This post might be long. So let me summarize first. For a loco with pulling force of 4oz, it can pull a long train (each car is 8” long, 4oz weight),
On a straight track with 2% grade, one loco can pull about 50 cars.
On a curved track (R=40”) with 2% grade, one loco can pull about 33 cars.
If we use two locos to pull at the front, the number of cars is about 53 cars.
To maximize the pulling power, we may want to put the second loco in the middle to pull 66 cars.
There were some interesting discussions about how long a train can be on a 2% graded track when a Kadee couplers would break. In that discussion, I calculated the force that is needed to pull a hypothetical train with 1000 cars on a 2% grade track. Then the question is what about if the track also has a curve (say 40” radius)? In this post, I will try to include the curve into the calculations.
The 1000 car train is hypothetical but makes the calculation relatively easier. We can also easily scale down to 100 cars, or one loco. I will start with 1000 cars. So we have
Grade: G=2%
Radius: R=40”
Number of cars: N=1000
Length of a car: L=8"
Weight of car: W=4oz
Pul
I changed the Crr to 0.02, which I think might be too high for rolling. But anyway, I got the numbers that are very close to what you listed that table, especially when the radius becomes large. Great!
Why do you say it matters where in the train you pull it?
Would the same locomotives be unable to push a train from the rear that they could tow from the front? If placing any power at the midpoint of the train was more effective why not put all of it there? After all 1/87 engineers are unable to see anyway.
Prototype put “pushers” at the front quite frequently when coupler strain was not an issue. Over Canada’s Westetn Mountain grades in the steam era pushers were routinely just coupled onto the road locomotive. Occasionally cut in behind if the road locomotive engineer had sufficient seniority and the schedule provided the time. It was unusual to put pushers at the rear of the train. Even in the steam era it was not rare to see additional power cut in somewhere down the middle of a train but perhaps that reflected the technique of doubling up a grade cutting the train in two halves, why run them separately if you had sufficient locomotives.
Distributed power is more about coupler strain than rolling resistance.
Curves increase rolling resistance but in that case speed and superelevation are important variables. At the ideal speed there should be much less flange resistance. In the prototype with such broad curves at modest speeds the flanges should not touch the rails. Superelevation also has an ideal speed where flanges don’t ride the rail.
We have two level 180 degree curves at double track spacing and the flange squeal round the 22" radius is evident but absent around the 24" curve. Our well cars are the noisy ones.
When we place two cars on a curved track, these two cars are not aligned. There is a small angle between them. We can see this more clearly if we place two passenger cars on a 24" curve. Because of this small angle, the front car cannot pull the car after with 100% of its pulling force. It actually only uses F*cos(t), where t is the angle between the two cars. For the example I gave, on a 40" radius curve and car length of 8", cos(t)=0.98.
So if we have one loco with 4oz pulling force at the front, assuming the loco is also 8" long, the force that pulls the first car is 3.92oz; we lose 0.08oz force. The force that pulls the second car is 3.92*0.98=3.84, we lose another 0.076oz, and so on.
But if we put two locos at the front, the pulling force from the locos is 8oz. The force that actually pulls the first car is 80.98=7.84oz, so we lose 0.16oz force. The force that pulls the second car is 7.840.98=7.68oz, so we lose another 0.16oz force. So the more power we put at the front, the faster we lose it (in absolute value). Because of this, one loco can pull 33 cars on 40" curved track with 2% grade; but two locos at the front can only pull 53 cars (because the force is lost so fast by its absolute value).
In prototype trains, the curve radius is so large so that we barely see any angle between two cars. So there is no such issue. This is more of a consideration in model railroad when our curves are sharper (as compared to our cars).
We need a vector analysis to see why stringlining takes place after certain limits on curves of a certain radius and with the draw/resistance calculated for different weights of trailing scale cars…trailing tonnages.
And don’t look at me. I’m retired and getting ready to plant seeds for germination in my greenhouse.
Jerry, thanks for taking the time it took to consider this and to post it. Nice work!
I agree. If you want to pull a particular train on a straight or curved track, and/or up a grade, you simply put a locomotive on it, and see how it performs. If it does the job, write down the details of the test, so you can repeat it at any time.
If that locomotive can’t do the required work, you add another one, either with it or on the rear of the train or somewhere in the middle. I’ve done this with all of my locomotives (I call the results “tonnage ratings”).
My cars aren’t all the same length, nor the same weight…some open cars, without loads, weigh only 2oz., but with a full load (usually loose material) they’re just slightly over 8oz.
For me, it’s a lot more fun doing the experimental work in determining the amount of power needed to move the train, rather than sitting around with a pencil and trying to make calculations where car weights and lengths are all the same, because for most of us they’re not…just like the real ones.
Definitely not a math class major, that’s for sure.
However, I think what Jerry is doing and posting for us can help some people, if that’s their thing.
My take is based on his numbers, if I read it correctly, that every freight car must match in weight, height, length and who knows what else has to be taken into consideration. I don’t think most of us (there may be some) have perfect, matching, rolling and more freight or passenger cars.
I think in the modeling world, we have to look at our own layouts to see how we can make trains run at certain lengths, eliminating drags, coupler issues and more.
On my layout, I have a 2 track helix with a 2% grade that is 95% the same all the way around the 3 1/2 turns. Depending on the engine and cars, I can pull with a pair of matching GP38-2 engines and 16 freight cars of various lengths up my helix. I’ve done more, but added a helper at the end.
Are there many home layouts that run 100 cars or more on a grade or helix? I can possibly see this helpful for a club layout.
Yes, that is correct. I think the reason is that your calculation used a fix number (effective grade) to account for the curves. Mine depends on the radius and the lenght of a car. The fixed number you used might be an average…
So moving all the motive power to, say, the middle of the train transfers half of the net vector force at the front of the train (some of which is wasted pulling “sideways” on the front of the second locomotive) to act directly on hauling the train? Less wasted sideways force?
How about putting both locomotives in the center of the train, pulling the back half and pushing the front half? Net gain?
Since we can’t get away from sharp curves on our layouts and we like to follow prototype that has no such sharp curves it helps to understand how our models operate.
We have found that splitting two diesel locomotives into a lead and a mid train power works better than double heading. Looks better too in our tiny world.