## 2 The Mechanical Model of the Railway Vehicle-Track System  
Figure 1 displays the mechanical model of a passenger railway vehicle running at constant velocity on a straight track affected by random vertical irregularities. The mechanical model of the vehicle comprises an equivalent beam for the carbody, two rigid bodies for the chassis of the bogies, and four rigid bodies for the vehicle wheelsets. These bodies are coupled to each other by Kelvin-Voigt systems, through which the secondary suspension and the primary suspension of the vehicle are represented. The carbody model exhibits three vibration modes: bounce, pitch, and the first vertical bending mode. The carbody inertia in relation to the bounce and pitch modes is represented by the mass \(m_{c}\) and the moment of inertia \(J_{c}\). In Figure 1, the displacements and the rotation of the carbody in the vertical plane are highlighted. The bounce motion is denoted as \(z_{c}\), and the rotation angle of the pitch motion is denoted as \(\theta_{c}\). The displacement of a section of the carbody, denoted as \(u_{c}\), results from the superposition of the three vibration modes mentioned above. Figure 1: The mechanical model of the vehicle–track system. At a given instant of time \(t\), the vertical displacement of the carbody at section \(x\) is given by the following relation:  
\[u_{c}(x,t)=z_{c}(t)+\bigg{(}x-\frac{l_{c}}{2}\bigg{)}\theta_{c}(t)+Y_{c}(x)T_{c}( t), \tag{1}\]  
where \(T_{c}\) is the time coordinate of the first vertical bending mode, \(Y_{c}\) is the eigenfunction of the vertical bending, and \(l_{c}\) represents the length of the carbody. The \(Y_{c}\) eigenfunction is calculated with the following relationship:  
\[Y_{c}(x)=\sin\alpha x+\sinh\alpha x-\frac{\sin\alpha l_{c}-\sinh\alpha l_{c}} {\cos\alpha l_{c}-\cosh\alpha l_{c}}(\cos\alpha x+\cosh\alpha x), \tag{2}\]  
where  
\[\alpha=\sqrt[4]{\frac{\omega_{c}^{2}\rho_{c}}{E_{c}l_{c}}} \tag{3}\]  
is the solution to the equation  
\[\cos\alpha l_{c}\cosh\alpha l_{c}=1; \tag{4}\]  
\(\omega_{c}\) is the natural angular frequency of the carbody vertical bending; \(\rho_{c}=m_{c}/l_{c}\) is the carbody mass per unit length; \(E_{c}l_{c}\) represents the bending stiffness of the carbody; with \(E_{c}\) represents Young's modulus; and \(l_{c}\) represents the second moment of aria of the cross section of the carbody. For bogies, two degrees of freedom are considered: the bounce denoted as \(z_{b1,2}\) and the pitch denoted as \(\theta_{b1,2}\). \(m_{b}\) denotes the bogie chassis mass, and \(l_{b}\) denotes the moment of inertia. The wheels, with mass \(m_{w}\), have vertical motion denoted by \(z_{w1\dots 4}\). Above each wheelset, the track model is an oscillator with one degree of freedom, having mass \(m_{r}\), stiffness \(2k_{xr}\), and damping constant \(2c_{xr}\). This is justified because the coupling between the wheels via the rail bending waves can be neglected at low frequencies. Oscillator displacement is denoted by \(z_{r1\dots 4}\). The vertical irregularities of the track in front of each wheel are denoted with \(\eta_{1\dots 4}\). When the track irregularity is a harmonic function, the vertical irregularities \(\eta_{1\dots 4}\) can be calculated as follows:  
\[\eta_{1,2}(x)=\eta_{0}\cos\frac{2\pi}{\Lambda}(x+a_{c}\pm a_{b})=\cos\omega \bigg{(}t+\frac{a_{c}\pm a_{b}}{V}\bigg{)}, \tag{5}\]  
\[\eta_{3,4}(x)=\eta_{0}\cos\frac{2\pi}{\Lambda}(x-a_{c}\pm a_{b})=\eta_{0}\cos \omega\bigg{(}t-\frac{a_{c}\pm a_{b}}{V}\bigg{)}, \tag{6}\]  
where \(x=Vt\) gives the position of the carbody center (\(V\) represents the velocity of the vehicle), and \(\omega=2\pi V/\Lambda\) is the angular frequency induced by the track irregularity of the amplitude \(\eta_{0}\) and wavelength \(\Lambda\). To model the wheel-rail contact, a linear elastic element of stiffness \(2k_{H}\) is inserted in each wheel-rail pair. The primary suspension model consists of a single Kelvin-Voigt system for each wheelset with the stiffness of \(2k_{zb}\) and the damping constant \(2c_{zb}\).