Train-resistance formulae are usually empirical and are based on one of two forms: in which R is the resistance per ton, f is a coefficient to be determined, V equals the velocity in miles per hour, and c is a constant also to be determined. Formulae of the second class, which include some power of V represented by the exponent n, usually employ the second power, but there are some variations even from this. These formulae disregard grade and curve resistances, inertia resistance, and the active resistance (or assistance) of the wind as distinct from mere atmospheric resistance. In short, they are supposed to give the resistance of a train moving at a uniform velocity over a straight and level track, there being no appreciable wind. It may readily be seen that, since grade and curvature resistances and the active pressure of the wind furnish resistances which are indefinitely variable, all general formulae must necessarily ignore these elements. The quantity c represents those elements of resistance which are supposed to be constant, or so nearly constant that their variation with velocity may be ignored. The journal friction and the rolling friction are generally considered as belonging to this class. The velocity resistances are usually assumed to vary as the square of the velocity, and for such formulae it only becomes necessary to determine the value of the coefficient f to obtain the value of this term. Very few resistance formulae take any account of any variation in train-load, whether the cara are loaded or empty, and whether (in freight-trains) they consist entirely of box cars, of flat cars, or a combination of all kinds. It is well known that all these elements have a very material influence on the actual train resistance, so much so that a formula which ignores such influences must be considered as very approximate. Out of a great multitude of formulae which have been proposed, a few have been selected for discussion, the formulae having been modified (if necessary) to bring them to a uniform basis for comparison, in which case R equals the total resistance in pounds per ton of 2000 pounds.
This formula has the merit of extreme simplicity, but since, as shown above, extreme simplicity is incompatible with accuracy, the most that can be claimed for such a formula is that it is approximately accurate for ordinary trains and for a considerable range of velocity. Evidently appreciating the fact that the formulae is not applicable to high velocities, the following modification has been suggested for velocities between 47 and 77 miles per hour:
A very simple formula ascribed to Wellington is as follows:
Although this formula is more simple than the formulae immediately following, it is evidently impossible for it to be accurate for all conditions. Wellington devised a series of formulae which should distinguish between the character of the loading, whether it was carried in box cars or flat cars, or whether the cars were loaded or empty. The formulae also allow for the variation due to the weight of the train. Assuming that the constants have been properly chosen, these formulae ought to give very much closer results than are obtainable by any other formula here quoted.
* The Baldwin Locomotive Works deny the authorship of formula 7. The real author is at present unknown.
3.9 +.0065V + (57V/W) ... for loaded flat cars,
. 3.9 + .0075V + (.64V/W) ... for loaded box cars,
. 6.0+.0083V + (57V/W)-- ... for empty flat cars,
6.0 + .01067 + .(64V/W) ... for empty box cars.
It should, however, be noted that a train consisting partly of box cars and partly of flat cars will have a higher resistance than is shown by any of the above formulae (and not a mean value), on account of the increased atmospheric resistance acting on the irregular form of the train.
It may be noted that Barnes's formula is identical with Wellington's simpler formula when the velocity is 29 miles per hour, but gives higher values for lower velocities and lower values for higher velocities.
R= 2.23+ [V5/8/(56.9+0.0311L)] ........ (11)
This formula declares that the resistance varies as the 3/5 power of the velocity, and also inserts the extra term L, which denotes the length of the train in feet. This constitutes another method of allowing for the variation in the resistance due to the loading of the train. There is reason, however, to doubt the correctness of the form of this equation, since, if the train were comparatively long (as it might be with a train of empties), the denominator of that fraction would be increased, the fraction itself would be decreased, and the resistance per ton would be less. It is well known that the contrary would be the case, since of two trains with the same actual gross weight, one consisting of loaded cars and therefore comparatively short, and the other a comparatively long train of empty cars, the short train of loaded cars will have a less total resistance and therefore (in this case) a less resistance per ton. In addition, a long train will have a somewhat greater atmospheric resistance than a short train of equal weight, and this will increase still more the unit resistance per ton. As Aspinall's tests were made on the basis of English rolling-stock, their values are hardly applicable to American practice. (e) Searles's formula:
R = 4.82 +0.00536V2
+0.00048V2 (weight of engine and tender)2 gross weight of train (12)
This formula does not take account of differences in the form of the train (whether box cars or flats) which would affect the atmospheric resistance. Neither does it take into account the relation of length to weight, or whether the cars are loaded or empty. Considering as before two trains, one of which is short and heavily loaded, and the other a long train of empties, the weight of engine and tender and the gross weight of the train might be the same in both cases, and yet the resistance per ton for the train of empties would be considerably higher than for the train of loaded cars, although this formula gives them the same figure.
For the purposes of comparison, we will compute the train resistance per ton according to the above formulae for a locomotive weighing 130 tons and with 2043 tons behind the tender moving at the rate of seven miles per hour. The resistances would be as follows:
(a) Baldwin: R = 3 + V/6 = 4.16.
(6) Wellington: R = 4 + .0055V2 = 4.27.
(c) Barnes: R=4 + .16V = 5.12.
(d) Aspinall: The formula is not strictly applicable, as before stated, but the comparison will be interesting. We will assume that the 2043 actual tons behind the tender, loaded two contents to one tare, consist of 44 cars. Assuming that the cars have a total length between coupler ends of 37 feet, the length of the train would be 1628 feet. Adding 62 feet for the engine, we would have L = 1690 feet. The formula then becomes:
R=2.23 + (25.6/56.9+52.6) = 2.23+.234 = 2.46
(e) Searles: R=4.82+0.262+0.183=5.265.
Applying Wellington's more accurate formula, and assuming first that the cars were loaded box cars, we would have:
R = 3.9 +0.367 + 0.014 =4.28.
If the cars were loaded flat cars, the resistance would be:
R=3.9 +0.318 +0.013=4.231.
It may be noted that these last two values agree fairly closely with Wellington's more general formula. The actual results obtained during Dennis's experiments were 4.7 pounds, which is a fair average between the low values given by Baldwin and Wellington and the higher values given by Barnes and Searles. The value given by Aspin-all's formula is apparently inapplicable.
Comparing these formulae for a fast passenger-train, the results will be given below. Assume that the train consists of six cars weighing 60,000 pounds each and that it is drawn by a locomotive whose total weight is 280,000 pounds. The train has a total length of 430 feet. We will compute, according to these formulae, its resistance at 50 miles per hour.
(a) Baldwin, Eq. 6: R=3+8.3 = 11.3. (6) Wellington: R=4+13.75 = 17.75.
(c) Barnes: R = 4+8 = 12.
(d) Aspinall: R = 2.23 +9.64 = 11.87.
(e) Searles: R = 4.82+13.40 + 73.5=91.72.
It may be noted that the first three formulae agree about as closely as they did for the slow freight-train, and that even Aspinall's formula, although based on English practice, gives a result which is very close to the average. It is seen, however, that Searles's formula gives a result which is out of all proportion to the other results. Although the formula is stated by its author "to give the resistance per ton for all trains, whether freight or passenger, and at any velocity under ordinary circumstances," it is evidently inapplicable to the assumed case unless we admit that the other formulae are worthless.