# Analysis, Evaluation and Simulation of Railway
## Materials and Methods  
### General Equations of the Train Movement  
The acceleration, a, of a train in its running direction can be described by a single scalar Newton equation and depends on the external and internal forces produced in the train:  
\[(m+J/R^{2})a=\sum_{k=1}^{n}F_{k}, \tag{1}\]  
\[m(1+k)a=\sum_{k=1}^{u}F_{tk}-\sum_{k=1}^{r}F_{rk}-F_{c}-F_{g}, \tag{2}\]  
where:  
\(n\)--overall number of effects being considered.  
\(M\)--mass of the train  
\(a\)--acceleration of the train  
\(J\)--moment of inertia of the different rotating masses (wheels, cog wheels, motor rotors) in the transmission, which causes an apparent increase of mass.  
\(R\)--wheel radius.  
\(F_{R}\)--propulsive force, tractive effort of locomotive/EMU k of u units in the train.  
\(F_{rk}\)--resistance forces, due to the element r of the train, locomotive, coach, generator, etc.  
\(F_{c}\)--curve resistance  
\(F_{g}\)--grade resistance, due to the slope of the track (positive or negative for the acceleration)  
\(k\)--increment of mass, due to rotating inertia, a coefficient in the range (0, 0.30) depending on the type of vehicle. It is in the range (0.06, 0.10) for a complete train.  
Resistance forces are the sum of all forces acting on the train at a given time or place. Some of these forces change directly as the axle loading does: For example, journal friction, rolling resistance, or track resistance. Other forces vary with speed and are known as flange resistance. Finally, some forces vary with the square of the speed: Air and wind resistance. These resistance forces are examined in detail with different models [40; 41; 42], but they are not the objective of this paper. A quadratic formula has been used for decades to approximate rail vehicles resistance:  
\[F_{r}=A+Bv+Cv^{2}, \tag{3}\]  
where \(v\) is the speed of the vehicle (m/s, mph, km/h), and \(A\) (N), \(B\) (N s/m) and \(C\) (Ns\({}^{2}\)/m\({}^{2}\)) are regression coefficients obtained by fitting run-down test of modern passenger and freight units to the Davis equation (e.g., different modifications were developed in 1970 by AREA association, or in 1992 by Canadian National [43]). For example, the modified version of 1970:  
\[F_{r}=0.6+\frac{20}{w}+0.01v+\frac{Kv^{2}}{wn}, \tag{4}\]  
where \(w\) is the weight per axle in tons; \(n\) the number of axles, \(K\) the air resistance (drag) coefficient, \(v\) the speed in miles per hour and \(F\) the resistance in pounds per ton.  
This paper uses different data from manufacturers (EMD, Bombardier, Alstom, Patentes Talgo, Vossloh, Krauss-Maffei, Siemens, etc.), associations (UIC) and railway administrations (SNCF, DB, SBB-CFF, FS, RENFE, etc.) for cars (Eurofima, Corail), locomotives (BoBo, such as Re4/4 III, or CoCo,such as Re6/6), and compositions of homogeneous and composite materials (SNCF). For example, Fr (in N), for a high-speed train (Train a Grande Vitesse, TGV-SE) from SNCF (French operator), or also for SNCF locomotives with n axles, both included in reference [42]:  
\[\begin{array}{l}F_{r}=2540+33.44v+0.572v^{2};TGV\\ F_{r}=mg(\frac{0.65}{1000}+13\frac{n}{m}+\frac{0.036}{1000}v+0.39\frac{v^{2}}{m} );\text{{locomotive}}\end{array} \tag{5}\]  
where the force is done in N, and \(v\) (speed) in km/h.  
The grade resistance or gravitational force (slope \(\beta\), from which calculate \(i=\sin\beta\)):  
\[F_{g}=mgi, \tag{6}\]  
and curve resistance by means of empirical formulae:  
\[F_{c}=\frac{k_{e}}{r_{c}}mg, \tag{7}\]  
where \(k_{e}\) is the gauge coefficient (750 m for 1435 mm gauge), and \(r_{c}\) is the curve radius in meters.  
All these forces are considered in simulations, taking into account traction and braking curves from manufacturers, and also the limits and constraints of the RPS, being described in next sections.