## 2 The Model: Governing Equations and Irregular Situations
A different treatment must be implemented to make this value distributed, because the condition of uniform distribution must be applied on the inverse value. Therefore, if for a particular track a value of \(\overline{k}_{s}\) with unit N/m is used, then \(k_{s}=\overline{k}_{s}l_{s}\) with the correct unit of N.  
In Equation (3), \(w_{01}(t)\) and \(w_{02}(t)\) are the masses displacements. Owing to the assumption of the tight contact, \(w_{0j}(t)=w(x_{j},t)\), \(j=1,2\) holds with \(x_{1}=vt\) and \(x_{2}=vt+d\). Therefore, the relevant derivatives using the chain rule are  
\[w_{01,tt}(t)=v^{2}w_{,xx}(x,t)+2 vw_{,xt}(x,t)+w_{,tt}(x,t)\ \text{with}\ x=x_{1} \tag{4}\]  
\[w_{02,tt}(t)=v^{2}w_{,xx}(x,t)+2 vw_{,xt}(x,t)+w_{,tt}(x,t)\ \text{with}\ x=x_{2} \tag{5}\]  
Subsequently, the loading term reads as follows:\[ \begin{array}{c}\big{(}P_{1}-M_{1}\big{(}w_{,tt}(x,t)+2 vw_{,xt}(x,t)+v^{2}w_{,xx}(x,t)\big{)}\big{)}\delta(x-vt)\\ +\big{(}P_{2}-M_{2}\big{(}w_{,tt}(x,t)+2 vw_{,xt}(x,t)+v^{2}w_{,xx}(x,t)\big{)} \big{)}\delta(x-vt-d)\end{array}  \tag{6}\]  
Nevertheless, the solution will be presented in moving coordinates \(r=vx-t\), \(t=t\), leading to the following system of equations (compare with [24; 25; 42]):  
\[ \begin{array}{c}EIw_{,rrr}+Nw_{,rr}+m\big{(}w_{,tt}-2 vw_{,rt}+v^{2}w_{,rr} \big{)}+c_{p}(w_{,t}-vw_{,r}-u_{,t}+vu_{,r})\\ +k_{p}(w-u)=(P_{1}-M_{1}w_{,tt})\delta(r)+(P_{2}-M_{2}w_{,tt})\delta(r-d)\end{array}  \tag{7}\]  
\[ \begin{array}{c}M_{s}\big{(}u_{,tt}-2 vu_{,rt}+v^{2}u_{,rr}\big{)}-c_{p}(w_{, t}-vw_{,r}-u_{,t}+vu_{,r})\\ -k_{p}(w-u)-k_{s}u_{,rr}+k_{s}u_{,rr}+c_{p}(u_{,t}-vu_{,r})=0\end{array}  \tag{8}\]  
In the following text, the term \"active point\" will be used to spatially locate a moving object on a beam. The moving object can be either a force or a mass upon which the force acts. The designation \"critical velocity\" will be used for the critical velocity of a moving constant force. In this sense, it corresponds to a resonance effect because if the force is moving with its critical velocity in the absence of damping, then the steady-state solution is never achieved because the deflections tend to infinity and never stabilize. The critical velocity also indicates the separation between two distinct types of the steady-state beam shape. Up to the lowest critical velocity, approaching the lowest value from below, the maximum deflection of the steady-state solution over the entire beam is achieved at the active point. The minimum deflection (in absolute value) is significantly lower than the maximum. As already mentioned, there is no solution for the critical velocity. For higher velocities, zero deflection is achieved at the active point, and maximum and minimum deflections across the entire beam are the same. Similar properties can be found at higher critical velocity; in particular, the jump to zero deflection at the active point when crossing the critical velocity is preserved. Since the model has two layers, it is expected to have two critical velocities. However, by analyzing the equations, it can be shown that there will always be one or three resonances. In the case of three resonances, after sorting them by numerical value, the value in the middle will identify the so-called \"false critical velocity\" [42]. This means that resonance will occur in the sense that a steady-state solution will never be reached, but the expected properties listed above will not be met. Additionally, it will be shown that the false critical velocity has no effect on stability. The regular case is therefore defined as the case with three resonances and the irregular case is therefore defined as the case with only one resonance. In the case of one resonance, the missing critical velocity has to be supplemented with the so-called \"pseudocritical velocity\", which will always be lower than the critical velocity. To access the instability conditions, either for one or two moving masses, instability lines will be traced in the velocity-moving mass plane.