the finite element models of the scale demonstrator of the carbody without/with anti-bending bar system. The damping of the first three bending modes of the scale model of the carbody increases due to the anti-bending bar system which introduces some structural damping, while the fourth is not modified. \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Mode**} & \multicolumn{2}{c}{**Frequency [Hz]**} & \multirow{2}{*}{**Error**} \\ \cline{2-2}  & **Experimental** & & **FEM** \\ \hline
1 & 9.012 & 9.019 & 0.0777 \\ \hline
2 & 44.06 & 45.32 & 2.860 \\ \hline
3 & 78.69 & 78.24 & \(-\)0.572 \\ \hline
4 & 172.3 & 170.7 & \(-\)0.929 \\ \hline \end{tabular}
\end{table} 
Table 1: Vibration modes frequency of the scale model of the carbody without anti-bending bar system.