As already pointed out, we are not attempting to develop a complex model that takes account of all the driving phenomena. The main focus is to study the dynamic and energy phenomena associated with train traction or braking. For the interested reader, there is a wide biography on the subject (Iwnicki, 2006).

Let us now consider the steel wheel shown in Figure 3.-, moving along a longitudinal plane at a speed "V", in contact with a steel rail and to which we apply a traction torque "T". Also coming into play is the gravitational force "Mg", where "M" is the mass suspended on the wheel and "g" is the acceleration due to gravity, the vertical reaction of the rail "N", which balances the vertical forces, and the forces opposing the train's forward motion, "ZFr", already mentioned in the previous section. Finally, thanks to the adhesion in the wheel-rail

contact zone, the tangential adhesion force "Ft" appears, which satisfies the following expression:

Ft = I^N (5)

Where "p" is the so-called adhesion coefficient, whose rolling values for wheel and steel rail is dependent on temperature, humidity, dirt, etc, and particularly on speed. An expression that will enable us to find the value of the adhesion coefficient according to speed is the following:

u= - (6)

l+0,01V

Where "V" is the speed of the train in Km/h, and "po" is the adhesion coefficient for zero speed. The value of "po" depends on the atmospheric conditions of temperature, humidity, dirt, etc, which under optimum conditions reaches a value of 0'33. Under such conditions and V=300 km/h, we obtain: u =0.082. Therefore, we can see that the wheel-steel rail contact has limited possibilities when it comes to transmitting the tangential tractive or braking efforts "Ft",.

Below is the expression used by RENFE in Spain for the adhesion coefficient, where, as is customary, the speed is expressed in km/h:

Right from the beginnings of the railway the major challenge of railway traction has been to increase the adhesion coefficient in the wheel-rail contact. Fig. 4 shows some of the adhesion coefficient values "po", used. What is surprising is the high value achieved for this coefficient in the United States of America.

 Values of "flo" SNCF (France) Electric monophase locomotives, multimotor bogies 0.33 Electric monophase locomotives, monomotor bogies 0.35 DB (Germany) Diesel locomotives 0.30 Electric monophase locomotives 0.33 RENFE (Spain) Diesel locomotives 0.22 - 0.29 Classic electric locomotives 0.27 Modern electric locomotives 0.31 USA SD75MAC diesel and electric locomotives 0.45

Fig. 4. Some values used for "po".

Some of the methods used to improve the values of the adhesion coefficient, especially during start up or acceleration, for reasons which will be made clear further on, are:

a. Introduction of sand in the wheel-rail contact. This is the traditional method using devices called "sandboxes", which are still frequently in use. However, this is a very aggressive system regarding the wear of wheel and rail materials.

b. Monomotor bogies that spread the tractive efforts evenly between all their shafts lead to an optimisation of the adhesion coefficient used.

c. Drawbars that connect the locomotive chassis to the bogies in a way that the locomotive's weight falls on the lowest part of the bogie at a level that is as close as possible to the wheel-rail contact. Apparently, the accelerations between the locomotive chassis and the bogie generate a force torque that aids the tractive or braking efforts, thus improving wheel-rail adhesion.

d. Electronic anti-slip, braking and traction control systems. These are similar systems to ABS, (Anti-lock Braking System), or ASR, (Anti-Slip Regulation) used in the automotive industry. The speed of the wheels is controlled and regulated by electronic devices so that there is no slip between the wheels and the rails.

In traction, the ideal situation is to have the maximum effort according to speed. As we know, the power developed is the product of force and speed:

Power = Ft • V (8)

Since the power supplied by the motor is approximately constant, the tractive effort available "Ft", (given by equation (8)), dependent on the train's speed "V", complies with a hyperbolic-type ratio, like that shown in Figure 5.

Apart from the "constant power hyperbola", (in blue), Figure 5, also shows the maximum tractive effort curve constrained by the adhesion conditions, (in red), obtained from equation (5) and equations (6) or (7)). If we look carefully at the constant power hyperbole, it can be seen that at very low speed the tractive effort would tend towards infinity. However, for technical reasons, the motors are only able to supply a limited tractive effort. This is called "continuous tractive effort", and appears in green.

Fig. 5. Effort-speed curves