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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 iound for a Division M. miles long operating N, trains per day and it is desired to ascertain what the performance will be if the length of division were changed to M, miles, the number of trains increased to N, 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 di am 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 M, and M, represent the miles per division and T, and T. represent the average road time of all trains and N, 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.

T M1,
That is
T. ./.

Ti=to +

and T:=to'+

Substituting to ior to'

Ti M,

T, M. That is, the values of n will be inversely proportional to the length of divisions.


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

If T, and T, represent the average road time of all trains corresponding to N. and N, trains per day it can be assumed that the delays 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 Ni and T; 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 T, when there are N, trains on the road, as shown in the following table. The expression for T, 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 Aver

age Road Time.

If Ti 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.

to S2

to' Si 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 curye will remain the same




or n2 = ni

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

Ti=to + and T:= to'+

21o'n tv

Ti 2toni + Va Substituting


to for to'
2S,ton, + S2


T. 2S,ton, + S VA 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 hat the cond ons of operation

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 M=31.3 miles, v= 125 trains and S= 25.93 M.P.H from which to find to, n, and T corresponding : Known factors, see Figs. 1 and 4....... P-R R-S

S-V Miles (M) (M.) (M)...


30.7 Number of trains (N3) (N2) (N.). 125


135 Best speed M.P.H. (S3) (S.) (S). 25.93 18.47 17.68 to


.7955 T average road time.

2.850 First derived factors (correction for dis

tance Art. V) Ni


135 to' = (M/M) to and (M./M) to... 1.770 2.754 it' (M/M3) n and (M/M) n.


.5015 T' average road time = to'+

2.905 4.520 Second derived factors (correction for

number of trains, Art. VI) V, and N,


136 same as to'

1.770 2.754 11" = (N/N1) n' and (Nz/N) n.


.498 T" average road time = to" +

2.822 4.535 Third derived factors (correction for

speed of trains, Art. VII) to"" (S2/53) to" and (S/S.) to"

1.207 2.638

1.736 "" same as n".


.498 2.850

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2 11'


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T'" average road time - to t

2.250 4.420 2.850

211" Compare last two lines with the follow

ing from Table 2: 11 from table 2..


.7955 T from table 2.

2.227 4.405 2.850 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.



WM. G. RAYMOND, Chairman;



The work assigned to Sub-Committee No. 3 was the study of the
effect of speed on the cost of operation.
As was done last year the work was divided into three parts:

1. Effect of speed on the cost of maintenance of way.
2. Effect of speed on the cost of transportation.

3. Effect of speed on the cost of maintenance of equipment.
No new data have been obtained nor do there appear to be any data
available that have not been used. Certain tentative conclusions have
been drawn from such data as were available and these conclusions are
given to the Association that they may be proven or disproven by such
experiment or analysis of accounts as the several members may wish, or
be able to make. In the absence of disproving evidence, the conclusions
are thought to be worthy of acceptance.

TENTATIVE CONCLUSIONS 1. Other conditions remaining unchanged, the cost of maintenance of way increases with increased speed of operation.

Exact relation has not been established nor can it be established alike for all conditions of track, roadbed, and traffic.

For minor differences in cost due to minor changes in speed not requiring changes in maintenance standards it may be assumed, until experiment proves otherwise, that the cost of maintenance of way varies with speed of operation as set forth in the report of this Committee in the Proceedings of 1921 at pages 760 to 772. For convenience the numerical values found in that discussion are repeated below in Table 1.



For Passenger Trains For Freight Trains

Straight Track

0. T. M.
Track Laying and Surfacing.
Bridges, Trestles and Cul-


0.0011 CV
0.0010 CV
0.0008 CV
0.0010 CV
0.0022 CV
0.0022 CV

Curved Track Straight Track Curved Track 0.0150 CV 0.0014 CV 0.0180 CV 0.0082 CV 0.0031 CV 0.0082 CV 0.0038 CV 0.0008 CV 0.0040 CV 0.0038 CV 0.0012 CV 0.0045 CV 0.0190 CV 0.0027 CV 0.0210 CV 0.0190 CV 0.0029 CV 0.0210 CV 0.00067 CV 0.0009 CV 0.0009 CV

0.00067 CV

Note-In this table “C” varies with the several items and in each line of the table is the main line cost of the particular item of its tabular line for any particular road or division at an assumed 'or the usual average speed; and “V” is any change in speed under consideration in miles per hour. It must be remembered that the table is not good for changes in speed requiring or permitting a change in maintenance standards.

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