For some reason I've been stuck on this. I've seen it with my own eyes and through video evidence, but still can't grasp it. How does one relate the tractive effort of a locomotive to the actual number of railcars the locomotive is capable of moving?
First lets assume that you've got dead straight and level track, and the railcars that you're pulling are loaded 4750 cu ft covered hoppers (most I've seen are max weight of 200,000lbs). Lets say that the locomotive you're pulling with is a GP38-2 that's been recently (within 6 months) shopped and checked over. Any locomotive can be substituted for the sake of ease of finding TE or CTE figures, but I'm just wondering what the equation is (if there is such) that determines how many cars the locomotive can pull.
Any help here?
BamaCSX83 For some reason I've been stuck on this. I've seen it with my own eyes and through video evidence, but still can't grasp it. How does one relate the tractive effort of a locomotive to the actual number of railcars the locomotive is capable of moving? First lets assume that you've got dead straight and level track, and the railcars that you're pulling are loaded 4750 cu ft covered hoppers (most I've seen are max weight of 200,000lbs). Lets say that the locomotive you're pulling with is a GP38-2 that's been recently (within 6 months) shopped and checked over. Any locomotive can be substituted for the sake of ease of finding TE or CTE figures, but I'm just wondering what the equation is (if there is such) that determines how many cars the locomotive can pull. Any help here?
The Davis equation, which calculates the rolling resistance of a train is what you need.
-Don (Random stuff, mostly about trains - what else? http://blerfblog.blogspot.com/)
As a practical matter - each carrier has developed Tonnage Ratings for their territories. When assigning power to a train, the tonnage of the train is know and those assigning power pick a group of engines whose combined tonnage rating just exceed the tonnage of the train on the ruling grade on the territory the train will traverse.
Never too old to have a happy childhood!
BaltACD As a practical matter - each carrier has developed Tonnage Ratings for their territories. When assigning power to a train, the tonnage of the train is know and those assigning power pick a group of engines whose combined tonnage rating just exceed the tonnage of the train on the ruling grade on the territory the train will traverse.
Generally, they are developed using computer simulations - usually called train performance calculators - that use train, grade and curve resistance along with train consist and loco performance data to simulate a train running from A to B. They output includes run time and energy consumed, among other things. They are fed and validated with lots of road testing. (there were some pretty terrific articles in Trains in recent years about CSX and GE testing "heavy" AC units)
For simple calculation, you can add 20#/ton/%grade for grade resistance and 0.7#/ton/degree for curve resistance. Typically, though, ruling grade curves are "curve compensated" - that is the grade on the curve is flattened a bit to compensate for the added curve resistance such that the sum of grade + curve resistance is constant for the whole climb up the ruling grade. So, for purposes of figuring out whether your train will stall en route, you can usually ignore curve resistance.
All right, so by the Davis equation, my math formula set-up would be as follows:
w=50,000lbs (200,000lb max weight by 4 axles)
n=4 (4 axles on a covered hopper)
V=10mph
k=0.07
With that information, this should give me: R=0.6+20/50,000+.01(10)+0.07(10)`2/50,000(4)
Which should mean that to move one covered hopper at 10mph R=0.70096lbs of resistance, or did i do my math wrong?
And if I did my math right, then it only takes about 70.096lbs of tractive effort for a single locomotive to move a 100-ton covered hopper at 10mph... so something like a GP38-2 which makes 61,000 lb-f of TE could theoretically move 87 hoppers all by its lonesome. Again, if I did my math right.
Bama,
I did not check it but your math looks correct. Your 200,000# per hopper is not correct since you forgot about the 63,000 typical tare weight. 200,000# is the net weight, 63,000# tare, 263,000# gross for a loaded "100 (net) ton hopper.
Mac
You've got a point there, I did forget the tare weight...
So my updated equation is as follows:
w=65,750lbs (263,000lb rail car by 4 axles)
n=4 axles
K=0.07
So with that, we have R=0.6+20/65,750+.01(10)+.07(10)*2/65,750(4)
Which means that R=0.70073lbs per ton
We have a 263000lb (or 131.5 ton) railcar, giving us an effort of 92.146lbs of effort
Again, with a GP38-2 making 61,000 lb-f of TE, it could theoretically move 66 covered hoppers all by itself (of course, again, if I did all my math right)
BamaCSX83 All right, so by the Davis equation, my math formula set-up would be as follows: w=50,000lbs (200,000lb max weight by 4 axles) n=4 (4 axles on a covered hopper) V=10mph k=0.07 With that information, this should give me: R=0.6+20/50,000+.01(10)+0.07(10)`2/50,000(4) Which should mean that to move one covered hopper at 10mph R=0.70096lbs of resistance, or did i do my math wrong? And if I did my math right, then it only takes about 70.096lbs of tractive effort for a single locomotive to move a 100-ton covered hopper at 10mph... so something like a GP38-2 which makes 61,000 lb-f of TE could theoretically move 87 hoppers all by its lonesome. Again, if I did my math right.
Weight per axle should be in tons, so your answer is 1.5#/ton, or 150# per car.
61K of TE will get you 407 cars (really 406 plus the locomotive...)
But, RRs aren't flat. Even a 1/2% grade requires 10# ton additional grade resistance, bringing the total up to 11.5#/ton or 1150#/car.
61K of TE will get you 52 cars plus a locomotive.
1% grade drops it back to 27 car, 2% 13 cars.
Grades are killers. That's why no Saluda and no Tennessee Pass (and why the Raton route is a loser)
But if you're on level track--
then there are two separate questions:
With how many cars can the engine keep moving at low speed, say 15 mph, or
How many cars can the engine start from a standstill?
On the level a GP38 might well be able to maintain 15 mph with 15000 tons or so. But can it start such a train? That we don't know.
timz But if you're on level track-- then there are two separate questions: With how many cars can the engine keep moving at low speed, say 15 mph, or How many cars can the engine start from a standstill? On the level a GP38 might well be able to maintain 15 mph with 15000 tons or so. But can it start such a train? That we don't know.
Yeah, talking about level track is almost a theoretical exercise - kinda like having massless rope and frictionless pulleys.
Note that if the resistance formula above is correct, then an ES44 with 140000? lb of continuous tractive effort can keep 90000+ tons moving on the level at 9 mph or so.
Put a train of 143-ton four-axle cars on a long 0.2% grade and it will accelerate downgrade to 60+ mph, if the formula is correct.
Try it this way. At low speed the last item of the formula is not much, so look at the first three which resolve to 1.3 + .9 + .45 or 2.65 pounds per ton. At 61,0000# TE that is a tonnage rating of 23,000 tons or 175 cars of 131 tons each. As a practical matter, yes we can start that cut, may have to take slack, but can start it.
As to speed on .2% downgrade; the grade supplies 4 pounds per ton of accelerating force, without ANY locomotive, subject to only 2.65 pounds resistance at 10 MPH. As speed increases air resistance will increase, but on a several mile long tangent downgrade of .2% the train would accelerate to 60 MPH sooner or later since acceleration due to gravity is more than resistance due to all forces. Assuming a 100 square foot cross section at 60 MPH I came up with 1.37 pounds per ton of air resistance, which makes 60 MPH the approximate terminal velocity, assuming no locomotive power.
For whatever it is worth, relating to the example of a GP38-2 on level track...
I have a MILW tonnage ratings book that gives tonnages at arbitrary minimum continuous speeds (in other words, selected by the railroad to ensure locomotives don't stay below their actual rated minimum speeds.) In this book, the GP38-2 is rated 9880 tons at 14 MPH for level track, 8110 tons for 18 MPH. These tons are, to be clear, trailing tons.
This translates to the fact that on the MILW at least you could never assign a GP38-2 more than 9880 tons trailing. Using your 131.5 ton covered hoppers, that's 75 of them... or if you'd like, 74 and a cabin car.
This gives a 'real world, real railroad' parallel to the calculations that might be of interest. I'm sure others can find other ratings for other roads that might differ slightly.
-Will Davis
Railroad Locomotives - My Blog
PNWRMNMAt low speed the last item of the formula is not much, so look at the first three which resolve to 1.3 + .9 + .45 or 2.65 pounds per ton.
Tim,
You are correct in that Don's version gives far lower values for journal resistance than the version I am using which is from "Management of Train Operation and Train Handling", Fourth Edition, published 1977 by the Air Brake Association.
The one I used is called "1970 modified Davis Eq." Don't recall where I found it.
Thanks to all for the answers, I believe me, I know that straight and level track is NEVER straight and level...
And again, I have seen a single locomotive start a cut of cars that seems "impossible" for it to move, but has indeed, moved it;, but I have always wondered how the "powers that be" have determined exactly what each locomotive is "rated" for as far as car numbers. Now I've got something to not only work with for grins and giggles, but gives me a better idea of what the "real guys" do.
Now, if I could only find a CSX tonnage ratings book for the area I'm in (CSX Dothan Subdivision), I think I'd be set...
As others have mentioned real world ratings are in tons between two points, not cars. For each train there will be a consist that will include somewhere on it loads, empties, tons, and feet. These are the important numbers for train dispatching.
Mac, you're right. I've heard those exact things broadcast on various scanner feeds when a train is coming out, or heading into a yard. But with those tonnage limits, (and some rough calculations/guesstimates) I could more, or less, come up with a max # of cars based on the power on the head end.
BamaCSX83 Now, if I could only find a CSX tonnage ratings book for the area I'm in (CSX Dothan Subdivision), I think I'd be set...
DEPART - ARRIVE - SD402 SD50 C40-8 ES44D SD70MA CW44C CW44AH NASHVL - BIRMIN - 2650 3150 3400 3700 4350 4600 5000 BIRMIN - MONTGL - 4200 5000 5450 5850 6900 7350 7950 MONTGL - DOTHAN - 2800 3300 3600 3900 4600 4900 5300 DOTHAN - WAYCRO - 4050 4800 5250 5650 6650 7050 7650
WAYCRO - DOTHAN - 3250 3850 4200 4550 5350 5650 6150 DOTHAN - MONTGL - 2850 3350 3700 3950 4700 4950 5400 MONTGL - BIRMIN - 3350 3950 4350 4650 5500 5850 6350
Forum formating doesn't provide accuracy in tabular data.
The railroad operates first and foremost on tonnage. Conversion from tons to cars is imprecise since mixed freight cars have various weights depending on whether loaded or empty. Car count does become a limiting factor in terms of siding length. It is very handy to have your trains short enough to fit in the sidings.
PNWRMNM Bama, The railroad operates first and foremost on tonnage. Conversion from tons to cars is imprecise since mixed freight cars have various weights depending on whether loaded or empty. Car count does become a limiting factor in terms of siding length. It is very handy to have your trains short enough to fit in the sidings. Mac
And even that can be 'fudged', by setting policy that trains in one direction must fit the sidings and trains in the other direction can exceed siding length. Makes it a little more difficult for the Train Dispatcher but it is a workable strategy.
BamaCSX83 if I could only find a CSX tonnage ratings book for the area I'm in (CSX Dothan Subdivision), I think I'd be set...
timz BamaCSX83 if I could only find a CSX tonnage ratings book for the area I'm in (CSX Dothan Subdivision), I think I'd be set...Best not to trust "tonnage ratings" that much. Some RRs try to be realistic and some don't bother; none of us here is much of an expert on which is which.
What I posted is what is used for the 'Bow Line' from Waycross to Birmingham that includes the Dothan Sub
Found 2 cents, here they are....
For DC locomotives, empirically determined, but based on calculations, a territory is determined to need a supply of horse power per ton of trailing load. The territory may include momentum grades, where an over-tonnage train may not be able to start but in a continuous train movement the train will lose speed but not stall before getting over the grade's summit. The phrase "making a run for the hill," comes to mind.
Hp/T ratings average out all the results of the more intricate calculations, and may include or exclude local factors like the mile of 1 per cent at the Martinez Bridge on the way from Oakland to Roseville, where exploiting short time amperage ratings allowed 3-tenths of a horse power per ton to let me get 5800 tons over the 'road.
Where Windy Point, near Palm Springs, and Midway on SP's Altamont trackage made tonnage ratings based on the formulas seem fictional, Hp/T usually worked. Not always.....most of the trips.
Many formulaic conclusions may never fail. Hp/T may never fail but also...as the weight of cars increases the effects of train resistance diminishes.
On Cuesta, the grade from San Luis Obispo toward San Francisco, I fired trains that were limited to 4250 tons with only power on the point.
Now, on the point, 5400 tons is the limit....roller bearings....no solid bearings....heavier cars...with less surface area for wind to affect...
As to Cuesta I suspect the difference is due to grade E couplers and knuckles rather than grade C.
Should not the values in th modified Davis formula be modified depending on revular journals or roller bearings?
Tonnage ratings my carrier uses are based upon how much tonnage the locomotive can keep moving just above its Minimum Continuous Speed on whatever the ruling grade is.
Personally, I take exception to this, because like it or not a train with max tonnage for it's engine consist will have some form of mechanical trouble and stop on the ruling grade for the territory. While the locomotive consist may be able to keep the max tonnage moving - there is not enough power to START the train on the grade. To get the train moving again, assistance will have to be secured. Assigning tonnage rating based upon MCS is a gamble, the gamble being whether the train stops on the ruling grade or not.
daveklepper Should not the values in th modified Davis formula be modified depending on revular journals or roller bearings?
No.
A well designed plain bearing only has a higher coefficient of friction when first starting. Higher torque is needed only to break the the oil film. Once running, the oil film becomes hydrodynamic in nature, and the friction is similar to a roller bearing.
The main advantage of a roller bearing on a rail car is the reduced maintenance vs. a plain bearing.
OK, now that we've settled all this, what about weather? (Always glad to add a kink to a discussion )
Hump yard computers, even in my day (early 70s), took journal temperature into account when calculating the retardation required for each car. I seem to recall a Davis variation that had an adjustment for temperature and maybe humidity but the details are hazy (no pun intended). But I clearly remember assigning extra power when snow was in the forecast.
Of course, on the Central in those days, it was unthinkable to assign only as much power as the tonnage required unless it was a low priority train. SVs, for example, always had four, five, or six of the newest GP40s available. Nobody ever got fired for overpowering an SV but a late SV could get somebody terminated.
ChuckAllen, TX
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