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AN EXPERIMENTAL INQUIRY INTO THE ADVANTAGES ATTENDING THE USE OF CYLINDRICAL WHEELS ON RAILWAYS; WITH AN EXPLANATION OF THE THEORY OF ADAPTING CURVES FOR THOSE WHEELS, AND ITS APPLICATION TO PRACTICE ; AND AN ACCOUNT OF EXPERIMENTS, SHEWING TIIE EASY DRAUGHT AND THE SAFETY OF CARRIAGES WITH CYLINDRICAL WHEELS. BY W. J. MACQUORN RANKINE, CIVIL ENGINEER. “ Sacadas do la mesma Experience, Madre de las CiencSas todas.” Cervantes, EDINBURGH: R. GRANT & SON, 82 PRINCE’S STREET. 1842 .
PRINTED UY STEVEXSON & CO. THISTLE STREET, EDINBURGH.
TO JAMES D. FORBES, ESQ. F.R. SS. L. &E., F. G. S,, PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF EDINBURGH, CORRESPONDING MEMBER OF THE FRENCH INSTITUTE, &C. &C, &C. THIS ESSAY IS RESPECTFULLY DEDICATED, BY HIS FORMER PUPIL, THE AUTHOR,
CONTENTS. Section First. . . Page General Account and Objects of the Inquiry_Evils attending the Use of Conical Wheels, l Section Second. Theoretical Investigation of the Rule for the Adaptation of Curves to the Use of Cylindrical Wheels; and Explanation of its Application to Practice, 3 Section Third. Account of Experiments on the Resistance of Railway Carriages with Cylindrical Wheels, ........ j Section Fourth. On the Effects of the Use of Cylindrical Wheels on Railways ; and their Advantages as compared with Conical Wheels, . . . 13
AN EXPERIMENTAL INQUIRY, &c. SECTION FIRST. GENERAL ACCOUNT AND OBJECTS OF THE INQUIRY-EVILS ATTENDING THE USE OF CONICAL 'WHEELS. The object of the following Essay is to describe a simple and practical method of removing a source of inconvenience, expense, and danger in the use of Railways, which, although it is not only well known to every engineer, but obvious to almost every passenger, has yet been hitherto, so far as I am aware, looked upon as a necessary and unavoidable evil. To enable Railway Carriages and Locomotive Engines to pass with ease round curves, the wheels are made of a slightly conical form in the tire, which enables them, by shifting spontaneously to one side or the other, and thereby presenting a larger diameter of wheel to the outer rail, to adapt themselves to different curvilinear tracks; and such wheels are now exclusively used on all the great public railways of the kingdom. The track of the curve is adapted to the use of those wheels, by giving to the outer rail a slight elevation, sufficient to counteract the centrifugal force of the carriages. The use of conical wheels produces the following evils, which are generally known and admitted. First, a continual oscillation of the carriages from side to side; so that even while travelling on a straight track, they describe in reality a series of small curves. This vibratory motion generates a resistance, increasing with the velocity, so great, that the power required to draw: a given load on straight lines, and on curves, is nearly the same; and is more than double the power which draws the same load on a straight line with cylindrical wheels. Secondly, the oscillation of the carriages renders them liable to be thrown off the rails by a very slight obstruction or irregularity; and has been undoubtedly the cause of serious accidents. Thirdly, it gives rise to a series of lateral shocks, which greatly increase the wear of the track, carriages, and machinery. Fourthly, it is disagreeable and inconvenient to the passengers. Fifthly, conical wheels, from the unequal pressure and friction upon different parts of their surface, become worn into irregular shapes, and. are soon unfitted alike for curves and straight lines. B
2 Cylindrical wheels have none of those defects. The motion of carriages running upon them is, when the track is in good order, perfectly steady at all velocities, and quite free from lateral oscillation: the saving of power which their use effects on straight lines, amounts, as I shall hereafter prove, to about one-half the resistance of the same load upon conical wheels ; and their greater safety and durability are necessary consequences of the first-mentioned advantages. Of those advantages of cylindrical wheels every engineer must be aware; hut an opinion has hitherto prevailed, that it is not possible to make them pass easily round curves; and this belief is the only obstacle to their general introduction. The experiments which have convinced me of the error of that opinion, were made on the Edinburgh and Dalkeith Railway, where none but cylindrical wheels are used. This line of railway is horse- worked. It was not originally intended for passengers; nor had the Company, by their original Act, power to convey them; although now between 250,000 and 300,000 are carried on it annually. It is by no means well suited for obtaining the minimum resistance to draught; for its surface is much disturbed by the falling in of coal wastes beneath; and it is so indirect in its course, that in a distance of seven miles and a half, there are so many as twelve curves, of radii less than half a mile, and several of them less than an eighth of a mile. Such defects, however, make this railway the more suitable for the present investigation, and have served to confirm its results; for the saving of power effected by the use of cylindrical wheels under circumstances so disadvantageous, will of course be greatly increased on railways of a better description. The curves on this railway were originally laid so as to have an elevation of the outer rail above the inner, sufficient to counteract the centrifugal force at the low velocity used, but still quite inadequate to prevent the flanges of the cylindrical wheels from rubbing against the outer rails with such force as to treble or quadruple the resistance. Instead of laying aside cylindrical for conical wheels, however, it was determined to adapt the track of the curves to the use of the former, by increasing the elevation of the outer rail; — but as it would have been hazardous to undergo the expense and inconvenience of such an alteration without first ascertaining how it was likely to work, the improvement had to be put off until a suitable opportunity for making experiments presented itself; which did not occur until the year 1837, when a Branch Railway was being carried to the harbour of Leith. The elevation of the outer rail of some of the curves on this branch was adjusted by trial, until carriages with cylindrical wheels passed round with smoothness and ease, and without any tendency of the flanges to graze either rail. With the assistance of the theoretical investigation contained in the second section of this essay, a general rule for the adaptation of curves to cylindrical wheels has been deduced from those experiments; and nearly all the curves on the Edinburgh and Dalkeith Railway have been
3 adjusted accordingly. The result has been an unequalled smoothness of motion and ease of draught, as the experiments detailed in the third section of this essay prove. Such is an outline of the inquiry, the details of which are given in the sequel. SECTION SECOND. THEORETICAL INVESTIGATION OF THE RULE FOR THE ADAPTATION OF CURVES TO THE USE OF CYLINDRICAL WHEELS ; AND EXPLANATION OF ITS APPLICATION TO PRACTICE. A Railway Carriage, in passing round a curve, has, it is well known, a certain tendency to deviate outwards, depending on the radius of the curve and the velocity of motion. A slight elevation of the outer rail, determined by a well-known rule, is sufficient to counteract this centrifugal tendency. When the wheels are cylindrical, an additional tendency to deviate outwards (requiring an additional elevation of the outer rail) is generated by the following circumstance. The outer wheel of each pair has, in passing round a curve, to travel a greater distance than the inner one in the same time ; the outer wheel has, consequently, to slip over a distance equal to the difference in length of the outer and inner rail; and the additional resistance thus produced by sliding friction on the tread of the outer wheel, obviously tends to twist the fore end of the carriage outwards. The additional elevation of the outer rail, of which I have spoken, acts by throwing the centre of gravity of the carriage inwards, and thereby increasing the proportion of the load which the inner wheel has to bear. The rolling resistance at the inner wheel thus becomes greater than at the outer ; and by a proper method of calculation, the elevation may be so adjusted that the surplus of rolling resistance at the inner wheel shall exactly balance the sliding resistance at the outer wheel. The carriage will then move steadily, without any tendency to twist or deviate from the curved track. The proper elevation of the outer rail for different radii was at first ascertained, as I have mentioned, by trial, in several particular cases ; but to connect those experiments -so as to obtain a general rule applicable to all cases, it was necessary to use mathematical analysis. Supposing, then, for the sake of simplicity, the load resting on a pair of wheels to be represented by unity ; the entire rolling resistance will be a certain fraction, which we shall call p. If the outer rail is elevated merely enough to counteract the ordinary centrifugal force, the load, and consequently the rolling resistance, will be equally divided between the outer and inner wheels ; but if, by an additional elevation of the outer rail, the centre of gravity is thrown inwards by a distance x ( the distance between the wheels being denoted by a), then the proportions
4 of the load which the outer and inner wheels have to hear, will be respectively 1 x , 1 . x -and — 4 - — 2 a 2 a and their proportions of rolling resistance shewing a surplus of rolling resistance at the inner wheel of 2 p.— , . , ( No. 1.) a This surplus must be made equivalent to the slipping resistance at the outer wheel. Now the load on the outer wheel is represented by 1 x a—2 x 2 a 2 a and the friction of iron slipping on iron is a certain fraction, which we shall call q. of the load. It must also be kept in mind, that while the rolling resistance travels over the entire length of the curve, the slipping of the outer wheel takes place merely over a distance equal to the difference in length of the two rails, which is to the entire length in the ratio of the guage to the radius ; and that consequently in comparing the mechanical effect of those two forces, the slipping resistance must he multiplied by this ratio: hence we have for its expression (calling the radius of curvature ?•), a a — 2 x a — 2 x ~ • —= c l- (No. 2.) r. 2 a 2 r The condition of steady motion will of course be found by making (No. 1) equal to (No. 2), hence n x a — 2x 2 P — ~q — 5— a 2r This equation, having been reduced, gives for the inward displacement of the centre of gravity ... (No. 3.) 4 p r + 2 q a This formula is applicable to carriages only; but the same reasoning may be applied to Locomotive Engines, substituting only a backward slipping of the inner wheels for a forward slipping of the outer wheels. The load on the slipping wheels of the locomotive is of course a + 2 x 2 a instead of - - 2 x 2a and this alteration having been made, the value of the displacement of the centre of gravity inwards becomes (No. 4.) 4 p r — 2 q a which differs from (No. 3) in having the term 2 q a in the denominator, negative instead of positive. But in practice, this term is very small,
5 and may safely be neglected; so that the following may be taken as an intermediate formula, applicable both to engines and carriages : — , . (No. 5.) 4 p.r Now representing by h the height of the centre of gravity of the load above the rails, and by e the elevation of the outer rail necessary to displace the centre of gravity inwards by the distance x : — a q. a 3 e — _. x — _ 1 _ h (No. 6.) 4 .p. r. h. So that this elevation is independent of the velocity, and varies inversely as the radius of curvature. By practical trials, I have found that when e and r are both expressed in feet, the value of the coefficient ^ ' a , 4 4 p.ti 8i inches, is 600 : — hence in those circumstances for a guage of 4 feet e in inches = 7200 (No. 8.) . . . (No. 7.) r m teet , To the elevation necessary to prevent the twist arising from cylindrical wheels, must be added that required to counteract the centrifugal force, which is found by the well-known formula, a x (velocity 2 in feet per second) r X Ifr foofc A table is annexed, exhibiting the results of these formulae for curves of different radii, and for velocities of 10, 20, 30, and 40 miles an hour. The surfaces of the two rails ought of course to he in the same plane. The proper elevation ought to be given, partly by depressing the inner rail, and partly by raising the outer, so that the centre line between the two rails may preserve the proper gradient. The raising and depressing of the rails ought to commence on the straight line beyond each end of the curve, at such a distance from the curve that the rails may not slope, in attaining their full elevation and depression, at a greater rate than that of the steepest gradient allowed on the railway. An experience of five years has proved that there is no practical difficulty in laying the rails of curves according to these rules. * The adhesion of the driving-wheels of the locomotive is of course affected (though slightly) by the inward displacement of its centre of gravity. The effective adhesion on a curve is twice that of the outer driving-wheel, on which the pressure is reduced in the ratio of a : a — 2x> which is equivalent to that of 1 : 1 — 2 he a 2 Consequently the adhesion of the driving-wheels must be considered as diminished in the same ratio. This, however, is a point of little importance in practice, as the fraction ~ l - is very a 2 small; and as there ought always to be a considerable surplus of adhesion above the power of the engines on a well-regulated line.
6 TABLE I. Elevations of the Outer above the Inner Rail of Curves, to suit Cylindrical Wheels, for a Guage of 4 feet 85 inches. Radius. Elevation constant at all Velocities. (Formula, N0.7.) ^Elevations for Velocities or—, 10 20 30 40 Miles an Hour. Miles- Feet. Inches. . Inches. Inches. Inches. Inches. 1200 6.0 6.2 i 5.5 5.7 6.0 1440 5.0 5.2 5.5 6.3 1800 4.0 4.1 4.4 4.8 5.5 f 3.6 3.7 4.0 4.4 5.1 2400 3.0 3.1 3.3 3.6 4.1 JL 2.7 2.8 3.0 3.3 3.8 2880 2.5 2.6 _ 2.8 3.1 3.5 5 8 2.2 2.3 2.4 2.7 3.1 3600 2.0 2.0 2.2 2.5 2.8 | 1.8 1.8 2.0 2.2 2.5 1 1.6 1.6 1.8 2.0 2.2 1 1.4 1.4 1.5 1.7 2.0 i| 1.2 1.2 1.3 « • 1.5 1.7 H 1.1 1.1 1.2 1.4 1.6 if 1.0 1.0 1.1 1.2 1.4 ii 0.9 0.9 1.0 1.1 1.3 if 0.8 0.8 0.9 1.0 i.i 2 0.7 0.7 0.8 0.9 1.0 H 0.6 0.6 0.6 0.7 0.8 3 0.5 0.5 0.5 0.6 0.7 The guage of 4 feet 8-J- inches, for which the foregoing table was calculated, is that which prevails on the principal English Railways. A guage of six feet, however, is that to w'hich the preference is now sometimes given, and is that which has been adopted on the Irish Railways. Now the coefficient in the formula No. 6, varies as the cube of the guage; and the ratio of the cube of 4 feet 8-| inches to that of 6 feet is very nearly that of 1 to 2 ; hence, taking the value of the coefficient as 600 for the former guage, it will be 1200 for the latter; therefore for a guage of six feet e m inches = — ; —-— .... ( JNo. 9.) r in feet Table II is calculated from this formula. The carriages for which both those tables and formulae are calculated
7 have the centre of gravity, on an average, about four feet above tbe rails. To suit carriages of a different construction, the coefficient will of course have to be varied in the inverse ratio of the height of the centre of gravity. TABLE II. Table of the Elevations of the Outer above the Inner Rail of Curves, to suit Cylindrical Wheels, for a Guage of six feet. Radius. Elevation constant at all Velocities. Elevations for Velocities of-^ 10 20 30 40 Miles an Hour. Miles. Feet. Inches. Inches. Inches. Inches. Inches. 2400 6.0 6.1 X 5.5 5.6 5.9 2880 5.0 5.1 5.4 5.7 5. 4.4 4.5 4.7 5.0 5.6 3600 4.0 4.0 4.3 4.5 5.1 | 3.6 3.6 3.9 4.1 4.5 l 3.1 3.1 3.3 3.6 3.9 4800 3.0 3.0 3.2 3.5 3.8 1 2.7 2.7 2.9 3.1 3.4 2.4 2.4 2.5 2.8 3.0 H 2.2 2.2 2.3 2.6 2.8 li 1.8 1.8 1.9 2.1 2.3 if 1.6 1.6 1.7 1.8 2.0 2 1.4 1.4 1.5 1.6 1.7 2£ 1.2 1.2 1.3 1.4 1.5 2 i 1.1 1.1 1.2 1.3 1.4 2| 1.0 1.0 1.0 1.1 1.2 3 0.9 0.9 0.9 1.0 1.1 ^2 0.8 0.8 0.8 0.9 1.0 4 0.7 0.7 0.7 0.8 0.9 5 0.5 0.6 0.6 0.7 0.7 SECTION THIRD. ACCOUNT OF EXPERIMENTS ON THE RESISTANCE OF RAILWAY CARRIAGES WITH CYLINDRICAL WHEELS. The objects of the experiments on resistance which I have made on the Edinburgh and Dalkeith Railway were two-fold:— First, to ascertain the resistance to the motion of a carriage with cylindrical wheels on a straight line.
8 Secondly, To find tlie additional resistance to the same wheels on moving round curves of different radii, laid according to the formulae given in the preceding section. The method pursued in all the experiments was the only one which appears to me to be satisfactory in such inquiries :—it consists in bringing the carriage up to a certain speed, and then allowing it to run loose until it stops of its own accord. The instants at which the carriage passes several points at given distances apart, are then noted; and from these data the velocity at given instants may be calculated; and consequently, the loss of velocity in a given time, which is the most accurate measure of the retarding force. The initial velocity of the carriage in all the experiments, when first allowed to run loose, was comparatively low; seldom exceeding 10 or 12 miles an hour; and I could detect no sensible diminution of the resistance as the velocity diminished. Hence it must be concluded, that these experiments shew simply the resistance arising from friction ; and that the resistance of the air, which increases rapidly when the velocity becomes high, did not sensibly affect them, except when wind interfered with the results. The following is an abstract of several experiments on the resistance of cylindrical wheels on straight lines of railway. No. 1 .—Level Straight Line. The carriage having been set in motion, and allowed to run loose, travelled 288 feet in the first 30 seconds. 573 feet in the next 125 seconds. Mean velocity during the first 30" zz 9’6 feet per second = velocity at the end of 15". Mean velocity during the next 125" zr 4-6 feet per second zr velocity at the end of 30"+62-J-"zz92^".. Loss of velocity in 92-g"—15"=77-^'' 5-0 To find the ratio of the retarding force to the weight of the carriage, g divide the loss of velocity in one second, or , by the accelerating 77-j force of gravity in a second, or 32 feet. Hence 5 Resistance in fractions of the load zz ——— T zz -00202. 32X77^ This amounts to about 4\ pounds per ton of load, or somewhat more than 1-5 00th. The carriage stopped of itself at the end of the 125 seconds ; hence in the last 62-g seconds, a velocity of 4-6 feet per second was lost. 4-6 This would indicate a resistance of -zr -00230 of the load : 32X621 ’■ but as it is difficult to ascertain with minute accuracy the instant of the carriage stopping, I do not regard this last result as of any value, ex-
9 copt in so far as it proves that the resistance in this experiment did not sensibly diminish with the velocity, and consequently that the resistance of the air was inappreciable. Experiment No. 2.— On the same portion of Railway, the motion being reversed. The carriage running loose, travelled 333 feet in 30": — mean velocity 1T1 feet per second = velocity at the end of 15". 834 ft. in the next 157":—meanvel. 5'3 feet per second — velocity at the end of 30"+ 78-Jj" = 108J". Loss of velocity in 108^"—15"=93j":—5'8 feet per second. ttetardmg force in fractions of the loa = 4-41 lbs. per ton, or rather less 500. Experiment No. 3.— Ascending a Gradient of 1 in 9000. 432 feet in 30": —mean velocity 14-4 feet per second = velocity at the end of 15". 675 ft. in the next 60":—meanvel. 1T25 feet per second = velocity at the end of 60”. Loss of velocity in 60"— 15"= 45": — 3-15 feet per second. 0,1 t* Gross retarding force in fractions of load,-—— 6 ’ 32 x 45 Deduct, effect of gravity = —_— 1 9000 •00212 •00011 Retarding force on a level, = 4^ lbs. per ton nearly, or 500’ as before. •00201 Experiment No. 4 —Ascending the same Gradient. 444 ft. in 30": — mean velocity 14-8 feet per second = velocity at the end of 15". 384 ft. in the next 30" : — mean vel. 12-8 feet per second = velocity at the end of 45". 582 ft. in the next 60": — mean vel. 9*7 feet per second = velocity at the end of 90". First result.—Loss of velocity in 45" — 15"= 30" 2-0 feet per second. 2 - 0 ” Gross Retarding force-- = - 00209 32 X 30 Deduct for the effect of gravity 1 9000 = -00011 Retarding force on a level = ■00198
10 Second Result_Loss of velocity In 90"—45" = 45":—3.1 ft. per sec. Gross Retarding force ——— = *00215 6 32 x 45 Deduct effect of gravity, —-— = *00011 S 9000 Retarding force on a level, •00204 These experiments, although not numerous, are very satisfactory, from their remarkable agreement. The following is an abstract of their results : — No. of Experiment. 1 . 2 . 3. 4 First Result, Second Result, Retarding Force in Fractions of Load, *00202 - *00197 •00201 - *00198 •00204 Mean, - .002004 = 4.489 lbs. per ton. The smallness of the number of experiments in this table was owing to the difficulty of getting weather sufficiently calm. In the following table are contained the results of several experiments which were rendered inaccurate by wind ; but which, on an average, confirm the fore going experiments in a remarkable manner. Resistance in Frac- t- tions of Load. No. 5. against a slight breeze, - *00250 » 6. slight breeze favourable •00188 » 7. do. do. adverse, - *00250 » 8. do. do. do. •00261 » 9* moderate wind favourable, - *00084 „ 10. do. do. adverse, •00310 „ 11. slight breeze favourable, - - *00182 „ 12. do. do. do. .00144 „ 13. do. do. do. - *00183 „ 14. do. do. do. - •00150 „ 15. do. do. do. •00174 „ 16. do. do. adverse, - •00210 Mean of 12 experiments, •00199 This mean result is somewhat less than that of the accurate experiments ; which evidently arises from the favourable winds having preponderated. These, experiments establish the fact beyond a doubt, that the resistance of the carriages, on a level straight line, is rather less than 4-J lbs. par ton of gross load; and that this resistance does not vary sensibly for velocities less than 10 or 12 miles an hour.
11 Now the average resistance of the very best carriages with conical wheels, at like velocities, has been established by a great variety of experiments to be at least 9 lbs. per ton, or ^l^th part of the load. The superiority of cylindrical wheels on a straight line of railway is thus placed beyond a doubt; and it is almost self-evident that the advantage will increase rapidly with the velocity; for it is now admitted by almost every person conversant with the subject, that the resistance of the railway carriages generally used increases with the velocity at a rate which cannot be accounted for by the resistance of the atmosphere, nor in fact by anything except the vibration occasioned by the conical form of the wheels, which gives rise to a rubbing friction on the treads and flanges of the wheels, and on the ends or shoulders of the journals. It now remains to be ascertained to what amount the resistance of carriages with cylindrical wheels is increased on travelling round curves, the rails of which have been laid so as to counteract the tendency to deviate outwards. The experiments I shall quote are few in number, owing to the rare occurrence of weather calm enough to allow of their being made with a sufficient degree of delicacy. From the great care and precision, however, with which those few experiments were made, I consider them to be quite conclusive. Experiment No. 1 was made on a curve of 1304 feet radius, having the outer rail elevated 5-7 inches above the inner, to suit a velocity of ten miles an hour. On this curve there is a gradient of 1 in 225, on which the carriage remained stationary when undisturbed; but the slightest force was sufficient to set it in motion downwards. Having been set in motion down the slope at a rate of ten miles an hour, it preserved that velocity of its own accord, without sensible diminution, as far as the bottom of the gradient. Hence, the retarding force was exactly equal to the accelerating force arising from gravity; that is to say, -i— of the load, or '00444. = 9'95 lbs. per ton. Experiment No. 2. The carriage was set in motion on a level curve of 1900 feet radius (the elevation of the outer rail being 3 - 9 inches), and the spaces which it traversed observed at intervals of ten seconds. The results were as follow : — Times, - 10" 10" 10" 10"- Spaces, - - 109|- 97-2 85-J 73J feet. Mean velocities, 10-95 9'75 8-55 7‘35 feet per second. Retardation in each 10" T2 T2 1-2 feet per second. 1 - 2 • Uniform Retarding force, —--— =r -00375 of load. b 10 X 32 Experiment No. 3 was also on a curve of 1900 feet radius, having the outer rail elevated 3.9 inches. The spaces in this experiment were observed at intervals of 5 seconds, as follow : —
12 Times, Spaces, Mean velocities, Retardation in each 5" Uniform retarding force, 5" 5" 5" - 60 57 54 feet. 12 1T4 10-8 feet per second. 0-6 0-6 feet per second. = -00375 of load, 5X 32 = 8.4 lbs. per ton. The following is an abstract of the results of these experiments : — Radii, ... 1304 feet 1900 feet Resistance, - -00444 -00375 of load Deduct resistance on a straight line, -00200 -00200 Surplus resistance due to curvature, -00244 -00175 Multiply by Radii, - - 1304 1900 Products, - - 3-18 3-32 These products are sufficiently near equality to justify a conjecture, that the excess of resistance on curves properly laid, above that on a straight line varies in the inverse ratio of the radius ; and this conjecture is rendered extremely probable by its being what theory would lead us to expect; for the amount of sliding friction on the outer rail, as we have seen in the Second Section, varies in the inverse ratio of the radius ; and the friction of the ends of the journals against the inside of their boxes being proportional (cceteris paribus) to the elevation of the outer rail, varies in the same ratio. The friction of the ends of the journals, indeed, in the carriages on which my experiments were made, is reduced to a very small amount, by their being terminated in an accurately turned segment of a sphere, which touches the end of the box in a very small central space only; and this small amount may be considered as compensated by the diminished pressure on the sides of the axles. These are the only elements in the surplus resistance on curves ; for, as I have stated, when the rails are properly laid, there is no oscillation, nor rubbing of the flanges of the wheels against the rails. The product of the radius of 1900 feet by the corresponding surplus resistance being taken as the most near the truth, the following formula will represent the probable resistance on a curve of any radius, in fractions of the load: — -002 + ——■■■ ^ ? ——— Jxadius mjeet. which is equivalent to {^ + R adu^t^ Me s } ^ ** ^ ' The following are some of the results of this formula for different dii: — Radius in "*0 Resistance in lbs. Radius in Resistance in lbs. miles. per ton. miles. per ton. i 10 1 5-9 - 8-2 k “ 5-4 2 7-3 2 5-2 i - - 6-4 3 - - 5-0
13 This formula is suited to a guage of 4 feet 8-g- inches. For other guages the surplus resistance on curves will probably increase in the simple ratio of the guage. As it was apprehended that an elevation of the outer rail of curves greater than six inches might cause discomfort to the passengers, this was the greatest elevation allowed in laying the rails of the Edinburgh and Dalkeith Railway; consequently the curves of radii less than 1200 feet, of which there are several, are not laid exactly according to the rules in the second section ; and the resistance on them exceeds what it would be if they were set to the correct theoretical elevation. The experiments on these curves, however, afford an approximate verification of the rule I have just given. The following are the results of two series of experiments : — Radii, 635 670 feet Resistances, "00784 "0075 The surplus resistances, deducting -002 as the resistance on a straight line, are respectively •00584 -0055 And their products by the respective radii 3-7 3-68, which, though they are greater than those deduced from the experiments formerly quoted (because the elevation being only 6 inches, is not sufficient to counteract the tendency to lateral motion), are yet sufficiently near them to afford an auxiliary confirmation. I have only to observe, in concluding the detail of experiments, that the carriages used were not pattern carriages, made for the purpose of having little resistance (as has been the case in some inquiries), but were chosen at random from among those used in the daily traffic of the Railway. SECTION FOURTH. ON THE EFFECTS OF THE USE OF CYLINDRICAL WHEELS ON RAILWAYS ; AND THEIR ADVANTAGES AS COMPARED WITH CONICAL WHEELS. It is generally understood that the average resistance of the best carriages with conical wheels, at velocities less than twelve miles an hour, is about "004 of the load, or nine pounds per ton nearly; and that there is no material increase of resistance on curves above the resistance on straight lines: that is to say, on curves of the radii commonly used on well-constructed railways, which are scarcely ever less than a mile. The experiments I have quoted in the preceding section therefore prove,— First, that the resistance of carriages with cylindrical wheels, on straight lines of railway, at velocities less than twelve miles an hour,
14 being ‘002 of the load, or 4J pounds per ton nearly, is only one-half of the resistance of carriages with conical wheels. Secondly, That the resistance of carriages with cylindrical wheels on curves laid with a proper elevation of the outer above the inner rail, according to the rules given in the Second Section, is very little greater than the resistance with conical wheels, when the radius of the curve is a quarter of a mile; and less than the resistance with conical wheels when the radius is 3-8ths of a mile; and that it continues to diminish rapidly, as the radius of the curve increases. These results are confirmed in a striking manner in the daily traffic on the Edinburgh and Dalkeith Railway, the passengers on which are drawn by horses at a speed of ten miles an hour. The journey between the foot of the fixed engine plane at Edinburgh and the station at Dalkeith, a distance of 11,740 yards, is regularly performed by the horses in 40 minutes ; each horse drawing a carriage with from 30 to 40 passengers, weighing altogether about five tons. Of this distance, 8349 yards are straight, and 3391 yards curved ; and as many of the curves are very quick, I have found the average resistance on them to be about 10 lbs. per ton. The total effect produced by one horse is shown in the following calculation. Distance. IjOAD. Resistance—, Effect, Yards. Feet. Tons. Per ton. Total. in Lbs. raised < Foot high. Straight 8349 = 25,047 5 41- lbs. 221 lbs. 563,557 Curved 3391 = 10,173 5 10 lbs. 50 lbs. 508,650 Divide by time of journey (= 40 minutes) 1,072,207 total. Pounds raised one foot per minute by one horse, 1 ^g gQg at 10 miles an hour. J ■ This result certainly proves that the resistance has not been underrated. Its approaching so near to the value which the best authorities have assigned to the maximum, effect of a horse, although the speed is so much greater than that at which the maximum effect is held to be produced, may be accounted for, partly by the short time of working and partly by the breeding- of the horses ; for the rule that the maximum effect is produced at 2 or 3 miles an hour, is evidently applicable to heavy draught horses only. The horses on the Edinburgh and Dalkeith Railway are not at all distressed by the work they perform. Indeed passengers have often re ■ marked, that at the end of the journey the horses seem little more fatigued than if they had run the distance at the same speed unloaded. The following calculation shows the mechanical power which would have been necessary to perform the journey above described with a load of 5 tons, had conical wheels been used; taking the average re sistance on curves and straight lines to be 9 pounds per ton; —
15 Distance. Load. Resistance— Lbs. raised one Yards. Feet. Tons. Per ton. Total. Foot high. 11,740 = 35,220 5 9 lbs. 45 lbs. 1,584,900 This amount of mechanical power is nearly half as much again as the amount at present expended in performing the same journey; so that if conical wheels had been used instead of cylindrical wheels, either the velocity of travelling, or the loads conveyed, must have been very much diminished, or three horses must have been employed for every two that are used at present; and the cost of locomotive power—which is at present y'jyjy of a penny per passenger per mile, being less than on any railway in Britain except the London and Birmingham—must have been increased nearly in the same proportion. Thus a saving of power, in the ratio of 2 : 3 nearly, is effected on the Edinburgh and Dalkeith Railway, by using cylindrical wheels, with curves adapted for them, as herein explained ; although, from the great number and excessive quickness of the curves on that railway, it presents one of the most unfavourable cases possible for displaying their advantages. Supposing a line of railway to be straight throughout, the saving of power by the use of cylindrical instead of conical wheels would be in the ratio of 1 : 2, the resistance being reduced from 9 to 4 % pounds per ton of load. Hence we may state generally, that the power required to produce a given effect, at a velocity of 10 or 12 miles an hour, is diminished by the use of cylindrical instead of conical wheels in a ratio varying from that of 2 : 3 to that of 1 : 2, according to the number, extent, and quickness of the curves on the line. We have no experiments to enable us to ascertain the exact amount of saving at higher velocities ; but it may safely be concluded beforehand, that the saving will increase with the speed ; for it is known, that the oscillation of conical wheels causes the resistance to increase rapidly with the velocity; whereas cylindrical wheels, having no tendency to oscillate, ought, from all that is known on the subject, to move with an uniform resistance at all velocities. In confirmation of this principle, the experiments I have quoted show, that there is no perceptible variation of resistance at velocities varying from two or three miles an hour to ten or twelve. The saving of power effected by using cylindrical wheels, however, is a minor advantage, in comparison with their superiority over conical wheels in point of safety. It is well known that locomotive engines moving at a high speed are liable to be thrown off the rails by trifling obstacles ; and, indeed, that they sometimes leap off spontaneously, without having met with any obstacle that can be detected. This evidently arises from the circumstance, that a carriage, and especially a locomotive engine, with conical wheels, never moves straight forward but for an instant at a time; so that whenever a small accidental obstruction, or an increase of speed beyond a certain limit, causes it to leap higher than the depth of the flanges, it is almost certain to alight off the track. This source of danger, which has been the cause of many accidents, is entirely removed by the use of cylindrical wheels.
16 I am fortunately able to produce a remarkable proof of this fact, in an accident described in a paper which I some time since laid before the Institution of Civil Engineers. Thirteen empty waggons ran amain from the summit of an inclined plane 1160 yards long, sloping at the rate of one in thirty ; and notwithstanding the weight of some of their breaks, acquired a high velocity which it was impossible to estimate exactly. They met with no obstruction until they reached the bottom of the plane, where a three-inch plank was laid across the rails, for the purpose of throwing them off; they all leapt over the plank in succession, alighted, without a single exception, on the rails beyond it, and continued their course at a velocity of about twenty miles an hour. This occurrence is attributable solely to the use of cylindrical wheels; and, remarkable as it may appear, is only one out of innumerable instances in which carriages have leapt over stones and other obstacles without being thrown off the rails. POINTED BY STEVENSON AND CO. THISTLE STREET, EDINBURGH.
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