r 
 
 REESE LIBRARY 
 
 __n n ix 
 
 UNIVERSITY OF CALIFORNIA. 
 
 
 I 
 
ROPE-DRIVING: 
 
 A TREATISE ON THE TRANSMISSION 
 
 OF POWER BY MEANS' OF 
 
 FIBROUS ROPES. 
 
 BY 
 
 JOHN J. FLATHEB, Pn.B., M.M.E., 
 
 Professor of Mechanical Engineering, Purdue University. 
 
 FIRST EDITION. 
 
 FIRST THOUSAND. 
 
 NEW YORK : 
 
 JOHN WILEY & SONS. 
 LONDON: CHAPMAN & HALL, LIMITED. 
 
 1895. 
 

 Copyright, 1895, 
 
 BY 
 
 JOHN J. FLATHER. 
 
 ROBERT ORUMMOND, ELECTROTYPER AND PRINTER, NEW YORK. 
 

 PKEFACE. 
 
 THE following treatise lias been prepared to supply the 
 existing need of a comprehensive manual of practical in- 
 formation concerning rope-driving and the principles npon 
 which the practice rests. 
 
 In its preparation free use has been made of whatever 
 literature could be found relating to the subject, and refer- 
 ences for further investigation are given in foot-notes 
 throughout the work. 
 
 Most of the data, however, have been collected by the 
 writer, who desires to acknowledge his indebtedness to 
 those who have assisted in the work by furnishing infor- 
 mation and drawings. Especially to be mentioned are Mr. 
 C. W. Hunt, the well-known authority upon this subject; 
 and Mr. Spencer Miller, long identified with the practice 
 of rope transmission. 
 
 J. J. FLATHER. 
 
 LAFAYETTE, INC., Oct. 1895. 
 
 OF THE 
 
 UNIVERSITY 
 
CONTENTS. 
 
 CHAPTER I. 
 INTRODUCTION 1 
 
 Leather belts and spur-gears Early use of ropes Advan- 
 tages of ropes Losses in various systems of transmission. 
 
 CHAPTER II. 
 
 MULTIPLE ROPE SYSTEM 10 
 
 Rope-wells Distribution of power to several floors Influ- 
 ence of arc of contact Rope-splicing. 
 
 CHAPTER III. 
 
 CONTINUOUS ROPE on WOUND SYSTEM 25 
 
 Vertical transmission through several floors Tension-car- 
 riages Rope-tightener Contraction of ropes Transmission 
 at an angle Side lead Multiple idlers Dynamo driving 
 Jack-shafts Coil- friction Winder-pulleys Application of 
 ropes to water-wheels Rochester plant. 
 
 CHAPTER IV. 
 
 LONG DISTANCE TRANSMISSION 62 
 
 Hirn's use of steel band Wire ropes Use of shafting 
 Limit of length for shafting Draw-rods Efficiency of 
 power transmissions. 
 
 CHAPTER V. 
 FIBROUS ROPES 75 
 
 Rawhide Leather Manilla Cotton Structure of cotton 
 fibre Strength of cotton ropes Manilla fibre. 
 
 CHAPTER VI. 
 
 MANUFACTURE OF ROPES 05 
 
 Size of yarns Degree of twist Lambeth rope Lubrication 
 of ropes Stevedore rope Effect of tar on ropes Area of 
 rope Strength of manilla ropes Factor of strength and 
 wear Working strain Size of ropes. 
 
 T 
 
VI CONTENTS. 
 
 CHAPTER VII. PAGE 
 
 WEAK OF ROPES 103 
 
 Influence of pulley diameter Internal wear External wear 
 Harmonic vibration Life of ropes Weight of ropes. 
 
 CHAPTER VIII. 
 
 HORSE-POWER TRANSMITTED BY ROPES Ill 
 
 Coefficient of friction Difference in tensions Influence of 
 centrifugal force Graphic representation Relative cost of 
 rope-driving. 
 
 CHAPTER IX. 
 
 DEFLECTION OF ROPES 128 
 
 The catenary Approximate equations Table of deflections 
 Method of laying out curve Inclined transmissions Ten- 
 sion-weight for given deflection. 
 
 CHAPTER X. 
 
 LOSSES IN ROPE-DRIVING 141 
 
 Engine- friction Effect of temperature Friction of shafting 
 Power absorbed by shafting. 
 
 CHAPTER XI. 
 
 LOSSES IN ROPE-DRIVING Concluded 159 
 
 Resistance due to bending Wedging in the grooves Effect 
 of groove angle Wood rims Differential driving Creep 
 of ropes. 
 
 CHAPTER XII. 
 
 CONSTRUCTION OF ROPE PULLEYS 1 77 
 
 Least diameter of pulley Uniformity of pitch Milled 
 grooves Cast grooves Light pulleys Wood-filled pulley 
 rims Rim sections Proportions for groove Idlers Hubs 
 Double arms Split pulleys Diameter of bolts Sections 
 of arms Stresses in pulley Methods of joining arms and 
 rim Built-up rope-pulleys. 
 
CONTENTS. yii 
 
 TABLES. 
 
 PAGE 
 
 1. Limit of Length for Steel Shafting 70 
 
 II. Limit of Length for Wrought-iron Shafting 71 
 
 III. Strength of Cotton Transmission Ropes 81 
 
 IV. Strength of Manilla Transmission Ropes 95 
 
 V. Executed Rope Transmissions 96 
 
 VI. Greatest Revolutions per Minute for given Diameter of 
 
 Rope 104 
 
 VII. Weight of Ropes 110 
 
 VIII. Values of 1-2 for a Working Stress Equivalent to 200d 2 
 
 pounds 117 
 
 IX. Angle embraced by Rope 117 
 
 X. Frictiou and Stress Moduli 118 
 
 XI. Horse-power Transmitted by Ropes 121 
 
 XII. Relative First Cost of Rope-driving 123 
 
 XIII. Relative Wear and Cost of Rope per Horse-power Trans- 
 
 mitted 125 
 
 XIV. Deflection of Rope 134 
 
 XV. Friction Per Cent under Varying Loads 143 
 
 XVI. Power-absorbed by Friction in Line-shaft 152 
 
 XVII. Power absorbed by Friction in Jack or Head Shafts 154 
 
 XVIII. Power absorbed by Lead-shaft carrying High-speed 
 
 Ropes 156 
 
 XIX. Angle of Grooves for Equal Adhesion 169 
 
 XX. Values of #' and ^V , 180 
 
 XXI. Least Diameter of Pulley for given Diameter and Speed 
 
 of Manilla Rope 180 
 
 XXII. Rope-pulleys for General Work 181 
 
OF THE 
 
 UNIVERSITY 
 
 ROPE -DRIVING 
 
 CHAPTER I. 
 
 ALTHOUGH toothed gearing and belts are the most fa- 
 miliar mediums for the transmission of power in mills and 
 factories, systems of rope transmission for this purpose 
 have been in use for many years, but it is only within the 
 last ten years that they have given promise of being gen- 
 erally recognized in this country as a convenient and effi- 
 cient means of accomplishing the ends for which they were 
 designed. Until about fifty years ago it was generally 
 thought by engineers that cotton- and woollen-mills, and 
 all others requiring a considerable amount of power, could 
 not be run effectively without large and ponderous lines of 
 upright and horizontal shafts of either cast or wrought iron 
 and heavy trains of gear wheels. 
 
 When large leather belts began to be introduced as a 
 substitute for gears it was thought to be an experiment of 
 very doubtful result, if not altogether impracticable;* but 
 when high speeds and lighter shafting were used in con- 
 junction with the wide belts, the marked success which at- 
 tended their general adoption in America during the next 
 twenty years attracted considerable attention in England. 
 It was not, however, until about thirty years ago that the 
 
 * Journal Franklin Inst., 1837. 
 
ROPE- DRIVING. 
 
 American system began to replace the old-fashioned gear- 
 ing. The belt system made very slow progress in England, 
 however, and before it had been at all extensively adopted 
 a newer method was introduced and quickly came into 
 prominence, making such rapid progress as to almost en- 
 tirely supplant the old wheels. 
 
 The use of hemp rope for power transmission had been 
 revived about 1860 by the Messrs. Combe & Barbour of 
 Belfast, who introduced it successfully into small mills in 
 the north of Ireland. This was followed by its speedy 
 adoption in the jute-mills of Dundee, and subsequently in 
 the cotton-mills of England. 
 
 Previous to this, fibrous ropes had been in use for trans- 
 mission purposes, but their application had been limited. 
 
 At the Colonial Eope Factory at Great Grimbsy, in Lin- 
 colnshire, from 1830 to 1837, ropes had been employed for 
 taking off the power from the engine and communicating 
 to the first motion shaft. The plan was very commonly 
 adopted in connection with rope-works, where the driving 
 rope employed was known as the fly-rope for working the 
 equalizer from end to end of the ropewalk. The machin- 
 ery in connection with the flax-spinning mills at the same 
 place was also driven by means of rope gearing.* 
 
 Ropes were also used in this country many years before 
 their more general adoption in England which followed the 
 movement introduced by Messrs. Combe & Barbour. In a 
 recent communication, Mr. Samuel Webber states that he 
 remembers the occasional use of rope-driving for temporary 
 purposes ''back in the forties/' In one case he mentions, 
 power was carried from a small engine outside through a 
 window into a mill to grind cards before the wheels and 
 main belts were ready. 
 
 The use of ropes, however, was not common, and it has 
 
 * Mr. W. Smith, Proc. I. M. E. 1876, p. 393. 
 
ROPE-DRIVING. 3 
 
 only been in recent years that rope-driving has come into 
 prominence as a factor in power transmission. 
 
 By this system, according to one of its earlier promoters,* 
 Mr. Jas. Durie, large powers were transmitted "by means 
 of round ropes working on grooved wheels, which in some 
 parts of this country [England] have been largely sub- 
 stituted for toothed gearing. In this mode of driving, the 
 fly-wheel of the engine is made considerably broader than 
 the fly-wheel of an engine having cogs on its circumference; 
 and, instead of cogs, a number of parallel grooves for the 
 ropes are turned out, the number and size of which are 
 regulated by the power to be taken oft' the fly-wheel. The 
 power which each of the ropes will transmit depends upon 
 their size and the velocity of the periphery of the fly-wheel/' 
 
 As rope-driving has, until recent years, been a matter 
 largely of experiment, the results which have been obtained 
 from its use have not always been of uniform excellence, 
 mainly, however, because designers have failed to properly 
 recognize the requirements. 
 
 As the conditions under which the systems were installed 
 have been so varied, it is not surprising to find many cases 
 where the ropes have been rapidly worn out and replaced 
 by leather lelting, or other methods of transmission; but 
 where rope-driving has been tried and has failed, investi- 
 gation will invariably show the absence of suitable condi- 
 tions or a disregard of correct principles of design or con- 
 struction. In many applications too great a strain is put 
 upon the rope, and the stretch and wear are rapid; in ad- 
 dition to this, the pulleys are often of unsuitable size, and 
 the rope is unnecessarily weakened through the action of 
 the fibres upon one another. Both of these causes are a 
 constant source of annoyance in a rope-drive which has 
 been poorly designed or constructed. A case in point can 
 
 *Proc. I. M. E. 187$, 
 
4 ROPE-DRIVING. 
 
 be shown where a rope would stretch to such an extent 
 that it had to be taken up every few days when new, and 
 later every two or three weeks a rope under these condi- 
 tions lasting less than four months. Investigation showed 
 that a tension-weight of 600 pounds had been placed 
 upon the rope, which was of f inch diameter. A weight 
 of 50 pounds was substituted, and the rope has since been 
 running satisfactorily for three years, and has only been 
 taken up once in that time. 
 
 Cases where ropes have suddenly broken are few in num- 
 ber, the risk in this respect being reduced to a minimum 
 by the fact that any defect in a rope, arising either from 
 wear or other cause, will show itself long before the point 
 of danger is reached. 
 
 The ropes by which the power is transmitted consist of 
 an elastic substance, and their lightness, elasticity, and 
 comparative slackness between the pulleys are highly con- 
 ducive to their taking up any irregularity that may occur 
 in the motive power. 
 
 Their quiet working and convenience in application, 
 much more so than wide belts, are also features which 
 caused ropes to be looked upon with great favor. An- 
 other reason for its rapid progress in England, which was 
 considered a great advantage by the millowners, was the 
 adoption of the multiple rope, now known as the English 
 system, in which a number of single ropes are -spliced and 
 run side by side. 
 
 The entire freedom from any risk of a breakdown or 
 stoppage of the works which might occur with gearing was 
 an important factor in replacing the latter by the newer 
 system; the working stress in the ropes being but a fraction 
 of their breaking strength, any signs of weakness in an in- 
 dividual rope would allow it to be removed, and the engine 
 run with the remaining ropes until a convenient opportu- 
 nity offered for the replacement of the weak member. 
 
One of the great advantages of rope-driving over gearing 
 lies in the steady motion produced, but this has been at- 
 tributed more to an accidental combination of heavy fly- 
 wheel and high velocity than to any inherent advantage in 
 the system itself. 
 
 In spur gears diameters of 20 to 24 feet were usual in 
 large engines, while for ropes the pulleys are 25 to 35 feet 
 in diameter; and, whereas the gears weighed from 20 to 25 
 tons, we find rope pulleys used for the same work weighing 
 from 60 to 75 tons.* 
 
 When we consider the speed at which these heavy wheels 
 are run from 3000 to GOOO feet per minute it is not 
 surprising that uniform rotation is obtained; and whether 
 it be that the energy stored in the moving mass prevents 
 fluctuation, or whether the elasticity and other properties 
 of the rope perform the same office, or which is more 
 reasonably the case all of these factors act together, the 
 truth remains that steadier running and a greater output 
 are now obtained with rope-driving than was formerly the 
 case with toothed gearing. 
 
 The general experience is not altogether in favor of ropes, 
 for, while the advantages of smooth running and easy 
 handling are conceded, it is also acknowledged that the 
 extra weight and greater width of pulleys increase the 
 journal friction over that found with toothed gearing, and 
 that otherwise a greater loss of power occurs, the causes of 
 which will be discussed subsequently. 
 
 * The Walker Manufacturing Co. of Cleveland recently made four 
 large rope pulleys for the Broadway Cable Railway Company, New- 
 York, which weighed 104 tons each. These were 32 feet in diameter 
 and were grooved for thirty-four 2-inch ropes. 
 
 Another example of heavy rope wheel is given in London Engin- 
 eer, January 11, 1884. This wheel was made by Hick, Hargraves & 
 Co , Bolton, England, for a cotton mill in India, and is 30 feet iu 
 diameter, 15 feet face, grooved for 60 ropes to transmit 4000 horse- 
 power. Its weight is 140 tons. 
 
6 ROPE-DRIVING. 
 
 It is difficult, however, to determine the relation between 
 the power absorbed by ropes and gears, for in nearly all 
 cases where rope-driving has been substituted for gears, 
 other changes have been made at the same time, or the 
 engines were, after the alteration, driven at an increased 
 speed, so that there has been little opportunity for com- 
 parison. There is, however, a generally -accepted opinion 
 among engineers that the loss in rope-transmission is from 
 5 to 10 per cent greater than with gearing. 
 
 Mr. A. G. Brown* states that in the older cotton-mills of 
 England, where the main drives are by gearing, and belting 
 is used for the intermediate and machine driving, the fric- 
 tion of the engine, shafting, gearing, and belting averages 
 about 20 per cent of the whole power, the engine develop- 
 ing at full loads from 500 to 800 I. H. P. 
 
 These engines were compound condensing, and consisted 
 in each case of an overhead beam engine which had been 
 converted into a compound by the addition of a high-press- 
 ure cylinder between the crank and beam centre. 
 
 In the newer mills, for doing practically the same work, 
 and where the main and some of the intermediate drives 
 are by ropes, the friction of the engine, ropes, shafting, and 
 belting averages 23 to 25 per cent, of the whole power, the 
 engines developing from 800 to 1500 I. H. P. It is fair to 
 assume that the newer engines have at least no greater per- 
 centage of friction than the older ones, except that due to 
 an increased journal friction attributable to the larger 
 journals and greater weight of the rope-pulleys ; also the 
 friction of the intermediate shafting and small belts should 
 be the same in each case ; therefore it is reasonable to con- 
 clude that the increase of power absorbed by the use of 
 rope-driving is chargeable to the system itself. The writer 
 has assumed in similar cases that the loss at the engine in 
 
 * American MacJiinist, July 21, 1888. 
 
ROPE-DRIVING. 7 
 
 rope driving is about 10 per cent of the indicated horse- 
 power, that an additional 10 per cent is absorbed by the 
 mill-shafting, and that from 5 to 8 per cent may be at- 
 tributed to losses in the rope itself due to resistance to 
 bending, wedging in the grooves, differential driving effect, 
 and creep, all of which affect the loss to a greater or less 
 extent. As compared with the above, the friction of shaft- 
 ing and engines in American cotton-mills, where belting is 
 used exclusively, indicates that the percentage of loss 
 where large belts are employed is probably a trifle less than 
 that obtained with ropes in English mills ; but the condi- 
 tions of practice are so varied that it is difficult to compare 
 the two systems from published results of tests. As shown 
 on page 155 for similar installations, the loss absorbed in 
 shaft-friction will not materially differ in the two systems ; 
 for large transmissions the engine -friction should be less 
 with ropes, but the losses in the ropes themselves due to 
 islip, differential driving, bending, creep, and other causes, 
 may, without special precautions, exceed to a small degree 
 the losses due to the belt. 
 
 Mr. J. T. Henthorn, in a paper read before the American 
 Society of Mechanical Engineers, states that the friction of 
 the shafting and engine in a print- mill should not exceed 19 
 per cent of the full power; but out of fifty-five examples of 
 a miscellaneous character which he has tabulated only seven 
 cases are below 20 per cent, 20 vary from 20 to 25 per 
 cent, fifteen from 25 to 30 per cent, eleven from 30 to 35 per 
 cent, and two above 35 per cent. We note that the greater 
 number lies between 20 and 25 per cent; allowing a varia- 
 tion of 5 per cent each side of these limits we shall obtain 
 values from which a fair average may be determined. This 
 will include those cases under 20 per cent, but not those 
 over 30, from which we find the mean loss to be 23.9 per 
 cent of the total power. 
 
 Mr. Barrus, speaking of this subject, quotes eight cases, 
 
8 ROPE- DRIVING. 
 
 the data of which were obtained from tests made by him- 
 self in various New England cotton-mills, in which the 
 minimum percentage was 18 and the maximum 25.7, the 
 total average being 22 per cent. 
 
 Mr. Samuel Webber states that 16 per cent of the total 
 power of a mill is sufficient to overcome the friction of 
 shafting and engine 10 per cent for the shafting and 6 
 per cent for the engine. But in this estimate Mr. Webber 
 does not include the loss due to the belts running upon 
 loose pulleys, which he does not consider to be part of the 
 shafting, as they are not so running while the machinery 
 is in operation ; and when it is not, they may be thrown off 
 as well as not, except for convenience. 
 
 He further estimates, both from his own experience and 
 the observations of others, that the power consumed by the 
 machine belts on the loose pulleys in a large cotton-mill 
 is about 8 per cent of the whole. This 8 per cent 
 added to the 16 per cent loss due to shafting and engine 
 will give 24 per cent of the total power a result which 
 agrees closely with the average values given above.* 
 
 Considering the greater loss which occurs in the use of 
 ropes and belts for main drives, the recent revival of gear- 
 ing for this purpose in England has much in its favor. 
 Of this Mr. Geo. Eichards states :f "The advantages of 
 rope transmission for main drives in large plants would 
 not be as apparent if compared with modern gearing. 
 The kind of gear used for this purpose to-day is not 
 the rough cast gear used formerly, whose uneven motion 
 produced a rumbling which could be heard a mile or two 
 from the mill. The present gears are often machine-cut, 
 made to bear equally on each tooth, and with a contact 
 
 * See "Dynamometers and the Measurement of Power," John 
 Wiley & Sons, New York. 
 
 f Richards's "Mechanical Progress," 1891. 
 
ROPE-DRIVING. 9 
 
 across the whole face of the gear, causing little more noise 
 than the ropes. 
 
 With greater speed and stronger and heavier gears steady 
 running is insured, and by using properly-proportioned 
 machine-cut teeth comparatively little noise results. The 
 saving of space is also an important factor advanced in 
 favor of gearing." At the present time steel gears 30 feet in 
 diameter are being made, with all the teeth cut, for trans- 
 mitting the powers of mill-engines in the Old ham district, 
 while in this country machine-cut gears from 30 to 50 feet 
 in diameter are in use. 
 
 However, although such gearing may be very superior 
 to the former slow-running cast gears, it is questionable 
 whether it can ever produce the same steadiness of running 
 which is so largely a distinguishing feature of rope-trans- 
 mission. Any shock or sudden fluctuation of load must 
 necessarily be transmitted through the gear teeth, whereas 
 with rope transmission such shock is partially absorbed by 
 the more or less elastic ropes and subsequently given out 
 by their recoil. 
 
 For this reason when uniformity of speed is desired 
 ropes are generally to be preferred to gears, even when the 
 latter are working under their most advantageous con- 
 ditions. 
 
10 ROPE-DRIVING. 
 
 CHAPTER II. 
 
 WHERE ropes have been used to replace gearing in the 
 English mills the plan adopted has been to put in a new- 
 grooved fly-wheel, or to place grooved segments upon the 
 existing fly-wheel, when the speed could be increased suffi- 
 ciently to allow of a limited number of ropes being em- 
 ployed, and the width of the wheel-pit was also sufficient 
 for the purpose ; but if this plan could not be adopted 
 grooved pulleys were put on the intermediate shaft, and the 
 ropes carried to the different stories of the mill. It has 
 sometimes been necessary to put in a countershaft, so as to 
 gain speed and obtain a sufficient distance between the 
 centres of the shafts on which the pulleys are placed. 
 
 Where rope-driving has been installed in new factories 
 special provision has been made for the ropes, and we find 
 in such cases rope-wells or chambers built in, suitably fitted 
 with platforms and staircases to give access to pulleys and 
 bearings on the various shafts, as shown in Figs. 1 and 2. 
 
 The majority of drives are arranged so that the ropes are 
 horizontal or inclined rather than vertical, and with the 
 driving or tight side of the rope on the lower side of the 
 pulleys ; then, when transmitting power, the two sides ap- 
 proach each other, and the arc of contact is increased. 
 
 An additional advantage is that obtained by the weight 
 of the rope acting on both pulleys, thus allowing a low 
 initial tension to be maintained. This does not hold for 
 short distances between centres, as under such conditions 
 the weight of the rope adds little to the total tension; on 
 the other hand, where the distance between the pulleys in 
 
12 ROPE-DRIVING. 
 
 vertical drives is small, the relative weight of the rope be- 
 ing small as compared with its tension, there will be little 
 tendency for the rope to leave the bottom sheave. 
 
 With these conditions the efficiency of vertical drives 
 will approach that of a horizontal or inclined arrangement 
 of ropes. 
 
 The manner of distribution of the power to the several 
 floors of a mill is shown in Fig. 1, which represents a 
 plant designed by Messrs. Lockwood & Greene of Boston 
 for the Lanett Cotton Mills, West Point, Ga., in which 
 1100 h.p. is delivered from the engine by means of twenty- 
 six If-inch ropes. The fly-wheel is 26 feet in diameter, and 
 makes 60 revolutions per minute, corresponding to which 
 the velocity of the rim is 4900 feet per minute. As will be 
 seen from the figure, the driving-sheaves are placed in a 
 well in the middle of the factory, and the line-shafts ex- 
 tend to the right and left, as shown. 
 
 There is no line-shaft on the second floor, as the various 
 machines may be driven from below. 
 
 The distribution is as follows : 
 
 ROPE-DRIVES IN LANETT COTTON MILLS. 
 
 
 Number 
 of 
 Ropes. 
 
 Dia. of 
 Rope. 
 Inches 
 
 Dia. of 
 Pulley. 
 Inches. 
 
 Revolutions 
 of Pulley 
 per minute. 
 
 Horse- 
 power. 
 
 1st floor 
 
 8 
 
 If 
 
 81 
 
 231 
 
 336 
 
 3d floor . ... 
 
 7 
 
 1 
 
 62 
 
 302 
 
 294 
 
 4th floor 
 
 11 
 
 If 
 
 62 
 
 302 
 
 462 
 
 
 
 
 
 
 
 A similar plant, also designed by Messrs. Lockwood & 
 Greene, has been recently erected at the Naumkeag Cotton 
 Mills, Salem, Mass., in which 1800 horse-power is distrib- 
 uted to five floors by means of forty-one If -inch manilla 
 ropes.* The fly-wheel is 26 feet in diameter, by about 9 
 
 * Power, March 1895. 
 
ROPE-DRIVING. 
 
 13 
 
 OF THE 
 
 XTNIVERSITt" 
 
14 
 
 ROPE-DRIVING. 
 
 feet face, and weighs 150,000 pounds. The velocity of the 
 ropes is the same as in the previous instance, namely, 4900 
 feet per minute. The distribution is as follows: 
 
 
 Number of 
 Ropes. 
 
 Dia of 
 Rope. 
 
 Speed of Rope. 
 
 Horse- 
 power. 
 
 1st floor 
 
 13 
 
 1 ji inches 
 
 4900 ft p m 
 
 58 T 
 
 2d " 
 
 14 
 
 
 
 fjqn 
 
 3d ".-..;. 
 
 4th " 
 
 4 
 5 
 
 " 
 
 
 
 180 
 99*; 
 
 5th " . . . 
 
 5 
 
 t( 
 
 tt 
 
 99ni 
 
 
 
 
 
 
 Total 
 
 41 
 
 
 
 1845 
 
 
 
 
 
 
 A system of main driving-geap designed and erected by 
 John Musgrave & Sons at the Atlas Mills, Bolton, Eng., is 
 shown in Figs. 2 and 3. This mill is 300 feet long by 135 
 feet wide, with a shed on one side 325 feet long by 45 feet 
 wide, and contains 84,000 spindles. 
 
 The engines are tandem compound, 24 and 46 by 6 ft. 
 stroke, and run at 50 revolutions per minute. The average 
 horse-power is 1050. The rope-wheel is 32 feet in diameter 
 and is grooved for thirty-two 11 -inch cotton ropes, which 
 run at 5026 feet per minute. 
 
 The arrangement of shaft is as follows : On the ground- 
 floor there are five lines of shafting, the main shaft being 
 driven from the rope-drum by means of ten ropes If inches 
 diameter running on a pulley 9 feet 4| inches in diameter, 
 which runs 170 revolutions per minute. 
 
 On the main shaft, close to the wall of the rope-well, is 
 a pulley 6 feet diameter, grooved for four If-inch ropes, 
 which drives, through a similar-sized pulley, the line-shaft 
 on the right. These ropes run at a velocity of 3205 feet 
 per minute; also on the main shaft, but on the opposite 
 side of the rope-well, is another pulley, 8 feet diameter, 
 grooved for six If-inch ropes, driving on to a pulley 64^ 
 
ROPE-DRIVING. 
 
 15 
 
16 KOPE-DKIVI^G. 
 
 inches diameter on the first line-shaft to the left of the 
 main shaft. This shaft runs 250 revolutions per minute, 
 corresponding to which the speed of the ropes driving it is 
 4274 feet per minute. 
 
 The second line-shaft on the left is driven from the first 
 one by means of a pair of pulleys 4 feet diameter, grooved 
 for five If -inch ropes. This second shaft runs 250 revolu- 
 tions per minute, and the ropes driving it have a velocity 
 of 3140 feet per minute. 
 
 The second line-shaft, mentioned above, drives the line- 
 shaft in the shed by means of a pair of pulleys 4 feet di- 
 ameter, grooved for four li-inch ropes, driving the counter- 
 shaft shown on plan, on which is a pulley 3 feet 4 inches 
 diameter, grooved for five H-inch ropes, driving on a 3-ft. 
 pulley on the shed line-shaft. These ropes have a velocity 
 of 2640 feet per minute and give to the shaft 278 revolu- 
 tions per minute. 
 
 All of the shafts described are on the ground -floor of the 
 mill. Of the shafts above this, on the next two floors, the 
 line-shafts each have pulleys 6 feet in diameter, grooved for 
 seven If-inch ropes, driven from the main rope-drum. 
 These shafts run at 266 revolutions per minute. The 
 shaft on the upper floor also runs 266 revolutions per min- 
 ute and is driven from the main rope-drum through a pul- 
 ley 6 feet in diameter grooved for eight If-inch ropes. 
 
 The distance from the centre of upper shaft to the cen- 
 tre of crank-shaft is 89 feet, and the length of each rope re- 
 quired for this drive is about 250 feet. 
 
 Another arrangement of rope-drive for cotton-mills is 
 shown in Fig. 4, which represents a section through the 
 engine-room at the thread-mills of the Nevsky Cotton- 
 Spinning Co., St. Petersburg.* In this drive there is no 
 rope-chamber, as the whole of the rope gear is contained 
 
 * John Musgrave & Sons, engineers. 
 
ROPE-DRIVING. 17 
 
 in the engine-room situated in the centre of the mill, which 
 is 680 feet long and 90 feet wide. 
 
 There is a short staircase from the engine-room floor to 
 the first landing, and the landings above this are reached 
 by an ornamental spiral stairway as shown. 
 
 There are two shafts, one from each side of the engine- 
 room ; these are driven by a pair of right- and left-hand 
 tandem compound engines, 30 and 52 by 6 ft. stroke, run- 
 ning at 50 revolutions per minute. 
 
 The average power developed by each engine is 1100 
 horse-power. 
 
 The rope-drums are each 30 feet in diameter and weigh 
 62 tons; these are grooved for twenty-eight If cotton ropes, 
 which run at 4700 feet per minute. 
 
 The first-and second-floor shafts make 300 revolutions 
 per minute, and are each driven by nine If -inch ropes from 
 the rope-drum running over pulleys 5 feet in diameter on 
 the shafts. 
 
 The shafts on the third and fourth floors run 200 revo- 
 lutions per minute, and are driven by five ropes, each If 
 inches diameter, running over 7 feet 6 inch pulleys on the 
 shafts. 
 
 In each of the two upper rooms there is a second line- 
 shaft, driven from the main line-shaft on each floor by 
 means of 54-inch pulleys, grooved for four l^-inch ropes, 
 which have a velocity of 2826 feet per minute. The 
 distance from the centre of the upper shaft to the 
 centre of the crank-shaft of engine is 56 feet 6 inches. 
 This is a short drive for a mill of this size; in fact, all of 
 the drives are short, the lower one especially so, being only 
 80 feet between centres, the peculiar arrangement of the 
 engine-room not admitting of a greater length; but the 
 plant is said to work extremely well. 
 
 In these examples the multiple-rope system is used, each 
 wind consisting of a separate rope stretched around the fly- 
 
18 
 
 ROPE-DRIVING. 
 
 wheel and its individual shaft-pulley, then spliced. The 
 degree of tightness will depend upon the material of the 
 rope and the amount of tension in the slack part necessary 
 for adhesion. In the majority of cases the initial tension 
 is very small compared with the strength of the rope es- 
 pecially so where the horizontal distance between driving 
 and driven pulleys is great, as, under such conditions, the 
 tension in the slack side due to the weight of rope in the 
 hanging catenary is often sufficient to prevent slipping; a 
 
 slack upper rope in horizontal 
 or inclined drives will also 
 increase the arc of contact, 
 thereby increasing the grip 
 of the rope. 
 
 This is shown in Fig. 5, 
 which represents two pulleys 
 of equal diameter, arranged, 
 as in the upper figure, to 
 drive with the slack side uppermost, and, in the lower 
 figure, with the slack side below. 
 
 The difference in the arc of contact, as shown in the 
 figures, is 60 degrees, which would under similar con- 
 ditions, with a velocity of 4000 feet per minute, produce a 
 difference of over 25 per cent in the amount of horse-power 
 transmitted by the two ropes under the usual working 
 tension. 
 
 New cotton ropes are often stretched as taut as possible 
 on account of their extensibility, as they will soon become 
 slack enough for good working, and may even have to be 
 respliced before becoming permanently set. 
 
 It is the practice of some engineers to strain both manilla 
 and cotton ropes as much as possible and unite the ends 
 with a temporary short splice when first put over the pul- 
 leys; after running a few days a permanent stretch is given 
 to the rope, which is then respliced with a long splice, the 
 
 FIG. 5. 
 
ROPE-DRIVING. 19 
 
 strain on the rope being very much reduced in this latter 
 case. 
 
 The splice in a transmission rope is not only the weakest 
 part of the rope, but is the first to fail when the rope is 
 worn out. If the joint is not strong the rope will fail by 
 breakage or pulling out of the splice, the projecting parts 
 will wear on the pulleys, and the rope fail from the cutting 
 off of the threads. Formerly much trouble was experienced 
 in this way on account of improper splicing. One form of 
 joint, according to Cromwell,* was made by pressing the 
 ropes firmly together and winding about with stout small 
 rope. The spliced part is taken as long as possible in order 
 to bend properly over the pulleys and give the required 
 strength. As this form of joint made the rope larger in 
 diameter at the splice, the effect produced was to run 
 faster when passing over the driving-sheave and slower 
 over the follower; the resulting motion was very irregular, 
 and the wear at the splice rapidly destroyed the rope. 
 
 A very simple splice is sometimes used with rope-driving 
 formed by opening out the ends of the rope for 12 or 15 
 inches and tying together the individual rope-yarns one 
 by one, allowing the ends to lie straight, and serving the 
 whole with spun yarn. 
 
 Similar joints wrapped with raw-hide belt-lacing give a 
 very smooth splice which lasts well. 
 
 Some engineers favor a short splice, in that it is easily 
 made and holds well, and offers a lesser length of enlarged 
 portion for surface contact with the pulley. 
 
 If properly made, however, there need be no enlarged 
 portion, and since a long splice is stronger we find such 
 joints preferred in most cases. 
 
 There are several kinds of long splices varying in length 
 from GO to 80 diameters of rope, but the one which seems 
 
 * J. H. Cromwell, " Belts and Pulleys," John Wiley & Sons. 
 
20 ROPE-DRIVING. 
 
 to give the best results in practice is the " English splice/' 
 directions for which are given in various trade publications. 
 The successive operations for splicing a If -inch rope by 
 this method are as follows : * 
 
 1. Tie a piece of twine, 9 and 10, Fig. 6, around the rope 
 to be spliced about six feet from each end. Then unlay 
 the strands of each end back to the twine. 
 
 2. Butt the ropes together and twist each corresponding 
 pair of strands loosely, to keep them from being tangled, 
 as shown at (a), Fig. 6. 
 
 3. The twine 10 is now cut, and the strand 8 unlaid and 
 strand 7 carefully laid in its place for a distance of four 
 and a half feet from the junction. 
 
 4. The strand 6 is next unlaid about one and a half feet 
 and strand 5 laid in its place. 
 
 5. The ends of the cores are now cut off so they just 
 meet. 
 
 6. Unlay strand 1 four and a half feet, laying strand 2 
 in its place. 
 
 7. Unlay strand 3 one and a half feet, laying in strand 4. 
 
 8. Cut all the strands off to a length of about twenty 
 inches, for convenience in manipulation. The rope now 
 assumes the form shown in (b), with the meeting-points of 
 the strands three feet apart. 
 
 Each pair of strands is now successively subjected to the 
 following operations : 
 
 9. From the point of meeting of the strands 8 and 7 
 unlay each one three turns; split both the strand 8 and 
 the strand 7 in halves, as far back as they are now unlaid, 
 and the end of each half strand " whipped " with a small 
 piece of twine. 
 
 10. The half of the strand 7 is now laid in three turns, 
 and the half of 8 also laid in three turns. The half strands 
 
 * From " Manilla Rope," C. W. Huut Co., New York. 
 
HOPE DKIVIKG. 
 
 21 
 
 FIG. 6. SPLICE FOR If -INCH 4- STRAND ROPE. 
 
22 ROPE-DRIVING. 
 
 now meet and are tied in a simple knot 11, (c), making the 
 rope at this point its original size. 
 
 11. The rope is now opened with a marlinspike, and the 
 half strand of 7 worked around the half strand of 8 by pass- 
 ing the end of the half strand through the rope, as shown, 
 drawn taut, and again worked around this half strand 
 until it reaches the half strand 13 that was not laid in. 
 This half strand 13 is now split, and the half strand 7 
 drawn through the opening thus made, and then tucked 
 under the two adjacent strands, as shown in (d). 
 
 12. The other half of the strand 8 is now wound around 
 the other half strand 7 in the same way. After each pair 
 of strands has been treated in this manner, the ends are 
 cut off at 12, leaving them about four inches long. After 
 a few days' wear they will draw into the body of the rope 
 or wear off, so that the locality of the splice can scarcely 
 be detected. 
 
 For a three-strand rope of the same size the foregoing 
 method is slightly modified. After tying the twine 9 and 
 10 around the rope about 6 feet from each end, unlay the 
 strands back to the twine, bring the butts together, and, 
 as in Fig. 7, twist the corresponding strands loosely together. 
 Now cut twine 10, and unlay strand 8 for a distance of 
 four and a half feet from the junction, and lay in strand 
 7. Unlay strand 1 four and a half feet, lay in strand 2, 
 and cut all the strands off to a length of about 20 inches, 
 as before explained for convenience in handling. The 
 splice now assumes an appearance similar to (&) with the 
 exception that there are only three meeting-points of the 
 strands, and these are 4J feet apart. 
 
 Each pair of strands is now subjected to the series 
 of operations described for the 4-strand splice in steps 9 
 to 12 inclusive. 
 
 In splicing a Lambeth cotton rope the operation is modi- 
 fied to a still greater extent, 
 
ROPE-DRIVIXG. 22a 
 
 Although considered as a troublesome rope to splice, the 
 following instructions,* if carefully followed, will enable 
 one to make an excellent joint without difficulty. 
 
 1. Tie a piece of twine 9 and 10 around the rope to be 
 spliced about 6 feet from each end. . Then unlay two 
 strands together of each end back to the twine. Butt the 
 
 FIG. A. 
 
 ropes together and tie each set of strands temporarily, as 
 shown at Fig. A. 
 
 2. The twine 10 is now cut, and the strands 6 and 8 
 unlaid together, and the strands 5 and 7 carefully laid in 
 their places together for a distance of 18 inches. Then 
 unlay strands 6 and 8, also 5 and 7, and tie strands 6 and 
 5 together temporarily. Next unlay strand 7 and lay in 
 strand 8 in its place for a distance of 3 feet from strands 6 
 
 * Prepared for this work by the Manufacturers' Engineering Co., 
 Boston, Mass, 
 
Z'-0 ROPE-DRIVING. 
 
 and 5. Theii tie strands 8 and ? temporarily. Next cut 
 off the ends of the core so that they will butt together. 
 Strands 1, 2, 3, and 4 are next laid in the same manner as 
 
 FIG. B 
 
 strands 5, 6, 7, and 8, but in the opposite direction. Care 
 must be taken to keep the turns in- the strands, or otherwise 
 they will be soft and bulky. Next cut off all the strands 
 
 FIG. C. 
 
 to a length of about 24 inches, for convenience in handling. 
 At this point the splice should be as shown in Fig. B, 
 
llOPE-DRtVIN-G. 22tf 
 
 The tension strand of a Lambeth cotton rope is the soft 
 white yarn running through the centre of the strand, 
 and is called the tension strand through its having to 
 bear the strain put upon the rope in the transmission of 
 power. 
 
 The friction bands of a Lambeth rope are the twisted 
 outside yarns which are tubed around the tension strands 
 to protect them from wear and contact in the grooves of 
 the pulley. 
 
 3. Take strand 2 and unlay it two turns and remove the 
 ten friction bands, then lay in tension strand 2 back again 
 
 FIG. D. 
 
 one turn, split out J of tension strand 2 and lay in the re- 
 maining | of tension strand 2 for one turn. This will 
 bring it to its former position. Remove the ten friction 
 bands from strand 1, and tie tension strand 1 and } of 
 tension strand 2 in a simple knot. At this point of the 
 knot the rope will be its original diameter, as shown in 
 Fig. 0. 
 
 4. Divide the friction bands removed from strand 1 in 
 two parts, and take J of tension strand 2, put it between 
 the two parts and over tension strand 1 and through the 
 
HOPE-DRtVIKG. 
 
 centre of the rope with the marlinspike. Next take ten- 
 sion strand 1 and work it around the } of tension-strand 2 
 in the manner as shown at Fig. D. 
 
 5. Draw it taut and continue to work it around J of ten- 
 sion strand 2 until it reaches the of tension strand 2; 
 at this point of tension strand 1 must be removed, and 
 continue to work f of tension strand 1 around tension 
 strand 2 until it reaches the friction bands removed from 
 strand 2; divide these friction bands in two parts, and take 
 f of tension strand 1, put it between the two parts and 
 
 FIG 
 
 over tension strand 2, and through the centre of the rope 
 with the marlinspike. Next take the quarter of tension 
 strands 1 and 2 and pass them through the centre of the 
 rope on opposite sides with the marlinspike. Then half 
 of the friction bands should be passed through the centre 
 of the rope at each end with the spike. At this point 
 the splice is complete, with the exception of cutting oft' 
 the ends, and should be as shown at Fig. E. 
 
 G. The strands 3, 4, 5, 6, 7, and 8 should be worked in 
 the same manner as 1 and 2. 
 
 Instead of using the ordinary marlinspike it will be 
 found very convenient to drill out the body as shown in 
 
OP THE 
 
 UNIVERSITY 
 
 Fig. 8, leaving only a thin shell four or five diameters 
 deep. 
 
 By inserting the end of a strand iii the bore of the mar- 
 
 . 7. ROPE-SPLICING. 
 
 linspike the latter, with the strand, may be passed through 
 and around the other strands as desired with much less 
 trouble than ordinarily attends the operation. 
 
 FIG. 8. IMPROVED FORM OF MARLINSPIKE. 
 
 For small braided ropes which cannot be spliced, a 
 very convenient method of joining the ends is by the 
 use of copper ferrules as shown in Fig. 9, which repre- 
 
 FIG. 9. COUPLING FOR BRAIDED ROPE. 
 
 sents a form of joint devised by Mr. B. Frank Barnes, 
 Rookford, Til. 
 
24 ROPE-DRIVING. 
 
 Samples of tins splice were furnished by Mr. Jo^. 
 Burnett, which showed an efficiency of about 85 per cent. 
 Thus in two samples the average breaking strength of the 
 rope was 380 and 375 pounds respectively, while the splice 
 pulled out under a strain of 320 pounds. 
 
 In this case the rope was a f-inch braided cotton cord 
 which had been in use about three years. The coupling 
 consists of a piece of copper tubing 1 inch long, into one 
 end of which the rope is inserted about half way. A 
 groove is then compressed around the tube and rope, by 
 means of a special tool; the open end of the tube is then 
 filled with sealing-wax and heated until the wax boils, 
 then the other end of the rope js inserted, and the tube 
 compressed. The melted wax fills the end of the rope, 
 making a solid joint between the shoulders. With the 
 large pulleys adopted (22 to 48 ; inches in diameter) no 
 trouble is experienced, and the ropes last from two to 
 three years, but the copper ferrules are changed about 
 every four months. 
 
 Wood pulleys are used, and the grooves are filled with 
 leather. Some of these ropes run as high as 5300 feet per 
 minute. All the ropes are operated on the American or 
 continuous wind system. 
 
ROPE-DRIVING. 25 
 
 CHAPTER III. 
 
 A GOOD exam pie of this system of rope transmission is 
 shown in Fig. 10, which represents a plant designed by 
 Mr. T. Spencer Miller, and erected for the Western Electric 
 Company, New York, by the Link Belt Machinery Co. 
 In this case vertical ropes are used, which are arranged to 
 transmit the power of two 200-h. p. Russell engines, 
 cylinders 18 by 27 inches, making 125 revolutions per 
 minute; fly-wheels 10 feet diameter, each turned with 
 eight grooves for 1^-inch rope. The ropes are of raw- 
 hide and wound continuously around the pulleys. As the 
 rope leaves the fly-wheel at the left-hand side it runs over 
 an idler, and from thence to a tension-pulley, or tightener, 
 which is suspended in such a manner as to be drawn back 
 by the weights, as shown. The arms which support the 
 tighteners are hung from rollers, which are grooved to fit 
 the surface of a section of extra heavy wrought-iron pipe, 
 upon which they roll. From this tightener the rope 
 passes direct to the right-hand groove of the pulley on the 
 main shaft above the engine, the tightener-pulley being 
 inclined sufficiently to make the bottom come in line with 
 the left-hand, while the top comes in contact with the 
 right-hand groove. 
 
 The main pulleys which drive the shaft above the 
 engines are mounted upon and keyed to sleeves 10 inches 
 diameter, which extend out on each side far enough to 
 form journals, by which they are supported in pedestals 
 independently of the shaft (see Fig. 33). Through the 
 sleeve is a hole considerably larger than the shaft which 
 passes through them and which is supported by separate 
 
26 
 
 ROPE-DRIVING. 
 
 FIG. 10. AMERICAN OR WOUND SYSTEM 
 OF HOPE TRANSMISSION, 
 
ROPE-DRIVING. 27 
 
 pedestals. One end of each sleeve is so formed as to 
 make, in connection with a sliding collar which is on the 
 shaft, a positive interlocking clutch, which can be thrown 
 in or out by a lever. In this way both engines may be 
 working at the same time, or the shaft may be run by 
 either engine alone, the other pulley standing and impos- 
 ing no friction upon the shaft. All the bearings of this 
 shaft are adjustable laterally by set-screw and vertically by 
 wedges. The other large pulleys upon this shaft are 
 driven by friction-clutches, and are used for driving dyna- 
 mos, each pulley driving two dynamos arranged tandem, 
 one belt running over the other. This shaft runs at 220 
 revolutions and is 44 inches diameter. At one end of the 
 main shaft is a pulley having twelve grooves, in which run 
 two ropes side by side to the top of the building and 
 around the various pulleys down again. Either of these 
 ropes is calculated to be amply strong for the work, but 
 two are used to avoid the necessity for stopping should 
 one break. Each of them winds three times around the 
 pulleys, thus giving six driving-strands. To avoid crowd- 
 ing, the tightener for one of these is placed upon the floor 
 above the other. The ropes pass from the main pulley 
 three times around the pulley above and then go to the upper 
 floors, as indicated. From the shafts, the wood-working 
 machinery, blowers for the foundry, and some of the eleva- 
 tors are operated; the other lines being used for driving 
 light machine tools, such as are used in making electrical 
 apparatus. 
 
 Each floor has a cut-off coupling, which is so arranged 
 that in case of accident it can be cut off at a minute's 
 notice, or when running overtime any floor can be cut off. 
 thus saving the cost of running any more machinery than 
 is necessary. 
 
 The use of the tension -carriage plays an important part 
 in the American system of rope transmission. As usually 
 
28 
 
 ROPE-DRIVING. 
 
 made it is automatic in its operation and so weighted as 
 to give a constant tension to the rope, as indicated in 
 Fig. 11. In this arrangement an initial tension is given 
 
 FIG. 11. AUTOMATIC TIGHTENER FOR ROPE TRANSMISSION. 
 
 and maintained by the automatic tension-carriage, which 
 is free to move backward or forward on a horizontal track 
 
 FIG. 12. TENSION-CARRIAGE. 
 
 as the load changes or the rope stretches, always taking 
 up the slack and maintaining the proper tension. An- 
 other form of tension -carriage is that shown in Fig. 12. 
 
ROPE-DRIVING. 
 
 29 
 
 In tliis case the carriage is mounted on gas-pipe or solid 
 shafting, and is provided with ball bearings arranged with 
 cast pockets so that the balls are allowed to circulate 
 lengthwise of the bearings. 
 
 An arrangement of light angle-iron tracks supporting a 
 four-wheeled carriage is used to a considerable extent and 
 makes an excellent tightener where it can be 
 employed, as it is cheap and readily set up. 
 
 It is obvious that vertical tensions may be 
 arranged in a similar manner to those shown 
 in Figs. 11 and 12. In such cases the weight 
 may be suspended directly from the carriage or 
 even the sheave, as in Fig. 13, or it may be led 
 off in any desired direction either above or 
 below the tightener-pulley. The tightener 
 pulley is often inclined from the vertical, so 
 that its projection is equal to the width of the 
 driving pulley, in which case it not only serves 
 to maintain a constant tension in the rope, but FIG. 13. 
 it thus acts as a guide to conduct the rope from one groove 
 to another. 
 
 FIG. 14. ROPE-TIGHTENER. 
 
 A modification of the usual belt-tightener is sometimes 
 used for rope drives, as shown in Fig. 14, but this, it will 
 
30 ROPE-DRIVING. 
 
 be noticed, is positive in action and is only used to increase 
 the tension on the ropes as the latter become extended 
 with use. 
 
 In the same way the tightener shown in Fig. 15 * is 
 used to take up any slack that may occur in the rope. 
 
 In this case the tightener-pulley T is mounted on 
 a standard free to slide in the bottom guides G. A 
 weighted lever L is connected to the pinion-shaft S by 
 means of a ratchet-wheel and pawl (not shown in the 
 figure). On the other end of this shaft and rigidly con- 
 nected to it a pinion meshes with the rack 11 upon one of 
 
 FIG. 15. DYBLIE'S ROPE-TIGHTENER FOR DYNAMOS. 
 
 the guides and causes the standard and tightener-pulley 
 to move along the base and thus automatically take up the 
 slack as it occurs. A detent, D, in the carriage catches 
 in the teeth of the rack and prevents the tightener from 
 slipping back. It is evident that any form of adjuster 
 which will not allow the tension-pulley to move in both 
 directions either back and forth or up and down does 
 not maintain a uniformity of tension; as the humidity in 
 the air may cause a rope to shrink very materially in a 
 short time the tension on the rope will be greatly increased 
 if the tightener-pulley is prevented from moving in.f 
 
 * Patented by J. A. Dyblie, Jan. 14, 1890. 
 
 f Mr. Louis I. Seymour states that in a certain out-door drive at 
 
ROPE-DRIVING. 
 
 31 
 
 It is a well-recognized fact that atmospheric changes 
 affect the length of a rope, which in the presence of mois- 
 ture always contracts. Experiments have shown that a 
 dry hemp rope 25 feet long will shrink to 24 feet upon be- 
 
 FIG. 16. TENSION-CARRIAGE WITH VARIABLE ARM. 
 
 ing wet.* It is for this reason that, in addition to taking 
 np the slack caused by variation in load, provision should 
 
 the Plymouth Cordage Company's Works in which l-inch rope is 
 used to transmit 125 h. p. the tension carriage is drawn in about 
 eight feet by the shrinkage of the ropes during a severe storm ; that 
 is, the rope is shortened about sixteen feet. The total length of rope 
 in this case is approximately 1600 feet, so that the contraction is thus 
 about one per cent of its total length. The rope used was of 
 superior quality manilla, laid with plumbago and tallow, otherwise 
 the shrinkage would have been greatly in excess of the amount 
 stated. 
 
 * Indian En.r/ineer, 1888. 
 
32 EOPE-D RIVING. 
 
 be made for maintaining a proper tension in the rope 
 when the latter is variable in its length, due either to 
 atmospheric changes, or permanent elongation as the rope 
 loses its elasticity. This variation in length is particularly 
 noticeable in rope-drives which are subjected to exposure 
 from the weather. 
 
 An arrangement sometimes adopted with horizontal car- 
 riages as a substitute for pulley and hanging weight is 
 shown in Fig. 16. In this arrangement the tail-rope, 
 usually of wire, is wrapped around and secured to a 
 grooved pulley-sheave keyed to a shaft which is fixed in 
 position but free to rotate. A weighted lever is secured 
 to the shaft by means of a set-screw and maintains a ten- 
 sion on the rope by virtue of its moment. It will readily 
 be seen that this tension will not be constant, for the 
 effective lever-arm of the weight, and hence the pull on 
 the tail-rope, will vary with the position of the lever; thus 
 in its normal position with the lever horizontal the tension 
 
 PR 
 
 in the tail-rope will be T - ; but if the tension-car- 
 riage moves either in or out on its guides the lever will 
 assume a new position as shown in dotted lines, in which 
 
 75 ~nf 
 
 case the tension will now be T = - . 
 
 As the initial tension which gives adhesion to the slack 
 side of the rope varies with the weight supported by the 
 tension-carriage, it is obvious that an increase of this 
 weight will increase the power which may be delivered by 
 the rope. As, however, the horse-power is proportional to 
 the difference in stress in the driving and slack sides of 
 the rope, the less weight on the tightener consistent with 
 obtaining sufficient frictional resistance to slipping, the 
 better will the ropes work. 
 
 An example of rope-driving, in which tension-carriages 
 
ROPE-DRIVING. 
 
 33 
 
 are arranged to work vertically, is shown in Fig. 17. In 
 this case the engine develops about 45 horse-power and 
 runs at 90 revolutions per minute, corresponding to which 
 the velocity of rope is about 1700 feet per minute. The 
 fly-wheel is six feet in diameter, and is grooved for five 1- 
 inch *opes The main sheave on the jack-shaf ' is also 6 
 
 FIG. 17. ROPE-DRIVE WITH VERTICAL TENSION-CARRIAGES. 
 
 feet in diameter, and is grooved for five ropes. The rope 
 passes continuously from the fly-wheel to the main sheave, 
 making five wraps; then over the deflecting-sheave to the 
 horizontal tension-carriage, and back to the fly-wheel. 
 
 The jack-shaft sheaves are grooved for four 1-inch ropes, 
 and are each provided with a friction-clutch, giving the 
 line two shafts and, practically, the advantages of inde- 
 
34 ROPE-DRIVING. 
 
 pendent motors, so that in case of accident to either the 
 other can be run independently. 
 
 The sheaves on these shafts are five feet in diameter, and 
 are grooved for five 1-inch ropes, four wraps being used 
 for the transmission of power; the other groove is for the 
 return from the vertical tension-carriage, which is shown 
 just beneath the pulley on the line-shaft. 
 
 Although used extensively for main drives and in many 
 cases for intermediate driving, ropes have not come into 
 use for general work throughout the factory. In certain 
 cases the whole of the driving is done by ropes, no cross- 
 shafts, gears, nor belts being used; but the greater conven- 
 ience of flat belts for conveying power to the machine in 
 ordinary shop-transmissions is so thoroughly acknowledged, 
 that any attempt to substitute rope for belting in general 
 would be met with little favor; nor would this be practi- 
 cable under the conditions which now obtain in our mills 
 and factories. One reason for this is the difficulty in 
 shifting a rope from a tight to a loose pulley. 
 
 Numerous devices have been employed for this purpose, 
 and in some few cases with satisfactory results; but the sys- 
 tem does not readily lend itself to such work, nor would 
 the arrangements that have been adopted be generally ap- 
 plicable. Another reason for its non-employment for 
 general work is the size of pulleys, and distance between 
 shafts which is necessary for satisfactory working with 
 ropes, and which, obviously, would exclude its use in many 
 instances. 
 
 Where the power is transmitted to a shaft whose axis is 
 in a different plane from that of the driver, it is often in- 
 convenient or impracticable to use bevel-gears or universal 
 couplings. For such special cases rope-driving is partic- 
 ularly satisfactory, as a flexible rope will readily conform 
 to any direction, and transmit its full power when arranged 
 with suitable idlers. 
 
ROPE-DRIVING. 
 
 35 
 
 An example of such transmission is given in Fig. 18. 
 This drive transmits 150 h. p. for the main shaft at a to 
 another at b, whose axis makes an acute angle with the 
 first, and which is several feet lower. The drive-sheave a 
 is 5 feet 3 inches in diameter and runs at 160 revolutions 
 per minute; it is grooved for six IJ-inch ropes, which have 
 a velocity of 2800 feet per minute; c and d are idlers, the 
 faces of which are nearly parallel to the drive-sheave; e 
 
 FIG. 18. ROPE-DRIVE WITH SHAFTS AT AN ANGLE. 
 
 and / are double idlers, there being two sheaves on each 
 shaft, one 1| inches less in diameter than the other. This 
 arrangement of idlers in quarter twist drives of this class 
 has been introduced where there are more than four ropes 
 in the system, with the object of reducing the wear conse- 
 quent to the friction produced by the side lead of the rope. 
 In drives where there are more than eight roes a cone 
 has been used to a limited extent 
 
 f OF THE 
 
 UNIVERSITY^ 
 
36 BOPE-DRIVING. 
 
 Such a practice is, however, very unsatisfactory, as the 
 trouble encountered by the differential driving exerted by 
 each rope on a different diameter of the cone is greater 
 than that which it attempts to obviate. However, by mak- 
 ing the several steps on the cone separate, so that they 
 form a series of independent idlers of different diameters,* 
 the difficulty is overcome. 
 
 It is evident that the employment of multiple idle 
 sheaves of equal diameter possesses many advantages over 
 a multiple-grooved pulley when used as guide-pulleys. 
 
 In this case each sheave is independent of the others, 
 and thus prevents in a large measure the evils due to dif- 
 ferential driving and slip which would otherwise occur 
 with fluctuations of load. 
 
 With ordinary transmissions where the vertical distance 
 between shafts is as great as 100 times the diameter of rope 
 no trouble is experienced from the side lead of the rope, 
 and, usually, no provision is made to obviate it. 
 
 An application of rope-driving to shafts at right 
 angles, embodying several excellent features, is shown 
 in Fig. 19. 
 
 The plant is designed to transmit 250 horse-power from 
 a 14-ft. fly-wheel, a, which is grooved for twelve 1-inch 
 ropes. The line-shafts, m and n, are driven independently, 
 and each drive has its own tension-carriage. The rope- 
 sheave 1} is 72 inches in diameter, and is grooved for five 
 ropes. At the side of the 72-inch sheave is a single- 
 grooved idler, i, loose on the shaft, which serves as a guide 
 for the rope to the tension-carriage. 
 
 The substitution of a loose idler for an extra groove on the 
 driven pulley in rope transmissions is due to Mr. Spencer 
 Miller, although it has long been in use in cable-railway 
 
 * This feature is the subject of a patent granted to Mr. Jghn 
 Gregg, March 11, 1890. 
 
ROPE-DRIVING. 
 
 practice; the advantage in its use is evident when we con- 
 sider that the tension-carriage, drawing out the stretch of 
 the rope, must necessarily drag the first rope through the 
 
 FIG. 19. TRANSMISSION AT RIGHT ANGLES. 
 
 groove of the pulley, which will require an excessive weight 
 on the tightener pulley and a greater length of time before 
 
38 ROPE-DRIV1KG. 
 
 equilibrium is restored. By having this groove made into 
 an individual wheel free to rotate on the shaft, this diffi- 
 culty is overcome, and the transmission responds very 
 freely to changes of load, so that when heavy machines 
 are thrown on and off the ropes are not set in vibration, 
 but the tension-carriage sheave K slides back and forth 
 on the track, taking up the shock, with a minimum 
 amount of wear on the ropes. 
 
 In the present case the rope runs continuously around 
 the fly-wheel and sheave from groove to groove. As it 
 leaves the fly-wheel at the left hand it passes over the idler 
 * to the tension-carriage sheave K, which is suspended on 
 adjustable hangers from a single-pipe track. This sheave 
 is tilted by means of the adjustable hangers, so that the 
 top is in line with the centre of the groove of the idler, 
 and the bottom is in line with the centre of the groove of 
 the guide-sheave j, which serves to carry the rope back to 
 the right-hand groove of the fly-wheel. 
 
 The employment of multiple idlers instead of a multiple- 
 grooved idle pulley is also shown in this figure, where power 
 is transmitted from the shaft n to another o at right 
 angles to the first. By engaging one or the other of the 
 driven pulleys e or f by means of a clutch, the shaft o may 
 be driven in either direction. 
 
 More recently ropes have been introduced to drive dyn- 
 amos and special isolated machines, and where the dis- 
 tance between dynamos and engine or driving-shaft is 
 sufficiently great to allow a moderate sag in the ropes, such 
 drives have been found to work very satisfactorily, provided 
 other conditions are favorable. Where the distance be- 
 tween shafts is limited, more wraps should be given to the 
 rope in order to lessen the tension in each member. One 
 great fault with dynamo drives is the use of too small a 
 pulley on the armature-shaft. We can point to a score of 
 plants using rope transmissions from a jack-shaft to dyn- 
 
ROPE-DRIVIKG. 
 
 39 
 
40 ROPE-DRIVING. 
 
 amo in which the rope is overstrained and the dynamo 
 pulley is only half as large as it should be, in consequence 
 of which the ropes are a constant source of trouble. 
 
 With cotton ropes the pulley may be somewhat smaller 
 than that used for a similar size of manilla rope; but in 
 any case there is a certain minimum diameter of pulley 
 which should be used for any given rope. (See page 179.) 
 
 Where the required number of revolutions cannot be at- 
 tained with the size of pulley imposed by the various con- 
 ditions, if the jack-shaft cannot be speeded up nor a larger 
 driving-pulley used, in such cases it would be better to take 
 out the rope and put in a good leather belt. 
 
 The simplest arrangement of rope transmission for dyn- 
 amos is that in which the rope is carried direct from the 
 engine fly-wheel to the grooved pulley on the armature 
 shaft, as shown in Fig. 20, which represents the system 
 used in the station of the Liverpool Overhead Railway.* 
 
 There are four horizontal compound condensing engines 
 with cylinders 15^ and 31 inches in diameter, 36 inches 
 stroke, each of which is connected to a separate generator 
 by means of 19 ropes 1J inches diameter. Each engine is 
 rated at 400 h. p. when running at 100 revolutions per 
 minute with 120 pounds initial pressure; as the fly-wheels 
 are 14 feet in diameter, the rope velocity will thus be about 
 4400 feet per minute. 
 
 In general it is more desirable to drive the machines 
 through an intermediate jack-shaft, especially so in those 
 cases where a varying amount of current is required, as, 
 for instance, in the lighting of public buildings. Such an 
 arrangement, when the jack-shaft is provided with suitable 
 friction or jaw clutches, will permit machines to be thrown 
 on or off as desired. The use of an intermediate shaft also 
 permits the attainment of the requisite speed of the dyn- 
 
 * Power, May, 1893. 
 
ROPE-DRIVING. 41 
 
 amo with moderate proportions of pulleys. In fact with 
 many of the smaller engines in use a jack-shaft is essential 
 if we wish to use rope-driving. 
 
 Take, .for example, a 75-h.-p. Corliss engine, running at 
 85 revolutions per minute; the diameter of fly-wheel for 
 this engine is 10 feet, and if we wish to drive the dynamo 
 direct through a f-inch rope, the pulley on the armature- 
 shaft should be, preferably, not less than 24 inches diam- 
 eter; the speed of the armature would then be only 425 
 revolutions per minute. To obtain a suitable speed, the 
 driven pulley could not be more than about 12 inches in 
 diameter, and with larger ropes this difference would be 
 still more pronounced. 
 
 A similarly rated high-speed engine runs at 230 revolu- 
 tions per minute, and is provided with a fly-wheel 5 feet in 
 diameter; with the same 24-inch pulley on the d}mamo the 
 speed of the latter when driven direct would be not more 
 than 570 revolutions per minute, so that in this case also 
 the driven pulley would have to be reduced very much 
 below that size which has been found best adapted to the 
 work. 
 
 It is true that the diameter of driving-wheel on the en- 
 gine-shaft could be increased, and this is sometimes done. 
 In the cases quoted the diameters necessary to give the re- 
 quired speed would be about 20 feet for the Corliss and 7 
 feet for the automatic engine. As these sizes give a cir- 
 cumferential velocity within the limit of safety from the 
 action of centrifugal force of the metal in the rim, it would 
 be highly desirable to use such driving-wheels if other prac- 
 tical considerations did not preclude their use. A driving- 
 wheel 7 feet in diameter could readily be used on the high- 
 speed engine without materially augmenting the journal 
 friction, and the increased rim speed would be a beneficial 
 factor in preventing momentary fluctuations due to change 
 in load. 
 
42 ROPE-DRtVlKG. 
 
 With the Corliss engine, however, the increased weight 
 due to a large built-np fly-wheel 20 feet in diameter would 
 usually debar its use on an engine of this capacity; in ad- 
 dition to this the large diameter would prevent its use in 
 many locations, even if the increased weight and loss of 
 power were no hindrance. Under these conditions the use 
 of a jack-shaft is the most suitable arrangement. 
 
 In many cases it is desirable to use two engines so ar- 
 ranged that either or both may furnish the power to any 
 one of several dynamos, as shown in Fig. 21, which repre- 
 sents a 175-h.-p. rope-transmission plant erected by the 
 Link-belt Engineering Company in the Virginia Hotel, 
 Chicago, where two Corliss engines are each connected to 
 a jack-shaft, having five counter-drives to the dynamos. 
 Both the driven and driving sheaves on the jack-shaft are 
 loose on the shaft, and are connected to it by means of fric- 
 tion or jaw clutches, thus permitting either or both engines 
 to be run, or any one of the five dynamos to be thrown in or 
 out of use. The positive jaw-clutch is used on the driven 
 sheave, as it does not readily get out of order and is 
 preferred by many engineers to the average friction- 
 clutch, especially in those cases where much power is 
 transmitted. 
 
 If it is desired to couple one engine to the shaft while 
 the other is running, the former is speeded up until the 
 loose driven sheave comes up to the speed of the shaft, 
 when the dog-clutch may be readily thrown in gear with- 
 out shock. With this arrangement there is a strong ten- 
 dency for the bearings of the driven sheaves to heat when 
 not coupled to the shaft; for this reason provision should 
 be made to reduce the friction between the loose pulley and 
 the shaft by relieving the tension on the rope when not in 
 use, or, what is much better, the loose sheave should be 
 mounted upon a hollow sleeve supported in pedestals inde- 
 pendently of the shaft, as noted in description of plant 
 
ROPE-DRIVIXG. 
 
 43 
 
 shown in Fig. 10; see also Fig. 33. In this way either 
 pulley may be at rest and impose no friction on the shaft. 
 
 FIG. 21. ROPE TRANSMISSION WITH JACK-SHAFT. 
 
 If T l is the maximum stress in a rope, T 9 the stress in 
 the slack part, and P(=T l T t ) is the driving force, then, 
 
44 
 
 ROPE-DRTVIXG. 
 
 as we shall show subsequently, the ratio of the maximum 
 
 T 
 
 stress T l to the driving force P, or -~, will vary from 
 
 about "1.5 to 5.5 depending upon the speed of rope, the 
 coefficient of friction, and the angle embraced by the rope 
 on the circumference of the pulley. In order, then, to 
 have the transmitting force P as large as possible for a 
 maximum tension T l9 the tension in the slack part of the 
 rope necessary for adhesion must be reduced to a mini- 
 mum. To a certain extent this can be obtained by de- 
 creasing the angle between the sides of the groove, but if 
 carried too far this is a detriment rather than an advan- 
 tage, for if the angle is sufficiently acute the rope will 
 wedge and require more or less force to pull it out of the 
 groove. The remaining expedient is to increase the arc of 
 contact. 
 
 It can be shown that the friction of a cord or rope 
 wrapped upon a fixed cylinder is independent of the di- 
 ameter of the cylinder, and that it increases very rapidly 
 with an increased arc of contact.* If the conditions are 
 such that the coefficient of friction = one third, a ten- 
 sion of one pound at the end T 9 of the rope, Fig. 22, will 
 support a strain at the end T l of: 
 
 1.69 pounds foi 
 
 .-. 2.85 
 
 8.12 
 
 65.94 
 
 530.43 
 
 4,348.53 
 
 an arc of contact equal to coil. 
 
 Therefore by increasing the number of wraps around the 
 cylinder it is possible to increase the difference between the 
 tensions in each part of the rope almost indefinitely. It 
 will be noticed here that the rope is not in flying motion, 
 
 *Weisbach, vol. i. p. 360. 
 
ROPE-DRIVING. 45 
 
 which would cause an equal centrifugal force to be set up 
 in each member, thus altering the ratio of stress. As the 
 centrifugal force varies with the square of the velocity, 
 there is with an increasing speed of the rope a decreasing 
 useful force and an increasing total tension 01? the slack 
 side; but up to a given limit, which we shall subsequently 
 show lies between 4000 and 6000 feet per minute, the 
 total power transmitted for a given maximum tension in 
 t-he rope will increase with the velocity. 
 
 The great advantage thus obtained by increasing the ad- 
 hesion was very early applied to numerous mechanical de- 
 vices, chiefly for winding and hoisting purposes, and later 
 for haulage systems. We are indebted to Willis* for the 
 following quaint account and sketch (Fig. 23) of an ar- 
 rangement for obtaining a continuous motion with a con- 
 tinuous travelling-coil, first suggested by the author of the 
 article, Sir Christopher Wren, over two hundred years ago. 
 
 "A DESCRIPTION AND SCHEME OF DK. WREN'S INSTRU- 
 MENT FOR DRAWING UP GREAT WEIGHTS 
 FROM DEEP PLACES." 
 Read May 5, 16~0. 
 
 " Having considered, that the ways hitherto used in all 
 Engina for winding up Weights by Eoaps have been but 
 two, viz. the fixing one end of a roap upon a cylinder or 
 Barril, and so winding up the whole coyle of roap ; the 
 other by having a chain or a loose roap catching on teeth, 
 as is usual in clocks: but finding withall that both these 
 wayes were inconvenient the first, because of the riding of 
 much roap in winding one turn upon another; the other, 
 because of the wearing out of the chain or roap upon the 
 teeth, I have, to prevent both these inconveniences, devised 
 another to make the weight and its counterpoyse bind on 
 
 * " Principles of Mechanism," p. 439 et seq. 
 
46 
 
 ROPE-DRIVING. 
 
 the cylinder, which it will doe if it be wound three 
 about. 
 
 "But because it will then in turning, scrue on like a 
 worm, and will need a Cylinder of a very great length, 
 therefore if there be two cylinders each turned with three 
 notches and the notches be placed alternately, the convex 
 
 FIG. 22. 
 
 FIG. 24. 
 
 FIG. 23. 
 
 edges to the concave as in the figure here adjoyned, the 
 roap being wound three times about both cylinders, will 
 bind firmly without slyding and work up the weight with 
 a proportionable counterpoyse at the other end of the 
 Koap." 
 
 The method of obtaining increased adhesion for a given 
 wrap by increasing the number of coils in contact with 
 
ROPE-DRIVING. 
 
 47 
 
 each pulley has long been in use in rope-driving. In some 
 of the earlier applications the grooves of the pulleys were 
 semicircular in section, and of sufficient size to allow the 
 rope to embrace the entire circumference of each pulley as 
 represented in Fig. 24.* 
 
 A better arrangement is that shown by Over man, f in 
 which the advantage of the angular groove is obtained, 
 and the ropes are not worn by rubbing against each other. 
 This is shown in Fig. 25, and is thus described: 
 
 FIG. 25. 
 
 "If the pulley A is grooved, of which at least two are 
 fastened to the same shaft, the rope is directed on one of 
 these pulleys, and passing around it goes to J3, which re- 
 volves on an inclined axis, such that the rope will be re- 
 ceived from A' and delivered to A in the plane of the 
 grooves. The number of pulleys may be multiplied to 
 gain adhesion. This method of augmenting friction is 
 preferable to the tension-roller, as no increase of tension 
 is required; and it has the additional advantage of bend- 
 ing the rope in the same direction, which makes it more 
 durable." 
 
 * Willis. Principles of Mechanism. 
 | Overman's " Mechanics," 1851. 
 
48 
 
 ROPE-DRIVING. 
 
 A similar arrangement was introduced in the San Fran- 
 cisco Cable Railway in 1877. In this case, Fig. 26, the 
 endless hauling-cable is passed alternately backward and 
 forward over two grooved drums a sufficient number of 
 times to obtain the necessary driving adhesion. 
 
 Positive motion is imparted to the drum A, which hauls 
 in the cable, while a second pulley, B 9 acting as an idler, 
 
 FIG. 26. COIL-FRICTION, STREET-RAILWAY CABLE. 
 
 increases the f notional grip of the cable on the drum, by 
 virtue of the increased arc of contact due to the number 
 of wraps.* 
 
 When, however, the cable permanently lengthens by 
 stretching, the drum B may be moved further back by 
 means of the sliding-base so as to take up the resulting 
 slack. 
 
 By the use of this winder pulley the property of fric- 
 tional adhesion produced by successive coiling is perfectly 
 
 *" Cable or Rope Traction," J. Bucknall Smith, C. E.; Engi- 
 neering, London, 1887. 
 
effectual, for, although each coil is only in contact with a 
 semi-circumference, the accumulation of frictional resist- 
 ance is produced precisely as if entire circumferential 
 grooves were employed. 
 
 However desirable such winder pulleys may be for cable 
 haulage or hoisting purposes, their advantage is greatly 
 overestimated when applied to continuous rope-transmis- 
 sion. A little consideration will show that the frictional 
 adhesion produced by a tension weight acting on a running 
 rope with numerous wraps is entirely different from coil- 
 friction. In the latter we have an accumulation of friction 
 by which a small resistance applied at one end of the 
 rope is able to hold an enormously greater load at the other 
 end. 
 
 In the continuous-rope system of power transmission, 
 however, the load is distributed among all the wraps, so 
 that when properly adjusted each wrap carries an equal 
 proportion of the load and is subjected to an equal resist- 
 ance on its slack side. There are few cases where the com- 
 bination of the winder-pulley with the continuous- rope 
 system offers any decided advantage over other methods. 
 
 There is an incidental advantage in using a winder, 
 especially in those cases where the difference in diameters 
 between the driver and follower is quite appreciable; under 
 such conditions the adhesion of the rope on each pulley 
 may be made more nearly uniform by the employment of a 
 winder, and there is less liability of the rope slipping in its 
 groove, but this may usually be obtained more satisfactorily 
 by other means (see page 168). 
 
 In numerous cases ropes using winder-pulleys have been 
 installed without regard to the work to be done or strain 
 put upon the ropes, and many of the evils of rope-driving 
 are directly traceable to this cause. 
 
 Many engineers ar,e opposed to using the winder-pulley 
 
50 HOPE-DRIVING. 
 
 in any form whatever, but occasionally it may be used to 
 advantage. 
 
 In outdoor or long-distance transmissions, or in special 
 cases where it is desired to transmit a maximum power 
 without undue stress in the rope, or in particular cases 
 where the ordinary working stress may be exceeded, a 
 winder-pulley may frequently be used to advantage, if the 
 slack-side tension be reduced accordingly. Using a winder- 
 pulley and increasing the back tension will permit a very 
 large increase in the power transmitted; but since this im- 
 poses an excessive strain on the rope, it soon wears out, and 
 is a constant source of trouble. 
 
 The gain in power by increasing the adhesion will be at 
 the expense of journal-friction, which is thus augmented 
 by the employment of wider-faced driving and driven pul- 
 leys, in addition to that due to one or two more winder- 
 pulleys; the wear of the rope, both external and internal^ 
 will also be greatly increased on account of the greater 
 number of flexures given to the rope in passing over the 
 winder-pulleys. 
 
 The use of a winder-pulley at each end of a long drive 
 in which only a single strand runs from the driver to driven 
 pulley is an example of the application of coil-friction to 
 the continuous-rope system; in this case both the working- 
 load and the back tension is carried by one rope instead of 
 being distributed among several wraps, as usually happens 
 in this system. 
 
 That the percentage of gain is not as great as might be 
 expected from the employment of coil-friction, will be seen 
 from the following considerations : 
 
 The ratio of tensions in the tight and slack sides of a 
 rope running over two pulleys is dependent upon several 
 factors, and may be determined from 
 
 a*. = 
 
ROPE-DRIVING. 51 
 
 in which T^ = tension in tight side of rope, 
 jP 2 = tension in slack side of rope, 
 e = base of hyp. log = 2.7183, 
 = coefficient of friction, 
 a = arc of contact (circular measure). 
 
 In this case the influence of centrifugal force is neg- 
 lected. 
 
 Since the power transmitted by a wrapping connector 
 is dependent upon the difference of tensions in the tight 
 and slack sides, it is evident that with an assumed total 
 tension T t the available force for transmitting power will 
 increase as T z decreases. 
 
 But T 9 may diminish as e^ a increases, that is, since e 
 is a constant and is constant for a given pulley and ma- 
 terial, T 9 decreases as a increases; hence if we increase 
 the arc of contact the tension in the slack side of the pul- 
 ley may be decreased in a ratio greater than unity, depend- 
 ing upon the factors involved; in which case the net force 
 P available for transmission will be increased, while the 
 original assumed allowable tension remains the same. 
 
 For example, if T l 200 pounds, = 0.3, a = 2.88 
 (a = 165), we shall have 
 
 900 
 
 .,uu _ _ 
 
 *! /O ryi Q\0.3X2.88 ua: > 
 
 and the net force P = 200 84 = 116 pounds. 
 
 Under the same conditions, if we pass the rope from the 
 driver over a winder-pulley back and forth twice, and then 
 to the driven pulley and its winder in the same way, we 
 should be able to transmit a little more than one and a half 
 times as much power at the same speed without increasing 
 the working tension in the rope. 
 
 T 
 
 In this case a 12.56 and ~ 43.1; hence 
 
 * 3 
 
52 HOPE-DRIVING. 
 
 r a = 4.6 and P = 200 - 4.6 = 195 + pounds. 
 
 Horse-power in second case _ 195 _ 
 * Horse-power in first case 116 "~ 
 
 Now, if it were practicable to maintain the same slack- 
 side tension T^ in these two instances and increase the 
 driving-side tension under the conditions of the second 
 
 T 
 case, viz. -^ = 43.1, we should have 
 
 P=T l -T 9 = (43.1 X 84) - 84 = 3536 pounds, 
 
 3536 
 
 and the ratio of power transmitted will now be or 
 
 116 
 
 thirty times as great as before. 
 
 These results indicate that while we may vastly increase 
 the driving-side tension for a given slack-side tension, by 
 using a winder in the manner indicated, that is, with a 
 single wrap connecting driver and follower, yet if we wish 
 to maintain an assumed maximum working tension for a 
 given-sized rope the percentage of gain will not be very 
 great under the usual requirements of rope-driving. 
 
 For a temporary drive the working strain may be in- 
 creased to about twice the usual value, but for a permanent 
 installation the usual working value should not be ex- 
 ceeded. 
 
 Where a number of ropes are employed on a short-drive 
 it is questionable whether the winder-pulley possesses suffi- 
 cient advantages to warrant its employment in place of the 
 continuous-wrap or individual-rope systems. In any case 
 the conditions should be carefully considered, and the ac- 
 tual gain compared with the various losses involved. 
 
 A recent example illustrating the application of the 
 winder-pulley is shown in Fig. 27, which represents the 
 system of rope-driving installed by Messrs. Hoadley Bros, 
 in the Fifty-second Street electric power-house of the 
 Chicago City Railway Company. 
 
54 ROPE-DBIVItfG. 
 
 The plant is designed for 10 generators of the Westing- 
 house No, 6 type, running 300 revolutions per minute. 
 There are also to be ten 24-inch by 48-inch engines of the 
 improved Wheelock type, arranged in five pairs, two oi 
 which are now in operation. These run at 100 revolutions 
 per minute with 100 pounds boiler-pressure. The power 
 usually varies from about 200 to 1000 h. p., but [during the 
 heavy traffic throughout the summer each pair of engines 
 has frequently transmitted 1500 h. p. 
 
 These engines have a built up fly-wheel (10 segments) 18 
 feet in diameter, 39 inches face, which weighs about 
 50,000 pounds. The rim is grooved for 21 wraps of 1- 
 inch manilla rope. The driven pulleys are 6 feet in 
 diameter, and contain 32 grooves for the rope, which runs 
 about 5600 feet per minute. Between the driven pulleys 
 and the engine fly-wheel there is placed a 6-foot winder, 
 containing 11 grooves, around which the rope is car- 
 ried before passing to the tension-sheave (Fig. 28), which 
 in the present arrangement is placed horizontally above 
 the engine near the ceiling, as shown in Fig. 29. Thus the 
 rope is wound around the engine fly-wheel and the driven 
 pulley, making 20 wraps; then it is carried from the 
 driven pulley to the winder back and forth 11 times, 
 thence it is led over vertical guide-pulleys, 7 feet in diam- 
 eter, to the horizontal tension-sheave 54 inches in diameter, 
 then down over another vertical guide-pulley to the fly- 
 wheel, where it started. By this means the arc of contact 
 of each member of the driving-rope is increased practically 
 180 degrees when all the ropes have adjusted themselves 
 to the load, so that the power transmitted with the same 
 tension in the rope will be about forty per cent more, if we 
 neglect friction, than would be transmitted by the twenty- 
 one wraps over the fly-wheel without the use of a winder. 
 
 The net gain will be considerably less, owing to the vari- 
 ous losses which this system entails. 
 
55 
 
56 
 
 The application of ropes to transmit the power from a 
 water-wheel to a line-shaft 40 or 50 feet or more above the 
 axis of the wheel has lately received considerable attention, 
 and offers many advantages over the ordinary method in 
 which a vertical shaft is used. The extreme weight of the 
 latter in many cases makes it a difficult matter to provide 
 a suitable bearing to support it. In such cases a horizon- 
 tal turbine is used, and the wheel-shaft carrying the rope- 
 pulley is extended and suitably supported, as shown in 
 Fig. 30. 
 
 The station of the Brush Electric Light and Power Co. 
 at Niagara Falls is driven in this way. 
 
 A line-shaft runs, through the building, with one end ex- 
 tending over the wheel-pit; to this are belted the gener- 
 ators in the usual manner. Seventy-five feet below this 
 shaft is located a 15-inch horizontal Victor turbine in a 
 case of boiler-iron, its shaft extending to bearings supported 
 by bridge-trees, which in turn are carried by the foundation 
 I beams that support the wheel-case. This shaft carries an 
 iron pulley 40 inches in diameter, grooved for 12 f-inch 
 manila ropes. (Cotton was tried, but was not satisfactory 
 here.) 
 
 The pulley on the driven shaft above is of wood, 70 
 inches in diameter. The driving side of the ropes hang- 
 perpendicularly, and are free from the driver to the driven 
 pulley. 
 
 The slack side has two idlers or guide-pulleys, one of 
 which is situated immediately below the driven pulley, and 
 the other is about 20 feet above the driver. A tightener is 
 adjusted in a running frame, in line with the driven and 
 upper guide-pulleys. 
 
 In putting on the rope the following course is taken : 
 " Commencing at one side of the pulleys the rope is passed 
 around from the driver to the driven pulley in every alter- 
 nate groove until the opposite side is reached, thence 
 
around the tightener-pulley in the running frame, which is 
 hung on an incline in such a manner that its discharg- 
 ing side is in line with the side of the driver pulley 
 whence we started . The remaining grooves are then filled, 
 and the ends of the rope are spliced in position around the 
 idler. Thus it is readily seen that there are two strands of 
 the rope on the idler at all times. The object of this is to 
 have one solid piece of rope on the idler at the same time 
 that the splice is, so as to relieve the spliced piece of the 
 strain of the idler. This system is giving far better satis- 
 faction than has the upright shaft to those who have tried 
 both."* 
 
 The plant designed by Mr. Robert Cartwright for the 
 electric station of the Citizens' Light and Power Company 
 of Rochester, N. Y., is worthy of careful study, and may 
 be considered a representative modern plant, adapted to 
 use steam or water-power, and employing both ropes and 
 belting, f 
 
 Fig. 31 represents a cross-section of the station, and 
 shows the general arrangement of the plant. The water- 
 wheels are twin Poole-Leffel central-discharge turbines, 
 23 inches in diameter, and at a speed of 560 revolutions 
 per minute, under a head of 92 feet 6 inches, develop 500 
 horse-power each, with a discharge of 3800 cubic feet of 
 water per minute. The wheels proper are made of phos- 
 phor-bronze, with buckets of Otis steel, tinned. The wheel 
 bed-plates are heavy cast-iron box sections, machined and 
 bolted together with heavy bolts fitting reamed holes. The 
 wheel-shaft is 4f inches diameter, running in adjustable 
 babbitt-lined bearings. A rope-wheel 4 feet in diameter is 
 keyed on the shaft, and is grooved for fifteen 1^-inch 
 manilla "Stevedore " ropes, made with four strands and a 
 core, worked in with plumbago in the process of making. 
 
 * F. E. Pritchard, Elect. World, April 16, 1892. 
 f See Trans. Am. Soc. C. E., vol. xxx., 1894, 
 
FIG. 31. SECTION THROUGH POWER STATION, 
 
59 
 
 From the 4-ft. wheel 15 ropes run to a rope-wheel on the 
 line-shaft above, 76.8 inches in diameter, and grooved for 
 sixteen 1^-inch ropes. The ropes being endless, the idler- 
 strand is passed over a 5-ft. single-grooved wheel, placed 
 in a movable frame. The frame traverses in iron guides 
 and maintains by its weight a constant tension on all the 
 ropes. This is made adjustable for the amount of tension 
 
 FIG. 32. PLAN OF POWER STATION. 
 
 by the application of counter-weights to the frame. The 
 speed of the line-shafts is 350 revolutions per minute, and 
 the rope travel is 7037 feet per minute. The water-wheels 
 are supplied from a steel flume 7 feet in diameter. From 
 the horizontal portion of the flume a 4-ft. pipe leads down 
 to each wheel, and has a geared 48-inch Chapman valve at 
 the lower end, between pipe and penstock, as shown in 
 
60 
 
 ROPE-DRIVIKG. 
 
ROPE-DRIVING. 61 
 
 Fig. 31. These valves are fitted with a 12-inch by-pass, 
 for the purpose of equalizing the pressure on both sides of 
 the large valve in opening or closing. 
 
 A horizontal " Woodbury " compound condensing slide- 
 valve engine, with extra heavy bed-plate, is set in the 
 power-room at point marked in Fig. 32 " Engine No. I." 
 Steam-cylinders are placed with the large cylinder out- 
 side, so that pistons and rod may be easily removed. Cylin- 
 ders are 19 inches and 31 by 24 inch stroke. At 167 revolu- 
 tions, with a boiler-pressure of 110 pounds per square inch, 
 vacuum 22 to 24 inches and cutting off at T 5 stroke, the 
 engine is rated at 500 horse-power, and is guaranteed to 
 produce a horse-power on an evaporation of 20 pounds of 
 water per hour. The crank-shaft is a steel forging in one 
 piece. Journals are 11^ inches diameter by 21 inches 
 long. Crank-pin 11^ inches diameter by 8 inches long. 
 The end carrying the rope-driving wheel has an outboard 
 bearing. Governor balance-wheel is 8^ feet diameter by 
 25-inch face. Rope-driving wheel is cast in halves 10 feet 
 6 inches diameter, and grooved for fifteen If-inch ropes, 
 These ropes lead to a 5-ft. rope-wheel on the line-shaft 
 above, with same arrangement for tightener as is applied 
 to the water-wheels. Rope speed of engine-drive is 5500 
 feet per minute. 
 
 The line-shafts are of hammered iron 5 inches in diameter, 
 and arranged with heavy floor pedestals, fitted with self- 
 adjusting, ring-oiled, babbitt-lined bearings. The rope- 
 wheels are placed on heavy cast-iron quills, furnished with 
 Hill friction-clutches of 500-horse power capacity each. 
 By a series of jaw-clutches, pulleys, and belts any line- 
 shaft can be operated from any water-wheel or engine, all 
 the line-shafts making the same number of revolutions. 
 Fig. 33 shows in detail the quill, clutch, and bearings. 
 
62 ROPE-DRIVING. 
 
 CHAPTER IV. 
 
 use of ropes in connection with portable tools and 
 travelling-cranes has long been established, and their con- 
 venience and adaptability to a wide range of work make 
 them a necessity in many shops. The advent of the small 
 electric motor in our machine-shops will, however, proba- 
 bly replace to a large extent all other forms of special 
 transmission for portable tools, as it is already replacing 
 countershafts and belting for machine-driving in many 
 cases. 
 
 One of the greatest fields of usefulness for rope-driving 
 is in the transmission of power to a moderate distance, 
 under conditions which are unfavorable to the use of belts 
 or shafting. 
 
 With rope-driving one is enabled at a comparatively 
 small cost to transmit power in any direction to a building 
 remotely situated from the source of power, which would 
 otherwise require a long and expensive line-shaft or an in- 
 dependent engine or other motor. The facility with which 
 it may be carried in any direction across rivers, canals, and 
 streets, above or under ground, up hill and down, over 
 houses and into buildings, is a feature very favorable to 
 the further extension of rope transmission; but the rapid 
 progress which has been made in the development of elec- 
 trical transmission has limited the economical application 
 of ropes to moderate distances. There are, however, cer- 
 tain limits between which the transmission of power by 
 ropes is yet more efficient than by any other known method. 
 
 The employment of ropes for this purpose i.e., trans- 
 mission of power to a distance is not a recent application. 
 
ROPE-DKIVING. 63 
 
 The first method of transmission of power to any consider- 
 able distance was made in 1850 by C. F. Hirn at Logel- 
 bach, near Colmar, Alsace.* " The works consisted of a 
 large number of buildings separated at some distance from 
 one another, which were required to be changed into a 
 weaving factory. As there was but one steam-engine on 
 the works, the expense of transmitting power to the various 
 buildings by ordinary shafting (the shortest length of which 
 was 84 yards), or of erecting separate engines, would neces- 
 sarily have been great; and the desire to obviate this ex- 
 pense resulted in the adoption of the telodynamic system. 
 
 The first plan adopted was the use of a band of steel 172 
 yards long, ^ inch thick, and 2 inches broad. This was 
 slung as an endless band over two wooden rollers or 
 pulleys, 6 feet 6 inches diameter, which were placed 84 
 yards apart and made 120 revolutions per minute, giving a 
 speed of 28 miles an hour in the band. In practice this 
 plan was found to be open to two objections : the lightest 
 wind agitated the baiK 1 ., and the pulley-guides tore it at the 
 points of riveting, whilst the guides themselves were 
 rapidly worn out. Notwithstanding these objections this 
 plan rendered valuable service, and continued in operation 
 for a year and a half, transmitting 12 horse-power to one 
 hundred looms. 
 
 "The difficulties of the flat band suggested round wire 
 ropes ^ inch diameter ; these were accordingly substituted 
 and were placed upon the same wooden pulleys, which, 
 however, were first grooved to the depth of half an inch.f 
 
 This plan answered every expectation, and experience 
 having fully sanctioned its use a second wire rope was soon 
 put in operation, transmitting the power to a distance of 
 
 * Mr. H. M. Morrison, Proc. Inst. M. E. 1874, p. 57. 
 
 f Prof. Unwin, in his Howard lectures (see Electrician, Feb. 3, 
 1893), states that an English engineer, Mr. Tregoning, suggested the 
 substitution of the wire rope. 
 
 OF THE 
 
64 ROPE-DRIVING. 
 
 about- 770 feet. The two pulleys were each 9 feet 6 inches 
 diameter, making 91| revolutions per minute, and a steel 
 rope | inch diameter was employed transmitting 50 h. p. 
 at a speed of 31 miles per hour. In this instance it was 
 found necessary to have supporting pulleys to prevent the 
 rope from trailing upon the ground. These carrying pul- 
 leys were placed half-way between the transmitting pulleys, 
 or 128 yards apart, and in the first instance they occasioned 
 very great difficulties by the rapidity with which they were 
 worn out in the groove. They were constructed succes- 
 sively of copper, wood, and polished cast iron, and were also 
 faced with leather, horn, india-rubber, lignum-vitae, and 
 boxwood. All these failed, however; the facings were soon 
 worn out, and when the groove was of metal or hard wood 
 and did not itself wear it destroyed the rope. After re- 
 peated experiments a dovetailed groove was formed in the 
 bottom of the pulley groove and filled with gutta-percha, 
 (as shown in Fig. 65, page 186.) 
 
 " This turned out a perfect success, and carrying pulleys 
 thus faced have an almost unlimited amount of durability/' 
 Fibrous ropes were used in the United States for long dis- 
 tance transmissions a few years later ; thus we find a com- 
 munication in the Scientific American* in which a cor- 
 respondent from Winsted, Conn., speaks of several rope- 
 drives in his vicinity, one of which had been in use since 
 1858. "It transmits the power for a manufactory, em- 
 ploying several circular saws, across the river, 225 feet dis- 
 tant, by a |-inch rope running over two pulleys six feet in 
 diameter, at a speed of 5600 per minute. The pulleys are 
 sheltered, but the rope runs exposed in all kinds of 
 weather, needing no attention except at times to be rubbed 
 with grease having a very small amount of rosin mixed 
 with it." 
 
 * Vol. HI, 1861, p. 315, 
 
ROPE-DRIVING. 
 
 65 
 
 < 
 
66 ROPE-DRIVING. 
 
 When power is taken from a water-wheel in locations 
 where laud is not available for buildings, the use of 
 ropes as a means of transmitting the power from the 
 wheel to the mill or factory forms a most economical ar- 
 rangement if the drive is properly designed for the work. 
 It is a great advantage in many other cases to have the 
 power plant and the several buildings of a works isolated 
 from each other; this is especially desirable in sawmills 
 and wood -working establishments, where the risk from fire 
 is greatly reduced by such an arrangement. 
 
 An example of rope-driving in which the conditions are 
 particularly adapted to this form of transmission was 
 erected a few years ago at Portland, Ore. The mill is built 
 on piles situated in the Willamette River, while the engine 
 and boiler room are upon the solid ground, some distance 
 away. The engines are a pair of Wheelocks, with cylinders 
 32 by 60 inches, intended to be speeded to 70 revolutions 
 per minute. The fly-wheel is 24 feet in diameter by 66 
 inches face; it is built up of ten segments grooved for 
 thirty-three 1^-inch manilla ropes, and fitted to a shaft 20 
 inches diameter; its weight is 40 tons. 
 
 Two ropes are taken from this wheel to jack-shafts 35 
 and 45 feet distant; the driven pulleys one with 16 and 
 the other with 17 wraps are each 76 inches in diameter, 
 and are keyed to 10-inch shafts. From the end of each 
 jack-shaft a 600-h.-p. transmission is arranged and 
 carried to the mill. Each shaft is fitted with a friction 
 clutch to allow either of the mill transmissions to be thrown 
 out if desired. 
 
 The general arrangement, showing location of driving 
 and driven sheaves, is represented in Figs. 34 and 35, from 
 which it is seen that power is delivered to two shafts 7 
 inches in diameter in the one case at a diagonal distance 
 of about 200 feet, and in the other at a distance of 185 feet 
 from their respective drivers both the driven shafts being 
 
ROPE-DRIVING. 67 
 
 at right angles with the jack-shaft. Each drive was 
 designed to transmit 600 h. p. with a rope velocity of 7550 
 feet per minute, but it has since been found advantageous 
 to reduce this speed to about 6000 feet per minute. 
 
 The arrangement of ropes in these transmissions is 
 similar to that used in the Chicago City Railway Company, 
 illustrated on page 53. In the present case only three wraps 
 of 1^-inch rope are used to convey the power from the jack- 
 shaft to the mill, but in order to prevent slip and decrease 
 the tension in the slack part of the rope the driving and 
 driven pulleys have each nine grooves, six turns being 
 carried around another pulley or winder, thereby increas- 
 ing the arc of contact and, hence, the adhesion. In this 
 plant the stress in the rope is very much greater than that 
 ordinarily used. 
 
 It is evident that the whole strain must be borne by the 
 three strands, as it is only the difference in tension of the 
 tight and slack sides of the ropes that can be used to trans- 
 
 PV 
 
 mit power; since = h. p., we find the difference in 
 
 ooOOO 
 
 33000 X 600 
 
 tension, Jr = ^7Tn ~ 3300 pounds ; and as there 
 
 oOOO 
 
 are three wraps, the difference in the stresses in the two por- 
 tions of the rope will be 1100 pounds. As we shall find 
 later the total stress will be greater than this, due both to 
 the action of centrifugal force in the rope and to the force 
 necessary for adhesion. 
 
 As the maximum working tension for a 1^-inch rope is 
 usually only about 450 pounds, it is evident that each 
 wrap carries nearly three times its proper load, taking the 
 wear and life of the rope into account. The engine has 
 only developed about 700 horse-po- er as yet, so the total 
 stress in each rope has been very much less than the 
 above, probably not more than 300 to 350 horse-power on 
 
68 HOPE-DRIVING. 
 
 each drive; but even with this reduced stress one rope was 
 replaced inside of fifteen months. 
 
 For short distances shafting is often employed for such 
 transmissions, but with this latter the friction of the jour- 
 nal-bearings is a very important consideration, and effectu- 
 ally debars its use for long distance transmission. 
 
 This will be seen from the following considerations : 
 
 Let = distortion of shaft (circular measure) per unit 
 
 length; 
 
 = distortion in degrees; 
 I = unit length of shaft; 
 L = length of shaft in feet; 
 
 r = distance of outer fibres from axis = -; 
 
 Z 
 
 d diameter of shaft; 
 PR = twisting moment on the shaft; 
 JV" = revolutions of shaft per minute; 
 v = velocity of circumference of shaft = ndN\ 
 G = modulus of torsion of the material 
 
 = two fifths of the modulus of elasticity; 
 f = maximum torsional stress in the outer fibres 
 IQPIt 
 
 TTd 3 ' 
 
 W = weight of shaft 3.36 pounds per foot per square 
 
 inch of section ; 
 F load due to friction. 
 
 ft WPRl m 
 
 Then * = = __ (i) 
 
 If the angle of torsion is given in degrees, then 
 
 _ * X 2* , 
 ~360~ 
 
 of length will be 
 
 X %7C 
 
 6 = ; therefore the angular distortion per foot 
 
 360 
 
ROPE-DRIVING. 69 
 
 0o _ 360 X /X12_360/ 1ZL 
 
 ' Tn X ~Gr~ -~^G ' ~T- ' ' (2 > 
 
 The working limit of the angle of torsion for steel shaft- 
 ing ought not to exceed 10 degree per foot in length of 
 the shaft; that is, 6 = 0.10Z, and for wronght-iron shaft- 
 ing fl = 0.075 L-* assuming the shaft to be of steel and 
 substituting the corresponding value for 8 in (2), we obtain 
 
 WL X nGd = 360/ X 12Z; 
 hence /- 800<Z if we assume that G 11,000,000 pounds. 
 
 Since the horse-power transmitted by the shaft equals 
 
 PR x ~ rpr, if we substitute the value of PR, (= -r^-/), 
 ooOOO \ 16' I 
 
 Ttd* 2 
 
 there is obtained h.p. = - -- f ; but the velocity at 
 
 16 ' ooOOO 
 
 the circumference of the shaft is v = ndN, also/= 800^; 
 hence 
 
 h.p. = 0.0095<Fv ....... (3) 
 
 If the bearing is well worn and fitted to its shaft the 
 resistance due to friction will probably lie between the 
 
 limits -(pW and ~<f>W,\ or between 1.570 Tf and 1.28 
 
 & Tt 
 
 01 X F, where is a coefficient, which in the present case we 
 
 shall assume equal to 0.06. 
 
 4. 
 Taking the lesser value, we shall have F= ~0W, 
 
 where .Pis the force at the circumference of shaft i eces- 
 sary to overcome the journal friction. If there are no 
 
 pulleys on the shaft W= ~^L X 3.36; the horse-power 
 
 exerted to overcome the friction will then be 
 
 Fv 4 , TT d*L X 3.36?' 
 h 'P" = 33000 = n* X 4 ~- 
 
 * Reuleaux : Der Konstrukteur. 
 f Unwin. 
 

 
 ROPE-DRIVING. 
 
 pressed as a ratio, the percentage of power required to 
 overcome friction will be 
 
 & (4) 
 
 Mb = o. 
 
 h.p. 0.0095dV 
 from which there is obtained 
 
 h.p. = 0.00063^ X h.p. - 
 d 
 
 That is, for a steel shaft whose diameter is one inch the 
 horse-power required to overcome the friction in a length 
 of 1585 feet will be equal to the total allowable transmit- 
 ting capacity of the shaft. 
 
 For wrought iron 6" ~ .075 L and h.p. .0075T0, from 
 which may be determined the value of the ratio 
 
 = 0.00( 
 
 or hp..=fx^ .. -(5) 
 
 The following tables, calculated from these formulae, 
 give the limits at which the power transmitted by a shaft 
 is absorbed by the friction of the bearings, the assumption 
 being that the factor of journal-friction equals 0.06, and 
 that the allowable twist shall not exceed 0.10 degree per 
 foot of length for steel, nor .075 degree for wrought iron. 
 The shaft is supposed to be free from all pulleys and gears. 
 
 TABLE I. LIMIT OF LENGTH FOR STEEL SHAFTING. 
 No pulleys on the line. 
 
 Diameter of 
 shaft in 
 inches. 
 
 Length in feet 
 when total 
 power is absorbed. 
 
 Length when 
 efficiency = 
 50 per cent. 
 
 Length when 
 efficiency = 
 75 per cent. 
 
 1 
 2 
 3 
 4 
 5 
 
 1,585 
 3,170 
 4,755 
 6,340 
 7,925 
 
 792 
 1,585 
 2,377 
 3,170 
 3,962 
 
 396 
 792 
 
 1,188 
 1,585 
 1,981 
 
ROPE-DRIVING. 
 
 71 
 
 TABLE II. LIMIT OF LENGTH FOR WROUGHT-IRON SHAFTING. 
 
 No pulleys on the line. 
 
 Diameter of 
 shaft in 
 inches. 
 
 Length in feet 
 when total 
 power is absorbed. 
 
 Length when 
 efficiency 
 50 per cent. 
 
 Length when 
 efficiency = 
 75 per cent. 
 
 1 
 2 
 3 
 4 
 5 
 
 1,250 
 2,500 
 3,750 
 5,000 
 6,250 
 
 625 
 1,250 
 
 1,875 
 2,500 
 3,125 
 
 312 
 625 
 937 
 1,250 
 1,562 
 
 From the foregoing it will be seen that shafting is alto- 
 gether unsuitable for conveying power any considerable 
 distance, and as belting is not adapted to this work choice 
 must be made of some other method. 
 
 For a mere dead pull, such as the alternate strokes 
 needed to operate a pump, work is, and has long been, 
 transmitted to very great distances; as by the long lines of 
 draw-rods, ropes, or wires used in mining regions, quarries, 
 and elsewhere, for transmitting the power of a water-wheel 
 by means of a crank on its main axis, pulling during half 
 its revolution, against a heavy weight at the end of the 
 line, and thus storing up energy for the return stroke. 
 
 Wooden pump-rods were used in this manner about 1865 
 near Petroleum, W. Va. A large condensing engine was 
 located in a central position, and the rods transmitted the 
 power to a number of oil-wells, twenty-seven in all, situated 
 at various distances and in different directions from the 
 source of power. The greatest distance was about four 
 miles. 
 
 Posts with crank-arms were used to change the direction 
 of the pull. The rods were of hickory, connected end to 
 end by means of iron straps. 
 
 The transmission of power from the famous 72-foot di- 
 ameter overshot wheel at Laxey, on the Isle of Man, is by 
 means of similarly connected trussed rods, which in this 
 
72 ROPE-DRIVING. 
 
 case are supported at regular intervals on small wheels 
 running on iron ways. About 150 horse-power are trans- 
 mitted in this way. 
 
 This method was adopted on a very large scale in the 
 mines of Devonshire for the transmission of power from 
 large overshot water-wheels to pumps fixed in the shaft of 
 the mine at a considerable distance higher up the valley. 
 
 In one case* the water-wheel was 52 feet diameter, 12 
 feet breast, and its ordinary working speed was 5 revolu- 
 tions per minute. The length of stroke given by the crank 
 to the horizontal or "flat " rods was 8 feet; the rods were 
 3|-inch round iron, and were carried on cast-iron pulleys. 
 
 At Devon Great Consols, near Tavistock, there are alto- 
 
 lec. World 
 
 Fra. 36. ROPE-DRIVES WITH BENT CRANKS AT 120 DEGREES. 
 
 gether very nearly three miles of 3-inch wrought-iron rods, 
 carried on bobs, pulleys, and stands, whereby power for 
 pumping and winding is conveyed along the surface to dif- 
 ferent parts of these extensive mines, from 11 large water- 
 wheels ranging up to 50 feet in diameter, to which the 
 water is brought along eight miles of leats 18 feet in width. 
 
 Rods and wire ropes have also been used to transmit ro- 
 tary motion to a considerable distance in a similar manner 
 by placing the cranks at 120 degrees, as shown in Fig. 36. 
 
 It is evident that the distance of transmission by this 
 contrivance will be subject to the sag of the ropes, unless 
 
 * " The Old Wheal Friendship Mine, near Marytavy." Proc. lust. 
 M. E. 1881, p. 100. 
 
OP THE 
 
 ROPE DRIVIK>!>* 73 
 
 intermediate shafts are employed. The motion must also 
 be comparatively slow, owing to the severe strains which 
 would be thrown upon the bearings and pins by the surg- 
 ing and swaying of the ropes during the rapid changes of 
 motion to which they would be liable. In order, then, to 
 transmit much power, heavy rods or large ropes would be 
 necessary, and under these conditions economical transmis- 
 sion would be limited to short distances. 
 
 Among the various means in use at the present time for 
 conveying power to a distance we find steam, water, gas, 
 compressed air, electricity, and rope systems. Each of 
 these has its own applications and advantages, but it must 
 be borne in mind that with the exception of rope trans- 
 mission, of which numerous examples have already been 
 given, all other forms usually require a generator at the 
 one end and a motor, with separate attendants, at the other. 
 
 Other things being equal, the relative merit of various 
 methods of transmitting power will be indicated by the 
 cost of transmitting a certain amount of power to any 
 given point, as compared with the cost of this power at 
 the generating station, while their absolute merit will be 
 shown by comparing the cost of the transmitted power at 
 the receiving station with the cost of producing the re- 
 quired power directly at this point.* Such determinations 
 are materially affected by variations in the amounts of 
 power and in the distance of transmission; the other prin- 
 cipal factors to be considered being the efficiency of the 
 system, the number of working hours per annum, the 
 price of 1 h. p. per hour at the generating and receiving 
 stations, and the convenience and applicability of the sys- 
 tem to each special case. 
 
 The efficiency of any system of transmitting power is 
 expressed by the ratio of the power obtained at the receiv- 
 
 *Stahl, "Wire-rope Transmission." 
 
74 ROPE-DRIVING. 
 
 ing station to the power given out at the generating sta- 
 tion. In all systems losses of power of greater or less 
 magnitude occur, and the most efficient system is that in 
 which those losses are reduced to a minimum. We shall 
 not attempt here to lay down rules governing the choice oi' 
 any particular method, for the requirements and condi- 
 tions are so varied that every individual case must be 
 decided upon by the engineer separately with a knowledge 
 of all the facts before him. Our present object is to ascer- 
 tain the principles governing the use of ropes, and to 
 determine those conditions best suited to their economic 
 working. 
 
ROPE-DRIVING. 75 
 
 CHAPTER V. 
 
 THE subject of rope-driving may properly be placed 
 under two heads, according to the nature of the material 
 composing the ropes, whether metallic or non-metallic. 
 With few exceptions metallic or wire ropes are used almost 
 exclusively on long-distance or telo-dynamic transmission, 
 while non-metallic ropes are employed for intermediate and 
 comparatively short drives, the consideration of which con- 
 stitutes the present subject-matter.* 
 
 Among the materials employed in this method of power 
 transmission we find special forms of leather belting used 
 as ropes working in V grooves; fibrous ropes, including flax, 
 hrmp, cotton, and manilla, are, however, chiefly employed. 
 
 Rawhide ropes, which are made from f inch to 2 inches 
 in diameter, are used to a limited extent. Where the 
 stress in the rope is not great and the accompanying slip 
 is small, rawhide works very well, and will last from three 
 to six, and in some cases ten, years. Under ordinary cir- 
 cumstances it is not necessary to use any dressing, as suf- 
 ficient lubrication is furnished by the rope itself; if the 
 rope slips in its groove the leather will be burned, and lose 
 its flexibility, and also its adhesive qualities, to a certain 
 extent. A rawhide rope has very little tendency to rotate 
 
 * Concerning wire-rope transmission the reader is referred to the 
 following: 
 
 "Wire-rope Transmission" (A. W. Stnhl); "Elektrischen Kraft 
 iibertragung" (A. Beringer); "Drahtseiltriebs" (D H. Ziegler); 
 " Constructeur" (F. Reuleaux); "Machine Design" and "Central 
 Stations" (W. C. Uuwin). Also trade pamphlets published by W. 
 A. Roebliug & Sons; Cooper, Hewitt & Co.; and others. 
 
76 HOPE-DRIVING. 
 
 on its axis; for this reason the wear is not always uniform, 
 and with a heavy tension it is liable to take the set of the 
 groove in which it runs. This is rather an advantage for a 
 straight drive, where the rope always runs in the same di- 
 rection; but in those cases where a rope is led on to the 
 pulleys at an angle this will be a disadvantage, as under 
 such conditions the rope often slips, and wear is excessive. 
 Where the rope is subject to wet or dampness, rawhide is 
 an excellent material to use, as it is very little affected by 
 dampness. The cost of rawhide rope will average about 
 six times that of a good quality of manilla transmission- 
 rope, and although it is to be preferred in certain cases, its 
 greater cost will limit its application. 
 
 Round-leather ropes, formed by twisting narrow strips of 
 leather into a continuous spiral, are used for light driving, 
 and are very desirable for some classes of work. 
 
 Solid round-leather ropes, made from several thicknesses 
 of belting cemented together and secured with screwed 
 wire forced into the leather, are made in various sizes up 
 to 2^ inches in diameter, but sizes larger than f or 1 inch 
 in diameter are seldom used. 
 
 Steel ropes with leather washers closely threaded on 
 have been tried with considerable success, but the expense 
 of such a rope would necessarily limit its application. 
 
 Other special forms of leather belting used as ropes are 
 found in the various modifications of the square and angu- 
 lar belts which have been used for a number of years for 
 both light and heavy drives. 
 
 Leather ropes as large as If inches square, made up of 
 layers of leather cemented together so that the whole is 
 uniform and continuous, have been used to replace quar- 
 ter-turn flat belts, and also for main driving. 
 
 These run in V grooves so that the adhesion is 
 greatly in excess of that produced by a flat belt on a 
 smooth pulley under the same tension. In the same 
 
ROPE-DRIVING. 77 
 
 way triangular belts built up from various thicknesses of 
 leather possess the advantages characteristic to all forms 
 of rope-driving which use a V groove, viz., greater ad- 
 hesion for a given tension, and the facilit}- with which such 
 transmitters lend themselves to the communication of 
 power between shafts at an angle with each other. 
 
 It is evident that several of these leather-rope belts may 
 be used side by side in a manner similar to the various ap- 
 plications of fibrous ropes. Such ropes have proven satis- 
 factory in those cases where the pulleys are of approxi- 
 mately the same diameter; but on pulleys whose diame- 
 ters vary considerably each portion of the leather rope in 
 contact with the driver tends to rotate the follower at a 
 different velocity, necessarily producing slip and wear, to 
 an extent depending upon the ratio of the diameters em- 
 ployed. 
 
 In England manilla is now being used very largely, but 
 cotton ropes were formerly preferred to the exclusion of all 
 others for all kinds of driving; but the most probable cause 
 of this was not that cotton was the best or most economi- 
 cal material for the purpose, but that rope-driving is most 
 common at cotton factories, and cotton ropes were made 
 in the locality by men who were familiar with the local 
 product, and had been for years making spindle and rim 
 bands of small size. When the demand for large sizes 
 arose these rope-makers applied themselves to the newer 
 industry, and shut out other materials.* 
 
 In the mills of Dundee and vicinity, and in the North of 
 Ireland, where flax and hemp are worked, we find ropes of 
 hemp, a local product, used entirely. 
 
 In many cases ropes of cotton are to be preferred, as they 
 are generally softer and more pliable than the ordinary 
 manilla ropes, thus allowing smaller pulleys to be used 
 
 * W. H. Booth, Am. Machinist, January, 1891. 
 
78 ROPE-DRIVING. 
 
 with less injury to the fibres. In fact, cotton ropes of 
 small diameter have been used for years in cotton ma- 
 chinery bandings over pulleys, and under conditions which 
 would wear out a manilla rope in one third the time. 
 There is also an advantage in that there is less internal 
 chafing and wear when the rope is bent over a pulley, on 
 account of the smoothness of the fibres and the great elas- 
 ticity of the yarns. 
 
 The cotton fibre is not, as it appears to the eye, a solid 
 cylindrical, gossarnerlike hair, but when fully ripe is shown 
 under the microscope as a flattened hollow ribbon or col- 
 lapsed cyliudric tube twisted 
 several times throughout its 
 length, as shown in Fig. 37;* 
 it is of equal size for about 
 three fourths of its length, and 
 it then gradually tapers to a 
 point. This point is a section 
 almost perfectly cylindric, and, 
 unlike the rest of the fibre, 
 often composed of solid matter. 
 
 Covering the outside mem- _ 
 
 , f . FIG. 37. COTTON FIBRE, OR- 
 
 brane of the fibre is an oleagi- LEANS VARIETY (Oossypium 
 
 nous coating generally known Hirsutum). 
 as cotton-wax. This wax amounts to about two per cent 
 of the fibre, and in the spinning of the material it requires 
 to be reduced to a certain point of liquefaction by the 
 heated temperature of the room before it can be made to 
 work properly without lapping on the drawing rollers. 
 
 These fibres vary in size from 0.00084 inch mean di- 
 ameter, and about \ inch long to 0.000635 inch mean 
 diameter and 2 inches in length, depending upon the 
 variety of the cotton; but for a given variety the differ- 
 
 * See "The Cotton Fibre," by Hugh Monie, Jr. Published by 
 Hey wood & Sou, Manchester, Eng. 
 
ROPE-DRIVING. 79 
 
 eoe is very small: thus in the Sea Island cotton the 
 maximum length of fibre is 2 inches, while the minimum 
 is If inches; in the same way in the Orleans variety 
 shown in Fig. 37 the maximum length is 1^ inches and 
 the minimum -}-|inch. 
 
 As a rule, those cottons which have the longest fibres 
 are also the smallest in diameter: they possess the natural 
 twist in a more perfect and highly developed form, and 
 are much stronger and more elastic. 
 
 In all good commercial fibres of cotton there is neces- 
 sarily (1) a very small percentage of solidified oleaginous 
 matter distributed over the internal surface of the fibre 
 deposited when the vital fluids were in active circulation; 
 and (2) a certain percentage of moisture known as water 
 of hydration. 
 
 These together with the twisted structure impart to the 
 fibres that suppleness, tenacity, and elasticity without which 
 they would be almost useless for manufacturing purposes. 
 
 The cotton fibre is thus naturally well adapted to the 
 work of being twisted into yarns; the presence of the 
 natural convolutions and comparative smoothness of the 
 surface of the unit filament permits considerable elonga- 
 tion, and the wax on its surface serves as a natural lubri- 
 cant and prevents the fibres from becoming brittle. 
 
 Thus it will be seen that ropes made from fibres pos- 
 sessing these characteristics are particularly well adapted 
 to the transmission of power in which the rope is con- 
 stantly undergoing a varying strain and is subjected to 
 much flexion. 
 
 The strength of cotton ropes is, however, relatively small 
 when compared with other fibrous ropes, and although the 
 weight is about one fifth less than manilia, for equal di- 
 ameters, the actual first cost is from fifty to seventy five 
 per cent greater than for the latter. 
 
 Nystrom gives the breaking strength of three-strand 
 
80 ROPE-DRIVING. 
 
 cotton ropes at less than one tenth that of similar manilla 
 rope, but this is apparently too low for a good quality of 
 transmission rope. 
 
 Tests made at Watertown on a number of Lambeth 
 ropes varying in size from 1 inch to 2 inches in diameter 
 indicate that the breaking strength is equal to about 
 4000tr pounds, while the extension varies from twenty to 
 twenty-five per cent, corresponding to a reduction in 
 diameter of about fifteen per cent. The weight of these 
 ropes is very closely 0.26(f pounds per foot of length. 
 
 A series of tests carried out by Kircaldy * on cotton 
 ropes ranging in size from ^-f inch to 2 T \- inches in di- 
 ameter give the breaking strength as 3700d 2 pounds for 
 minimum value and 5800eP pounds as a maximum. The 
 weight per foot of these ropes varied from 0.25d' 2 to 0.29cP ; 
 the extension under a stress of about 85 per cent of the 
 breaking load varied from 17 to 27 per cent. 
 
 Reduced to a common basis in which the strength is 
 made proportional to the weight, and averaging the results, 
 we find that the breaking strength may be represented by 
 4600cT pounds. 
 
 The data on cotton ropes are too meagre to determine 
 whether their strength decreases as the diameter increases, 
 but this is probably the case. 
 
 Reuleaux gives 7500 pounds per square inch of section 
 for cotton transmission-ropes, which agrees very closely 
 with the above values. 
 
 From the formula, breaking strength, S = 4600^ 3 pounds 
 the values given in Table III have been calculated, and 
 may be considered as representing approximately the 
 strength of cotton transmission-ropes of good quality. 
 
 The working strength of cotton transmission-rope may 
 be taken higher, in proportion to its ultimate strength, 
 
 *See also Kent's " Mechanical Engineer's Pocket Book." 
 
HOPE-DRIVING. 81 
 
 TABLE III. STRENGTH OF COTTON TKANSMISSION-ROPES. 
 
 Ultimate Brea^in^Strength, 
 
 * 1,150 
 
 | 1,800 
 
 f 2,600 
 
 % 3,500 
 
 1 4,600 
 li 7,200 
 11 10,400 
 If 14,000 
 
 2 18,400 
 
 than is used with manilla, for the latter is weakened by 
 the grease with which it is lubricated; and, moreover, a 
 larger factor must be allowed for wear on account of the 
 character of the manilla fibre, which breaks more easily 
 under bending strains. 
 
 As compared with manilla, then, the advantages of cot- 
 ton ropes of the same diameter are: Greater flexibility, 
 greater elasticity, less internal wear and loss of power due 
 to bending the fibres, and the use of smaller pulleys for a 
 given diameter of rope. Its disadvantages are: Greater 
 first cost, lesser strength, and, possibly, a greater loss of 
 power due to pulling the ungreased rope out of the groove 
 in any case this is usually small with speeds over 2000 
 feet per minute, 
 
 As we have already noted, manilla rope is used very ex- 
 tensively for transmission purposes, but its application has 
 not always met with that success which would follow u 
 more thorough knowledge of its requirements. Inefficient 
 rope-drives are erected and run for a few months, or per- 
 haps only days, and are replaced with larger ropes if the 
 sheaves will permit, or, as in many cases, tne ropes give 
 way to leather belting, and henceforth rope-driving is con- 
 demned. The true cause is not so much the inefficiency 
 of the ropes as it is the lack of knowledge governing their 
 
ROPE-DRIVIKG. 
 
 use and application; in order to obtain a proper concep- 
 tion of this a study of the structure of the rope will be 
 found advantageous. Manilla, or, more prop- 
 erly, manilla hemp (abaca) rope, is made from 
 the fibres of the Musa text His, a plant closely 
 allied to the banana, growing near Manilla, 
 in the Philippine Islands. The fibres are a 
 part of the outer covering of the leaf-stalk, 
 which attains a length sometimes as great as 
 15 feet. To obtain fibres of suitable size for 
 manufacturing rope the leaf stalks are sub- 
 divided, and in the process of segregation 
 the fibre assumes an appearance somewhat 
 similar to that produced in splitting a piece 
 of wood it is rough and uneven, and more 
 or less splintery throughout its length. These 
 fibres, although in themselves not very large, 
 are composed of very fine and much elongated \ 
 bast-cells, which overlap each other as shown 
 in Fig. 38. The cells are irregular in outline 
 and vary considerably in size. The length 
 of the cells is about one fourth of an inch. 
 A series of tests on mauilla fibre carried out 
 by Dr. Stanley M. Coulter of Purdue Univer- 
 sity, shows that the cells are not, as com- 
 monly supposed, held together by an inter- 
 cellular tissue or mucilaginous substance. A 
 cross-section of a portion of the fibre, (a), 
 Fig. 39, enlarged 450 times, shows that there 
 are no intercellular spaces; and various reac- 
 tions to determine the presence of mucilage CELLS 
 or other vegetable glue revealed no traces of NILLA FIBRE, 
 
 ENLARGED. 
 
 its presence. 
 
 The characteristic roughness possessed by the manilla 
 fibre is due entirely to mechanical causes, such as, for 
 
 FIG. 38. 
 
ROPE-DRIVING. 
 
 83 
 
 instance, the laceration of a cell in the separation from the 
 leaf-stalk, or the subsequent opening out of the ends of 
 the cells. 
 
 The contour of the perimeter is also rough, as noted in 
 the figure, 396, as it retains the form impressed upon it by 
 the contiguous cells when in the plant. 
 
 These fibres have great strength in the direction of their 
 length, but are weak transversely;* when made into rope 
 they are compelled, in bending over the sheave, to slide on 
 
 FIG. 39a. MANILLA FIBRE, 
 
 MAGNIFIED 450 TlMES. 
 
 0.016* 
 
 FIG. 396. CROSS-SECTION 
 OF MANILLA FIBRE. 
 
 each other while under pressure from the load. This 
 causes the internal chafing and grinding which, if not pre- 
 vented, soon wears out a rope when subjected to bending 
 strain. 
 
 In addition to the action of the fibres upon each other, 
 the strands and the yarns of which the strands are com- 
 posed also slide a small distance upon each other, causing 
 friction, and hence internal wear. 
 
 By opening out an old dry rope which has been used over 
 a sheave, a fine powder will be disclosed, showing that 
 where the rope was bent over the sheaves the strands, in 
 sliding on each other, ground some of the fibres to powder. 
 Another reason for this is that the fibres in an old rope be- 
 come brittle and weak when dry, so that the constant 
 
 * The tensile strength of manillu fibres will average over 30,000 
 pounds per square inch of section. 
 
84 ROPE-DRIVING. 
 
 flexure to which they are subjected rapidly disintegrates 
 the cell congeries. 
 
 Aside from the external wear which a rope suffers from 
 contact with its sheave (part of which is the differential 
 driving effect), these two are the principal causes of rapid 
 wear in a rope-drive, to remedy which we must in the first 
 case lubricate the fibres, and in the second, prevent undue 
 flexure of the rope. How this is effected in practice will be 
 seen presently. 
 
BOPE-DRIVIOT. 85 
 
 CHAPTER VI. 
 
 Itf manufacturing manilla rope the fibres are first spun 
 into a yarn,, this yarn being twisted in a direction called 
 right-hand. From 20 to 80 of these yarns,* depending 
 on the sizo of the rope, are then put together and twisted 
 in the opposite direction, or left-hand, into a strand. 
 Three of these strands for a 3-strand, or four for a 4-strand, 
 rope are then twisted together, the twist being again in 
 the right-hand direction. It will be noticed that when the 
 strand is twisted it untwists each of the threads, and when 
 the three strands are twisted together into rope it untwists 
 the strands, but again twists up the threads. It is this 
 opposite twist that keeps the rope in its proper form. 
 When a weight is hung on the end of a rope the tendency 
 is for the rope to untwist and become longer. In untwist- 
 ing the rope it will twist the threads up, and the weight 
 will revolve until the strain of the untwisting strands just 
 equals the strain of the threads being twisted tighter. In 
 making a rope it is impossible to make these strains exactly 
 balance each other, and it is this fact that makes it neces- 
 sary to take out the turns in a new rope that is, untwist 
 , it when it is put at work. 
 
 * A three-strand rope one inch in diameter is the key to the sizing 
 of the yarns. Yarns of 20s are of such a size as to require 20 to fill 
 a tube half an inch in diameter, or to make one strand of an inch 
 rope; 26s requires 26 to fill the same size tube ; and so on. The size 
 of cotton yarns, on the other hand, depends upon the number of 
 hanks per pound ; thus 20's, or number 29 cotton, requires 20 hanks 
 (340 yards each) of this size to weigh one pound. 
 
86 ROPE-DRIVING. 
 
 The fibres of manilla which are thus twisted into ropes 
 will average over 6 feet in length, varying from 3-J- to 12 
 feet. If they were long enough, the most advantageous 
 method of using them would be to lay the fibres side by 
 side, and secure them at the two ends; each fibre would 
 then bear its own share of the strain, and the strength of 
 the bundle would be that of the sum of the strengths 
 of the separate fibres. As a long rope could not be 
 formed in this way, the fibres are secured by twisting 
 so as to produce sufficient compression to prevent them 
 from moving upon each other when a strain is applied ; but 
 in attaining this amount of compression their strength is 
 greatly reduced; this very compression acts as a constant 
 weight on the fibre, and must be deducted therefrom before 
 the available strength can be applied. 
 
 The weakening effect produced by twisting varies consid- 
 erably among the fibres of the same rope according to their 
 distance from the centre or heart of the bundle. If a cer- 
 tain amount of twist be given to a bundle of fibres, the 
 outer ones will be strained more, and will act with less use- 
 ful effect than those on the inside, which will have to bear 
 the greater part of the strain while the rope is being used. 
 It will therefore be evident that if the fibres were twisted 
 at once into a thick rope the outer fibres would be so much 
 strained as to be of little or no use in contributing to the 
 strength of the rope, but by making the rope as we have 
 indicated by first twisting into yarns, then into strands, and 
 finally combining these into a rope, the strain is more 
 equalized, and the important properties of length and 
 strength are secured without too great a sacrifice of the 
 strength of the individual fibres. 
 
 The degree of twist in the rope may be determined by 
 constructing a right-angled triangle, the base of which is 
 the circumference, and the height the length of one turn 
 of the strand measured parallel to the axis. The difference 
 
ROPE-DRIVING. 87 
 
 between this height and the hypothenuse is the quantity by 
 which the rope is twisted. The ropemaker's ordinary rule 
 for a three strand rope is to have one turn to as many 
 inches as are contained in the circumference of the rope; 
 but the degree of twist is variable and more or less de- 
 pendent upon the judgment of the maker. 
 
 Experiments by Reaumur to determine tiie effect of twist 
 upon a rope showed that a small well-made hemp cord 
 broke in different places with a mean weight of 65 pounds; 
 while the three strands of which it was composed bore 29^, 
 33^, and 35 pounds respectively, so that the total absolute 
 strength of the strands was 98 pounds, although the aver- 
 age real strength was only 65 pounds, thus showing a loss 
 of 33 per cent. 
 
 More recently the test of a small rope showed an average 
 strength of 4550 pounds, while the aggregate strength of 
 its 72 yarns was 6480 pounds each yarn bearing about 90 
 pounds; thus there is a loss of 1930 pounds, or about 30 
 per cent. 
 
 Ropes made of the same hemp and the same weight per 
 foot, but twisted respectively to two thirds, three fourths, 
 and four fifths of the lengths of their component yarns, 
 supported Uie following weights in two experiments made 
 by Duhamel:* 
 
 Pounds. Pounds. 
 
 Two thirds 4,098 4,250 
 
 Three fourths ... 4,850 6,753 
 
 Four fifths 6,205 7,397 
 
 The results of these experiments led Duhamel to make 
 ropes without twist by placing the yarns together and 
 wrapping them round to keep them together. The rope 
 had great strength and pliability, but not much durability 
 
 * Tomlinson, vol. vn. p. 574, 
 
 OF THE 
 
83 ROPE-DRIVING. 
 
 on account of the outer covering wearing away or opening 
 when bent, thus admitting moisture to the interior, which 
 rotted the yarns. 
 
 In general, the greater the twist the more hard and rigid 
 the rope is, and the better it will keep its form; but it is 
 not as strong, weight for weight, as the more loosely 
 twisted rope; moreover, the hard-twisted rope is more rigid 
 than the other, and is not ;is suitable for transmission pur- 
 poses, owing to the rapid wear which constant flexure 
 produces. 
 
 A very excellent transmission-rope, known as the " Lam- 
 beth," is made in a somewhat similar manner to that just 
 described, but it is not open to the same objections. In 
 this case the rope, which is of cotton, is made of four 
 strands twisted together in the usual manner, but the 
 strands are themselves composed of a bundle of fine yarns 
 and have scarcely an appreciable twist. Each bundle, 
 comprising many hundred yarns, is wound spirally with 
 smaller bundles of about 100 yarns each. In this way 
 the outside of the rope acts as a shield or covering to 
 the cores which do the work. By this means a certain 
 amount of the natural elasticity of the cotton is re- 
 tained, and its pliability is much greater than in ordinary 
 hawser-laid ropes. 
 
 Lubrication of transmission-ropes is provided for in 
 various ways. Frequently the rope is laid up dry and a 
 coating or dressing is given to the exterior, which is sup- 
 posed to penetrate to the interior and lubricate the fibres. 
 With some dressings this may occur, but with others 
 the effect is merely local; the interior of the rope re- 
 mains dry, and much bending soon wears it out. With 
 cotton ropes, as we have already noted, the internal 
 chafing and wear is very much reduced, and for this 
 reason cotton ropes are laid up dry and are not usually 
 
ROPE-DRIVING. 89 
 
 lubricated; they are, however, generally coated with some 
 form of dressing to prevent the fibre from rising or the 
 rope fraying, and to protect it from an undue amount of 
 moisture when exposed to the weather; it also assists in 
 retaining the natural moisture in the fibres, without 
 which they would become brittle and weak. 
 
 According to the nature of the dressing, the interior 
 may or may not be affected by the outer coating. Bees- 
 wax and black lead, with a little tallow, forms an excellent 
 moisture-proof covering for ropes; it fills in the spaces 
 between the strands, and the rope soon assumes a perfectly 
 round and smooth appearance, like a bar of iron ; with this 
 coating the interior retains its natural condition, and should 
 therefore not be used where a lubricant is desired. Pine 
 tar is much used on cotton ropes for the same purpose. 
 Mixtures of tallow and black lead; molasses and black 
 lead; equal parts of resin or beeswax, with black lead, 
 tallow, and molasses melted together, and applied hot; and 
 various other compounds are in use for this purpose. 
 Tallow, lard, and other greases are used separately, and are 
 fairly satisfactory as a lubricant. When the rope runs out- 
 of-doors a water-proof coating is necessary to preserve it 
 from decay, and for this purpose, if it is also desired to 
 lubricate the rope at the same time, there is probably noth- 
 ing better than tallow and black lead, or graphite, pro- 
 vided the rope is not twisted too hard, which would pre- 
 vent the dope from penetrating. In certain drives that 
 are subjected to hard service and are more or less exposed 
 to the weather boiled linseed-oil is used very successfully. 
 According to Mr. A. D. Pentz,* the rope is treated every 
 two weeks to about two quarts of the oil, dripped one drop 
 at a time upon one of the sheaves, which is uncovered on 
 top; the rope runs on the bottom of the sheave and slowly 
 
 * Eng. Magazine, Nov. 1893, p. 256. 
 
90 HOPE-DRIVING. 
 
 absorbs the oil. Although the rope is very materially 
 weakened by this process, yet the greater freedom of the 
 fibres permits a heavier working strain to be carried, for it 
 is the relative wear of the fibres that determines the life of 
 the rope. A manilla rope with the fibres properly lubri- 
 cated will, under the same conditions, outlast from two to 
 four similar dry-laid ropes which are allowed to run dry. 
 Manilla transmission-ropes are generally laid up in tallow ? 
 paraffine, soapstone, or a mixture similar to the above 
 preparations. 
 
 A superior manilla transmission-rope is that known as 
 the " Stevedore/' or black rope, which is made with both 
 three and four strands, the latter being laid up about a 
 central core. The yarns of this rope are each coated with 
 a mixture of graphite and tallow, so that when twisted 
 into strands the coating lodges in the hollows and uneven 
 places among the fibres, and thoroughly lubricates the 
 strands and individual fibres composing the rope, which is 
 thus made practically as nearly water-proof as possible. 
 After it has been in use a short time its appearance is that 
 of a black rod of iron, smooth and round, similar to the 
 beeswax-coated rope previously mentioned, but perfectly 
 flexible. A manilla rope thus made will last from three 
 to eight years if not overstrained; when running indoors 
 under favorable conditions the latter limit may be attained, 
 but when exposed to the weather, or when working under 
 less favorable conditions, its life will be shortened. 
 
 Hemp ropes intended for outdoor service are sometimes 
 treated by passing the yarns through boiling-hot tar, suit- 
 able machinery being used to regulate the amount of tar 
 retained in the yarns so that the fibres may be coated over 
 and thus preserved from decay. Tarring protects rope 
 from injury by exposure to rain and immersion in water, 
 but it makes its fibre rigid and impairs its strength; for 
 
HOPE-DRIVING. 91 
 
 this reason it is unsuitable for ordinary transmission pur- 
 poses. 
 
 It has been shown by experiment: * 
 
 . 1. That white or untarred rope in continual service is 
 one third more durable than tarred. 2. That it retains its 
 strength much longer when kept in stock. 3. That it re- 
 sists the ordinary injuries of the weather one fourth longer. 
 
 With the exception of the outside yarns of large haw- 
 sers, manilla ropes are not tarred. 
 
 The breaking strength of a rope depends both upon 
 the quality of the material and the degree of twist given 
 to the strands \ for a loosely twisted rope of a given diame- 
 ter the strength is less than that in a hard twist of 
 the same diameter, but compared weight for weight, the 
 rope with the lesser degree of twist is the stronger. 
 
 In discussing the strength of ropes, which formerly was 
 always given in terms of the circumference, there is a lack 
 of uniformity among writers in the relation between the 
 diameter and area of a rope. The circumference, as meas- 
 ured by a tape, depends upon the number of strands in the 
 rope and their compression upon one another. If the 
 strands still retained their circular section when twisted 
 into a rope, the circumference of a 3-strand rope would be 
 2.86 times the diameter of the circumscribing circle, as 
 given by Nystrom; if the strands completely fitted the 
 circle its measure would be n times the diameter as given 
 by Unwin and others. 
 
 As neither of these conditions obtains in practice the 
 true value lies between 2.86 and 3.14, and we shall assume 
 3 as the most suitable factor. 
 
 In the same way the area of the cross-section of the rope 
 is variously estimated as that of the area of the circular 
 strands, or as the area of the full diameter of the rope. 
 
 *Duhamel : Traite de la fabrique des manoeuvres pour les vaisseaux, 
 
92 ROPE-DRIVING. 
 
 By assuming the rope to be made of three strands, the 
 cross-section of each of which is a circle, the area of the 
 
 rope would evidently be Sf-rO* 2 )* where d is the diameter 
 
 of each strand. If d represents the diameter of rope, i.e., 
 the diameter of circumscribing circle, the area in terms of 
 d will become 
 
 The section of the strands, taken at right angles to the 
 axis of the rope, is, however, not a circle, as can be seen 
 from Fig. 40. The degree of twist given to the strands, 
 
 FIG. 40. ACTUAL SECTION OF ROPE. 
 
 and the compression of the latter upon one another, will 
 evidently affect the area of the section; for the longer the 
 spiral the more nearly will the cross-section of each strand 
 approach a circle. It is obvious that the true value must 
 lie between 0.52<f and 0.7854eP . In determining this area 
 the writer made a number of plaster casts at different 
 
ROPE-DRIVING. 
 
 93 
 
 points of several 3-strand manilla ropes, varying in size 
 from f inch to If inches diameter. With these casts as 
 dies, which were covered with printer's ink, impressions 
 were made and the area obtained by using a planimeter. 
 The area at several sections was found to be practically 
 constant for each rope, and varied between O.GloP and 
 0.65d 2 the mean value being 0.63tf, from which we 
 obtain the ratio 
 
 Area of section of rope 0.636^* 
 
 Area of circumscribing circle " 0.7854^ a 
 
 = 0.81; 
 
 that is, the actual area of a 3-strand rope equals eight 
 tenths of the area of the circumscribing circle. 
 
 FIG 
 
 CROSS-SECTION OF ROPE. 
 
 A print obtained with a plaster cast, showing the distor- 
 tion of the strands from a true circle, is reproduced in 
 Fig. 40. Fig. 41 is a similar print, showing the relation 
 between the actual area of the cross-section and the area 
 included between circular strands and the tangents joining 
 them. It will be noted that the excess of area outside of 
 
94 ROPE-DRIVING. 
 
 the lines drawn tangent to the inner circles is but slightly 
 greater than that between the tangents and the inner 
 circles. 
 
 An inspection and tabulation of the results of numerous 
 tests on manilla rope* shows that the strength per square 
 inch increases as the diameter of the rope decreases. 
 Formulas for the strength of rope based upon the circum- 
 ference and a constant multiplier as, for instance, S = 
 800c a , where S is the breaking strength and c the circum- 
 ference must be regarded as giving only an average value 
 for diameters approximating those experimented upon. 
 As the strength of good manilla rope varies from 10,000 
 pounds per square inch for a 2-inch rope to over 12,000 
 pounds for a half-inch rope, it can be seen that a more 
 accurate determination may be made if a variable is used 
 in the formula. With the above assumption of area ratios, 
 obtained by trial, the following expression has been de- 
 duced, which will give a very fair value of the breaking 
 strength for new manilla ropes: ^ = WQd*x, in which x 
 is a variable depending upon the diameter of rope; for 
 manilla rope we may assume the empirical value 
 x 81 9<tf, where d is the diameter in inches. 
 
 The ultimate strength per square inch of actual section 
 will then be 
 
 S = 
 
 From these formulas Table IV has been computed : 
 The above values are for new manilla ropes made of 
 selected stock; ropes that are greasy or wet will be reduced 
 
 * Major Parker's report of tests made at Watertown Arsenal, 
 1885. Ex. Doc., No. 36. Experiments of M. Doboul in "Bulletin 
 de la Societe d'Encouragemcnt des Arts," Paris, 1888. Riehle Bros,' 
 laboratory tests ; and otheir , 
 
HOPE-DRIVING. 95 
 
 TABLE IV. STRENGTH OF MANILLA TRANSMISSION-ROPES. 
 Dia. of rope = d. Breaking Strength, strength per squar e inch, 8. 
 
 % ' 1,900 12,200 
 
 2,900 11,950 
 
 4,100 11,750 
 
 i 5,500 11,525 
 
 1 7,100 11,350 
 H 8,800 11,125 
 li 10,900 10,975 
 1| 15,000 10,625 
 If 19,800 10,300 
 
 2 25,100 10,000 
 
 in strength from 20 to 30 per cent, but when ropes are 
 used for transmission of power the lubrication of the 
 fibres is of more importance than the actual breaking 
 strength of the rope, as in any case the apparent working 
 strain, as calculated from the power transmitted and the 
 speed of the rope, should not be taken greater than 5 per 
 cent of the ultimate strength; in many cases not more 
 than 2 per cent is used. 
 
 If we wished to obtain the working strength in 
 any given case we would usually divide the breaking 
 strength by an assumed apparent factor of safety, but 
 for a flying rope, in addition to this, it is necessary 
 to provide a factor of wear; moreover, the actual strains 
 are generally much greater than the normal as calcu- 
 lated from the power transmitted, due to vibrations in 
 running, imperfect tension mechanisms, defects in con- 
 struction, and other causes. As the strength at the splice 
 is only 70 to 75 per cent of the strength of the rope, the 
 actual margin for wear and unknown strains is not as large 
 as would at first seem apparent. In the first place, the 
 strength of a lubricated rope is weakened about 25 per 
 cent by the grease, then 25 per cent more must be deducted 
 for the strength of the splice; this leaves only about 50 
 per cent of the original strength of the rope which is 
 
96 
 
 ROPE-DRIVING. 
 
 I 
 
 8 a 
 
 bfl 
 
 * a to a s 
 
 S E B B ^ 
 
 GO s- 02 CO 
 
 M 
 
 51 
 
 a , .73 
 
 a-r^a 
 
 1 61 
 
 -T= & S3 c "S 53 
 
 a*^Sr^-c^'>fl 
 
 I6a8 
 
 (??C<iC50O(??GQOO(?^O-'JC-7OTHO7<MC5 
 1 l-iti !C?~H-i lOir-l-! i I 1C T- (i 1 i-H i l 
 
 O 
 fM 
 
 CO JO (N JO O3 JO JO 1O O O O CO O3 
 
 COCOCOi iCOOrHCQC3COi>QO!OCOO5COO 
 
 r-i O) O? (75 03 W <M C? <tt' (M* T-i rH T-! ^H rH rH (^ 
 
 
 
 O OS O 00 O? CO O 
 
 
 fctf 
 
 CO <M CO CO <W C5 CO 
 
 
 O ^5 O O O O >O O 1O iO ' 
 
ROPE-DRIVING 
 
98 EOPE-DRIVIKG. 
 
 available for transmission of power. To allow for possible 
 imperfections in the rope, due to its manufacture or the 
 material used, we must allow a further reduction of, say, 
 one fifth (that is, 10 per cent of the original strength of an 
 unlubricated, unspliced rope); thus we have only 40 per 
 cent of the original breaking strength, which we may con- 
 sider as the actual working strength. Allowing 10 per 
 of this as the apparent working strain (equal 4 per cent of 
 original strength), we obtain 36 per cent as the actual 
 margin for wear and unknown stresses which may be set 
 up in the rope; that is, we have practically only an actual 
 factor of safety of ten, used in its ordinary acceptation, 
 instead of twenty-five, as would appear from the low work- 
 ing stress. 
 
 The working strain in executed rope transmissions will 
 be found to vary considerably, as shown in Table V ; 
 but plants which have been successful, as well as 
 those in which the wear of the rope was destructive, indi- 
 cate that 200^ a pounds is an economical working strain.* 
 
 As this value is such a small percentage of the breaking 
 strength, it is unnecessary to use a coefficient varying with 
 the diameter of rope, as the difference is not worth con- 
 sidering. From data furnished by the Messrs. Pearce 
 Brothers, of Dundee, who have erected rope-belting exten- 
 sively, Prof. Unwin shows that in different cases in practice 
 the driving force, or difference in tension, on the two por- 
 tions of the rope is equal to 75 to 80 d* pounds; that is, 
 
 T l T^ = P = 75 to 80 d\ 
 
 where T, = tension on driving side; 
 T 9 = tension on slack side; 
 P = driving force. 
 
 * See paper by C. W. Hunt, Trans. A. S. M. E., vol. xn. 
 page 230. 
 
KOPE-DBIYING. 99 
 
 T 
 
 It is probable that the value of the ratio ~ varies from 
 
 !- to 2-J iu ordinary practice, depending upon the speed of 
 the rope, the coefficient of friction, and the arc of contact 
 between rope and pulley. 
 
 T 
 
 Unwin assumes - = 1.2 when the belt embraces 0.4 of 
 
 the circumference of the smaller pulley; hence the greatest 
 tension would be 
 
 T l = 1.2 P = 90tf a to 96d 2 . 
 
 This is based upon a coefficient of friction = 0.7 for a 45 
 groove, which we believe to be greatly in excess of its aver- 
 age value for running ropes. Moreover, the rope velocity 
 was not considered in determining the above ratio; at 
 speeds over 2000 feet per minute the influence of cen- 
 trifugal force cannot be neglected, as it produces a very 
 considerable force in the rope, and for these reasons the 
 
 T 
 
 value of the ratio -~ will be greater in average practice 
 
 than 1.2. 
 
 If we assume a speed of 4000 feet per minute, the value 
 of this ratio for greasy ropes may be taken equal to 2, from 
 which there is obtained T l = 2 x 75d 2 to 8Qd* = 150rf* to 
 IQOd' 2 a value somewhat less than that which we have as- 
 sumed as a suitable working strain, viz., 200tP pounds. In 
 a recent communication from Messrs. Combe, Barbour & 
 Combe of Belfast, who have been engaged in furnishing 
 rope transmission for thirty years, it is stated that their 
 basis of calculating the horse-power is to assume that u 
 rope 5^ inches circumference (If inches diameter) work- 
 ing on a four-foot pulley going 100 revolutions per minute 
 will drive 8 h. p. under medium circumstances; and by 
 '' medium circumstances" they mean " that the ropes must 
 work at a distance of at least 20 feet from centre of shafts 
 
100 HOPE-DRIVING. 
 
 and at a less inclination than 40 from the horizontal, at a 
 speed not under 2000 feet per minute. Should the ropes be 
 working vertically or at an angle greater than 45, instead 
 of taking 8 h. p. as the basis, you should take 7 or 6 respec- 
 tively, according as the conditions grow worse and worse. 
 On the other hand, should the ropes work horizontally 
 and at a greater distance than 20 feet from centre to centre, 
 and with a speed up to 3600 feet per minute, and the pul- 
 leys be fairly large, say from 5 to 7 feet in diameter, you 
 may take 10 instead of 8 as the basis.'' From these consid- 
 erations the working strain may readily be determined. If, 
 
 T 
 
 as before, we take ' = 2, we find that under average con- 
 ditions the allowable working strain will be T^ = 135d 3 
 pounds. Under more favorable conditions and a higher 
 
 T 
 velocity -=j will be greater and T^ will approach 200*1*. 
 
 Although we have assumed the normal working load not 
 to exceed 200d' 2 pounds, this must be considered as the 
 economical load for the lasting qualities of the rope; in 
 many cases, however, the first cost, the more convenient 
 adaptation of smaller ropes, and the use and lesser cost of 
 smaller pulleys outweigh the greater economy obtained by 
 the larger ropes, and loads are carried far in excess of that 
 given. By the use of a greater number of wraps the work- 
 ing load on each will be reduced; but this adds to the first 
 cost of the plant, and many concerns prefer to put in a new 
 rope every year or two rather than put in more or larger 
 ropes every six or eight years. 
 
 The ropes most commonly found in use vary from f inch 
 to 2 inches in diameter, although other sizes are frequently 
 employed; in one case, cited by Mr. T. S. Miller,* a rope 
 
 * Trans. A. S. M. E. 1891. 
 
ROPE-DRIVIM. 101 
 
 no larger than -f^ inch diameter is used very satisfactorily 
 to transmit 20 h. p. 
 
 The largest rope in use for this purpose, so far as the 
 writer is aware, is 3| inches in diameter: it is of cotton 
 (Lambeth), and is used to drive the cable drums in the 
 Washington Street power station of the North Chicago 
 Street Railway Company. 
 
 In England and Germany ropes smaller than 1 inch 
 in diameter are seldom employed except for machine 
 driving ; in ordinary cases the sizes most frequently 
 adopted vary from 1-J to 2 inches in diameter. 
 
 In main drives where a number of ropes are used, the 
 tendency, as judged from recent practice, seems to favor a 
 diameter about If or If inches. 
 
 With a given velocity and working tension the weight of 
 rope required for transmitting a given horse-power will be 
 the same, irrespective of the diameter of rope ; the 
 smaller rope will require more parts, but the weight will 
 be the same. As we have stated, many engineers prefer 
 to use a greater number of ropes over wide-faced pulleys; 
 but in order to reduce the expense incident to a large 
 number of grooves in the pulleys a closer margin is allowed 
 on the smaller ropes, and in consequence the normal work- 
 ing strain on each rope is usually increased far in excess of 
 that which would be necessary if the same weight of rope 
 were employed as for. the larger diameter. As a result of 
 this, the small ropes are rapidly worn out, and frequent 
 renewals become necessary. 
 
 We are aware tlmt small ropes are advocated and in- 
 stalled by many engineers, but we believe this to be wrong 
 both in principle and practice. In rope-driving the first 
 cost and erection of small driving-pulleys may influence a 
 designer to use small ropes; but as ropes are sold by the 
 pound, and as the weight necessary to transmit a given 
 power should be the same, irrespective of the diameter, it 
 
102 ROPE-DRIVING. 
 
 is evident that the first cost of the rope itself will be the 
 same, whether large or small ropes are employed. More- 
 over, if the pulley is proportioned to the size of rope in 
 each case, the smaller rope will last only about one third as 
 long as a rope twice its size, under similar conditions. 
 
HOPE-DRIVING. 103 
 
 CHAPTER VII. 
 
 HOWEVER desirable it may be to use a given diameter of 
 rope, the conditions of the problem frequently prohibit the 
 employment of such ropes, and the designer must deter- 
 mine whether he shall use a smaller diameter or a different 
 material or method of transmission. 
 
 If we assume that there is a minimum diameter of pulley 
 which may be safely used for any given diameter and speed 
 of rope, it will be evident that the number of revolutions 
 of the pulley imposes conditions which limit the choice of 
 rope diameter. 
 
 Thus if the maximum speed of rope be taken at 5000 
 feet per minute for a permanent installation, in which the 
 working load is 200^ 2 pounds, and the least diameter of 
 pulley D = d ( V F) + 12" (page 179), then the greatest 
 number of revolutions which can be obtained under these 
 conditions with a 1-inch manilla rope will be 550. 
 
 If the least diameter of pulley for cotton ropes be taken 
 equal to O.&Z), then the greatest speed will be approxi- 
 mately 700 revolutions per minute for the same diameter 
 of rope. If a greater rotative speed be desired, it is evident 
 that a smaller rope must be used. 
 
 In any case, the greatest number of revolutions which 
 may be obtained without excessive wear for a given diame- 
 ter of rope will be found in Table VI, which has been de- 
 termined from the formula N= =:, in which 
 
 TtJJ 
 
 XT revolutions per minute of smaller pulley; 
 V = velocity of rope in feet per minute; 
 D = least permissible diameter of pulley. 
 
104 
 
 ROPE-DRIVING. 
 
 For machine-driving greater speeds are obtained by the 
 use of smaller ropes than those given in the table; but it 
 is not advisable in most cases to use a rope less than f inch 
 diameter for general transmissions. 
 
 TABLE VI. GREATEST REVOLUTIONS PER MINUTE FOR GIVEN 
 DIAMETER OF ROPE. 
 
 Diameter 
 of 
 Rope. 
 
 Maximum Revolutions per Minute of Smaller Pulley 
 corresponding to a Linear Velocity of 
 5000 Feet per Minute. 
 
 Manilla. 
 
 Cotton. 
 
 i 
 
 ii 
 
 H 
 
 i 4 
 
 710 
 550 
 430 
 350 
 
 280 
 240 
 
 890 
 670 
 530 
 440 
 350 
 290 
 
 The wear of a rope is both internal and external. As we 
 have previously noted, the internal wear is due to the bend- 
 ing of the fibres and their sliding upon one another, which 
 produces a grinding action, very much increased when the 
 strands are not lubricated or when a hard twist is given to 
 the rope, thus preventing by the greater compression of the 
 fibres upon one another that freedom of action which is so 
 essential to the life of the rope. It is evident that a similar 
 compression of the fibres will occur when a rope under ten- 
 sion is wrapped around a pulley: the greater this tension 
 the greater also will be the compression and distortion of 
 the fibres. 
 
 The external wear is due to the contact between the rope 
 and the sides of the groove in which it runs, and is greatly 
 increased when slip occurs; roughness in the groove also 
 increases the wear, and for this reason the rim should be 
 turned smooth and polished, as the outer fibres, rubbing 
 
ROPE-DRIVIKG. 105 
 
 011 a rough- turned or cast surface, will gradually break, 
 fibre by fibre, and thus give the rope a short life. 
 
 Contact between different ropes, or between a rope and 
 some obstructing surface, such as a partition wall, post, or 
 floor-beam, is frequently the cause of a large portion of the 
 external wear of a rope : this may be due to faulty construc- 
 tion or erection; pulleys designed with too small a pitch 
 between the grooves, or running out of true, causing the 
 ropes to vibrate and flap against each other, or, as in out- 
 side work, a swaying, producing contact, may be set up in 
 the ropes, due to the action of the wind. 
 
 Excessive swaying will also tend to cause the rope to 
 jump its groove. In order to prevent this and reduce the 
 side motion as much as possible it is often customary in 
 outdoor drives to place idlers for both tight and slack 
 sides of the ropes so as to guide each portion as it enters 
 upon or leaves the groove. 
 
 A characteristic uniform surging sometimes occurs in 
 flying ropes due to the harmonic vibration which is set up 
 when the speed and distance between shafts bears a cer- 
 tain relation to the tension in the ropes. Cases of exces- 
 sive vibration due to this cause have been remedied by 
 slightly increasing or decreasing the speed of the ropes. 
 Where such vibration causes the rope to beat against an 
 obstruction, as a floor or ceiling, the external wear is of 
 course increased. 
 
 With the same total stress in a rope, it may be assumed 
 that the wear, both internal and external, increases directly 
 with the number of flexures, the slip, and the surface in 
 contact; and also, that a reverse bending is more injurious 
 to the rope than single bends in a constant direction. 
 For a given speed the number of flexures and the actual 
 surface in contact with the pulleys will decrease as the 
 distance between centres increases, and hence the wear 
 will vary inversely with the distance between centres of 
 
106 HOPE-DRIVING. 
 
 pulleys, but it must be noted that with imperfect con- 
 struction an increased distance between shafts will favor 
 swaying and rubbing of the ropes against each other and 
 the edges of the grooves. 
 
 The number of flexures and the surface in contact will 
 evidently increase directly as the velocity, and therefore 
 the wear may be assumed to vary directly as the velocity of 
 the rope. If we assume that two IJ-inch ropes are neces- 
 sary to transmit a given horse-power, it will require eight 
 f-inch ropes to transmit the same power at the same speed 
 and tension per square inch of section. If suitable pul- 
 leys are used in each case the wear will be considerably 
 greater with the smaller ropes. For the total external 
 surface of the eight ropes in contact with the pulley each 
 revolution is twice as great as that obtained with the 1J-- 
 inch ropes; moreover, the distance between centres of 
 shafts will generally be considerably less with the smaller 
 ropes, and as the number of revolutions of the smaller 
 pulleys should be more than twice as great for the same 
 speed of rope (since the pulleys are less than half the size 
 of those used for the larger rope; see page 180), the number 
 of flexures and the wear of the rope on the pulleys will be 
 greater with the f-inch rope ; for not only is each rope 
 bent more than twice as many times per minute, thus pro- 
 ducing eight times the bending in the smaller ropes, but, 
 as the slip is independent of the diameter of the rope, it 
 will be evident for the same proportional stress and arc of 
 contact that the slip will be four times greater in the case 
 of the smaller ropes, even if we neglect the differential 
 driving effect which may be assumed to increase with the 
 number of ropes. 
 
 Thus it will be seen that under similar conditions and 
 proportional stress, we should expect the smaller rope to 
 wear out more than twice as fast as one double its size; 
 and when the stress is proportionally greater in the smaller 
 
ROPE-DRIVING. 107 
 
 rope, as we ordinarily find it, the wear will be still 
 greater. 
 
 These conclusions are borne out in practice, for in trans- 
 missions using small ropes, f-inch in diameter and under, 
 the life of manilla ropes is usually only from six to twelve 
 months; in many cases such ropes will last only three 
 months, although others have been in active service for 
 periods varying from one to two years. 
 
 On the other hand, the larger-sized ropes, one inch to 
 two inches in diameter, will last from two to six years, and 
 under favorable conditions large ropes have lasted eight 
 and even ten years. 
 
 The same is true regarding cotton ropes. 
 
 The comparatively short life of small ropes used in 
 machine-driving has led to the belief that cotton ropes 
 wear out rapidly; but such an impression, at least regard- 
 ing the larger sizes, is altogether erroneous, as these ropes 
 when properly put on and cared for will give good service 
 for ten or twelve years. 
 
 Messrs. John Musgrave & Sons, Bolton, Eng., who have 
 had a large experience with rope-driving, state that some 
 of their ropes have been in use for seventeen years and were 
 still in good order. 
 
 The life of a rope, whether of manilla or cotton, will de- 
 pend altogether upon the work it has to do and the atten- 
 tion it receives. 
 
 Two ropes cut from the same coil can be put to work on 
 different drives, and one will last only six months while 
 the other will be in good condition after continual service 
 for ten years. 
 
 From an investigation of numerous examples of rope- 
 driving under a variety of existing conditions the writer 
 is led to believe that ropes less than f inch diameter 
 should not be used if it is at all practicable to employ tke 
 larger sizes, and that ropes one mch in diameter and over 
 
108 ROPE-DRIVING. 
 
 are to be preferred where it is possible to use the larger 
 pulleys which are necessary for such ropes. 
 
 With larger ropes the wear is not only much less, but, 
 where the nsual multiple-wrap system is used, when a, 
 number of yarns give way the rope does not part at once 
 if subjected to a greater or sudden tension, but may run 
 until a convenient opportunity offers to shut down; where- 
 as with the small rope, having a greater number of wraps, 
 when a strand or a number of yarns give way, any in- 
 creased stress due to additional load or imperfections in 
 the system is liable to still further rupture the yarns in the 
 weaker rope, and a sudden break or pulling out occurs. 
 
 Besides the greater life of the rope, and consequent less 
 cost, the saving of time on account of fewer breakdowns 
 and stoppages is a factor worth considering; and although 
 this feature does not cut as much of a figure with rope- 
 driving as it does with factory belting, yet it is of such 
 importance that many men would rather use some other 
 form of transmission than suffer the annoyance incident to 
 respl icing a broken rope every few months. While the 
 time'lost in repairing the rope or in laying down a new one 
 may not be great in itself, the stoppage of a department in 
 a busy season may prove to be a serious loss. Where 
 foundries and isolated shops have been driven by small 
 ropes this has happened so frequently that the ropes have 
 been taken out and replaced with shafting, or other more 
 expensive method of transmission. 
 
 In order to avoid any serious loss of time or inconven- 
 ience due to a sudden rupture two independent ropes 
 should be used, each having its own tension sheave and 
 weight. This does not involve any more wraps than would 
 otherwise be used for a single wind, for the normal work- 
 ing stress is in any case so much less than the actual 
 strength that for a temporary run of a few hours, or even 
 days, one rope could readily carry double its working load 
 
ROPE-DRIVING. 109 
 
 in case the other should give out. When two independent 
 ropes are thus used they may be wound separately, each 
 wrap occupying successive grooves on the pulleys ; or, 
 which is more frequently the case, the ropes may be wound 
 in parallel, thus bringing each rope in alternate grooves- 
 in either case it is preferable to use an independent 
 tightener, although a single-tension carriage, provided 
 with two sheaves, may be used; but with this latter, owing 
 to local causes and the difficulty of splicing both ropes of 
 an equal length, the load is not as well distributed between 
 the two ropes. 
 
 In subsequent considerations of the driving power of 
 ropes the relation between the ultimate strength, weight 
 per foot of length, normal working strain, and the diame- 
 ter of rope will be represented by the following equations 
 which have been determined for manilla transmission rope : 
 
 Let d = diameter of rope in inches; 
 
 w = weight of rope in pounds per foot; 
 S t = breaking strain in pounds; 
 t = normal working strain in pounds; 
 x = an empirical coefficient. 
 Then w ='.0.3216^; 
 8 = 
 
 x = 81 - 9& 
 
 The weight w per foot of length varies considerably m 
 different makes of rope, depending upon the amount of 
 twist and the foreign matters in the rope. 
 
 It is well known that much of the cheaper manilla rope 
 
 * On account of the relatively great difference between & and t, it 
 is not thought advisable to consider the increase in strength of the 
 smaller ropes, as in any case the difference would be very slight, and, 
 moreover, it must be noted that t is the normal estimated strain, and 
 may vary considerably from the actual strain. 
 
110 
 
 ROPE-DRIVING. 
 
 on the market is largely adulterated with weighing mate- 
 rial, such as gelatine size, French clay, and white lead. 
 Thus some manilla ropes will weigh not more than 0.26d~ 
 pounds when dry and very loosely twisted; in other cases 
 the weight will be as much as 0.46<:f pounds per foot of 
 length. The value we have given, viz., 0.32d 2 pounds, cor- 
 responds very closely to the average weight of good quality 
 lubricated transmission ropes. 
 
 Cotton ropes are about twenty per cent lighter for equal 
 diameters, and will vary from 0.20d 2 to 0,29d 2 pound per 
 foot. In the following table (VII) 0.32d 2 has been used 
 for manilla and Q.ZGd 2 for cotton ropes. 
 
 TABLE VII. WEIGHT OF ROPES 
 
 
 Weight in Pounds per Foot. 
 
 Diameter 
 
 
 of 
 
 
 
 Rope. 
 
 Manilla. 
 w = 0.32d. 
 
 Cotton. 
 w = 0.26d*. 
 
 f 
 
 0.18 
 
 0.15 
 
 
 .32 
 
 .26 
 
 H 
 
 .50 
 
 .40 
 
 H 
 
 .72 
 
 .58 
 
 If 
 
 .98 
 
 .79 
 
 2 
 
 1.28 
 
 1.04 
 
ROPE-DBIVItfG. Ill 
 
 CHAPTEE VIII. 
 
 IN determining the horse-power which a rope will trans- 
 mit under given conditions the centrifugal force due to the 
 velocity and weight of rope is an important factor, and its 
 influence should be considered in all cases where the 
 speed is greater than 2000 feet per minute; for at high ve- 
 locities this force diminishes the pressure exerted between 
 the rope and the circumference of the pulley, thus reducing 
 the friction between rope and pulley. When, therefore, a 
 rope has to transmit a given force, P, it must be subjected 
 to a greater tension the greater the centrifugal force F . 
 At a speed of about 90 feet per second the centrifugal 
 force increases faster than the power from increased veloc- 
 ity of the rope, and at 140 feet per second this force equals 
 the assumed allowable back tension in the rope; and since 
 the transmitting force is equal to the difference in tension 
 in the two parts of the rope, it will be seen that no power 
 will be transmitted at this speed unless the assumed allow- 
 able tension be exceeded. 
 
 It is evident that for a given total tension the less back 
 tension required to prevent the slip of the rope on the 
 pulley the greater will be the power transmitted at a given 
 speed. 
 
 The determination of this back tension is, however, at- 
 tended with a degree of uncertainty, as there are no con- 
 clusive experiments which give reliable data for its calcu- 
 lation; the coefficient of friction, 0, as stated by various 
 authorities, varying all the way from 0.075 up to 0.88.* 
 
 * The probable reasou for such widely divergent values lies in the 
 fact that the coefficient of friction varies with the percentage of 
 slip, and those tests made with very little slip would show a small 
 
.112 ROPE-DRIVING. 
 
 Reuleaux quotes the experiments of Leloutre and others 
 as indicating a value of 0.075 for cylindrical pulleys with 
 new hemp rope, 0.088 for semicircular grooves, and 0.15 
 for a wedge groove of 60. 
 
 Experiments by the Messrs. Pearce Brothers, of Dundee, 
 give a value of equal to 0. 57 to 0.88 for ropes on ungreased 
 pulleys, and <p 0.38 to 0.41 when the pulleys are greased. 
 
 Unwin states that the coefficient of friction for ropes on 
 a flat-metal pulley is equal to 0.28, from which the actual 
 coefficient for a grooved pulley is obtained by multiplying 
 0.28 by the cosec. of half the angle of the groove. For an 
 angle of 45 this would give = 0.72. These latter values 
 are probably very much higher than is ordinarily found in 
 actual practice with well-lubricated ropes and moderate 
 slip. From a consideration of the above and various other 
 experiments, and the conditions under which they were 
 carried out, it would appear that for ropes which are partly 
 worn and sufficiently greased to wear well with a low per- 
 centage of slip a value of 0.12 for a flat-surfaced, smooth- 
 metal pulley will approach very closely to those conditions 
 which obtain in average practice, from which the following 
 working coefficients are deduced: 
 
 = 0.12 cos 
 
 /Angle of groove^ 
 
 - c v 
 
 2 
 
 
 / 
 
 
 
 
 Angle of groove . 
 
 30 
 0.46 
 
 35 
 
 .40 
 
 40 
 .35 
 
 45 
 .31 
 
 50 
 
 .28 
 
 55 
 .26 
 
 60' 
 .24 { 
 
 Coefficient of friction, 0... 
 
 Besides varying with the angle of groove, as shown, the 
 
 coefficient; on the other hand, it is safe to assume that the larger 
 values were obtained under conditions in which the slip was greatly 
 increased. Varying atmospheric conditions and different degrees of 
 lubrication are also largely responsible for these divergent results. 
 
 See paper by Prof. Lanza on ""Friction of Leather Belting," in 
 Trans. A. S. M.E., vol. vn. p. 347; also "Experiments on Power 
 Transmitted by Belting," by Wilfred Lewis, in Trans. A. S, M. E. 
 vol. vn. p. 549. 
 
ROPE-DRIVING. 113 
 
 coefficient of friction is affected by the condition of the 
 rope, and for dry ropes may be taken somewhat greater 
 than the above value; will also be increased with an in- 
 creased percentage of slip. 
 
 If the arc of contact on smaller pulley, the coefficient 
 of friction between rope and sheave, and the total tension 
 in the rope be known, the tension on slack side of the 
 pulley, and hence the horse-power transmitted, can be 
 readily determined from the following considerations:* 
 
 Assuming, as before, that the driving force P is equal 
 to the difference in tension 2\ on the driving side of the 
 rope and T y on the driven side, and noting that the driv- 
 ing force must equal the friction F between the surfaces, 
 we obtain 
 
 T l - T^ = P = F. 
 
 The friction F depends upon the arc of contact a between 
 the rope and its sheave, the coefficient of friction 0, and 
 upon the centrifugal force F set up in the rope, due to 
 its velocity and weight; it is, however, independent of the 
 diameter of pulley. To determine the values of F 9 T lf 
 and T y it will be necessary to assume a given tension in 
 the rope; .also its speed and weight, coefficient of friction, 
 and arc of contact. 
 
 Let/ = friction between element of rope and pulley; 
 g = acceleration due to gravity = 32.16; 
 p = normal pressure exerted by element on pulley; 
 t = allowable working tension = 200^ 2 pounds; 
 v = velocity of rope in feet per second; 
 w = weight of rope per unit length and area 
 = 0.3216d 2 pounds per foot; 
 
 z = abbreviation for -: 
 
 *Weisbach, vol. in. p. 254. See also Eeuleaux. 
 
114 ROPE-DRIVING. 
 
 A = area of cross-section of rope; 
 
 F = centrifugal force due to speed and weight of 
 
 rope; 
 
 P = driving force = T l T^ 
 R = radius of pulley; 
 T = tension in rope at any point; 
 TI = tension in rope on tight side; 
 T^ = tension in rope on slack side; 
 a = least arc of contact between rope and pulley 
 
 circular measure 0.0175 X arc in degrees; 
 = coefficient of friction. 
 
 If in Fig. 42 T is the tension in the rope at any point 
 D, then the tension at the point E y whose distance from 
 D is ds, will be T + dT. Assuming / to represent the 
 
 >T+ cfT 
 
 FIG. 42. 
 
 friction on the element DE, we shall have/= dT when 
 the several forces are in equilibrium. Since the friction 
 is dependent upon the normal pressure p exerted by the 
 element upon the rim of the pulley, and since this normal 
 pressure is diminished by the centrifugal force due to the 
 weight and velocity of the element, we shall have 
 
KOPE-DKIVING. 115 
 
 dT=<l>(p-F.) ...... (4) 
 
 Now ^ is the resultant of the two forces Tand T -\- dT\ 
 hence 
 
 which may be assumed equal to Tda on account of the 
 smallness of da and dT. 
 
 The centrifugal force of the element of the rope is 
 
 A v* , TV* 
 F = wA Rg ds = W lRg dS > 
 
 and since ds = Rda, we have 
 
 F 9 = w~da. 
 
 t 9 
 
 Substituting these values of p and F in dT = <p(p F ), 
 we obtain 
 
 -;/; - 
 
 For convenience, let z ~T\ then 
 
 dT 
 dT = cf>T(l - z)da, or ^- = 0(1 - z)rf*. 
 
 Integrating, 
 
 /a\ ^r / 
 / -^ = / 0(1 - ^Jrfflf 
 
 A 7 
 
 log^ = 0(l-^)a; ... (6) 
 
 hence 
 
 in which e is the base of the hyp log = 2.7183; therefore 
 
116 HOPE-DRIVING. 
 
 /TT rn D /7' [~ </>a(l z)~] . ^ 
 
 J-i j. t JT - 2 i\_e ^ 2 
 
 (7) 
 
 If we assume -~ = r, there is obtained 
 
 T, 
 
 r - 1 "" T, - T, ~ P' 
 
 T 
 
 These ratios, r and -, Reuleaux calls the friction 
 
 modulus and the stress modulus, respectively. 
 
 As the common logarithm of e is 0.43430, the value of 
 r may be more readily obtained from common log r 
 0.4343 0(1 z) ; if the arc of contact is given in degrees, 
 a = 0.01750-, which gives the common log ( 
 
 r = 0.4343 X .0175a0(l - z) = 0.0075780a (1 - z). (8) 
 
 As the weight of a manilla rope one foot long = 0. 
 pounds, the value of z for varying speeds can be deter- 
 mined from 
 
 . 
 
 Z = :- = - ...... (9) 
 
 gt t 
 
 If now we assume a constant working stress t = 
 pounds, then 
 
 In the work which follows we shall assume that the ten- 
 sion T l in the rope on the tight side (driving tension) equals 
 the allowable tension t. 
 
 From these assumptions the following table (VIII) of 
 the values of 1 z has been computed ; 
 
117 
 
 TABLE VIII. VALUES OF 1 z FOR A WORKING STRESS EQUIVA- 
 LENT TO 200d 2 POUNDS. 
 
 ue S of oy Values of 
 
 per Minute. per Minute. 
 
 1000 0.98 5500 0.58 
 
 2000 0.94 6000 0.50 
 
 2500 0.91 6500 0.41 
 
 3000 0.87 7000 0.32 
 
 3500 0.83 7500 0.22 
 
 4000 0.78 8000 0.11 
 
 4500 0.72 8500 0.0 
 
 5000 0.65 
 
 It will be seen from the above that when the velocity of 
 the rope is as great as 8500 feet per minute, 1 2 = 0, 
 
 T 
 hence log r = and -^ 1; that is, T l T^ = 0, and 
 
 A 
 
 therefore no power will be transmitted unless the assumed 
 working tension t be exceeded. 
 
 In average work the lesser arc of contact embraced by 
 the rope generally on the smaller pulley will be about 
 165, and this value may be assumed for approximate cal- 
 culations with a working degree of accuracy. If the angle 
 in degrees is known, its value, a, in circular measure, can 
 be obtained from Table IX, in which a = 0.0175or. 
 
 TABLE IX. ANGLE EMBRACED BY ROPE. 
 
 Fraction of 
 
 Degrees, Circular Measure, Circumference, 
 
 a. 360/a. 
 
 105 1.83 0.29 
 
 120 2.09 0.33 
 
 135 2.35 0.37 
 
 150 2.62 0.42 
 
 165 2.88 0.46 
 
 180 3.14 0.50 
 
 195 3.43 0.54 
 
 210 3.66 0.58 
 
 240 4.19 0.66 
 
 If we now assume the coefficient of friction to be 0.31 
 for a 45 groove, we may obtain the value of the ex- 
 pressions 
 
118 BOPE-DRIVIKCK 
 
 = =r . 
 
 In order to simplify calculation, the following table (X) 
 
 contains values of r and -- -, which will enable the horse- 
 
 power transmitted by a rope to be determined with a degree 
 of accuracy depending upon the assumption of the coeffi- 
 cient of friction : 
 
 TABLE X. FRICTION AND STRESS MODULI. 
 
 $ ^i- 
 
 0.1 1.11 10.41 
 
 0.2 1.23 5.40 
 
 0.3 1.35 3.86 
 
 0.4 1.49 3.02 
 
 0.5 1.65 2.54 
 
 0.6 1.82 2.22 
 
 0.7 2.01 1.99 
 
 0.8 2.22 1.82 
 
 0.9 2.46 1.69 
 
 1.0 2.72 1.58 
 
 1.1 3.00 1.50 
 
 1.2 3.32 1.43 
 
 1.3 3.67 1.87 
 
 1.4 4.06 1.33 
 
 1.5 4.48 1.29 
 
 The following application will show the use of the tables. 
 Let it be required to determine the horse-power transmitted 
 by a rope 1 inch in diameter running at a velocity of 4000 
 feet per minute over a pulley with 45 grooves. Assuming 
 an arc of contact of 165, we find from Table IX a = 2.88; 
 for the required velocity, 4000 feet per minute, Table VIII 
 gives 0.78 as the value of 1 z; therefore, assuming the 
 coefficient of friction = .31, we obtain 
 
 <pa(l -z) = 0.31 X 2.88 X .78 = .69. 
 
 T 
 From Table X the value of -, corresponding to 0.69, 
 
HOPE-DRIVING. 
 
 119 
 
 is about 1.99; and as 3P = 200rf a pounds = 200 in the 
 
 T 
 
 present case, we find P = ~ = 100 pounds. Since 
 
 PV 
 33000 
 
 1.99 
 
 there 
 
 100 X 4000 
 33000 
 
 12.15, represented by the ordinate Im in Fig. 43. 
 The loss due to centrifugal force may now be obtained 
 
 FT. PER MINUTE lee.WU 
 
 FIG. 43. CENTRIFUGAL EFFECT IN ROPES. 
 
 by assuming the latter reduced to zero, in which case the 
 factor 1 z is equal to unity; therefore log r = .4340#, 
 from which, with the previous conditions, we obtain P = 
 120 and the corresponding h. p. = 14.5. 
 
 This value is represented on the diagram. Fig. 43, by the 
 ordinate In : the difference between In and Im mn will 
 
120 
 
 ROPE-DRIVING. 
 
 then be tke loss due to the centrifugal force set up in the 
 rope = 14.5 12.15 = 2.35 horse-power. 
 
 For any special case where the data are known or may 
 
 EUe.World 
 
 FIG. 44. HORSE-POWER TRANSMITTED BY MANILLA ROPES. 
 
 be determined, the formulas and tables already given should 
 be used to ascertain the horse-power transmitted, or the 
 diameter and number of ropes required for a certain work, 
 as the case may be. For average work, however, it will be 
 
ROPE-DRIVING. 
 
 found that the assumed values of a and 0, previously 
 noted, -will give very satisfactory results, and upon these 
 assumptions the writer has computed the following table of 
 horse-power (Table XI) for various-sized ropes, running at 
 speeds from 1000 to 7500 feet per minute: 
 
 TABLE XI. HORSE-POWER TRANSMITTED BY ROFES. 
 
 Working Strain = 200<2 2 pounds. 
 d = diameter of rope in inches. 
 
 Velocity of 
 Rope in Feet 
 per Minute. 
 
 Diameter of Rope. 
 
 % 
 
 H 
 
 1 
 
 94 
 
 Itt 
 
 m 
 
 2 
 
 1000 
 
 1.24 
 
 2.25 
 
 3.57 
 
 5.59 
 
 8.02 
 
 10.85 
 
 14.20 
 
 2000 
 
 2.70 
 
 3.84 
 
 6.84 
 
 10.68 
 
 15.39 
 
 20.93 
 
 27.36 
 
 2500 
 
 3.30 
 
 4.71 
 
 8.38 
 
 13.10 
 
 18.86 
 
 25.66 
 
 33.54 
 
 3000 
 
 3.83 
 
 5.46 
 
 9.80 
 
 15.39 
 
 21.87 
 
 29.74 
 
 38.88 
 
 3500 
 
 4.30 
 
 6.23 
 
 11.09 
 
 17.33 
 
 24.94 
 
 34.03 
 
 44.35 
 
 4000 
 
 4.74 
 
 6.83 
 
 12.15 
 
 18.98 
 
 27.33 
 
 37.17 
 
 48.59 
 
 4500 
 
 5.01 
 
 7 24 
 
 12.89 
 
 20.15 
 
 29.00 
 
 39.45 
 
 51.57 
 
 5000 
 
 5.20 
 
 7.'47 
 
 13.29 
 
 20.76 
 
 29.89 
 
 40.65 
 
 53.15 
 
 5500 
 
 5.29 
 
 7.60 
 
 13.53 
 
 21,14 
 
 30.43 
 
 41.39 
 
 54.11 
 
 6000 
 
 5.08 
 
 7.32 
 
 13.10 
 
 20.36 
 
 29.32 
 
 39.77 
 
 52.12 
 
 6500 
 
 4.74 
 
 6.83 
 
 12.13 
 
 19.00 
 
 27.34 
 
 87.21 
 
 48.63 
 
 7000 
 
 4.12 
 
 5.93 
 
 10.54 
 
 16.47 
 
 23.72 
 
 32.26 
 
 42.18 
 
 7500 
 
 3.25 
 
 4.67 
 
 8.32 
 
 13.00 
 
 18.73 
 
 25.42 
 
 33.23 
 
 The graphic representation of these values, Fig. 44, 
 shows the effect of centrifugal force in diminishing the 
 power transmitted under an assumed working tension, and 
 would indicate that with tensions of 200^ 2 pounds the 
 speed should not exceed 5500 feet per minute. The in- 
 creasing effect and loss of power due to centrifugal force 
 in the rope can also be seen in the diagram, Fig. 43, which 
 represents the horse-power transmitted by an inch rope 
 under an assumed constant tension of 200 pounds. The 
 straight line AB shows the power which would be trans- 
 mitted if centrifugal force were neglected, and is obtained 
 by making z = in the general equation 
 
122 ROPE-DRIVItfG. 
 
 log r = 0.4343^0(1 -z); 
 
 the curve AC represents the power transmitted when cen- 
 trifugal force is taken into account; and the curve AB 
 shows the power absorbed by centrifugal force: this latter 
 curve is obtained by subtracting the vertical ordinates be- 
 twejen the straight line AB and the curve AC. By a con- 
 sidation of the diagram it will be seen that at speeds 
 less than 2000 feet per minute the power absorbed by 
 centrifugal force is very small, and may be neglected as 
 far as practical results are concerned. Beyond this speed, 
 however, the loss from this cause increases very rapidly, 
 until, as previously shown, at a speed of about 8500 feet 
 per minute the whole of the allowable tension is absorbed. 
 
 Assuming that the maximum power is transmitted by a 
 rope at a velocity of about 5500 feet per minute, it is evi- 
 dent that the first cost of the rope will be a minimum for a 
 given power when running at this speed. The ratio of the 
 first cost of the rope running at any other speed may be 
 obtained by dividing the horse-power at 5500 per minute 
 by the horse-power at the required speed.* 
 
 Thus, if the first cost of a IJ-inch rope which will trans- 
 mit 21.14 h. p. at 5500 feet per minute be represented by 
 unity, the cost at 3000 feet per minute will be 
 
 21.14 _ 
 15^9 * 
 
 since st\ J-inch rope running at 3000 feet per minute will 
 transmit 15.39 h. p. 
 
 The relative first cost for a given diameter of rope to 
 transmit the same horse-power at varying speeds is shown 
 in the accompanying Table XII. 
 
 Although the first cost of a rope to transmit a given horse- 
 power is a minimum for a speed of about 5500 feet per 
 
 * C, W. Hunt, in Trans. A. S. M. E., vol. xii. 
 
ROPE-DRIVING. 
 
 123 
 
 minute, yet the economy is not as great as would appear from 
 the foregoing table, for the effect of wear must be considered. 
 The causes of wear, internal and external, have been pre- 
 viously discussed; it will be sufficient to note here that the 
 
 TABLE XII. RELATIVE FIRST COST OF ROPE-DRIVING. 
 
 Velocity of Rope 
 in Feet 
 per Minute. 
 
 Relative First 
 Cost per 
 Horse-power. 
 
 Velocity of Rope 
 in Feet 
 per Minute. 
 
 Relative First 
 Cost per 
 Horse-power. 
 
 1,000 
 
 3.78 
 
 4,500 
 
 .05 
 
 2,000 
 
 1.89 
 
 5,000 
 
 : .02 
 
 2,500 
 
 1.62 
 
 5,500 
 
 .00 
 
 3,000 
 
 1.38 
 
 6,000 
 
 .03 
 
 3,500 
 
 1.22 
 
 6,500 
 
 .12 
 
 4,000 
 
 1.10 
 
 7,000 
 
 .28 
 
 internal destructive effect produced by bending and distort- 
 ing the fibres and the wear due to external contact, slipping, 
 or wedging in the grooves of the pulleys, may, within the 
 limits of ordinary practice, be considered as directly propor- 
 tional to the velocity of the rope. What this wear is in 
 terms of the velocity there is not sufficient data to deter- 
 mine, but if the coefficient be represented by c, the relative 
 wear for a given diameter may be determined by multiply- 
 ing the velocity by this coefficient; that is, the relative 
 wear cv, in which the wear increases directly with the 
 Telocity, but not, however, directly with the horse-power 
 transmitted. 
 
 If we assume the coefficient to be such that the wear on 
 a rope at 1000 feet per minute is equal to unity, then the 
 wear on the rope at any other speed will be 
 
 T. , , . required speed 
 
 Eela tl vewear = - - -. 
 
 To determine the relative wear per horse-power trans- 
 mitted by a given rope at varying speeds, it will be neces- 
 
124 
 
 sary to determine the wear per horse-power for the rope 
 running at any required speed, and then divide this valuo 
 by the wear per horse-power when running at 1000 feet per 
 minute. Let it be required to determine the wear of a rope 
 transmitting a given horse-power at 5500 feet per minute, 
 as compared to what it would be at 1000 feet per minute. 
 
 The horse-power transmitted at 1000 feet per minute is 
 found to be 3.57 for a one-inch rope, at which speed we 
 have assumed that the wear is equal to unity, hence the 
 wear per horse-power may be considered equal to 
 
 57 - -' 
 
 At a speed of 5500 feet per minute the horse-power trans- 
 mitted by the same rope is 13.53, but the wear at this 
 speed is assumed to be five and a half times greater than 
 at 1000 feet, other conditions being the same, therefore 
 
 5 5 
 the wear per horse-power = =-$-$ = .406; that is, the 
 
 lo.Oo 
 
 wear of any rope transmitting one horse-power at 5500 feet 
 
 per minute is - = 1.45 times the wear which would 
 .28 
 
 occur at 1000 feet per minute. 
 
 From this it will be seen that although the first cost of 
 a rope is cheaper at the higher speeds, the rope lasts longer 
 while running at the lower speeds conditions remaining 
 constant. 
 
 Taking the case we have been considering, it is found 
 that the relative first cost of the rope is inversely as the 
 horse-power transmitted, or 
 
 Cost of rope at 1000 feet _ 13.53 _ 
 Cost of rope at 5500 feet ~ 3.57 
 
 that is, the first cost is 3.78 times greater for the slower 
 speed. But it is shown above that the rope will wear out 
 
EOPE-DRIVItfG. 
 
 125 
 
 nearly 50 per cent faster at the increased speed; therefore, 
 taking the life of the rope as well as the first cost into con- 
 sideration, the relative cost to transmit a given horse-power 
 
 q ryo 
 
 at the speeds noted will be ' =2.6 times greater for the 
 
 1.45 
 
 lower speed. This can be determined more conveniently 
 by assuming H = horse-power transmitted at V feet per 
 minute, H l = horse-power transmitted at V l feet per min- 
 ute, where H t > H and V. > V. The wear per horse- 
 
 V V 
 
 power in each case will then be proportional to and ; 
 
 H rt j 
 
 the relative first cost of the rope per horse-power running 
 at the lesser speed, compared to that when running at the 
 greater, will be as H l is to H\ hence the ratio of relative 
 cost, taking wear into account, is 
 
 V 
 
 H 
 
 (ii) 
 
 TABLE XIII. RELATIVE WEAR AND COST OF ROPE PER HORSE- 
 POWER TRANSMITTED. 
 
 Velocity 
 in Feet 
 per Minute. 
 
 Relative Wear 
 per Horse-power 
 transmitted. 
 
 Relative Cost of Rope 
 per H. P. transmitted, 
 considering Wear. 
 
 1,000 
 
 1. 
 
 1. 
 
 2,000 
 
 1.03 
 
 .54 
 
 2,500 
 
 1.06 
 
 .45 
 
 3,000 
 
 1.10 
 
 .40 
 
 3,500 
 
 1.13 
 
 .36 
 
 4,000 
 
 .18 
 
 .347 
 
 4,500 
 
 .25 
 
 .345 
 
 5,000 
 
 .34 
 
 .36 
 
 5,500 
 
 .45 
 
 .38 
 
 6,000 
 
 .64 
 
 .44 
 
 6,500 
 
 .93 
 
 .52 
 
 7,000 
 
 2.40 
 
 .80 
 
 7,500 
 
 3.22 
 
 1.36 
 
126 
 
 ROPE-DRIVING. 
 
 Table XIII has been calculated upon the above basis of 
 comparison, namely, that the wear in a rope at 1000 feet 
 per minute is equal to unity and increases directly as the 
 speed; and also, that the cost of ropes for a permanent 
 
 _ 3. a 
 
 VFI OCITY IN FEET PER MINUTE lt*. World 
 
 FIG. 45. RELATIVE WEAR AND COST OF ROPE PER H. P. 
 
 installation is proportional to the square of the ratio of the 
 power transmitted at different speeds multiplied by the in- 
 verse ratio of the corresponding speeds. To ascertain the 
 relative wear per horse-power of a rope running at any 
 given speeds, it will only be necessary to form a ratio 
 
(UNIVERSITY) 
 
 ^x^CAUFfWNUj^^ 
 
 ROPE-DRIYINQ. 127 
 
 between the values in the table corresponding to the given 
 speeds. Thus the wear per horse-power at 5000 feet per 
 
 1 34 
 
 minute, as compared to that at 2500 feet, will be -^ = 1.26. 
 
 In the same way the relative ultimate cost per horse- 
 
 nn 
 
 power of a rope running at these speeds will be ' = 0.8, 
 
 or 20 per cent less for the greater speed. The accompany - 
 h]g diagram, Fig. 45, represents these relative values 
 graphically, to which is added the curve of relative first 
 cost. It will be noticed that although the first cost of a 
 rope is a minimum for a speed of 5500 feet per minute, 
 when wear is considered the minimum cost occurs at a 
 speed of 4500 feet per minute. 
 
128 ROPE-DRIVINQ. 
 
 CHAPTER IX. 
 
 As previously noted, it is desirable in all cases of rope 
 transmission to so arrange the drive that the slack side of 
 the rope shall be on the upper part of the pulley, thus in- 
 creasing the arc of contact, as the two sides will then ap- 
 proach each other when in motion. 
 
 In order that the desired tensions T^ and T t shall be 
 attained in the two parts of a rope, the deflections or sag 
 must be of predetermined values. The centre line of the 
 rope will lie in a curve, which may be determined with no 
 appreciable error by assuming the rope to have no elastic- 
 ity and to be of constant cross-section, under which condi- 
 tion the curve will be that known as the catenary, the trans- 
 cendental equation of which is 
 
 y = o 
 
 Let the form of curve in which the rope hangs be repre- 
 sented by PO'P' (Fig. 46), in which 0' is the lowest point 
 of the rope. Take 0' as the origin of coordinates, and let 
 x = 0' A' and y' = A'P be the abscissa and ordinate of 
 any point P in the curve, and let I be the distance between 
 the points of support. 
 
 Since we have assumed the rope to be perfectly inelastic, 
 the tension at any point of the curve must be in the direc- 
 tion of the rope. Let T be the stress in the rope at P, the 
 vertical and horizontal components of which are repre- 
 sented by V and H respectively. Assume the length of 
 curve O'P = s. Since the weight of a unit length of rope 
 = w, the vertical component of the tension T is evidently 
 
ROPE-DRIVING. 
 
 129 
 
 equal to one half the weight of the rope between the points 
 P and P', or V = iw PO'P' MS. To determine the 
 length of curve, produce the tangent through the point P 
 and rectify the curve O'P on this tangent, making s = PQ. 
 
 FIG. 46. 
 
 Erect a perpendicular to PQ at the point Q, meeting the 
 vertical from P at A ; then will 0.1 drawn through A par- 
 allel to O'A' be the directrix to the catenary, and QA will 
 equal A A' = c.* Hence 
 
 - QA\ 
 
 Therefore 
 
 PQ* = 
 
 f/) 2 -' 2 ; 
 
 V=ws = w V (ij'Y H- Icy'. 
 
 = V (y'Y + icy'. 
 
 * For geometry of this curve see ' ' The Funicular Polygon w in 
 Bowser's "Analytic Mechanics," p. 216 et seq.; also Price's "Me- 
 chanics," vol. i. 
 
130 ROPE-DRIVING. 
 
 At the vertex the tension is horizontal and equal to the 
 weight of a length, c, of the rope; but this tension is the 
 same as the horizontal component at the point P; hence 
 H = we. Since T= 1/F a -j- H* 9 we obtain the following 
 value for the tension in the rope : 
 
 , . . (13) 
 
 In order to determine the parameter, c, let the equation 
 e-f the curve y = - \e c -|- 6~ <y be developed into the following 
 series : 
 
 Since the character of the curve is such that the quotient 
 
 />* 
 
 is a proper fraction, the series will be converging. Stop- 
 c 
 
 ping at the third member as giving sufficient accuracy, we 
 have 
 
 therefore x" 2 c (y c) ....... (14) 
 
 Substituting the -value of y = y' -\-c in this equation, we 
 obtain x* = %cy', which is the equation of a parabola re- 
 
 ferred to its axis and the tangent to its vertex. Now let - 
 
 
 
 (Fig. 47) equal the half distance between supports = x> 
 and h = sag of the rope = y' ; then from the previous 
 
 / 1 Y 
 
 equation we obtain I j = 2ck: 
 
 J 
 
 8k' 
 
 F 
 
 hence the parameter c = - Substituting this value of 
 
KOPE-DKIVItfG. 
 
 131 
 
 c in equation (13), we have 
 
 T = (+*) 
 
 in which w equals the weight of a unit length of rope = 
 0.32f? 2 for manilla ; that is, the tension at any point in the 
 rope is equal to the weight of a portion equivalent in length 
 to the parameter plus the ordinate y f of the point. From 
 this equation we may obtain the sag of the driving or driven 
 portions of the rope by substituting for T the values of T l 
 and T^, the deflections corresponding to which may be rep- 
 resented by h l and h^ (Fig. 47). T, will evidently be con- 
 
 DRIVEN 
 
 FIG. 47. 
 
 stant and equal to 200d 2 if our preconceived conditions 
 are maintained, but the value of T^ will be variable, in- 
 creasing with the speed. 
 
 Practically, it will be impossible to maintain a constant 
 tension in the rope, so that the amount of sag obtained by 
 calculation is liable to vary with the conditions of service. 
 The tension may, however, be approximately determined 
 by the deflection. Assuming the distance between support- 
 ing points of the rope equal to the distance between centres 
 of pulleys, and solving for li, we obtain 
 
 h - IT * + 1 A/~T* V 
 3 ~~ 2 w % * "" o~ ' 
 
132 KOPE-DKIVING. 
 
 The positive and negative signs before the radical in these 
 equations indicate two values for h, the lesser of which 
 only is to be used. As pointed out by Reuleaux, between 
 
 1 T 
 
 the two lies a value h = - -, which is obtained when the 
 
 2 w 
 
 quantity under the radical = 0; that is, when T . 
 
 V 2 
 
 This deflection is interesting, as it denotes the minimum 
 stress which may exist in the rope. 
 
 Since the sum of the tensions increases with the speed, 
 the sag of the rope when at rest is not directly obtainable 
 from the previous values of h l and 7* a . 
 
 The tension necessary for adhesion, which constitutes a 
 part of the stress, T 7 ,, is dependent upon the speed at 
 which the rope is intended to be run, so that in order to 
 determine the sag 7/- when at rest, for a given maximum 
 tension T J9 the initial tension must be obtained for any 
 given speed, and this value substituted in the general for- 
 mula 
 
 h = -- - \ j . 
 
 Assuming the tension on the tight side of the rope to 
 be made up of three parts, namely, the driving force P, 
 the centrifugal force F n9 and the tension T 3 , necessary to 
 balance the strain for adhesion, we obtain 
 
 In like manner the tension on the slack side of the rope 
 may be assumed to be produced by the strain necessary for 
 adhesion, plus the strain due to centrifugal force; that is, 
 
 T T 4- F 
 
 * i * 3 i * fl- 
 it is evident that if the normal tension T l be diminished 
 by the centrifugal force the remaining stress will be equal 
 to the tension necessary for adhesion plus the driving 
 
ROPE- 1) RIVING. 
 
 133 
 
 force; hence if we obtain the value of the ratio ~~ from 
 
 e<K, in which the stress due to centrifugal force is neg- 
 lected, we shall have in T 2 the initial tension necessary for 
 adhesion, or 7' 2 T 3 . If r l\ is assumed to be constant, then 
 7 ' 3 will be constant for a given coefficient of friction and 
 
 arc of contact; therefore 
 
 T j__ /TT 
 
 - - 3 will equal the tension in 
 & 
 
 the rope when the latter is at rest. 
 
 The following simpler method for obtaining the sag, 
 though less exact, is sufficiently accurate for any practical 
 case that may arise, for it must be borne in mind that any 
 theoretical calculation for the deflection of a running rope 
 can at best be only an approximation, as it is exact only 
 when the rope is running at its normal speed, transmitting 
 its full load and strained to its normal tension. Let / be 
 the distance between two shafts which are at the same 
 
 FIG. 48. 
 
 level (Fig. 48), and let li be the deflection of the rope; also, 
 let CPD = be the inclination of the rope at P, from 
 which 
 
 CD 2k 4h 
 
 sin = 
 
 and hence, since Fis the vertical component of the tension 
 
134 
 
 ROPE-DRIVIXG. 
 
 w 
 
 Tin .the rope and equal to , where Wi& the weight of 
 rope between supports, we shall have 
 
 ~F~~T7 
 
 sin 
 
 As h is small compared to I, we may neglect --W, as it 
 
 will have no appreciable influence on the result; then if 
 we assume the length of rope to be equal to the distance 
 between the shafts, we shall have, approximately, 
 
 Wl_wV_ 
 = 87* ~ = 87*' 
 
 where wl W\ hence the sag of the rope at the centre 
 will be 
 
 *=!!> LI*] 
 
 in which w = 0.33d 2 for manilla ropes. 
 
 From this formula the following table (XIV) has been 
 
 TABLE XIV. DEFLECTION OF ROPE. 
 
 Calculated from h ^ r 
 
 8.Z 
 
 
 Deflection in Feet in Slack Side of Rope 
 when r 3 = T l P = 
 
 Deflection 
 
 Deflection 
 in Both 
 
 Distance 
 between 
 Pulleys 
 in Feet. 
 
 Corres 
 
 91d a 
 
 >ponding to 
 per Mir 
 
 lOOd* 
 
 Velocity ir 
 ute of: 
 
 112d2 
 Feet 
 
 Tight Side 
 of Rope 
 when 
 
 Pounds. 
 
 Sides of 
 the Rope 
 when 
 Initial 
 Tension 
 
 
 2000 
 
 3000 
 
 4000 
 
 5000 
 
 
 o- 
 
 30 
 
 0.42 
 
 0.39 
 
 0.36 
 
 0.32 
 
 0.18 
 
 0.25 
 
 40 
 
 0.74 
 
 0.70 
 
 0.64 
 
 0.57 
 
 0.32 
 
 0.45 
 
 60 
 
 1.67 
 
 1.58 
 
 1.44 
 
 1.28 
 
 0.72 
 
 1.02 
 
 80 
 
 2.97 
 
 2.81 
 
 2.56 
 
 2.28 
 
 1.28 
 
 1.82 
 
 100 
 
 4.65 
 
 4.40 
 
 4.00 
 
 3.57 
 
 2.00 
 
 2.84 
 
 120 
 
 6.70 
 
 6.33 
 
 5.76 
 
 5.14 
 
 2.88 
 
 4.10 
 
 140 
 
 9.12 
 
 8.61 
 
 7.84 
 
 7.00 
 
 3.82 
 
 5.58 
 
 160 
 
 11.90 
 
 11.25 
 
 10.24 
 
 9.14 
 
 5.12 
 
 7.27 
 
ROPE-DRIVING. 
 
 135 
 
 computed and the curves plotted, as shown in Fig. 49, T l 
 being assumed as constant and equal to 200^ 2 pounds. 
 The tension T^ which varies with the speed, has been 
 separately determined for the several cases considered from 
 
 T 
 
 -L = eWi-), 
 
 * 2 
 
 in which the coefficient of friction, 0, =0.31 and a = 2.88; 
 hence 
 
 12 Ft. 
 
 40 60 80 100 120 140 160 Ft. 
 
 DISTANCE BETWEEN CENTERS OF PULLEYS. Elec. World 
 
 FIG. 49. DEFLECTION OF ROPES. 
 
136 ROPE-DRIVING. 
 
 log ^ = 0.4343 X 0.9(1 - z). 
 f 
 
 The deflection on the tight side will then be 
 
 ' - ST ~ 8X00 
 and on the slack side & 2 = -?p> in which T 7 , has the values 
 
 -^ 2 
 
 given in the table. 
 
 As previously pointed out, the initial tension in the rope 
 when at rest may be obtained by neglecting F^ , in which 
 
 T 
 
 case we have log ~ = 0. 43430**, from which we find 
 
 * 2 
 
 ^ = 2.46, or T; = 82d 2 ; and since T 3 = T t when there is 
 
 * * 
 
 no centrifugal force acting, the initial tension T % will 
 
 equal 
 
 hence T 9 = 141^ 2 . From this we may obtain the deflec- 
 tion when the rope is at rest, as noted in the last column 
 of Table XIV, which has been computed from 
 
 ft = 
 
 where I is the distance in feet between centres. 
 
 To draw the curve of the rope in order to determine 
 the space it will occupy, assume it to hang in a parabola 
 with origin at vertex and lay off the X and Y axes, as in 
 Fig. 50. Let 05 = %l = -J- distance between points of sus- 
 pension of the rope, and let 5H = h, the sag of rope. Di- 
 vide 5H into a convenient number of equal parts, also 
 divide 05 into the same number of equal parts; erect per- 
 pendiculars from the points of division 1, 2, 3 ... and join 
 
ROPE-PRTVIKft. 
 
 13? 
 
 1', 2', 3'. . . with the origin 0. The curve drawn through 
 the points of intersection will represent one branch of the 
 parabola desired. 
 
 In outdoor drives, where the configuration of the ground 
 would prevent the proper amount of sag being used, or, in 
 
 o 
 FIG. 
 
 1 23 
 
 50. METHOD OF LAYING OUT CURVE. 
 
 4 5 , 
 
 Eke. World 
 
 TV iV 
 
 FIG. 51. INCLINED TRANSMISSION, 
 
 general, where obstructions prohibit the proper deflection 
 for a given distance between centres of shafts, the rope 
 should be carried on supporting pulleys or sag-carriers, 
 examples of which have already been given. 
 
138 HOPE-DRIVING. 
 
 Where the pulleys are placed at" different heights we 
 have an inclined transmission, and the curve in such cases 
 is unsymmetrical, as in Fig. 51.* 
 
 This curve is best solved by an approximation similar to 
 that already given. 
 
 It will be noticed that the curve of the rope ABC is 
 made up of two unequal parts, AB and BC, whose hori- 
 zontal projections are I' and I" and their corresponding 
 deflections h' and h". The given difference in height be- 
 tween the points of support of the rope is ED = h li f 
 li" , and their known horizontal distance is 1=1'-}- I" t 
 From the preceding formula for the deflection (equation 
 [15] ) we have 
 
 /" and A" = ? = /*-, 
 
 in which fi is taken as a coefficient depending upon the 
 
 value of the tension in the rope -5-^. 
 
 o J. 
 
 Since h = /*' - li" and I = V + 1", we have 
 
 therefore I' - I" = ~. But V = I - I" and I" = I - I'-, 
 hence 2/311' 01* = h, and the required distance I' = 
 
 Toy . In the same way V 9 = -^^ , and the deflections 
 lipl Zpl 
 
 are 
 
 in each of which equations fi has a separate value depend- 
 ing upon the tension in the rope. It is evident that the 
 tension T' at A will be greater than that at G, on account 
 of the greater weight of rope lying in the upper branch of 
 
 * Weisbach, vol. m. p. 243. 
 
ROPE-DRIVING. 
 
 130 
 
 the parabola. If T' is known, the lesser tension T" am 
 be determined by assuming the length of the, rope to be 
 the same as if the points A and C, Fig. 52, were at the 
 same height and I feet apart. From previous considera- 
 
 FIG. 52. 
 
 tion it is seen that the tension at any part of the rope in 
 a parabola is equal to the weight of rope equivalent in 
 length to the parameter c added to the ordinate of the 
 point; hence if h" equals the ordinate of Q li' h, we 
 shall have 
 
 T" = w (c + h") and T' = w(c + //). 
 
 Since c=, 
 
 and 
 
 therefore T" - wli" = T' - wli' . Substituting for h' its 
 value, h -\- Ji", there is obtained 
 
 T" = T - wh. 
 
 When a tension-carriage is used the necessary weight 
 can be ascertained from the formula for back tension, 
 
140 ROPE-DRIVIHG. 
 
 ?\ = T t -\-F 9 ; if we assume F = 0, we shall have the 
 initial tension necessary for adhesion equal to the ten- 
 
 T 
 
 sion in slack side of rope, that is, T 3 = T t . But T y = L - 
 
 /w.4o 
 
 T 
 
 (from log -= = 0.4340O', assuming previous values of 
 
 * 2 
 
 and a); therefore the initial tension in the rope will be 
 
 T, _ 200flf 
 3 ~ 2.46 ~ 2.46 ' 
 When diameter of rope equals 
 
 f inch the initial tension = 31 pounds. 
 
 3 {( (( __ Aft 
 
 " " 62 " 
 
 ^ (( <( a __ go (( 
 
 11 f( (( (( __ 19 I 7 *f 
 
 1 " " " " = 184 " 
 
 If " " " = 250 " 
 
 2 a ' ( C( (l ^ 3^5 4< 
 
 The actual weight to be placed on the tension-pulley will 
 depend upon the arrangement of drive, and in order to 
 obtain the best results should be determined for each par- 
 ticular case. It is to be noted that with a vertical tension- 
 carriage the weight of carriage and sheave must be consid- 
 ered as a portion of the weight producing 
 3 tension in the rope. 
 
 If W equals the total weight necessary 
 to maintain the tension T t in each part 
 of the rope leading off from the tension- 
 pulley, then, when these two portions are 
 parallel, we shall have W=2T Z -, but if 
 the rope leads from the tension-pulley so 
 that it includes an angle 2# between its 
 sides, as shown in Fig. 53, then the weight W will be less, 
 and may be found from W= 2jP 3 cos 0. 
 
ROPE-DRIVING. 
 
 CHAPTER X. 
 
 IT has been previously stated (see page 7) that where 
 rope-driving is used the loss at the engine in friction may 
 be taken in a general way as about 10 per cent of the rated 
 horse-power of the engine, that an additional 10 per cent 
 is absorbed by the shafting, and that from 5 to 8 per cent 
 may be attributed to losses in the rope itself, due to resist- 
 ance to bending, wedging in the grooves, differential driv- 
 ing effect, and creep, all of which affect the loss to a greater 
 or less extent. 
 
 According to the accepted laws of solid friction we should 
 expect that with an increased load on the engine the fric- 
 tion would be increased in direct proportion to the load, 
 but in nearly all experiments to determine the friction of 
 steam-engines we have the anomaly that the work neces- 
 sary to overcome friction is practically constant, and, in 
 fact, in many cases it is a little greater when running with- 
 out load than when the engine is fully loaded. 
 
 However, it is probable that the ordinary laws of friction 
 obtain here as in other cases of sliding and rolling con- 
 tact; but instead ol having a constant coefficient of friction 
 for the surfaces in contact, the coefficient may be con- 
 sidered as a variable depending upon the degree and distri- 
 bution of lubrication. The lubrication of engine bearings 
 (both rolling and sliding contact), approaches more nearly 
 to the condition which obtains when the bearings are sub- 
 jected to an oil bath, and although the lubrication is 
 restricted, yet the result is similar in action. 
 
 Experiments* on the friction of a well-lubricated journal 
 
 *Proc. I. M. E. November, 1885; also Kent's MecUauiculEugiiieer's 
 Pocket Book, 
 
142 ROPE-DRIVING. 
 
 (oil bath), show that the absolute friction, that is, the ab- 
 solute tangential force per square inch of bearing, required 
 to resist the tendency of the brass to go round with the 
 journal, is nearly a constant under all loads within ordinary 
 working limits. Most certainly it does not increase in 
 direct proportion to the load, as it should do according to 
 the ordinary theory of solid friction. The results of these 
 experiments seem to show that the friction of a perfectly 
 lubricated journal follows the laws of liquid friction much 
 more closely than those of solid friction. They show that 
 under these circumstances the friction is nearly indepen- 
 dent of the pressure per square inch, and that it increases 
 with the velocity, though at a rate not nearly so rapid as 
 the square of the velocity. 
 
 The experiments on friction at different temperatures 
 indicate a great diminution in the friction as the tempera- 
 ture rises. Thus in the case of lard-oil, taking a speed of 
 450 revolutions per minute, the coefficient of friction at a 
 temperature of 120 is only one-third of what it was at a 
 temperature of 60. In regard to engine friction, whatever 
 be the cause, it is a well-known fact that the coefficient of 
 friction decreases as the load increases, so that at all ordi- 
 nary speeds the internal resistance of the engine may be con- 
 sidered sensibly constant, in which case the so-called fric- 
 tion-card of the engine represents practically the friction 
 of the machine when fully loaded the indicated power 
 without load being sensibly the measure of the wasted work 
 of the engine when in operation under load of whatever 
 amount. 
 
 That is, the engine friction is independent of the load 
 and is a function of the characteristic of the engine itself, 
 of the speed of piston and rotation, and, to a slight extent, 
 of the steam-pressure and of the method of steam-distri- 
 bution : so that while we may speak of the friction as being 
 a certain percentage of the horse-power of an engine, it 
 
ROPE- DRIVING. 
 
 143 
 
 must be understood to refer to the rated indicated horse- 
 power ;* at less than rated power the percentage of loss due 
 to friction will be greater, and at maximum power the per- 
 centage will be less. This is shown by way of illustration 
 in the following table (XV), which is taken from a paper 
 presented by Prof. Thurston before the American Society 
 of Mechanical Engineers in 1886 :f 
 
 This engine, a " Straight Line," was 8 inches in diameter 
 of cylinder by 14 inches stroke; it had a balanced valve 
 with stroke varying from 2 to 4 inches according to position 
 of governor and eccentric, a fly-wheel 50 inches in diam- 
 eter, weighing 2300 pounds. Its rated load was 35 to 40 
 horse-power. 
 
 TABLE XV. FRICTION PER CENT UNDER VARYING LOADS. 
 
 Revolutions 
 per 
 Minute. 
 
 Steam 
 Pressure. 
 
 Indicator 
 Horse-power. 
 
 Friction 
 Horse-power. 
 
 Friction 
 per cent. 
 
 232 
 
 50 
 
 7.41 
 
 3.35 
 
 45 
 
 229 
 
 65 
 
 7.58 
 
 2.60 
 
 34 
 
 230 
 
 63 
 
 10.00 
 
 4.00 
 
 40 
 
 230 
 
 73 
 
 11.75 
 
 3.65 
 
 32 
 
 230 
 
 75 
 
 14.02 
 
 4.02 
 
 28 
 
 230 
 
 80 
 
 15.17 
 
 3.17 
 
 21 
 
 230 
 
 75 
 
 16.86 
 
 2.86 
 
 17 
 
 230 
 
 75 
 
 28.31 
 
 3.36 
 
 11.75 
 
 229 
 
 60 
 
 33.04 
 
 3.16 
 
 9.5 
 
 229 
 
 58 
 
 37.20 
 
 2.34 
 
 6.3 
 
 229 
 
 70 
 
 43.04 
 
 3.19 
 
 7.4 
 
 230 
 
 85 
 
 47.79 
 
 2.75 
 
 5.8 
 
 230 
 
 90 
 
 52.60 
 
 2.60 
 
 4.9 
 
 230 
 
 85 
 
 57.54 
 
 2.54 
 
 4.4 
 
 Examining the above table, it will be seen that the fric- 
 tion of the engine varies somewhat with varying steam- 
 pressures and total power, but in such a manner as to indi- 
 cate the controlling cause, as, for instance, imperfect lubri- 
 
 * Thurston, Trans. A. S. M. E., vol. x. p. 110. 
 f Trans., vol. vm. p. 0, 
 
144 KOPE-DBIVING. 
 
 cation, to be irregular iu action, and, possibly, to some 
 extent, due to errors of observation and to accident. The 
 average friction horse-power is 3.11 h. p., and the variations 
 from this value are distributed throughout the whole series, 
 showing that the work necessary to overcome the friction 
 is practically constant, and independent of the load. The 
 friction of this engine is quite low, as at its normal rating 
 the percentage of loss is less than 7 per cent. According 
 to D. K. Clark,* the frictional resistance of steam-engines 
 varies from 8 to 20 per cent of their normal indicator h. p. 
 the size of engines experimented upon ranging from 13 
 to 350 h. p. 
 
 In recent tests of American engines it has been shown 
 that-with first-class workmanship and balanced valves the 
 percentage of loss at normal working load may be reduced 
 to about 6 per cent, both with Corliss and with high-speed 
 single-cylinder automatic engines; but this is exceptional, 
 and we may expect with ordinary lubrication that the fric- 
 tional resistance will vary from 8 to 13 per cent of the 
 normal load. 
 
 Compound engines of the better class should not absorb 
 more than about 10 per cent, and triple-expansion engines 
 no more than 12^ per cent, under full load. 
 
 For small engines, either single or compound, there is 
 probably little difference between the internal resistance, 
 whether geared with ropes or flat leather belts, for the 
 weights of fly-wheel and grooved pulley and the diameter 
 of shaft would be essentially the same in each case. With 
 larger engines, however, the belts would require a wider- 
 faced fly-wheel, which, on account of the greater distance 
 between bearings, would necessitate a larger shaft, and 
 hence increased work in overcoming journal-friction as- 
 suming the same weight of wheels and speed of rotation. 
 
 * " The Stecam Eugiiie," vol. n. p. 616, 
 
ROPE-DRIVING. 
 
 145 
 
 With the rope-driving, also, the speed of the rim will be 
 greater, and a somewhat lighter pulley may be used to 
 insure the same degree of steadiness in running; in many 
 cases the ropes are delivered from the fly-wheel in a nearly 
 vertical direction, so that a certain portion of the weight 
 on the hearings is neutralized by the upward pull of the 
 ropes. Moreover, the elasticity and recoil of the ropes act 
 
 n 
 
 FIG. 54. 
 
 in the same manner as mass in the rim, and for this reason 
 a lighter wheel may be used with rope-driving provided 
 the construction is such as will permit it. The journal- 
 friction is therefore presumably less in the larger class 
 of engines employing ropes, when compared with those 
 using belts. The difference is, however, not great ; and 
 since the actual resistance in any case is also dependent 
 upon peculiarities of type and construction, the lower 
 values of engine-friction previously given, viz., 8 to 10 per 
 cent of normal horse-power, may 
 
 S T^- OF THE 
 
 TQTNIVERSITT 
 
146 
 
 ROPE-DRIVING. 
 
 good for rope-driving plants of medium size, while 9 to 12 
 per cent of the normal horse-power may be taken as suit- 
 able for the larger class of engines running under favor- 
 able conditions. 
 
 In the ordinary transmission of power by shafting we 
 find the shaft loaded with pulleys and the power taken off 
 in varying amounts throughout its entire length; it is un- 
 usual except in short lengths to receive the power at one 
 end and transmit it at the other. Moreover, in long shaf t- 
 
 FIG. 55 
 
 ing the head or receiving shaft is usually situated midway 
 between the ends, and the power distributed more or less 
 uniformly from this head shaft to either end ; therefore, 
 in estimating the power absorbed by friction in ordinary 
 mill or factory shafting loaded with pulleys the previous 
 formulas (page 70) do not apply, as these relate only to 
 those cases where power is taken off at the end of the shaft. 
 The conditions of practice as we find them in actual 
 transmissions are so various, that it is difficult to lay down 
 any general rule by which the power absorbed by friction 
 
ROPE-DRIVING. 147 
 
 may be determined: the number and weight of pulleys 
 and couplings, the intensity and direction of belt-pull, the 
 condition of bearings and their lubrication, till affect the 
 amount of work lost in friction. 
 
 For the ordinary factory shafting, from which power is 
 taken fairly uniformly throughout its length and dis- 
 tributed horizontally to counter- or auxiliary shafts situ- 
 ated on one or both sides of the main shaft, there will be 
 three general cases to be considered, as shown in Figs. 54, 
 55, and 56, and eacli of these cases will be modified, depend- 
 ing upon the direction of the belt to and from the main 
 shaft. 
 
 For our present purposes it will be sufficient to take that 
 case in which the shaft friction is a maximum for the 
 assumed direct-ion of main belt-pull corresponding to the 
 arrangement shown in Fig. 54. 
 
 The friction will evidently be proportional to the weight 
 of the shaft and the unbalanced belt-pull acting on the 
 shaft. 
 
 The weight of pulleys, belts, clutches, and couplings 
 carried by the line-shaft will vary from about one and one- 
 half to three times the weight of shaft, so that the total 
 weight on the bearings will vary from two and one-half to 
 four times the weight of shaft; for head and jack shafts 
 the total weight will probably vary from three to five times 
 the weight of shaft. 
 
 In addition to this weight there is the unbalanced belt- 
 pull, which increases the load on the bearings. Although 
 the tension on the tight side of the belt should not ordi- 
 narily exceed about twice the tension in the slack side 
 necessary for adhesion, yet it is probable that belts are fre- 
 quently run with a ratio of tension equal to one to three, 
 and occasionally one to four; on the other hand, it is a 
 very common thing for belts, especially short ones, to be 
 laced so taut that the initial tension is greatly in excess of 
 
148 
 
 ROPE-DRIVING. 
 
 that required for adhesion, iu which case the sum of the 
 
 tensions approaches twice that in the tight side of the belt.* 
 
 With ordinary shop-worn belting it will be safe to assume 
 
 that the tension f l\ on the slack side of the belts is one half 
 
 T 
 
 the tension T 7 , on the tight or driving side, that is, r l\ = '; 
 
 & 
 
 hence, since T 7 , T 9 = P, the driving force, we have 
 
 I'X 33-000- 
 The velocity of intermediate belting is so variable that 
 
 FIG. 56. 
 
 any assumption of speed must be regarded as applying to 
 a particular case or representative of a certain type of 
 factory, and cannot be taken as general. In many ma- 
 
 *This is often a source of much trouble, as the increased tension 
 not only increases the loss due to friction but in many instances the 
 useful power transmitted is not sufficient to drive the machine. In 
 such cases, by slacking out the lacing or inserting a short piece of belt 
 so as to reduce the tension, heavy cuts can readily be taken when it is 
 practically impossible to run the machine empty with the tight belt, 
 
140 
 
 chine-shops the average speed of intermediate belts is not 
 more than 500 feet per minute ; in others the average 
 speed is more than twice as great, and in wood-working 
 shops it is still greater. 
 
 For our present purpose we shall assume an average 
 speed of 660 feet per minute for belts running from the 
 main shaft to a secondary or counter shaft. 
 
 T V 
 
 Substituting this value in HP = - X , there is 
 
 6 ooOOO 
 
 obtained T. = --- HP = 100 X HP', but since the 
 060 
 
 horse-power which the shaft is capable of transmitting 
 
 (FN 
 may be considered equal to , where d is the diameter of 
 
 shaft in inches and ^Vthe number of revolutions per minute, 
 we have the tension on the tight side of all the belts 
 
 d' A Y 
 ^ = 100 X ~^ = d*N m , therefore the sum of tensions 
 
 + TJ = ld 3 N, and the pull per foot of length of 
 
 Q 
 
 Zd'N 
 shaft =-.- nr . 
 
 In the present case it will be noted by reference to Fig. 
 54 that there is an additional pull on the bearings due to 
 the tensions in the belt from fly-wheel to main-line shaft. 
 If the ratio of tight to slack side tension remains the same 
 as before, and we consider that the velocity of main belt is 
 four times as great as that of the intermediate belting, the 
 additional belt-pull will equal approximately one fourth of 
 the sum of the belt-pulls from the main to the counter 
 shafts or machinery. The resultant of these tensions, 
 combined with the weight of shafting and pulleys, will be 
 the effective load on the bearings. 
 
 Assuming an angle of 30 with the horizontal for the 
 line of action of the resultant pull on the bearings due to 
 
150 ROPE-DRIVING. 
 
 the tensions in the tight and slack sides of the main belt, 
 the combined horizontal pull on the bearings will be 
 
 COS 
 
 and the vertical pull will be, when W 8 = weight of loadeu 
 shaft, 
 
 Therefore the resultant of both horizontal and vertical 
 forces acting on the bearings will be 
 
 W= w sin 30 X - 
 
 As previously shown, the horse-po"wer necessary to over- 
 
 come journal-friction will be HP = , where F is the 
 
 ooUUU 
 
 force of friction at the circumference of shaft and v is the 
 speed in feet per minute of a point on the circumference. 
 If the bearing is well worn and fitted to its shaft, the resist- 
 ance due to friction will probably lie between the limits 
 
 0JFand 0TF, where is a coefficient which, from the 
 
 M 7t 
 
 results of experiments on shafting with ordinary lubri- 
 cation, we have assumed equal to 0.06, and W is the 
 resultant load, in pounds, on the bearings. 
 From the lesser of these values there is obtained 
 
 (16) 
 
 But we have assumed that the weight of a loaded shaft 
 varies from two and one-half to four times the weight of 
 shaft; taking an average value of three for line-shafting, 
 and noting that the weight of shaft per foot of length 
 
ROPE-DRIVING. 151 
 
 equals (3.36cf), we have the friction on a loaded shaft L 
 feet long, due to its weight = 
 
 A I fT \ 
 
 -031- X 3.36f/ a ]Z, 
 n \4 / 
 
 Substituting the value of W in formula (16) when the 
 belt tensions are taken into account, and noting that 
 
 W s 3 ^ (3.36d 2 .), we have for the total friction load 
 
 su 
 
 From the formula for the power absorbed by friction we 
 obtain 
 
 lience the ratio of power absorbed by friction to the horse- 
 power which the shaft is capable of safely transmitting 
 will be 
 
 HP 0.0 5 8dNF 0.08F 
 
 0612^ = -^-Percent. . . (1,) 
 
 , 
 
 From this expression the following table (XVI) has been 
 computed for the given diameters and lengths of shafting 
 running at 100 and 250 revolutions per minute, the belt 
 speed for the secondary belts being assumed at an average 
 of G60 feet per minute. 
 
 For intermediate belts having a greater average velocity 
 than that assumed, viz., 6GO feet per minute, the friction 
 horse-power for a given number of revolutions will be less 
 than that given in the table. Thus if the average velocity 
 of cross-belts equals 2640 feet per minute, the horse-power 
 transmitted being the same, it follows that the tensions in 
 the secondarv belts will be one- fourth of that obtained with 
 
152 
 
 ROPE-DRIVING. 
 
 the lesser speed; if the main driving-belt have the same 
 velocity, the tension in this belt may be considered equal to 
 that existing in the intermediate belts: therefore, as the 
 velocity of the belt increases for a given speed of rotation 
 the sum of the tensions acting on the bearings will decrease, 
 and the maximum horse-power transmissible by the shaft 
 will be exerted with a decreasing friction loss. 
 
 TABLE XVI. POWER ABSORBED BY FRICTION IN LINE- SHAFT. 
 
 Diameter of 
 Shaft 
 in Inches. 
 
 Revolutions 
 per 
 
 Minute. 
 
 Percent) 
 100 
 
 ige of Loss w 
 200 
 
 hen Length ii 
 300 
 
 i Feet = 
 400 
 
 2 
 
 2* 
 3 
 3* 
 
 (100 
 
 1250 
 j 100 
 |250 
 100 
 ' 250 
 100 
 ' 250 
 
 5.5 
 
 7.8 
 5.8 
 8.9 
 6.1 
 9.9 
 6.8 
 11.4 
 
 10.1 
 11.6 
 10.3 
 12.4 
 10.4 
 13. 
 10.6 
 14,2 
 
 is" 
 
 16.6 
 15 
 17.2 
 15.4 
 
 18 
 
 ti',6 
 21.5 
 20 
 22 
 
 Line of action of resultant of main-belt tensions = 30. 
 
 Velocity of main belt 2640 feet per minute. 
 
 Belts from line shaft are horizontal and run at an average of 660 
 feet per minute. 
 
 All the belts are assumed to drive from one side of the shaft toward 
 the engine. 
 
 Weight on bearings three times weight of bare shaft. 
 
 If in any case the shafts are belted vertically or at any 
 other angle than that assumed, the formula for F will be 
 modified accordingly. 
 
 For a head or jack shaft carrying heavier pulleys the 
 weight acting on the bearings may be taken equal to four 
 times the bare weight of shaft; in which case, other condi- 
 tions remaining the same, we obtain 
 
 sin 30 x ^ 
 
 Since, however, the extra weight of pulleys on a jack- 
 shaft is liable to produce a greater bending moment, it is 
 customary to assume a larger shaft to transmit a given 
 
KOPE-DRIVIKG. 153 
 
 horse-power; therefore, instead of using as the working 
 
 100 
 
 horse-power transmissible by the shaft, it is better to use 
 
 tTN 
 under these conditions HP = TT>~~* 
 
 From this value of the power transmitted we obtain 
 
 F= n^V [4|(3.36d 2 )L + sin 30 x i(|x.8dW)]V[l.s(|x.8cPJV)] 8 
 
 = tf\Q.8d*L -f 0.0114eZ 3 jy] 2 + [O.lleM] 2 . 
 
 The ratio of power absorbed by friction to the horse- 
 power which the shaft is capable of safely transmitting will 
 now become 
 
 HP Q.tfSdNF Q.WF 
 
 HP 
 
 from which Table XVII has been determined. In calcu- 
 lating the values given in this table it has been assumed 
 that the belt speeds are the same as those previously con- 
 sidered, namely, 2640 feet per minute for main belt to 
 head shaft, and one fourth of this, or 660 feet per minute, 
 from head shaft to auxiliary shafting. This may be low in 
 many cases, but, as already pointed out, the force F will 
 decrease under the assumed conditions as the belt speed 
 increases, so that we may expect the friction loss obtained 
 by the above formula to be somewhat less than the actual. 
 In the foregoing discussion it has been assumed that the 
 shaft transmits its allowable maximum power, that is, for 
 
 d*N 
 
 line-shafting the power transmitted = 77^-, and for head- 
 
 100 
 
 d*N 
 shafts the power transmitted = -. 
 
 1/vO 
 
 As a general thing, the actual average power transmitted 
 by a shaft is not more than about three-fourths of its as- 
 sumed working capacity; and since the weight and speed 
 remain practically constant, the percentage of loss under 
 conditions approximating those we have assumed will be 
 somewhat greater than that given in the table. But while 
 
154 
 
 BOPE-DRIVIKG. 
 
 the power transmitted may be diminished 25 per cent, the 
 percentage of increase in friction loss will vary between wide 
 limits, depending upon the speed of rotation and length of 
 shaft. Thus for a 3" head-shaft 100 feet long, delivering 
 three-fourths of its allowable capacity at 100 revolutions 
 per minute, the loss increases from 9.0 to 11.5 per cent, 
 which represents a gain of 28 per cent; at 250 revolutions 
 the loss is now 14.0 per cent, corresponding to an increase 
 of 14.3 per cent; while at 400 revolutions the loss is 17.8 
 per cent a gain of only 10.6 per cent. 
 
 TABLE XVII. POWER ABSORBED BY FRICTION IN JACK OR 
 HEAD SHAFTS. 
 
 Diameter of 
 Shaft 
 in Inches. 
 
 Revolutions 
 per 
 Minute. 
 
 Percentage of Loss w 
 ^ 50 
 
 hen Length in Feet = 
 100 
 
 
 (100 
 
 4.8 
 
 8.5 
 
 2 
 
 ^'250 
 
 7.2 
 
 10.1 
 
 
 (400 
 
 10.0 
 
 12.5 
 
 
 ( 100 
 
 5.1 
 
 8.7 
 
 2* 
 
 ^250 
 
 8.3 
 
 11.0 
 
 
 400 
 
 12.2 
 
 14.3 
 
 
 ( 100 
 
 5.5 
 
 9 
 
 3 
 
 ^250 
 
 9.5 
 
 12 
 
 
 (400 
 
 14.2 
 
 16.1 
 
 
 (100 
 
 5.8 
 
 9.1 
 
 8* 
 
 ^250 
 
 11.5 
 
 12.9 
 
 
 (400 
 
 16.6 
 
 17.8 
 
 
 (100 
 
 6.3 
 
 9.5 
 
 4 
 
 hso 
 
 12.1 
 
 14.2 
 
 
 400 
 
 18.4 
 
 20.5 
 
 For these determinations it is supposed that the shafting 
 is properly supported, with hangers sufficiently close to 
 each other to prevent undue deflection under working con- 
 ditions, and that the shaft is in line, having good bearings 
 lubricated as in common practice.* Departures from these 
 
 *In the above discussion it was assumed that the coefficient of 
 friction is constant and that the friction varies directly as the .load. 
 While recent experiments on machine journals running in oil indi- 
 cate that the coefficient of friction varies inversely with the load 
 there seems no good reason to doubt the truth of Morin's laws for 
 
HOPE-DRIVIKG. 155 
 
 assumptions will further increase the friction loss; but, 
 on the other hand, this loss will be decreased if lighter or 
 fewer pulleys be used throughout the length of the shaft, 
 if the bearings be continuously lubricated,, or if the ma- 
 chines be belted directly from the shaft below. "Where 
 shafting is employed there will generally be an additional 
 loss due to the friction of the auxiliary shafting and coun- 
 ter-shafts, which is extremely variable. 
 
 It is outside the province of the present subject to discuss 
 the losses in these secondary shafts: the losses which we 
 have here been considering are those which exist in main 
 line-shafting, jack- and head-shafts receiving their power 
 presumably by leather-belting or ropes, with either of which 
 for similar drives there should be no appreciable difference 
 in the total weights of pulleys and shafting and the friction 
 involved. 
 
 The diameter of grooved shaft-pulleys will be larger and 
 the rim will be thicker; but for the same horse-power trans- 
 mitted the width will be less and the weight not materially 
 increased. In any case the total weight of grooved pul- 
 leys compared with that of belt-pulleys used in the same 
 system is very small, and any individual differences may 
 be neglected when taken as a whole. In those cases where 
 ropes are used exclusively, as, for instance, in dynamo rooms 
 and other power-stations, the pulleys are frequently heavier, 
 and the shafting usually is fitted with a number of friction- 
 clutches, thus materially increasing the weight on the bear- 
 ings; in cotton-mills also, where the ropes are geared di- 
 rect from the engine to the various floors of the mill, there 
 is frequently a heavy stress on the shaft, especially on the 
 upper floors, due to the weight of the ropes: under such 
 
 such comparatively rough and imperfectly lubricated bearings as we 
 have been considering in which the friction between the rubbing 
 surfaces in contact and not the viscosity of the lubricant is a measure 
 of the resistance. See paper by Prof. Denton in American MacJiinist^ 
 Oct. 23, 1890, 
 
156 
 
 ROPE-DRIVIKG. 
 
 conditions the shafting should be considered as head-shafts. 
 In work of this nature the velocity of the ropes is usually 
 much greater both from the engine to the first shaft, and 
 from the latter to the machine or secondary shaft. Assum- 
 ing a speed of rope double that used for the previous tables, 
 and letting the weight of pulleys, ropes, clutches, and coup- 
 lings equal four times the weight of shaft, it can be shown 
 that the formula for the friction load will be reduced to 
 
 0.0057<f A'] 2 -f (0.033flLV r )'. 
 
 TABLE XVIII. POWER ABSORBED BY HEAD-SHAFTS CARRYING 
 HIGH-SPEED ROPES. 
 
 Diameter of 
 Shaft in inches. 
 
 Revolutions 
 
 Percentage of Loss when Length of 
 Shaft in Feet = 
 
 
 " 
 
 50 
 
 100 
 
 
 
 100 
 
 4.2 
 
 8.1 
 
 2* 
 
 
 250 
 
 4.8 
 
 8.6 
 
 
 . 
 
 400 
 
 5.6 
 
 9.2 
 
 
 100 
 
 4.3 
 
 8.2 
 
 3 
 
 ^250 
 
 5.2 
 
 8.7 
 
 
 (400 
 
 6.0 
 
 9.5 
 
 
 
 100 
 
 4.3 
 
 8.3 
 
 Si 
 
 
 2oO 
 
 5.3 
 
 8.8 
 
 
 
 400 
 
 6.2 
 
 9.6 
 
 
 
 100 
 
 4.4 
 
 8.3 
 
 4 
 
 
 250 
 
 5.6 
 
 9.1 
 
 
 
 400 
 
 7.1 
 
 10.1 
 
 Line of action of resultant of tensions in main drive = 30. 
 Velocity of ropes from engine 5280 feet per minute. 
 Ropes from shaft are horizontal, and run at an average of 2640 feet 
 per minute. 
 
 All ropes drive from one side of the shaft toward the engine. 
 Weight on bearings four times weight of bare shaft. 
 
 From this formula Table XVIII has been calculated, 
 and may be considered to represent the percentage of loss 
 in the first shaft when working under full load; with 
 lighter loads the percentage will be greater. 
 
 It must be noted that with any other arrangement of shafts 
 the friction loss will vary. In the present case the ropes 
 from the engine to the jack-shaft make an angle of 25 with 
 
ROPE-DRIVING. 157 
 
 the horizontal, and the ropes from the jack-shaft back to 
 the machine or secondary shaft are horizontal, as in Fig. 54. 
 
 When rope- wells are used, each successively higher shaft 
 will have an increased friction percentage, since the belt- 
 pull becomes more nearly vertical, and the resultant load 
 on each shaft is thereby increased. 
 
 It is worthy of remark that in long lines of shafting 
 with high rim velocity the influence of belt-pull on the 
 bearings is very slight compared to the weight of shaft 
 and pulleys, so that the loss in friction is but little more 
 than that due to weight alone. We see in this an addi- 
 tional argument for high rotative speeds in shafting, for, 
 while the percentage of loss increases, from 8.1 to 9.2 in a 
 2-inch shaft 100 feet long running at 100 and 400 revolu- 
 tions per minute respectively, the power transmitted by the 
 
 d*N . 
 
 shaft, as calculated from HP = - V increases from 21^ for 
 
 125 
 
 100 revolutions per minute to about 87 h. p. for 400 revolu- 
 tions per minute, so that while the friction percentage 
 
 9.2 
 increases in the ratio ^ - = 1.14, the power transmitted 
 
 o. I 
 
 increases in the ratio of the number of revolutions per 
 minute, or 4 to 1. In the first case the friction loss is 
 21-J X .081 = 1.74 h. p.; and in the second, the loss is 
 87 X .092 = 8.00; therefore the net power transmitted by 
 the shaft running at 100 revolutions per minute is 19 h. p. 
 whereas by increasing the speed to 400 revolutions per 
 minute the net power transmitted will be 79 h. p. With 
 higher belt velocities and increased rotative speeds in our 
 factory shafting the friction loss, instead of being from 30 
 to 50 per cent of the total power transmitted, ought not to 
 exceed one-half of these percentages; for with higher speeds 
 narrower and lighter pulleys could be used, the belts could 
 be run slacker, and lighter shafting could be employed. 
 Although it has been previously considered in a general 
 
158 ROPE-DRIVING. 
 
 way that the friction due to the shafting may be taken as 
 about 10 per cent of the full load transmitted to the shaft, 
 yet, in the light of further investigation, it will be seen 
 that, owing to the various conditions under which the 
 shafting is run no general value for friction loss can be 
 assigned. 
 
 While the friction absorbed by large engines is reason- 
 ably less for rope-geared fly-wheels when compared with 
 engines using flat belts, driving in a.similar manner, there 
 seems to be no good reason for supposing that the friction 
 in ordinary mill-shafting should be appreciably different 
 when rope-driven. 
 
 On account of the larger diameter of pulleys used with 
 rope-driving, the velocity of the rope may be, and usually 
 is, greater than that in a belt used in the same place, and 
 for this reason the pull on the shaft due to the tensions in 
 the rope may be less; but with long lines of rope-driven 
 mill-shafting the main drives only are of rope, and any 
 difference in pull on the bearings which might exist in 
 favor of the rope-driven main shaft must necessarily be 
 small when compared with the total friction load due to 
 the pull of the numerous cross or machine belts which, 
 running at a greatly reduced speed, produce by far the 
 greater effect on the shaft. With short lines of shafting, 
 however, there will generally be a small saving in favor of 
 a rope-driven plant. Under these conditions the effect of 
 the ropes to and from the first motion shaft is usually in 
 excess of that due to the belts which may be used to trans- 
 mit power from the main or jack shaft to secondary shafts 
 or machines; and therefore, since the rope-pull is less than 
 would be produced by belts used in the same place, we 
 may expect the friction to be less. When ropes are used 
 entirely, as in electric and other power stations, we should 
 expect the friction loss to be somewhat less, assuming that 
 the ropes are run at a higher speed than would be used 
 for belts in the same place. 
 
KOPE-DRIVIXG. 159 
 
 CHAPTEK XL 
 
 IT has been stated that a further decrease of from 5 to 
 8 per cent of the power transmitted by a rope may be at- 
 tributed to losses in the rope itself due to resistance to 
 bending, wedging in the groove, differential driving effect, 
 and creep, all of which affect the loss to a greater or lesser 
 extent. 
 
 Various formulas have been proposed by several eminent 
 authorities by which the resistance of a rope to bending 
 might be determined. Eytelwein's formula assumes that 
 the resistance of a rope is directly proportional to the ten- 
 sion and the square of the diameter, and inversely propor- 
 tional to the radius of curvature of the pulley; in which 
 case the stiffness of a hemp rope for each winding and un- 
 winding is given by 
 
 where c is a constant equal to 0.23, d is the diameter of 
 rope in inches, r is the radius of pulley in inches, and T 
 is the tension in the rope. If the ratio of the diameter of 
 rope to the diameter of pulley over which it runs equals 1 
 to 30, the above formula becomes, for a rope running over 
 two pulleys, 
 
 a- = O.QSdT. 
 
 Reuleaux states that, since transmission-ropes are usu- 
 ally quite slack, the coefficient of stiffness should be taken 
 somewhat less than Eytelwein's value, and suggests that 
 
1 60 KOPE-DKIVING. 
 
 two-thirds would represent a fair approximation; this 
 would give 
 
 2 fP d* 
 o-=~X 0.23- T = 0.15- T 
 
 3 r r 
 
 for each pulley in the system. 
 
 If the tension on the tight and slack sides of the rope be 
 represented by T l and T^ respectively, the average load 
 on the rope may be considered equal to \(T^-\-T^\ if, 
 further, the conditions be assumed such that the slack-side 
 tension equals one half that in the tight or driving side, 
 we shall have T J(f T 7 ,). Hence for two pulleys, when the 
 diameter of the latter equals 30 times the diameter of rope, 
 
 = 0.02^x17;. 
 
 Since 2", has been taken in our previous work as equal 
 to 200d 2 pounds, the stiffness in the rope will now become 
 
 X 150<P 
 
 Now under these relations of tension the driving force 
 may be obtained from 
 
 hence the ratio of loss due to bending will be 
 
 For a f-inch rope running over two pulleys the loss 
 equals 2.25 per cent, while for a 2-inch rope under similar 
 conditions the loss becomes 6 per cent. This is what we 
 
HOPE-DRIVING. 161 
 
 might expect; for it is reasonable to suppose that the per- 
 centage of loss should increase with the diameter of rope. 
 
 It will be noticed that the work done in bending a rope 
 over its pulley is directly proportional to the number of 
 bends, and therefore in designing a rope transmission 
 every effort should be made to restrict the number of 
 bends, as this is not only a large factor in the wear, but, as 
 just shown, the power transmitted for a given tension is 
 constantly reduced as the number of bends increases. 
 
 This feature is a decided disadvantage with installations 
 on the continuous-rope system where one rope is bent in 
 both directions around a number of pulleys on the several 
 floors of a factory. Under such conditions, and also in those 
 special cases where it is absolutely necessary to run pulleys 
 smaller than that obtained from the formula* D = d 17 X 
 
 2", the user should put in the very best quality 
 of loosely twisted rope and run it at a less strain than 
 would otherwise be adopted; for under such conditions the 
 flexibility and elasticity of the rope are more desirable than 
 a high breaking strength. 
 
 While the foregoing formula of Eytelwein may give a 
 measure of the force required to bend a rope over a 
 pulley under a certain set of conditions, it will be 
 evident, since the conditions vary considerably in dif- 
 ferent installations, that, in order to be generally appli- 
 cable to any given case, a formula must contain other 
 factors than those included in Eytelwein's and other simi- 
 lar formulas. In a flying rope running in V grooves, in 
 addition to the bending of the fibres there is a permanent 
 reduction in cross-section, due to the uniform compression 
 of the rope, as will be noticed by measuring a rope that 
 has been running some time; besides this there is a tem- 
 porary deformation due to the distortion of the rope as it 
 passes over the pulley. 
 
 * See page 179. 
 
162 KOPE-DRIVIXG. 
 
 The flexibility and elasticity of the rope undoubtedly 
 have much to do with the resistance to bending, and the 
 degree of twist put into a rope is an important factor in 
 tliis connection; for although hard-twisted ropes are very 
 much stronger than those loosely twisted, the internal 
 wear is much greater, as the fibres are held more rigidly 
 and do not slide as freely upon each other. The advan- 
 tages of lubricating the fibres of a manilla rope have been 
 already discussed; it is sufficient to state here that the 
 degree of lubrication affects the flexibility of the rope, and 
 hence enters as a factor in determining the loss due to 
 bending. The varying angle of contact and, to a lesser 
 extent, the angle of the groove, must also have a certain 
 influence upon the resistance. In view of these considera- 
 tions it will be seen that any deductions from existing for- 
 mulae are of doubtful utility when applied to transmission- 
 ropes in use, and shoult? be considered only as relative, and 
 not absolute. 
 
 Another source of loss is that due to the wedging of 
 the rope in the groove. Although this action exists to a 
 greater or less extent in all rope transmissions where the 
 shape of the groove is such that the rope does not bottom, 
 yet it is undoubtedly true that its effect in a well-con- 
 structed plant has generally been over-estimated. That it 
 does exist to a harmful degree can be seen in many instal- 
 lations from the way in which the tight side of the rope 
 follows upon the driven pulley. With the single-rope 
 method this is especially true in new installations, where 
 the tension is purposely made rather high to allow for 
 stretch in the rope; it is, however, frequently found in the 
 continuous-rope system where the tension-carriage is over- 
 weighted. The factors which enter into the consideration 
 of this loss are: Tension on the driving and slack sides of 
 the rope, the angle of groove, and the velocity, weight, 
 and condition of the ropes. As we have previously shown 
 
(page 132), the driving-side tension T : is made up of three 
 parts, namely, the driving force P, the centrifugal force 
 F Qi and the tension jT 3 , necessary to balance the strain for 
 adhesion; that is, 
 
 In like manner the tension in the slack side of the rope 
 consists of the strain necessary for adhesion plus the 
 strain due to centrifugal force, that is, 
 
 T t = T F. 
 
 It has been claimed that no loss can occur in pulling the 
 rope out of the groove, since the centrifugal force set up 
 in the rope is many times greater than any caused by the 
 tension on the slack side when leaving the pulley; but it 
 is obvious that a part cannot be greater than the whole, 
 and therefore the centrifugal force, while greatly reducing 
 the wedging force, cannot altogether eliminate it. 
 
 An abnormal degree of slack-side tension has a direct 
 effect upon the wedging of the rope in the groove, for if 
 sufficient sag is not allowed and the slack-side tension is, 
 therefore, needlessly great, it follows that the power trans- 
 mitted will be reduced, since the driving force, P, is equal 
 to the difference of the tension in the two portions of the 
 rope T l T^ on the other hand, the force drawing the 
 rope into the groove will be increased, since the force 
 equals 
 
 It is obvious, therefore, that the slack -side tension 
 
 * Although the centrifugal force increases the tension in the rope, 
 its teudency is to cause the latter to leave the pulley, and therefore 
 it should not be considered as a part of the forces drawing the rope 
 into the groove. 
 
164 ROPE-DRIVING. 
 
 sliould be no greater than just sufficient to give adhesion 
 to the ropes and prevent undue slip at the desired speed. 
 
 As to the best angle for the groove in the pulley, opinion 
 is still somewhat divided; but in England the general prac- 
 tice seems to favor an angle of 45 as the most suitable.* 
 
 In the earlier installations a more acute angle was used, 
 as evidenced by the discussion of Mr.' Durie's paper on 
 Rope-driving, presented to the British Institution of Me- 
 chanical Engineers.! Grooves having an angle of 30 had 
 been tried, but it was found that the wear on the rope was 
 altogether too great; 40 was a very satisfactory angle, and 
 is still preferred by many engineers, while others use as 
 large an angle as GO . Occasionally half-round grooves 
 are used; but with semicircular grooves on cast-iron 
 pulleys either the tension in the rope must be increased 
 or a greater number of ropes must be employed: in any 
 case the advantage seems doubtful. With wooden-rimmed 
 pulleys, however, the semicircular groove is the better 
 form; for since the coefficient of friction on the wooden 
 pulley is from thirty to fifty per cent greater than for a 
 similarly shaped groove on an iron pulley, it follows that 
 the tension in the rope need not be so great; moreover, 
 wooden pulleys that have been in use for some time would 
 indicate that the semicircular groove is better adapted to 
 the work, for with anything but very light loads the angu- 
 lar groove is soon cut out by the rope, producing a rim 
 somewhat similar to the one shown in Fig. 57, which rep- 
 resents an angular-grooved wooden pulley that had been in 
 use but a few months. 
 
 With a semicircular groove in the first place the latter 
 will retain more nearly its original form, and the wear on 
 the rope will be greatly reduced. 
 
 The pliability of the rope has considerable influence on 
 
 * M. E. iu American Macliinist, Nov. 10, 1893. 
 f See Proc. lust. M. E. 1876. 
 
HOPE-DRIVING. 
 
 165 
 
 the shape of the groove; while a 30 angle may give the 
 correct shape for a soft, loosely twisted cotton-rope, a harder 
 twist may require an angle of 40 or even 50; in the same 
 way a 40 groove may be all right for some makes of ma- 
 nilla rope, while others of a less yielding nature would give 
 better results if an angle of 50 or even 60 were used. 
 
 At the present time there are very few pulleys used in 
 this country, except for machine bands, having grooves 
 with an angle less than 40. Formerly the Yale & Towne 
 
 Elec. World 
 
 FIG. 57. RIM OF WOODEN PULLEY SHOWING WEAK. 
 
 Mfg. Co. employed grooves of 30 for small cotton ropes 
 (about f inch in diameter) driving their travelling-cranes; 
 but these were subsequently changed to 50, and at the 
 same time larger ropes, 1 inch, were adopted. 
 
 For manilla rope-drives one large manufacturer uses an 
 angle of 60 on all his pulleys, whether of wood or iron ; 
 this angle was arrived at after much trial, and represents an 
 experience with manilla-rope transmission, covering a great 
 many years. 
 
 The most suitable angle of groove is that which affords 
 the greatest frictional adhesion without undue slip, and at 
 the same time offers the least resistance to the rope in leav- 
 ing the groove: with much slip the rope is rapidly worn 
 out, while with an excessive grip the wear is also rapid, and 
 a relatively large amount of force is absorbed in overcoming 
 the wedging. The usual practice in this country for both 
 cotton and manilla ropes is an angle of 45; in many cases 
 
166 
 
 ROPE-DRIVIXG. 
 
 FIG. 58. 
 
 the section of the groove is formed by arcs of circles, hav- 
 ing a radius equal to from 3 to 4 diameters of rope, in which 
 case the included angle is constantly changing, and the co- 
 efficient of friction, and hence the grip, will vary with the 
 diameter of rope used. Thus in Fig. 58, if AB and A'E' 
 
 represent two 
 ropes of slightly 
 different diame- 
 ters running over 
 a grooved pulley, 
 the one which 
 sinks deeper into 
 the groove will 
 include tho 
 greater angle of 
 contact ABC,i\,\\(\. 
 hence the lesser 
 coefficient of friction. This will hold true of two ropes hav- 
 ing the same diameter but different degrees of twist: the 
 harder rope will not sink as deeply into the groove, and its 
 coefficient of friction will be greater than that of the smaller 
 rope, other things being equal, on account of the lesser 
 angle in the groove at the point of tangency of the harder 
 rope. 
 
 With the excellent ropes that are now being made for 
 transmission purposes this form of groove possesses many 
 advantages, even with the continuous- wound system. 
 
 In this system the tension is assumed to be the same in 
 each wrap; but there is unquestionably a variation of pull 
 due to momentary fluctuations, which must either be ab- 
 sorbed by the elasticity of the rope or transmitted through 
 the rope until the strains are equalized. If the rope used 
 is uniform in structure, that portion which receives the 
 greater strain will be drawn more deeply into the groove; 
 and if the latter be of the curved form, the coefficient of 
 
ROPE-DKIVItfG. 167 
 
 friction will be reduced: so that the resultant adhesion pro- 
 duced in this portion of the rope will be less than that in 
 some other part which, under a lighter strain, occupies a 
 position on the pulley such that its coefficient of friction, 
 and consequent adhesion, is greater for a given back-tension. 
 With the usual arrangement of pulleys, namely, that in 
 whicli the larger wheel is the driver, the tendency of an 
 increase in tension is to increase the velocity of the driven 
 pulley. If several wraps of a series occupy positions in the 
 grooves such that with a given ratio of pulley diameters 
 the velocity of the smaller pulley is twice that of the driver, 
 any decrement, x, of the eifective radii of another wrap, 
 
 which is drawn more deeply into the grooves of the two 
 
 o o />. 
 
 pulleys, will alter the velocity ratio from T to - , a quan- 
 
 J_ JL OC 1 
 
 tity which must of necessity be greater than 2. The ten- 
 dency of this wrap then is to produce a greater velocity in 
 the driven pulley, which cannot occur without some slip in 
 the other wraps; but these wraps have a greater adhesion, 
 and therefore tend to drive the pulley at a less number of 
 turns per minute, which will produce slip in the more 
 heavily strained member: hence the effect of any change 
 in position due to sudden increased strain on one wrap 
 will tend to quickly adjust the tensions in all positions of 
 the rope and neutralize any inequalities in driving effort. 
 This form of groove, as we shall show subsequently, is even 
 more desirable for the individual rope system, where the 
 evils of differential driving are frequently so pronounced. 
 As noted previously, the frictional grip depends both 
 upon the arc of contact of the rope with the pulley and 
 the coefficient of friction, which latter varies with the angle 
 of the groove. In order, therefore, to produce the same 
 friction on each pulley, the product of the arcs of contact 
 by their respective coefficients of friction must be equal. 
 If, as before, we take the coefficient of friction of a well- 
 
168 ROPE-DRIVING. 
 
 lubricated rope on a smooth, flat metal pulley equal to 0.12, 
 
 the coefficient for the same rope in a groove whose angle 
 
 /j 
 is 6 degrees will be = 0.12 cosec ; hence with arcs of 
 
 contact a and a' we should have for an equal grip on each 
 pulley 
 
 (pa = 0'or', 
 
 n nj 
 
 0.12 cosec - x a = 0. 12 cosec X a', 
 
 & 
 
 assuming that the multiplier 0.12 will give the correct co- 
 efficient of friction on each pulley, since the percentage of 
 slip is to be the same. 
 
 As the numerical value of the cosecant of an angle varies 
 inversely with the angle, it will be obvious that the pulley, 
 having the lesser arc of contact, should also have the more 
 acute angle in the groove. This property of rope friction 
 is frequently taken advantage of in designing a plant, and 
 it is not uncommon to find the grooves in the large wheel 
 more obtuse than those in the smaller, especially when 
 there is considerable difference in the diameters of the 
 pulleys.* 
 
 From the above equation there is obtained 
 
 6 0' a' 
 
 cosec = cosec X > 
 
 6 (X 
 
 in which and a represent the angle of groove and arc of 
 contact, respectively, on the larger pulley, and 0' and a' 
 similar values for the smaller pulley. With the least angle 
 of groove equal to 35, 40, or 45, the corresponding angle 
 in the larger pulley should be as indicated in Table XIX, 
 when the ratio of the arcs of contact is known. 
 
 *Mr. T. Spencer Miller, of the Lidgerwood Mfg. Co., has been 
 granted a patent (U. S. patent No. 444919), upon the application of 
 this principle to continuous- rope transmissions, in which with sheaves 
 of different diameters more obtuse grooves are given to a larger wheel. 
 
ROPE-DRIVING. 169 
 
 TABLE XIX. ANGLE OF GKOOVE FOR EQUAL ADHESION. 
 
 Arc of contact on small pulley a' 
 
 0.9 
 
 0.8 
 
 0.75 
 
 0.7 
 
 0.65 
 
 0.6 
 
 Arc of contact on large pulley a 
 
 Angle of groove in large pulley 
 when groove in small pulley 35 
 
 40 
 
 44 
 
 74 
 
 51* 
 
 55 
 
 60 
 
 Angle of groove in large pulley 
 
 
 
 
 
 
 
 when groove in small pulley = 40 
 Angle of groove in large pulley 
 
 45 
 
 50 
 
 54 
 
 58 
 
 64 
 
 70 
 
 when groove in small pulley =45 
 
 50 
 
 55< 
 
 60 
 
 66 
 
 72 
 
 80 
 
 Of course an idler or binder pulley may be used to in- 
 crease the arc of contact on the smaller pulley, and thus 
 maintain an equality of grip on each pulley; but this device 
 produces an objectionable reverse bend in the rope, which 
 should be avoided as much as possible in rope transmission. 
 Both of these arrangements, as well as the winder-pulley 
 previously discussed, are intended to prevent slip and at 
 the same time obtain the maximum adhesion for the least 
 amount of back-tension without increasing the losses due 
 to wedging in the grooves, journal-friction, and wear in 
 the rope. 
 
 As pointed out by Mr. W. H. Booth,* although there 
 may be some loss and wear due to wedging in the groove, 
 by far the greater loss is occasioned by the deeper wedging 
 of one rope as compared with another, so causing them to 
 grip upon a different circumference, in which case each 
 rope tends to impart a different velocity to the driven pulley; 
 the actual resultant velocity will be a mean of the several 
 velocities of the individual ropes, so that slipping and wear 
 of some or all of the ropes must occur, due to the differen- 
 tial driving thus set up. 
 
 With continuous rope transmissions this effect is not 
 so apparent, although it exists to a certain extent on account 
 of inequalities in the rope and various mechanical imper- 
 fections in the system; but in the English or independent 
 
 * American Machinist, Dec. 8, 1888. 
 
170 HOPE-DRIVING. 
 
 rope transmission its effect is very marked, and is generally 
 considered as the principal source of loss of efficiency in 
 this system.* 
 
 While there is undoubtedly a considerable loss which 
 may be charged to this cause, it is also true that its effect 
 may be greatly reduced by a careful study of the require- 
 ments of the problem and an intelligent application of 
 correct principles to the case in hand. 
 
 It is evident that in order to prevent slip and the loss of 
 power incident thereto it will be necessary to obtain a uni- 
 form velocity in the several ropes running over a pair of 
 pulleys; to approach this desideratum each set of ropes 
 should be of the same make and degree of hardness, of 
 uniform diameter, evenly spliced, having the same amount 
 of sag in both members, and run over grooves of uniform 
 diameter, shape, and smoothness. 
 
 Obviously it would be impossible in practice to maintain 
 all these conditions, even if it were practicable to overcome 
 the mechanical difficulties and install a plant under the 
 given requirements ; yet much may be done to effect the 
 desired end. 
 
 Uniformity of length is an important requirement, for, if 
 one rope of a set be allowed a less amount of sag than the 
 others, the sum of the tensions will be greater in this rope, 
 and in consequence it will be drawn more deeply into the 
 groove; its pitch diameter will therefore be reduced and 
 its velocity will be different from the others in the set, in 
 which case if the driving and driven pulleys are of unequal 
 diameters the tendency will be to give the driven pulley a 
 different velocity, and slip must necessarily occur. 
 
 The following from the American Machinist is pertinent 
 to the subject: 
 
 "It will be readily seen that in a set of ten or a dozen 
 
 * American Machinist, Dec. 1, 1892. 
 
ROPE-DRIVING. 171 
 
 ropes, each of a somewhat different length, the loss of 
 power from this cause may easily become a very serious 
 item, and to this there is to be added the correspondingly 
 diminished life of the ropes. A similar action occurs 
 when worn ropes are allowed to work conjointly with new, 
 even though the deflections, and therefore the tensions, on 
 the several ropes are practically equal. In this case the 
 loss due to abnormal tension and wedging of some of the 
 ropes into the grooves is avoided, but the differential driving 
 effect due to the ropes virtually running on pulleys of 
 different diameters still exists, and is equally objectionable. 
 It may here be noted that the effect of the differential 
 driving upon the ropes depends to a very large extent 
 upon the relative diameter of the two pulleys. 
 
 "It is evident that, when the driving and driven pulleys 
 are of the same diameter, any variation in the effective 
 pitch diameters of the several grooves will have no appre- 
 ciable effect upon the transmission, provided that the 
 diameter and shape of the corresponding grooves in the two 
 pulleys are the same. It may be noted, however, that the 
 worn ropes which run deeper in the grooves, having a 
 slightly less velocity, are subjected to a somewhat greater 
 stress than their newer and larger companions." 
 
 With a form of groove similar to that shown in Fig. 58, 
 in which the sides of the groove are circular arcs of large 
 radius, there will be a tendeocy to correct the differential 
 driving, especially so when the driving pulley is smaller 
 than the driven. In this case, if we assume that all the 
 ropes in a set are put on with the same amount of sag, so 
 that the tensions are practically equal for both old and new 
 ropes, any difference in diameter of rope will cause the 
 larger to carry more than, its share of the load, since the 
 effective radius of the pulley, and therefore the velocity of 
 the rope, is greater; moreover, since the coefficient of fric- 
 tion varies inversely with the depth of contact in this form 
 
172 
 
 ROPE-DRIVING. 
 
 of groove, the larger rope is acted upon by a more intense 
 grip, by reason of which more work is imposed upon it, 
 while a smaller rope will be relieved of some of its work. 
 
 The tendency of the larger rope is to turn the driven 
 pulley at an increased velocity. Thus in Fig. 59 if D be 
 
 R= -3" ' F Elec. World 
 
 FIG. 59. 
 the driving-pulley and 7^ the follower, with respective diam- 
 
 T> 
 
 eters such that the velocity ratio -- is 1 to 3, then the 
 
 smaller ropes o will tend to drive .Fin the ratio , while 
 the larger ropes, n working at a distance x further away 
 from the centre of pulleys, will tend to drive .Fin the ratio 
 
 , and therefore at a higher velocity. The resulting 
 
 1 + x 
 - -- 
 
 O ~T~ */ 
 
 velocity of the follower will depend upon the work done by 
 each rope, so that some slip must necessarily occur; on 
 account of the greater load taken by the larger ropes they 
 will be rapidly compressed and worn: hence any initial 
 variation in turning effort will be speedily reduced to a 
 minimum by this equalizing process, which, although it 
 incurs loss, ultimately insures a better distribution of the 
 work with the least wear on the ropes. 
 
 On the other hand, when the driving-pulley is larger 
 than the follower the smaller ropes are drawn further into 
 the grooves and tend to impart an increased velocity to the 
 driven 'sheave. Thus in Fig. 60 the large pulley is the 
 
 7? ^ 
 driver and the velocity ratio is = - with the smaller 
 
ROPE-DRIVING. 173 
 
 ropes 0; with the large ropes n, working at a distances 
 nearer the circumference,, the tendency will be to produce 
 
 a ratio equal to z- - : hence the effect will be retardation. 
 1 + x 
 
 Since the larger ropes acting higher in the groove have an 
 increased grip and speed they will exert a greater influence 
 upon the smaller pulley; and although the smaller ropes 
 may be drawn more deeply into their grooves in attempt- 
 
 FOLLOWER 
 
 _ Elec.World 
 
 R=3r 
 
 FIG. 60. 
 
 ing to drive the follower at a greater number of revolutions 
 per minute their effect is lessened on account of the dimin- 
 ished grip, due to the form of groove, in consequence of 
 which the larger rope acts with a greater effect and the 
 smaller with a lesser effect than would obtain with an ordi- 
 nary V-groove under like conditions. 
 
 To equalize the driving efforts of a number of ropes, and 
 to prevent the slip which must inevitably occur with a 
 solid-grooved rim, Mr. John Walker has devised and pa- 
 tented* a "differential" driving-pulley, since it allows the 
 ropes to travel at different speeds suited to the conditions 
 imposed upon each rope. 
 
 Originally intended for cable-railway machinery, where 
 the wear on the drums due to the wire cable is excessive, 
 the differential principle has been extended to other uses, 
 notably elevator sheaves and rope-transmission pulleys. 
 
 In the latter the rope is led over a number of separate 
 
 * Feb. 23, 1893. 
 
174 
 
 ROPE-DKIVING. 
 
 FIG. 61. 
 
 rings, Fig. 61, adapted to turn loosely and independently 
 
 of each other on the 
 smooth circumfer- 
 ence of the drum. 
 
 While the rope is 
 passing over the pul- 
 ley the tendency of 
 the rings will be to 
 adjust themselves to 
 the strain in each 
 member by moving 
 around the circumference of the drum. Thus the driving- 
 tension is equalized, and each rope is brought to do its own 
 share of the work without slipping in its groove. These 
 rings have a diametrical friction, due to the pressure of the 
 rope in the groove transferred to the flat surface of the 
 drum. In addition to this an adjustable rubber washer is 
 inserted between the rim of pulley and loose flange, so that 
 by tightening this adjustable joint the separate rings are 
 caused to exert enough pressure upon each other to pro- 
 duce a certain amount of friction on the side surfaces, the 
 combined friction of the several rings being sufficient to 
 drive the pulley or ropes, as the case may be. In this way 
 each rope bears its due share of the work, as the adjust- 
 ment is such that the friction between the several parts 
 is brought into equilibrium. 
 
 In practice the axial rotation of the ropes will frequently 
 exert a modifying influence on the differential driving, 
 since, other conditions being unchanged, a rotating rope 
 tends to maintain its circular form, and therefore will work 
 less deeply into the groove. An additional advantage is 
 that such rotation promotes the durability of the rope, as 
 the wear is more uniform. 
 
 The loss due to the elastic slip or creep of the belt has 
 some influence upon the efficiency of transmission, but in 
 
ROPE-DRIVING. 175 
 
 any case its effect is small. When an elastic body, such 
 as a rope, is placed under tension, it stretches, and the 
 elongation, within the limit of elasticity, is proportional to 
 the strain in the rope. When power is transmitted from 
 one pulley to another the driving side is subjected to a 
 greater tension than the slack side, in consequence of which 
 the velocity in the driving side will be slightly greater than 
 that in the slack side, and the circumferential velocities of 
 the two pulleys will not be the same. This will be evi- 
 denced from the following considerations: 
 Let V = circumferential velocity of driver; 
 
 V = " " follower; 
 
 Tj = tension in driving side of rope; 
 
 TI= " " slack side of rope; 
 
 A cross-section of rope in square inches; 
 
 L = original length of a piece of the rope in either 
 = side in its normal condition; 
 
 e = elongation in driving side due to tension T ; 
 
 e' = l< " slack side due to tension T^\ 
 
 E = modulus of elasticity of the rope. 
 
 T T 
 
 When the rope is at work e = ^ and e' = -^, so that 
 
 -tii A Mi A 
 
 the length of each member will now be L -j- e and L -j- e', 
 respectively. The length of rope running on to the driving- 
 pulley in a unit of time will therefore be greater than that 
 
 delivered to the driven pulley in the proportion y-^ and 
 
 L -j- e 
 
 the velocity of the two pulleys will now be 
 
 ^|i ; hence T<= Kxf 
 
 T T 
 
 Calling L unity, and assuming j f = 2, - = 320 pounds 
 
 * - 
 
 (since T^ = 200d 2 pounds, and A = 0.8 Xj^ 2 ), and E 
 
176 ROPE-DKIVING. 
 
 40000,* we have 
 
 320 , _ 320 
 
 ~ 40000 ~ 2X40000 ; 
 
 therefore 
 
 ' A = 0.996F; 
 
 L + 40000 
 
 that is, the loss due to creep when all the ropes are working 
 under normal conditions is about one half of one per cent. 
 With different ratios of driving to tight-side tension and 
 different intensities of stress it is evident that this loss will 
 vary in the different ropes; in any case the loss ought not 
 to exceed one and a half per cent, as will be seen if we as- 
 sume extreme conditions. 
 
 For instance, let the stress on the rope be 600 pounds 
 per square inch of section and assume the ratio of tensions 
 equals one to five; with the most elastic long-staple cotton 
 rope it is possible that the modulus of elasticity may be re- 
 duced to 30,000 pounds; therefore, under these conditions, 
 the velocity of the circumference of the driven pulley will 
 be 
 
 30000 
 
 thus representing a loss of 1.6 per cent. It is probable 
 that one per cent is an ample allowance, even under un- 
 favorable conditions. 
 
 * See Reports of Tests on Manilla Bope, Ex. Doc. No. 36, 1885. 
 
177 
 
 CHAPTER XII. 
 
 THE construction of rope pulleys is a matter of consider- 
 able importance, for it is evident from what has preceded 
 that the size, shape, and condition of the pulley all exert a 
 marked influence upon the efficiency of rope transmission. 
 
 It has been shown that the size of the pulley materially 
 affects the wear of the rope: the larger the sheaves the less 
 the fibres of the rope are flexed, and the less they slide on 
 each other; consequently there is less internal wear of the 
 rope. 
 
 The minimum diameter of the pulley, as given by differ- 
 ent authorities, varies from thirty to forty times the diame- 
 ter of rope to be used. Forty times the diameter of rope 
 for manilla is excellent practice, and experience has shown 
 that a larger multiplier would be still better, as the larger a 
 pulley, the better for either belt or rope passing over it; 
 but such a rule, although convenient to use, was evidently 
 founded upon largo-sized ropes running at a high velocity, 
 and a little consideration will show that while forty times 
 the diameter of rope may be all right for a two- inch rope, 
 it is also true that a lesser proportion will give suitable di- 
 ameters of pulley for a one inch rope. If we take two ropes 
 dj and J 2 , whose diameters are one and two inches, respec- 
 tively, and bend them over pulleys of the same diameter, 
 the fibres in d l will be stressed an amount equal to x due to 
 the stretch and sliding of the fibres one upon another. On 
 the other hand, since the outer fibres in c/ 2 are twice as far 
 from the neutral axis, these fibres will stretch and slide 
 upon each other to an extent equal to twice that produced 
 
178 ROPE-DRIVIKG. 
 
 in the smaller rope ; but the relation of the areas of the 
 two ropes is 
 
 under the given conditions ; hence the total slip in the 
 larger rope, due to the sliding of the fibres upon each other, 
 will be eight times greater than in the smaller one, or as 
 
 -y-f. Therefore, for equal extension and slip of the fibres in 
 
 bending a rope over a stationary pulley, it would appear 
 that the pulleys should have diameters proportional to the 
 cube of the diameters of the ropes. 
 
 In the above consideration no account has been taken of 
 the external wear, produced by slipping in the groove, 
 wedging, rubbing contact, and other causes of loss which 
 affect a running rope; for a given velocity these losses will 
 increase with an increased speed of rotation, and hence will 
 be greater with a smaller pulley, which would indicate that 
 the diameter of the latter should not be directly propor- 
 tional to the cube of the diameter of the rope. It is also 
 evident that a rope running at 5000 feet per minute will 
 be subjected to a greater number of bends and a greater 
 external wear than it would if running over the same pul- 
 leys at 2000 feet per minute. The first of these influences 
 is recognized by many engineers who use a rule approxi- 
 mating the following: 
 
 For the least diameter of pulley, D, multiply the circum- 
 ference of rope by ten times its diameter and divide by 
 two. * This is practically equal to D = 15tT inches. 
 
 While we believe in using as large a pulley as possible in 
 
 *Mr. Jas. Gamble in Textile Recorder, 
 
HOPE-DRIVING. 179 
 
 any given case, conditions will arise when it is desirable to 
 use the smallest possible diameter without excessive injury 
 to the rope. In such case the individual circumstances 
 should be considered by the designer. 
 
 From the foregoing it is obvious that for a given rope 
 and tension the least diameter of pulley which may satis- 
 factorily be used under known conditions should be de- 
 pendent upon the velocity of the rope, and should vary 
 with the size of the latter in such a manner that for any 
 given speed the pulley diameter will be proportional to 
 some power, greater than unity and less than the cube of 
 the diameter of the rope. 
 
 From an investigation of numerous examples in operation 
 under varying conditions, some of which work satisfactorily 
 and others very poorly, it would seem that while the value 
 15d 2 may give a suitable diameter of pulley for a soft cotton 
 rope of small diameter, such a value is entirely too small 
 for an equal-sized manilla rope; but, on the other hand, 
 40d is somewhat larger than absolutely necessary for these 
 ropes, although it is always desirable to use such a pulley 
 if conditions permit. 
 
 If for any reason it is necessary to adopt a small pulley, 
 the least pitch diameter for a sheave to be used with 
 manilla rope working under an assumed tension of 20(k/ 2 
 pounds may be determined from the following empirical 
 formula, which is believed to represent the requirements of 
 good practice: 
 
 D = d 1 - 7 X &V+ 12", 
 
 in which D = pitch diameter of pulley in inches; 
 
 d = diameter of rope in inches; 
 
 V= velocity of rope in feet per minute. 
 In order to simplify the use of this formula the follow- 
 ing values of d 1 - 7 and ^Fhave been calculated as given 
 in Table XX, 
 
180 
 
 ROPE-DRIVING. 
 
 Table XXI gives values of D for ropes varying from f 
 inch to 2 inches in diameter, running at 2000 to 5000 feet 
 per minute. When the speed of the rope is not known the 
 diameter given in the last column should be used for the 
 minimum size of pulley; in fact it would be better to use 
 these diameters, or even larger ones, in all cases, provided 
 the constructive features in the plant will permit their ap- 
 plication. With an increased tension in the ropes the diam- 
 eter of pulley should be increased also. In the same way 
 if the working tension should be less than 200^ 2 pounds, 
 then the diameter of pulley may be less than here given. 
 If cotton rope be used, the least diameter of pulley may 
 also be taken somewhat less. 
 
 TABLE XX. -VALUES OF d>* AND 
 
 D i a of ro pe d 
 
 3 
 
 1 
 
 U 
 
 U 
 
 If 
 
 2 
 
 
 
 
 
 
 
 
 Value of d 1 - 7 
 
 61 
 
 1 
 
 1 46 
 
 1 99 
 
 2.59 
 
 3.25 
 
 Velocity of rope, V, in feet 
 
 1000 
 
 1500 
 
 2000 
 
 3000 
 
 4000 
 
 5000 
 
 Value of y'F 
 
 10 
 
 11 4 
 
 12.6 
 
 14.4 
 
 15.87 
 
 17 1 
 
 
 
 
 
 
 
 
 TABLE XXI. LEAST DIAMETEK OF PULLEY FOR GIVEN DIAME- 
 TER AND SPEED OF MANILLA ROPE. 
 
 Diameter 
 
 
 nf 
 
 
 Rope. 
 
 2000 
 
 3000 
 
 4000 
 
 5000 
 
 3 
 
 20 
 
 21 
 
 22 
 
 224. 
 
 1^ 
 
 251 
 
 264 
 
 28 
 
 29 
 
 U 
 
 
 33 
 
 35 
 
 37 
 
 u 
 
 37 
 
 404 
 
 43^ 
 
 46 
 
 if 
 
 44 
 
 49 
 
 584 
 
 564 
 
 2 
 
 53 
 
 58| 
 
 634, 
 
 67 
 
181 
 
 TABLE XXII. ROPE PULLEYS FOII GENEKAL WORK. 
 
 Diameter of Rope. . 
 
 I 
 
 1 
 
 H 
 
 n 
 
 If 
 
 2 
 
 Diameter of Pulley 
 
 24 
 
 36 
 
 48 
 
 60 
 
 72 
 
 84 
 
 It is well to remember that the cause of many failures 
 and much trouble experienced in rope-driving is due to 
 the use of too small a pulley for the size of rope and ten- 
 sion carried, but we have yet to hear of a case where the 
 diameter of pulley has been too great. When it is not ab- 
 solutely necessary to restrict the diameter to the smallest 
 possible which may be used, the least diameter should con- 
 form to that given in Table XXII, which has been arranged 
 for general work, and gives least diameters, to the nearest 
 half foot, for rope-pulleys suitable for all speeds within the 
 limits of good practice. 
 
 When a close velocity ratio between driver and follower, 
 greater or less than unity, is required, the pitch diameter 
 of each pulley should be measured from the point of tan- 
 gency of the rope in the groove, and not from the centre 
 of the rope. For, if D represent the diameter of driver 
 measured to centre of rope (Fig. 62), and F the corre- 
 sponding diameter of follower, the velocity ratio will be 
 
 -JFJ , where x is the common vertical distance between 
 
 * &X 
 
 centre of rope and point of contact. Where the pulleys 
 are the same size it is obvious that the velocity ratio will 
 not be changed, but as the difference in diameter increases 
 the influence of x will be more marked; thus if D = 3F, 
 
 the velocity ratio will be -^ , a result manifestly 
 
 *\ Ti 1 
 
 greater than -. In any case, the smaller the diameter of 
 
182 
 
 fcOPE-DRIVIKG. 
 
 pulleys for a given velocity ratio, the greater the effect of 
 the quantity x. 
 
 In ordinary single transmissions it will be sufficiently 
 close to assume the diameter of pulley as measured from 
 centre of rope. 
 
 APPARENT DIAMETER=D 
 REAL. " =0-2* 
 
 FIG. 62. DIAMETER OF PULLEY. 
 
 It is evident irom the previous considerations that the 
 condition and shape of the groove is a matter of much 
 importance: so generally is this recognized, that manufac- 
 turers now almost universally turn their rims to special 
 gauges and templets, which insure uniformity in diameter 
 and shape of groove. Special tools have been devised for 
 turning the grooves, and one prominent manufacturer has 
 constructed a special machine in which the rims are milled 
 out. By this process some of the smaller pulleys are 
 machined in one operation as many cutters being used as 
 there are grooves to be milled. Uniformity of pitch, diam- 
 eter, and contour are thus insured independently of the 
 operator. Formerly the bottom of the groove was fur- 
 
HOPE-DRIVING. 183 
 
 nished with spikes, or the sides were cut into angular teeth 
 in order to prevent the rope from slipping;* but a greater 
 experience with rope-driving has shown that the whole sur- 
 face of the groove must be perfectly smooth, and should 
 be carefully polished as well as machined, since the fibres 
 of the rope, if allowed to rub on a rough-turned or cast 
 surface, will gradually break, fibre by fibre, and thus give 
 the rope a short life. It is also necessary to avoid using 
 any pulley with sand or blow holes in the groove, as they 
 are very destructive to a rope: when blow-holes occur, if 
 not honeycombed excessively, they should be filled in with 
 lead, or, preferably, Babbitt's metal; otherwise the pulley 
 should not be used. 
 
 Some rope-pulleys are simply cast and the rims smoothed 
 up by holding a piece of abrasive material, such as a bro- 
 ken emery-wheel or grindstone, against the surface of the 
 groove while the pulley revolves at a high speed. While 
 this may produce a smooth surface, such an expedient is 
 doubtful economy; for no matter how carefulty a multiple- 
 grooved pulley may be cast, it is almost impossible to ob- 
 tain a rim all the grooves of which are true and of the same 
 diameter: the result is that, with such pulleys, the ropes 
 tend to vibrate and sway from side to side, rubbing against 
 each other, and frequently against side-posts, walls, floors, 
 and other obstructions, which rapidly destroy the rope. 
 
 Attempts have been made to produce a finished surface 
 in the groove by casting the rim in an accurately turned 
 chill, but such pulleys have not been as successful as an- 
 ticipated. A form of pulley made by the Link-Belt Engi- 
 neering Co. of Philadelphia consists of an iron sheave cast 
 in rings, some with and some without arms and hub, from 
 which a complete pulley is built up having the requisite 
 number of arms for strength, and at the same time being 
 
 * Willis : " Principles of Mechanism," 
 
184 
 
 ROPE-DKIVIKG. 
 
 light and free from excessive shrinkage strains liable to ex- 
 ist in wheels having light arms and heavy rims and hubs. 
 The sheaves have a slight projection 
 on one side and corresponding re- 
 cess on the other, with bolt-holes at 
 the circumference, so that a multiple- 
 grooved pulley with any number of 
 grooves may be readily built up by 
 bolting the sections together as shown 
 
 FIG. 63. BUILT-UP ROPE PULLEY FOR LIGHT WORK. 
 in Fig. 63. These wheels are made from metal patterns and 
 moulded in a three-part iron flask. The core which makes 
 the groove is continuous and of green sand, producing a 
 very smooth casting. The groove is finished with an em- 
 ery-wheel swinging in a frame like a cut-off saw, the final 
 finish being given by the use of emery and oil. While this 
 process produces a single-grooved wheel more cheaply than 
 by turning, it is doubtful whether from a manufacturing 
 standpoint there is any economy in the built-up wheel 
 when we consider the degree of accuracy now obtained in 
 moulding multiple-grooved wheels, and the consequent re- 
 duction of labor in turning them. 
 
 Another special form of rope-driving pulley is that made 
 by Jofyn Mnsgrave & Song, Bolton, England. This pulley 
 (Fig. 64), it will be noticed, is extremely light, but is suffi- 
 ciently strong for the requirements. This lightness is at- 
 tained by the use of steel arms turned tapering, and firmly 
 secured to the rim and the hub, which latter is split in 
 
HOPE-DRIVING. 185 
 
 three segments and ringed with steel. By the use of this 
 form of pulley the shafts are relieved to a considerahle 
 extent of the weight and consequent friction entailed by 
 the ordinary pulley, and there are no excessive shrinkage 
 strains to contend with., as is usually the case with the 
 
 FIG. 64. ROPE PULLEY WITH TURNED STEEL ARMS. 
 
 common form of single-casting pulley. In pulleys of this 
 form, where wrougbt-iron and steel rods are used for arms, 
 the ends of the rods should be dipped in acid and tinned 
 before setting in the mould. In addition to this the rods 
 are frequently headed up and 
 The bottom of the groove in 
 
186 
 
 ROPE-DRIVIHG. 
 
 FIG. 65. 
 WOOD-PILLED PULLEY RIM. 
 
 times filled with wooden blocks dovetailed into a channel 
 cast to receive them, in which case the rope runs on the 
 bottom and the shape of the groove approximates that 
 
 shown in Fig. 65; after 
 being fitted and secured 
 the groove is trued up and 
 turned out to the desired 
 shape. Gutta-percha, rub- 
 ber, leather, tarred hemp, 
 and other materials have 
 been used for the same pur- 
 pose; but in the best mod- 
 ern practice a smooth cast- 
 iron surface is preferred to 
 any other, and we find these 
 groove linings confined chiefly to pulleys used in wire-rope 
 transmissions and hoisting-machinery. 
 
 Of late years an all-wood rim with V or U grooves has 
 been used to some extent, as it makes a cheap pulley, and 
 is very satisfactory for light work when a semicircular 
 groove is adopted. Wooden rims, however, should not be 
 used for heavy work, and the slip of the rope should be re- 
 duced to a minimum ; otherwise the grooves and ropes will 
 be rapidly cut out and the whole system will be very unsat- 
 isfactory. 
 
 The proportions of rope-pulley rims depend somewhat 
 upon the shape of groove adopted. Some manufacturers 
 have a standard groove, with straight sides, which may be 
 used for several different-sized ropes, as shown in Fig. 66. 
 For light work such a pulley is very satisfactory, and has 
 many advantages in regard to constructive features not 
 possessed by any other form. The groove shown in Fig. 
 67* is very commonly used in England, but in this country 
 
 Low and Bevis, " Machine Design," p. 156, 
 
ROPE-DRIVItfG. 
 
 187 
 
 manufacturers of large rope-pulleys usually prefer a form 
 in which the abrupt change in profile (as at a) does not 
 occur. Of these the more common form is a modification 
 
 FIG. 66 RIM SECTIONS STRAIGHT GROOVE. 
 
 of the English section, in which the straight sides forming 
 the angular groove are connected to the rib between the 
 grooves by curves as in Fig. 68. 
 
 Another form which has much in its favor, especially 
 for the independent rope system, is the circular arc groove, 
 in which the sides are formed by arcs of circles, 
 
188 
 
 Working proportions for this groove are given in Fig. 
 69, in which the unit d equals the diameter of rope. It 
 
 FIG. 67. RIM SECTION ENGLISH FORM 
 
 FIG. 68. RIM SECTION IMPROVED ENGLISH FORM. 
 will be noticed that the centre for the curve is located at 
 the intersection of a line drawn through the centre of the 
 
ROPE-DRIVING. 
 
 189 
 
 rope at an angle of 22J with the horizontal and a line 
 drawn through the tops of the dividing ribs; the angle of 
 the groove embraced by the rope is thus dependent upon 
 the position of the latter in the groove : in its normal posi- 
 tion the angle is one of 45. Tho Walker groove, as now 
 used in rope-pulleys made by Eraser & Chalmers of Chicago 
 and the Walker Mfg. Co. of Cleveland, has its sides formed 
 with circular arcs similar to Fig. 69, but the angle of the 
 
 'Elec. World 
 I UNIT=d 
 
 ANGLE OF GROOVE 45WHEN ROPE IS IN ITS NORMAL POSITION 
 
 FIG. 69. RIM SECTION GROOVES WITH CIRCULAR ARCS. 
 
 groove is more acute; at the point of tan gen cy when the 
 rope simply rests in the groove the angle measures 33. 
 
 Rope-pulleys for ordinary transmissions are usually flat 
 on the inside of the rim, or slightly tapered as shown by 
 the full lines in Fig. 69. In large wheels, however, the 
 rim is frequently swept up, and where the weight with a 
 flat rim would be greater than required for strength or 
 steady running, the inside is hollowed out as shown by 
 dotted line. This gives a more nearly uniform section 
 and makes a stronger and lighter wheel, but the expense of 
 construction is greater. Guide-pulleys or idlers should be 
 made with a semicircular groove, so that the rope runs 
 upon the bottom instead of being wedged between the 
 sides, as in grip-pulleys. Some engineers maintain that 
 
190 
 
 HOPE-DRIVING. 
 
 the wedge groove should be used in all cases, and many 
 plants are in operation having V-shaped grooves in the 
 idlers; but we believe the most satisfactory results will be 
 obtained by using a groove in which the rope runs on the 
 bottom. Of these there are two general forms the one 
 in which the rope has considerable play in the groove, and 
 the other of such form that the rope is embraced by a 
 portion of the surface of the groove: the first is used more 
 particularly for single ropes, while the latter is adapted to 
 any number of wraps. 
 
 A modification of the first form is frequently used for a 
 number of routes as shown in Fig. 70. 
 
 FIG. 70. FORM OF GROOVE FOR GUIDE PULLEYS. 
 
 The proportions shown in Fig. 71 will be found very 
 satisfactory for single or multiple groove guide-pulleys, in 
 which the pitch is equal to that used with the grip-pulleys 
 represented in Fig. 69; the general design conforms closely 
 to the latter, brt the pulley is somewhat lighter. 
 
 For shaft-pulleys and the smaller sizes of fly-wheels not 
 exceeding about nine feet in diameter the casting is usually 
 made in one piece, unless a "split" pulley is required. In 
 order to relieve the wheel in cooling the arms were formerly 
 
ROPE-DRIVING. 
 
 191 
 
 given a curved or S shape, as this form yields more readily 
 and is supposed to conform to the unequal contraction of 
 the rim and hub of the pulley in cooling. With properly 
 proportioned wheels, however, and with due care in the 
 foundry, straight- armed pulleys may be cast as strong as 
 those of curved outline, and as the former are lighter, neater, 
 and cheaper than the latter they are now almost universally 
 employed. In the larger single-casting pulleys the hubs 
 are frequently split in order to favor the arms in cooling: 
 
 FIG. 71. FORM OF GROOVE FOR GUIDE PULLEYS. 
 
 when split by a diametral plane the two portions at the 
 hub are generally secured by bolts and nuts as in Fig. 72, 
 or less frequently by pins and cotters; when split in three, 
 wrought-iron or steel rings are shrunk onto the ends of the 
 hub, which is turned to receive them, as represented in Fig. 
 73. Occasionally rings are shrunk on and bolts used as 
 well, but this precaution is not common practice. 
 
 There is no general rule by which the number of arms 
 maybe determined. For small rope-pulleys the number is 
 usually six, while for the larger sizes six or eight arms are 
 used for pulleys cast in one or two parts ; but these, in some 
 
192 
 
 ROPE-DRIVING. 
 
 cases, have two sets of arms as shown in Fig. 74, which 
 represents a 78-inch pulley made by the Robert Poole & 
 Son Co. for the Providence Cable Tramway Co. The usual 
 
 FIG. 72. 
 
 FIG. 73. 
 
 run of wheels from 9 feet to 15 feet in diameter are cast 
 in halves, six or eight arms being employed. 
 
 Occasionally, however, narrow rope fly-wheels, of diam- 
 eters up to 18 feet, are made in two pieces only, with eight 
 arms. 
 
ROPE-DKIVING. 
 
 193 
 
 Moderate-sized wheels arc frequently cast in one, but 
 arranged so as to be readily split in two afterwards. 
 
 Cottoh Ropes 
 
 Pulleys cast in halves having light rims should have 
 double arms along the line of separation, as shown in Fig. 
 75, which represents a form of split pulley made by Fair- 
 
HOPE-DRIVING. 
 
ROPE-DRIVING. 195 
 
 banks, Morse & Co. ;_ this prevents undue bending action, 
 and is otherwise the better able to resist the effect of cen- 
 trifugal force due to the added weight at the point of 
 connection. 
 
 For diameters ranging from 15 feet to 20 feet the usual 
 practice is to make the pulley in six or eight segments with 
 as many arms, although some makers prefer ten segments; 
 from 20 to 26 feet ten segments are usually adopted. For 
 wheels from about 26 to 32 feet twelve segments and as 
 many arms are generally used, while larger-sized wheels are 
 built up of fourteen, sixteen, and even more segments 
 sometimes as many as twenty-four being employed.* 
 
 When rope-pulleys are made in halves or are built up it 
 is important that the connecting bolts and flanges in the 
 rim should be strong enough to resist the maximum stresses 
 that may occur in the joint. 
 
 Ordinarily in rope-pulleys the net section of bolt area is 
 about 12 per cent of the section at the joint, but this ranges 
 from 25 per cent to about 6 per cent in different cases. 
 Various shop rules are used by designers for obtaining the 
 bolt section, many of which give the bolt area directly in 
 terms of the cross-section of the rim. While such formulae 
 mny give satisfactory results for an assumed average rim 
 speed, it is better to design the joint in large wheels from 
 a consideration of the particular conditions in each case. 
 
 In very slow-moving heavy wheels the principal stress 
 may be that due to the weight of the wheel itself, but in 
 rope-pulleys and fly-wheels the rim has usually a high 
 velocity, and the strain due to centrifugal force in the rim 
 is the principal factor in determining the bolt section; it is 
 obvious, however, that in high-speed heavy wheels both 
 influences should be taken into consideration. 
 
 The tension in the rim produced by centrifugal force is 
 
 * American Machinist, Feb. 16, 1893. 
 
196 
 
 KOPE-DK1V1NG. 
 
 equal to one half the centrifugal force due to the weight 
 and velocity of the rim multiplied by the ratio of diameter 
 to semi-circumference, or equal to 
 
 F 2r F 
 
 ^-2 V __ 
 c\ ^ - ~~ -- 
 
 2 nr n 
 
 As this force is resisted at each end of a diameter, the 
 strain T h , or hoop-tension, acting at either end will be one 
 half the above; hence 
 
 in which F has the usual value 0.00034 WRN*, 
 where W = entire weight of rim ; 
 
 R = effective radius of rim in feet; 
 N = revolutions of wheel per minute. 
 The tension at each end of a diameter due to the weight 
 
 of the rim is evidently equal to 
 
 1fW\ 
 
 - -- 
 
 iS\ 
 
 in the rim due to F and W will be 
 
 therefore the stress 
 
 If the bolts could be placed at the point 
 of application of this force, their effective 
 
 o 
 
 section A would be -, where /is the allow- 
 
 able stress in the bolts. 
 
 In the actual construction the centre of 
 bolts must be placed at some distance 
 from the point of application, in which 
 case the bolts are subjected to an addi- 
 tional strain due to the bending moment 
 at the joint, as shown in Fig. 76. If this bending 
 moment is taken up by the rim, the bolt section will be 
 
ROPE-DRIVIKG. 197 
 
 ^as before; but if the rim and bolt flanges are rigid and 
 
 resist any deformation, the total strain comes upon 
 the bolts. Under these conditions the net section A 
 
 <y 
 
 may be obtained by multiplying j by the ratio of the 
 
 leverage y of the force 8 to the leverage x of the resist- 
 ance in the bolt; hence if we assume that x = \y, a com- 
 
 mon value, we shall obtain A = 2-.. 
 
 By using studs with nuts at each end the bolt may be 
 placed nearer the rim, in which case, for the same depth of 
 
 flange, the ratio - may be made less than 2. In order to 
 
 obtain ample strength at the rim, the bolt section should 
 be determined on the supposition that the full bending 
 moment will be thrown upon the bolts. In this case the 
 bolt-flanges or lugs must be able to resist any bending due 
 to the force S. The bolt-flanges will be as strong as the 
 rim if we make the respective moments of resistance equal 
 to each other; hence 
 if b = face of wheel; 
 
 b' breadth of bolt -flange minus the width of bolt- 
 holes; 
 
 t mean thickness of rim; 
 
 t' thickness of bolt-flange or lug, 
 then 
 
 or = ,. 
 
 This thickness t' may be reduced somewhat if strength- 
 ening ribs are carried from the rim to the lower edge of 
 the bolt-flange, as shown in Fig. 76. In the above, t 
 was taken as the mean thickness of rim reduced to a rec- 
 tangle, and no account has been taken of the strength- 
 ening ribs and flanges. If the leverage of the bolts is 
 
198 ROPE-DRIVING. 
 
 assumed as one half, we may obtain the total net section 
 of the bolts as follows : 
 
 Since F = 0. 00034 WRN\ 
 
 .OOOlOS WRN* ~ 
 
 = -ifo. 
 
 In the above formula / may be taken equal to 6000 
 pounds, for bolts up to 1 or 1-J- inches diameter, but for 
 larger sizes 8000 to 9000 pounds may be used. 
 
 If t = the mean thickness and b breadth of rim in 
 inches, the weight W may be determined from 
 
 W=%7rJtx 1Mb X 0.26 
 = IQ.GRtb. 
 
 If the rim is properly bolted, the principal strains to 
 which the hub-bolts are subjected are those due to the 
 weight of hub and arms and the tension produced by key- 
 ing to the shaft. 
 
 As the weight of hub and arms in large pulleys is usu- 
 ally much less than the weight of rim, it will be seen that 
 the strain on the hub-bolts due to the weight and centrif- 
 ugal force of these parts will be less than that on the rim- 
 bolts. 
 
 In practice some makers design the hub-joint so that the 
 net section of all the bolts is equal to the bolt section in 
 one edge of the rim-joint; this, however, is not usual 
 practice. An inspection of a great many pulleys made by 
 various manufacturers shows that the hub-bolt section is 
 often twice as great as the rim-bolt section; but in many 
 of these cases the rim-bolting is very weak. 
 
 It is safe to make the bolt area the same in each case; 
 but if the rim-joint is made according to the method 
 
HOPE-DRIVING. 
 
 199 
 
 indicated, the bolts in the hub will usually be sufficiently 
 strong if their total effective section is equal to that in 
 one rim joint: for light pulleys, however, this should be 
 increased.* 
 
 The section of arm for those pulleys in which the arms 
 and hub are cast together is usually elliptical or segmental, 
 as shown in Fig. 77. 
 
 FIG. 77. SECTIONS OF ARMS. 
 
 The elliptical form as here given is so proportioned that 
 its minor axis is one half its major; in the segmental form 
 the minor axis is four tenths the major. 
 
 The cross-section of the arm may be determined by con- 
 sidering it as a beam fixed at one end and loaded at the 
 other with the force P due to the pull of the ropes. The 
 bending moment on each arm will then be 
 
 Pr 
 iw 
 
 b ~ ~N~' 
 
 -LV a 
 
 assuming that the load is divided equally among the num- 
 ber of arms, N a . 
 
 * For complete analysis of strains in fly-wheel rims, see Prof. Un- 
 win's discussion in his "Machine Design," vol. ir. Also. Mr. 
 Stanwood's paper in vol. xiv. Trans. A S. M. E., with Prof. 
 Lanza's discussion. 
 
200 ROPE-DRIVIKG. 
 
 Since the bending moment is equal to the modulus of 
 the section, Z, multiplied by the stress,/, in the material, 
 we have 
 
 Jf t =fZ; 
 
 For an elliptical section whose major axis is h and minor 
 axis OAh, 
 
 Z=^XM; 
 hence 
 
 If we let T lf the maximum tension in the rope, equal 
 
 T 
 
 200d 2 pounds, and -~ = 2, we shall have the load acting on 
 
 * 
 each arm 
 
 P = lOOd'n, 
 
 where n is the number and d the diameter of ropes. 
 Therefore by substituting this value of P in equation (18) 
 
 there is obtained h = 
 
 0.04/2VV 
 
 Assuming that/= 2500 pounds per square inch for cast 
 iron and that N a equals 6, we have 
 
 h = 0.44 
 
 where D = pitch diameter of pulley and li is the major 
 axis of arm produced to centre of wheel, both in inches. As 
 the smaller pulleys require a larger margin of strength, on 
 account of their greater liability to breakage in handling 
 and casting, the above formula should be modified by intro- 
 ducing a constant; if we let this constant equal -J inch, a 
 suitable value for the width of arm will be obtained from 
 
 .5" ..... (19) 
 
HOPE-DRIVING. 201 
 
 As the centrifugal force set up in the rim of the pulley 
 causes an additional stress in the metal, the value of / 
 should be chosen with reference to the speed at which the 
 pulley is to run. The higher speed not only induces a 
 greater stress in the material, but the liability of failure 
 due to vibration or shocks is greatly increased. The stress 
 in the rim due to centrifugal force may be obtained by 
 considering the pulley as a cylinder subjected to a force 
 p( = F ) per square inch. The thickness of a thin cylinder 
 to resist rupture may be obtained from 
 
 ._pD"_pr 
 
 ~w~ r 
 
 in which t = thickness in inches; 
 
 p = pressure per square inch; 
 r = radius of cylinder in inches; 
 f = allowable stress in pounds. 
 
 In this case 
 
 W= weight in pounds = 0.261 pounds per cubic inch; 
 v = velocity of rim in feet per second ; 
 g = 32.16; 
 
 T 
 
 R = radius of rim in feet = ; 
 
 1/w 
 
 hence 
 
 Fr W r Wtfr 
 
 and 
 
 12TFV 
 
 7 ^ ' 
 
202 
 
 If F Q act on one square inch of pulley whose thickness 
 is unity, we shall have 
 
 < 
 = ~, very nearly; 
 
 this is the stress in rim per square inch of section due 
 to centrifugal force alone. 
 
 When v in feet per second = 
 
 50 
 
 60 
 
 70 
 
 80 
 
 90 
 
 100 
 
 150 
 
 200 
 
 " F" " " minute 
 
 3000 
 
 8(500 
 
 4200 
 
 4800 
 
 5400 
 
 6000 9000 
 
 12000 
 
 Stress /' in pounds due to centrif- 
 
 
 
 
 
 
 
 
 
 ugal tension 
 
 250 
 
 360 
 
 490 
 
 640 
 
 810 
 
 1000 2250 
 
 4000 
 
 The stresses due to the pull of the ropes, and those due 
 to contraction in cooling, are additional to those here given, 
 hence / should be taken sufficiently low to allow for the 
 various stresses which may be set up in the pulley. If we 
 assume that the working stress should not ordinarily exceed 
 3500 pounds per square inch for cast-iron pulley arms, and 
 that it should be less for high speeds where the dynamic 
 effect of shock and vibration is greater, a suitable value 
 for cast iron may be obtained from the empirical formula 
 
 50000 
 
 5+ fF 
 
 From this formula the annexed values of / have been 
 calculated; it will be noticed that the value of / used in 
 formula (19), namely, 2500 pounds per square inch, corre- 
 sponds to a velocity of rim equal to about 3500 feet per 
 minute. 
 
 Velocity of rope in feet per minute F = 
 Allowable stress / = 
 
 1200 
 9-300 
 
 1800 
 2900 
 
 2400 
 2700 
 
 3600 
 2450 
 
 4800 
 2300 
 
 0000 
 2150 
 
ROPE-DRIVING. 03 
 
 The arms should taper toward the rim T V inch per inch 
 of length, that is, -j s on each side, but in no case should 
 the width of arm at the rim be less than two thirds its 
 width at the centre of shaft. 
 
 Very wide pulleys in which the proportions for single 
 arms would be inconveniently large may be made with two 
 or three sets of arms; in such cases they may be considered 
 as two or three separate pulleys combined in one, except 
 that the proportions of the arms should be 0.8 to 0.7 times 
 that of single -arm pulleys.* 
 
 In designing the hubs of wheels practice varies consider- 
 ably. Some authorities give the thickness of metal around 
 the eye in terms of the pulley diameter only, others take 
 into account the diameter of pulley and also the diameter 
 of shaft or the breadth of face, while the length is variously 
 given in terms of the diameter of shaft or the face of 
 pulley, or both. 
 
 For rope-pulleys if the thickness of metal in the hub is 
 made proportional to the diameter of pulley and also to the 
 diameter of shaft, and if the length of hub is made to vary 
 with the number and size of ropes and the diameter of 
 shaft, we believe the requirements of strength and good 
 proportions will be best attained. 
 If D diameter of pulley, 
 d s = " " shaft, 
 d h = " " hub, 
 d = " " rope, 
 n number of ropes, 
 L h length of hub, 
 
 then the diameter and length of hub may be obtained 
 from the following formulae, which have been deduced from 
 the proportions of a large number of rope pulleys made by 
 representative manufacturers : 
 
 * Reuleaux, " Constructeur." 
 
204 
 
 ROPE-DKIVIHG. 
 
 and 
 
 L h = 0.6^1 
 
 For loose or idle pulleys the diameter of hub may be 
 made less than that given by the above formula, which 
 allows for keying. In general the thickness of metal 
 around the eye in an idle pulley may be taken as about 
 two thirds as great as that in fixed pulleys of the same 
 diameter and face. The length of hub in idlers should be 
 sufficient to give a good bearing surface, and may vary 
 from two to three times the diameter of shaft depending 
 somewhat upon the speed of rotation. 
 
 n 
 
 FIG. 78, METHOD or JOINING ARMS AND RIM. 
 
 In the construction of large rope-pulleys which are 
 made in segments, the usual method is to bolt the rim sec- 
 tion to the arms at the ends of the segment, as shown in 
 Pig. 78. 
 
 When the rim segments are joined midway between the 
 arms as in Fig. 79 the several connections are simplified to 
 some extent, but with this method of connection there is a 
 decided tendency for the joint to open under the influence 
 of centrifugal force, which is increased somewhat by the 
 weight of the connecting flanges and bolts. With the rim- 
 joints at the junction with the arms, however, a rigid con- 
 
ROPE- DRIVING. 
 
 205 
 
 nection between adjacent arms is obtained, and the centrif- 
 ugal effect of the rim tends to increase the tension in the 
 arm without opening the joint. 
 
 Examples of both methods of connection are given in 
 Figs. 80 to 92, which also represent some of the details of 
 construction in various built-up rope-pulleys. 
 
 As will be noticed, these built-up wheels are made with 
 a large central boss or hub, usually in one piece, but some- 
 times in two, provided with seats for the arms, or bored, 
 
 F IGj 79. METHOD OF JOINING ARMS AND RIM. 
 
 either tapering or straight, to receive the ends of the arms, 
 which are turned to fit the holes. 
 
 The arms are usually of round or elliptical section, cast 
 hollow; but other forms are common. Thus in Figs. 85 
 and 90 the arm is of cruciform section, while in Fig. 81 a 
 modification of the H section is used. 
 
 In Fig. 80 we have an interesting form of wheel patented 
 some years ago by Mr. James Barbour, of Combe, Barbour 
 & Combe, Belfast.* The rim of the wheel is made in seg- 
 
 * Engineering, Sept. 7, 1888.; 
 
206 
 
 ROPE-DRIVING. 
 
 nients, and is attached to the boss or hub by means of the 
 bolts or tie-rods d, extending from the rim to the boss as 
 shown, and passing through the arms, which are made hol- 
 low for the purpose. The rim and hub are recessed to re- 
 ceive projecting pieces a' a' on the ends of the arms. The 
 
 FIG. 80. BUILT-UP WHEEL. (Barbour.) 
 
 head of the tie-rod d' can be made to form part of the 
 periphery of the wheel. The tie-rods have slots formed in 
 their inner ends to receive keys b p , which keys are passed 
 into the hub through holes parallel to the axis of the shaft, 
 and in this way the rim of the wheel is secured to the hub. 
 
FIG. 81. THIRTY-FOOT ROPE-WHEEL. (Hick.) 
 
 FIG. 81 (a). 
 SECTION OF GROOVE. 
 
 FIG. 81 (5) and (c). 
 SECTION AT Boss. 
 
208 ROPE-DRIVING. 
 
 The object of the arrangement is to obviate the use of 
 a number of bolts for connecting the rim to the arms; 
 moreover, the tie-rods withstand the centrifugal force of 
 the rim, and the wheel can be driven with safety at a 
 high rate of speed. The usual projections from the 
 principal surfaces are also eliminated, and the currents 
 of air generated by the high velocity of the rim are less- 
 ened. This latter feature is one the value of which is be- 
 coming more recognized lately, especially in rope-driving, 
 when a high velocity of rim is especially advantageous. 
 
 To lessen the fan action set up in a large fly-wheel, we 
 frequently find the arms boarded up, provision being made 
 for this in the wheel. 
 
 Fig. 81 represents a fly-wheel 30 feet in diameter made 
 by Hick, Hargreaves & Co., Bolton, for a 1000 horse-power 
 condensing-engine.* The weight of the wheel is 54 tons. 
 The rim is 6 feet wide, and is grooved for 27 cotton ropes 
 5 inches in circumference, or nearly If inches in diameter, 
 at 2-J inches pitch. The grooves are shown in detail 
 at (a). 
 
 The rim is constructed of twelve segments with twelve 
 arms. The segments are planed at the joints, and are 
 bolted together with eight bolts and nuts, of which the 
 two next the arms are If inches in diameter and the others 
 are 1 inches. Each arm is secured with four 2-inch T- 
 head bolts and nuts to the rim segments, two to each of 
 two segments. The boss, nave, or centre, shown in sec- 
 tion at (), is 6 feet i inch in diameter, or 7 feet across the 
 platforms, or slightly raised plane surfaces, on which the 
 arms take their bearings. It is 18 inches wide at the rim 
 and 2 feet wide at the bearing on the shaft. Twelve sock- 
 ets are bored out to receive the ends of the arms. The 
 arms are formed approximately of H section, as at (c), and 
 
 * D. K. Clarke, " Steam-Engine," vol. in. 
 
ROPE-DRIVING. 209 
 
 measure 14 inches by 9 inches at the centre and 9-J inches 
 by 6| inches at the rim. They are turned conically to fit 
 
 FIG. 82. TWENTY-FOUR-FOOT ROPE- WHEEL. (Walker.) 
 
 the holes in the centre, tapering from 12 inches in diam- 
 eter at the outer ends of the sockets to 7-J inches at the 
 
210 HOPE-DRIVING. 
 
 inner ends in a total length of 2 feet 5 inches. The arm 
 is keyed into the centre with two cotters, each 24 inches 
 long, 1 inch thick, tapering in width from 3f inches to 3^ 
 inches, or -J inch in 24 inches. They are driven in one 
 from each side of the nave reversely, and make up a 
 united width of 6f inches. To join the rim the arm is ex- 
 panded into a flat flange 21 inches by 18 inches and 4 
 inches thick, through which the four holts already named 
 are passed. The opening at the centre is 25 inches in di- 
 ameter, or 2 inches larger than the shaft. The wheel is 
 fixed on the shaft with six keys 5 inches wide and If 
 inches thick, bearing on six flat seats formed on the shaft, 
 with a taper of inch to the foot. 
 
 Pig. 82 represents a somewhat similar method of con- 
 nection employed by the Walker Manufacturing Company, 
 Cleveland. In this pulley the arms are of round or pipe 
 section, and present a much neater appearance and better 
 proportions than obtain in the previous pulley. 
 
 This wheel was made for the Baltimore City Passenger 
 Railway Station, and is 24 feet in diameter, 30 inches face, 
 grooved for ten 2 -inch ropes at 2J inches pitch. The de- 
 tails are shown in Figs. 83 and 84. 
 
 The centre is a single casting 5 feet 6 inches in diam- 
 eter, bored to receive the arms, of which there are ten. 
 Each arm is 10J inches in diameter near the hub and 8 
 inches in diameter near the rim, where it is expanded into 
 a flange and bolted to the rim segments. The bolts con- 
 necting the arms to the segments and the segments to 
 each other are all turned 1|| inches in diameter, and fit in 
 drilled and reamed holes. 
 
 The shoulder on the arms at the hub is 15 inches in di- 
 ameter, and is faced to fit the machined seats on the boss. 
 By this means perfect alignment is obtained, and no 
 motion of the parts is possible unless the key shears. 
 
 Fig. 85 represents a rope-pulley designed by Mr. F. Van 
 
KOPE-DEIVIKG. 
 
 211 
 
 Bolt holes drilled and reamed 
 Bolts turned 
 
 . 83, DETAILS OF PULLEY SHOWN IN FIG. 83. SECTION OF ARM, 
 
212 
 
 HOPE-DRIVING. 
 
 DETAIL OE ARM AT CENTER 
 
 FIG. 84, DETAILS or PULLEY SITOWN IF FIG. 83, 
 
214 
 
 Vleck for the San Diego Cable-railroad Power-station.* It 
 is 25 feet in diameter, 42| inches face, grooved for twelve 
 2-inch cotton ropes at 3^ inches pitch. This drum is com- 
 posed of ten separate segments and ten arms, bolted and 
 keyed midway in each segment. The central hub is 4 
 
 feet in diameter across flats, and is proportioned, as in fact 
 the whole wheel is, with special regard to lightness. 
 
 The designs shown in Figs. 86 to 89 are for the large 
 rope-pulleys used in the Fifty-first Street and Houston Street 
 
 * Trans. A. S. M. E., vol. xn. p. 77. 
 
j? IG 87 THIRTY-TWO-FOOT ROPE- WHEEL. (Walker Mfg. Co.) 
 
216 
 
 ROPE-DKIVLtfG. 
 
 SECTION ON LINE 
 A B 
 
 SECTION O;N LINEJ 
 
 C D 
 
 HALF SECTION 
 ON LINE E F 
 
 FIG. 88. THIRTY-TWO- FOOT ROPE-WHEEL. (Walker Mfg. Co.) 
 
ROPE-DRIVIKG. 
 
 stations of the Broadway Cable Road. The first wheel, 
 Fig. 86, is 32 feet in diameter, 37^ inch face, and is grooved 
 for 13 cotton (Lambeth) ropes 2 inches in diameter, 2f 
 inches pitch. The speed of the ropes is very low only 
 1877 feet per minute. Mr. M. W. Sewall * gives the fol- 
 lowing particulars regarding the details of this wheel, as 
 at first designed. It will be seen by reference to the figure 
 that the arms of the drum are secured to the centre by 
 clamping two turned portions on the inner end of each 
 arm, between heavy cast-iron disks, one of which is cast 
 
 FIG. 89. SECTION OF RIM OP PULLEY 
 SHOWN IN FIGS. 87 AND 88. 
 
 as a flange on the hub and the other acts as a fol- 
 lower. These are bolted up a slight distance apart, and 
 the holes for the arms are bored accurately to fit the turned 
 portions of the same. The bolts are then loosened, the 
 arms put in place, and the follower bolted hard up to the 
 hub flange. A taper pin is then driven into a reamed hole 
 through each arm and the parts of the centre, and held in 
 place by a nut. The arms and the centre are then practi- 
 
 * For complete description of the cable-driving machinery see 
 American Machinist, May 24 and 31, 1894. 
 
HOPE-DRIVIKG. 
 
 frothy $ Poales, 
 
 FIG. 90. HEAVY ROPE-WFIEEL, TWENTY-SEVEN FEET DIAMETER. 
 (Wetherill & Co.) 
 
819 
 
 FIG. 91. SECTION OF TWENTY-SEVEN-FOOT WHEEL, 
 
220 
 
 cally one piece, and no working of the parts among them- 
 selves is possible. None of the bolts secure an individual 
 arm, and no stress transmitted to the centre through the 
 arms brings any of them into direct tension ; several bolts 
 might be broken without detriment to the structure. This 
 drum would be improved, so far as convenience of manu- 
 facture is concerned, by a shoulder on the arm to deter- 
 mine its distance from the centre when assembling in the 
 shop. The arms are attached to the rim in the usual man- 
 ner, but are rather light in proportion to the rest of the 
 design. These pulleys were redesigned to suit the contrac- 
 tor for the work, The Walker Manufacturing Co. of Cleve- 
 land, and were constructed in a manner similar to that 
 shown in Fig. 87, except that single arms and centre were 
 used. 
 
 Bradlev <f Poates. Ennr's. N. Y. 
 
 FIG. 92. SECTION OF GROOVES FOR WHEEL SHOWN IN 
 FIGS. 90 AND 91. 
 
 Figs. 87, 88, and 89 represent the 32-foot drums used in 
 the Houston Street station, and are much wider and 
 heavier than those used in the Fifty-first Street station. 
 As will be noticed, these drums are 8 feet 4 inches face, 
 grooved for thirty-four 2-inch ropes at 2J inches pitch, and 
 arranged with two sets of arms and two centres. The weight 
 of each puHey is 104 tons. 
 
 As constructed, the arms have heavy flanges bolted to the 
 centre, into which a turned projection on the end of the 
 
KOPE-DRIVItfG. 221 
 
 arm is accurately fitted. In addition to this two of the four 
 bolts securing each arm to the centre is turned to fit reamed 
 holes in the boss ; these bolts thus act as dowels and pre- 
 vent any working of the parts. The bolts are calculated 
 to resist the maximum tension in the arms, and are 2 T \ and 
 2 inches in diameter ; in the same way the arms are bolted 
 to the rim, two of the four bolts fitting in reamed holes. 
 The arms are of hollow elliptical section 15 by 10 inches at 
 the hub, and 10 X 7-J inches at the rim. Each centre is 8 
 feet diameter across flats ; they are bored 26 inches in di- 
 ameter, and are 3 feet long in the bore. 
 
 Large rope-pulleys from 20 to 32 feet in diameter are, 
 when extra wide, frequently made with two centres as well 
 as two sets of arms, as just noted in Fig. 87. This is also 
 seen in Figs. 90 to 92, which represent a rope-driving wheel 
 26 feet 9 inches pitch diameter, 8 feet face, grooved for 
 twenty-four 3-inch ropes, at 3J inches pitch. This pulley 
 was designed and built by Robt. Wetherill & Co. of Ches- 
 ter, Pa., and consists essentially of two separate and inde- 
 pendent drums, flanged and bolted together at the rim. 
 Each centre is made of two separate disks 6 feet 6 inches 
 in diameter, bored and faced on the inside. The arms, of 
 which there are twelve, are of cruciform section between 
 the boss and the rim, where they are flanged and bolted to 
 the rim segments in the usual manner. At the centre the 
 arms are wedge-shaped, 8 inches thick, and are so propor- 
 tioned that when accurately planed and fitted they form a 
 complete circle. 
 
 These arm segments are then bolted between the two 
 centre disks, and make a strong and compact hub. 
 
INDEX. 
 
 Actual section of ropes, 92 
 
 Adhesion of ropes, 168 
 
 Advantage of high rotative speeds, 157 
 
 Advantages of rope-driving, 4, 5 
 
 American system of rope-driving, 24, 25 
 
 Angle embraced by rope, 117 
 
 Angle of groove, 164 
 
 Area of ropes, 92 
 
 Arras, number of, in pulleys, 190 
 
 Arms, shape of, 199 
 
 Arms, taper of, 203 
 
 Atlas Mills, 12 
 
 Atmospheric changes, 31 
 
 Automatic tension-carriage, 28 
 
 Axial rotation of ropes, 174 
 
 Barbour, James, built-up pulley, 205 
 
 Barrus, G. H., friction in mills, 7 
 
 Beeswax on ropes, 89 
 
 Belts, leather, 1, 34, 40 
 
 Blow-holes in pulleys, 183 
 
 Bolt area for pulleys, 195 
 
 Braided rope-joint, 23 
 
 Broadway Cable Road, built-up pulleys, 217 
 
 Brown, A. G., friction loss, 6 
 
 Brush Electric Light and Power Co., Niagara Falls, 56 
 
 Built-up pulleys, 183, 195, 205 to 221 
 
 Catenary curve, 8, 12 
 Cast grooves, 183 
 Centrifugal force, 45 
 
 Centrifugal force, influence of, 111, 119, 163 
 Combe & Barbour, early rope-drives, 2 
 
 Combe, Barbour & Combe, basis for calculating horse-power, 99 
 
 223 
 
224 INDEX. 
 
 Cotton ropes, 77, 81 
 
 Cotton fibre, 78 
 
 Cotton wax, 78 
 
 Continuous- wind system, 25 
 
 Cone-pulleys for ropes, 35 
 
 Coil-friction, 44, 48 
 
 Corliss engine and use of jack-shaft, 41 
 
 Coefficient of friction, 112 
 
 Cost of ropes, 101, 122 
 
 Coulter, Dr. S. M., tests on inanilla fibre, 82 
 
 Coupling, cut-off, 27 
 
 Coupling for braided rope, 23 
 
 Creep of ropes, 174 
 
 Cross-section of rope, 92 
 
 Cut-off coupling, 27 
 
 Deflection of rope, 131, 170 
 Degree of twist in ropes, 86 
 Details of rope-pulleys, 205 to 221 
 Diameter of bolts for pulleys, 195 
 Diameter of pulleys, 103, 161, 177 
 Differential driving, 36, 170 
 Differential pulley, Walker's, 173 
 Double arms in pulleys, 193 
 Double ropes recommended, 108 
 Draw-rods, 71 
 
 Durie, James, on rope-driving, 3 
 Dyblie's rope-tightener, 30 
 Dynamo-driving, 38 
 
 Early use of ropes for driving, 2, 64 
 
 Effect of centrifugal force, 45 
 
 Effect of tar on ropes, 91 
 
 Effect of wedging in groove, 162 
 
 Efficiency in any given case, 73 
 
 Elasticity of ropes, 4, 162 
 
 Elastic slip of ropes, 174 
 
 Engines, friction in, 6, 142 
 
 English rope system, 4 
 
 Executed rope transmissions, 96 
 
 Experiments on friction, 141 
 
 Examples of large rope- wheels, 205 to 221 
 
INDEX. 225 
 
 Fairbanks, Morse & Co., split pulley, 193 
 
 Fibrous ropes, 75 
 
 Fibre, cotton, 78 
 
 Fibres of manilla, 82 
 
 Flax ropes, 77 
 
 Fly-wheels, heavy, 5 
 
 Frictional grip, 167 
 
 Friction and stress moduli, 118 
 
 Friction-clutch, 33, 42 
 
 Friction, coil, 44, 48 
 
 Friction, coefficient of, 112 
 
 Friction loss, 141, 158 
 
 Friction of engines, 6 
 
 Friction of shafting, 6, 68, 145 
 
 Gear wheels, 1, 5, 8 
 Graphite on ropes, 89, 91 
 Gregg, multiple sheaves, 36, 38 
 Groove, angle, 164 
 Groove, shape of, 186 
 Groove, surface of, 183 
 Groove, wedging of rope in, 162 
 Guide-pulleys, 189 
 
 Harmonic vibration, 105 
 
 Heavy liy-wheels, 5 
 
 Hemp ropes, 2, 77, 90 
 
 Henthorn, J. F., friction in mills, 7 
 
 Hick, Hargreaves & Co., built-up pulley, 208 
 
 Him, C. F., transmission of power, 63 
 
 lloadley Bros., Power-house, Chicago City Ry. Co., 53 
 
 Horse-power of ropes, 111 to 121 
 
 Hubs of pulleys, 191, 203 
 
 Hunt, C. \V., form of splice, 20 
 
 Idlers, 34, 36 
 
 Inclined transmissions, 139 
 
 Influence of belt pull, 157 
 
 Influence of centrifugal force, 111, 119, 163 
 
 Introduction of rope driving, 2 
 
 Jack-shaft, use of , for dynamo-drives, 38, 42 
 
226 INDEX. 
 
 Jaw clutch, 42 
 
 Joint for braided ropes, 23 
 
 Kircaldy, tests on ropes, 80 
 
 Lambeth ropes, 80, 88 
 
 Lanett Mills, 12 
 
 Large rope-wheels, 205 to 221 
 
 Laxey, large overshot wheel at, 71 
 
 Least diameter of pulleys, 103, 161, 179 
 
 Leather belts, 1 
 
 Length of manilla fibre, 86 
 
 Life of ropes, 107 
 
 Limit of length in shafting, 09 
 
 Linseed-oil on ropes, 89 
 
 Link-belt Co., Western Electric Co.'s Plant, 25 
 
 Link-belt Co., Virginia Hotel Plant, 42 
 
 Liverpool Overhead Railway, 40 
 
 Lockwood & Greene, Lanett Mills, 12 
 
 Lockwood & Greene, Naumkeag Mills, 12 
 
 Long-distance transmission, 62 
 
 Loss due to bending, 159 
 
 Loss due to friction, 141, 158 
 
 Loss due to winder-pulleys, 50 
 
 Losses in ropes, 141, 159 
 
 Lubrication of ropes, 79, 88 
 
 Manilla fibre, 82, 86 
 Manilla ropes, 77 
 
 Marlin-spike, improved form of, 23 
 Method of joining arms and rim, 204 
 Miller, T. S., plant of Western Electric Co., 25 
 " " varying angle of groove, 168 
 " " use of small ropes, 100 
 " " use of loose idler, 36 
 Multiple idle sheaves, 36, 38 
 Multiple-rope system, 4, 17 
 Musgrave & Sons, Atlas Mills, 12 
 " " " , Nevsky Mills, 16 
 " " " , rope-pulley, 185 
 " " " , life of cotton ropes. 107 
 
IKDEX. 
 
 Naumkeag Mills, 12 
 
 Nevsky Mills. 16 
 
 Normal working load, 98, 100 
 
 Outdoor transmission, 50, 66 
 Overshot wheel at Laxey, 71 
 
 Pine tar as lubricant, 89 
 
 Pitch diameter of pulley, 181 
 
 Power absorbed by friction in shafting, 152 
 
 Power absorbed by ropes and gears, 6 
 
 Power transmitted by shafting, 153 
 
 Pulleys, diameter of, 38, 103, 161, 177, 181 
 
 Pulley, tightener, 48 
 
 Pulleys, supporting, 64 
 
 Pulley, winder, 47 to 54, 66 
 
 Pulleys, wood, 56, 164 
 
 " , with double arms, 193 
 
 " , with steel arms, 185 
 
 " , very wide, 203 
 
 Rawhide ropes, 25, 75 
 Relative cost of ropes, 122 to 127 
 Relative wear of ropes, 122 
 Rim sections, 166, 187 
 Ropes, cotton, 77, 81 
 
 " , cost of, 101, 122 
 
 ' ' , early use of, 2 
 
 " , elasticity of, 162 
 
 " , fibrous, 75 
 , flax, 77 
 
 " , hemp, 2, 77, 90 
 
 " , horse-power of, 121 
 
 " , Lambeth, 80, 88 
 
 " , life of, 107 
 
 " , lubrication of, 79 
 
 " , manilla, 77 
 
 " , rawhide, 25, 75 
 
 " , round leather, 76 
 
 " , shrinkage of, 31 
 
 ' ' , speed of, 1 03 
 
 " , splicing of, 18 
 
228 INDEX. 
 
 Ropes, steel and leather, 76 
 
 " , stevedore, 90 
 
 " , strength of, 79, 91 
 
 " , square leather, 76 
 
 " , wear of, 101, 104, 123 
 
 " , weight of, 109 
 
 " , wire, 75 
 Rope wells, 10 
 Rotation of ropes, 174 
 
 Sag of ropes, 131, 170 
 Sections of arms, 199 
 
 " " rim. 187 
 Section of rope, 92 
 Semicircular grooves, 164 
 Sewall, M. W., built-up pulleys, 2tt 
 Shafting, for long-distance transmissions, 68 
 
 ' ' friction of, 145 
 
 limit of length, 69 
 loss due to friction, 68 
 
 " power transmitted, 153 
 
 Shafts, jack, use of, for dynamo-drives, 38, 42 
 
 " at an acute angle, 3^ 
 
 ' ' at right angles, 36 
 Shrinkage of ropes, 31 
 Side lead of ropes, 35 
 Size of pulley, 38 
 
 " " ropes in use, 100, 107 
 Slack-side tension, til 
 Small ropes, 24 
 Speed of ropes, 103 
 Split hubs, 191 
 Splicing of ropes, 18 
 Stevedore, transmission rope, 90 
 Stress in rim-bolts, 196 
 
 " " ropes, 67 
 Strength of ropes, 79, 91 
 Supporting pulleys, 64 
 Surface of groove, 183 
 
 Taper of arms, 203 
 
INDEX. 229 
 
 Tallow on ropes, 89 
 Tar, effect of, on ropes, 91 
 Telodynamic transmissions, 63 
 Temporary installations, 52 
 Tension carriage, 27 
 
 *' weight, 28, 32, 140 
 " in ropes, 111 to 118, 132 
 Tests on ropes, 80 
 Tightener, 29. 30 
 
 " pulley, 48 
 Transmissions at an angle, 34 
 outdoor, 50, 66 
 
 " of power to a distance, 62 
 
 " telodynamic, 63 
 
 Turbines, Victor, 56 
 Twist in ropes, 86, 88 
 
 Use of ropes with portable tools, 62 
 Use of water-wheels, 56, 66, 71 
 
 Van Vleck, built-up pulley, 214 
 Vibration in ropes, 105 
 Victor turbine, 56 
 
 Walker, differential pulley, 173 
 Walker Mfg. Co., built-up pulley, 210, 220 
 Watertown tests on ropes, 80 
 Water-wheels and rope-driving, 56, 66 
 Wear of ropes, 83, 101, 104, 122 
 Wear due to side lead, 35 
 Weakening effect due to twisting, 86 
 Webber, Samuel, early use of ropes, 2 
 
 " " friction of shafting, 8 
 
 Wedging action, 162, 169 
 Weight for tension-carriage, 139 
 Weight of ropes, 109 
 Wells, rope, 10 
 
 Western Electric Co.'o plant, 25 
 Wetherill & Co., built-up pulley, 221 
 Wide pulleys, 203 
 Willamette Mills, 66 
 Winder-pulley, 47 to 54, 6.6 
 
230 
 
 INDEX. 
 
 Wire ropes, 75 
 
 Wood pulleys, 56, 164, 186 
 
 Wood-filled pulley riin, 186 
 
 Working strength of ropes, 95, 98 
 
 Wound system of rope-driving, 25 
 
 Wren's "Instrument for drawing up great weights," 45 
 
RETURN CIRCULATION DEPARTMENT 
 TO* 202 Main Library 
 
 LOAN PERIOD 1 
 HOME USE 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS 
 
 Renewals and Recharges may be made 4 days prior to the due date. 
 
 Books may be Renewed by calling 642-3405. 
 
 DUE AS STAMPED BELOW 
 
 AUTO DI^P 
 
 
 
 HIM 7 n 1QAQ 
 
 
 
 uUii J U I/O" 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 UNIVERSITY OF CALIFORNIA, BERKELEY 
 FORM NO. DD6 BERKELEY, CA 94720 
 
BERKELEY LIBRARIES