The rail vehicle resistance (or train resistance or simply resistance) is the total force necessary to maintain a rail vehicle in motion. This force depends on a number of variables and is of crucial importance for the energy efficiency of the vehicle as it is proportional to the locomotive power consumption. For the speed of the vehicle to remain the same, the locomotive must express the proper tractive force, otherwise the speed of the vehicle will change until this condition is met.
Davis equation A number of experimental measurements of the train resistance have shown that this force can be expressed as a quadratic equation with respect to speed as shown below:
R = A + B V + C V 2 {\displaystyle R=A+BV+CV^{2}}
Where R {\displaystyle R} is the resistance, V {\displaystyle V} is the speed of the rail vehicle and A {\displaystyle A} , B {\displaystyle B} , and C {\displaystyle C} are experimentally determined coefficients. The most well-known of these relations was proposed by Davis W. J. Jr. and is named after him. The Davis equation contains mechanical and aerodynamic contributions to resistance. The first formulation assumes that there is no wind, however, formulations that do not make this assumptions exist:
R = A + B 1 V + B 2 v + C v 2 {\displaystyle R=A+B_{1}V+B_{2}v+Cv^{2}} , where v {\displaystyle v} is the speed of the air with respect to the vehicle while B 1 {\displaystyle B_{1}} and B 2 {\displaystyle B_{2}} are experimental coefficients that separately account for mechanical and aerodynamic (viscous) phenomena respectively.
The coefficients for these equations are determined with experiments by measuring the tractive effort from the locomotive at different constant speeds or with a coasting experiments (the rail vehicle is set in motion at a certain speed and then the traction is disengaged, causing the vehicle to stop due to resistance). Most methods for determining these coefficients do not consider the effect lateral forces on the vehicle. Lateral forces can be caused by the centripetal acceleration of the vehicle following the curving of the tracks, by lateral tilt of the rails, or by aerodynamic forces if crosswind is present. These forces affect the resistance by pushing the vehicle laterally against the rail causing sliding friction between the wheels and the rails. In case of crosswind, the resistance is also affected by the change in the aerodynamic contribution as a consequence of changes in the flow.
Physical interpretation of the Davis equation
Speed-independent term
The first term in the Davis equation ( A {\displaystyle A} ) accounts for the contributions to the resistance that are independent from speed. Track gradient and acceleration are two of the contributing phenomena to this term. These are not dissipative processes and thus the additional work required from the locomotive to overcome the increased resistance is converted to mechanical energy (potential energy for the gradient and kinetic energy for the acceleration). The consequence of this is that these phenomena may, in different conditions, result in positive or negative contributions to the resistance. For example, a train decelerating on horizontal tracks will experience reduced resistance than if it where travelling at constant speed. Other contributions to this term are dissipative, for example bearing friction and rolling friction due to the local deformation of the rail at the point of contact with the wheels, these latter quantities can never reduce the train resistance. The term A {\displaystyle A} is constant with respect to vehicle speed but various empirical relations have been proposed to predict its value. It is the general consensus that the term is directly related to the mass of the vehicle with some observing an effect of the number axles as well as the axle loads.
Speed-linear term The coefficient in the second term of the Davis equation ( B {\displaystyle B} ) relates to the terms linearly dependent on speed and is sometimes omitted because it is negligible compared to the other terms. This term accounts for mass-related, speed-dependent, mechanical contributions to the resistance and for the momentum of the intake air for cooling and HVAC. Similarly to A {\displaystyle A} , empirical formulas have been proposed to evaluate the term B {\displaystyle B} , and again a mass dependence is present in all major methods for determining the rail vehicle resistance coefficients, with some also observing a dependence from number of trailers and locomotives or a dependence from length.
Speed-quadratic term
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