828 Broadway aiitorma Withdrawn REESE LIBRARY UNIVERSITY OF CALIFORNIA. UMH/FPSITV OF HA? A TREATISE BELTS AND PULLEYS. EMBRACING FULL EXPLANATIONS OF FUNDAMENTAL PRINCIPLES; PROPER DISPOSITION OF PULLEYS; RULES, FORMULAS, AND TABLES FOR DETERMINING WIDTHS OF LEATHER AND VUL- CANIZED-RUBBER BELTS AND BELTS RUNNING OVER COVERED PULLEYS; STRENGTH AND PROPORTIONS OF PULLEYS, DRUMS, ETC. TOGETHER WITH THE PRINCIPLES OF AND NECESSARY RULES FOR ROPE-GEARING AND TRANSMISSION OF POWER BY MEANS OF METALLIC CABLES. BY J. HOWARD CROMWELL, Pn.B., AUTHOR OF A TREATISE ON TOOTHED GEARING. NEW YORK: JOHN WILEY AND SONS. 1888, Copyright, 1885, BY TQHN WILEY & SONS. PREFACE. IN the manufacture of modern machinery, which in the great majority of cases embodies a vast deal of careful study and precise calculation, there is probably no one element which enters as largely into the calculations and forms as important a part in the daily operations in the machine-shop as the end- less belt for the transmission of power. The lathe, the drill, the planer, the shaping-machine in short, almost without ex- ception, all machine-tools are commonly driven from the shop-shaft by means of belts and pulleys; and we can scarcely glance into a shop or factory of any description without en- countering a mass of belts which seem at first sight to mo- nopolize every nook in the building and leave little or no room for anything else. Notwithstanding the countless thousands of belts for trans- mission in use and constantly being replaced in the shops and factories of America; notwithstanding the fact that many thousands of dollars are consumed every year by the rapid wear and destruction of our machine-belts, and the immense field thereby opened for the practical study and application of the principles of economy in this connection there is no branch of machine-construction which is to-day in as crude and unsatisfactory a state of development as this all-important transmission by belt and pulley. Strange as it may seem, it is IV PREFACE. nevertheless true, that there is scarcely a machine-shop in America which can definitely and correctly calculate the proper width of a leather belt which will safely transmit a given horse- power. Nor are the manufactures of machine-belting in any degree in advance of the shops, for I have never yet seen the manufacturer who has any better solution for this apparently simple problem than his own "judgment." Having taken the pains to write to a large number of the best-known machine-shops and belt-manufacturers throughout the country, asking for information concerning belting, and in every case having received an answer to the communication, I am compelled to assert that among all the letters received not a single one contained any definite information on the subject. As specimen answers to these letters I may quote the following extracts : " We have no particular method of calculating widths of belts aside from tables found in books of reference." "There is no rule for the width of belting that we know of : it is always determined by the width of the pulley upon which it is to run." " We determine the width of belts more by experience than by any fixed rule." " We always try and make the strain as light, in pounds per inch of width, as possible, and when we are limited for room we use double belts. 100 pounds per inch of width is ^about the ultimate strength of transmission, and if you can reduce the working strain to 50 pounds, it means long life to the belt." " It is difficult to give any positive rule about belting that would apply to all cases." From one of the largest and best-known belt- manufacturing concerns in the country comes the following : " We have no rules or formulas for esti- mating the power of belts other than those given in works on mechanical engineering, nor do we apply these strictly. It is PREFA CE. V very much a question of judgment. . . . You will consider this letter very indefinite, but we do not know how to make it less so." Here are extracts from a letter received from another well-known belt-maker: "We wish to express the fear that what we have to say will be disappointing to you, to say the least. ... As to the horse-power, we have no rule. . . . We have made no tests of the tensile strength of leather, for the reason that we do not consider it a matter of any importance. . . . We have made no efforts to obtain the coefficient of fric- tion. . . . When we can obtain a homogeneous material which will be easily workable and a perfect substitute for leather, the manufacture, sale, use, and study of belting may begin to be a matter of satisfaction ; in the meanwhile they are puzzling, if not indeed exasperating." These extracts (many more of similar nature might be given) show almost no knowledge at all, on the part of our great belt- manufacturers and machine-shops, concerning the subject; and worse still in some cases, that little or no effort has been made to obtain any knowledge other than that of rough guesswork and rule of thumb. Small wonder is it, then, that the ordinary mechanic's practical knowledge of the subject is infinitely small. Several of the parties above referred to state that they use the rules found in the various books of reference ; let us look over some of these works and endeavor to reach fair conclusions concerning the rules and formulas in common use to-day. Arnold, in his " Mechanical Principia," gives the rule for belt- widths: "Multiply 36000 by the number of horse - powers ; divide the amount by the number of feet the belt travels per minute ; divide this quotient by the number of feet in length of belt contact with the smaller drum or pulley, and divide this by 6 : the result is the required width of belt in inches." VI PREFACE. Professor Reuleaux offers the formula b = 18 \/P, b represent- ing the width of the belt in millimetres and P the force in kilograms transmitted by the belt. Unwin, in " Elements of Machine Design," gives the formula 2 P /3 = in which ft is the belt-width in inches, P the force transmitted in pounds, and f the safe working tension per inch of width, which he takes at 70 pounds for a belt -fa of an inch thick. The formula is to be used only when the belt embraces about 0.4 of the smaller pulley-circumference. In Nystrom's Mechanics we find b = -~- , b denoting the belt width in inches, H the horse-power transmitted, d the diameter of the smaller pulley in inches, and a the number of degrees occupied by the belt on the circumference of the smaller pulley. Let us now assume an example which will serve to determine the variations in the results of calculations from the above rules and formulas. Suppose we wish to determine the proper width for a belt which will transmit a force of 25 horse-power; the smaller pulley having a diameter of 5 feet = 60 inches, and the velocity being 10 feet per second = 600 feet per minute. The belt embraces 0.4 of the pulley-circumference = 0.4 x 15.7 = 6.28 feet = 360 x 0.4 = 144 degrees. For the force trans- mitted, in pounds, we have P = 1375 pounds. With these quantities as data, Arnold's rule, given above, gives 36000 x 25 us for our required belt-width ^- - = 39.8 inches. 600 x 6.28 x 6 If we divide the force 1375 pounds by 2.2, we obtain 625 kilo- grams, and Reuleaux's formula gives b = 18 1/625 45 milli- metres = 450 x 0.04 = 1 8 inches. From Unwin's formula we PREFACE. Vli obtain ft = = 39.3 inches, and from the formula of *7 ^OO X 2 ^\ Nystrom b = -2 =21.7 inches. Haswell in his " Engi- 60 x 144 neer's and Mechanic's Pocket-book," gives a rule by which our belt-width would be 42 inches. Summing up our results will show that, for the same belt, under the same circumstances, the width is according to the authorities named as follows : Haswell 42 inches. Arnold 39.8 " Unwin 39.3 Nystrom 21.7 Reuleaux 18 Of these different values the greatest is 2j times the least. Probably Arnold, Haswell, and Nystrom are in use in our shops more than the others, and these give results, for the belt-width in question, differing from each other by more than 20 inches. According to a list of prices for double, white-oak tanned belt- ing, which is before me, the difference in cost for the above-cal- culated belt, supposed to be double and 100 feet long, between Nystrom and Haswell would be six hundred and sixteen dol- lars, to say nothing of the difference in the cost of the pulleys, shafts, etc. These great differences between the results from the rules of different authors are apparently due to the difference of opinion concerning the value of the coefficient of friction, which is taken all the way from 0.22 to 0.40, and to the fact that each writer on the subject has striven to obtain simple rather than accurate rules. At best we are dealing with an uncertain ma- terial when we attempt to deduce rules for the strength of leather belts, and if the elements of belt-thickness, method of Vlll PREFACE. lacing or fastening, etc., are entirely or partially neglected, the uncertainty of accurate results must be very greatly increased. In the matter of joint-fastening alone, a glance at the table on page no will show that a 2o-inch leather belt ^ inch thick run- ning over two equal cast-iron pulleys will transmit a force of over 1800 pounds with a riveted joint or 1250 pounds when fastened with a double raw-hide lacing, while with a single leather-lac- ing the same belt will transmit but 976 pounds. In other words, to transmit a force of 1000 pounds over two equal cast- iron pulleys by means of a leather belt -% inch thick we will need a belt-width of 12 inches for riveted joint, 16 inches for double raw-hide lacing, and 22 inches for single leather-lacing. I believe that it is utterly impossible for any man to write an entirely simple work on the subject of belting, which will be of any practical use to the mechanical world. The subject is complicated by difficulties far greater than are ordinarily met with in dealing with mechanical questions, and to attempt to simplify it beyond a reasonable limit is simply to omit certain necessary considerations, and thereby render the investigation worthless. My object in writing this work on belts and pulleys is, therefore, to present to the mechanical public a small yet comprehensive and, above all, an accurate book on the subject. I have constantly endeavored to have due regard for simplicity, yet when I have found it necessary to sacrifice either simplicity or accuracy, I have invariably chosen the former. All measure- ments and dimensions are given in English units in order to avoid the confusion sometimes resulting from the use of the Metric System, and I have endeavored by numerous simple ex- amples throughout the book to fully illustrate the use of the various rules and formulas. In translating the part devoted to metallic cables from Reuleaux, the formulas and tables have PREFACE. IX been transformed from the metric system into English measures, which will, I trust, satisfactorily explain the unusual numbers which have resulted in a few instances. In the hope that my humble endeavors to furnish accurate information on the subject of belt-transmission to those whom it may concern may be in a measure, if not entirely, successful, and trusting that in the present instance I may receive from mechanical men the same generous support and encourage- ment that have attended my previous efforts in the field of mechanical literature, I present to the public my " Treatise on Belts and Pulleys." J. H. C. NEW YORK, May i, 1885. TABLE OF CONTENTS. SECTION I. PAGE Introduction Absence of early Mechanical Records Uncertain Origin of the Belt and Pulley Probable Origin I SECTION II. Fundamental Principles Direction of Rotation Relations be- tween Circumference, Diameter, and Radius Velocity Revo- lutions Power Horse power 9 SECTION III. Rules for the Proper Disposition of Pulleys Axes which coin- cide geometrically Parallel Axes Axes which intersect each other Axes which cross without intersecting 28 SECTION IV. Transmissions by Belts without Guides Half-crossed Belt Conditions necessary for maintaining the Belt on the Pul- leys Distance between Pulleys 29 SECTION V. Transmissions by Belts with Pulley-guides Half-crossed Belt with Pulley guide Half-crossed Belt with Movable Pulley- guide General Case of Crossed Arbors Arbors at Right An- gles 32 Xll TABLE OF CONTENTS. SECTION VI. PAGE Length of Belts Open Belt Open Belt, approximate Formula Crossed Belt Belts with Guides and intricately arranged 45 SECTION VII. Speed cones Stepped Cones Open Belt Crossed Belt Graph- ical Method Continuous-speed Cones 51 SECTION VIII. Materials used for Belting Leather Vulcani/ed Rubber Intes- tines of Animals Rawhide Hemp and Flax Leather and Metallic Wire 65 SECTION IX. Lacing and other Modes of Fastening Shortening Single and Double Lacing Belt-hooks Cleat-fastening 68 SECTION X. Strength of Leather Belts Resistance to Slipping Coefficient of Friction Tensions on Belts Breaking-strength Width for Different Kinds of Fastening Width necessary to transmit certain Powers 75 SECTION XI. Leather Belts over Leather-covered Pulleys Coefficient of Fric- tion Tensions Width for Different Kinds of Fastening- Width necessary to transmit certain Powers 115 SECTION XII. Vulcanized Rubber Belts Number of Layers of Duck Thick- ness Breaking-strength Coefficient of Friction Width for Different Kinds of Fastening Width necessary to transmit certain Powers Rubber Belts over Leather- and Rubber cov- ered Pulleys 14.0 TABLE OF CONTENTS. xiii SECTION XIII. PAGE Rim, Nave, and Fixing-keys for Pulleys Rounding of the Rim Flanged Rim Rim of Pulley for Belt with Circular Cross- section Split Pulleys Approximate Weight of Pulleys 159 SECTION XIV. Arms of Pulleys Oval Cross-sections Number of Arms Strength of Arm s Straight Arms Single and Double Curved Arms 166 SECTION XV. Shafts Safe Shearing Stress Steel Wrought-iron Cast-iron Diameter necessary to transmit certain Powers 171 SECTION XVI. The Tightening-pulley Fast and Loose Pulleys Reversing by m?ans of Fast and Loose Pulleys Fast and Loose Pulleys for Belts with Circular Cross-sections 179 SECTION XVII. Rope belts Tension almost entirely due to the Weight Pulley for several Rope-belts Proper diameters for Rope-belts Di- ameters of Pulleys for Rope-belts 185 SECTION XVIII. Jointed Chain-belts Rouiller's Chain-belt Metallic Belt of Go- din Jointed Chain-belt of Clissold Coefficient of Friction Dimensions '. 192 SECTION XIX. Tensions of Metallic Cables Number of Strands and Wire Co- efficient of Friction 196 xiv TABLE OF CONTENTS. SECTION XX. PAGE Calculation of Diameters of Cables Formulas and Tables of Di- ameters of Cables for Different Numbers of Wires 200 SECTION XXI. Deflections in the Cable of a Horizontal Transmission Deflec- tion of Cable in Motion Deflection in a State of Repose De- flection in the Driving and Driven Parts 207 SECTION XXII. Transmission by Cable with Increased Tension Increased Di- ameters of Cable and Wires 212 SECTION XXIII. Transmission by Inclined Cable Tensions in Inclined Cables Deflections Height above the Ground 217 SECTION XXIV. Method of Tracing the Curves of Cables Approximately Para- bolic Curves 221 SECTION XXV. Transmission by Cable with Pulleys near together Small Value oi Si 222 SECTION XXVL Rim of Cable-pulleys Single Cable Several Cables upon one Pulley 223 SECTION XXVII. Arms and Nave of Cable-pulleys Number of Arms Oval Cross- sections Flanged Cross-sections Straight Arms Curved Arms Reserve Cables 226 TABLE OF CONTENTS. XV \ SECTION XXVIII. PACK Pulley-supports and Intermediate Pulleys Stations at the Ex- tremities Intermediate Stations Changing the Direction of the Cable 230 SECTION XXIX. Dimensions of Pulley supports Ratio between the Radius of the Pulley-support and Diameter of the Wires 234 SECTION XXX. Pressure upon Axes of Pulley supports Weight of Large Pulleys 235 SECTION XXXI. Station Pillars Brick and Stone Piers Pedestals Two Pulleys side by side 238 APPENDIX I. Experiments for determining various Coefficients of Friction Leather over Cast-iron Pulleys Leather over Leather-covered Pulleys Vulcanized-rubber Belts over Cast-iron and Covered Pulleys 243 APPENDIX II. Special Applications of Principles of Belts and Pulleys Devices for changing Motion and Direction of Rotation Increasing and Decreasing Speeds Intermittent Motion Different Meth- ods of arranging Principal Pulley and Shop Shafts in Mills. . . 252 BELTS AND PULLEYS. I. Introduction. Says Thomas Ewbank in his famous " Hydraulics and Mechanics :" "Tradition has scarcely preserved a single anecdote or circumstance relating to those meritorious men with whom any of the useful arts originated ; and when in process of time History took her station in the temple of Science, her professors deemed it beneath her dignity to record the actions and lives of men who were merely inventors of machines or improvers of the useful arts ; thus nearly all knowledge of those to whom the world is under the highest obligations has perished forever. ... A description of the foundries and forges of India and of Egypt, of Babylon and Byzantium, of Sidon and Carthage and Tyre, would have imparted to us a more accurate and extensive knowledge of the ancients, of their manners and cus- toms, their intelligence and progress in science, than all the works of their historians extant, and would have been of infinitely greater service to mankind. " Had a narrative been preserved of all the circum- stances which led to the invention and early applica- tions of the lever, the screw, the wedge, pulley, wheel 2 BELTS AND PULLEYS. and axle, etc., and of those which contributed to the discovery and working of metals, the use and manage- ment of fire, agriculture, spinning of thread, matting of felt, weaving of cloth, etc., it would have been the most perfect history of our species the most valuable of earthly legacies. Though such a work might have been deemed of trifling import by philosophers of old, with what intense interest would it have been perused by scientific men of modern times, and what pure de- light its examination would have imparted to every inquisitive and intelligent mind !" Rollin, writing of u The Arts and Sciences of the Ancients" many years ago, finds fault with the world for neglecting the great inventors and admiring the military heroes of antiquity. " Of what utility to us at this day," he asks, " is either Nimrod, Cyrus, or Alexander, or their successors, who have astonished mankind from time to time? With all their magnifi- cence and vast designs they are returned into nothing with regard to us. They are dispersed like vapors and have vanished like phantoms. But the inventors of the arts and sciences labored for all ages. We still enjoy the fruits of their application and industry; they have procured for us all the conveniences of life ; they have converted all nature to our uses. Yet all our admira- tion turns generally on the side of those heroes in blood, while we scarce take any notice of what we owe to the inventors of the arts" In like manner, Robertson, in his work on India, laments the loss of, or rather absence of, early records concerning the useful arts and sciences. He says : " It is a cruel mortification, in searching for what is in- INTRODUCTION. 3 structive in the history of past times, to find the ex- ploits of conquerors who have desolated the earth, and the freaks of tyrants who have rendered nations un- happy, are recorded with minute and often disgusting accuracy ; while the discovery of useful arts and the progress of the most beneficial branches of commerce are passed over in silence and suffered to sink into ob- livion." The origin, age, first application, and use of the mechanism known to us as the " endless belt and pulley" are entirely unknown ; as far back into the history of the ancients as we can see by means of the earliest mechanical records, we find the endless belt running continuously around the pulley precisely as it does to-day. We may theorize, and assume a probable origin ; we may bring up, in support of our assumption, all the reason and logical conclusions at man's dispo- sal ; we may even convince mankind that we have cor- rectly traced and explained the path over which the mechanism has come down to us from the dim ages of the past. But here we must stop ; we can go no far- ther : and the fact will yet remain that the real age and origin for which we are searching are still undiscovered and unknown. If, however, we cannot know with certainty the real age and origin of belts and pulleys, it is nevertheless a satisfaction to us to be able to trace out, by analogy, by reason, and by the known existence of things which must have necessitated the use of pul- leys, what seems to us to have been the origin, the successive modifications, and the line of improvement by which this most useful contrivance has been handed down to us. 4 BELTS AND PULLEYS. In searching for an uncertain origin or beginning of anything, we most naturally start by determining up- on the very simplest and most rudimentary form (knowing that simplicity almost always precedes com- plexity, and that a thing must of necessity have a skele- ton before it can have a form), and then strive to fix upon its exodus from the conception to the tangible thing itself. In order then to trace the growth of the thing in question from its origin to its present much altered and improved form, we strive to imagine the slightest possible change, in the right direction, which can be given to the original. Having successfully achieved the first transformation or alteration, we con- tinue to pick out each slight alteration and improve- ment in proper order, until we have reached the present most improved form. If we assume, as is claimed by some writers, that the mechanism of the belt and pulley was among the first mechanical contrivances of primitive man, we must search for its origin among what we judge to be the first necessities of the human race and the modes of obtaining these necessities. Although many claim that the human race, in the beginning, passed through a fireless period, that men lived without the use of fire or artificial heat, we must nevertheless conclude that this element was one of the first necessities of human life, and that the first effort made by prehistoric man in the line of invention was for the purpose of produc- ing fire. It is very generally admitted that the first " fire-machine" (Reuleaux concludes that this was the first machine of any description. See Kenedey's Eng- lish translation u Kinematics of Machinery," London, INTRODUCTION. 5 1876,9.204) consisted of an upright piece of wood, having one end pointed. This, fitted into a hollow in another piece of wood and being twirled rapidly back- wards and forwards with the hands, generated sufficient heat to set fire to some small fragments of dry wood or other combustible material (Fig. i). Here we have the first belt and pulley hardly recognizable, it is true, but none the less the probable origin. The upright piece of wood here constitutes the pulley and FIG. i. FIG. 2. the human hands the belt. The first transformation seems to have been the substitution of a cord wound several times around the upright piece (as shown in Fig. 2) in place of the direct application of the hands. This rude contrivance, though it produced only an oscillating motion, was used for other purposes than that of producing fire; the primitive drill, lathe, pot- ter's wheel, etc., were driven to and fro in this manner, the work being done only on the forward turn, and the 6 BELTS AND PULLEYS. backward turn serving only to place the work in such a position that the operation of cutting could be again continued. The change from this contrivance to the rope and pulley used for drawing water from deep wells, and for lifting the vast blocks of stone, columns, etc., used by the ancients in building, was indeed slight, and may reasonably have taken place not long after the first introduction of the improved form of " fire-machine.'* For how long a period this oscillating motion suf- ficed for the rough manufacturing purposes of the age, or at just what era in the life of man the change was made to the endless belt, which transformed the oscil- lating into a continuous rotary motion, is indeed a mystery. Whole generations even centuries may have been needed to impress upon the primitive mind the advantages of continuous rotation and to accom- plish the necessary change in the mechanism. It seems most probable to us that the loss of time in- curred by the useless backward motion in lathes, drills, etc., and the natural desire on the part of these an- cient artisans to accomplish more and more work in less and less time, must have led to the adoption of the two pulleys and the endless belt. Gradually, very gradually, the slight but all-important change was made. Some early thinker now unknown even in the uncertain histories of the past ages connected the loose and separated ends of the single cord, passed the now endless cord over two cylindrical sticks, fitted roughly into a frame to hold them apart, and caused both to rotate by turning one with a crank. Next some primitive inventor obtained the friction neces- IN TR OD UC TION. 7 sary for the transmission of considerable forces by- winding the cord several times around each pulley ; and so in process of time, in his attempts to obtain and transmit greater powers, the man of the ages long since forgotten at last discarded the round cord for the broad flat band or belt of the present era. Reuleaux says "the crossed belt appears to be the alder;" but to us it seems most probable that the flat band was first used in its simplest form, i.e., open, and that the crossed belt was afterwards introduced in order to pre- vent (by its additional embracing of the pulleys) slip- ping, and to produce a rotation of the driven pulley in a direction contrary to that of the driver. As to the material of the primitive cord and belt, we can prove nothing : it is, however, reasonable to sup- pose, since the skin of wild animals was the easiest material to obtain, and since, from the earliest reeord.s of history, skins have been used for clothing, bow- strings, etc., that the material of the primitive belt differed from the leather of to-day only in that it was untanned and unfinished, and perhaps taken from ar different animal. Doubtless the fixing together or lacing of the ends of belts was the source of considera- ble difficulty to the ancients, for in all cases where such a belt could be made to perform the necessary work, round cords tied together at the ends seem to have been used. It is supposed, and very reasonably, from certain known circumstances, that the first idea of continuous rotary motion which was developed in the mind of man took the form of an undershot water-wheel, driven by the current of a stream or river. The Chi- 8 BELTS AND PULLEYS. nese have doubtless used these water-wheels, for pur- poses of irrigation and drawing water, for many centu- ries, and, according to tradition, they were also used at an early date in ancient Assyria, Mesopotamia, and other countries of Asia Minor. These pristine water- wheels consisted of a rough axle and two or more long blades, usually built up of sticks and bamboo, some- times with rough buckets formed out of mud or clay. It is not at all unlikely that the first attempts to con- FIG. 3. struct large pulleys were founded upon the principles of construction seen in the water-wheels, and that the pulleys were used without rims, as shown in Fig. 3. From the fact that wagon-wheels with entire rims and fellies are known to have been in existence in the earliest Greek and Egyptian times, we may very fairly conclude that the use of the complete wooden, if not also iron, pulley reaches far back into antiquity, and that its advent into the world probably took place not long after the discovery of the endless belt. History informs us that the ancient city of Nineveh FUNDAMENTAL PRINCIPLES. 9 was surrounded by a massive stone wall over 100 feet high, and that the city was fortified with 1500 towers, each 200 feet in height. Babylon, " the noblest city ever built by man/' had a fortified wall which reached to the incredible height of 360 feet, and her famous hanging-gardens were built of " flat stones of amazing size." The Tower of Babel is said to have been "40 rods square at the bottom, and upwards of 600 feet high." These gigantic structures supposed to have been built about the year 2200 B.C. could not have been erected without the aid of strong ropes and pul- leys, or similar contrivance. Thus for over four thou- sand years have been known and used successfully the cord and pulley which we use to-day. For how many centuries in the unknown ages of the prehistoric period men toiled and labored with their crude " fire-ma- chines," perhaps even lived and died without reaching that much of "the machine," we must leave for future investigation and development to decide. 2. Fundamental Principles. The mechanism known in modern mechanics as the " endless belt and pulleys" is, primarily, a device, the object of which is to transmit a continuous rotary mo- tion from one shaft or arbor to another parallel shaft, and the first fundamental principle of the mechanism may be clearly expressed as follows: If two drums or pulleys be placed in certain positions relative to each other, each being allowed the motion of rotation about its fixed axis, and no other, and if an endless band be passed tightly over the circumferences of the pulleys IO BELTS AND PULLEYS. as represented in Fig. 4 ; then, if a continuous rotary motion be given to one of the pulleys, the friction be- tween it and the band will cause the latter to move around the circumference, and the second pulley will FIG. 4. (because of the friction between it and the band) therefore be caused to rotate continuously about its fixed axis that is, the continuous rotary motion of the driving-pulley will be directly transmitted through FIG. 5. the endless band to the second pulley. In this defini- tion it is presupposed that the friction between the band and pulleys is sufficiently great to overcome the resistance of the pulleys ; otherwise the driving- pulley will simply slide around upon the band without FUNDAMENTAL PRINCIPLES. II causing it to move, and consequently the second or driven pulley will remain motionless. (a) Direction of Rotation. Belts maybe either open, as shown in Fig. 4, or crossed, as in Fig. 5 : in the former case the two pulleys rotate in the same direc- tion, while in the latter case the driven pulley rotates in a direction contrary to that of the driver. (/;) Relations between Circumference, Diameter, and Radius. The circumference C of a circle, the diameter of which is represented by D, is given by the expres- sion C=nD, (I) in which n represents the constant quantity 3.14159. RULE. To determine the circumference of a circle in inches or feet, multiply the diameter in inches or feet by the constant 3.14159. Since the radius of a circle is equal to one half its diameter, if we denote the radius by R, we shall have R , or D =. 2R, and formula (i) becomes by sub- stitution C = 2nR (2) RULE. To determine the circumference of a circle in inches or feet, multiply the radius in inches or feet by the constant 2n = 6.28318. From formula (i), by transposing the quantities, we may write 12 BELTS AND PULLEYS. \ RULE. To determine the diameter of a circle in inches or feet, divide the circumference in inches or feet by the constant 3.14159. In a similar manner from formula (2) we may obtain R = . . f4) 27T RULE. To determine the radius of a circle in in- ches or feet, divide the circumference in inches or feet by the constant 27t = 6.28318. If we let C and C denote the circumference of two circles, D and D' , R and R', the respective diameters and radii, we shall have, from formulas (i) and (2), C = nD = 2nR, and C = TtD' = 2nR r ; and we may write the proportions C : C :: nD : nD f :: 2nR in the form of an equation, C nD 2nR which, by cancelling the equal constants in numerator and denominator, becomes D R RULE. The ratio of the circumferences of any two circles is equal to the direct ratio of their diameters or radii. \ FUNDAMENTAL PRINCIPLES. 13 (V) Velocity. The circumferential velocities of two pulleys which are connected by one and the same belt (supposing there is no slipping of the belt on either pulley) must obviously be the same, each being equal to the velocity of the belt. For the belt must unroll from the driving-pulley just as fast as it is developed from the pulley-circumference ; it must also roll upon the circumference of the driven pulley with the same velocity, else the belt would constantly tend to become tighter on one side and looser on the other, and sliding or rupture would necessarily ensue."* The circumferential velocity of the driven pulley and the velocity of the belt are entirely independent of the pulley-diameters, and depend solely upon the circum- ferential velocity of the driving-pulley. Thus, if the circumferential velocity of the driver is 10 feet per second, 10 feet of circumference, and no more or less, can be developed per second upon the belt, be the driver ever so large or ever so small. In the same manner, just 10 feet of belt can roll per second upon * The tensions on the two s,ides (or parts) of the belt are not the same (as will be seen farther on); consequently the circumferential velocities of the two pulleys are not absolutely the same. According to Professor Reuleaux, if v and v denote the circumferential velo- cities of the two pulleys, t and T the tensions on the two parts of the belt, E the coefficient of elasticity o,f the belt, and S the strain on the driving part of the belt, the true velocities will be given by the ex- t pression = . Reuleaux says, " The loss of velocity due ~S to the sliding has for a mean value about \ percent; it is accompanied by a loss of work, which is transformed into, heat and produces wear of the belt and pulleys." 14 BELTS AND PULLEYS. the circumference of the driven pulley, without refer- ence to its size or diameter. (//) Revolutions. Since the circumferential velocities of any two pulleys, which are connected by one and the same belt, are the same without regard to the diameters of the pulleys, and since the circumferences of the two pulleys are directly proportional to their diameters (formula 5) ; if one of the pulleys has a diameter equal to twice that of the other, the circum- ference of the former will also be equal to twice that of the latter, and the former will need just twice as much time in which to perform one entire revolution as the latter. In other words, the larger pulley will make just one half as many revolutions in a given time as the smaller. In a similar manner, if the diameter of the larger pulley is three or four times that of the smaller, the former will need three or four times as much time for each revolution as will the latter, or the larger pulley will make only one third or one fourth the number of revolutions in a given time as the smaller. In formula, denoting by n and ri the num- bers of revolution of the two pulleys, and by C and C', D and D\ and R and R the respective circumferences, diameters, and radii, we shall have C" D RULE. The ratio of the numbers of revolutions of two pulleys, which are connected by one and the same belt, is equal to the inverse ratio of their circumfer- ences, diameters, or radii. FUNDAMENTAL PRINCIPLES 15 If we represent by n the number of revolutions per minute, by v m the velocity in feet per minute, and by R f and C f , respectively, the radius and circumference of the pulley in feet, we shall have for the velocity the expression v m = 27tR f n = C f n (7) RULE. To determine the velocity, in feet per min- ute, with which a pulley rotates, multiply the circum- ference of the pulley, in feet, by the number of revolu- tions per minute. If R and C denote respectively the radius and cir- cumference- of the pulley in inches, we shall have, be- r> tween R and R f , C and C f the relations R f = -- and c C f = . These values, substituted in formula (7), give 27tRn Cn = -^ = 0.5236^. ... (8) RULE. To determine the velocity of a pulley in feet per minute, multiply the circumference of the pulley in inches by the number of revolutions per minute, and divide the product by 12, or multiply 0.5236 times the radius in inches by the number of revolutions per minute. Let v represent the velocity of the pulley in feet per second ; we shall then have the expression 7; = 6ov l6 BELTS AND PULLEYS. and formula (8) becomes, by substitution, 277 Rn Cn 6 -- 2~ --7t> which reduces to v = = 0.00873^. . . . (9) RULE. To determine the velocity of a pulley in feet per second, multiply the circumference of the pul- ley in inches by the number of revolutions per minute, and divide the product by 720, or multiply 0.00873 times the radius in inches by the number of revolutions per minute. If we substitute the value v m = 6ov in formula (7), we shall obtain the expression 6ov = which reduces to C ' n v = -~- = o.iotfR f n. . . . (10) RULE. To determine the velocity of a pulley in feet per second, multiply by the circumference of the pulley in feet by the number of revolutions per minute, and divide the product by 60; or multiply 0.1047 times the radius in feet by the number of re- volutions per minute. FUNDAMENTAL PRINCIPLES. IJ By transposing formula (7), we may obtain, for the number of revolutions per minute, the formula v m v m n = -- -. . . . . (II) 2nR f C f RULE. To determine the number of revolutions per minute with which a pulley turns, divide the velocity of the pulley in feet per minute by the pulley-circum- ference in feet. In a similar manner, by transposing formulas (8), (9), and (10) we may obtain the following formulas for the- number of revolutions per minute : n = --^ = ^ (12) C 0.5236^ RULE. To determine the number of revolutions per minute, divide 12 times the velocity in feet per minute by the circumference of the pulley in inches ', or divide the velocity in feet per minute by 0.5236 times the radius of the pulley in inches. J2OV V /M _ ' _ I T O \ fi 7^ ':: 7^. . I I s I C 0.008737? RULE. To determine the number of revolutions per minute, divide 720 times the velocity in feet per second by the circumference of the pulley in inches, or divide the velocity in feet per second by 0.00873 times the radius of the pulley in inches. , n ~ -FT- ^- ..... (14) C f 18 BELTS AND PULLEYS. RULE. To determine the number of revolutions per minute, divide 60 times the velocity in feet per second by the circumference of the pulley in feet, or divide the velocity in feet per second by 0.1047 times the radius of the pulley in feet. The numbers of revolutions per minute of two or more pulleys, which are fixed upon one and the same shaft, must plainly be the same, for the shaft at each revolution will carry each and all of the pulleys just once around without reference to the diameters of the pulleys. If, therefore, we denote by n the common number of revolutions, and by v and v f the circum- ferential velocities of two pulleys, which are fixed upon one and the same shaft, we shall have, from formula (9), the equations Cn v = - - 0.008 7 T>Rn y 720 J~>f and v' -- = o.oo873^X C, R, C', and R f denoting respectively the circumfer- ences and radii of the two pulleys. From these two equations we may write the proportion /~* /~*t v : v f :: : - - :: 0.00873^/2 : 0.00873^^. 720 720 By cancelling out the equivalent quantities, and writ- ing the proportion in the form of an equation, we have C FUNDAMENTAL PItlNCIPLES. IQ -The ratio of the velocities of two pulleys which are fixed upon one and the same shaft is equal to the direct ratio of the pulley circumferences, radii, or diameters. (e) Power. By the power of a pulley we mean the force with which the circumference of the pulley turns: it is equal to that force which, if applied to the pulley- circumference in a direction opposite to that in which the pulley rotates, would be just sufficient to stop the FIG. 6. motion of the pulley. The powers of two pulleys which are connected by one and the same belt are equal ; for the driving-pulley transmits all its circum- ferential force to the belt, and the belt in turn trans mits the same force to the driven pulley (less a very slight amount which is consumed in the stretching of the belt). Let the circles of which the radii are R, R , r, and A (Fig. 6) represent four pulleys, connected by belts as shown in the figure, A being the driving-pulley and R 20 BELTS AND PULLEYS. and r being fixed upon one and the same shaft. The power P of the driving-pulley is transmitted directly to the pulley R through the belt xy. We may consider the imaginary line abc as a simple lever, the fulcrum of which is at the point a, and the arms of which are ac and ab. If now we let P represent the power of the pulley r, which is transmitted directly to the pulley R! through the belt x 'y' , we shall have, from the principles of the simple lever, the relation or PR = Pr, P r_ P ~~~ ~R (16) RULE. The ratio of the powers of two pulleys which are fixed upon one and the same shaft is equal FIG. 7. to the inverse ratio of the pulley-radii (diameters or circumferences). Let the circles of Fig. 7 represent a number of pulleys, connected by belts as shown in the figure, and together constituting a " pulley-train." Let A be the driving-pulley, and let r' f be arranged to lift the weight FUNDAMENTAL PRINCIPLES. 21 P fl by means of a cord wound around its circumfer- ence, as shown in the figure. From formula (16) we shall have the expression P r PR P = R or p = T-- Also, we shall have P. - ?L P" - FRr P' ~ R ' r' ' Substituting, in the last-found equation, the value of P determined above, gives _ PRR_ rr' From formula (16) again we may write the equation T)'f // P" R ff ' nr n'ff ____ . P" R n r" J and by substituting in this the last-found value of / >/; , we shall finally obtain the formula _ = PRR'R" rr'r" P' !f rr'r fr Then, inversely, P = R R'R fr ( l8 ) RULE. To determine the power of an increasing pulley-train (one in which the powers of the pulleys . 22 BELTS AND PULLEYS. constantly increase from the driver), multiply the power of the driver by the continued product of all the larger pulley-radii (diameters or circumferences) except that of the driver, and divide the result by the continued product of all the smaller pulley-radii (diameters or circumferences) except that of the driver. To determine the power of a decreasing pulley- train (one in which the powers of the pulleys con- stantly decrease from the driver),* multiply the power of the driver by the continued product of all the smaller pulley-radii (diameters or circumferences) ex- cept that of the driver, and divide the result by the continued product of all the larger pulley-radii (diame- ters or circumferences) except that of the driver. From formula (15) we know that the circumferential velocities of two pulleys which are fixed upon one and the same shaft vary directly as the pulley radii, diameters, or circumferences. We may therefore ob- tain, by combining formulas (15) and (16) and denoting the circumferential velocities of the pulleys R and r (Fig. 6) oy Kand v respectively, ('9) RULE. The ratio of the powers of two pulleys which are fixed upon one and the same shaft is equal * If the pulley-train represented in Fig. 7 were a decreasing in- stead of an increasing train, the "direction" of the train would be reversed. That is, the pulley /v>" would be the driver and the pulley A the one which lifts the weight. FUNDAMENTAL PRINCIPLES. 23 to the inverse ratio of the circumferential velocities of the pulleys. A glance at formula (19) will show that the increased power which we obtain by means of an increasing pul- ley-train necessitates a loss of time corresponding to the gain in power. For since the power varies in- versely as the velocity, if we increase the power two, three, or four fold we necessarily decrease the velocity two, three, or four fold also. Thus, if by means of the train represented in Fig. 7 we can lift a weight of 1000 pounds with a circumferential force on the driving- pulley amounting to say 200 pounds only, we will need just ^y = 5 times as much time as if we apply the force of 1000 pounds directly to the pulley which lifts the weight. Nevertheless there is a real gain repre- sented in the increasing pulley-train ; because, without it or a similar contrivance, we might tug, with our 200 pounds of power, for a lifetime, and still be unable to lift the 1000 pound weight one inch from its resting- place. (/") Horse-power. The term " horse-power," as com- monly used, is equivalent to 33,000 foot-pounds: it is that amount of force or power which will lift a weight of 33,000 pounds one foot high in one minute, or a weight of one pound 33,000 feet high in one minute. If we represent the horse-power of a pulley by 77, and the circumferential force or power in pounds byP, then H X 33,000 pounds lifted one foot high per minute will represent the power of the pulley. If therefore we denote by v m the circumferential velocity of the pulley in feet/^r minute, w r e shall have, for the power in pounds, the expression 24 BELTS AND PULLEYS. * And inversely, H = -^-. (21) 33000 RULE. To determine the power of a pulley in pounds, divide 33000 times the horse-power by the cir- cumferential velocity of the pulley in feet per minute : to determine the horse-power, multiply the power of the pulley in pounds by the circumferential velocity in feetjfor minute and divide the product by 33000. If v denote the circumferential velocity of the pul- ley in feet per second, we shall have the relation v m = 6oz>, and formula (20) becomes, by substitution, p= 3300Q//" 6ov ' P=. ...... V Pv Inversely, #==- ........ (23) RULE. To determine the power of a pulley in pounds, divide 550 times the horse-power by the cir- cumferential velocity in feet per second ; to determine the horse-power, multiply the power of the pulley in pounds by the circumferential velocity in feet per second, and divide the product by 550. The size of a pulley is usually given in terms of its diameter: thus a "36-inch pulley" is a pulley the FUNDAMENTAL PRINCIPLES. 2$ diameter of which is 36 inches; a "4-foot pulley" is one the diameter of which is 4 feet. Example i. The diameter of a pulley is 10 inches; it is required to find the circumference. From formula, (i) we have C = nD = 3.14159 X 10 or C = 31.4159". Also we have R -- = 5", and formula (2) gives 2 C = 2?tR = 6.28318 X 5 or C 31.4159". Example 2. The circumference of a pulley is C = 314.159"; it is required to find the diameter. We have, from formula (3), D = ~ = m59 = 100". n 3.14159 Example 3. The diameters of two pulleys, which are connected by one and the same belt, are D = 30" and D r = 10" ; the larger pulley makes n =. 120 revo- lutions per minute. It is required to determine the number of revolutions per minute of the smaller pul- n D' 120 10 ley. rrom formula (6) we have = -^ or r = n D n 30 30 X 120 rrom this, n = 360. 10 Example 4. A pulley, the radius of which is 2 feet, makes 100 revolutions per minute; it is required to de- termine the circumferential velocity in feet per minute. We have, from formula (7), v m = 27tR f n, or v m 6.28318 X 2 X 100 = 1256.6. Example 5. The radius of a pulley is 24 inches, and the number of revolutions per minute 100; it is re- quired to determine the circumferential velocity of the pulley in feet per minute. From formula (8) we have 2nRn 6.28318 X 24 X loo v m - or v m -- = 1256.6. 12 12 26 BELTS AND PULLEYS. Example 6. The radius of a pulley is 24 inches and the number of revolutions per minute loo ; it is re- quired to determine the circumferential velocity of the pulley in feet per second. Formula (9) becomes, by sub- stituting the numerical data, v = 0.00873 X 24 X 100, or v 20.95. Example 7. The circumferential velocity of a pul- ley is 1256.6 feet per minute, and the radius 2 feet ; it is required to find the number of revolutions per v minute. From formula (11) we have n ^ = 1256.6 6.28318 X 2 = Example 8. It is required to determine the number of revolutions per minute of a pulley of which the radius is 24", and the circumferential velocity, in feet per second. 20.95. From formula (13) we have v ' _ 20.95 ' 0.00873^ ~ 0.00873 X 24 = Example 9. A shaft which makes 100 revolutions per minute bears two pulleys of which the radii are R = 36 inches and R' = 24 inches ; it is required to determine the circumferential velocities of the two pulleys in feet/^r second. From formula (9) we have, for the circumferential velocity of the pulley R ', v' = o 00873 X 24 X IOO = 20.95 feet per second, and from v 36 20.95 X 36 formula (lO we have - - = , or v = = 20.95 24 24 31.425 test per second. Example 10. In an increasing pulley-train we have the following data: Power of the driving-pulley P=. IOO pounds, radii of the pulleys (of which there are six FUNDAMENTAL PRINCIPLES. 2/ besides the driver, and arranged as shown in Fig. 7), R : = R' = R" = 36" and r = V = r",= 12" ; it is required to determine the power of the pulley-train. By substituting the above values in formula (17) we 100 X 3 6 X 3^ X 36 obtain P" = - - = 2700 pounds. 12 X 12 X 12 Example n. Suppose the circumferential velocity of the driving-pulley in Example 10 is 1200 feet per minute ; it is required to determine the circumferential velocity of the pulley r". From formula (19) we TOO From this, ^ = 1200 X TOO - = 44.44 ieet /> which is very frequently seen in practice. i js FIG. 9. This disposition allows us to dispense with all ex- terior guides, if we are careful to place the pulleys in such a manner that the line of intersection of their middle planes shall be tangent to the circles contained in TRANSMISSIONS BY BELTS WITHOUT GUIDES. 31 these planes at the points in zvhlch tJie belt leaves the pulleys. In Fig. 9, in which a and b l are these points, the belt must run in the direction indicated by the arrows. If we wish to run the belt in a contrary direc- tion it is necessary to move the pulleys upon their arbors until the line of intersection of their middle- planes becomes tangent to the circles at the points a l and b. This condition is fulfilled when, with reference to the crossing K of the pulley-axes, the new positions occupied by the pulleys are found to be symmetrical with the positions of the pulleys before the change. The transmission represented in Fig. 9 may be con- sidered as the general solution of transmissions by belts without guides. It gives, in fact, the transmission by open belt, when the angle /3 included between the middle planes of the pulleys is equal to o, and the transmission by crossed belt when this angle is equal to 1 80. In all intermediate positions the belt is only partially crossed : for /3 = 90, we have a half-crossed belt, for fi = 45 a crossing of one fourth, etc. In short, partially crossed belts, the tendency to run off the pulleys is very great. According to Redten- bacher, in order that this accident may be avoided, the distance between the centres of the pulleys should not be less than twice the diameter of the largest pulley ; that is, the angle of deviation of the belt should not exceed 25. Moreover, in order that the wear of the belt may not be excessive, the distance between the centres of the pulleys should not be less than 10 VbD, b representing the width of the belt and D the diameter of the driving-pulley. It is evident that, in each particular case, it is advantageous to take, for 3 2 BELTS AND PULLEYS. the separation of the pulleys, the greater of these two values. 5. Transmissions by Belts with Pulley-Guides. RULE. In a transmission by belt with pulley-guides, in order that the belt may run properly upon the pulleys and pulley-guides, the point in which the belt leaves each pulley must be the point of tangency be- FlG. FIG. ii. tween the pulley and the line of intersection of its middle plane with that of the following pulley. Figs. 10 and II represent transmissions of this kind for pulleys with parallel axes. In Fig. 10 the middle planes of the two pulley-guides are tangent to the two pulleys of transmission A and B, and their common diameter is equal to the distance between the middle TRANSMISSIONS BY BELTS WITH GUIDES. 33 planes of these pulleys. This disposition of pulleys permits of the movement of the belt in either direction. When, as is most commonly the case, a movement of the belt in one direction is sufficient, we may make use of the simpler disposition of pulleys represented in Fig. II, in which the axes of the pulley-guides coincide geometrically. A and B are the pulleys of transmis- FlG. 12. sion ; the middle planes of the pulley-guides are par- allel, and are tangent respectively to the pulleys A and B at the points in which the belt leaves the latter pulleys. The common diameter of the pulley-guides is equal to the distance between the middle planes/ of the pulleys of transmission. As indicated in the figure, the pulleys of transmission A and B rotate in opposite directions. 34 BELTS AND PULLEYS. If we consider B as a pulley-guide (in which case it may run loose upon the arbor of A), the two pulleys 7 and D may be taken as pulleys of transmission, and fixed upon two separate arbors, the directions of which are the same. If the pulley-guides C and D are placed between the arbors of A and B, as is indicated in Fig. 12, they will rotate in the same direction, and may consequently be FIG. 13. fixed upon one and the same arbor. The pulleys of transmission A and B will also rotate in the same direc- tion. In this case the belt can move in one direction only, and remain properly upon the pulleys and guides. The two pulley-guides C and D may be replaced by a single pulley, provided it is placed obliquely so as to run on both sides of the belt without causing displace- ment. BY BELTS WITH GUIDES- 35 Fig. 13 represents a transmission by belt for two pulleys, the axes of which intersect each other. In this disposition, which differs from that of Fig. 11 only in the inclination of the axis of the pulley B, the movement of the belt can take place only in one direc- tion. To obtain a movement in the other direction, it is necessary to move the pulley-guides along their FIG. 14. common axis until the condition necessary for main- taining the belt in position is fulfilled for this particu- lar case. It must be remembered that the two pul- ley-guides rotate in contrary directions, and therefore cannot be fixed to the arbor upon which they run. From the arrangement shown in Fig. 12, that of Fig. 14 may be devised ; this disposition corresponds to the case in which there is a very slight angle between the arbors, and the pulley-guides rotate in the same direction. BELTS AND PULLEYS. The disposition represented in Fig. 15 is still more simple, and may be used for a greater angle between the axes as great as 25. FIG. 1 6. Half-crossed belt ivith pulley-guide. In this case the re- lative positions of the pulleys of transmission are such that the dis- position represented in Fig. 9 could be used, except that the separation of the pulleys is too slight, and the belt would there- fore tend to run off. To deter- mine the arrangement of the belt, we begin by giving to the part 55 the direction of the line of intersection of the middle planes of the pulleys A and B ; then from the point c, chosen arbitrarily upon the line 55, we draw, to the circumferences of the pulleys, the tangent lines ca and FIG. 15. FIG cb. The plane of these tangents determines the middle plane of the pulley-guide C, to which the lines are also tangents. Rotation may take place equally well in either direction. Because of the cramped position of TRANSMISSIONS BY BELTS WITH GUIDES. 3/ the pulleys and the consequent difficulty in placing the arbor of the pulley-guide in proper position, this arrangement is very rarely seen in practice. FIG. 17. Another disposition for transmission by half- crossed belt with pulley-guide. In this figure the pulleys of transmission are so placed that the line of intersec- tion 55 of their middle planes is the common tangent to the circles contained in the planes, and the middle plane of the pulley-guide C coincides with that of the FIG. ,7. pulley of transmission A. The portion of belt which leaves the pulley A is inclined (as shown in the figure) as in the crossed belt in order that it may properly roll upon the pulley B, while the portion which leaves the pulley .# is guided by the pulley-guide C. The pulley- guide is in contact with the line of intersection 55, and with a tangent to the circle A drawn from an arbitrary point upon the line 55. In this disposition the direc- tion of rotation must be as indicated in the figure, 3 BELTS AND PULLEYS. This mode of transmission is very convenient when we wish to drive a series of vertical arbors from one hori- zontal shaft ; it also finds frequent employment in mills for grinding various materials, and when the separa- tion of the pulleys of transmission is necessarily slight. FlG. 1 8. Half-crossed belt with movable pulley-guide. In this disposition, which is used for a greater separa- tion of the pulleys of transmission than in that of Fig. 17, we may, by moving the pulley-guide from the posi- FlG. I tion C to the position C Q (shown by the dotted lines), cause the belt to pass from the fixed pulley B to the idle pulley B^. in a similar manner, the pulley-guide may be used for running the belt off the pulleys en- tirely. The position C Q should be so chosen that the tensions upon the belt for the two positions will be the same or slightly less for C Q than for C. General case of crossed arbors. When the pulleys of transmission cannot be so placed that the line of inter- 7'RANSMISSIONS BY BELTS WITH GUIDES. 39 section of their middle planes is a common tangent to the circles contained in the planes, it becomes necessary to make use of two pulley-guides. Fig. 19 represents an arrangement which may be adopted in such cases, and which may be regarded as the general solution of the problem of transmission by belts with pulley- FIG. 19. FIG. 20. guides. Fig. 20 represents a special application for the case in which the line of intersection 5S of the middle planes passes through the centre of the middle circle of one of the pulleys of transmission ; in this figure the axis of the pulley B is supposed to be situ- ated in a plane parallel to the pulley A. After having obtained the line of intersection 55, we choose upon 40 BELTS AND PULLEYS. it two arbitrary points c and , through which we draw, to the middle circles of the pulleys of transmission, the tangent lines ca, cb, ,# and cjb^. The planes cab and c \ a J ) \ which are thus determined are those of the two pulley-guides, which should be placed respectively in contact with the above-named tangent lines. With Si FIG. 21. this disposition, rotation may take place equally well in either direction. The mode of transmission represented in Fig. 19 may be simplified by giving to the axes of the two pulley-guides a common direction mm parallel to the two pulleys of transmission (Fig. 21). In this figure SS represents the intersection of the middle planes of the TRANSMISSIONS BY BELTS WITH GUIDES. 4! two pulleys of transmission, ac and & 1 c 1 the intersec- tions of planes perpendicular to 55 with the middle planes of the pulleys of transmission A and B respec- tively. In the perpendicular planes, tangentially to the right lines ac and bj^ we place the two pulley-guides C and C r The arrows indicate the directions of rota- tion ; to obtain a movement of the belt in a direction contrary to the one indicated, it is necessary to give to FIG. 22. the pulley-guides C and C l the positions indicated at C and C\ by the dotted lines. It may be remarked here that the belt, instead of passing from c to a and from c l to #,, may be made to pass from c to a l and from c l to a, which causes a change in the direction of rotation. The pulley-guides, instead of being horizontal, as in the figure, may be placed vertically that is, respectively in the planes of the pulleys of transmission A and B ; in this case, how- BELTS AND PULLEYS. ever, it becomes necessary to take account of the angle of deviation (see 4). When the pulleys of transmission can be so placed that the intersection 55 of their middle planes is tan- gent to one of the pulleys, and the distance between the parallel planes containing the axes of the pulleys A and B is sufficient, we may substitute, for the dis- position shown in Fig. 20, the one represented in Fig. FIG. 23. 22. This arrangement is often seen in practice ; the axes of the pulley-guides are parallel to that of the pulley of transmission A. The middle planes of the pulleys A and B may make any desired angle with each other. If the distance AC is great compared with the width of the belt, the pulley-guides, instead of being the one above the other, may be placed upon the same axis, as shown in Fig. 23. If the distance between B and C is sufficiently great, the arbor B may be provided with two pulleys, one fixed and the other idle. TRANSMISSIONS BY BELTS WITH GUIDES. 43 When, on account of lack of space, it is impossible to make use of one of the dispositions which we have de- scribed above, we ought to seek at least to place the axes of the pulley-guides in the middle plane of one of the principal pulleys and the pulley-guides themselves parallel to each other, as, for example, in Fig. 24. In this case we first draw the tangent line ab\ then in a plane drawn through this line normally to the plane of FIG. 25. the figure we place the pulley-guide C in such a man- ner that it is tangent at the point a to the line of in- tersection of the middle planes of the pulleys A and C. Through the point a^ we then draw the line a^ paral- lel to ab, and in a plane drawn through this line parallel to the plane of the pulley-guide C, we place the second pulley-guide tangent to the intersection of the middle planes of the pulleys ^4 and C l and to the middle plane of the pulley^. In this manner the axes mm and injn^^ 44 BELTS AND PULLEYS. of the pulley-guides are found parallel to each other, and also situated in a plane parallel to that of the pulley B. By making the belt of Fig. 23 pass over a fourth pulley we may obtain an arrangement by which we may drive two pulleys B and C by means of a single driving-pulley A. Fig. 25 represents a disposition of this kind much used in spinning-mills. The arbors B and Care in dif- ferent stories of the building, and each bears two pul- leys, one fixed and the other loose; we use, in this case, the permissible deviation of the belt from its exact position mentioned in 3. Fig. 26 represents another mode of transmission by belt, in which the two parallel arbors B and C are driven by a single pulley A. The axes of these arbors are both perpendicular to that of the arbor A ; the first intersects it, while the second crosses it without inter- secting. In the machinery of spinning-mills a great number of transmissions are found in which three, four, or even a greater number of pulleys are driven by means of a single driver. It may be remarked here, that in all cases of transmission by leather belt in which pulley-guides are used which are in contact with LENGTH OF BELTS. 45 the upper surface of the belt, it is advantageous to place the belt so that the contact of the pulleys is always upon the same surface the flesh or wrinkled side. 6. Length of Belts. It is often necessary in practice to calculate the proper length of a belt for a given separation of the axes of the pulleys upon which the belt is to run and for known pulley radii or diameters. Thus when we have two pulleys, the bearings and positions of which FIG. 27. are already fixed, if we can determine the proper length for the belt, we can save time and prevent waste of belt in cutting too long or too short. Open Belt. Let us denote by L the total length of the required belt; by L l the distance between the centres of the two pulleys upon which the belt is to run ; by R the radius of the larger pulley, and by r that of the smaller. Let Fig. 27 represent the pulleys con- 46 BELTS AND PULLEYS. nected by an open belt. In the figure the lines ob and o'c are parallel, because each is perpendicular to the line be ; hence the angles xob and yoc are equal. Let us denote each of these angles by (p. It is evident from the figure that the total length of the belt must be L = 2(bc -f- arc ab -f- arc cd\ Draw the line ck parallel to oo' \ we shall have ck = L v because ob and o'c are parallel. In the triangle bkc, in which the angle kbc is a right angle, we shall have be = cl- b& or be = Ll -- 6Jf- But ok = o'c r and bk = ob ok = R r\ hence be = VL* -(R- r)\ The arc ab is equal to the arc ax -f- the arc xb\ arc 27tR n Rep ax - - i.57, an d arc 4ri = - - = 4 ibo Therefore arc ab i.tfR + 0.0175^ = (1.57 + 0. Also the arc a/ is equal to the arc dy the arc yc\ 2nr arc ay = -- = \.$jr and arc yc = ^ = 0.017 $rq>. Hence we shall have arc cd = i.$/r 0.017 $r

= I 4 -<== 14-5 Formula (27) therefore gives for the proper length of the belt L 2[|/I20* (20 + I0) a + I.57( 2 + 10) + 0.0175 X 145 (20+ 10)] = 2(ll or L = 341.80" = 28J feet. In transmissions by belts with pulley-guides, and in all cases where the intricate arrangement of the belt renders arithmetical calculation long and tedious, the proper length of the belt may be determined more easily and with sufficient exactness graphically, by measurement with the rule. To illustrate: Suppose we have an arrangement of pulleys such as is represented in Fig. 12, which figure is a sketch (containing two projections) of the transmission, drawn to a scale of ^. SPEED-CONES. 5 1 The separation of the pulleys A and B is 5 feet = 60", and the diameters respectively 2 \" and 13". From the figure it is evident that the total length of the belt is L = arc xy+yD + Dx' + arc x'y' + y'C + Cx. By measuring with the compasses the above arcs, we find xy = 34i" and x'y' = 20". The line yD in the left- hand projection is given in its true length by the line ND in the right-hand figure ; hence, by measuring ND, we obtain for the true length yD 30". The distance Dx' is given in its true length in the left-hand figure, and therefore, by direct measurement, we ob- tain Dx' = 34 /x . In a similar manner we obtain by measuring KC the true length y' C = 36", and by di- rect measurement, Cx = 26". We have consequently L = 34* + 30 + 34 + 20 + 36 + 26 = 1 80*" =15' ". In a similar manner in Fig. 18, by measuring the arcs xy and x'y' ^ and the length, NX, Ky ', and yz, we may obtain the length necessary for a belt which will properly run on the pulleys represented in the figure. 7. Speed-cones. The contrivance known to mechanics as " speed- cones" consists of two stepped pulleys arranged as shown in Fig. 29. The object of speed-cones is to ob- tain different speeds for the driven arbor from the con- stant speed of the driving-shaft. To illustrate : Suppose in Fig. 29 we assume between the radii of the pulleys the relations R = 37-, R' r' , and R' = ^r" . We have seen from formula (6) that the ratio of the revolutions of two pulleys which are connected by one and the same belt is equal to the inverse ratio of the pulley- 52 BELTS AND PULLEYS. radii. Hence, if we assume that the driving-shaft xy makes 100 revolutions per minute (N= 100) when the belt is on the pulleys R and r, we shall have for the revolutions of r (and consequently of the shaft x'y r ) r> n N 100 X 3 = 300. When the belt is on the pulleys R and r' we shall have for the revolutions of R' T* (and consequently of the shaft x'y f ) n' = N f = 100 X I = IOO. Similarly, when the belt is on the pulleys X~ L, FIG 29. R" and r" we have for the revolutions of R" and the r" shaft x'y' N" = N ~^r = 100 X \ = 33f Such differ- ent speeds for the driven arbor are necessary in many machine-tools, as the lathe, drill, etc., because the speed of the mandrel and spindle must vary with the SPEED-CONES. S3 material to be worked and with character of the work to be done. Open Belt. From formula (25A) we have for the lenth of the belt 1.572 +- in which 2 = R -f- r and A = R r (see Fig. 29). Since now the length of the belt must be the same for each pair of pulleys in the cone, we shall have 2 ( ^Z7= in which 2' = R + r' and A' = R r - r f . By means of the binomial formula we may extract the square roots of the quantities under the radical signs as follows : and J' 2 J' 4 J /B But since Z t is usually very large compared with //, z/ 4 /I 6 ^y-j and ,-j 6 are very small quantities, and may with- 54 BELTS AND PULLEYS. out serious error be neglected. Similarly, we may A" A r * neglect the quantities ^r- 3 and 5 . Hence we shall have which reduces to If we represent by JVthe constant number of revolu- tions per minute of the driving-shaft (corresponding to R), and by n the number of revolutions per minute of the driven shaft when the belt is on the pulley r, we shall have, from formula (6), R n n _ = -, or **?r^ R n' n' , ri Also -^ = w , = ^, or R=*^ in which ;^ r represents the revolutions per minute of the driven shaft when the belt is on the pulleys^" and r' , and N' the revolutions per minute of the driving- shaft. which being constant is equal to N. Hence we shall have SPEED-CONES. 55 which substituted in formula (28) gives We shall also have (as above for the quantity 2') 4> = R' - r' = r f - - /, or 4 = r'(-jf -- ij (30) Example I. Suppose we have two shafts, the dis- tance between which is L l = 100": the revolutions per minute of the driving-shaft is N= 100, and we wish to construct a pair of speed-cones such that the revolutions per minute of the driven shaft correspond- ing to the pulleys r, r ', r" , and R" shall be ;/ == 300, ri = 200, n" 100, and N'" 50. From formula (6) we shall have n 300 -\ T 1 1 O1* 1\. 3 ?*. r N 100 We may choose any convenient value for r, and find from the above expression the corresponding value of R. Suppose we take r = 4" ; hence R = ^r = 3 X 4 12". Then 2 = 12 -f 4 = 16 and J = 124 = 8. From formula (30) we shall have A' = r'(-- Vioo 5 6 BELTS AND PULLEYS. and formula (29) becomes .(200 \ 64 r'* ,-64 r'\ r'[ --- \- i = 16 + ,pr3r'= 16 + - \ioo / r 3. 14x100' 314 From this by reducing we shall have r' 2 + 942?-' = 5024 + 64. Adding ( -- j = 47i 2 to each side of this equation gives r" + 942r'+ 221841 = 5024 + 64-)- 221841 = 226929. Extracting the square root of this expression gives r 1 + 471 = ^226929. From this r 7 = 1/226929 471, or ' r' = 476.38 471 = 5.38". Then -pr rrr ^r = -, Or ^ = 2/ = 10.76^. In the same manner for the pulleys T?" and r" we shall have from formula (30) and formula (29) becomes 2r" = 16 -1 = 16.204, 100 ' / 314 or Also ^ = 3 , or /? // = r // = 8.i02 // . r 100 SPEED-CONES. For the pulleys r" and R f " we shall have 57 Hence formula (29) gives y>" = 16 + ^^ , 942^ - 5024 + 64 - r"' 2 . Hence r ///2 + 942^" = 5088. As before, adding (~") to each side, and extracting roots, we shall have r ! " = 1/5088 + 221841 471 = 5.38". Then * = . /== ! = 2f or jr=2r"'=io76". Crossed Belt. The calculation of the radii of the speed-cone pulleys becomes very much simpler when 58 BELTS AND PULLEYS. crossed belts are used. If, in Fig. 30, we assume the relations 2 = R + r = R f + r' = R" + r" , etc., we shall have for the corresponding angles, cp, g>', g>", etc.; R + r 2 . , R+r' 2 . sin cp = j -- = y-, sin cp = -- j --- = sin f/ , etc. r, L^ The conditions that the length of the belt must be the same for each pair of pulleys, and that the belt must bear the same tension for each pair of pulleys, will therefore be fulfilled if we take the sums of the radii of each pair of pulleys equal to each other. Or, which is the same thing, we shall have R' = ^-r r ....... (31) Letting R -f- r f = 2', we shall have from above 2 =2'. From formula (6) we may write R n n - = Tr or K r^~r, r N N R n f n f ,ri ~'='=- =r * Hence 2 = = r(~ + i), SPEED-CONES. 59 '+= + Example 2. Taking the data of Example i, it is required to calculate the radii of the speed-cone pul- leys for crossed belt. We obtain, as in Example i, R 12", r 4", and 2 = 16". From formula (32) we shall then have OO\ 4 = 4 x 3 = 5>33 ' Formula (31) then gives R' = 16 - 5.33 = 10.67". For the third pair of pulleys formula (32) gives _,, _ J n + N \ _ (300+ ioo\ _ _ r U" + JW 4 \ioo + ioo/ and from formula (31) we shall have For the fourth pair of pulleys from formula (32) we shall have Formula (31) then gives r x// = ^ - R'" = 16 - 10.67 = 5.33^. 60 BELTS AND PULLEYS. Suppose now that we wish to add to the speed-cones another pair of pulleys (R iv and r iv ) having such radii that the number of revolutions per minute of the driven shaft, when they are in use, shall be JV iv = 33^. We shall have from formula (32) 300 + ioo\ =4x3- 12", and from formula (31) We have now two speed-cones, which are made up of pulleys as follows : First Cone. Second Cone. R = 1.2" r = 4" R' = 10.67" r' = 5.33" R" = 8" r" = 8 /r r'" = 5.33^ R"' = 10.67" r = 4" ^ iv = I2 7/ A glance at this table will show that the two cones are similar and equal, but so placed on their shafts that they taper in opposite directions. We may there- fore write the following: Rule for Speed-cones, Crossed Belt. Use two equal and similar stepped cones tapering in opposite direc- tions. Mr. C. A. Smith, in the American Machinist, Feb- ruary 25, 1882, gives a very neat graphical method for determining the radii of speed-cone pulleys for open belt, as follows: Lay off (Fig. 31) AB equal to the SPEED-CONES. Vc,, given distance between the two shafts (AB -=. Z,), drawn to any convenient scale. Strike the circles repre- senting the pulleys R and r (the radii of which are deter- mined, as in Examples I and 2 of this section, from the given revolution-ratio -^.j, and draw the portion of belt ab. Lay off (from the smaller pulley-centre) BC = AB X 0.496 0.496^, and erect the perpendicular CD = - >. Then from D as a centre strike the cir- 3.1416 cle x tangent to ab. Divide AB = L l into as many FIG. 31. equal parts as the shaft B is to revolve, less one, while the shaft A makes one revolution, when the belt is on the required pulleys R' and r '. Lay off, from the cen- tre of the smaller pulley, BO equal to one of these parts (BO = L 1 -r- -^ i), and from o draw the line oa' tangent to the circle x. The circles drawn from B and A as centres and tangent to oa' give the required radii r' and R'. When we wish to have the revolutions 62 BELTS AND PULLEYS. of the driven shaft B less than those of the driving- shaft A, or when the smaller pulley is to be on the shaft A, we lay off (for r'" and R' f/ ) the distance Ao' = N L l -r- T^777 I, draw o'b' tangent to the circle x, and the circles r" and R'" give the required radii. Crossed belts are not so often used for speed-cones as open belts, and the speed-cones for the former are so easily calculated from formula (32), that it is un- necessary to give graphical methods for determining the radii. Continuous Speed-cones. Sometimes (especially in cotton machinery and in machines requiring gradu- ally increasing or decreas- ing speeds for the driven arbors) continuous speed- cones are used instead of the stepped speed-cones already described. It may, however, be remarked that in ordinary shop machin- ery, such as lathes, planers, drills, etc., etc., continuous speed-cones are very rarely seen. To construct a pair of continuous speed-cones for open belt we may proceed as follows : Having given several of the different numbers of revolutions re- quired of the driven shaft (for example, n 300, #'= , FIG. 32. SPEED-CONES. ri f = 100, N"' , N iv = 50, and the revolutions of the driving shaft being 'N= 100), lay off (Fig. 32) ab == a!b f = the width of the belt -f- the proper clear- ance X the number of changes in the speed of the driven shaft: in this case there are five changes. Then calculate, from formulas (29) and (30), the radii R, r, R ', r f ', r iv , and K lv , corresponding to the known FIG. 33. FIG. numbers of revolutions, and draw the pulleys of which R, r, etc., are the radii, and which are represented by the dotted rectangles in the figure. Through the cen- tres of the step-widths (x, y, z, x ', etc.) draw the curves xy%i x'y'z', and the outlines of the cones are complete. Rankine gives for continuous speed-cones for open belt the rule, " Use two equal and similar conoids taper- 6 4 BELTS AND PULLEYS. ing in opposite ways and bulging 'in the middle, accord- r _i_ r l r _ r - - 2 + t ^ 2 in to the formula r Q in which r is the radius in the middle, r, and r 2 the radii of the larger and smaller ends respectively, and c the distance between the centres of the shafts. Fig. 33 represents a pair of continuous speed-cones, open belt, calculated from this rule, taking r^ = 10", r 2 = 4", c = loo", - = 7-057", and ab = a'V = 14". To construct a pair of continuous speed-cones for FIG. 35. crossed belt, calculate from formula (32) the radii R, r, R", r", r iv , R [v (Fig. 34), and connect the centres of the step-widths by the curves xyz, x'y ' z' ', in the same manner as in Fig. 32. Or we may use two equal and similar cones tapering in opposite directions (Fig. 35). An example will best explain the mode of calcula- tion for a pair of continuous speed-cones by which we wish to obtain a given gradual change in the speed of the driven arbor. Suppose our driver makes 100 revolutions per minute, and that we wish, by slowly MATERIALS USED FOR BELTING. 65 sliding the belt along the cones, to obtain for the driven arbor a speed varying from 100 to 10 revolu- tions per minute. According to the rapidity with which we wish the changes to take place we choose the number of changes let us say in this instance 10. Of these changes, the number of revolutions per minute of the first is 100. With the 9 remaining changes we must therefore gain 100 10 = 90 revolu- tions per minute, or 10 each. The revolutions of the changes are therefore as follows: 1st, 100 ; 2d, 90; 3d, 80; 4th, 70; 5th, 60; 6th, 50; 7th,4o; 8th, 30; gth, 20 ; loth, 10. We may now calculate the diameters as for stepped cones, and by drawing curves through their face-centres obtain the outlines for the required con- tinuous cones.* 8. Materials used for Belting. Belts are most commonly made of leather, cut into strips of the required width, and riveted together at their ends to make up the required length. Strips taken from the back part of the hide, and oak or hem- lock tanned, are generally considered the best, although some kinds of patent-tanned leather are said to have greater adhesive power. Cow's hide is almost invari- ably used for the leather of belts ; the skins of horses, elephants, and other animals have, however, been util- * In designing continuous speed-cones it is always best to make the curves as gradual in taper as possible for the given changes, in order to avoid the excessive stretching and wear of the belt which would otherwise occur. 66 BELTS AND PULLEYS. ized for this purpose, in some cases with very good results. For very heavy work, belts made of two or more thicknesses of leather are used, in which case the strips are fastened together with cement or rivets, and the joints carefully " broken/' In order to gain strength and prevent stretching, leather belts are sometimes edged on the upper side with narrow strips of leather, which are riveted, laced, or cemented fast to the belts. It has also been proposed (and to our knowledge in one case at least tried) to strengthen belts by riveting along their edges thin strips of brass, steel, or other metals. Of late years vulcanized- rubber belts have been very successfully introduced in this country. They are usually made continuous, thus avoiding the use of rivets, and consist of one or more layers of cotton-duck placed between layers of vulcanized rubber, the rubber covering the edges in order to protect the seams from injury. Rubber belts are now made in widths about the same as leather; they weigh nearly the same, and are said to be equally strong and pliable. The intestines of sheep, cats, and other animals have been used to a considerable extent for belts ; they are exceedingly strong and tough, and can be obtained, it is said, thirty, or forty feet in length. Gut belts are either round, to run in grooved pulleys, or woven into flat bands for use on ordinary flat-faced pulleys. Raw- hide possesses, it is claimed, fifty per cent more strength than tanned leather ; but belts of this material, unless constantly oiled, soon become stiff and ungov- ernable, and are not to be depended upon for general purposes of transmission. Belts of hemp, flax, canvas, MATERIALS USED FOR BELTING. 6/ sheet-iron and steel, and several combinations of leather and metallic wire, have been proposed, and in some cases used ; but these at present offer no practical advantages over leather and vulcanized rubber. For all practical purposes, then, we have two kinds of belting leather and rubber, between which we may offer the following comparison : Those who favor leather belts claim that they are in the main stronger than rubber, and that they will wear much longer, especially when used for cross or half-cross pulleys ; that leather belts cease to stretch after once or twice shortened and relaced, while those of rubber do not ; and that leather will bear contact with oil and grease without harm, while rubber thus exposed will soften, and stretch out of shape. Wide leather belts can be cut up into narrow ones, while rubber belts cannot be cut without injuring the finished edges ; also, leather can be more easily repaired when injured than rubber. On the other hand, rubber belts do not need to be riveted, but are made continuous ; they do not slip so easily on the pulley-faces as leather, and are cheaper at first cost for the same sizes. It is also claimed that 'rubber belts endure exposure to cold and wet much better than leather, retain their flexibility better, and do not lose strength so rapidly from wear. Leather and vulcanized belts both are good. Thousands of each perform well their arduous duties all over the civilized world. Each has hundreds of admirers and champions. We therefore deem it best to express no preference on our own part, preferring rather to have each purchaser choose for himself, assuring him that either good leather or ijood vulcanized rubber will do 68 BELTS AND PULLEYS. his work as faithfully and well as any reasonable man should desire. 9. Lacing and other Modes of Fastening. Endless belts, of whatever material they are made, when subjected to a considerable strain for any length of time become lengthened or stretched. As a result of this lengthening, the belts hang loosely upon their pulleys, and consequently slip and slide. It is there- fore necessary to have some ready means of shortening belts to their proper lengths, and thus make them again fit tight upon the pulley-faces. This is very generally done by leaving the belt with two ends (i.e., not end- less), and then lacing together the free ends with leather thongs or cords. When a laced belt becomes stretched, it is unlaced, cut off to the proper length, and laced up again, new holes having been punched at the cut end."* Lacing-thongs are commonly made of leather or good clean rawhide, softened and stretched somewhat to render it firm and even ; they vary in width from one quarter to three quarters of an inch, and in thick- ness from one sixty-fourth to nearly one eighth of an inch, according to the width. We may say very simply, in lacing belts, punch the holes just large enough to easily admit the lacing-thong f inch to I inch from the ends of the belt (no more material than is necessary * Sometimes belts of considerable length are shortened to take up the stretch by simply running off one pulley and twisting the belts until the proper lengths are obtained. This practice is, however, a very bad one, because the twists cause the belts to become cracked and to wear out rapidly, and should never be indulged in except in cases of immediate necessity. LACING AND OTHER MODES OF FASTENING* 69 should be cut out, because this tends to weaken the belt) ; use for small belts a -J-inch thong ; for belts from 4 inches to 8 inches wide, a f-inch thong ; for belts from 8 inches to 15 inches wide, a J-inch thong; and for belts over 1 5 inches in width, a f-inch thong. The first requi- site in lacing together the free belt-ends is to have the ends square that is, at right angles with the sides of the belt ; if the ends are not square the belt will not lie straight on the pulleys, and will tend, consequently, to FIG. 36. FIG. 37. run off the pulleys, and otherwise interfere with the proper motion of the machine. The simplest mode of lacing belts, which is repre- sented in Fig. 36, consists in starting at one side, and lacing over and over through all the holes until the other side of the belt is reached. This does well enough for small belts not to be subjected to any severe strain, although even they will do more satis- BELTS AND PULLEYS. factory work if laced differently ; but for larger belts better and safer methods must be used. Fig. 37 shows a style of lacing quite common among machinists, and which combines quickness of operation with strength about as well as any of the simpler methods. Begin at the side a in the figure, and lace with both ends of the thong, as shown, fastening the ends at b in a knot or other convenient manner. A still better lacing is represented in Fig. 38. The thong is here crossed on one side of the belt only the upper side, and care should be taken not to cross unevenly the double parts on the pulley-side. In heavy-driving belts, and in all belts where the strain is severe, double rows of holes should be punched, and the joining thus rendered doubly secure against breakage. Messrs. J. B. Hoyt & Co., manu- facturers of leather belting, New York, inform me that all their belts are laced according to the double method represented in Fig. 39, in which a is the side to be placed next the pulley. This lacing has the advantage that all its parts on the outside of the belt are parallel to the di- rection of motion, and the ten- dency is therefore to keep the ends of the belt at all times in their proper positions. The above-mentioned gentlemen, after many years of experience with leather belting, have come to believe this method the best in ordinary use. FIG. 38. LACING AND OTHER MODES OF FASTENING- 7 1 An excellent style of lacing for large belts is given by Mr. John W. Cooper in his " Use of Belting/' which FIG. 39. we represent in Fig. 40. Begin with one end of the lacing-thong at the point a, and lace successively through the holes i, 2, 3, 4, 5, and so on, all around the rows of holes until the point a is again reached, where the thong is fastened off as ,shown in the figure. Although in this case the parts of the thong are not parallel to the direction of mo- tion, yet they are so slanted on the pulley-side in one direction and on the outside equally in the other that the result is practically the same, and the lacing is, beyond doubt, one of the best in existence. Several kinds of metallic belt- hooks or fasteners have been from FIG. 40. time to time con- BELTS AND PULLEYS. trived and introduced never, however, to our knowl- edge, with any great degree of success. For small belts the best of these hooks do well enough, and lessen the work of relacing and shortening ; but large driving- belts, and those used to transmit large powers, must, for good results, be strongly laced by one of the methods already given, or an equally good one. Among the d (MM FIG. 41. various metallic belt-hooks we may give the following as probably the best in use : Fig. 41 represents a kind of belt-hook which is quite extensively used for light belts. Figure a is the hook itself. To fasten, proceed as follows : Cut slits in the belt-ends parallel in length to the length of the belt ; place the ends as shown in Fig. b ; force through the slits the belt-hooks as in the figure, turn them, and flatten out the belt as in figure c. LACING AND OTHER MODES OF FASTENING. 73 Figure d represents the pulley-side of the belt and figure c the outside. In Fig. 42 the hook (figure a) has a double hold on the belt through the two rows of holes, and is there- fore a stronger fastener than the preceding hook. Figure b represents the outside of a belt fastened with n n n n n FIG. 42. these hooks, figure c the pulley-side, and figure d a section through the two ends of the belt showing one hook. An ingenious buckle for fastening together the belt- ends is given in Mr. Cooper's " Use of Belting," and credited to a Canadian inventor. The fastener consists 74 BELTS AND PULLEYS. of two separate parts, one containing a series of parallel metallic tongues (represented by the dotted lines in figure 43 a) which are inserted through holes in the belt-ends, and the other a rectangular cover which is slipped over the projecting ends of the tongues after they have been forced through the belt. Figure 43 a represents the outside and figure b the pulley-side of the belt. Figure c is a sectional drawing showing a pair of tongues and the cover. a FIG. 43. All belt-hooks and metallic fasteners used for belts to be run over pulleys should be of brass, copper, or other soft metal, in order to prevent scratching the surface of the pulley, and the consequent additional wear of the whole belt. A very simple, if not very firm and secure, method of fastening, without the use of lacing thongs or hooks of any kind, is shown in Fig. 44. One end of the belt STRENGTH OF LEATHER BELTS. 75 is cut into cleat-shaped pieces, shown in figure b at y, y-> y> an d the other punched with oblong slots, figure a, x, x, x. The cleats are forced through the slots, the belt-ends hammered out flat, and the joining is complete. Figure c shows a section through the ends of the belt, FIG. 44. with the cleat and slot fastening. Such a fastening as this is at best weak and uncertain, and should not be used at all in practice, except for some exceptionally light work, where lacing-thongs or belt-hooks are not easily to be obtained. 10. Strength of Leather Belts Resistance to Slipping. The discussion of the strength and resistance to slipping of leather belts is attended with well-nigh in- 76 BELTS AND PULLEYS. surmountable difficulties, from the fact that the sub- stance with which we have to deal is almost wanting in homogeneity. We are able by means of standard rules and formulas to calculate closely the strength of a cast-iron column or wrought girder, because within reasonable limits cast-iron and wrought-iron are homo- geneous ; in other words, if we know the breaking strength and safe-working strength of a certain kind of iron, we know these strengths of other iron of the same kind: they are approximately the same. Other metals also are even in texture and homogeneous in nature ; many kinds of wood possess this valuable homo- geneity to a remarkable extent. But this is by no means true of leather. Few substances, if any, with which mechanical men have to deal show such widely varying results under apparently similar circumstances as the leather which furnishes for us the countless number of transmission-belts seen in nearly every shop and factory in the land. In a series of tests made by a prominent firm of leather-belt manufacturers in New York City, strips of leather two inches wide were cut from one of the ordinary sides used for belting, and carefully tested in the same testing-machine and under precisely similar circumstances. These strips were broken at strains varying all the way from 1400 pounds to 3475 pounds ; which result elicits the strange fact, that one strip of leather may be nearly two and a half times as strong as another strip equal in width and thickness, and taken from the same side of leather. The strips in question when in their original positions in the skin were but 15 inches apart at their nearest points. Nor is this all : in two strips which, in the STRENGTH OF LEATHER BELTS. 77 side of leather, joined each other, lay immediately side by side, the difference in breaking strength was 675 pounds, or 33/2" pounds per inch of width ; a variation of 32 per cent of the greater strength and of nearly 47 per cent of the smaller. A gentleman for many years engaged in the manu- facture of leather belting has informed the author that he once cut off twelve inches of solid part (i.e., without rivets or splicing) from a roll of two-inch belting ; cut the piece longitudinally into two parts ; tested them in a correct machine ; and found that one part with- stood 400 pounds greater tensional strain than the other. The gentleman also said that he had tested with a good dynamometer two eight-inch belts, made from similar leather in his own factory, running over pulleys equal in size, doing the same kind of work, and carefully stretched over their pulleys with as nearly as possible the same tensions, and found that one would transmit nearly a horse-power more work without slip- ping than the other. Many other similar examples from practice might be cited to show with how much pf uncertainty and variation from averages the investi- gator of belt-transmissions is compelled to deal. Let the examples already given, however, suffice for this purpose ; and let us, keeping always well on the safe side, endeavor to calculate, as simply as the compli- cated nature of the subject will allow, the proper strengths and sizes for the various transmission-belts in use in practice. The strain brought to bear upon an ordinary endless belt running continuously over its pulleys, leaving out of the question considerations due to centrifugal force, 78 BELTS AND PULLEYS. etc., etc., is one of simple tension ; and were it not for other complicating elements which enter into the cal- culations, the proper strength for a belt to withstand a certain strain could be quite easily calculated. For example, if we represent by P the actual strain on the belt in pounds, by A the cross-section of the belt in square inches, and by f the safe working tensional stress in pounds per square inch for the material of the belt, we can write the formula P=Af, and, by transposing, A = -^. From this simple formula, were the tensional strain all which we must take into account, we could easily calculate our belt widths and thicknesses. But, un- fortunately for the simpli- city of our calculations, other considerations must be looked into before we can correctly obtain the necessary rules and formu- las. In the first place, probably nine belts out of ten in ordinary use will slip around on their pulleys FIG before they will break ; that is, the resistance of the belt to slipping is not equal to its strength. It there- fore becomes necessary to embody in our calculations STRENGTH OF LEATHER BELTS. 79 for strength considerations which will prevent slipping of the belt upon its pulleys. Let ACB (Fig. 45) represent a band or cord drawn over an angle of a solid, as shown in the figure. Let forces, represented by T and /, act at the ends of the cord in the directions shown, and let a represent the angle DCB. In drawing the cord over the angle or corner the friction between the block and cord must be overcome. By the principles of the parallelo- gram of forces, the resultant normal pressure R of the forces T and / is given by the expression R = Vr + ?2~Ttcos sin -\* -r- /, 2 / F= 2F sin -, 2 2 J a a r CDF sm = 2q)t sin , 2 2 STRENGTH OF LEATHER BELTS. 8 1 . a sm - and finally F= - (34) i (p sin The force, then, which is required to draw the cord over the angle in the direction of T is a sm . a' l - cpsm- or T=t When the angle a is very small we may say correctly enough a I (p sin = i, and formula (35) becomes + 2 S in^ (36) Suppose now instead of one angle over which to draw the cord we have several, as shown in Fig. 46, the angles being equal each to each. Let / be the tension at one end of the cord, t l that at the first angle, / a that at the second angle, etc., to the tension T = t n at the other end. From what precedes, we shall have for the force necessary to draw the cord over the first angle 82 BELTS AND PULLEYS. t i = t (l -f- 2cp sill -). For the force necessary to draw the cord over the second angle we shall have Hence or sin i + 2cp sin / \ a = / ^i + 2y sm -J . FIG. 46. In a similar manner =^ I a \ sin -j, sm - J ^/ \ ^&/ \ */ STRENGTH OF LEATHER BELTS. And finally a t n = T t ( i -f 2cp sin - (37) By means of this formula we are able to calculate the forces which tend to cause an endless belt to slip upon its pulley, the tensions in the belt necessary to prevent slipping, and consequently the strength and width of the belt itself. Let K, Fig. 47, be a pulley, A over which, embracing a centre angle BCA = EDB = a, a belt tABT passes as shown in the figure. We can assume the arc AB to be composed of an infinite number (n) of in- finitely small sides ; each will then be expressed by . it From formula (37) we have for the force T the ex- pression FIG. 47. / a \n =. t i 4- 2

04 in formula (41) gives T log - = 0.007578 X 040:, or, when a is in degrees, T log - = 0.00303^ (45) f See Appendix I. STRENGTH OF LEATHER BELTS. 8/ Similarly, by substituting in formula (42), T log 7- = 2 -729 X 0.4^, or, where a is a fraction of the circumference, T log - = 1.0916** (46) t The following table, calculated from formulas (44), T (45), and (46), gives values of -- for different values of the arc a from 30 to 300 corresponding to from 0.524 to 5.236 in circular measure, and from T ^- = 0.083 to -| = 0.833 in fractions of the circumference. To illustrate the application of the table, suppose we have a pair of cast-iron pulleys over which we pro- pose to run a leather belt. Suppose the arc embraced by the belt, upon the pulley over which it is most likely to slip (the pulley having the smaller amount of contact with the belt, or the smaller pulley), is 75 = 1.309 in circular measure ^ 0.208 in fraction of the circumference. We look along the column of degrees until we find the value 75, along the column of circular measures until we find 1.309, or along the column of fractions of the circumference until we find -/^ = 0.208, and, op- posite to these values we find the required value for T the ratio of the tensions, = 1.689. BELTS AND PULLEYS. TABLE OF TENSIONS FOR LEATHER BELTS OVER CAST-IRON PULLEYS. a = T In degrees. In circular measure. In fractions of the circumference. t 30 0.524 tV = ' 8 3 1.233 45 0.785 i = 0.125 1.369 60 1.047 \ = 0.167 1.521 75 1.309 ft 0.208 1.689 90 I-57I i = 0.250 1.874 105 1.833 A = 0.292 2.082 1 20 2.094 * = 0.333 2.312 135 2.356 1 = 0.375 2.565 150 2.618 A = 0.417 2.849 165 2.880 tt = 0.458 3.163 180 3.142 | = o 500 3.5I4 195 3.403 M- 0.541 3.901 210 3.665 A = 0-583 4-333 240 4.189 | = 0.667 5.340 27O 4.712 f = 0.750 6.589 300 5-236 f = 0.833 8.117 The greatest strain brought to bear upon an endless belt, or the strain tending in the greatest degree to cause breakage, is the tension in the driving part of the belt, that is T. This tension acts in one direction and the lesser tension / in a contrary direction. Con- sequently it is the excess of the greater over the lesser tension which overcomes the resistance of the pulley and causes rotation. If we represent the force of re- sistance in pounds at the circumference of the pulley (which is the force transmitted by the pulley) by P 9 we shall have the expression P= T - t. (47) STRENGTH OF LEATHER BELTS. 89 Hence T = P + t, which may be put in the form By substituting for P within the parenthesis its value from formula (47), we obtain But T- t" T T Hence T = Pi l + ~ - \; .T ----! + ! T T=P{ -}, (48) 9 o BELTS AND PULLEYS. by means of which and the preceding table the ten- sion T for different values of a may be determined. The following table, calculated from formula (48), T_ gives values of T - for different values of the arc a. TABLE OF GREATEST TENSION FOR LEATHER BELTS OVER CAST-IRON PULLEYS. In degrees. In circular measure. In fractions of the circumference. J /'X 30 0.524 rV = 0.083 5.29 45 0.785 i = 0.125 3.71 60 1.047 \ = 0.167 2.92 75 1.309 -^ = 0.208 2.45 90 I-57I I = 0.250 2.14 105 1.833 A = 0.292 1.93 120 2.094 i = 0-333 1.77 135 2.356 1 = 0.375 1.64 150 2.618 A = 0.417 1.54 I6 5 2.880 4i = 0.458 1.47 1 80 3.142 i - 0.500 1.40 195 3.403 H = 0.541 1.35 2IO 3.665 T V = 0.583 1.30 24O 4.189 f = o.66 7 1.23 27O 4.712 I = 0.750 1.18 300 5-236 1 = 0.833 1.14 To illustrate the use of the table : Suppose the force transmitted by a pulley is P 500 pounds and the angle embraced by the belt a = 105. In the table opposite to the value a = 105 we find the value 1.93. Hence T P X 1.93 = 500 X 1.93 or T 965 pounds. STRENGTH OF LEATHER BELTS. gi We have now developed rules by which the actual strain upon the belt may be determined : we have still to determine the strength of the belt, or, in other words, the amount of material necessary in the belt to safely sustain the given strain. We have said that the strain T upon an endless belt is a tensional strain. If, therefore, we represent by b the breadth of the belt in inches, by 8 its thickness, also in inches, and by /the greatest safe-working stress in pounds per square inch, we shall have, for the relation between the strain and the strength, the expression (49) T and consequently bS = TT ........ (50) Because of the great variations in the strength of leather the quantity f can be only approximately de- termined. Experiments and tests upon the strength of leather, be they ever so numerous and carefully made, serve only to impress more strongly upon the mind of the experimenter this unfortunate lack of ho- mogeneity in the substance with which he is dealing. In this predicament he who would investigate the sub- ject of leather belts must be satisfied with an average value taken from a great many widely differing values for his coefficient of strength ; and until our manufac- turers are able to produce leather which shall be to a reasonable extent -uniform, the subject of strength of belting must remain as it is now the most uncertain and indefinite one with which mechanical men have to deal. 92 BELTS AND PULLEYS. The weakest part of an endless belt is obviously at the joint: the value of the safe-working stress /must therefore be taken for this part. The author has dur- ing the last three years tried a great many experiments with the view of obtaining the average strength of laced and riveted joints. These average breaking strengths he has found to be about as follows : For ordinary single leather-lacing, 950 pounds per square inch ; For ordinary single rawhide-lacing, 1000 pounds per square inch ; For good double leather-lacing, 1 200 pounds per square inch ; For good double rawhide-lacing, 1400 pounds per square inch ; For ordinary riveted joints, 1750 pounds per square inch. We may therefore take for our safe-working stress in pounds per square inch the following values : Single leather-lacing, / 325 ; Single rawhide-lacing, f= 3 So; Double leather-lacing, f = 375 ; Double rawhide-lacing, f = 400 ; Riveted joints, f= 575. By substituting these values successively in formula (50), we obtain the following formulas : STRENGTH OF LEATHER BELTS. 93 f For single leather-lacing, 66 = ;. . . . (51) f For single rawhide-lacing, 66 = ; . . . . (52) T For double leather-lacing, 66 = - ; . . . . (53) T For double rawhide-lacing, 66 = ; . . . . (54) T For a riveted joints, 66 = -- (55) Example. Required the width of a leather belt J inch thick, which will safely transmit a force of P = 600 pounds when laced according to each of the above-men- tioned methods, the pulleys over which the belt is to run being of the same diameter that is, the angle em- braced by the belt being a 180. From the table on page 90 we have, T = P X 1.40 = 600 X 1.40 = 840 pounds. From formula (5.1), therefore, we have i 840 4 X 840 * X 4 == 3*? -355-' or, for single leather-lacing, b = 10.34" = IOU". 94 BELTS AND PULLEYS. From formula (52), I 840 4 X 840 or, for single rawhide-lacing, b = 9.6" = 9H". From formula (53), i 840 t _ 4 X 840 ^=375' ~37T~' or, for double leather-lacing, * = 8.96" = 8ff' . From formula (54), i _ 840 4 X 840 U X ' , 6? - - 4 400 400 or, for double rawhide-lacing, * b = 8.40" = 8i| r/ . From formula (55), /, ^ l - 8 4 T, _ 4 X 840 ^4 57? ^?T or, for a riveted joint, * = 5.84" = sir- STRENGTH OF LEATHER BELTS. 95 The following tables of formulas have been calculated from the table on page 90 and formulas (51), (52), (53), (54), and (55), respectively. The above example may be calculated from these tables as follows : We have for our data, P = 600 pounds, a = 180, and d = J". From formula (66), for single leather-lacing, bS = 0.0043 1 X 600 ; b = 0.00431 X 600 X 4 = 10.34". From formula (82), for single rawhide-lacing, b$ 0.004 X 600 ; b 0.004 X 600 X 4 = 9.60". From formula (98), for double leather-lacing, bd 0.00373 X 600 ; b 0.00373 X 600 X 4 8.952". From formula (114), for double rawhide-lacing, bd 0.0035 X 600 ; b 0.0035 X 600 X 4 = 8.4o" From formula (130), for a riveted joint, bd = 0.00243 X 600 ; b 0.00243 X 600 X 4 = 5.832". 9 6 BELTS AND PULLEYS. TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS. Single Leather Lacing. a. in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 A- = - 8 3 bd = O.OI628/* 56 45 0.785 i = 0.125 bd = O.OU42/* 57 60 1.047 \ = 0.167 bd = o.ooSgS/ 3 58 75 1.309 & = 0.208 bd = 0.00754/* 59 90 I-57I J = 0.250 bd = o.oo658/> 60 105 1.833 & = 0.292 b8 = 0.00594^ 6l I2O 2.094 i = 0.333 3<5 = 0.00545^ 62 135 2.356 t = 0.375 bd = 0.00505/ 5 63 150 2.618 ^ = 0.417 bd = 0.00474^ 64 I6 5 2.880 tt = 0.458 3<5 = 0.00452/* 65 180 3.142 -J- = 0.500 3<5 = 0.0043I/ 3 66 J 95 3.403 tt = -54i 3<5 = 0.00415/ 5 67 210 3.665 S = o.583 ^6 = 0.00400^ 68 240 4.189 t = o.66 7 bd = 0.0037S/ 5 69 270 4.712 f = 0.750 ^5 = 0.00363/ 3 70 300 5.236 1-0.833 ^5 rr 0.0035I.P 71 TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS^ Sing le Ra wh ide La dug. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 ^ = 0.083 35 = O.OI5II/* 72 45 0.785 i = 0.125 bd = o. 01060 P 73 60 1.047 J = 0.167 bd = 0.008347* 74 75 1.309 t = 0.208 bd = o.oojooP 75 90 I.57I } = 0.250 bd = o.oo6n/> 76 105 1.833 7 T = 0.292 bd 0.005 5 1/ 3 77 120 2.094 * = 0-333 35 = o. 00506 /> . 78 135 2.356 f = 0.375 bd = 0.00469^ 79 150 2.618 A 0.417 bd = O.OO44O/* 80 165 2.880 =0.458 bd = O.OO42O/* Si 180 3.142 i = 0.500 bd = 0.00400/ 3 82 *95 3.403 J} = 0.541 bd = 0.00386/ 5 83 210 3.665 T = 0.583 bd = O.OO37I/ 3 84 240 4.189 = 0.667 3(5 = 0.0035I/ 3 85 270 4.712 f - 0.750 36 X = 0.00337/* 86 300 5.236 1 = 0.833 3<5 = O.OO326/* 87 f. STRENGTH OF LEATHER BELTS. N7- 97 TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS. Double Leather-Lacing* a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 Y 1 ^ = 0.083 35 = O.OI^llP 88 45 0.785 i = 0.125 bd 0.00989^ 89 60 1.047 t = 0.167 35 = 0.007 79 /* 90 75 1.309 & = 0.208 35 = O.OO653/* 91 QO I-57I i = 0.250 35 = 0.0057I/ 3 92 105 1.833 fa = 0.292 35 = 0.005I4/* 93 120 2.094 i = 0.333 35 = o. 00472^ 94 135 2.356 f = 0.375 35 = 0.00437^ 95 150 2.618 -fz = 0.417 ^ = 0.0041 \P 96 165 2.880 tt= 0.458 bd = 0.00392/* 97 180 3.142 | = 0.500 b8 = 0.00373/ 5 98 195 3.403 if = 0.541 35 = o. 00360 P 99 210 3.665 X= ,9.583 bd = 0.00347/* IOO 240 4.189 % 0.667 bd = 0.00328/* 101 270 4.712 f = 0.750 ^5 = O.OO3I5/* 102 300 5-236 f = 0.833 ^5 =i 0.00304/* 103 TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS. Double Rawhide- Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 T V = 0.083 bd = O.OI323/ 3 104 . 45 0.785 i - 0.125 bd = o. 00928^ 105 60 1.047 i = 0.167 bd = o. 00730^ 1 06 75 1.309 -ff = 0.208 bd = o. 00613 P 107 90 I-57I J = 0.250 bd 0.00535^ 108 105 1.833 ff = 0.292 M = o. 00483 P 109 120 2.094 i = 0.333 bd = O.OO443/* no .135 2.356 1 = 0.375 ^5 O.OO4IO/* in 150 2.618 fV = 0.417 bd = 0.00385/ 3 112 I6 5 2.880 fl- = 0.458 35 = 0.00368/ 5 H3 1 80 3.142 | 0.500 35 = O.OO35O/ 5 114 195 3.403 Jf = 0.541 35 0.00338/ 5 U5 2IO 3-665 A = 0.583 35 = o. 003257* 116 240 4.189 f = 0.667 35 = 0.00308/* 117 270 4.712 I = 0.750 35 = 0.00295/ 3 118 300 5.236 f -0.833 35 = 0.00285^ 119 9 8 BELTS AND PULLEYS. TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS. Riveted Joint. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 ^ == 0.083 bd = 0.009207* 1 20 45 0.785 t = 0.125 M = 0.00645/* 121 60 1.047 t = 0.167 ^5 = o.oosoS/* 122 75 1.309 ff = 0.208 ^5 = 0.00426P J23 90 I-57I i = 0.250 bd = 0.00372/* 124 105 1.833 ^=0.292 bd = 0.00336/* 125 1 20 2.094 t = 0-333 ^d = 0.003087* 126 135 2.356 1 = 0-375 M = o. 002857* 127 150 2.618 A = 0.417 ^^ = 0.00268/ 3 128 165 2.880 tt = -458 ^5 = 0.002567* 129 180 3.142 | 0.500 b8 = 0.00243 P 130 195 3.403 H = 0.541 bd 0.002357* 131 210 3-665 & = 0.583 b8 = 0.002267* 132 240 4.189 f = o.c6 7 b8 = 0.002147* 133 270 4.712 t = 0.750 ^5 = 0.002057* 134 300 5-236 1 = 0.833 b8 = 0.001987* 135 STRENGTH OF LEATHER BELTS. 99 Often, when we know the horse-power to be trans- mitted, it is convenient to calculate belt-widths from this, without rinding the circumferential force. From formula (20) we have, when v m represents the velocity in feet per minute, and H the horse-power, ( I36 ) ^ * } and from formula (22), when v represents the velocity in feet per second, By substituting this last value of P in formulas (56) to (135), and reducing, we may obtain the following tables of formulas for calculating belt-widths from the horse-power transmitted and the velocity in feet per second :* ' * By substituting the value of P given in formula (136) in formulas (56) to (135), we may obtain formulas for belt-widths in terms of the horse-power and velocity in feet/ 240 4 187 f = 0.667 bd = I .931* 167 Z/ 270 4 712 f = 0.750 bd = I 85 V 168 300 5 236 f = 01833 bd = I H 793- 169 102 BELTS AND PULLEYS. TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST-IRON PULLEYS. Double Leather-Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 O 524 *=o 083 bd = 7 ? 6l f 170 45 O .785 *=0 125 bd = 5 H 440- 171 60 I .047 * = o 167 bd = 4 -< 172 75 I .309 & = o 208 bd = 3 59V 173 H 9 I 571 i = 250 bd = 3 V 174 105 I .833 ft = o 292 bd = 2 .827 175 I2O 2 .094 *0 333 bd = 2 stff I 7 6 135 2 .356 | = o 375 bd = 2 H 404- 177 150 2 .618 A 417 bd = 2 .*lf I 7 8 I6 5 2 .880 it 458 bd 2 4 179 180 3 .142 | = o 500 bd = 2 H .052 180 195 3 .403 H = o 541 bd = I .980 z/ 181 210 3 .665 &-c 583 bd = i H .909- 182 240 4 .189 f=0 667 bd = I .804- tf 183 27O 4 .712 f = 750 bd = 1 .733^ 184 300 5 .236 -| = o 833 bd = I 4 185 STRENGTH OF LEATHER BELTS. 103 TABLE OF FORMULAS FOR LEATHER BELTS OVER CAST- PULLEYS. Double Rawhide- Lacing. IRON a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 T V = 0.083 217 STRENGTH OF LEATHER BELTS. IO5 Example. Required the width for a leather belt -f^ inch thick which will transmit a force of 15 horse- power at a velocity of 10 feet per second, the angle a TT _ ^ ^ being 105. In this case = - = 1.5 and d = ~. v 10 16 Hence from formula (143) we have * X ;= = 3-267 g = 3.267 X 1.5- Therefore b = 3*267 X 1.5 X y, or, for single leather-lacing, b= 15.682"= 15H" nearly. From formula (159), fX= 3-031 x 1.5, 16 = 3.031 X 1.5 X y, or, for single rawhide-lacing, b - 14.549" = From formula (175), b X j- 6 = 2.827 X 1.5, 16 b 2.827 X 1.5 X , 106 BELTS AND PULLEYS. or, for double leather-lacing, 6= 13.570" = I3IT- From formula (191), bX ^ 6 = 2.657 X 1.5, 16 * = 2.657 X 1.5 X --, or, for double rawhide-lacing, b = 12.754" = I2f". From formula (207), *X = 1-848 X i.S, 6= 1.848 X 1.5 X > or, for a riveted joint, _ O Q^f^ff _ QT// ^7 - 0.070 = O-^ . In the majority of cases leather belts (single) are ap- proximately -^ inch thick. Very often the arc em- braced by the belt is 180 that is, the pulleys are equal ; and, perhaps, more often the arc a is about 135 = f the circumference. For these cases, then, we may obtain formulas which will prove very useful in practice. STRENGTH OF LEATHER BELTS. TO/ By substituting d -% inch in formulas (66), (82), (98), (114), (130), (148), (164), (180), (196), and (212), successively, and reducing, we obtain the following formulas : When a 180 and 6 = ^ inch, Single leather-lacing, b O.OI9/P; .... (218) Single rawhide-lacing, b = 0.0183^; .... (219) Double leather-lacing, = o.oi7iP; .... (220) Double rawhide-lacing, b o.oi6oP', .... (221) Riveted joint, b = Q.oinP. .... (222) TT Single leather-lacing, b = 10.839 ; ... (223) TT Single rawhide-lacing, =10.057 I ( 22 4) TT Double leather-lacing, b 9.381--; . . . (225) TT Double rawhide-lacing, b= 8.800 ; . . . (226) TT Riveted joint b = 6.112 . . . . (227) By substituting $ = -^ inch in formulas (63), (79), (95), (in)' ( T2 7)> (145), (161), (177), (193). and (209), successively, we obtain the following formulas : When a = 135 and S ^ inch, Single leather-lacing, b 0.023 \P\ .... (228) Single rawhide-lacing, b = O.O2I4/ 5 ; .... (229) Double leather-lacing, b O.O2OO/ 5 ; .... (230) Double rawhide-lacing, b O.OiSfP', .... (231) Riveted joint, = o.oi3oP. .... (232) 108 BELTS AND PULLEYS. TT Single leather-lacing, #= 12.699 ; . . . (233) TT Single rawhide-lacing, b = 1 1.794 ; ; . . . (234) TT Double leather-lacing, b = 10.990 -- ; . . . (235) TT Double rawhide-lacing, b = 10.171 ; ... (236) TT Riveted joint, b 7.168. . . . (237) Example. Required the width of a -^-inch leather belt, single leather-lacing, to transmit a force of 600 pounds, the pulleys being equal. From formula (218) we have b 0.0197 X 600, b= 11.82" = I iff". Example. Required the width of a ^--inch leather belt, single rawhide-lacing, to transmit a force of 15 horse-power at a velocity of 10 feet per second, the pulleys being equal. From formula (224) we have b = 10.057 X JJ, b= 1 5.085" - Example. Required the width of a ^--inch leather STRENGTH OF LEATHER BELTS. 1 09 belt, double rawhide-lacing, to transmit a force of 600 pounds, the arc embraced by the belt being about 135. From formula (231) we have b 0.0187 X 600, b 11.22" = 1 1 3V'. Example. Required the width of a ^--inch leather belt, riveted joint, to transmit a force of 15 horse-power at a velocity of 10 feet per second, the arc embraced by the belt being about 135. From formula (237) we have = 7.168 X , 10 b = 10.75" = lof ". , The following tables give the forces in pounds (/>), and the values of the horse -power divided by the velocity in feet per second I 1, corresponding to dif- ferent widths of ^--inch leather belts from I inch to 30 inches for a 180 and a = 135 for each of the five methods of joint-fastening given above. For a great many cases which arise in practice the tables will prove convenient and labor-saving. 1 10 BELTS AND PULLEYS. TABLE OF WIDTHS OF LEATHER BELTS OVER CAST-IRON PULLEYS, WHEN a 1 80 AND d = y. From Formulas (2iS)-(222). Width in inches. />, single leather- lacing. P, single rawhide- lacing. />, double leather- lacing. P, double rawhide- lacing. P, riveted joints. No. I 50.76 54.64 58.48 62.50 90.09 I Ii 76.14 81.97 87.72 93-75 135.14 2 2 101.52 109.29 116.96 125.00 180.18 3 2| 126.90 136.61 146.20 156.25 225.23 4 3 152.28 163.93 175-44 187-50 270.27 5 3i 177-66 191 .26 204.68 218.75 315.32 6 4 203.05 218.58 233.92 250.00 360 . 36 7 4i 228.43 245.90 263.16 281.25 . 405.41 8 5 252.71 2/3.22 292.40 312.50 450.45 9 51 279.19 300 . 46 321 64 343-75 495.50 10 6 304.57 327.87 350.88 375-00 540.54 ii 7 355-33 382.51 409.36 437-50 630.63 12 8 406 . 09 437.16 467.84 500.00 720.72 13 9 456.85 491 .80 526.32 562.50 810.81 14 10 507-61 546.45 584.80 625.00 900 . 90 15 ii 558.38 601.09 643.27 687.50 990.99 16 12 608.57 655.74 701.75 750.oo i 08 i .08 17 14 710.00 765.03 818.71 875.00 1261.26 18 16 812.18 874.32 935.67 1000.00 1441.44 19 18 9I3.7I 983 61 1052.63 1125.00 1621.62 20 20 1015.22 1092 . 90 1169.59 1250.00 1801.80 21 22 1116.75 1202.19 1286.55 1375.00 1981.98 22 24 1218.27 1311.48 1403.51 1500.00 2162.16 23 26 1319.79 1400.77 1520.47 1625.00 2342.34 24 28 1421.31 1530.05 1637.43 1750.00 2522.52 25 30 1522.84 I639-34 1754.44 1875.00 2702.70 26 STRENGTH OF LEATHER BELTS. II] TABLE OF WIDTHS OF LEATHER BELTS OVER CAST-IRON PULLEYS, WHEN a = 180 AND = -%" . From Formulas (223)-(227). Width in inches. H ~ , single leather- lacing. H , single rawhide- lacing. H , double leather- lacing. H , double V rawhide- lacing. H , riveted joints. No. I 0.0923 0.0994 O.IO66 0.1136 0.1636 I Ii 0.1384 0.1491 0.1599 0.1705 0.2454 2 2 0.1845 0.1989 0.2132 0.2273 0.3272 3 2i 0.2307 0.2486 0.2665 0.2841 O . 4090 4 3 0.2768 0.2983 0.3198 0.3409 0.4908 5 3* 0.3229 0.3480 0.3731 0.3977 0.5726 6 4 0.3690 0.3977 0.4264 0.4546 0.6544 7 4* 0.4152 0.4474 o 4797 0.5II4 0.7362 8 5 0.4613 0.4972 0-5330 0.5682 0.8181 9 5* 0.5074 0.5469 0.5863 0.6250 0.8999 10 6 0.5536 0.5966 0.6396 0.6818 0.9817 ii 7 0.6458 o . 6960 0.7462 0-7955 I.I453 12 8 0.7381 0-7954 0.8528 0.9091 1.3089 13 9 0.8303 o . 8949 0-9594 1.0228 1.4725 14 10 0.9226 0.9943 1.0659 1.1364 1.6361 15 ii .0149 1.0937 1.1726 1.2500 i 7997 16 12 .1071 1.1932 1.2792 1.3637 1-9633 17 14 .2916 1.3920 1.4924 1.5910 2 . 2905 18 16 .4762 1.5909 1.7056 1.8182 2.6178 19 18 .6607 1.7897 1.9188 2-0455 2.9450 20 20 .8452 1.9886 2.1318 2.2728 3.2722 21 22 2.0297 2.1875 2.3452 2.5001 3-5994 22 24 2.2142 2.3863 2.5584 2.7274 3.9266 23 26 2.3988 2.5852 2.7716 2.9546 4.2539 24 28 2.5833 2.7840 2.9848 3-1819 4.58H 25 30 2.7678 2.9829 3.1977 3.4092 4.9083 26 112 BELTS AND PULLEYS. TABLE OF WIDTHS OF LEATHER BELTS OVER CAST-IRON PULLEYS, WHEN a = 135 AND d = -/%". From Formulas (228)-(232). Width in inches. P, single leather- lacing. P, single rawhide- lacing. P. double leather- lacing. /, double rawhide- lacing. />, riveted joints. No. I 43.29 46.73 50.00 53.48 76.92 I I* 64.94 70.09 75.00 80.21 115.38 2 2 86.58 93.46 IOO.OO 106.95 153.85 3 2* 108.23 116.82 125.00 133.69 192.31 4 3 129.87 140.19 150.00 160.43 230.77 5 3* 151.51 163.55 175.00 187.17 269.23 6 4 173.16 186.92 200.00 213.90 307.69 7 44 194.81 210.28 225.00 240.64 346.15 8 5 216.45 233.65 250.00 267.38 384.62 9 54 238.10 257-01 275.00 294 . I 2 423.08 10 6 259.74 280.37 300.00 320.86 461.54 ii 7 303.03 327.10 350.00 374-33 538.46 12 8 346.32 373.83 400 . oo 427.81 615-38 13 9 389.61 420.56 450.00 481.28 692.31 14 10 432.90 467.29 500 . oo 534.76 769.23 15 ii 476.19 514.02 550.00 588.24 846.15 16 12 519.48 560.75 600.00 641.71 923.08 17 14 606 . 06 654-21 700.00 748.66 1076.92 18 16 692 . 64 747.66 800.00 845.62 1230.77 19 18 779-22 841.12 900.00 962.57 1384.61 20 20 865.80 934.58 1000.00 1069.52 1538.46 21 22 952.38 1028.04 1100.00 1176.47 1692.31 22 24 1038.96 II2I.5O I 200.00 1283.42 1846.15 23 26 H25.54 1214.95 1300.00 1380.38 2000 . 00 24 28 1212.12 1308.41 1400.00 1497.33 2153.84 25 30 1298.70 1401.87 1500.00 1604.28 2307.69 26 STRENGTH OF LEATHER BELTS. TABLE OF WIDTHS OF LEATHER BELTS OVER CAST-IRON PULLEYS, WHEN a = 135 AND d = 3 7 ". From Formulas (233H.237). Width ia inches. , single leather- lacing. single rawhide- lacing. , double V leather- lacing. , double V rawhide- acing. , riveted V joints. No. I 0.0787 0.0879 0.0910 0.0983 0.1395 I i* o. 1181 0.1272 0.1365 0.1475 0.2093 2 2 0.1575 0.1696 0.1820 o. 1966 0.2790 3 2 o. 1969 O.2I2O 0.2275 0.2458 0.3488 4 3 0.2362 0.2544 0.2730 0.2950 0.4185 5 3i 0.2756 0.2968 0.3185 0.3441 0.4883 6 4 0.3150 0.3392 0.3640 0-3933 0.5580 7 4* 0-3544 o 3816 0.4095 0.4424 0.6278 8 5 0.3937 0.4239 0.4550 0.4916 0.6976 9 5^ 0.4331 0.4663 0.5004 0.5108 0.7673 10 6 0.4725 0.5087 0-5459 0.5899 0.8371 ii 7 0.5512 0.5935 0.6396 0.6882 0.9766 12 8 0.6300 0.6783 0.7279 0.7866 1.1161 13 9 0.7087 o 7631 0.8189 0.8849 1.2556 14 10 0.78/5 0.8479 o . 9099 0.9832 I.395I 15 ii 0.8662 0.9327 J . OOO9 .0815 1.5346 16 12 0.9450 .0175 1.0919 .1799 1.6741 17 14 . 1024 .1870 1-2739 3765 1-9532 18 16 .2600 .3566 1.4558 5731 2.2322 19 18 .4174 .5262 1.6378 .7698 2.5112 20 20 5749 .6958 1.8198 1.9664 2.7902 21 22 .7324 1.8654 2.0018 2. 1630 3.0692 22 24 .8900 2.0349 2.1838 2-3597 3.3482 23 ' 26 2.0475 2 . 2045 2.3057 2.5563 3.6273 24 28 2 . 2048 2.3741 2.5477 2.7530 3.9063 25 30 2.3623 2.9676 2.7297 2.9496 4-1853 26 Example. Required the force in pounds which can be safely transmitted by a leather belt 20 inches wide and 3^ inch thick, running over two pulleys of equal diameters (a = 180), the joint being fastened by a double rawhide-lacing. In the table on page no, column of belt-widths, line 21, we find our width of 20 inches, and corresponding 114 BELTS AND PULLEYS. to this, in the column for double rawhide-lacing, we find the required force P = 1250 pounds. Example. Required the width of a leather belt -^ inch thick, which will safely transmit a force of 1000 pounds running over two pulleys of equal diameters, the fastening being a riveted joint. In the table on page no, column for riveted joints, line 17, we find P = 1081.08 pounds, the nearest value, not less than 1000 pounds, and, in the column for belt-widths, we find the value corresponding to P= 1081.08, b = 12 inches. Example. Required the horse-power which can be safely transmitted by a leather belt 12 inches wide and -% inch thick, running over two pulleys of equal diame- ters at a velocity of 15 feet per second, the fastening being a single rawhide-lacing. In the table on page ill, column of belt-widths, line 17, we have b = 12 inches, and, in the column for single rawhide-lacing, the corresponding value H - 1.1932. V TT Hence = 1.1932, H 15 X I.I93 2 > or H 17.90. Example. Required the velocity at which a leather belt 12 inches wide and ^ inch thick can be driven over two pulleys of equal diameters, in order to transmit a force of 17.90 horse-power, the fastening being a sin- gle rawhide-lacing. LEATHER-COVERED PULLEYS. 115 In the table on page in, column for single rawhide- lacing, we find, corresponding to a belt-width of 12 inches, H - = 1. 1032. V 17.90 17.90 Consequently - - = 1.1932, v = - , v 1.1932 or v 15 feet per second. Example. Given the data a = 135, d = % inch, H ~ 30, v = 15, double leather-lacing, required the belt-width. In this case =30 = v 15 The table on page 113, column for double leather-lac- ing, line 22, gives H - 2.0018, v and a corresponding belt-width of b = 22 inches. ii. Leather Belts over Leather-covered Pulleys. As we have demonstrated in the foregoing pages, the average leather belt will not transmit a force equal to its strength, for the reason that it will slip upon its pulley before it will break. If we can conveniently in- crease the adhesion between the belt and pulley, i.e., increase the coefficient of friction, and in this way pre- Il6 BELTS AND PULLEYS. vent slipping, the belt can be made to do more work without increasing its size. Various methods have been from time to time proposed for obtaining a greater coefficient of friction, such as coating the pulley-faces with gum, rosin, etc.; but these methods have more often than otherwise proved useless, from the fact that the belt is soon rendered stiff and clumsy by the sub- stance placed upon the face of the pulley. Probably the best of all contrivances in use for this purpose is the pulley with a leather-covered face. The leather is easily fastened securely upon the pulley, and we have then practically a leather belt running over a leather pulley. A series of carefully tried experiments has given the coefficient of friction for leather belts over leather-covered pulleys equal to 0.45-0.05 greater than that for leather belts over cast-iron pulleys.* If we substitute q> = 0.45 successively in formulas (40), (41), and (42), and reduce, we shall obtain for leather belts over leather-covered pulleys the follow- ing expressions : T logy = 0.1953^; .... (238) when a is expressed in circular measure, T log-- = 0.00341^; .... (239) T * Reuleaux says : "For a covering entirely new the value of is between 6 and 7; after some service this value decreases, but still does not become less than 4 to 5 ; the arc embraced by the belt being equal T to it. The smaller value, i.e., = 4, corresponds to 0.44 for the coefficient of friction." See also Appendix I. LEA THE 2^- CO VERED P ULLE VS. when a is expressed in degrees, log-- = 1.2280:; 117 (240) when a is expressed in fractions of the circumference. The following table, calculated from the above T formulas, gives values of for different values of a from 30 to 300. The arrangement is similar to that of the table on page 88. TABLE OF TENSIONS FOR LEATHER BELTS OVER LEATHER-COVERED PULLEYS. In degrees. In circular measure. In fractions of circumference. t 50 0.524 ^ = 0.083 1.266 45 0.785 i = 0.125 1.424 60 1.047 = 0.167 1.601 75 1.309 jfe = 0.208 1.802 90 I-57I i = 0.250 2.027 105 1.833 A = - 2 92 2.281 120 2.004 i = 0.333 2.566 IS5 2.356 1 = o.375 2.886 150 2.618 A = -4i7 3-247 165 2.880 H = 0.458 3-653 180 3.142 J = 0.500 4.110 193 3.403 if = 0.541 4-623 210 3.665 A = 0.583 5.201 240 4.189 t = 0.667 6.583 270 4.712 f = 0.750 8.331 3OO 5-236 1-0.833 12.655 T By substituting the successive values of - from the above table in formula (48), we obtain the following table, similar to the one on page 90 : US BELTS AND PULLEYS. TABLE OF GREATEST TENSION FOR LEATHER BELTS OVER LEATHER- COVERED PULLEYS. a = T=PX In degrees. In circular measure. In fractions of circumference. 30 0.524 rV = 0.083 4.76 45 0.785 i - 0.125 3.36 60 1.047 i = 0.167 2.66 75 1.309 ^ = 0.208 2.25 90 I.57I J = 0.250 1.97 105 1.833 ^ = 0.292 1.79 120 2.094 i = 0.333 1.64 135 2.356 1-0.375. 1-53 150 2.618 h 0.417 1.44 I6 5 2.880 it -0.458 1.38 1 80 3.142 i = 0.500 1.32 IQ5 3.403 if 0.541 1.28 210 3.665 A -0.583 1.24 240 4.189 f = 0.667 Z.ll 270 4.712 f = 0.750 1.14 300 5-236 1-0.833 1.09 Example. A leather belt running over a leather- covered pulley transmits a force of 500 pounds. It is required to determine the greatest tension on the belt, assuming that the belt embraces f the circumference of the pulley. From the table we find, corresponding to a = the circumference, T = P X 1.18 = 500 X 1.18, or T 590 pounds. Example. The greatest tension on a leather belt, running over a leather-covered pulley and embracing the circumference, is T 792 pounds. Required the force in pounds which it can transmit. The table gives T=PX 1.32, LEATHER-COVERED PULLEYS. as the greatest tension corresponding to a = cumference. Hence the cir- or 792 == P X 1.32, P = 600 pounds. 79 2 . 1.32' By substituting the values of T'from the above table successively in formulas (51), (52), (53), (54), and (55), the following tables of formulas have been obtained. The application of these formulas will be easily un- derstood from the explanation of the similar tables on pages 96-98. TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER- COVERED PULLEYS. Single Leather-Lacing. a. in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 A = 0-083 bd = O.OI464/ 7 241 45 0.785 \ = 0.125 bd = O.OI034/* 242 60 1.047 i = 0.167 tf = o.ooSiS/ 3 2^3 75 1.309 -% = 0.208 bd = 0.00692^ 244 90 I-57I { = 0.250 bd = o. 00606 P 245 105 1.833 A = 0-292 bd 0.005 5 1/* 246 120 2.094 i = 0.333 bd o. 00503 P 247 135 2.356 -1 = 0.375 bd 0.0047I/ 7 248 150 2.618 T2 = 0.417 bd = o. 00443 P 249 I6 5 2.880 tt = 0.458 bd 0.00424/ 5 250 180 3.142 4- = 0.500 bd = o. 00406 P 251 195 3.403 if = 0.541 bd o. 00394^ 252 2IO 3-665 A = 0.583 bd 0.003827' 253 240 4.189 * = 0.667 bd = 0.00363^ 254 270 4.712 f = 0.750 bd = 0.00351^ 255 300 5-236 1 = 0.833 bd = 0.00335^ 256 I2O BELTS AND PULLEYS. TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED PULLEYS. Single Rawhide- Lacing. o. in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 T V = 0.083 5 = 0. 01 360/ 5 257 45 0.785 i = 0.125 M = 0.009607* 258 60 1.047 | = 0.l67 bd = 0.007607* 259 75 1.309 ^ = 0.208 t>8 o. 00643 P 260 90 I-57I : 0.250 is 0.005637* 261 105 .1.833 7 T =. 0.292 ltd = 0.005 1 iP 262 120 2.094 0.333 bd = 0.00469^ 263 135 2.356 I = 0.375 bd 0.004377^ 264 150 2.618 T% = -4I7 bd 0.0041 iP 265 I6 5 2.880 H- = 0.458 bd 0.00394/ 3 266 180 3.142 j- = 0.500 bd = 0.003777^ 267 195 3.403 M = 0.541 bd = 0.003667-' 268 210 3.665 A = -5S3 bd 0.003547* 269 240 4.189 \ = 0.667 bd = 0.0033 7 P 270 270 4.712 f = 0.750 bd 0.003267* 271 3OO 5-236 \ = 0.833 bd = 0.003117* 272 TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED PULLEYS. Double Leather-Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 T V = 0.083 bd = 0.012697* 273 45 0.785 i = 0.125 <5 = 0.008967* 274 60 1.047 J- - 0.167 bd = 0.007097* 275 75 1.309 -f% = 0.208 ^<5 o 00600 P 276 90 I.57I J = 0.250 bd = 0.005257* 277 105 1-833 A = 0-292 bd = 0.004777* 278 I2O 2.094 i = 0.333 M = 0.004377* 279 135 2.356 1 = 0.375 bd = 0.004087* 280 150 2.618 A = 0.417 /;<5 = 0.003847* 281 I6 5 2.880 tt = -458 bd = 0.003687* 282 ISO 3.142 -- O.5OO bd = 0.003527* 283 195 3-403 if = 0.541 /^5 = 0.003417* 284 210 3.665 A = -583 bd = 0.003317* 285 240 4.189 f^o.667 /65 = 0.003157* 286 270 4.712 I = 0.750 bd =1 0.003047* 287 300 5-236 1 = 0.833 bd = 0.0029 1 7* 288 LEATHER-COVERED PULLEYS. 121 ^^ TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED PULLEYS. Double Raw hide- Lacing. a in degrees. a in circular measure. a. in fractions of circumference. Formula. No. 30 0.524 T V = 0.083 bd = o.ongoP 289 45 0.785 i = 0.125 bd = o.oo$4oP 290 60 1.047 i = 0.167 M = 0.00665^ 291 75 1.309 -% = 0.208 M = O.OO563/ 3 292 90 I-57I J = 0.250 5 = 0.00493/ 5 293 105 1.833 = 0.292 ** = o. 00448^ 294 I2O 2.094 t = 0-333 M = O.OO4IO/* 295 135 2.356 -1 = 0.375 M = o. 00383^ 296 150 2.618 T 5 = 0.417 M = 0.00360/* 297 165 2.880 ii = 0.458 S O.OO345/' 298 1 80 3.142 -J- = o . 500 bd = 0.00330/ 3 299 195 3.403 M = 0.541 bd = O.OO32O/* 300 2IO 3-665 A = 0.583 bd = O.OO3IO/* 301 240 4.189 | = 0.667 bd o. 00295 P 302 270 4.712 f = 0.750 bd = 0.09285/ 5 303 300 5.236 I = 0.833 bd = O.O0273/* 304 TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED PULLEYS. Riveted Joint. a. in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 A = 0.083 bS = 0.00828/ 5 305 45 0.785 i = 0.125 bS = 0.005847' 306 60 1.047 i = 0.167 bS = o. 00463 P 307 75 1.309 ^ = 0.208 dd = o.oo39i/ > 308 90 I.57I i = 0.250 b$ = 0.00343^ 309 105 1.833 ^ 0.291 1>S = 0.003 u /* 310 I2O 2.094 i = o-333 it o. 00285 P 311 135 2.356 f = 0.375 b8 = o. 00266 P 312 150 2.618 fs = 0.417 M = 0.002507* 313 I6 5 2.880 tt- 0.458 bd = o. 00240 P 3H 180 3.142 \ = 0.500 b8 = 0.0022qP 315 J 95 3.403 if = 0.541 b8 = 0.00222P 316 210 3-665 A = 0-583 bd = O.OO2I6P 317 240 4.189 1 = 0.667 bS = o. 00205 P 318 270 4.712 f = 0.750 bd o.ooigSP 319 300 5.236 1 = 0.833 bd = o.ooigoP 32O 122 BELTS AND PULLEYS. Example. A leather belt J inch thick, running over a leather-covered pulley, transmits a force of 500 pounds. Required the width of the belt for single leather-lacing and single rawhide-lacing, taking a = 45. From formula (242) we have /; X i = 0.01034 X 500, b = 0.01034 X 500 X 4, or, for single leather-lacing, b = 20.68" = 20^-" nearly. From formula (258) we have b X i = 0.00960 X 500, b 0.00960 X 500 X 4, or, for single rawhide-lacing, b = 19.20" = I 9 |f.* Example. With the data a= 1.833, circular meas- ure, d = % inch, and b 20 inches, required the forces in pounds which the belt can transmit for each of the * If we take the above data, P = 500, a 45, d \ inch, and cal- culate the width of a leather belt running over a cast-iron pulley, we shall have, from formula (57), for single leather-lacing, b 0.01142 X 500 X 4 = 22.84 inches. The difference between the widths of the belt necessary for transmission over cast-iron and leather-covered pul- leys is therefore 22. 84 20.68 = 2.16 inches, which shows a gain for the leather-covered pulley of nearly 10 per cent over the cast-iron pulley. LEATHER-COVERED PULLEYS. 12$ above methods of joint-fastening, supposing the belt to run over a leather covered pulley. From formula (246) we have 20 X 0.25 20 X = 0.0055 iP, P-- -- or, for single leather-lacing, P 907.44 pounds. From formula (262), 20 X 0.25 20 X J 0.005 I IP, P or, for single rawhide-lacing, P 978.47 pounds. From formula (278), or, for double leather-lacing, /*= 1048.22 pounds. 124 BELTS AND PULLEYS. From formula (294), 2O X O X i = o. or, for double rawhide-lacing, /> = 1116.07 pounds. From formula (310), or, for a riveted joint, P= 1607.71 pounds. The formulas of the following tables, obtained by TT substituting P= 550 in formulas (241) to (320), and similar to the formulas on pages 100-104, will prove convenient in calculating widths of leather belts over leather-covered pulleys from the horse-power trans- mitted and the velocity in feet per second : LEA 7^HER-CO VERED P ULLE YS. 125 TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER COVERED PULLEYS. Single Leather Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 A = 0.083 bd = 8 H V 321 H 45 0.785 4 = 0.125 bd = 5 687- 322 V 60 1.047 1 6" = 0.167 bd = 4 H 499- 323 H 75 1.309 * = 0.208 bd 3 806- V 324 H 90 I-57I i = 0.250 bd = 3 333- 325 105 1-833 A = 0.292 bd = 3 95f 326 120 2.094 i = 0-333 bd = 2 . 7 6 7 f 327 135 2.356 f = 0-375 bd = 2 H 59!- 328 H 150 2.618 A = 0.417 bd = 2 437- 329 I6 5 2.880 H = 0.458 bd = 2 H ' 332 ~ 330 180 3.142 * = 0.500 bd = 2 H 2 33- 331 195 3.403 = 0.541 bd = 2 167- 332 V 210 3.665 _7_ =0.583 bd = 2 H . IOI 333 240 4.189 f = 0.667 bd = , H 997- 334 27O 4.712 f = 0.750 bd = I H 93i- 335 H 300 5-236 f =0.833 bd = I 843 f 336 126 BELTS AND PULLEYS. TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-I PULLEYS. Single Raw hide -Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 T V = 0.083 TT bd = 7.480 337 45 0.785 t = - I2 5 bd = 5-280^ 338 60 1.047 i = 0.167 TT bd = 4.180 339 75 1.309 -fz 0.208 b$ = 3.537- 340 90 I-57I i = 0.250 68 = 3-097- 341 105 1.833 A = 0.292 bd = 2.8ll~ 342 120 2.094 i = 0-333 bd = 2.580^ V 343 135 2.356 * -0.375 LX H bo = 2.404 V 344 150 2.618 T = .4I7 T r b$ = 2.261 V 345 I6 5 2.880 tt = -458 TT bd = 2.167 346 180 3.142 -J- = 0.500 TT bd = 2.074 '^ V 347 195 3-403 if = 0.541 bd = 2.013 V 348 210 3.665 A = 0.583 bd = 1-947- 349 24O 4.189 t = 0.667 bd = 1.854^ 350 270 4.712 l = 0.750 bd = i . 793 - 35i 3OO 5.236 1 = 0.833 H &o = i. 701 f 352 LEA THER- CO VERED P ULLE YS. 127 TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER COVERED PULLEYS. Double Leather-Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 fg = 0.083 TT t>5 = 6.980 t 353 45 0.785 i = 0.125 ^ dS = 4.928 ^ v z/ 354 60 1.047 i = 0.167 H bS = 3.900 355 75 1.309 ^ = 0.208 H b = 3 . 300 J J v 356 90 I-57I = 0.250 H 35 = 2.888 357 105 1.833 ^ 0.292 H b 2.624- ' z/ 358 120 2.094 i = 0.333 ^ ^5 = 2.404- 359 135 2.356 1 = 0.375 ^ M -2.244- 360 150 2.618 fs = 0.417 II b 2.II2 361 165 2.880 = 0.458 H b8 = 2.024 ^ 362 180 3.142 \ = 0.500 TT bS= 1.936- 363 195 3.403 H = -54i ^5 - 1.876^ 364 210 3.665 A = 0.583 z> 365 240 4.189 f = 0.667 ^ = 1.733- 366 270 4.712 $ = 0.750 /^5 i .672 367 300 5-236 f -0.833 35 1.601 z/ 368 128 BELTS AND PULLEYS. TABLE OF FORMULAS FOR LEATHER-BELTS OVER LEATHER-COVERED PULLEYS. Double Rawhide-Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. ff 30 0.524 A- = 8 3 bd = 6.545- 369 H 45 0.785 I = 0.125 bd = 4.620 370 H 60 1.047 i = 0.167 371 ff 75 1.309 -ff = 0.208 bd -. 3-097- 372 ff 90 T-57I i = 0.250 bd = 2.712 373 H 105 1-833 ^ = 0.292 ** = 2.464- 374 H 120 2.094 i = 0.333 bd = 2,255 375 // 135 2.356 1 = 0.375 bd = 2.107 ' V 3/6 H 150 2.6I8 A- = 0-417 bd = 1.980- V 377 77 165 2.880 jj 0.458 bd = 1.898- 378 H 1 80 3.142 i = 0.500 bd = 1.815^- 379 H 195 3.403 tt = 0.541 bd = i .760 380 H 210 3.665 ft- =0.5*3 bd = 1.705- 38i ^7 240 4.189 f = 0.667 ^ = 1.623- 382 ^ 27O 4.712 f = 0.750 = 1.568 z/ 383 H 3OO 5.236 1 = 0.833 bd = 1.502 384 LEATHER-COVERED PULLEYS. 129 TABLE OF FORMULAS FOR LEATHER BELTS OVER LEATHER-COVERED PULLEYS. Riveted Joints. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 A = 0.083 H bd = 4.554- 385 V H 45 0.785 i = 0.125 bo = 3.212 V 386 60 1.047 i = 0.167 bd = 2. S 47- 387 V 75 1.309 % = 0.208 H bd = 2.151 388 V H 90 I-57I i = 0.250 bS = 1.887- V 389 H 105 T.833 -ff = 0.292 V 390 120 2.094 1 = 0.333 ff b8 = 1.568- V 391 H 135 2-356 1 = 0.375 bd = 1.463- 392 ff 150 2.618 T2 = 0.417 **= 1-375- 393 H 165 2.880 tt = 0.458 b8 =1.320- 394 180 3 142 1 = o 500 bS = 1.260^ ^ 395 195 3.403 H = -54i ,, H bS = i. 221 V 396 210 3.665 A = 0.583 r fr rRR 397 V ff 240 4.189 * - 0.667 bS = 1.128- 398 // 270 4.712 f = 0.750 ^5 = 1.089- 399 V ff 300 5-236 f = 0.833 68 = 1.045 400 ?y I3O BELTS AND PULLEYS. Example. A leather belt \ inch thick, running over a leather-covered pulley, transmits a force of 20 horse- power at a velocity of 15 feet per second. Required the width of the belt for single and double rawhide- lacing, assuming that the belt embraces an arc of the pulley equal to 2.880 circular measure. From formula (346) we have b X i = 2.167 X ~, 6 = 2.167 X ^y X 4, or, for single rawhide-lacing, *=; 11.557"= HiV- Formula (378) gives * X i = 1.898 X ~, b= 1.898 X ~ X 4, or, for double rawhide-lacing, b 10.123* = io|". Example. A leather belt running over leather-cov- ered pulleys is -% inch thick and 12 inches wide. Required the velocity at which the belt can transmit a force of 10 horse-power, assuming a =. 45, and that the belt has a double leather-lacing. We have from formula (354) 10 4.928 X 10 12 X A - 4-928 X -, fr Ux-^-, or v 18.77 ft. per second. LEATHER-COVERED PULLEYS. 131 Example. A leather belt (with a riveted joint) running over leather-covered pulleys is 16 inches wide and T 3 inch thick ; the arc embraced by the belt on the smaller pulley is 150, and the velocity of the belt 10 feet per second. It is required to determine the horse-power which can be transmitted by the belt. From formula (393) we have or H 21.82. By substituting # = % in formulas (251), (267), (283), (299), (315), (331), (347), (363), (379), and (395), successively, we obtain the following formulas : When a = 180 and H = 8.60 X 2.9070, v or 12. Vulcanized-rubber Belts. Vulcanized-rubber belts are usually made, as ex- plained in 8, by placing one or more layers of cotton duck between layers of vulcanized rubber. The num- ber of these layers is indicated by the term ply : thus a one-ply belt contains one layer of duck, a three-ply belt contains three layers, etc. The thickness of each layer of duck varies more or less according to the amount of material and the force with which the lay- ers are pressed together in the manufacture. We may, however, with sufficient correctness for ordinary pur- poses, take for the average thickness of a ply T V inch. A three-ply belt is therefore approximately J inch thick, a four-ply belt -J- inch thick, etc. VULCANIZED-RUBBER BELTS. 141 The strength of vulcanized-rubber belting seems to be about that of leather of the same thickness. A series of tests made for the author by Messrs. Fair- banks & Co., on their standard testing-machine, gave for superior new vulcanized-rubber belting an average strength of nearly 4000 pounds per square inch of sec^ tion. A great number of other tests made by the author on ordinary vulcanized-rubber belts which had been in practical use for a short time gave results es- sentially the same as for leather. We shall therefore use for the safe-working stress in pounds per square inch for vulcanized-rubber belting the following values, given in 10: Single leather-lacing, f= 325 ; Single rawhide-lacing, / = 350 ; Double leather-lacing, / = 375 ; Double rawhide-lacing, f = 400 ; Riveted joint, f= 575. The coefficient of friction for vulcanized rubber over cast-iron seems to be slightly greater than for leather over leather-covered pulleys.* Since, however, rubber belts are very seriously injured by slipping about their pulleys, and for this reason greater care should be taken to prevent slipping, we propose to neglect the ap- parently small difference and take the coefficient equal * See Appendix I. 142 BELTS AND PULLEYS. to that for leather over leather-covered pulleys. We have then

423 75 1.309 ff = 0.208 1)8 0.00692^ 424 90 I-57I i = 0.250 5 nr 0.00606P 425 105 1.833 A = 0.292 M = o-oossi/* 426 120 2.094 i = 0-333 ^5 = 0.00503^ 427 135 2.356 1 = 0.375 b$ =. 0.00471^ 428 150 2.618 A = 0.417 b$ = 0.00443/ 5 429 I6 5 2.880 = 0.458 <^5 O.OO424/ 3 430 180 3.142 % = o 500 b8 = o. 00406 P 431 195 3.403 H = 0.541 b8 = 0.00394^ 432 210 3.665 A = 0.583 ^5 = 0.00382^ 433 24O 4.189 f = 0.667 <^5 = 0.00363^ 434 270 4.712 f = 0.750 k$ = 0.0035I/ 3 435 300 5-236 I = 0.833 bd = o. 00335^ 436 VULCANIZED-RUBBER BELTS. 143 TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST- IRON PULLEYS. Single Rawhide Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 A = 0.083 bd = o.oi36oP 437 45 0.785 i = 0.125 bd = 0.00960^ 438 60 1.047 j = 0.167 bd = o.oo f j6oP 439 75 1.309 fa 0.208 bd = 0.00643/ 3 440 90 I-57I i = 0.250 bd = o. 00563^ 441 105 1-833 fa = 0.292 ts = 0.005 i i/ 3 442 120 2.094 $ = 0.333 6$ = 0.00469/ 3 443 135 2.356 1 = 0.375 bd = 0.00437^ 444 150 2.618 T 5 = 0.417 bS = o.oo4iiP 445 I6 5 2.880 ii = 0.458 bd = 0.00394^ 446 1 80 3.142 { = 0.500 bd = 0.00377/ 3 447 195 3-403 it = 0.541 bd = 0.00366^ 448 210 3-665 A -0.583 bd rz: O.OO354/ 5 449 240 4.189 1- = 0.667 bd = 0.00337/ 3 450 270 4.712 f = 0.750 bd =: 0.00326^ 451 300 5-236 1 = 0.833 bd = o.oo3iijP 452 TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST- IRON PULLEYS. Double Leather- Lacing. a. in degrees. a in circular measure. a in fractions of circumference. Formula. No. . 30 0.524 T V = 0.083 b$ = O.OI269/ 5 453 45 0.785 i 0.125 bd = 0.00896^ 454 60 1.047 1 = 0.167 b = o.oojogP 455 75 1.309 fa = O.2O8 bS = o.oo6ooP 456 90 I-57I i = 0.250 bd = 0.00525^ 457 105 1.833 ^ = 0.292 bd = 0.00477^ 458 120 2.094 i = 0.333 bd = 0.00437/ 3 459 135 2.356 l = o.375 bd = o. 00408 P 460 150 2 618 A = 0-417 bd = o. 00384^ 461 I6 5 2.880 H = 0-458 bd = o.ootfSP 462 1 80 3.142 i = 0.500 bd-= O.OO352/ 5 463 195 3.403 H = 0-541 bd = O.OO34I/ 5 464 2IO 3-665 A = 0.583 3^ = 0.0033 \P 465 240 4.189 1 = 0.667 bd = 0.003I5/ 5 466 27O 4.712 f = 0.750 bd = o. 00304 P 467 3OO 5-236 1 = 0.833 bd = O.OO2QIP 468 144 BELTS AND PULLEYS. TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST- IRON PULLEYS. Double Rawhide-Lacing. a. in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 T2 = 0.083 b8 o.ongoP 469 45 0.785 i = 0.125 bd = 0.00840^ 470 60 1.047 J = 0.167 bd = o. 00665 /* 471 75 1.309 ^ = 0.208 68 0.00563^ 472 90 I-57I i = 0.250 65 = 0.00493/ 3 473 105 1.833 -JT 0.292 bd = 0.00448^ 474 120 2.094 '* = 0-333 M = o.oo^ioP 475 135 2.356 f = 0-375 6d = 0.00383^ 476 150 2.618 TV = 0-417 5 = o.oo36o/> 477 I6 5 2.880 4i - 0.458 5 = 0.00345/ 7 478 180 3.142 \ = 0.500 <55 = 0.00330^ 479 195 3.403 if = 0.541 ^<5 0.00320^* 480 210 3.665 & = 0-583 <^5 = O.OO3IO/ 1 481 240 4.189 | = 0.667 bd = 0.00295^ 482 270 4.712 f = 0.750 ^5 = 0.00285/* 483 300 5.236 1=0.833 ^5 = o. 00273^ 484 TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST- IRON PULLEYS. Riveted Joints. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 A = 0.083 & = 0.00828P 485 45 0.785 i = 0.125 bd = o. 00584^ 486 60 1.047 i = 0.167 b8 o. 00463^ 487 75 1.309 ff = 0.208 b8 = 0.0039 iP 488 90 I.57I i = 0.250 bft 0.00343/* 489 105 1.833 ^ = 0.291 bd = o.oo^nP 490 120 2.094 i = 0.333 b = O.OO285/* 491 135 2.356 f = 0.375 b8 = O.Q0266P 492 150 2.618 T K = 0.417 bd = 0.00250/ 5 493 165 2.880 tt = 0.458 bd o.oo24oP 494 180 3.142 -J = 0.500 bd 0.00229/ 5 495 195 3-403 H = 0-541 bd = O.OO222/ 5 496 210 3.665 = 0.583 bd = 0.002I6/* 497 240 4.189 I = 0.667 bd o. 00205 P 498 270 4.712 f = 0.750 bd o.ooigS/* 499 3OO 5.236 f = 0.833 ^(5 o.ooigo/' 500 VULCANIZED-RUBBER BELTS. 145 TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER IRON PULLEYS. Single Leather-Lacing. CAST- a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 TV = 083 bo = 8 H .052 V 501 45 0.785 i = o 125 bo = 5 687 ? 502 60 1.047 i = o 167 bo - 4 H 499- 503 75 1.309 Jr = o 208 bd 3 .806^ V 504 90 I.57I i = o 250 bd = 3 H 333- 505 105 1.833 * = o 292 bd 3 H 031- 506 120 2.094 l-o 333 bd = 2 *? 507 135 2.356 l-o 375 bd = 2 H .591- 508 150 2.618 * = o 4*1 bd = 2 H 437- z/ 7" 509 165 2.880 tt = o 458 bd = 2 332- z> 510 180 3.142 |-o 500 bd = 2 .233^ 511 195 3-403 H=o 54i bd = 2 .^2 z^ 512 210 3.665 A = o 583 bd = 2 H .101 z> 513 240 4.189 = 667 bd I H 997- 514 270 4.712 f = Q 750 bd = I H 93i- 515 300 5-236 f = o 833 bd j 8 f 516 10 146 BELTS AND PULLEYS. TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST- IRON PULLEYS. Single Rawhide- Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 jV = 0.083 bd H V 517 45 0.785 i = 0.125 bd ^^ 5 * 280 518 60 1.047 = 0.167 bd TT = 4.I8O 519 75 1.309 -^ = 0.208 bd H = 3-537- 520 90 I.57I = 0.250 bd II = 3.097- 521 105 1.833 ff = 0.292 bd = 2.8ll| 522 120 2.094 * = 0.333 bd = 2.580^ 523 135 2.356 1=0.375 bd = 2.404- 524 150 2.618 * = 0.417 bd = 2.261 525 I6 5 2.880 H = o 458 bd 2.167 526 1 80 3.142 J^o^o bd V 527 195 3-403 if = 0.541 bd 2.013 528 210 3.665 & = 0.583 bd H = 1.947- 529 240 4.189 f = 0.667 bd = i.8 54 f 530 270 4.712 f = 0.750 bd H -1.793- 531 300 5.236 f = 0.833 bd = 1.701- 532 VULCANIZED-RUBBER BELTS. 147 TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER CAST- IRON PULLEYS. Double Leather-Lacing. a in degrees. a in circular measure. a in fractions of circumference. Formula. No. H 30 0.524 & = 0.083 bd = 6.980- V 533 H 45 0.785 i = 0.125 bd 4.928- 534 ff 60 1.047 - 0.167 bd = 3.900 535 75 1.309 ft = 0.208 #5 = 3.300 536 90 I.57I i = 0.250 <^5 = 2.888- 537 105 1.833 fa 0.292 bd 2.624 7/ 538 // 120 2.094 t = 0.333 o = 2.404 539 77 H 135 2.356 1 - 0.375 bd = 2.244- 540 jf 150 2.6*9 ^ = 0.417 541 ff , I6 5 2.880 tt= 0.458 bd = 2.024 542 ff 180 3.142 -J- = 0.500 bd = 1.936 543 195 3.403 M = 0.541 ff bd = 1.876- 544 // 2IO 3.665 h = 0.583 ^ = I.82I- 545 Z/ ff 240 4.189 f = 0.667 bd 1.733- 546 ^ 77 270 4.712 f - 0.750 M = 1.672- 547 ff 30O 5.236 f = 0.833 ^5 = i .601 548 148 BELTS AND PULLEYS. TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER IRON PULLEYS. Double Rawhide- Lacing. CAST- a in degrees. a in circular measure. a in fractions of circumference. Formula. No. 30 0.524 -1% = 0.083 bd = 6.545f 549 45 0.785 = 0.125 bd = A ^ 4.620 V 550 60 1.047 i = 0.167 bd = 3.658^ V 551 75 1.309 -f^ =. O.208 bd = V 552 90 I.57I i = 0.250 bd = 2.712^ 553 105 1.833 -fa = 0.292 bd = 2.464 554 1 20 2.094 i = 0.333 bd -^ H 2.225- 555 135 2.356 1 = 0.375 bd = H 2.107 556 150 2.618 A = 0.417 *** 1.980 557 165 2.880 tt= 0.458 M = TT 1.898- 558 1 80 3.142 i = 0.500 M = mif 559 195 3.403 ft = 0.541 ^ = t-j,6.f 560 210 3-665 T2 = 0.583 /^6 V = I 705 - 56i 240 4.189 | = 0.667 bd = 1.623 562 270 4.712 f = 0.750 bd = 1 . 568 S5> 563 300 5.236 f 0.833 bd = H 1.502- 564 VULCANIZED-RUBBER BELTS. 149 TABLE OF FORMULAS FOR VULCANIZED-RUBBER BELTS OVER IRON PULLEYS. Riveted Joints. CAST- a in degrees. o in circular measure. a in fractions of circumference. Formula. No. H 30 0.524 A- 0.083 bd = 4-554- 565 ff 45 0.785 I - 0.125 bd = 3.212 566 H 60 1.047 J = 0.167 bd = 2.547 OH" v 567 H 75 1.309 ^ = 0.208 bd 568 IT 90 I-57I i = 0.250 bd = 1.887- 569 V H 105 1.833 fa = 0.292 bd = 1.711- 570 V - i 68^ 120 2.094 i = 0.333 bd 571 135 2.356 1 = 0.375 bd = i. 4 6 3 f 572 ff 150 2.618 y 5 ^ '4 I 7 bd 1-375- 573 V If . I6 5 2.880 it -0.458 bd 1.320 574 V ff 180 3.142 i = 0.500 bd = 1.260 V 575 H 195 3.403 M = 0.541 bd = I.22I V 576 210 3.665 A = 0.583 bd = 1.188^ 577 V IT 240 4 189 i- = 0.667 bd = 1. 128- 578 z/ H 270 4.712 t = 0.750 bd = 1.089- 579 ff 300 5-236 1 = 0.833 bd 580 150 BELTS AND PULLEYS. The formulas for vulcanized-rubber belts -^ inch thick (say three-ply) over cast-iron pulleys are as follows : When a= 180, Single leather-lacing, b O.OI86P; .... (581) Single rawhide-lacing, b = O.OJ72/; .... (582) Double leather-lacing, b o.oi6iP; .... (583) Double rawhide-lacing, = o.oi5iP; .... (584) Riveted joint, b = 0.0105^. .... (585) TT Single leather-lacing, =10.208 ; . . . (586) v TT Single rawhide-lacing, = 9.481-; . . . (587) Double leather-lacing, b 8.850 ; . . . (588) v Double rawhide-lacing, b = 8.297- ; . . . (589) Riveted joint, b = 5.760. . . . (590) When a 135, Single leather-lacing, = 0.0215^; .... (591) Single rawhide-lacing, = O.O2OO/; .... (592) Double leather-lacing, = o.oi87/ > ; .... (593) Double rawhide-lacing, = 0.0175^; .... (594) Riveted joint, = 0.0122/1 .... (595) TT Single leather-lacing, =11.845 ; . . . (596) TT Single rawhide-lacing, = 10.990 ; . . . (597) VULCANIZED-RUBBER BELTS. TT Double leather-lacing, b = 10.258 ; TT Double rawhide-lacing, b = 9.632- ; Riveted joint, 6= 6.668 H v ' (598) (599) (600) TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON PULLEYS, WHEN a. = 180 AND 8 = fe". From Formulas (5 81)- (585). Width in inches. /*, single leather- lacing. P, single rawhide- lacing. P, double leather- lacing. P, double rawhide- lacing. P, riveted joints. No. I 53.88 58.04 62.15 66.27 95-51 I I* 80.82 87.06 93.23 99-40 143.27 2 2 107.76 116.08 124.30 132.54 191.02 3 *t I34-70 145.10 155.38 165.67 238.78 4 3 161.64 174.11 186.45 198.81 286.53 5 31 188.58 203. 13 217.53 231.94 334-29 6 4 215-52 232.15 248 . 60 265.08 382.04 7 4i 242 . 46 261.17 279.68 298.21 429.80 8 5 269.40 290. 19 310.75 331-35 477.56 9 5i 296.34 319-21 341.83 364.48 525.31 10 6 323.28 348.23 372.90 397.61 573-07 ii 7 377-16 406.27 435.05 463.88 668.58 12 8 431-03 464.31 497.20 530.15 764.09 13 9 484.91 522.34 559-35 59 6 -42 859.60 14 10 538-79 580.38 621. 50 662.69 955-11 15 ii 592.67 638 42 683.65 728.96 1050.62 16 12 646.55 696 . 46 745.80 795-23 1146.13 17 14 754-31 812.54 870. 10 927.77 1337.15 18 16 862.07 928.61 994.40 1060.31 1528.18 *9 18 969.83 1044.69 1118.71 1192.84 1719.20 20 20 1077-59 1160.77 1243.01 I325-38 1910.22 21 22 1185-34 1276.84 1367.31 1457.92 2101.24 22 24 1293. 10 1392-92 1491.61 I590-46 2292.26 23 26 1400.86 1509.00 1615.91 1723.00 2483.29 24 28 1508.62 1625.07 1740.21 1825.53 2674.31 25 30 1616.38 I74I-I5 1864.51 1988.07 2865.33 26 152 BELTS AND PULLEYS, TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON PULLEYS, WHEN a = 180 AND 6 = -%". From Formulas (586)- (590). Width in inches. H , single leather- lacing. H , single rawhide- lacing. H ~, double leather- lacing. H , double rawhide- acing. H , riveted joint. No. I 0.0980 0.1055 O.II30 0.1205 0.1736 I I* 0.1469 0.1582 0.1695 0.1808 0.2604 2 2 0.1959 0.2109 0.2260 0.2410 0.3472 3 *i 0.2449 0.2637 0.2825 0.3013 0.4340 4 3 0.2939 0.3164 0.3390 0.3616 0.5208 5 31 0.3429 0.3692 0-3955 0.4218 0.6076 6 4 0.3918 0.4219 0.4520 0.4821 0.6944 7 4* . 4408 0.4746 o 5085 0.5424 0./8I2 8 5 0.4898 0.5275 0.5650 0.6026 0.8681 9 54 0.5388 0.5801 0.6214 0.6629 0-9549 10 6 0.5878 0.6328 0.6779 0.7231 1.0417 ii 7 0.6857 0.7383 0.7909 0.8437 I.2I53 12 8 0.7837 0.8438 0.9039 o . 9642 1.3889 13 9 0.8817 0.9493 I .0169 .0847 1.5625 14 10 0.9796 1-0547 I .1299 .2052 1.7361 15 ii 1.0776 I. 1602 1.2429 .3258 1.9097 16 12 1755 1.2657 1-3559 .4463 2.0833 17 14 .3715 1.4766 I.58I9 .6873 2 . 4306 18 16 .5674 1.6876 1.8078 .9284 2.7778 19 18 .7633 1.8985 2.0338 2.1694 3-I250 20 20 9592 2.1095 2.2^98 2.4105 3-4722 21 22 2.1552 2.3204 2.4858 2.6515 3.8194 22 24 2.35H 2.53H 2.7II8 2.8926 4.1667 23 26 2.5470 2.7423 2-9377 3.1336 4.5139 24 28 2.7429 2.9532 3-I637 3.3747 4.86II 25 30 2.9389 3.1642 3o897 3.6157 5-2083 26 UN: J> ULCANIZED-R UBBER BEL TS 153 TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON PULLEYS, WHEN a = 135 AND d = -%". From Formulas (591)- (595). Width in inches. /*, single leather- lacing. /', single rawhide- lacing. P, double leather- lacing. />, double rawhide- lacing. />, riveted joint. No. I 46.45 50.05 53-62 57-11 82 24 I Ii 69.67 75.08 80.43 85.67 123.36 2 2 92.89 100.10 107.24 114.22 164.47 3 2| 116.12 125.13 134.05 142.78 205-59 4 3 139-34 150.15 160.86 I7I-33 246.71 5 3i 162.56 175.18 187.67 199.89 287.83 6 4 I35-79 200 . 20 214.48 228.44 328.95 7 4i 209.01 225.23 241.29 257.00 370.07 8 5 232.23 250.25 268.10 285.55 411.18 9 5^ 25S-4 6 275.28 294.91 314.11 452.30 10 6 278.68 300.30 321.72 342.66 493.42 ii 7 325.13 350.35 375-34 399-77 575-66 12 8 371-57 400 . 40 428.95 456.88 657.89 13 9 . 418.02 450.45 482.57 513.99 740.13 14 10 464.47 500.50 536.19 571.10 822.37 15 ii 510.91 550.55 589.81 628.21 904 . 60 16 12 557.36 600.60 643.43 685.32 986.84 17 14 650.26 700 . 70 750.67 799.54 1151.32 18 16 743-15 800.80 857.91 913-76 I3I5.79 19 18 836.04 900.90 965-15 1027.98 1480.26 20 20 928.94 1001 .00 1072.39 1142.20 1644.74 21 22 1021.83 IIOI. 10 1179.62 1256.42 1809.21 22 , 2 4 1114.72 1201 .20 1286.86 1370.64 1973.68 23 26 1207.62 1301.30 1394.10 1484.87 2138.16 24 28 1300 .51 1401.40 1501.34 1599.09 2302.63 25 30 1391-40 1501.50 1608.58 1713-31 2467.10 26 154 BELTS AND PULLEYS. TABLE OF WIDTHS OF VULCANIZED-RUBBER BELTS OVER CAST-IRON PULLEYS, WHEN a = 135 AND d = -fa". From Formulas (596)- (600). Width in inches. , single leather- lacing. , single rawhide- lacing. , double V leather- lacing. IT , double V rawhide- lacing. , riveted V joint. No. I 0.0844 0.0910 0.0975 o. 1038 0.1495 I I* O.I266 0.1365 o. 1462 0.1557 0.2243 2 2 0.1689 O.I82O 0.1950 0.2076 0.2990 3 2i O.2III o 2275 0.2437 0.2596 0.3738 4 3 0.2533 0.2730 0.2924 0.3H5 0.4486 5 3i 0-2955 0.3185 0.3412 0.3634 0.5233 6 4 0.3377 0.3640 0.3899 0.4153 0.5981 7 41 0-3799 o 4095 0.4387 0.4672 0.6728 8 5 0.4221 0.4550 0.4874 0.5191 0.7476 9 5 0.4643 0.5005 0.5362. 0.5710 0.8224 10 6 0.5066 0.5460 0.5849 0.6229 0.8971 ii 7 0.59TO 0.6370 0.6824 0.7267 I . 0466 12 8 0.6754 0.7280 0.7/99 0.8306 I. 1962 13 9 0.7598 0.8189 0.8773 0.9344 1-3457 14 10 0.8443 o . 9099 0.9748 1.0382 1.4952 15 ii 0.9287 I . 0009 1.0723 I . 1420 1.6447 16 12 I.OI3I 1.0919 1.1698 1.2458 I . 7942 17 14 I.I820 1.2739 1.3647 1.4535 2.0933 18 16 1.3508 1-4559 1-5597 1.6611 2.3923 19 18 I.5I97 1.6380 1-7547 i. 8688 2.6914 20 20 1.6885 1.8199 i . 9496 2.0764 2.9904 21 22 1.8574 2.0019 2 . 1446 2.2840 3.2894 22 24 2.0262 2.1839 2 3396 2.4917 3.5885 23 26 2.I95I 2.3658 2-5345 2.6993 3-8875 24 28 2 . 3640 2.5478 2.7295 2 . 9070 4.1866 25 30 2.5328 2.7298 2.9245 3.II46 4.4856 26 Example. Required the width for a vulcanized-rub- ber belt f inch thick which will transmit a force of 1200 pounds, the fastening being a single rawhide-lacing and the arc embraced by the belt on the smaller pulley being a =. 90. VULCANIZED-RUBBER BELTS. 155 Formula (441) gives b X - 0.00563 X 1 200. 4 Hence b 0.00563 X 1200 X -, o or b = 9". Example. Required the width for the above belt with riveted joint instead of single rawhide-lacing. We have from formula (489) b X - = 0.00343 X 1 200, 4 b 0.00343 X 1200 X -, or b = 5489" = Sir- Example.- A vulcanized-rubber belt ^-inch thick t embraces an arc equal to \ the circumference of its smaller pulley, and transmits a force of 20 horse-power at a velocity of 10 feet per second. Required the proper width for double leather-lacing. Formula (539) gives I 20 * X 4 = 2 '44 * , b 2.404 X 2 X 4, or b = 19.232" --= 1 944". I ?6 BELTS AND PULLEYS. Example. A three-ply vulcanized-rubber belt run- ning over two equal pulleys transmits a force of 1275 pounds. Required the proper width for single raw- hide-lacing. The table on page 151, column for single rawhide-lacing, line 22, gives, corresponding to P = 1276.84 pounds, b = 22". Example. Given the data H = 20, v = 20, a =. 135, d ^-inch. Required the proper width for the belt, for single rawhide-lacing. TT The table on page 1 54 gives, corresponding to = I , a belt-width of II inches. (Column for single rawhide- lacing, line 16.) Vulcanized-rubber belts are very rarely seen running, over leather or rubber covered pulleys. We may, how- ever, take for the coefficients of friction of rubber on leather and rubber on rubber, respectively, cp 0.50* and

*/ = 0.4 + + --,. . . . (604) in which d represents the diameter of the shaft upon which the pulley is keyed, and R the radius of the pulley. The length of the nave should not be taken less than L = 2.$ow (605) Often (in idle pulleys, for example) the length L is taken equal to the face-width B of the pulley. Example. A pulley of 36 inches diameter is keyed upon a shaft of 4 inches diameter ; required the nave dimensions. From formula (604) the thickness is A 18 w = o.4 + + = 1427", 1 62 BELTS AND PULLEYS. and from formula (605) we have for the length of the nave L = 2.$ox 1-427 = 3-5675". In idle pulleys the interior diameter of the nave, or the eye of the pulley, is taken slightly greater than the FIG. 51. diameter of the shaft upon which the pulley is to run ; often the eye of an idle pulley is furnished with a coat- ing of bronze or white metal, in order to diminish the friction. Keys. There are three kinds of keys which are used to fix pulleys upon their arbors : the hollow key (Fig. FIG. 52. FIG. 53. FIG. 54. 54), used for light pulleys; the flat key (Fig. 52), used for pulleys of medium size ; and the countersunk key (Fig. 53), used for very large and heavy pulleys. RIM, NAVE, AND FIXING-KEYS FOR PULLEYS. 163 The width s and thickness s 1 of the fixing-key are given by the expressions q> ' d Sl ~ ~ 10' . . (6o 7 ) and the inclination varies from y^- to - Example. Required the width and thickness of the fixing-key for the pulley of the preceding example, in which the diameter of the shaft is d = 4''. For- mulas (606) and (607) give for the required width and thickness, respectively, and = 0.96", ^ = 0.16 + - = 0.56". r 10 FIG. 55. Split pulleys (Fig. 55) are often used for light work. They offer the advantage of being easily put up and taken down without interfering with the shaft-hang- ing's. With oullevs of this kind fixing -kevs mav be 164 BELTS AND PULLEYS. dispensed with, the two parts of the pulley being pressed upon the shaft by means of the nuts a, a, with sufficient force to prevent slipping. For this purpose the eye of the pulley is made slightly less than the diameter of the shaft upon which the pulley is to be fastened. When the division passes through a pair of FIG. 56. arms, as in the figure, each half of the split arm must be as strong as an entire undivided arm, and conse- quently of the same dimensions as the entire arms.* Weight of Pulleys. The weights of pulleys can evi- dently be calculated from one formula only approxi- mately, since the arms, nave, etc., vary considerably in * A better and stronger form of split pulley is represented in Fig. 56. In this case all the arms are entire, and the pulley presents a better appearance, as well as a simpler form. According to Unvvin (see "Elements of Machine Design," 168), the net section of the bolt at the rim should be one quarter the section of the rim plus \ square inch, and that of the bolt at the nave one quarter the section of the nave plus square inch. RIM, NAVE, AND FIXING-KEYS FOR PULLEYS. different pulleys. We may. however, calculate the weights of pulleys with sufficient accuracy for ordinary purposes from the formula 7? / /A 2 / 7?\ 3 \ G = (0.163 ? + 0.015^ j + 0.00309^) )b\ . (6oS) in which G is the weight of the pulley in pounds, R and b respectively the radius of the pulley and width of the belt. 2 The following table gives values of yj for different r> values of -r- : TABLE OF WEIGHTS OF PULLEYS. R b G 3 R b G b* R b G b* R b G b* 1.0 0.181 2-5 0.550 5-0 1-579 8.25 4.111 .1 0.202 2.6 0.580 5-2 1.691 8.50 4.378 .2 0.223 2.7 0.612 5-4 1.807 8.75 4.657 3 0.244 2.8 0.642 5.6 1.929 9.00 4-947 4 0.266 2.9 0.675 5.8 2.057 9-25 5.250 5 0.289 3-o 0.708 6.0 2. 190 9-50 5.567 .6 O.3I2 3-2 0.777 6.2 2.329 9-75 5.895 7 0-335 3-4 0.850 6.4 2.473 IO.OO 6.237 .8 0.360 3-6 0.926 6.6 2.623 10.25 6.592 !-9 0.385 3-8 .007 6.8 2.780 10.50 6 961 2.0 0.4II 4.0 .oSq 7-o 2-943 11.00 7.742 2.1 0-437 4.2 .180 7-25 3-155 11.50 8.581 2.2 0.464 4 4 273 7-50 3-378 I2.OO 9.482 2-3 0.492 4.6 370 7-75 3.6II 12.50 10.446 2.4 0.520 4-8 .472 8.00 3.856 13.00 n.475 Example. The radius of a pulley is 16 inches, and the width of the belt which runs upon the pulley 4 inches ; required the approximate weight of the pulley. /? T 6 Here , = 4. From formula (608), l66 BELTS AtVD PULLEYS. G = (0.163 X 4 + 0.015 X 1 6 + 0.00309 X 64)64, G = (0.652 -f- 0.240 4~ 0.19776)64 = 1.08976 X 64 ; or, G = 69.74 pounds. Example. Required the approximate weight of a pulley for the data R = 36", b 4^". In this case T- = ~ 8, and tf 91.125. From the table we find R _ % = 3.856. Hence G = 3.856 X 91.125 = 351.378 pounds. 14. Arms of Pulleys* Ordinarily the arms of pulleys have oval cross-sec- tions, the diameter in the plane of the pulley being twice the smaller diameter. The profile of such a cross- K section may be drawn by circle- arcs as shown in Fig. 57. The dotted circle is drawn on the greater diameter h^ of the pul- \ C 1 l e y- arm > an d the arcs ab and a'b' have their centres respec- J\ ^SSBP^ /' tively in the points c and c f . ^rS! The arcs ab and a'b' are con- nected at their ends by small cir- cle-arcs as shown in the figure. The axes of pulley-arms may be straight as in Fig. ARMS OF PULLEYS. 167 58, curved as in Fig. 59, or double curved in the form of a letter 5. Single-curved arms may be drawn in the following manner: Take (Fig. 59) the arc AE equal to f the arc EF, determined by the centre s of the arms at the rim of the pulley, and draw A^O perpendicular to AO. From the centre D draw CD perpendicular to Jl:' FIG. 58. FIG. 59. OE, and the point C of intersection of DC and OC is the centre for the curved axis of the arm. The number of arms (TV) necessary for pulleys of different sizes may be determined by means of the formula or the following table calculated from it : R T- = i 2 3 4 5 6 7 8 9 10 11 12 13 5 1 68 BELTS AND PULLEYS. The formula . . (610) gives the greater diameter for the pulley-arms. The diameter or width h is taken at the nave as shown in Fig. 58, and the width /i l at the rim may be conven- iently taken equal to J/z. These expressions have been determined, with a certain approximation from the most accurate formulas ; for large and medium sized pulleys they are especially applicable, but for small light pulleys the dimensions should be slightly in- creased in order that the pulleys may be easily cast without taking special precautions. Example. Required the numbef of arms and the arm dimensions for a pulley having a radius of 1 8 inches, the belt for the pulley being 6 inches wide. Here R i% From the above table we find the number of arms to be N = 4, and formula (610) gives for the width of the arms in the plane of the pulley 6 18 h = 0.24 + - + - - = 2.19". 1 4 10 X 4 The width at right angles with the plane of the pulley is therefore A, = |- X 2.19= 1.46*. ARMS OF PULLEYS. 169 To trace the profiles of the arms proceed as follows: Straight arms (Fig. 60). Having drawn the diameter EOC, take ab = cC = Cd f //, and draw the lines ac and bd, which give the limits of the profile. Connect ac and FIG. 60. FIG. 61. bd with the rim and nave by small circle arcs, and the profile is complete. Curved arms (Fig. 61.) The centre C for the axis having been determined, draw the straight line ad, then take aE =. Eb = and Cc = Cd -? ; the 3 6 points c and d thus determined are the centres for the arcs which limit the profile, and cb and da are the radii. Double-curved arms.* Fig. 62 shows a simple method for drawing double-curved arms. Draw the radial line oA, making 30 with the horizontal. Take oc \oA, and through the point c draw the line pD, making 60 with the horizontal. Intersect the line *From the author's "Treatise on Toothed Gearing." 170 BELTS AND PULLEYS. pD by a horizontal line through the point A : the points D and/ are respectively the centres for the arcs oc and cA, which together form the axis of the arm. Lay off the arm-widths as shown in the figure. From the FIG. 62. point / as a centre strike the arcs ab and ef, and find upon the line oD the centres for the remaining arcs bd and fk f . Another very similar method for drawing double- curved arms is shown in Fig. 63. Draw the radial line MM oA, making 45 with the horizontal. Take oc = -oA, and through the point c draw the vertical line pD. SHAFTS. 171 Intersect the line pD by the horizontal line Ap. The points/ and D are the centres for the arcs of the axis. Lay off h and //, as shown in the figure, and proceed, as in Fig. 62, to strike the arcs ah, ef y bd, and fk'. 15. Shafts* When a shaft is so supported by its bearings as to be subjected to a torsional strain only, as is almost in- variably the case in pulley-shafts (the bending strain due to the weight of the pulley and the force trans- mitted by the belt being ordinarily slight enough to be safely neglected), the calculation of the proper strength for the shaft may be made as follows : FIG. 64. In Fig. 64, P represents the total force tending to twist the shaft, i.e., the total force transmitted by the belt ; R the distance from the centre of the shaft to the point at which the force acts, i.e., the radius of the pulley ; and d the diameter of the pulley-shaft. The *From the author's " Treatise on Toothed Gearing." 172 BELTS AND PULLEYS. greatest safe torsional strain which can be sustained by the shaft is given by the expression in which f is the greatest safe shearing stress in pounds per square inch for the material of the shaft. From this, . . , PR d O.I9535/" or, d= 1.720 \l-jr (6n) RULE. To determine the diameter of a pulley-shaft of any material multiply the total force transmitted by the belt by the radius of the pulley, divide this pro- duct by the greatest safe shearing stress in pounds per square inch for the material of the shaft, extract the cube root of the quotient thus obtained, and multiply the result by 1.720. Example. Required the diameter for an oak shaft upon which is a 6o-inch pulley transmitting a force of loco pounds, taking f 500 pounds. From formula (6 1 1) we have = 1.720 = 1.720x3-915 = 6.734"= We propose to take for steel f f = 12000 pounds; for wrought-iron/' = 8000 pounds ; and for cast-iron f = 4000 pounds. These values of f are nearly SHAFTS. 173 mean between those used by Stoney, Haswell, and Unwin, which differ far more than is conducive to any degree of accuracy. Substituting the above values of f successively in formula (611) and reducing, we obtain, For steel, d = 0.075 *V~PR (612) For wrought-iron, d 0.086 *V~PR (613) For cast-iron, d = o.io!& *V~PR (614) RULE. To determine the diameter for a pulley- shaft of steel, wrought or cast iron, multiply the total force transmitted by the radius of the pulley, extract the cube root of the product, and multiply the result by 0.075 for steel, 0.086 for wrought-iron, and 0.108 for cast-iron. Example. A 48-inch pulley transmits a force of 1000 pounds. Required the diameter for a steel shaft. .From formula (612) we have d 0.075 VTooo X 24 = 0.075 X 28.84, or, d 2.163" = 2 ii /x nearly. Example. Taking the data of the preceding ex- ample, required the diameter for a shaft of cast-iron. Formula (614) gives d 0.108 Viooo X 24 = 0.108 X 28.84. or, d = 3- 1 1 5" = 3-g-" nearly. Formulas for the diameters of pulley-shafts in terms 174 BELTS AND PULLEYS. of the horse-power transmitted and the revolutions per minute may be obtained as follows: As before explained, we have the expression P = 63000 Rif H representing the horse-power, R the radius of the pulley, and n the number of revolutions per minute. Substituting this value in formulas (611), (612), (613), and (614), and reducing, we obtain the following: General formula, d = 68 44 \ / For steel, For wrought-iron, d = 3.422 A / . .... (617) d= 2.984^7- (616) For cast-iron, d = 4.297 H (618) RULE. To determine the diameter for a pulley- shaft of any material from the horse-power and num- ber of revolutions per minute, divide the horse-power by the product of the number of revolutions into the greatest safe shearing stress in pounds per square inch for the material of the shaft, extract the cube root of the quotient thus obtained, and multiply the result by 68.44. SHAFTS. 175 To determine the diameter for a pulley-shaft of steel, wrought or cast iron, from the horse-power and number of revolutions per minute, divide the horse- power by the number of revolutions, extract the cube root of the quotient, and multiply the result by 2.984 for steel, 3.422 for wrought-iron, and 4.297 for cast-iron. Example. Required the diameter for an oak pulley- shaft which transmits a force of 10 horse-power and makes 40 revolutions per minute. If we take for the greatest safe shearing stress for oak f 500 pounds per square inch, we shall have, from formula (615), ~-^-> 12.60 3 / IO 3/1 dfr= 68.44 \ - 68.44A/- - = 68. vy 40x500 f V 2000 or, d = 5.432" = 5 T y nearly. Example. Taking the data of the preceding ex- ample, required the diameters for shafts of steel and wrought-iron. From formula (616). d = 2.984 \ 2.984 Vo.25 2.984 X 0.62996, V 4 or, for steel, d = 1.88" = i|". From formula (617), 3 /IO d == 3422 W = 3422 X 0.62990, or, for wrought-iron, BELTS AND PULLEYS. Pulley-shafts are most commonly of wrought-iron ; when, however, wrought-iron shafts, in order to give the necessary strength, become so large as to be incon- venient, steel shafts are used. Cast-iron shafts are, as a rule, unreliable and treacherous ; they are therefore seldom used except for the transmission of slight powers and in cheap, inferior machinery. The follow- ing tables, calculated from formulas (612), (613), (616), and (617) to the nearest -^ inch, will be found very convenient in designing pulley-shafts of steel and wrought-iron : TABLE OF SHAFT-DIAMETERS. d for steel. rtf for wrought-iron. PR d for steel. dior wrought-iron. 250 500 1000 1500 2000 25OO 3000 3500 4OOO 4500 5OOO 6000 7000 8000 1 0000 12500 15000 20000 25 oo 30000 35000 40000 45000 50000 'A 'H 2 1% 2M 60000 70000 80000 90000 I 00000 IIOOOO 120000 130000 140000 150000 175000 200000 250000 500000 750000 I 000000 1500000 2000COO 2500000 3000000 3500000 4000000 4500000 5000000 SHAFTS. TABLE OF SHAFT-DIAMETERS. H d for steel. ^ for wrought-iron. H d for steel. dior wrought-iron. 0.025 0.050 0.075 O.IOO 0.150 0.200 O.25O 0.300 0.350 O.4OO o 500 0.600 0.700 0.800 0.900 I. 1.25 1.50 1-75 2. 2.25 2.50 2.75 3- 3-25 3-50 If 2i 2| 2| 2| 2| 3|| 488> or 6' 2.126" = 2i". Because of the circular cross-sections of rope-belts and the character of the material generally used, it is necessary that the wear due to the bending of the ropes about the pulleys be reduced as low as possible. To this end very large principal pulleys are used from about 7 to 15 feet in diameter commonly. It is a safe rule, that the diameter of the smaller principal pulley should not be less than thirty times the diameter of the rope, and when small ropes are used we may con- veniently increase the durability by taking the diameter of the smaller pulley equal to 45 to 60 times the diame- ter of the rope. Thus in the three examples given IQ 2 BELTS AND PULLEYS. above we may have for the diameters of the smaller principal pulleys 30 X 2.44 = 73.20'' = 6J feet, 30 X i-9 = 57" = 5 ^et nearly, and 30 X 2.13 = 63.9" = Si feet. The ends of rope-belts are usually spliced together by pressing them firmly together and winding about with stout small rope. The spliced part should be as long as possible in order to bend properly over the pulleys and give the necessary strength. The weight per foot of length of rope-belts is approximately given by the formula G = o.3

= 0.28, a = 0.95^, 6) 30, t T T I / / ^ = 0.12, ^=1.15, --=1.27, ^, = 0.105. (626) By making use of these values we may obtain for the diameter of the joint pivots (d. Fig. 74) the ex- ~ff d = 0.0146 VP= 3.656 ; .... (627) rrr 0.04H -(PR) = 1.644 /T ~. (628) We should take for jointed chain-belts the following proportions (see Fig. 74) : / b c i e i h d = ^ d = *' d = i> d = s' d = 2 *' ( 62 9) For small pulleys it is convenient to take In practice d should not be taken less than 0.32 inch, even when a smaller diameter would be sufficient for strength. In jointed chain-belts the limit of the force * P = force in pounds transmitted, // = horse-power, n ~ revolu- tions pc*' minute. JOINTED CHAIN-BELTS. 195 P which maybe transmitted (supposed to be applied at the circumference of the pulley) is about 500 pounds, which would require a width cf about 1 1 inches in a simple leather belt. Example. Given the data H = 20, n = 50, n, = 100. Required the dimensions for a jointed chain-belt, sup- posing the radius of the smaller pulley to be R t = 5/. Formula (628) gives . , 20 d-.-. 1.644 X = i.6 4 4 \ -- = - = 0.5624" = A". V 2 5 2.924 From formula (629) then we obtain / = 3 X 0.5624 = 1.6872", b = 2| X 0.5624 = i 5466", c = 0.2", * = 0.12", h 2\ X 0.5624 = 1.2185", -ffj = s/ = 8.436", R = 2XR,= 16.872". Clissold has also invented a transmission by means of a thick belt with trapezoidal section. This, how- ever, has proved poorly because of its want of dura- bility.* * The experiments of Wedding of Berlin have shown that in an Angular groove, the angle being 30 (Fig. 50), the force necessary to produce slipping of the cable is twice that corresponding to a cable lying in a round groove. This confirms the preceding expressions, i i since sin 30 2 TRANSMISSION BY METALLIC CABLE. 19. Tensions of Cables. Transmission of forces by means of metallic cables was first introduced about the year 1850, by the Hirn brothers.f The use of metallic cables, by means of which we are able to transmit great forces at distances as great as several thousand feet without notable loss, depends essentially upon the principles of transmission by belt, the principal difference being that with a metallic cable the tension is due to its own weight. The two principal pulleys of a transmission by cable, as a general thing, have their axes parallel ; also the pulleys are in the same plane, so that the cable may be driven without guides. Moreover, the axes of the principal pulleys are ordinarily in the same horizontal plane, forming what is termed a horizontal transmis- sion. An inclination of the plane of the axes to the * From Reuleaux. f In this first application the axes of the pulleys were about 280 feet apart; the force transmitted was 42 horse-power, at 60 revolutions per minute. TENSIONS OF CABLES. 197 surface of the ground constitutes an oblique transmis- sion. Vertical transmissions by metallic cable are very rarely used. When the driven pulley transmits to a third pulley the force which it receives from the driver the transmission is said to be compound. In a simple transmission by cable the two pulleys are ordinarily of the same diameter. In order to prevent the cable from touching the ground, when the height of the pulleys above the ground is insufficient and the separation of the axes great, intermediate rollers are used to support the cable. By inclining the rollers more or less they may be used for guides when the axes of the pulleys cross or intersect each other. We meet, however, very few examples of transmission by cable in which the axes of the pulleys are not parallel. When it becomes necessary to give to the cable a considerable deviation, we can place between two vertical rollers a horizontal guide ; but it is preferable in such cases to rely upon a compound transmission, with pulleys placed obliquely to each other. The inferior limit for the separation of the pulley- axes in transmissions by metallic cable should be about 50 feet. , The distances between the rollers which support the cable are determined by the flexibility of the cable and its position above the ground. The transmission-cables ordinarily used are com- posed of 36 iron wires divided into six twists, each containing six wires twisted around a central core of hemp ; the six twists are likewise twisted around a larger core, also of hemp (Fig. 75). When it is neces- 198 BELTS AND PULLEYS. sary to strengthen the cable, we may, without serious disadvantage, replace the central hempen core by a twist of iron wire similar to the six others. It has also been proposed to replace by an iron wire the smaller hempen cores of the separate twists, in order to over- come the looseness of the cable, which may tend to produce a rapid wear. The value of such an arrange- ment yet remains to be established. It has the dis- FIG. 75. FIG. 76. advantage of destroying the elasticity of the cable. When the cores are of hemp, it is of first importance that first quality hemp be used in their manufacture, instead of the inferior qualities which have been hither- to extensively used for this purpose. The wires com- posing the cable should be forced firmly together, so that the diameter of the cable is not more than eight times that of the wire. In cables having more than 36 wires the number of twists is generally six, and the large and small cores of hemp. TENSIONS OF CABLES. 199 f While there is no absolute necessity of limiting the number of twists to six, this number is almost always used: in the different cables in use the total number of wires is therefore 36, 48, 54, 60, 66, 72, etc. Fig. 76 represents a cross-section of a cable of 60 wires. In these different cables the relations between the external diameter d and the diameter d of the wires are as follows : For the number of wires i = 36 48 54 60 66 72, - 8.00 10.25 11.33 12.80 13.25 14.20. In order to obtain the tensions T and t in metallic cables we make use of the formulas determined for tensions in ordinary belts. By substituting in these formulas a coefficient of friction cp 0.24, and an arc of the pulleys equal to -J- the circumstance, a iSo = n, we may obtain the relations T T+t t = z-02, - = 2.99, ,= 0.48; (630) ' or, in round numbers, T T+t t > = 2, r- =3, =0.5.* . . (631) * The loss of velocity due to the shipping of the cable does not ordinarily exceed -fa per cent; it may therefore be neglected alto- gether in our calculations. 20O BELTS AND PULLEYS. 20. Calculation of Diameters of Cables. In a transmission by metallic cable composed of i wires the tension T in the cable corresponds to a ten- sion 5 in the wires; this tension should not exceed 25601.4 pounds per square inch of section.* To de- termine the diameter d of the wires the following for- mulas may be used : For a resistance of P pounds acting at the circum- ference of the pulley, fP -- I - 62 7y^ (632) For a force of H horse-power, with a velocity of v feet per second at the circumference of the pulley, 7^%, . ; 37-86 7 y<-,. . in which v should not materially exceed 100 feet per second. For a force of H horse-power at n revolutions of the pulleys per minute, * = 4070.04^/5^. . . (634) If we represent by s = 25601.4 5 the tension pro- duced in the wires by the bending of the cable around the pulleys, and by (PR) the statical moment of rota- tion of the driven pulley, we shall have (635) * 18 kilograms per square millimetre. CALCULATION OF DIAMETERS OF CABLES. 2OI Finally, if in place of the moment (PR) we have the horse-power and revolutions per minute, d = 0.227 H Sni (636) It is, moreover, important that the ratio of the radius of the pulleys to the diameter of the wires be taken not less than the limit, R _ 14223000 (637) This relation serves to calculate the following table : s s R 8 s J R 8 711.15 24890.25 571 12800.70 12800.70 mi 1422.30 24179. 10 588 14223.00 11378.40 1250 2844.60 22756.80 625 15645.30 9956.10 1429 4266 . go 21334.50 667 17067.60 8533.80 1667 5689.20 19912.20 714 18489.90 7111.50 2000 7IH.50 18489.90 769 19912.20 5689.20 2500 8533.80 17067.60 833 21334.50 4266.90 3333 9956.10 15645.30 909 22756.80 2844.60 5000 11378.40 14223.00 1000 24179.10 1422.30 ICOOO For a constant value of S -\- s the minimum value of the radius of the pulleys is given by the table by making ~ = 2.* This minimum value corresponds to o 8 /T~ * We may obtain from formulas (636) and (637) R = Ky - ' S O The sum s -{- S being constant, the maximum value of the product obtained by making = 2. 202 BELTS AND PULLEYS. /? 7? S= 8533.8, s = 17067.6, 833. For values of y nearly equal to 833 the numerical value of R differs very little from the minimum value ; we may there- fore safely give somewhat greater values to R when, by so doing, we can make use of patterns and models already on hand. The two tables which follow have been calculated from formulas (632)-(634), and (635) and (636) respec- TT tively. In the first table we have given 1000 -zr~- in order to avoid the small numbers which result from H SRn' Diameter 8 for number of wires / = P S H Sv H JOOO hn 36 42 48 60 72 O.O2O 0.0184 o 0172 0.0156 0.0140 0.0054 O.COOOIO 0.000088 0.024 0.0220 0.0208 0.0184 0.0168 0.0078 0.000015 0.000123 O.O28 0.0260 0.0244 0.0216 0.0! 96 0.0107 O . OOOO2O O.OOOI75 0.032 0.0296 0.0276 0.0248 o 0228 0.0139 o . 000026 O.OOO229 0.036 0.0332 0.0312 0.0280 0.0256 0.0176 0.000033 0.000281 o 040 0.0368 0.0348 0.0308 0.0284 . 02 I 8 0.000040 0.000352 0.048 0.0444 0.0416 0.0372 0.0340 0.0313 o . 000060 o . 000492 0.056 0.0516 0.0484 0.0432 0.0396 0.0426 0.000079 o.ooo6S6" o 064 0.0592 0.0556 0.0484 0.0452 0.0557 0.000103 0.001072 0.072 0.0664 0.0624 0.0556 0.0508 0.0705 0.000131 0.001125 0.080 o . 0740 0.0692 0.0620 0.0564 0.0870 0.000160 0.001389 o 088 0.0812 0.0764 0.0680 0.0624 0.1331 0.000195 0.001688 0.096 0.0888 0.0832 0.0744 0.0680 o . i 408 0.000232 0.002004 o. 104 o . 0960 o . 0900 o . 0804 0.0736 0.1471 0.000272 0.002356 O.II2 0.1036 0.0968 0.0868 0.0792 0.1586 0.000306 0.002725 O. 12O 0.1108 0.1040 0.0928 0.08-18 0.1958 o . 000364 0.003329 In metallic transmission-cables, wires of less than O.O2 inch or more than 0.08 inch diameter are very seldom CALCULATION OF DIAMETERS OF CABLES. 2O3 used. The values of d given in these two tables, in the second to the fifth columns, are taken from values contained in the first column, and should in practice be taken in round numbers. The quality of the metal used for transmission-cables is of first importance, from the fact that only superior qualities can withstand for any length of time the rapid wear to which the cables are subjected. Swedish iron, which possesses at the same time a remarkable tenacity and great strength, is especially adapted for the wires of transmission-cables. In order to reduce as much as possible the number of joints, only long wires should be used. Experience has shown that for transmission-cables wires of steel offer no advantages over those of good iron. Diameter of wire 8 for number of wires i f(/v) s H S n 36 4* 48 60 72 0.020 0.0188 0.0180 o.oi6& 0.0160 1554 0.025 O.O24 0.0228 0.0220 0.0204 0.0192 2685 0.043 0.028 O.O264 0.0256 0.0236 0.0224 4264 0.068 0.032 o . 0304 O.O292 0.0268 0.0252 6365 O.IOI 0.036 o 0340 0.0328 o . 0304 0.0284 9062 0.144 0.040 0.0380 0.0364 0.0336 0.0316 12431 0.197 0.048 0.0456 0.0436 0.0404 0.0380 21481 0.341 0.056 0.0532 0.0508 0.0472 0.0444 34112 0.542 0.064 o . 0608 0.0580 0.0540 o 0508 50919 0.894 O.O72 0.0684 0.0656 o 0608 0.0572 72499 I.I52 0.080 0.0764 0.0728 0.0676 0.0636 99451 1.580 0.08S 0.0836 o . 0800 0.0744 0.0700 132369 2.103 0.096 0.09.12 0.0872 0.0808 0.0760 171851 2.730 0.104 0.0988 0.0944 0.0876 0.0824 218493 3-471 O. 112 0.1064 0.1016 0.0944 0.0888 272892 4-335 O. I2O 0.1140 0.1092 O.IOI2 0.0952 335646 5.332 In the formulas (632)-(634) the radius R of the pul- leys is supposed to be known ; the values of d given 204 BELTS AND PULLEYS. r> by them are admissible only when the ratio -~- gives for the tension s a value which, added to S, does not exceed 25601.4 pounds. In the case where j-f-^ ex- ceeds this limit, it is convenient to begin the calculation by giving to 7? a greater value. To make use of the preceding formulas and tables, we must begin by fixing upon a value for the tension *S. This may easily be done with the aid of the considerations contained in the following paragraph, and in the examples which we now give we shall suppose this preliminary opera- tion already accomplished. Example. It is proposed to transmit, by means of a metallic cable running over pulleys 9.84 feet in diame- ter, a force of 550 pounds : required the proper diameter for the wires of the cable, supposing the number to be i = 36. If we take 5 = 9956.1, we shall have -= = - j 995 - 1 0.0552, which in the first table (column 6, line 9) cor- responds to a diameter of $ 0.064 inch. From this we obtain -^ = '-? = 022, which, in the table on o 0.064 page 201, corresponds nearly to S = 9956.10, and is therefore admissible. If we had taken R = 48 inches, r> , o we should have had - F = ^ = 750 a value less o 0.004 than the limit mentioned above, and it would therefore be necessary to increase the value of fi. Example. The force transmitted by a metallic cable is 300 horse-power, and the velocity v = 82 feet per CALCULATION OF DIAMETERS OF CABLES. second; taking S = 11378.4, and consequently s = TT 25601.4 11378.4 = 14223, we shall have -^r- = ^TJ OQQ ; - = 0.000322. In the first table the near- 11378.4 X 82 TT est value of -~- is 0.000306 (column 7, line 15). The oZ' diameter for the wires is therefore d = 0.112 inch for i =36, d =. 0.0868 inch for i = 60. For the value s 14223, we have, for the radius of the pulleys, R = 14223000 X 0.0848 = 84.8 inches. 1 he expression v = 14223 2rtRn rr-f- gives for the number of revolutions per minute 1 2 X OO 82 X 12 X 60 : -6^T 4 :8- =UI - Example. It is required to calculate the horse-power which may be transmitted by a cable of thirty-six wires, the diameter of the wires being 0.08 inch, the diame- ter of the pulleys 9.84 feet, and the number of revolu- R 59.04 tions per minute go. In this case we have -$ =. _ tf 0.08 T A *) *} 'JOOO = 73^, which, from formula (637), gives s = - ^~ = 19272.3 and 5 6329.1. For d = 0.08 and i = 36, TT the first table furnishes the value 1000-^75- = SRn 0.001389; hence H _ 0.001389 _ 0.001389 X 6329.1 X 59-4 X 9Q 1000 1000 = 46.71 horse-power. With a pulley of 8 feet diameter 206 BELTS AND PULLEYS. R 48 14223000 we would have -? = TT = ooo, s = 7 = 23705, o 0.08 600 S = 1896.4. Consequently 0.001389 X 1896.4 X 4B X 90 H = = 1 1. 40 horse-power. 1000 Example. Upon the driven arbor of a transmission by cable a resistance of no pounds acts continuously with a lever arm of 40 inches. Required the proper diameter for the 36 wires of the cable, supposing we give to the pulleys the smallest admissible radius. In order to satisfy this last condition, we ought to take (from what precedes) s 17067.60 and 5 = 8533.80, which gives -^ (PR) = 2 X 1 10 X 40 = 8800. In the ^ second table (column 6, line 5) we find, for the nearest value of --(PR), d = 0.036 inch. From the table on p page 201, therefore, we obtain -~ = 833, R = 833 X 0.036 = 30 inches. Example. A cable of 42 wires transmits a force of 30 horse-power at a velocity of 100 revolutions per minute. Required the proper diameter for the wires of the cable, taking 5 = 8533.80. In this case s , s H 17067.60 30 17067.60, and -~ = - - X - - 0.6. The sec- 5 n 8533.80 100 ond table gives, for the nearest value of -~ to 0.6, o fl d 0.056 inch. From formula (637), then, we have for the radius of the pulleys R = 0.056 X 833 X 0.056 = 46.65 inches. DEFLECTIONS IN A CABLE. 2O/ 21. Deflections in the Cable of a Horizontal Transmission. In order that, in the two parts of a transmission- cable, the tensions T and/ have proper values (not too small, for then the cable will slip on its pulleys; nor too great, because the wear is then great), the deflec- tion which we give to each part, in a state of repose, must be a determined quantity. It is equally necessary that we know the deflections which are produced dur- ing the motion of the cable, in order to leave sufficient room for the passage of the cable. The deflection of a cable depends upon the tension of its wires. Let us represent by A the separation of the pulleys of a horizontal trans- mission in feet; h the deflection of the cable in feet (h l for the driving part, /z 2 for the driven part, and 7/ for the state of repose) ; S the tension per square inch in the wires (S l for the driving part, 5 2 for the driven part, and S for the state of repose). For a metallic cable of any number of wires we have the relations 3 = 0-3535 [o.369 - y (0-369 J* - i (638) and 1 = 3-8029^+^). . . . (6 39 ) By means of these formulas the following table has 2O8 BELTS AND PULLEYS. been calculated. As a first approximation we may take simply h A - = 04755 --. .... (640) In order to make use of the table, we begin by A determining from the given quantities the ratio -~- of the separation of the pulleys to the tension developed in the wires, and then find in the table the number nearest to this ratio. From this we obtain the value of j , which gives the amount of deflection h. The ^i tension 5 of the cable in a state of repose is not the arithmetical mean between 5^ and S^ ; we may, by a more complicated calculation, however, determine it from the length of the two cable parts. The value which we need to know is the deflection h in the two parts of the cable for a state of repose, and we have approximately 0.287*,. . (641) This expression gives for h Q a value slightly too great, but which approaches more nearly the true value as the tensions 5, and 5 2 become less. The error may be still farther decreased by using, instead of exact values of A, and h those furnished by formula (640). The driving part of the cable does not necessarily DEFLECTIONS IN A CABLE. 209 occupy the higher position, as is the case in Fig. 77 : it may be placed in the lower position, as in Fig. 78. In the latter, the space required by the deflection of FIG. 77. the cable is considerably less than in the former. The two parts of the cable do not intersect each other as long as A z h^ < 2R. With a cable in motion, we may place, at the lowest point of the curve, a gradu- ated rule, by means of which we may observe at any instant the tensions. The graduation of the rule may, moreover, be such as to give directly the tension 5. 14 210 BELTS AND PULLEYS. TABLE OF DEFLECTIONS IN METALLIC CABLES. h A A S h A A s h A A S h ~A A S 0.003 0.006 0.033 0.069 0.063 O.I2S 0.093 0.183 0.004 0.008 0.034 0.071 0.064 o. 130 0.094 0.185 0.005 O.OII 0.035 0.073 9-065 0.132 0.095 0.186 0.006 0.013 0.036 0.075 0.066 0.134 0.096 0.188 0.007 0.015 0.037 0.077 0.067 0.136 0.097 o. 190 0.008 0.017 0.038 0.079 0.068 0.138 0.098 0.191 O.OOg 0.019 0.039 0.081 0.069 0.140 0.099 0.193 0.010 0.021 0.040 0.083 to . 070 0.142 0.100 0.195 O.OII O.O23 0.041 0.085 0.071 0.144 O.IOI 0.196 0.012 0.025 0.042 0.087 0.072 0.145 O.IO2 0.198 0.013 O.027 0.043 0.089 0.073 0.147 ! 0.105 0.203 0.014 0.029 0.044 0.091 0.074 0.149 O.IIO 0.2II 0.015 O.03I 0.045 0.093 o 075 0.151 0.115 0.219 0.016 0.034 0.046 0.095 o 076 0.153 0.120 0.226 0.017 0.036 0.047 0.097 0.077 T55 ! 0.125 0.234 0.018 0.038 0.048 0.099 0.078 0.156 O.I3O 0.24T 0.019 0.040 0.049 O. IOI 0.079 0.158 0.135 0.248 0.020 0.042 0.050 0.103 0.080 0.160 O.T40 0-255 0.021 0.044 0.051 o. 105 0.081 0.162 0.145 0.261 0.022 0.046 0.052 o. 107 0.082 0.164 0.150 0.267 0.023 0.048 0-053 0.109 0.083 0.165 0.155 0.274 0.024 0.050 0.054 0. Ill 0.084 0.167 0.160 0.279 O.025 O.052 0-055 0.113 0.085 0.169 0.165 0.285 0.026 0.054 0.056 o. 115 0.086 0.171 o. 170 0.291 0.027 0.056 0.057 0.117 0.087 0.173 o.i75 0.296 0.028 0.059 0.058 0.119 0.088 0.174 0.180 0.301 O.O29 0.061 0.059 O.I2I 0.089 0.176 0.185 0.305 0.030 0.063 O.o6o 0.123 0.090 0.178 o. 190 0.310 0.031 0.065 0.061 0.125 0.091 0.179 0.195 0.315 0.032 0.067 0.062 0.127 0.092 0.181 0.200 0.319 Example. In the last example of 20 the separation A of the pulleys is 360.8 feet, and we take the tension S\ 8533.8 pounds per square inch. Required the deflections in the parts of the cable. For the driving A 360.8 part of the cable the relation ~ = ^777-5 0.0423 DEFLECTIONS IN A CABLE. 211 corresponds in the table (column 2, line 18) to the value 0.02. Hence we have A l = 360.8 X 0.02 7.216 feet. For the driven part of the cable we have from O .3 .3 O formula (631) 5 = = 4266.9, and consequently 2 A 260 8 A -~ = -^7 = 0.0845. For this value of -~ the table 5 4266.9 5 gives (column 4, line 9) -j- = 0.041, and we have 7z 2 = A 360.8 X 0.041 = 14.79 f eet - From formula (641) the deflection of the cable in a state of repose is A Q = FIG. 78. 0.67 X 14.79 + - 2 8 X 7.216 = 11.93 feet. We have also A. 2 A l 14.79 " 7-2i6 = 7-574 and 2R = 2 X 3.8875 = 77750 feet. Since 2R > h^ A 19 we may if necessary make use of the disposition of Fig. 78. (See first example of 22.) Example. In the third example of 20 the distance b f elow the line of centres of the pulleys is 9.84 feet ; it is required to determine the proper distance between the pulley-centres. Assuming that we can make use of the disposition represented in Fig. 78, the greatest admissible value for the distance of separation of the pulleys may be calculated from the deflection of the cable while in a state of repose. Making use of the approximate formula (640), and remembering the value 212 BELTS AND PULLEYS. 5, 6642.141 pounds per square inch, we shall have A* and h, = 2//,. Formula (641) then gives _ (0.67 X 2 + 0.28)^" X 0.4755 A. = 9.84 - 6647.747" / 0.84 = V 5^7S5^ x6642.141 .. , -- - = = i/8484 7' 2 3 = feet. 22. Transmission by Cable with Increased Tension. When the pulleys of transmission are very distant from each other the deflections given by the preced- ing formulas become so great that it is often necessary to place the pulleys at a great elevation, or to provide a deep trench for the cable when we wish to dispense with intermediate pulleys and guides (see 28). In a great many cases we may arrive at the same result by simply giving to the cable a greater tension than is necessary to prevent slipping, and taking care to give to the cable a diameter sufficient to withstand the ad- ditional strain. This artifice may be employed all the more easily when the transmission is to be used for moderate forces, and consequently a small diameter of the cable is sufficient. It is only necessary to examine carefully the rules which follow, to be convinced that a rational use of this method presents in reality little or no difficulty. A transmission by cable, established under the above conditions, constitutes, by way of distinction from ordinary cable-transmission, what we term a TRANSMISSION WITH INCREASED TENSION. 213 transmission with increased tension. We may distin- guish it from ordinary transmissions by giving the sign s to the forces and dimensions connected with it (T 8 , 4, S s , 3 S instead of T, t, 5, and 6). The tension T, in the ordinary mode of transmission, ought not to be less than 2P; in a transmission with increased tension the tension ought to be increased by a certain factor which we shall designate by m. We shall there- fore have T a = mT, t B =(2in-i)t, A = - 2 i I. (642) The tension 5, in the driving part of the cable is not changed, but in the driven part the tension S zs is no longer equal to '. We take instead The diameter 3 8 of the wire is deduced from the diameter 3 given by one of the formulas (632) to (634), by means of the relation 3 S = 3 Vm. ..... (644) If, however, 3 is calculated from formula (636) or (638), we must take 3 8 = 3tym. ..... (645) From these formulas the following table has been calculated. It is important to remark, that in cables with increased tension the strain in the wires is no 214 BELl^S AND PULLEYS. greater than in ordinary cables, because they have a proportionately greater diameter. The cable is heavier in the former than in the latter case, and should there- fore be strained more firmly over the pulleys in order to reduce the deflection in the driven part. T 8 m= T 7s P t t 8 S28 t ~ P ~ S 2 S 2S _ ts Si ~ 7 a SS A<- y = \m y - \fm 1.2 2.4 1.4 0.58 I. 10 .06 1.4 2.8 1.8 0.64 1.18 .12 1.6 3-2 2.2 0.69 1.26 17 1.8 3.6 2.6 0.72 1-34 .22 2.0 4.0 3-o 0-75 1.41 .26 2.2 4.4 3-4 0.77 1.48 3 2.4 4.8 3-8 0.79 55 34 2.6 5-2 4.2 0.81 .61 38 2.8 5.6 4.6 0.82 .67 .41 3-o 6.0 5-0 0.83 73 1.44 3-2 6.4 5-4 0.84 79 1.47 3-4 6.8 5.8 0.85 .84 1-50 3-6 7.2 6.2 0.86 .90 1-53 3-8 7.6 6.6 0.87 95 1.56 4.0 8.0 7.0 0.88 2.00 i-59 4.2 8.4 7-4 0.88 2.05 1.61 4.4 8.8 7.8 0.89 2. IO .64 4.6 9.2 8.2 0.89 2.14 .66 4.8 9.6 8.6 0.90 2.19 .69 5-0 10. 9.0 0.90 2.24 71 5-5 II. 10. 0.91 2.36 75 6.0 12.0 II. 0.92 2-45 .82 6.5 13.0 12,0 0.92 2.55 .87 7 o 14.0 13-0 o-93 2.65 .91 7-5 15-0 14.0 0-93 2.74 .96 8.0 l6.O 15-0 0.94 2.83 2.00 Example. In the first example of 21 the driven part of the cable has a deflection of /z 2 = 11.76 feet, and the diameter of the wire is 0.056 inch. If we wish to diminish the value of // 2 by using a cable with increased tension, the value of 8 must be increased ac- cordingly. If we take m = 2, the table gives (col- TRANSMISSION WITH INCREASED TENSION. 21$ umn 4, line 5) -f = 0.75, S 2S = 0.75 X 8533.8 = 6400.35 A pounds. Consequently -~- = ^ - - = 0.056, which, in the table of 21, corresponds to T- = 0.027 or ^i h = 0.027 X 360.8 = 9.74 feet. The tension of de- flection s has the same value as if for an ordinary ca- ble ; the quotient -~r-- does not change its value, and o i i & consequently d may be determined by means of formu- la (636). The preceding table gives, then, d s = 1.26^ = 1.26 X 0.56 = 0.07 inch. When, in calculating the diameter $ for an ordinary cable of 36 wires, we obtain a very small value, the cable itself may have such a small diameter that its manufacture involves as great an expense as for a cable of larger diameter. In such a case we cannot recom- mend too highly the use of a transmission by cable with increased tension, which has the advantage of re- ducing the deflection in the driven part of the cable without appreciably increasing the expense of manu- facture. As a general rule, we should never make use of wires of a less diameter than 0.04 inch, so that the minimum diameter of cable may be 0.32 inch. Example. For a transmission by cable, we have given H = 5.5, n = 100, and A 590.4. If we as- sume vS, = 14223 and s 11378.4, we have -~- - Oj 11 = 2i-3 x J>1 5 - 0.044, which, for i = 36 (table on 14223 ioo page 203) gives, for the diameter of the wire $ 0.024 2l6 BELTS AND PULLEYS. \\i t, A 59-4 A inch. We have also - = 0.0415, -- = ' - 0.0830, and consequently, from the table of page 210, /i, = 0.0198 X 590.4 = 11.69 feet, A a = 0.04 X 590.4 = 23.616 feet, A 9 h, 23.616 11.69 = 11.926 feet. But since R = - d = 1250 113704 X 0.024 30 inches, A 9 A, is greater than 2R. In this case, therefore, we cannot place the driven part of the cable above the driving part, and the axes of the pulleys must have a height above the ground at least equal to R + h* = 2.5 + 23.62 = 26.12 feet. Sup- pose now we take for the cable diameter 0.32 inch, instead of 8 X 0.024 = 0.192 inch; that is, we take 0.04 inch for the diameter of the wires. We have then <5 S 0.04 -~- - = 1.67, and the preceding table gives (col- umns 6 and 4, line 18) 5 2S = 14223 X 0.89 = 13058.47. Consequently ~^- = - - 0.0452 and A 98 = 0.0228 -> 2S 13055.47 X 590.4 = 13.46 feet, A,, A, = 13.46 11.69 = 1.77 feet. As before, R = 1250 X <$ s = S inches and 2R -= 8.33 feet: the inequality h^ A, < 2R is now sat- isfied, and we may give to the cable the desired ar- rangement. The maximum deflection in this case cor- responds to the state of repose, for which we have, from formula (641), A os = 12.28 feet. The height of the pul- ley-axes above the ground must be at least A 08 + R 12.28 + 4.165 = 16.445 feet; that is, less by nearly 10 feet than for the first calculated cable. TRANSMISSION BY INCLINED CABLE. 217 23. Transmission by Inclined Cable. Of the various transmissions by metallic cable, the one which has met with the greatest development cor- responds to the case in which the pulleys are not on the same level, one being higher than the other, and constitutes, therefore, what we call an inclined trans- mission. We give here the rules necessary for such transmissions. In the cable B CD, Fig. 79, which rep- ni a' resents a part of an inclined transmission, the summit of the curved axis is not in the middle of the distance between the points of suspension, and the deflections are therefore different from those in the cable of a horizontal transmission. The deflections may, how- ever, be easily determined in functions of the elements of a horizontal transmission, having the same separa- tion of pulleys and sensibly the same tensions. Let us represent by h and A, respectively, the deflection of the cable and the separation of the pulleys of a horizontal trans- mission ; 2l8 BELTS AND PULLEYS. S the tension corresponding to the point of sus- pension of the part of the cable under consideration ; // and ti ', respectively, the smallest and greatest de- flection (FC and EC) in an inclined transmission, in which the separation of the pulleys measured horizon- tally is equal to A ; a' and a" , respectively, the distances CB^ and CD l of the summit of the curve from verticals through the points of suspension ; S / and S", the tensions (at B and D) at the lower and higher points of suspension respectively ; H the difference between the levels (EF) of the points of suspension. The values of // and S may be determined by means of the rules already given. We have then H - , A" = H + K ; (646) / /7 - A), 5 /7 - S' - 3.804^. (648) In certain cases the value of a' maybe negative ; the summit of the curve of the cable prolonged is then sit- uated beyond the lower pulley. The tension of flec- tion s, and consequently the diameter of the pulleys, are determined when we have obtained the value of the tension S", which very often does not differ materially from 5. The difference between the two tensions be- TRANSMISSION BY INCLINED CABLE. 2 19 comes important only in cases where several inclined transmissions are taken from a single higher pulley. Example. A transmission by cable, the data of which are the same as in the fifth example of 20, has its pulleys placed at different heights ; taking for the difference in the levels of the pulleys H = 16.4 feet, it is required to determine the deflections and the posi- tions of the curve-summits. For the driving part of the cable we have 5*1 = 8533.8, 7*i = 7.216 feet, H 16.4 feet, A 360.8 feet. Starting at the lower pulley, we have, from formula 16.4 -- 1.35 feet, 2 //", 7-2i6 -f- 1.35 = 8.566 feet ; S6o.8/ I 16.4 \ a\ -- *\l - - - ^jgj = 1 80.4X0 432 = 77-93 feet, a'\ == 360.8 + 77.93 = 382.87 feet. For the driven part of the cable, S, = 4266.9, h, = 14.79 feet ; consequently / i 16.4" \ 16.4 k\ == I4-79V 1 + 16 T^TO?; " ~2~ = 7 * 73 feet > h'\ = 16.4 + 773 = 24.13 feet. 22O BELTS AND PULLEYS. For the state of repose, h Q = 0.67 X 14.79 + - 28 X 7.216 = 12.05 feet ; hence . 16.4 '"'+- ~- ^ 5-24 feet, a\ = ti f . 16.4 + 5.24 21.64 360.8 / i i6.4\ - --- i ^j = 119.06 feet, 4 I2.05/ 2 \ 4 I2.05 a'\ 360.8 -- 119.06 = 241.74 feet. The tensions in the driving part of the cable are as follows: S\ = 8533.8 - (7.216- 1.35)3.804 =8511.5, S", = 8533-8 + (8.566 - 7.216)3.804 = 8538.94; the values of S\ and S'\ differing so slightly from S l that we may neglect the difference. The heights which the calculations furnish for the deflections of an inclined transmission should be laid out in the drawing to a scale three or five times that of the horizontal lines ; we then trace the curve of the cable as an arc of a parabola (see the following paragraph), and try if the conditions of the ground will permit us to use the curve obtained. If this prove not the case, we must recommence the calculation by adopting new values for the tension until we have obtained a curve which will satisfy the conditions. With a little prac- tice, it is easy to determine by the eye the proper val- ues to be adopted, and the calculation may then be made without difficulty. METHOD OF TRACING THE CURVES OF CABLES. 221 24. Method of Tracing the Curves of Cables. The curve of a cable may be drawn with sufficient accuracy for ordinary purposes by assuming it to be an arc of a parabola. After having determined the summit C of the part of the cable BCD, Fig. 80, as explained in the preceding section, divide into two equal parts, at the points C, and C 2 , the two distances B^C and D^C l (B l D l being tangent to the curve of its summit), and through the points C^ and C 2 draw the lines BC l and DC.,, which give the directions in which the cable leaves the pulleys. Divide the distances CC V and into a certain number of small equal parts at the points i, 2, 3, etc., and I, II, III, etc.; by joining il, .2!!, 3!!!, etc., we obtain a series of lines tangent to the required parabola. By a similar method with CC^D we obtain the other part of the curve. When the sum- mit C of the curve falls outside of the pulleys, on the side of the pulley which occupies the lower level, apart of the parabola near the summit cannot be made use of. but the construction is still the same. 222 BELTS AND PULLEYS. 25. Transmission by Cable with Pulleys near together. When the distance between the pulleys of a trans- mission by cable is small, it is necessary, first of all, that the deflections have not too small values, in order that the cable may run properly upon the pulleys, and also that we may be able to shorten the cable without seriously increasing the tension. We adopt then for S 1 a very small value, and thus determine upon a value for the deflection ; then, by means of formula (638) and the table calculated from it, obtain S 1 ; / and R are then calculated as we have already indicated. For a small tangential resistance and a small separation of the pulleys, transmissions by cable may still be used with satisfactory results. Example. A metallic cable transmits a force of 6 horse-power at 150 revolutions per minute ; the separa- tion of the pulleys is 65.6 feet and the deflection in the driven part of the cable 3.28 feet. We have then r A A 0.05, which, from 21, corresponds to -~ 0.103, 65.6 and we obtain S. = - ~ = 637. In order to find the 0.103 value of tf, we must know that of s Assuming that s -|- 5*! is still equal to 25601.4, we have s = 25601.4 s H 24964.4 6 637 24064.4, which gives > = - - = S, n 637 150 1.57. The second table of 20 gives (column 7, line li), therefore, 8 0.08 inch, for i 36. From for- mula (637) we have for the radius of the pulleys R = RIM OF CABLE-PULLEYS. 22$ I4223OOO 0.08 ^ = 45.0 inches, rrom what precedes, we find that these values of 8 and R are perfectly admis- sible. If we wish to take for the diameter of the ca- ble, d = 8# = 0.48 inch, that is, & is reduced to 0.06 inch, it is only necessary to give to R a smaller value. In this case the table of 20 gives (column 7, lines 8 and 9) -~ = 0.718, hence s = 0.7 iSS^ 0.718 X o j r I ~1 637-^- = 11434.15, and formula (637) gives R = 14223000 o.oo- -- = 74 inches. In some cases pullevs of 11434.15 large radii cannot be conveniently used, and we are obliged to use pulleys of different radii in order to make the deflections great enough. For the transmis- sion of considerable forces, we obtain good results only on the condition of giving to the pulleys a certain ve- locity of rotation, the limits for which are indicated at the end of the following paragraph. 26. Rim of Cable-pulleys. When first used, the rims of cable-pulleys were made ,of wood covered with leather, but practice soon de- monstrated the fact that rims of metal are preferable, and at the present time the latter are used almost ex- clusively in all cases where durability forms an impor- tant factor. Figs. 81 and 82 represent two cast-iron rims, single and double. The sides of the groove in the single rim are inclined at an angle of 30 with the middle plane of the pulley. In the double rim such 224 BELTS AND PULLEYS. an inclination would necessitate too great a weight for the projection between the two grooves ; the in- clination of the sides of this projection is therefore less than 30. In Fig. 82 (which represents a portion of a large pulley) this inclination is 15. All the dimensions indicated in the figures are in terms of the diameter d FIG. 82. of the cable. Since cables of less than 0.4 inch diameter are seldom used, we may consider the value of d = 0.4 inch as the inferior limit of the unit for the construction of cable-pulleys. The grooves in the faces of the pul- leys are bottomed with gutta-percha driven into the dovetails, as shown in the figures ; or small pieces of wood, which are introduced into the dovetails through openings in the side of the rim. Fig. 82 shows two RIM OF CABLE-PULLEYS. 22$ openings of this kind covered up by pieces which are bolted in after the insertion of the wooden pieces. Of late years grooves with leather bottoms have come in- to use for very heavy cables ; to this end old belts cut into strips and wedged into the dovetails may be ad- vantageously used. Professor Fink has successfully employed bottoms formed by winding twine tightly around in the dovetails ; bottoms thus made give great resistance to slipping. Bottoms of cork have also been used, but while they offer the advantage of being in- expensive, they have not been tested sufficiently in prac- tice to determine their utility for transmission by cables where there is danger of slipping. When we wish to make use of bottoms of twine, the depth of the dove- tails need not be so great as that indicated in the fig- ures. In the first three modes of furnishing the grooves with bottoms which present more resistance to slip- ping than cast-iron (gutta-percha, wood, and leather), the profile of the groove upon which the cable rests may be hollowed out after the introduction of the ma- terial into the dovetails. Pulleys of 12 to 15 feet in diameter are ordinarily cast in two pieces, which makes them easier to handle and transport ; projections are cast upon the inside of the rim by means of which the two parts may be bolted together. In order that no harm may come to the rim through excessive centrifugal force, the velocity of rotation of the rim should not exceed 100 feet per second. The velocity of about 90 feet per second, which is now com- monly given to metallic cables, may be considered as without disadvantages in ordinary practic^ 15 226 BELTS AND PULLEYS. 27. Arms and Nave of Cable-pulleys. The body and rim of a cable-pulley are ordinarily of cast-iron, as is often the case with the entire pulley. We however sometimes find arms of wrought-iron set into cast-iron rims (see Fig. 96). In any case the num- ber of arms A is determined from the expression The cross-sections of cast-iron arms are oval or flanged ; in either case the width in the plane of the pulley is given by the formula T /? h = ^d + -j- ..... (650) 4 -^ In a flanged cross section the thickness of the prin- cipal flange (in the plane of the pulley) is e = , and that of the secondary flange e' = \e. In an oval cross- section the thickness is one half the width, as in pul- leys for transmission by belt. The width at the rim may be taken equal to -f the width at the nave. Arms with flanged cross-sections are generally straight (Fig. 83), and eight in number, while those having oval cross-sections are curved, either single, as explained in 14, or double, as in Fig. 84. To draw double-curved arms for cable-pulleys, we p begin by striking a circle with a radius OA = , then ARMS AND NAVE OF CABLE-PULLEYS. 227 take upon the circle the lengths AB and BC, correspond- ing to the division by the arms. Draw the arc OR representing one portion of the double curve, in the same manner as for single-curved arms. Through the centre of curvature C of this arc (which, for eight arms, is on the circumference ABC) draw the line CED, FIG. 83. FIG. : and taking ED EC, obtain the radius of curvature corresponding to the part EF oi the arm. To draw ' the curves which limit the profile, it is necessary only to follow the method of 14, remarking that the centres for the arcs are found upon the line CD. When straight arms are used the nave is sometimes cast with grooves, into which iron rings are afterwards placed ; by putting on the rings hot, and allowing them to cool, they are very firmly fixed, and add greatly to 228 KELTS AND PULLEYS. the strength of the pulley. The dimensions of the nave are determined, as already explained for pulleys for transmission by belt, in 13. Example. In a transmission by cable the radius of the pulleys is 50 inches, the diameter of the arbor is 4.8 inches, and that of the cable 0.48 inch ; it is re- quired to determine dimensions of the pulley. From formula (649) the number of arms is A = 4 -f- - 40 0.40 = 7. The width of the arms at the nave is, from formula (650), A = 4 X 0.48 -f - = 1.92 -f- 1.8 = 3.72 inches. Formula (604), in which d represents the diameter of the arbor, gives for the thickness of the 4.8 50 nave w = 0.4 -f- ~^ + 7" = -4 + O.8 + I = 2 - 2 inches. The length of the nave (L) ought to be at least equal to 2-J X 2.2 5.5 inches. For very important transmissions it is prudent to have a reserve cable ; that is, to divide the force to be transmitted between two cables, each having sufficient strength to transmit the whole force. An arrangement of this kind is in use at SchafThouse, in a transmission by metallic cable of 600 horse-power, of which we shall have occasion to speak farther on. In this transmis- sion the two pulleys are placed upon one driving arbor, as shown in Fig. 85. The pulleys which run loosely upon the arbor are fixed to the two gear-wheels B and D, which engage with the intermediate gears A and C. The latter gears run loosely upon their journals, which form a part of and rotate with the driving arbor. By means of this arrangement each cable is made to trans- ARMS AND NAVE OF CABLE-PULLl mit an equal share of the total force. If one of the cables breaks, the pulley over which it ran is free to rotate in the opposite direction, and the gears are thus put in motion. In order to prevent too rapid motion in the pulley, which by the breaking of a cable may be- come loose upon the arbor, the transmission at Schaff- house is provided with a powerful brake, by means of which the motion of the motive turbine-wheel may FIG. 85. be almost instantaneously arrested. Instead of the intermediate gears A and C, simple sectors, such as are represented in the figure on the right, might be used in this trrnsmission. In this case as soon as a break in one of the cables occurred, the sectors would be put in motion, and when the toothless parts came opposite the gears D and B the motion of the pulleys would be stopped, and danger of further accident avoided. 230 BELTS AND PULLEYS. 28. Pulley-Supports and Intermediate Pulleys When the principal pulleys of a transmission by cable are placed far apart, and especially when they are not high above the ground, it is often necessary to support the cable by other pulleys. In certain cases it is suffi- cient to support at a single point the driven part of the cable while the driving part is left free, as shown in FIG. 86. Fig. 86. When several pulley-supports are necessary, the driving part is also supplied with at least one, as shown in Fig. 87. In other cases the number of pul- ley-supports is the same for both parts of the cable ; it FIG. 87. is then best to place the pulleys of the driving part directly over those of the driven part, instead of juxta- positing them, as has been several times attempted, and which causes rapid wear of the cable, consequently produces a wearing friction upon the pulley-grooves, and also tends to make the cable run off the pulleys. PULLEY-SUPPORTSINTERMEDIATE PULLEYS. 2$l In the arrangement represented in Fig. 88 the pulley- supports of the driving part are placed under those of the driven part in order to gain space above the ground. FIG. In most cases when the distance between the princi- pal pulleys makes a great number of pulley-supports necessary, this arrangement may be advantageously replaced by a series of successive transmissions (Ziegler), Fig. 89. The pulley-supports of Fig. 88 are then re- placed by intermediate double-grooved pulleys placed at as near the same distances apart as possible, so that M, FIG. 89. in case of breakage in any of the cables a single reserve cable may be used to replace it.* *This has been done by Ziegler at Frankfort-on-the-Main, where a force of 100 horse power is transmitted at a distance of 984 metres nearly of a mile. 232 BELTS AND PULLEYS. The different points at which a cable is supported are called stations ; those which correspond to the principal pulleys of the transmission are called the sta- tions at the extremities and the others intermediate FIG. 90. stations. Sometimes it is necessary to change the directions of the cable at an intermediate station ; Him has proposed to accomplish this change of direction by means of a horizontal pulley, Fig. 90, while it has also been suggested to use a pair of bevel gears, Fig. 91. FIG. 91. The use of transmissions by cable is very convenient when we wish to divide between several establishments, belonging to different proprietors, the force derived from a single motor : to do this we have simply to PULLE Y-SUPPORTS INTERMEDIA TE PULLE VS. 233 make the intermediate stations the starting-points or stations at one extremity of supplementary transmis- sions. Stations of this kind are called division-stations. Pulley-supports are also used in the special case in which the driven arbor is placed almost vertically above or below the driving-arbor. There would be serious difficulty in making use of an inclined cable, connect- ing directly the two pulleys A and B, Figs. 92 and 93 ; it is preferable by far to use the pulley-supports 7", T, placed in such a manner that one part of the cable, TA or TB, may be horizontal. It is then sufficient to determine, by means of the preceding rules, the proper tensions to give to the horizontal part of the transmis- sion without reference to the inclined part. The use of cables for the transmission of forces to great depths into the shafts of mines, for example is still in a period of development. We may say, how- ever, from attempts already made in this direction, that satisfactory results have been obtained.* * Review of Society of German Engineers, 1866, p. 371. Werner, " Use of transmissions by metallic cables for the shafts of mines." 234 BELl'S AND PULLEYS. We meet with a remarkable example of this mode of transmission in the arrangement at Schaffhouse, where a force of about 600 horse-power, taken from the current of the Rhine, is received by turbines at the left bank, and is intended to be transmitted across the river to the right bank, there to be divided among several factories. This important application, credit for which is due to the Society of Hydraulic Engineers of Schaffhouse, is very nearly completed, and affords, in all its details, information of the greatest interest to engineers. 29. Dimensions of Pulley-supports. The pulleys intended to support the driving part of the cable ought properly to have the same diameter as the pulleys of transmission ; those supporting the driven part may, in normal transmissions, have smaller dimensions. The following table indicates the limits below which we should not take the radius R Q of the pulley-supports. The numbers contained in the table have been cal- culated by means of the formula R _ 28446000 6 "" 51202.8 S\' Si J *0 8 ^i j *o 8- 711-15 24890.25 563 12800.70 12800.70 741 1422.30 24179.10 571 14223.00 11378.40 769 2844.60 22756.80 588 15645.30 9956.10 800 4266.90 21334.50 606 17067.60 8533.80 833 5689.20 19912.20 625 18489.90 7HI.50 870 7111.50 18489.90 645 19912.20 5689.20 909 8533.80 17067.60 667 21334.50 4266.90 952 9956.10 15645.30 690 22756.80 2844.60 1000 11378.40 14223.00 714 24197.10 1422.30 1053 PRESSURE ON PULLEY-SUPPORT AXES. 235 The values contained in the table furnish excellent dimensions for ^ principally for large values of S r In transmissions with increased tension (see 21) the difference between R Q and R is so small that we may take, without disadvantage, R = R . In compound transmssions (see 28) there is no difference in size be- tween the principal pulleys at the extremities and the intermediate pulleys. 30. Pressure upon the Axes of Pulley-supports. In a transmission by cables, which we have taken care to calculate for its entire length, we should know the tensions at each station, and (from the curves of FIG. 94. FIG. 95. the cables traced according to 23) the directions of the different parts which are to be supported by in- termediate pulleys. For example, in Fig. 94, for an intermediate pulley we should know the values of T, 236 BELTS AND PULLEYS. t, TV and ^ and their directions. We can then deter- mine by means of formulas already given the approxi- mate weight of the pulley, which allows us to trace graphically (Fig. 95) the resultant Q of the different forces. To accomplish this we draw the lines A B, BC, CD, D E, and EF respectively equal and parallel to T, TV t, / and G. The line A F, which completes the polygon, represents in amount and direction the re- sultant Q. Pulley-supports are ordinarily in construction iden- tical with the principal pulleys for the same diameter of cable. By virtue of the rules of 26 and 27, the following formulas may be obtained for the approxi- mate weights of the pulleys : For single-grooved pulleys, G ry . 145-6 . ii5.52\/M , / 51 = 0.034375 [(45 + -^- + -JS-J bJ + (0.33 + For double-grooved pulleys, G VI , 265.6 , . , f = 0.034375 L(84 + -jr + -y- + 0.33 + 0.464 , o.n5\ [RY , / , o.oo28W^V ( 6 53) Example. In the fourth example of 20 for a radius of 30 inches the diameter of the wires (of which there are 36) is 0.036 inch. The diameter of the cable itself is therefore d 8 X 0.036 = 0.288 inch, which PRESSURE ON PULLEY-SUPPORT AXES. 237 gives , ^TO !O4- The weight of the pulley for a single groove is, from formula (652), =0.024 X FIG. 96. D 145.6 115. 52\ / r r 45 + 0^88 + ^083-] I04 + l' 33 + o^ ^ 1 1 <;\ / , o.ooaSN ~| Sfj 10816 + (0005 + ^^^^-J 1124864] = 2 4 pounds. 238 BELTS AND PULLEYS. Example. For the transmission of 300 horse-powei of the second example of 20 we have d 0.087, which for a cable of 60 wires gives = log -=- 0.007578^, which in the 0.57403 0.57403 present case becomes V = ^^- = -j.^ or cp = 0.42083. In this experiment, although tried with five different FIG. ioi. pulleys and as many different belts, the greatest value obtained for the coefficient was

0.61 for the coefficient of fric- tion. These belts, which were 2 inches wide and T 3 ^ inch thick, broke through the solid parts when tested for strength, at an average strain of 1088 pounds. This o would give for the ultimate strength 1088 X , or about 2900 pounds per square inch very little if any below that of ordinary belt-leather. Leather over Leather-covered Pulleys. Using the belting and pulleys of the first five experiments men- tioned in this Appendix, the author tried the following experiments with leather-covered pulleys : Experiment i. With apparatus of Fig. 100. a = 1 80, t = 60 pounds. The lever-arm be was 26.15 inches long when the belt began to slip. Hence T = 26.15 T 261.5 X 40 = 261.5 pounds, log - == log -^- = log 4.358 = 0.63929. Consequently