General traction equation, resistance forces

Railway traction is considered to be one of the main problems of longitudinal rail dynamics (Faure, 2004; Iwnicki, 2006). It is seen as a one-dimensional problem located in the longitudinal direction of the track, governed by the Fundamental Law of Dynamics or Newton's Second Law, applied in the longitudinal direction of the train's forward motion:

Where the term to the left of the equal sign is the sum of all the forces acting in the longitudinal direction of the train, "M" is the total mass and "a" is the longitudinal acceleration experienced by the train. The sum of forces consists of the tractive or braking effort "Ft" and the passive resistances opposing the forward motion of the train, "ZFr", (Fig. 1.-). The tractive or braking effort "Ft", in a final instance, is the resultant of the longitudinal adhesion forces that appear in the wheel-rail contact zones, either when the train's motors make the wheels rotate in the direction of forward motion, or when the braking forces act to stop the wheels rotating (in this case, these forces will obviously be negative or counter to the direction of the train's forward motion).

Fig. 1. Longitudinal train dynamics.

Let us now focus on the passive resistances that are opposing the train's forward motion, "ZFr". These basically consist of five types of forces:

a. Rolling resistances of the wheels.

b. Friction between the contacting mechanical elements.

c. Aerodynamic resistances to the train's forward motion.

d. Resistances to the train's forward motion on gradients.

e. Resistances to the train's forward motion on curves.

The phenomena associated with the appearance of the aforementioned resistances to the train's forward motion are widely known (Andrews, 1986 ; Faure, 2004; Coenraad, 2001 ; Iwnicki, 2006). For this reason, we will simply present one of the most common expressions for modern passenger trains:

Where "V" is the train's speed in (Km/h), "rg" is the inclination of the gradient as a (o/oo), "R" is the radius of the curve in (m), "L" is the weight of the locomotives in (Tm) and "Q" is the towed weight or the weight of the coaches in (Tm).

As can be seen from equation (2), the resistances to forward motion consist of three components: one constant, another that is linearly dependent on speed and another that depends on the speed squared. Moreover, when the train is running on a curve or a gradient, the final term that is not dependent on speed needs to be added.

We will now develop the Bond-Graph model, (Karnoppet alia, 2000), for the longitudinal train dynamics expressed in equation (1). In this model, shown in Figure 2, there are three basic elements:

a. The most important element is the train itself, whose longitudinal motion is represented by the Inertial port "I". The parameter of this port is the train's total mass, "M".

b. The tractive or braking efforts "Ft". In whatever case, this is an element that supplies energy according to a defined force. In Bond-Graphs, these energy sources with a

defined effort are represented by the Source of Effort port "SE". In this case, the parameter of the port is the defined effort.

c. The third element comprises the resistances to forward motion, which are represented by the resistance port "R" in the Bond-Graph. This port will have a variable parameter so that it can satisfy the equation (2).

The three ports comprising the Bond-Graph are brought together in a type "1" node, since all three phenomena are produced at the same speed, which is the speed of the train's forward motion. The inertial port satisfies the following: the input force "e1" is equal to the first temporal derivative of the quantity of motion "P":

In node "1", given that all the bonds connected to it have the same speed, the following is satisfied: the algebraic sum of the forces is zero. Therefore, the effort "e1" equals:

el = e2- e3 (4)

Where effort "e2" is the tractive effort "Ft", (or braking, if that is the case), and "e3" is the force of the passive resistances (given by equation (2)).

On the other hand, as we know, the Quantity of Motion "P" is equal to the product of the mass "M" and the speed "V", of the train in this case. By taking all this and entering into equation (3), it may be easily deduced that we will reach an expression that is identical to that shown in equation (1). In fact, it is the Inertial port "I" that resolves the fundamental equation of dynamics given by equation (1).