I’m on my way to explain the principle curvature equivalent gradient. As this is related to a discipline I enjoyed, I choose to go the long way and start with the FORCE.
But, before that, a few words of caution:
Take this post, and any other that I will write on this subject, with a bigger than usual pinch of salt. The subject of traction calculations and trailing loads is a part of the railway engineering discipline but not a core part of track design. Some of the national design standards impose alignment design rules driven from this subject, with little or sometimes with no background explanation. I am writing these posts to try to explain a few things I think I know and, hopefully, still remember. However, this disclaimer is required:
I’ve learnt about these things towards the end of the previous millennium as a side-info to the track design discipline. They are now a dear but very dusty memory. Some of the things I state might be over-simplified, incomplete, out of date or even wrong. (See Notes 0 & 1).
“Bending moment” and not “moment of doubt” (“crankshaft” and not “winched rod”)
Although the traction related technical terms I use might be correct in my mother tongue, I haven’t checked if all the English equivalents I translated them to are technically correct, as I haven’t been required to use them in English before writing these posts.
Because of these two reasons I’ll try to avoid being too technical. I will not speak about torque and work, only a bit about power, friction and a few other interesting stuff. Newton will probably be mentioned. I told you, you’ll like this!
Traction force (traction effort? tractive force? … the force that moves a train…)
In order to move a railway vehicle, the engine needs to develop a certain traction force that will overcome the initial resistance forces that oppose the movement and keep the vehicle at rest.
A traction force needs to be present to keep the vehicle in motion unless braking is required. Then the traction force is null.
Sometimes the vehicle has enough potential energy to keep moving without traction. Then also the traction force can be absent.
The traction effort generated by the engine is dictated by the engine’s power, P. The maximum power of the engine defines the maximum traction force the locomotive is capable of.
For a constant maximum power P, we will get a traction force F, depending on velocity (V). Higher the force F, proportionally lower the velocity, V. The direction of the force dictates the velocity direction (See note 2). The velocity value is the speed.
If we represent this force in a F(v) graph we will get a hyperbola curve (red in the graph below).
I was taught that the locomotives can in fact develop a higher power (and traction force) for up to an hour, to pass over difficult sections of track. I’m not sure if this is the case all around the world. But it is worth checking – an hour of higher power is something!
As you can see from the traction force graph, at very low speeds the traction force is very high – the locomotive can pull a bigger weight.
But also at low speed, especially when movement starts, the loco needs to overcome the adhesion between the wheels and the rails, otherwise the wheels will slip, rotating in place and damaging the rails, with no vehicle movement.
This adhesion condition defines another limit of the traction effort, which, to a certain value of the speed, is smaller than the effort produced by the maximum power of the engine.
The adhesion force formula is the following:
The wheel-rail adhesion coefficient, k, is around 0.2-0.35 (but don’t trust me with this).
The traction force is expressed in Newtons (N) and the weight in KN – I’ll discuss why later.
This adhesion-limited force defines a safe upper limit of the traction force. If the traction force is greater than this limit the wheels will slip.
I annotated the speed where the two limits intersect as Vc – let’s call this critical speed (I don’t remember the exact name).This is the speed at which the locomotive develops the maximum traction force.
The actual traction force a locomotive develops at any moment, is related to the speed the vehicle has and is always below these two limits, of adhesion and maximum power.
This is, in brief, the story of the Force!
Next, the Resistance!
0.) I am tempted to add here that one of my teaching principles was “If you can’t convince them, confuse them!”. My students never liked it. But it was good to see the difference between confusion and indifference on their faces. The confused ones could later be convinced. Dear reader, beware, some confusion might be purposefully (educativelly?) used by the author. But it would not be fair to reveal myself like this – please ignore this note. This isn’t the note you are looking for!
1.) Example of simplification:
The locomotive’s engine develops a traction force. A significant part of that force is transferred to the traction axles and is present at the contact between wheels and rails. Almost the same amount of force is active at the coupler to the wagons/coaches and it is the one that pulls them on track. All these forces are ”traction forces”. There are differences between them which I will not cover. I am simplifying them into one and will jump between one and another without necessarily highlighting the differences. This kind of simplification will happen for several other things in my posts. I try to keep the core principle right but some details will for sure be missed.
2.) I’m using “velocity” and not “speed”, because in this case both the force and the velocity are vectors – they are “direction aware”, they point somewhere in space. When the direction can be ignored, and we can refer only to a value, we can use “speed”. I’ll stick with “speed”, although sometimes that might be technically incorrect but the direction is obvious in all these discussions. I hope ...
3.) You understand now, dear reader, that Emperor Palpatine never needed unlimited power. He could have had, with any amount of Power, close to infinite Force! He just needed to move very slowly!