## 2 The Model: Governing Equations and Irregular Situations  
The track is modeled as an infinite beam standing for the rail, supported by linear spring-damper elements representing the rail pads, \(k_{p}\) and \(c_{p}\), point masses modeling the sleeper half-masses, \(M_{s}\), and supported further by linear spring-damper elements representing the vertical stiffness and damping of all other layers, \(k_{f}\) and \(c_{f}\), that is the ballast, sub-ballast, foundation and other layers that might be placed below the sleepers. The rail may be subjected to an axial force \(N\). The model is shown in Figure 1.  
The main assumptions for the analysis are listed below:  
* The beam is prismatic and straight, and its material is homogeneous and isotropic. * The beam may be subjected to an axial force acting on its axis (considered positive when inducing compression). Figure 1: The two-layer model of a ballasted railway track subjected to an axial force and traversing by two proximate masses acted upon by constant vertical forces. * The beam obeys linear elastic Euler-Bernoulli theory. Other researchers have found this beam theory adequate for modeling the steel rail because the beam height is small with respect to the length affected by the main vibrations. Differences in results obtained by other beam theories are noticeable only at very high frequencies [45]. In addition, the main simplification of the two-layer model affects the foundation model, and therefore, additional details regarding higher-order beam theories do not provide more realistic results. * The vertical displacements are measured from the equilibrium position determined by the weight of the model's components. * The initial conditions are assumed to be homogeneous; however, this does not affect either the critical velocity or the conditions for instability. * Mass horizontal position is determined by its velocity. * No friction is assumed to act at the contact point. * The load and vertical displacements are considered positive downward. * As usual in several applications, the acting force may or may not represent the moving mass weight. * It is assumed that the moving masses are always in contact with the beam, which means that the contact is assumed as tight, and, therefore, the mass displacement and the displacement of the corresponding beam axis position are the same at all times. An extension to a situation with a linear contact spring can be easily implemented, but it is not presented in this paper. The equations of motion of the model depicted in Figure 1 in fixed coordinates are  
\[Elw_{xxxx}+Nw_{xx}+mw_{tt}+c_{p}(w_{,t}-u_{,t})+k_{p}(w-u)=\ p(x,t) \tag{1}\]  
\[M_{s}u_{,tt}-c_{p}(w_{,t}-u_{,t})-k_{p}(w-u)+k_{f}u-k_{s}u_{,xx}+c_{f}u_{,t}=0 \tag{2}\]  
where \(x\) is the spatial coordinate and \(t\) is the time. In addition to the parameters defined in the first paragraph of this section, \(EI\) and \(m\) are the bending stiffness and mass per unit length of the beam, \(w(x,t)\) and \(u(x,t)\) are the unknown vertical displacement fields of the beam and discrete masses, respectively, and \(p(x,t)\) means the load. Derivatives are denoted by the respective variables in subscripts preceded by a comma. The load is given by  
\[p(x,t)=(P_{1}-M_{1}w_{01,tt}(t))\delta(x-x_{1})+(P_{2}-M_{2}w_{02,tt}(t))\delta (x-x_{2}) \tag{3}\]  
where \(\delta\) is the Dirac delta function, \(P_{1}\) and \(P_{2}\) are the moving forces, \(M_{1}\) and \(M_{2}\) are the moving masses, \(v\) is the constant velocity and \(d=x_{2}-x_{1}\) is the distance between the moving masses. Without affecting generality too much, all parameters are assumed to be distributed to simplify the solution. Thus, discrete values are divided by the distance between the sleepers \(l_{s}\), which will result in continuous supports. In Figure 1, there is also a shear component \(k_{s}\) because the model should have some shear stiffness. Placing it between the sleepers is not very realistic, but this is immaterial for the analysis.