This section is from the "The Economics Of Railroad Construction" book, by Walter Loring Webb, C.E.. Also see Amazon: The Economics Of Railroad Construction.

When the train has succeeded in acquiring a high velocity on a favorable down grade and then strikes an up grade, it will be unable to maintain that high velocity, and its velocity will gradually decrease. The distance required for the decrease from one velocity to another will be as given by the retardation curves shown in Figs. 27, 28, and 29. For example, if the train has somehow acquired a velocity of 30 miles per hour, the tractive power of that engine at that velocity is 10,400 pounds. Assume that it then strikes a +0.4% grade. The tractive resistance per rating ton is 8.0+2.6, or 10.6 pounds per ton. The tractive force required is therefore 28,200 pounds. The deficit at this velocity is 17,800 pounds, which must be considered as a retarding force.

Table XXII. Determination of Coordinates of Velocity-distance Curves for One Type of Locomotive.

As before, since the difference of velocity heads for 29 and 30 miles per hour equals 2.07 feet, the distance 5,320,000 x 2.07=619.

s = 17,800.

This gives the point on the +0.4% retardation curve in the ordinate over 29 miles per hour. The numerical work of computing all of these values for one curve can best be accomplished by a series of three columns, such as columns 5, 6, and 7 in Table XXII, following the preliminary set of columns from 1 to 4. Each retardation curve will require a similar set of columns. Figs. 27, 28, and 29 are copies of the diagrams prepared by Mr. A. C. Dennis in illustrating the article above referred to. It will be found that the distances given in columns 7, 10, and 13 of Table XXII agree substantially with the value of the ordinatcs given in these diagrams. Such discrepancies as do exist are due to the fact that the tractive power of the engine has been measured to scale from the diagram Fig. 26, which indicates the tractive power.

Borrowing the example given by Mr. Dennis, assume that our locomotive has been loaded with 2452 rating tons behind the engine, which is the requirement for a +0.4% grade at 7 miles per hour. Suppose that a start has been made on a level grade 5000 feet long, followed by 4000 feet of 0.6% grade, followed by 3000 feet of -0.2% grade. Since the train starts on a level and is loaded on the basis of a +0.4% grade, we must follow its course for 5000 feet on the acceleration curve marked "level" in the diagram of Fig. 27. This curve has an ordinate corresponding to 5000 feet when the train is moving at a velocity of 20 miles per hour. This is therefore the velocity of the train when it strikes the +0.6% grade. The course of the train must then be studied from the retardation curve for 0.6%. This curve has an ordinate of 3600 on the 20 miles per hour line. Incidentally we may remark that if this train had started from any point with a velocity of 30 miles per hour up this grade, its velocity would have dropped to 20 miles per hour in 3600 feet. 4000 feet further gives us 7600 feet. The point on this curve which has an ordinate of 7600 feet occurs at about 8 miles per hour. The train then passes the summit at this velocity and starts on a - 0.2% grade, which will evidently be an acceleration curve. Eight miles per hour on this grade corresponds to about 400 feet. Adding 3000 we have 3400 feet, which on this curve corresponds to a velocity of nearly 21« miles per hour.

The only apparent difficulty in the above demonstration is the fact that, when the train is starting, the resistance is far higher than the resistance which has been found to be so uniform at velocities above 7 miles per hour. Whether this would be compensated by the fact that at very slow velocities the tractive force may be largely increased by the use of sand is not very certain. Mr. Dennis's diagram showing the tractive force at velocities but little above zero do not show any marked increase in the tractive force at very low velocities. The above method cannot be considered as precise, except on the basis that at very low velocities the resistance is no greater than at somewhat higher velocities, which is certainly not the case. These diagrams are probably very reliable for variations of freight-train velocities between 7 and 30 miles per hour. They are useful in obtaining the behavior of a train through a sag or over a hump. They are probably not so reliable when considering the movement of a train which starts from rest. In the numerical case just considered the velocity of the train at the end of the level grade of 5000 feet would probably be less than 20 miles per hour, since the resistance at starting would be considerably greater. If, however, it had somehow acquired that velocity of 20 miles per hour at the beginning of the +0.6% grade, its behavior over that grade and down the following grade would certainly be about as computed.

The following formula has been proposed by Mr. G. R. Henderson, and has the merit of great simplicity combined with practical agreement with the more complicated formulae based on elaborate tests.

Let R = the resistance of the train or the pull at the tender draw-bar in pounds; T = the number of tons back of the tender, including cars and contents; C = the number of cars in the train; P=rate of grade in per cent.

For speeds up to 12 miles per hour R = T(3.5+20P)+50C.

Applying this formula to a numerical case, let us assume three trains, one a train of empties, the second half filled, and the third of full cars, a full load being assumed as twice the weight of the car. The first train has 45 empties, each weighing 20 tons; the second train has 28 cars, each weighing 20 tons and carrying 20 tons of freight; the third train has 20 cars, each weighing 20 tons and carrying 40 tons of freight. Then the draw-bar pulls on a level would be as follows:

R = (900×3.5)+ (50×45) =5400, R= (1120×3.5) + (50×28) =5320, R = (1200 × 3.5) + (50 × 20) = 5200.

The resistances per ton are 6, 4.75, and 4.33 respectively.

It may be noted that the above values per ton are not as high for empty cars as those given by Mr. Dennis's tests, although the values for loaded cars agree fairly well.

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