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To Find Partial Area of the Train Hour Diagram:1st- Calculate the area of the rectangular portion. 2nd- Total area under train hour curve. 3rd- Partial area between 0 and any point t-3 is found as follows assuming n.398, nt-.398x3-1.194. From curve 'B' the ratio corresponding to nt=1.194 (on top line) is.91. That is, the area under the curve between 0 and t-3 is 91% of the total area or . If overtime occurs after t-3, the over time area equals total area minus partial area.

91VIT

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FIG. 5.

IV-Effect of Various Operating Conditions Upon Dimensions of Train-Hour Diagrams.

Having determined the dimensions for a known set of operating conditions it is important to be able to forecast what the performance of trains will be, by constructing train-hour diagrams for different sets of conditions.

Suppose that Fig. 3 is the train-hour diagram found for a Division Mi miles long operating N1 trains per day and it is desired to ascertain what the performance will be if the length of division were changed to M1⁄2 miles, the number of trains increased to N2 trains per day or the speed of trains increased, assuming that other conditions remain constant. If trains could be operated perfectly, which is the limiting case, the train-hour diagram in every case would be a rectangle, hence increasing the length of division without changing the number or speed of trains would lengthen the train-hour diagram; increasing the number of trains without changing the length of division or speed of trains would increase the height of the train-hour diagram; and increasing the speed of trains without changing the length of division or number of trains would decrease the length of the train-hour diagram. Hence changing any one of these conditions affects only one dimension of the perfect train-hour diagram, which is the rectangular portion of the theoretical or actual train-hour diagram.

Inasmuch as the remaining portion of the train-hour diagram reflects the delays to trains it is logical to assume that the longer the division, or the greater the number of trains the greater will be the delays to each train and the following discussion is for the purpose of setting forth mathematical expressions to show the effect of these conditions upon the average road time.

V-Effect of Length of Division Upon Average Road Time.

If M1 and M2 represent the miles per division and T1 and T: represent the average road time of all trains and N1 represents the number of trains operated (being the same in both cases), it can be assumed that if the character of profiles and track arrangement are identical the running time over equal distances will be identical if there are no delays. Likewise the delays in equal distances will be identical, or the average road time will be proportional to length of divisions.

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That is, the values of n will be inversely proportional to the length

of divisions.

VI-Effect of Number of Trains Upon Average Road Time.

If T1 and T2 represent the average road time of all trains corresponding to N1 and N2 trains per day it can be assumed that the delays to trains going the same distance will be proportional to the number of trains. Or the areas under the curve portions of the train-hour diagrams will be similar, that is, they will be proportional to the square of their linear dimensions.

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EXAMPLE. If we assume that the straight lines in Fig. 2 show the relation between the number of trains and the average time on the road for the four divisions A, B, C and D, the intercepts of these lines with the horizontal axis will give the value of to. Then if N1 and T1 for any other point on the line are taken, the value of n at this point can be calculated, also the average time on the road T2 when there are N2 trains on the road, as shown in the following table. The expression for T2 gives the equations for these lines and it can be assumed that the reason why there are not more actual points on the line is because other conditions were variable.

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VII-Effect of Speed With a Given Number of Trains Upon Average Road Time.

If T1 and T. represent the average road time of all trains corresponding to Si and S. speed of trains then for a given length of division the minimum time on the road to and to', respectively, will be inversely proportional to the speeds.

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With a given arrangement of tracks and the same number of trains the delays will be the same in both cases or the area under the curve will remain the same

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The average time on the road in each case is as follows:

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EXAMPLES-Division E has been selected to show the application of the mathematical theory to actual operating results. As will be seen by reference to Fig. 1 this division has been divided into three sections P-R, R-S and S-V. The section P-R is practically level and the grade on the other two sections is nearly uniform about 0.4 per cent. The dimensions of the theoretical train-hour diagrams are given in Table 2 and as before explained were obtained from the actual train-hour diagrams made from data taken from the train dispatcher's sheets.

Let us suppose that we had obtained only the dimensions for the train-hour diagram for the section S-V, but that we knew the distances between P and R (31.3 miles), between R and S (48.7 miles), and between S and V (30.7 miles) and knew that the best speeds on the section P-R and R-S were respectively 46.6 per cent and 4.5 per cent faster than on section S-V, and that there were 125 trains operated over section P-R, 136 over section R-S and 135 over section S-V, would it then be possible to compute the dimensions of the train-hour diagrams for the sections P-R and R-S?

The procedure would be as follows (see example below): Begin by assuming that the conditions of operation for the three sections P-R, R-S and S-V are the same except that the sections are not the same length. Values of to, n, and T for sections P-R and R-S are then calculated, correcting for the difference in length of sections. The next step is made to correct for the difference in the number of trains operated over each section, and the third step is for the purpose of correcting for the difference in speed of trains over the various sections.

Thus we have given for section S-V M= 30.7 miles, N=135 trains, S = 17.68 M.P.H. t = 1.736 hrs., n = .7955 and T = 2.85 hrs. For section R-S we have given M31.3 miles, N = 125 trains and S 25.93 M.P.H from which to find to, n, and T corresponding :

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This comparison is interesting because it shows that the results obtained by the application of the theoretical laws are approximately the same as obtained in actual operation.

In most cases we look for much greater variations between the theoretical and actual results than shown above, because the procedure thus far only corrects for lengths of divisions, number of trains and train speeds. Other conditions, such as difference in track facilities, characteristics of motive power, supervision, etc., have their effect which we hope can be similarly analyzed and closer comparisons drawn between various methods of operation.

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