## 4 Experimental and Theoretical Results
### Determination of Rubber Support Stiffness  
Figure 13 shows the acceleration of the rail coupon resting on the rubber supports of the scale model of the carbody (see Figure 7). To reduce the influence of the first bending mode on dynamic response, rubber supports were positioned at the two nodes of the first bending mode. The accelerations were measured above the rubber supports, and the hammer blow was applied to the middle of the rail coupon. Figure 13 shows the accelerometer obtained from averaging the two measured acceleration. The resonant frequency of the rigid vibration mode is located at 58 Hz, while the first bending mode has its resonance at 1176 Hz. To determine the stiffness of rubber supports, an analytical model of the experimental system for measuring the accelerance of the rail coupon was developed (Figure 14). The rail coupon is modelled as a free-free Timoshenko beam, and the rubber supports are considered as elastic elements with hysteretic damping. Figure 12: Accelerance experimentally determined in the middle of the scale model of the carbody with anti–bending bar system. Figure 13: Rail coupon acceleration experimentally determined above to rubber supports. The equations of motion for steady-state harmonic behaviour can be deduced as follows  
\[ \begin{array}{l}GS\kappa\left[\frac{\mathrm{d}^{2}\overline{w}(x)}{\mathrm{d}x^{ 2}}-\frac{\mathrm{d}\overline{\theta}(x)}{\mathrm{d}x}\right]+\omega^{2}m \overline{w}(x)=-\overline{F}\delta(x-x_{F})+2k[\overline{w}(l_{1})\delta(x-l_ {1})+\overline{w}(l_{2})\delta(x-l_{2})],\\ EI\frac{\mathrm{d}^{2}\overline{\theta}(x)}{\mathrm{d}x^{2}}+GS\kappa\left[ \frac{\mathrm{d}\overline{\theta}(x)}{\mathrm{d}x}-\overline{\theta}(x) \right]+\omega^{2}\rho I\overline{\theta}(x)=0,\end{array}  \tag{1}\]  
where \(m\) and \(\rho\) are the mass per unit length and density of the beam material, \(S\) and \(I\) are the area and moment of inertia of the cross-section, \(E\) and \(G\) are the longitudinal and transverse modulus of elasticity, \(\kappa\)--shear coefficient, \(\overline{k}\)--complex stiffness of the rubber support (\(\overline{k}=k(1+i\eta)\), where \(\eta\) is the loss factor of the rubber support, and \(i^{2}=-1\)), \(\overline{w}(x)\), \(\overline{\theta}(x)\) are the complex amplitudes of displacement and angle of rotation of the beam in the \(x\)-section of the reference system \(Oxz\), \(\overline{F}\) the complex amplitude of the harmonic excitation force of angular frequency \(\omega\), \(l_{1}\) and \(l_{2}\) determine the position of the rubber supports, \(x_{F}=L/2\) is the position of the excitation force, where \(L\) is the length of the beam and \(\delta(. )\) is Dirac's delta function. The last term in Equation (1) represents the sum of the two elastic forces in the rubber supports. The following boundary conditions shall be considered:  
\[ \begin{array}{l}\frac{\mathrm{d}\overline{w}(0)}{\mathrm{d}x}-\overline{ \theta}(0)=0,\qquad\frac{\mathrm{d}\overline{\theta}(0)}{\mathrm{d}x}=0\\ \frac{\mathrm{d}\overline{w}(L)}{\mathrm{d}x}-\overline{\theta}(L)=0,\qquad \frac{\mathrm{d}\overline{\theta}(L)}{\mathrm{d}x}=0.\end{array}  \tag{2}\]  
The following equation is obtained from Equation (1) by removing the angle of rotation of the section  
\[ \begin{array}{l}EI\frac{\mathrm{d}^{4}\overline{w}(x)}{\mathrm{d}x^{4}}+ \omega^{2}\rho I\left(1+\frac{E}{G\kappa}\right)\frac{\mathrm{d}^{2}\overline {w}(x)}{\mathrm{d}x^{2}}+\omega^{2}m\left(\frac{\omega^{2}\rho I}{GS\kappa}-1 \right)\overline{w}(x)=\\ \left(1-\frac{\omega^{2}\rho I}{GS\kappa}\right)\left[\overline{F}\delta(x-x_ {F})-2\overline{k}(\overline{w}_{1}\delta(x-l_{1})+\overline{w}_{2}\delta(x-l _{2}))\right]-\\ \frac{EI}{GS\kappa}\left[\overline{F}\delta^{\prime\prime}(x-x_{F})-2\overline {k}\left(\overline{w}_{1}\delta^{\prime\prime}(x-l_{1})+\overline{w}_{2} \delta^{\prime\prime}(x-l_{2})\right)\right],\end{array}  \tag{3}\]  
where \(\overline{w}_{1,2}=\overline{w}(l_{1,2})\).