Victorian Institute of Engineers.
VERTICAL CURVES ON RAILWAYS,
t~ Paper Read by MR. W. R. RENNICS, 9th April, 1897.
YV
V~TITH sudden changes in grade in a railway line there are great variations in 1 the normal condition of tension in trains as they pass over, resulting sometimes when the couplings are slack in severe jerks, and in breakage of Ceuplings, with the accompanying danger to life and property. How to ease off these changes of grade by the introduction of suitable vertical curves so as to reduce the jerks to safe limits, or to entirely remove them, is an important gtiestion. As far as the author knows, this subject has been very much neglecte i I y railway engineers, and Wellington, in his valuable treatise on " The 'ecnomic Theory of Railway Location," has treated the subject in an entirely erroneous manner, as will be shown later on. The following paper is an attempt to elucidate some of the simplest cases occurring in practice, and the author does not claim that all his statements are mathematically correct, but that they are near enough for all practical purposes.Excluding bumps in station yards when shunting, all jerks are due to slack couplings, and if latter cannot be avoided then jerks in station yards can only be prevented by care on the part of train men ; Whereas those which occur on the line between stations are capable of being reduced to definite limits by the introduction of suitable vertical curves without that ' will be considered are
judgment
a occurring between stations. They occur under jerks tWO quite distinct sets of conditions, viz. :—(1) While under steam in sags ; (2) 'he running without steam over summits—and are in each instance induced by e presence of a sag in the line. A sag is a change of grade or a hollow curve, in
„Which f a en óg
a y be all rising or all falling, or a mixture. A summit is the Assume a train of uniform weight per lin. foot with slack couplings rolling over any Part of the road without steam. The velocity resistances--being the same per ton for each truck—will have no tendency to make one car travel faster or slower than another, or to alter their relative velocities in any way, and can therefore have no effect whatever on the jerks, and is therefore neglected.
(1) Assume the train roiling without steam over a sag in the line. Gravity acting on the care on different grades will bring the Whole train into compression if acting for sufficient time When it rolls over the sag and gets on to a grade of uniform rate it will roll on with the buffers in contact, there being no force within the train itself to bring it 'nto a state of tension except the re-action of the buffer springs, which will be
°cry small. Consequently, when rolling over a sag without steam, there can be he jerk worth taking notice of. (2) If the train, then, strikes a summit the head of the train is accelerated by gravity, and, owing to the slack, acquires a slightly higher velocity than the cars in the rear, which have their velocity thus suddenly increased when all the slack in front is taken up. This sudden increase in velocity
!felt as a jerk, and is necessarily greatest for the last car other things being equal.
Therefore, if a train rolling without steam, but having slack in it, strikes a summit there must be a jerk. (3) Assume the train steaming. When over a sag the action of gravity on cars on different grades still has exactly the same tendency to produce compression. If the tractive force of the engine be sufficient to over- come these compressive forces the whole train will be kept in tension, and there Can be no jerk. If somewhat too small to effect this, the front portion of train
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will be kept in tension and part of the rear will come into compression. As the train then steams on to the grade of uniform inclination the compressive force
'
will cease, and the traction of engine will draw oat the train to its original tension' but in doing so will give the head a higher velocity than the rear cars, which wilt thus have their velocity increased suddenly, the last car receiving the greatest increase other things being equal. Had there been no tractive force there would have been no jerk. Also, as iwe have just seen, if the tractive force be suifie't ently great there can be no jerk, but as it gets less, jerks occur, so that there roast be some particular tractive force which will produce the greatest jerk under given conditions of line and train. It will be shown that the greatest jerks occur when the tractive force is just able to keep the front half of the train in tension, and leaves the rear half to come into compression. Under these conditions it is seen that the jerk depends on the tractive force of the engine, which is subject very much at times to the control of the driver, and is at other times limited by high speeds. (4) When steaming over a summit with the train in tension there can he no jerk, because the increase in velocity in passing over is gradual and uniform throughout the whole train. (5) When steaming over a summit immediately preceded by a sag with part of the train in compression, the effect of the change of grade over the summit is to increase the traction of the engine by the additions' effect of gravity on it and ondhe front of the train as it passes over, and so to mage the front portion while drawing out the slack acquire a higher velocity than mull engine had continued on a uniform grade, so that the effect of the summit is t0 increase the jerk due to the sag.
COMMENTS.—From the preceding it is evident that there are several ways of manipulating the train so as to prevent jerks in passing over changes of grat{e (A) When steaming over sags the train may be kept in tension either by the engine exerting sufficient tractive force at the head to keep the whole in tension' or by the guard applying the van-brake at the rear to make good by a retarding force at the rear any deficiency in traction at the head, or the driver may nearly shut off steam on leaving the sag and draw out the slack gradually. In any case, the use of the van brake is beneficial. (B) When rolling without steam over summits if the brake be applied to the engine as it rolls over the summit it will prevent its increase of velocity, and so prevent jerks, or if the brake be applied in the van before reaching the summit it will bring the train into tension and so avoid the jerk. (C) The changes of grade may also be so improved by suitable vertical curves that without requiring the driver, or guard,to make any endeavour to reduce it the greatest jerk that can possibly occur under any given conditions cannot exceed some fixed limit ; but assuming that the driver steams fairly over the sag, or rolls over summit without altering his lever.
In the following investigations all sharp sags and summits will be assumed to be long enough to take the whole train and the opposing grade long enough for the jerk to occur before the head of train strikes any other change of grade. All curved sass will be assume"
least, as long as the train itself followed as above by a sufficient length o, uniform grade to take the jerk before head of train meets any other change, and cur ed summits long enough to take the jerk. The engine, when steaming, will v be assumed to be exerting, a c natant tractive force from the time it enters the sag until the jerk occurs. In these calculations—
L = length of train in feet.
1, = portion of train in rear of train.
12 = remaining portion in front.
w = weight of train in lbs. per foot.
T = tractive force of engine in lbs.
G = grade of intersection expressed in English method, viz., 1 in G.
r = " rate of change of grade" of a vertical curve in feet per chain' chord,
(Radius of curve — All being in same units of length.)
s =
ratio of maximum possible amount ofisiack in train to the length cit e train or the average amount per ft, run of train.W = weight of last car in tons, 2
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VERTICAL CURVES ON RAILWAYS. 3
ASSUMING TIGHT COUPLINGS.—To construct the diagram of compressions or tensions at any point in a train when rolling without steam over any sort of a sag. In Fig. l let A D represent the length of the train. Let B be any point in train. For convenience suppose the head of train D to be on ailevel. Let D E be the
c
algebraical sum of the horizontal components of the weights of all parts of train, and let E F be the similar sum for the part A B of train. Join A E and make F C parallel to A E cutting vertical through B in C. Then B C is the stress at B clue to the influences of gravity on train. For, as the couplings are close, all Parts of train are moving at any instant with the same velocity, so that the resulting accelerating force must be uniformly distributed throughout the train.
•' of the total accelerating force D E acting on the train the portion taken up by A B is B G. But the horizontal component of gravity acting on A B is F E = G C. .. the difference between G C and B G must be the force which the Part A B exerts on the rest of train. If G C be greater than B G this force is a compression and if less it is a tension. As B is any point, the locus of C is a Curve giving the state of tension or compression at all parts of the train for one particular position on the line. By taking the train on different portions of a sag, a curve may be drawn giving the maximum compressions or tensions that can occur at every point while passing over the sag. The above is a general statement
applicable to any sort of sag or train, but in the rest of this paper the sags will be
considered as consisting of two straight intersections or of a single circle and the train of uniform weight per foot, in which cases C will always be above A D.
SHARP SAGS.—If the sag be a sharp one, viz., the intersection of two straight grades whose angle of intersection is the same as a level meeting a grade of one in G. and if B be taken on the point of intersection, the compression at B will then
VERTICAL CURVES ON RAILWAYS.
be a maximum. Let w be the weight of train in lbs. per foot run and L its length in feet. Then in Fig. 1
w1, w1,
EF=— and ED=—
G G
so that F and E coincide.
Then in Fig. 2
The triangles A D E and D B C are similar.
BC DE BD DA
BD. DE 12 w1, w 11 12f (1).•. BC
=
D A LG LG
L
w L.(2) which is a max. when 11 — 12 = — and its value is then
2 4G
Join A C. Then ordinates from any points in A D to the lines A C and C 17 are the compressions at those points. If different points in train be taken i,,13 succession at the point of intersection, i.e , taking varying lengths l , the point C will describe a curve, as shown dotted. giving the max. possible compressions which can occur at any point in the train. As D E varies as 11 it will be seen that this construction for getting C is a modification of a well-known method of describing a parabola. .. the curve A C D is a parabola whose axis is a vertical through the centre of A D.
Example No, 1.—Taking a train 400 ft. long, including engine, and weighing 1680 lbs. (4 ton) per foot run rolling without steam over a sharp sag, whose angle of inclination is the same as a level meeting 1 in 50.
Then L — 400, w = 1680 lbs. and G = 50.
From equation (2) the max. compression is in centre of train and when centre is just over the intersection of the two grades, and is then
w L 1680 x 400
— 3360 lbs.
4 G 4 x 50
Thus in Fig. 3, L K = 3360. The max. compressions at other points varying as the ordinates to a parabola ; thus at the quarter points the max. possible compressions are 4 x 3360 = 2520 lbs., or from equation (1) the max. compression 100 ft. from tail is-
—Fic.3—
o
4
for.0 , feet
I
i.e, ten wil foi cto
w 1, 12 1680 x 100 X 300
L 100X50
, 4
K
00 - ----~
— 2520 lbs,
VERTICAL CURVES ON RAILWAYS. 5
CURVED SAGS.—In a vertical curve, if r be the rate of change of grade in feet for chords of 1 chain, i.e., in Fig. 4 if A B and B C are 1 chain chords C D = r
feet.
lengtb
It O be the centre of the circle.
CD OB The angle DBC=angle BOC and —=—
CB BO 2
CB
C D = — and as B 0 is radius of curve.
BO 2 C D varies as C B.
2 2
C B chord
(The radius of curve B O = — = all being in same units, say feet.)
CD r
!•e•, the rate of change of grade on the same curve varies as the square of the lonRth of the chord taken. In the following formulæ the rate of change of grade Will be r feet per chain, and if they are to apply to chords of 100 ft., r in
1002 r
totrnulae must be replaced by — 2.296 r, or a curve whose rate is 1 ft. per 662
°bain has a rate of 2296 ft. per 100 ft., ora curve whose rate is lf t. per 100 ft. has
--fI
G.S-
a rate of •4366 ft. per chain. In a curve whose rate is r feet per ch., the rate of d C D
en in Ant 0 asions n that od of
!Meal ghing angle
entre
ng 0
;Bible
salon
•'• il
(7) train CON s
equal (8) (9) there Esa
and it that eprtp eyrve muet
Es; and t.
1'h,
teusit
or a
resyll
aa is CII (10)
(il) and 1 (12)1 If tan (13)
~7;u•~ i z 5~e ,Jt<. f~\~t.SOS:'`~~t urnre-n rr- Ì:`._ ,.wr n.t.~açc:_z •
6 VERTICAL CURVES ON RAILWAYS.
Then in Fig. 5—
D E = '000115 r w L2
L 2 2
and EF= J 00023rlwdl = •000115rw(L-1ä)
DA L
and BC=CG—BG.EF—BG BC= .000115 rw(L22 2 —I—L11)
= .
000115r [(L + 12 ) (L — 12) — L 15]but L-12 =1,
BC'=•000115rw[(L+12)1,—L1,]
BC—•000115rw11 12 which is a max. when I1 = 12 = - L
2 (4) and max. value of B C is then •0000287 r w L . 2
Taking different values of 11, the point C will describe a curve A C D giving the compressions at all points of the train, which are the maximum which can occur at these points. This curve is a parabola.
Example No. 2, taking the same train as for sharp sag, 400 ft. long' 1680 lbs. per lin. ft., and suppose the curve to be such that tangente from head and tail of train intersect at a grade of 1 in 50 meeting e level. The rate of change of grade of this curve will be •2178 ft. pet chain, and the max. compression will be in centre of train and from equation (4) will be •0000287 r w L2 = •0000287 x 2178 X 1680 X 4002 = 1680 lbs. The compressions at other points in train will 1,e as the ordinates to a parabola, so that at the quarter points the compressions will be 1260 los., or from (3) •000115 X
•2178 x 1680 x 100 x 300 — 1260 lbs. It will be noticed that these compressions are just half those in example No. 1. which would have occurred at same pointe had there been no curve.
TRAIN STEAMING OVER SAGS.—Next assume a tractive force at the head
of
the train. The acceleration due to this is uniform throughout the train, but the tension varies uniformly from zero at the tail to T lbs. at the head.
(5) .•. at any point as B the tension is — T 11
SHARP SAGS.—In sharp sags the resulting tension in train at B is (5)—(1) L 11 T 12 11 w
(6)
L LG
r
change of grade for chords of 1 foot is — — •00023 r.
662
Let L be length of train and assume the whole train on the curve. Assume 09 short length d 1 at a distance 1 from head of train, the horizontal component 01 gravity acting on d 1 is •00023 I r wd 1, and sum of the horizontal components ie
fL •00023
whole train isDE —J 00023rw1d1= rwL2 =•000115rwLI 2
1 2 BG DE BA DA
BADE •000115rwL I, 2
BG= — •000115 r w LI,
(3)
7
me an9 tent er tents ín
VERTICAL CURVES ON RAILWAYS, f at any point the resulting tension is to be nil.
11 T 12 11 w
—
oL
LG 12w
(7 '• T = ~ and 11 will beiin compression.
tran
and consequently no jerk 1, must he- o1and
be 1 no compression
ust, at least equal L w~
G
T = - Lw
L w
G . G =
T
here is toe be no jerk h the tractive force fome butbseeat least over a
from equation sag.
L w 400 X 1680
— 13440 lbs.
G
-
arid
50
in Fig. 3 if D J = 13440 lbs. then D J = 4 times K L, but D J = 2 K M, and t D J= 4 K L, then KM
-
2 K L and ALD being a parabola, A J must be a , at
point due to the traction DJ and the ordinate to thecurve to LtD ais nthe
mr
e is the resultingtension. e tWhen A sag, is a tangent to curve, fromwhole train ust be in tension.
and the rear 4.—If
train will be in compression.
A J must cut the curve 13440
Illus, if D J (Fig. 7) _ _ 6720 lbs. taking 11 -- 100 ft., the resulting 411114 n at this point will be from (6) 2
11 T 12 11 w 100 X 6720 300 x 100 x 1680
L L G
-
400 400 X 50
or =1680-2520=-840
tesylt ng tension at centre r lbs. i
ns
e d of a tension. Also taking 1, = 200, the is
2u0 x 672i 2u0 X 20u x 1680
— 3360 — 3360 = nil.
400 400 x 50
aa v
s evident from Fig. 7.
'u VED BAGS.—In curved sags the tension at any point B is (3) — (2) (10) 13. T
--•000115rw1112
(11) and if this is to be nil. T = •000115 r w 12 L
and 11 will be in compression, so if there is to be no compression in train (12,11 must = o and 1, , L
and T must = •000115 r w 1;.
If tangents to curve from head and tail of train make an intersection a
tan 2T
a =_ '00023 L r = - wL
(13l wL
or the angle is equal to an intersection of a level meeting a grade of I in - 2T g the
occur long' gents ing
;. per n (4)
The that 15
%
usions ointe d of
the
(8)
(9)
8 VERTICAL CURVES ON RAILWAYS.
from (9) and (13) it follows that with the same tractive force at head of train the angle of intersection between two tangents from head and tail when et curve may be twice as great as the angle of intersection for a sharp sag withe0 there being a possibility of a jerk.
In am particular place it is almost impossible to predict what tractive force thG engine will be exerting.
Example No. 5.—The same train as before, but steaming over a curved sag. tb0 rate of change of grade of curve being •2178 ft. per chain as in example No. 2.
If there is to be no jerk
T must — from (12) '0001"5 r w 12
)0 115 x •2178 x 1680 x 4002
= 6720 lbs.
Ema___-_ 400e--- — FIC.7—
r•
50
- t7Ó-'l
----- 400' --
400 --'—_---~.
—
Flc.8—In Fig. 6 making D J = 6720 lbs. = 4 times K L, A J must be a tangent Go A L D as proved in example 3, and whole train must be in tension.
6720
If D N (Fig. 6) _ — = 3360, the tension 100 ft. ftom tail will be 2
11
(13) from — — '000115 r w 11.12 100 x 3360
•000115 x '2178 X 1680 X 100 x 300 400
= 840-1260 — — 420 or a compression of 420 lbs. instead of tension, and at 200 ft. from tail the tension will be
200 X 3360
'000115 x '2178 x 200 x 200 400
= 1680 — 1680 = nil. as is evident from Fig. 6.
Example No. 6.—lf in example No. 2 the intersection of grades had been takeil l 1 in 25 instead of 1 in 50, the rate of curve would be doubled, and in this example T would be doubled or D J in Fig. 6 would be 6720 X 2 = 13440 lbs. to just keep the train in tension
,
i.e., comparing this with example No. 1, it is seen that with the engine exerting the same tractive force the grade on which head of train stand may exceed that on which tail stands by twice as great an angle if there be , 1 curve between them as if grades had continued to form a sharp sag and stsll prevent all possibility of a jerk.LoosE COUPLINGS.—Now although the assumption so far has been that the trata is close coupled it has been only as a matter of convenience because when doge coupled the changes of tension and compression take place almost at once. W11 however, the couplings are loose the compressive forces require a little time 1f bring the buffers in contact in that part of the train subject to compression.
head of when p0 without force the
sag. the No. 2.
;ent to
I of e
akea aa mple T it keep 6th the staa&
be e d stsll ie Mile i c1000
wheu me
f
n. I
VERTICAL CURVES ON RAILWAYS. 9
the time taken to pass over the sag be sufficient all the buffers in that part will be in compression. When on vertical curves even if the curve be not long nough to bring all the possible buffers in contact the error made in assuming they are so will be small in any ordinary cases where the jerk is of any conse- quence. But making this assumption we shall not be able to compare the jerks resulting from widely different changes of grade on the same train. This, however is not the object of this paper which is rather to devise rales by which curves may be introduced so as to limit the maximum possible jerk to definite limits under varying lengths of train.
JERKS STEAMING OVER SAGS IN A TRAIN WITH LOOSE CouPLINGS.—.Let the ratio of total slack to total length of train be S. Assuming the rear of the train in compression the effect of compressive forces has been to shorten the rear, of the train by a length sl,. The traction of the engine after getting off the sag on to the straight mast move the length 12 a distance S I, before the last car can be picked up. The cars in 12 have to be moved a distance S 1, and cars in 12 each less and less until last car but one is moved a distance equal to slack between it and last car, Assume that the distance the train as a whole moves with respect to last car is j s 1. Now the train is rolling along at this timeiwith the initial velocity and the cars in the rear are separated from the engine by the slack and their velocity is being reduced by the velocity resistances while the front portion is also being retarded by velocity resistances, but at same time is being accelerated by the engine. Now the velocity resistances being the same for both portions can have no effect on their relative velocities, except that the front portion having its velocity slightly increased will have a very slightly greater resistance, but this being so small is neglected. Consequently the additional velocity imparted to the last car when picked up by the front of train is the same as if the whole train had started from rest without frictioi. in the given conditions of compression and tension and with the engine exerting the same tractive force, and the resulting increase in velocity to the whole train when the last car has just been picked up will be the same as ifitrain had been close coupled and the tractive force had acted on it for the same time. Consequently this increase in velocity of the train as a whole is the increase which is suddenly put into last car in the form of a jerk. The traction of the 'engine 'is ,T lbs., and ithe weight of train is w L lbs. and let t be the time in seconds required to draw out the slack.
T
The accelerating force on each lb. of the train is — and in t wL
gT
seconds it would move the train through a distance 4 — t2 feet where g = the wL
acceleration of gravity and — 32. This distance must equal } s 1, before last car is picked up.
1 3s1, w∎L
(14) 64 T
The increase in velocity under the influence of the accelerating force in t seconds is w Lt feet per second
g T I3s1,wL _ 148Ta1,
(15) — w L ✓ 64;T w L
which is the sudden increase in velocity of the last car.
g T t2 wL 16T t2
. — sl,
wL 3 sl, wL t2 -
64 T
T wL
.,,.. ^l 6 .T'-7 {/!.']i 1 `rRr ._ ~. •vï,a¢~, A-c w! K l': s: ]L~ yHfih2y.~~ F-_7- ia— .y~u. _ u ,
10 VERTICAL CURVES ON RAILWAYS.
Now the pull on the draw bar which produces this increase in velocity depends on the nature of the springs in the draw gear, their initial stress, the method of attachment to under frame ofcar, etc., but in any case it must vary approximately as the increase in velocity and as the weight of the car. Assuming that it varies directly as the weight W tons of the car and as the increase in velocity it must vary as
(16)
W ~ 48w
L 1,
SHARP SAGS.—In sharp sags from equation (7) T = increase in velocity in feet per sec. is
(17) 48sl,12 w (48s1, 1,
4 wLQ ✓ L
l~ w~
G
a
and pull varies as W 1 48 s 1, 1,
,/ ì
awhich is a maximum when 1, = 1, _ or max pull on last car varies as (18)
w J 12 L
Example No. 7. Taking same train as in previous examples but having slack couplings and steaming over sharp sag of 1 in 50. As in example No. 4 take T = 6720 lbs. then tension in centre of train is nil., and 20G ft. in rear is subject to compression.
The intensity of the accelerating force in the train is
T 6720
1 lb. on each lb. in train or the acceleration is •ul of
WL 1680 x 400 ' 100
gravity = •01 g = •32. Taking s = :02 the distance this accelerating force has to move the train is i a 1 = I X 02 x 200 = 3 ft., and from well known equation connecting accelerating force, time and distance, viz.,
a — (
ft. we have where s=aft.3 X •32 t'=•16t' t2 = 18.75
t = 4.33 seconds.
or from equation (14)
t=
1 3 s 1, w L=
1 3 x •02 X 200 x 1680 x 4004 64 T 1/ 64 x 6720
= s/ 18.75 = 4.33 seconds.
and the increase in velocity is from old equation V = ft.
= •32 X 4'33 = 1.385 ft. per second.
or from equation (15)
_ 148 Tsl i 1 48 X 6720 X• 02 X 200
w L ,/ 1680 x 400
= 4 1.92 = 1.38 ft. per second.
or from (18) the max. increase in vel is- 112eL= X12 X'02 x 400
50 4 1.92 = 1.38 ft per sec.
Example No. 8.—Had the grade of intersection been 1 in 100 the tractive force required to keep half the train in tension would have been from equation (7)
T 12 w — 200 x 1680 _ 3360
G 100
)f
y
.*~'JR'IF..í'74~/»'-YR'.t`LTRiRi?R~S~`?~717fiC
VERTICAL CURVES ON RAILWAYS.
and increase in velocity would be from (15)
I 48 T s li 1 48 X 3360 X •u2 x 200 = ✓ 96 = 1 ft. per sec.
w L 1680 x 400
-r from (17) increase in velocity when 1, = 200 is j 48 x •02 x 200 x 200 = J 96 = 1 .1 400 X 100
or from (18) max. increase in velocity is
I 12 sL 1 12 X '02 X 400 = V.96 = 1
y/ (,} V 100
CURVED SAGS.—In curved sags from (6) T = '000115 r w 1, L increase in velocity is from (16)
(19) V
148 X '000115 r w 1, L s 1, = X00552 r 1, 1, 1 w L
and pull varies as W V •00552 r 1, 1, s which is a max. when 1 1 L
2 1 2 —
max. pull varies as W V •00138 r L2 s (20) or as W L V .00138 r s
As the pull varies as V r it is seen that comparatively large variations may be made in r without materially altering the intensity of the pull on draw gear.
Example No. 9.—Taking same train, but having slack couplings, steaming over a sag, as in examples Noe. 2 and 5, having a curve whose rate is '2178 ft. per chain.
When the tractive force is just able to keep 300 ft. in front of train in tension -- and leaves 100 ft. in compression, the increase in velocity is from (19)
V •00552 r 1, 1, s
= V '00552 x '2178 x 100 x 300 x '02
= V •721 = •851ft. per sec.
When the tractive force is just able to keep half the train in tension the increase is in vel. is from (20)
LV.00138rs
= 400 V •00138 x •2178 x '02 '981 say = 1 ft. per sec.
or the same as in example No. 8 where the intersection is 1 in 100, while in this example the intersection of the tangents from head and tail of train is 1 in 50, and comparing examples Nos. 6 and 8 it will be seen that the tractive forces are the same, viz., 3360 lbs., and these are the greatest possible increases in velocity in these sags assuming a uniform tractive force daring whole passage of train over the sags.
LIMIT TO JERES.—It is proposed that rates be fixed for curves so that the max. possible pull cannot exceed definite fixed limits. Thas, suppose we limit the
jerk to that which occurs when a 16 ton truck has it velocity suddenly increased by 1 ft. per sec.
SHARP SAHs.—Equation (10) becomes .5 W ~G = 16 and as 16
1
1
12 VERTICAL CURVES ON RAILWAYS.
~ L = 2
V
G
or G = 4or G = 4
. the greatest intersection that would not require improving would be that of a level meeting 1 in G when G =
4 ,L being the length of train in feet.
Example No. 10.—For a 400 ft. train an intersection equal to level meeting 1 in 100 would not require improving but anything greater would.
CURVED SAGS.—With the same limit from equation (20), when s = •02 we get that
•0052WL VT
•0052 L VT
must not exceed 16 and W = 16.
must not exceed 1 V r must not exceed ,00521
L or 1 L J 37000
L2
Example No. 11—When L= 400 r mast not exceed •23 ft. per chain.
L = 800 r must not exceed •058 L = 1200 r must not exceed .026
etc. etc.
JERKS OVER SUMMITS RUNNING WITHOUT STEAM.
Assume that all thebuffers are in contact on account ;of having just deft a sag andfor convenience that the rear of train is on a level.
The forces taking out the slack are the accelerating forces of gravity on the portions of train on the down grades.
SHARP BIIMMIT.—The accelerating force in a horizontal direction of a portion] on w1
the grade is —. Phis distributed over whole train L makes the accelerating force G
w1 1
— on each lb. which acting for a short time S t gives the train an GwL GL
gl$t
additional velocity of feet per record.
• LG
Where g is accelerating force of gravity and = 32 321 $t additional velocity in feet per second —
281 LG 3V
Bat $ t =-- seconds, where V — initial velocity in miles per hour = --ft.
per. sec' 3 V 2
32 X 21$ .. Additional velocity in ft. per sec. _
3LGV 21318
LGV
as 1 increases from o to 1 the total increase in vel. (ft. per sec.) is 1
21.3 /Idl ---=10.612
LGV J LGV
r must not exceed
o
13
VERTICAL CURVES ON RAILWAYS.
10.6 12 LGV
Z 2 a a 3 3 3
10.6 x•354 S G L V LGV
makes the train move through a distance in ft.
10.612 2
81 —
_7'11s S1 LGV 3V LGV . as 1 increases from o to 1 the total distance in ft. moved through is1
7.1
12 d 1 = 2.37 12LG VZ , LGV2
o
This distance must = 2
L the distance the train as a whole moves with respect to last car before it is picked up.
2.371$
__
SLLG VZ 2
13 = S L2 G V2 = .211 s G L2 VZ 4.74
â 3 d I= •595 S G L V the increase in velocity
3.775 Si L V } G
and as before the pull on draw gear varies as W X increase in vel.
S 3 7
3775W S L V
But at any instant this increase of velocity acting for short time
8
t which2
8 l
3
V
(21) or pull varies as
G
If 1 becomes greater than L, i.e., if tail of train passes the summit before the jerk takes place it may be investigated on similar lines to foregoing.
Example No. 12.—If L = 400 ft. and V = 20 miles per hr.
and a =
•02 8 = '0737 and pull varies asIf W = 16 tons and pull is limited to 16 as before.
.2783 L} V must not exceed 1 1
and G must not be less than •0215 L V, i.e., G must not be less than 172.
of
tin get
ie
:e n
•2783 W L1 Vi G
a- s
G
1
G}must not be less than •2783 Li V}
velocity In moving a distance 1 on the curve a short length L' will increase the of train by
1
•0049 r L' / 1 d 1 _ 00245 r L' 12
LV f LV
o
the effect of all parts L' in 1 is to increase the velocity by 1
'00245 r 12 d 1 __ 000816 r 1'
LV LV
o
2 1'
3V will make the train At any time this velocity acting for short time t'
move a distance in feet of
'000816 r 1' 2 1' •000544 r 1' l'
14 VERTICAL CURVES ON RAILWAYS.
CURVED SUMMIT.—Suppose r the rate of change of grade in ft. ner chain, and that rear of train is on level and a portion I is on curve.
—
F c.g=—
A very short length of train L'
•00023 r 1 w L' lbs.
This force distributed over 00023 r1 w L' on each lb.
wL
short time t' 2 l'
= 3V an increase in velocity of
•00023 r1L' x 21' g
L 3V
__ •0049 r L' 1 l' LV
at head of train produces an accelerating force of whole train produces an accelerating force of 00023 r 1 L' which acting for a very
L
where 11 is distance moved in short time t' will give
LV 3V — LV2
. as 1 increases from o to 1 the distance becomes '000544 r
l' l'
L
V'
which is the distance centre of gravity of train car.
•000136r14 S L
L V2 2
14 = S L' V2 _ 3676'5 SL' V2
•000272 r 7'786 Si
Lz VI
=
ri
'000136 r 14 L V2
has moved with respect to last
sin,
e of of ery ive
ity
in
rew:37,,k.,rc7ran 57.4MOMFx .y,. x:r r met ri c <;ss *5^' n~47iu f iL1~~7 .;.:. >..
VERTICAL CURVES ON RAILWAYS.
Increase in vel. 000816Vr 13 becomes L3 2 3 2s
•000816 r x 7186 S L V
L V r4
it/ 2 _ •386 r S L V and pull on draw gear varies as
1 s â
(22) •386 W r4 S L V
15
It will be seen that as the pull varies as r very considerable changes may be made in r without material alteration in jerk. Also as jerk varies as S an alteration in the amount of slack in train has a much greater influence on jerk than a similar alteration in r.
If 1 becomes greater than L the increase in vel. may he investigated on similar lines to the foregoing.
I f jerk be limited to 16 and W = 16 tons, •386 W r 4 S 4 L 2 V Z must not 3 3 3
exceed 16 and •386 r S L V must not exceed 1
r$ most not exceed _ _ 1 must not exceed 2'59
•386 L L 4 V L S V
45.
2 ;t 2 L S V
3
Example No. 13.—When S = 2 S = •u532 and from (221 pull varies as
r b S ' 205wrL V 2
If L = 400 and V = 20 miles per hour and jerk be limited as above r must not exceed •t 88 per ch.
etc. etc.
GENERAL.—From tie foregoing calculations it is seen that careful driving may almost entirely eliminate jerks and that the effect of a given alteration in the amount of slack in trains has proportionally a greaten effect in reducing the jerk than a corresponding decrease in the rate of curvature so that it should be kept as low as possible to avoid expense in earthworks in construction of line and damage to trucks in shunting in station yards,
COMPARISON WITH WELLINGTON'S RULES.—For a sharp sag we have seen WL
(8) that when T= —there can be no jerk. Now suppose G, the grade of G
WL
repose for any particular velocity when T= —the engine is just overcoming G
velocity resistances. Conversely if the engine e be assumed to be just overcoming velocity resistances the intersection may be made equal to the grade of repose for
WL that velocity without the possibility of a jerk and G= -
T
For curved sags we have seen that (11) when there is no possibility of a jerk G the grade of intersection between tangents to curve from head and tail of train
WL
maF= or the erades on which the head and tail of train stand may be twice 21
st
r must not. exceed
16 VERTICAL CURVES ON RAILWAYS.
as steep on a curve as on a sharp sag. Now if the engine be exerting more tractive power than is sufficient to overcome velocity resistances these angles may be increased and if exerting less they must be reduced if there is to be no possibility of a jerk. Wellington's rule on P. 363 is "To obviate all danger of the train crowding upon the cars in front without the use of brakes at any sag in a grade line: The rate of grade on which the head of the train etandr must in no case exceed that on which the rear of the train stands by more than the grade of repose of the last car, otherwise the latter will crowd up upon the train." It will be seen then that this rule is correct only when the engine is steaming over a sharp sag and is just overcoming velocity resistances no more and no less. Re assumes also that the same rule applies when passing over curved sags, whereas we have seen that the rate of grade at the head of train may exceed the rate of grade at the tail by twice the grade of repose of the last car (assuming the last car has the same grade of repose as the rest of the train seeing that the train is to run "without the use of brakes") and Wellington's own method as applied to a sharp sag if applied to a curved one will result in showing that the grade between head and tail of train when on curve may be twice that of a sharp sag. On this assumption, however, he bases a rule for vertical curves given p. 365 which evidently gives curves of twice the radius actually required under these conditions and consequently of twice the length required. The sentence following the rule given is " With half this length of curve which is considerably, more than is usual in laying out vertical curves all danger of taking out the slack in the front half of the train where there is most danger of breaking in two will be avoided." Now it appears to the author that it is in the rear half of the train that jerks are most severe for the reasons given in the paper and that the greatest possible jerk is on the last car and occurs when the rear half only is subject to compression. Wellington would appear to be quite satisfied with this condition. Also we have seen (8) that tL there it no possibility of a jerk when steaming over a sharp sag when T=-- Now as the velocity gets greater, the traction engine can exert gets less and there- fore G gets greater or the angle of intersection gets smaller, whereas Wellington, p 364 §430 says that as velocity increases so may the angle. He entirely neglects jerks which occur when rolling down long inclines without steam.
dF to cc Il ti b t] tl R
s t
0
P c
.r:nrmw7,1W,1 a x, =T:r r7Rrr.4x- , xZVe ss ls,4- io` n- ~. 3
7
Dicussion on Vertical Curves.
Mr. Perrin (a visitor) asked if the slack mentioned in the paper (.02) was deduced from actual experiments.
Mr. W. R. Rennick, in reply said that value .02 was got by actual measure- ments, from a long train of empty waggons.
Mr. Stone enquired if the action of the springs had been taken into account ? It seemed to be a point that when the steam was shut off at the bottom of a sag a considerable amount of energy might be stored up in the stretched springs. Be the jerk. Did it only include the jerk in tension or the jerk at the bottom due to buffers coming into contact ? As far as he could see Mr. Rennick had only included the jerk of the tension on the drawbar.
Mr. Rennick had purposely left out the question of the effect of the springs of the draw gear as it would have made the whole matter very complicated. Also When the trucks bumped against oue another, he had left out the action of the springs in the buffers.
Mr. Stone asked if Mr. Rennick could say that the action of the springs was so small compared to the other forces that it could be ignored. He had noticed in travelling in long trains that, when the steam was shut off, there was a jerk. It might not be sufficient to cause any trouble but it added to the small bumps which Mr. Rennick's formulae did not regard as permissible.
Yr. Rennick had assumed that the effects of the spring could be ignored in comparison with the other forces. He had not taken into consideration the effect of shutting off steam while running over a sag, but had assumed the train steaming over the whole sag. The driver was assumed not to alter the engine whilst run- ning. Should he shut off steam on entering the sag and then put it on again on leaving, the whole train would come into compression instead of a portion only and then in steaming out of the sag there would be a terrific jerk ; that would be bad driving which he had not assumed.
Professor Kernot had not been able to thoroughly check Mr. Rennick's deduc- tions. Mr. Rennick, on page 3 of his paper speaks of the horizontal components of the weights of parts of train. Now weight being a vertical force, can have no horizontal component. No doubt railway men would understand exactly what he means which is the horizontal resolved patt of the weight the other resolved Part being a normal to the grade. There was one point that should be noticed, i.e. that a locomotive ran a good deal heavier than a carriage. In the locomotive there was a lot of mechanism to be kept in motion. There were a large number of working parts, all of which acted to a certain extent as brakes ; so that when an engine was running without steam it did not run so freely as the train which came down on the locomotive and it was in a state of compression. This point might have been referred to. He had been too busy to look into "Wellington" to- investigate Mr. Rennick's statements. Wellington was a good writer and a man who had gone thoroughly into this matter ; but no one would maintain that he was infallible. Regarding the general running of trains. In the transactions of of the Institute of Civil Engineers some time ago a paper from Japan was read as to the application of a kind of seismometer which recorded on a diagram all jerks in any direction. It was a simple instrument that could be fitted to any engine or carriage. The Dynograph was rather too combersome for this work. Was it necessary to have such a large amount of slack (8") in goods trains for the purpose of making engines start easily? Eight inches appeared too great. Passenger trains of course were tight coupled. The half of eight inches seemed to him to be sufficient.
Mr. Rennick said with respect to the Professor's criticism re the horizontal components that his paper was written principally for railway men of course and that in the early part of his paper he had disclaimed being mathematically exact in his statements partly to avoid having to use complicated expressions. In the Victorian Railway there was Oins or sins of slack in trucks, about 22f long buffer to buffer, but he believed in America they had 4ins. to 6ins. of slack in. trucks 35ft to 40f t long.
The discussion was adjourned to next meeting.
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DISCUSSION on "VERTICAL CURVES on RAILWAYS."
T TH
E President (Professor Kernot) remarked that although this paper . had been before the Institute for some time no flaws had been found with Mr. Rennick's mathematics. It was a subject the exact mathematical treatment of which was very difficult. The actual experience gained seemed to bear out the calculations made in this paper. If that was so, it was very satisfactory. It had often appeared to him that a convenient way for arranging the curves on a railway would be to adopt some uniform change per chain. They would get what would be polygons in reality, and the curves could be set out with very great ease. It did not seem right to make very sudden changes of grade at any given point. He did not know if anything had come of a suggestion of Mr. Moline's a long time ago re putting in curves 10,000ft. radius.Mr. Rennick, in reply, said that the result of the curves already put in had been very satisfactory. The more experience he had of vertical curve and the effect of train length, etc.. on the jerk, the more satisfied he was that the conclusions arrived at in his paper were substantially correct.
The curves recommended conformed to the natural features of the country, for where the ruling grade was very flat the train lengths was long and the curves were very easy and long, and where the ruling grade was steep the curves were sharp and short, and the actual lift in a hollow or lower in a summit in either case was not very great, so that the cost of putting in suitable curves when line is being built is small. In his paper he had assumed that the driver did nothing to prevent jerks ; but when a train was rolling down a succession of grades without steam it would nearly always happen that the brakes on engine have to be applied to prevent an unduly high speed. This would keep the train in compression, except when passing over a summit, when part of the rear might come into tension with a jerk, and the limiting value given to (r) on p. 15 would not apply, and at these times the fireman and driver would have very little to do, so that they could be easily expected to apply the brakes when going over a summit, even if they were not required on other parts of the road.
The author would therefore recommend that the same curves be put in for summits or for sags, separated by a piece of uniform grade at least the length of the train, and that they be omitted only when it is certain that the engine will be steaming over summits or running without steam in sags, when travelling either way on the line, or when the change of grade was not greater than given at the top of p, 12. When these curves cannot be put in with the required piece of straight between reverse curves of very much greater radius should be used.
Until lately the construction branch had not adopted Vertical Curves as a system. In some places the sudden change of grade had been avoided by putting in a piece of level about 4 chains long in hollows and summits.
In those cases the jerks had been very much reduced. The North Eastern line had been noted for "break-aways," but since the Vertical Curves had been put in there had been very few. The question of making the grades so that there could be no possibility of jerks depended entirely on assuming a certain tractive power of the engine when steaming, or a certain retarding force at the head of train when rolling without steam, either of which was a most uncertain element, The only positive cure was to avoid Slack couplings.
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