## 1 Introduction
### Calculation of the Frequency Response Functions of the Experimental Carbody Model
The solutions of these equations of motion were of the form:  
\[\overline{Z}_{co}=\frac{(-\omega^{2}M_{mc}+\overline{k}_{mc}+8\overline{k}_{xb} \psi^{2}h_{b}^{2}+4\overline{k}_{zc}\epsilon^{2})-4\overline{k}_{zc}\epsilon \underline{\xi}}{(-\omega^{2}M_{c}+4\overline{k}_{zc})(-\omega^{2}M_{mc}+ \overline{k}_{mc}+8\overline{k}_{xb}\psi^{2}h_{b}^{2}+4\overline{k}_{zc} \epsilon^{2})-16\overline{k}_{zc}^{2}\epsilon^{2}}\overline{F}_{o}, \tag{48}\]  
\[\overline{T}_{co}=\frac{(-\omega^{2}M_{c}+4\overline{k}_{zc})\underline{\xi}- 4\overline{k}_{zc}\underline{\epsilon}}{(-\omega^{2}M_{c}+4\overline{k}_{zc} )(-\omega^{2}M_{mc}+\overline{k}_{mc}+8\overline{k}_{xb}\psi^{2}h_{b}^{2}+4 \overline{k}_{zc}\epsilon^{2})-16\overline{k}_{zc}^{2}\epsilon^{2}}\overline{ F}_{o}, \tag{49}\]  
\[\overline{\Theta}_{co}=\frac{1}{-\omega^{2}I_{c}+4\overline{k}_{zc}d_{c}^{2}} \left(x_{o}-\frac{I_{c}}{2}\right)\overline{F}_{o}. \tag{50}\]  
With the relations (48)-(50), the frequency response functions of the acceleration of the experimental model of the carbody with anti-bending bars corresponding to the three vibration modes could be calculated  
\[\overline{H}_{z_{c}}(\omega)=-\omega^{2}\frac{\overline{Z}_{co}(\omega)}{ \overline{F}_{o}},\ \overline{H}_{T_{c}}(\omega)=-\omega^{2}\frac{\overline{T}_{co}(\omega)}{ \overline{F}_{o}},\ \overline{H}_{\Theta_{c}}(\omega)=-\omega^{2}\frac{ \overline{\Theta}_{oo}(\omega)}{\overline{F}_{o}}. \tag{51}\]  
The amplitude of vibration of the experimental model of the carbody at some section \(x\) was given by the relation  
\[\overline{w}_{c}(x,\omega)=\overline{Z}_{co}(\omega)+\left(x-\frac{I_{c}}{2} \right)\overline{\Theta}_{co}(\omega)+X_{c}(x)\overline{T}_{co}(\omega), \tag{52}\]  
and the acceleration amplitude of the experimental model of the carbody was  
\[\overline{a}_{c}(x,\omega)=-\omega^{2}\overline{w}_{c}(x,\omega). \tag{53}\]  
The acceleration frequency response function of the experimental model of the carbody was calculated with the relation  
\[\overline{H}_{c}(x,\omega)=\frac{\overline{a}_{c}(x,\omega)}{\overline{F}_{o} }=\overline{H}_{z_{c}}(\omega)+\left(\frac{I_{c}}{2}-x\right)\overline{H}_{ \Theta_{c}}(\omega)+X_{c}(x)\overline{H}_{T_{c}}(\omega), \tag{54}\]  
in which the contributions of the three modes of vibration could be identified,  
\[\overline{H}_{c}(x,\omega)=\overline{H}_{cz_{c}}(\omega)+\overline{H}_{c\Theta _{c}}(x,\omega)+\overline{H}_{cT_{c}}(x,\omega). \tag{55}\]  
The first term \(\overline{H}_{cz_{c}}(\omega)=\overline{H}_{z_{c}}(\omega)\) represented the contribution of bounce vibration, the second term \(\overline{H}_{c\Theta_{c}}(x,\omega)=({I_{c}}/{2}-x)\overline{H}_{\Theta_{c}}(\omega)\) is due to pitch vibration, and the third term \(\overline{H}_{cT_{c}}(x,\omega)=X_{c}(x)\overline{H}_{T_{c}}(\omega)\) came from the vertical bending of the experimental model of the carbody. After processing, Equation (54) was brought to the form:  
\[ \begin{array}{ll}\overline{H}_{c}(x,\omega)=&-\omega^{2}\frac{-\omega^{2}M_ {mc}+\overline{k}_{zc}+8\overline{k}_{xb}\psi^{2}h_{b}^{2}+4\overline{k}_{zc }\epsilon(\underline{\epsilon}-\underline{\xi})-X_{c}(x)[\omega^{2}M_{c} \underline{\xi}+4\overline{k}_{zc}(\underline{\epsilon}-\underline{\xi})]}{(- \omega^{2}M_{c}+4\overline{k}_{zc})(-\omega^{2}M_{mc}+8\overline{k}_{xb}\psi^ {2}h_{b}^{2}+4\overline{k}_{zc}\epsilon^{2})-16\overline{k}_{zc}^{2}\epsilon^{ 2}}-\\ &-\omega^{2}\frac{\left(x_{o}-\frac{I_{c}}{2}\right)\left(x-\frac{I_{c}}{2} \right)}{-\omega^{2}I_{c}+4\overline{k}_{zc}d_{c}^{2}}\end{array}  \tag{56}\]  
For the calculation of the frequency response function of the acceleration in the middle of the experimental model of the carbody, \(x={I_{c}}/{2}\) iwa considered, and above the elastic supports, \(x={I_{c}}_{1,2}\).