## 1 Introduction
### Theoretical Model of the Experimental Model of the Carbody with Anti-Bending Bars  
Experimental model of the vehicle carbody (Figure 13) could be assimilated to a uniform free-free Euler-Bernoulli beam of length \(l_{c}\), with rectangular cross-section of width \(b\) and height \(h\) (Figure 14). Figure 12: Scheme of the demonstrative experimental system. The mass of the experimental model of the carbody was \(M_{c}=\rho bhl_{c}\), where \(\rho\) is the density of the material, and the mass of the beam per unit length is \(m_{c}=M_{c}/l_{c}\). The bending stiffness of the beam is \(E_{c}l_{c}\), where \(E_{c}\) is the longitudinal modulus of elasticity, and \(l_{c}\) is the moment of inertia of the cross-section of the beam. Additionally, the loss factor due to structural damping was introduced, denoted by \(\eta_{c}\). Anti-bending bars of length \(l_{b}\) were fixed rigidly to the lower part of the experimental model of the carbody, at the distances \(l_{b1,2}=(l_{c}\pm l_{b})/2\) from the origin of the reference system (\(XOZ\)), respectively, at the distance \(h_{b}\) from the neutral axis of the experimental model of the carbody. The anti-bending bars were stressed in tension-compression, and each introduced a longitudinal stiffness given by the relation  
\[k_{xb}=\frac{\pi E_{b}D_{b}^{2}}{4l_{b}} \tag{1}\]  
where \(E_{b}\) is the longitudinal modulus of the bar material, and \(D_{b}\) is the diameter of the cross-section of bar. The experimental model of the carbody was laid on four elastic supports that modeled the suspension elements of the vehicle carbody, of stiffness \(k_{zc}\) and loss factor \(\eta_{zc}\). The longitudinal base of the elastic supports was \(2a_{c}\), and the distances \(l_{c1,2}=l_{c}/2\pm a_{c}\) fixed the support points of the experimental carbody model on the elastic supports. The main vibration modes of the experimental model of the carbody were bounce, pitch, and the first vertical bending mode. The vertical displacement of a section of the  
Figure 14: Dimensions of the experimental model of the carbody with anti-bending bars. Figure 13: Theoretical model of the experimental model of the carbody with anti-bending bars. experimental carbody model at \(x\) distance from the origin of the reference system, \(w_{c}(x,t)\), was the result of the superposition of the three modes of vibration,  
\[w_{c}(x,t)=z_{c}(t)+\bigg{(}x-\frac{l_{c}}{2}\bigg{)}\theta_{c}(t)+X_{c}(x)T_{c}(t) \tag{2}\]  
where \(z_{c}\) is the displacement of the experimental model due to the bounce, \(\theta_{c}\) is the pitch angle, and \(T_{c}\) is the time-coordinate of the first vertical bending mode; \(X_{c}\) is the eigenfunction associated to the vertical bending mode  
\[X_{c}(x)=\sin\beta x+\sinh\beta x-\frac{\sin\beta l_{c}-\sinh\beta l_{c}}{ \cos\beta l_{c}-\cosh\beta l_{c}}\ (\cos\beta x+\cosh\beta x) \tag{3}\]  
with  
\[\beta=\sqrt[4]{\omega_{vb}^{2}m_{c}/(E_{c}L_{c})} \tag{4}\]  
which verifies the characteristic equation  
\[\cos\beta l_{c}\cosh\beta l_{c}-1=0 \tag{5}\]  
where \(\omega_{vb}\) is the angular frequency of the vertical bending of the experimental model of the carbody without anti-bending bars. Vertical harmonic force \(F(t)=F_{o}\text{cos}\omega t\), of amplitude \(F_{o}\) and angular frequency \(\omega\), acts upon the experimental model of the carbody at the distance \(x_{o}\) from the origin of the reference system. Equation of motion of the experimental model of the carbody with anti-bending bars was obtained using the Euler-Bernoulli beam theory, of the form  
\[E_{c}l_{c}\frac{\partial^{4}w_{c}(x,t)}{\partial x^{4}}+m_{c}\frac{\partial^{2 }w_{c}(x,t)}{\partial t^{2}}=\sum_{i=1}^{2}\bigg{[}F_{zci}\delta(x-l_{ci})+F_{ zbi}l_{b}\frac{\text{d}\delta(x-l_{bi})}{\text{d}x}\bigg{]}+F(t)\delta(x-x_{o}) \tag{6}\]  
where \(\delta(.