Benjamin's Machine Design. By CHARLES H. BENJAMIN, Professor in the Case School of Science. Hoskins' Hydraulics. By L. M. HOSKINS, Professor in Leland Stanford University. Adams' Alternating-current Machines. By C. A. ADAMS, Professor in Harvard University. [In preparation. ] HENRY HOLT AND COMPANY NEW YORK CHICAGO MACHINE DESIGN BY CHARLES H. BENJAMIN Professor of Mechanical Engineering in the Case School of Applied Science NEW YORK HENRY HOLT AND COMPANY 1908 Copyright, 1906 BY HENRY HOLT AND COMPANY PREFACE. THIS book embodies to a considerable extent the writers experience in teaching and in commercial work. While the underlying mechanical principles of machine design are permanent, the application of them is continually changing. The researches of the ex- perimenter and the practice of the builder are always showing good reasons for the modification of design* Although the present work was prepared primarily for a text-book, it contains mainly what the writer has found necessary in his own practice as an engineer. As far as possible the formulas for the strength and stiffness of machine details have been fortified by the results of experiments or by the practical experience of manufacturers* Attention is called particularly to the experiments on cast-iron cylinders, pipe fittings, helical springs, roller bearings, gear teeth, pulley arms, and the bursting strength of fly-wheels. What the student needs to learn before graduation is also what he needs to remember after, and it is hoped that this book contains the necessary facts and principles and not too much else. C. H. B. iii 361349 TABLE OF CONTENTS. I. UNITS AND TABLES 1 I. Units. 2. Abbreviations. 3. Materials. 4. Notation. 5. Formulas. 6. Profiles of uniform strength. 7. Factors of safety. II. FRAME DESIGN 15 8. General principles of design. 9. Machine supports. 10. Machine frames. III. CYLINDERS AND PIPES 25 II. Thin shells. 12. Thick shells. 13. Steel and wrought iron pipe. 14. Strength of boiler tubes. 15. Pipe fittings. 16. Steam cylinders. 17. Thickness of flat plates. IV. FASTENINGS 54 18. Bolts and nuts. 19. Machine screws. 20. Eye bolts and hooks. 21. Riveted joints. 22. Lap joints. 23. Butt joints with two straps. 24. Efficiency of joints. 25. Butt joints with unequal straps. 26. Practical rules. 27. Riv- eted joints for narrow plates. 28. Joint pins. 29. Cotters. V. SPRINGS , 73 30. Helical springs. 31. Square wire. 32. Experiments. 33. Springs in torsion. 34. Flat springs. 35. Elliptic and semi-elliptic springs. VI. SLIDING BEARINGS 86 36. Slides in general. 37. Angular slides. 38. Gibbed slides. 39. Flat slides. 40. Circular guides. 41. Stuffing boxes. VII. JOURNALS, PIVOTS AND BEARINGS 96 42. Journals, 43. Adjustment. 44. Lubrication. 45. Fric- tion of journals. 46. Limits of pressure. 47. Heating of journals. 48. Experiments. 49. Strength and stiffness of journals. 50. Caps and bolts. 51. Step bearings. 52. Friction of pivots. 53. Flat collar. 54. Conical pivot. 55. Schiele's pivot. 56. Multiple bearing. V vi MACHINE DESIGN. CHAPTER. PAGE. VIII. BALL AND ROLLER BEARINGS 118 57. General principles. 58. Journal bearings. 59. Step bearings. 60. Materials and wear. 61. Design of bearings. 62. Roller bearings. 63. Grant roller bearing. 64. Hyatt rollers. 65. Roller step bearings. IX. SHAFTING, COUPLINGS AND HANGERS 130 ' 66. Strength of shafting. 67. Couplings. 68. Clutches. 69. Coupling bolts. 70. Shafting keys. 71. Hangers and boxes. X. GEARS, PULLEYS AND CRANKS 146 72. Gear teeth. 73. Strength of teeth. 74. Lewis' formula. 75. Experimental data. 76. Teeth of bevel gears. 77. Rim and arms. 78. Sprocket wheels and chains, 79. Silent chains. 80. Cranks and levers. XI. FLY-WHEELS 166 81. In general. 82. Safe speed for wheels. 83. Experiments on fly wheels. 84. Wooden pulleys. 85. Rims of cast-iron gears. 86. Rotating discs. 87. Plain discs. 88. Conical discs. 89. Discs with logarithmic profile. 90. Bursting speeds. XII. TRANSMISSION BY BELTS AND ROPES 184 91. Friction of belting. 92. Strength of belting. 93. Taylor's experiments. 94. Rules for width of belts. 95. Speed of belting. 96. Manila rope transmission. 97. Strength of Manila ropes. 98. Wire rope transmission. TABLES. PAGE. I. STRENGTH OF WROUGHT METALS 6 II. STRENGTH OP CAST METALS 7 III. VALUES OP Q IN COLUMN FORMULA 10 Ilia. VALUES OF S AND K IN COLUMN FORMULA 10 IV. CONSTANTS OF CROSS-SECTION 11 V. FORMULAS FOR LOADED BEAMS. 12 VI. SIZES OF IRON AND STEEL PIPE 31 VII. SIZES OF EXTRA STRONG PIPE 33 VIII. SIZES OF DOUBLE EXTRA STRONG PIPE 34 IX. SIZES OF IRON AND STEEL BOILER TUBES 36 X. STRENGTH OF STANDARD SCREWED PIPE FITTINGS 41 XI. BURSTING STRENGTH OF CAST IRON CYLINDERS 45 XII. STRENGTH OF REINFORCED CYLINDERS 47 XIII. STRENGTH OF CAST IRON PLATES 52 XIV. STRENGTH OF IRON OR STEEL BOLTS 54 XV. DIMENSIONS OF MACHINE SCREWS 57 XVI. DIMENSIONS OF RIVETED LAP JOINTS 65 XVII. DIMENSIONS OF RIVETED BUTT JOINTS 65 XVIII. STRENGTH AND STIFFNESS OF HELICAL SPRINGS 77 XIX. FRICTION OF PISTON ROD PACKINGS 93 XX. " " " " " 94 XXI. " " " " " 94 XXII. FRICTION OF JOURNAL BEARING 108 XXIII. FRICTION OF ROLLER AND PLAIN BEARINGS 126 XXIV. " " " " " " 127 XXV. DIAMETERS OF SHAFTING c 132 XXVI. PROPORTIONS OF GEAR TEETH 148 XXVII. SIZES OF TEST FLY-WHEELS 172 XXVIII. FLANGES AND BOLTS OF TEST FLY-WHEELS. 172 XXIX. FAILURE OF FLANGED JOINTS 173 XXX. SIZES OF LINKED JOINTS 173 XXXI. FAILURE OF LINKED JOINTS , 174 XXXII. BURSTING SPEEDS OF ROTATING Discs , 182 XXXIII. HORSE POWER OF MANILA ROPE 192 XXXIV. HORSE POWER OF WIRE ROPE 194 vii MACHINE DESIGN. CHAPTER I. UNITS AND TABLES. 1. Units. In this book the following units will be used unless otherwise stated. Dimensions in inches. Forces in pounds. Stresses in pounds per square inch. Velocities in feet per second. Work and energy in foot pounds. Moments in pounds inches. Speeds of rotation in revolutions per minute. The word stress will be used to denote the resistance of material to distortion per unit of sectional area. The word deformation will be used to denote the dis- tortion of a piece per unit of length. The word set will be used to denote total permanent distortion of a piece. In making calculations the use of the slide-rule and of four-place logarithms is recommended ; accuracy is expected only to three significant figures. 2. Abbreviations. The following abbreviations are among those recommended by a committee of the American Society of Mechanical Engineers in Decem- ber, 1904 ? an$ will be used throughout the book, MACHINE DESIGN. NAME. ABBREVIATION. Inches . in. Feet . ft. Yards . . yd. Miles. ... . . spell out. . Ib. Tons . spell out. Seconds . sec. Minutes ...... . min. Hours ...... . hr. Linear ...... . lin. Square ...... . sq. cu. Per . spell out. Fahrenheit ..... . fahr. Percentage ..... . % or per cent. Brake horse power .... . b.h.p. Electric horse power . e.h.p. Indicated horse power . i.h.p. British thermal units . B.t.u. Diameter . Diam. 3. Materials. The principal materials used in machine construction are given in the following tables with the physical characteristics of each. By wrought iron is meant commercially pure iron which has been made from molten pig-iron by the puddling process and then squeezed and rolled, thus developing the fiber. This iron has been largely sup- planted by soft steel. Ordinary wrought iron contains from 0.1% to 0.3% of carbon. Soft steel may contain no more than this, but is different in structure. The particles of iron in the puddling process are more or less enveloped in slag or earthy matter and as the bloom is squeezed and MATERIALS. 3 rolled the particles become fibers separated from each other by a thin sheath or covering of slag, and it is this that gives such iron its characteristic structure. The principal impurities in the iron are phosphorus from the ore and sulphur from the fuel. In making steel, on the other hand, the molten iron has had the silicon and carbon removed by a hot blast, either passing through the liquid as in the Bes- semer converter, or over its surface as in the open- hearth furnace. A suitable quantity of carbon and manganese has then been added and the metal poured into ingot molds. If the steel is then reheated and passed through a series of rolls, structural steel and rods or rails result. Bessemer steel contains from 0.1% to 0.6% of carbon and has a fine granular structure. This material has been much used for rails. Open hearth steel differs from Bessemer but little in its chemical composition but is usually more reliable in quality on account of the more deliberate nature of the process of manufacture. It is generally used for boiler plate and for steel castings. Two grades of boiler plate are commonly known as marine steel and flange steel, the latter being of the better quality. Steel castings are poured directly from the open hearth furnace and allowed to cool without any draw- ing or rolling. They are coarser and more crystalline than the rolled steel. Crucible steel usually contains from one to one and a half per cent of carbon, is relatively high priced and only used for cutting tools. It is made by melting steel in an air-tight crucible with the proper additions of carbon and manganese. Cast iron is made directly from the pig by remelt- 4: MACHINE DESIGN. ing and casting, is granular in texture and contains from two to five per cent, of carbon. A portion of the carbon is chemically combined with the iron while the remainder exists in the form of graphite. The harder and whiter the iron the more carbon is found chemi- cally combined. Silicon is an. important element in cast iron and influences the rate of cooling. The more slowly iron cools after melting the more graphite forms and the softer the iron. Two per cent of silicon gives a soft gray iron of a high tensile strength. Machinery iron contains usually from one and one half to two per cent of silicon. Malleable iron is cast iron annealed and partially de- carbonized by being heated in an annealing oven in con- tact with some oxidizing material such as haematite ore. This process makes the iron tougher and less brittle. All castings including those made from alloys are somewhat unreliable on account of hidden flaws and of the strains developed by shrinkage while cooling. The so-called high-speed or air -hardening tool steels are alloys of steel with various substances such as chromium (chrome steel), tungsten (Mushet steel), molybdenum, etc., etc. They are characterized by extreme hardness at comparatively high temperatures. Their other physi- cal characteristics are not of particular interest. The addition of nickel to steel increases its ultimate strength and also raises its elastic limit. The tensile strength is sometimes as high as 200,000 Ib. per sq. in. and the steel is also tough and well adapted to resist shocks. The bronzes are alloys of copper and tin, copper and zinc, or of all three. The copper-tin alloys usually contain 85 or 90 per cent of copper and are expensive. MATERIALS. 5 The copper-zinc alloys, or brasses as they are sometimes called, should have from 60 to 70 per cent of copper for maximum strength and ductility. Bronzes high in tin and low in copper are weak, but have considerable ductility and make good metals for bearings. Tin 80, copper 10 and antimony 10 is Babbitt metal, so much used to line journal bearings, the antimony increasing the hardness. The late Dr. Thurston's experiments on the copper- tin-zinc alloys showed a maximum strength for copper 55, zinc 43 and tin 2 per cent. The tensile strength of this mixture was nearly 70,000 Ib. per sq. in. Phosphor bronze is a copper alloy with a small amount of phosphorus added to prevent oxidation of the copper and thereby strengthen the alloy. Manganese bronze is an alloy of copper and man- ganese, usually containing iron and sometimes tin. A bronze containing about 84 per cent copper, 14 per cent manganese and a little iron, has much the same physical characteristics as soft steel and resists corrosion much better. The constants for strength and elasticity given in the tables are only fair average values, and should be determined for any special material by direct experi- ment when it is practicable. Many of the constants are not given in the table on account of the lack of reliable data for their determination. The strength of steel, either rolled or cast, depends so much upon the percentages of carbon, phosphorus and manganese, that any general figures are liable to be misleading. Structural steel usually has a tensile strength of about 65,000 Ib. per sq. in., while boiler plate usually has less carbon, a low tensile strength and good ductility. MACHINE DESIGN. 03 >> J ~ & H r'fl 8 H bO a 02 CO 1C O a o o O OO Oi o O a O* -2 _4 to 02 MATERIALS. g w o o o o o o CO 00 8 8 MACHINE DESIGN. 4. Notation. Arc of contact =0 radians. Area of section =A sq. in. Breadth of section =b in. Coefficient of friction =/ Deflection of beam =A in. Depth of section =h in. Diameter of circular section d in. Distance of neutral axis from outer fiber y in. Elasticity, modulus of, in tension and compression =E in shearing and torsion = Gr Heaviness, weight per cu. ft. w Length of any member = I in. Load or dead weight = W Ib. Moment, in bending =Jflb.-in. in twisting = Tib. -in. Moment of inertia rectangular =1 polar = J Pitch of teeth, rivets, etc. =p in. Radius of gyration =r in. Section modulus, bending ij twisting Stress per unit of area =S Velocity =v ft per sec. 5. Formulas. Simple Stress. Tension, compression or shear, = r (0 NOTATION. 9 Bending under Transverse Load. orr General equation, M = (2) Eectangular section, M= ^ . . . .(3) Rectangular section, bh 2 =~- (4) Circular section, M=^? . . . .(5) Circular section, d= r'^ . . . .(6) 'V >S Torsion or Twisting. General equation, Circular section, Circular section, Hollow circular section, Other values of - and - may be taken from Table 4. y y Combined Bending and Twisting. Calculate shaft for a twisting moment, Column subject to Bending. Use Rankine's formula, ~^ = ~ ~^ (12) The values of r 2 may be taken from Table IV. The subjoined table gives the average values of g, while S is the compressi ve strength of the material. 10 MACHINE DESIGN. TABLE III. VALUES OF q IN FORMULA 12. Material. Both ends fixed. Fixed and round. Both ends round. Fixed and free. Timber 1 1.78 4 16 Cast Iron 3000 1 3000 1.78 3000 4 3000 16 Wrought Iron 5000 1 5000 1.78 5000 4 5000 16 Steel 36000 1 36000 1.78 36000 4 36000 16 25000 25000 25000 25000 Carnegie's hand-book gives S = 50000 for medium steel columns and g=inrTRnr, rcfonr and inm for the three first columns in above table. In this formula, as in all such, the values of the constant should be determined for the material used by direct experiment if possible. Or use straight line formula, --r= S Jc- . . .(12a) .1 / TABLE Ilia. VALUES OF S AND k IN FORMULA (12a). (Merriman's Mechanics of Materials.) Kind of Column. S k Limit - Wrought Iron : Flat ends 42000 128 218 Hinged ends. . . ." 42000 157 178 Round ends 42000 203 138 Mild Steel : Flat ends 52500 179 195 Hinged ends 52500 220 159 Round ends.. 52500 284 123 Cast Iron : Flat ends 80000 438 122 Hinged ends 80000 537 99 Round ends 80000 693 77 Oak : Flat ends 5400 28 128 FORMULAS. 11 Carnegie's hand-book gives allowable stress for structural columns of mild steel as 12000 for lengths less than 90 radii, and as 1710057- for longer columns. This allows a factor of safety of about four. TABLE IV. CONSTANTS OF CROSS-SECTION. Form of Section and Area A. Square of Radius of Gyration Moment of Inertia Section Modulus I y Polar Moment of Inertia J Torsion Modulus J y Rectangle /i 2 bh 3 bh* bh*+b*h W+Vh bh 12 12 6 12 6 V6 2 +/i 2 Square d 2 d 4 d 3 d 4 d 3 d 2 12 12 6 6 4T24 Hollow Rectangle bh z bji 3 l bh 3 b l h 3 l 67i 3 fW, or 7-beam I2(bh b h ) 12 Qh bh-b l h l Circle d 2 TTd 4 d 3 TTd 4 d 3 5*. 16 64 10.2 32 5.1 Hollow Circle *+*, 7r(d 4 d 4 j) 10. 2d* 7r(d 4 -d 4 ,) d 4 d 4 ! 64 32 5. Id Ellipse a 2 *W 6 2 7r(ba 3 + ab 3 } &*+&' f* 16 64 10.2 64 10.2a Values of / and J for more complicated sections can be worked out from those in table. 12 MACHINE DESIGN. TABLE V. FORMULAS FOR LOADED BEAMS. Beams of Uniform Cross-section. Maximum. Moment. M Maximum. Deflection. A Wl Wl 3 Cantilever, uniform load Wl 3EI WP Simple beam, load at middle 2 Wl 8EI Wl 3 Simple beam, uniform load 4 Wl 4SEI 5W7 3 Beam fixed at one end, supported at other, load at middle 8 3WI ~W 384#7 .Q182W1 3 Beam fixed at one end, supported at other, uniform load Wl g .0054TFZ 3 Beam fixed at both ends, load at middle . . Beam fixed at both ends, uniform load. . . . Beam fixed at both ends, load at one end, Wl T Wl 12 Wl ~2~ Wl 3 1S2EI Wl* mEI Wl* 12EI 6. Profiles of Uniform Strength. In a bracket or beam of uniform cross-section the stress on the outer row of fibers increases as the bending moment increases and the piece is most liable to break at the point where the moment is a maximum. This difficulty can be remedied by varying the cross-section in such a way as to keep the fiber stress constant along the top or bottom of the piece. The following table shows the shapes to be used under different conditions. FACTORS OF SAFETY. 13 Type. Load. Plan. Elevation. Cantilever. Center .... Rectangle . . Parabola, axis horizontal. Cantilever. . Simp. Beam Uniform . . . Center Rectangle . . Rectangle . Triangle. Two parabolas intersecting Simp. Beam Uniform . . . Rectangle . . under load. Ellipse, major axis horizontal. The material is best economized by maintaining a constant breadth and varying the depth as indicated. This method of design is applicable to cast pieces rather than to those that are forged or cut. The maximum deflection of cantilevers and beams having a profile of uniform strength is greater than when the cross-section is uniform, fifty per cent, greater if the breadth varies, and one hundred percent greater if the depth varies. 7. Factors of Safety. A factor of safety is the ratio of the ultimate strength of any member to the ordinary working load which will come upon it. This factor is intended to allow for : (a) Overloading either inten- tional or accidental, (b) Sudden blows or shocks. (c) Gradual fatigue or deterioration of material. (d) Flaws or imperfections in the material. To a certain extent the term " factor of ignorance" is justifiable since allowance is made for the unknown. Certain fixed laws may guide one, however, in making the selection of a factor. It is a well-known fact that loads in excess of the elastic limit are liable to cause failure in time. Therefore, when the elastic limit of the material can be determined, it should be used as a basis rather than to use the ultimate strength. 14 MACHINE DESIGN. Furthermore, suddenly applied loads will cause about double the stress due to dead loads. These two considerations point to four as the least factor that should be used when the ultimate strength is taken as a basis. Pieces subject to stress alternately in opposite directions should have large factors of safety. The following table shows the factors used in good practice under various conditions : Structural steel in buildings . . 4 " " " bridges . . 5 Steel in machine construction . . 6 " " engine " 10 Steel plate in boilers ... 5 Cast iron in machines . . . 6 to 15 Castings of bronze or steel should have larger factors than rolled or forged metal on account of the possibility of flaws. Cast iron should not be used in pieces subject to tension or bending if there is a liability of shocks or blows coming on the piece. CHAPTER II. FRAME DESIGN. 8. General Principles of Design. The working or moving parts should be designed first and the frame adapted to them. The moving parts can be first arranged to give the motions and velocities desired, special attention being paid to compactness and to the convenience of the operator. Novel and complicated mechanisms should be avoided and the more simple and well tried devices used. Any device which is new should be first tried in a working model before being introduced in the design. The dimensions of the working parts for strength and stiffness must next be determined and the design for the frame completed. This may involve some modification of the moving parts. In designing any part of the machine, the metal must be put in the line of stress and bending avoided as far as possible. Straight lines should be used for the outlines of pieces exposed to tension or compression, circular cross sections for all parts in torsion, and profile of uniform fiber stress for pieces subjected to bending action. Superfluous metal must be avoided and this excludes all ornamentation as such. There should be a good 15 16 . . MACHINE DESIGN. practical reason for every pound of metal in the machine. An excess of metal is sometimes needed to give inertia and solidity and prevent vibration, as in the frames of machines having reciprocating parts, like engines, planers, slotting machines, etc. Mr. Oberlin Smith has characterized this as the " anvil " style of design in contradistinction to the " fiddle" style. In one the designer relies on the mass of the metal, in the other on the distribution of the metal, to resist the applied forces. A comparison of the massive Tangye bed of some large high-speed engines with the comparatively slight girder frame used in most Corliss engines, will em- phasize this difference. It may be sometimes necessary to waste metal in order to save labor in finishing, and in general the aim should be to economize labor rather than stock. The designers should be familiar with all the shop processes as well as the principles of strength and stability. The usual tendency in design, especially of cast iron work, is towards unnecessary weight. All corners should be rounded for the comfort and convenience of the operator, no cracks or sharp inter- nal angles left where dirt and grease may accumulate, and in general special attention should be paid to so designing the machine that it may be safely and con- veniently operated, that it may be easily kept clean, and that oil holes are readily accessible. The ap- pearance of a machine in use is a key to its working condition. Polished metal should be avoided on account of its tendency to rust, and neither varnish nor bright colors FIG. 1. OLD PLANING MACHINE. AN EXAMPLE OF ELABORATE OHNAM ENTATION. GENERAL PRINCIPLES OF DESIGN. 17 tolerated. The paint should be of some neutral tint and have a dead finish so as not to show scratches or dirt. Beauty is an element of machine design, but it can only be attained by legitimate means which are appro- propriate to the material and the surroundings. Beauty is a natural result of correct mechanical construction but should never be made the object of design. Harmony of design may be secured by adopting one type of cross-section and adhering to it throughout, never combining cored or box sections with ribbed sections. In cast pieces the thickness of metal should be uniform to avoid coolin'g strains, and for the same reason sharp corners should be absent. The lines of crystallization in castings are normal to the cooled surface and where two flat pieces come together at right angles, the interference of the two sets of crys- tals forms a plane of weakness at the corner. This is best obviated by joining the two planes with abend or sweep. Rounding the external corner and filleting the in- ternal one is usually sufficient. Where two parts come together in such a way as to cause an increase of thickness of the metal there are apt to be " blow holes " or "hot spots" at the junction due to the uneven cooling. " Strengthening " flanges when of improper propor- tions or in the wrong location are frequently a source of weakness rather than strength. A cast rib or flange on the tension side of a plate exposed to bending, will sometimes cause rupture by cracking on the outer edge. When apertures are cut in a frame either for core- prints or for lightness, the hole or aperture should be 18 MACHINE DESIGN. the symmetrical figure, and not the metal that sur- rounds it, to make the design pleasing to the eye. The design should be in harmony with the material used and not imitation. For example, to imitate structural work either of wood or iron in a cast-iron frame is silly and meaningless. Machine design has been a process of evolution. The earlier types of machines were built before the general introduction of cast-iron frames and had frames made of wood or stone, paneled, carved and decorated as in cabinet or architectural designs. When cast iron frames and supports were first introduced they were made to imitate wood and stone construction, so that in the Earlier forms we find panels, moldings, gothic traceries and elaborate decorations of vines, fruit and flowers, the whole covered with con- trasting colors of paint and varnished as carefully as a piece of furniture for the drawing-room. Relics of this transition period in machine architecture may be seen in almost every shop. One man has gone down to posterity as actually advertising an upright drill designed in pure Tuscan. 9. Machine Supports. The fewer the number of supports the better. Heavy frames, as of large en- gines, lathes, planers, etc., are best made so as to rest directly on a masonry foundation. Short frames as those of shapers, screw machines and milling machines, should have one support of the cabinet form. The use of a cabinet at one end and legs at the other is offensive to the eye, being inharmonious. If two cabinets are used provision should be made for a cradle or pivot at one end to prevent twisting of the frame by an uneven foundation. The use of intermediate supports is GENERAL PRINCIPLES OF DESIGN. 19 always to be condemned, as it tends to make the frame conform to the inequalities of the floor or foundation on what has been aptly termed the " caterpillar prin- ciple." A distinction must be made between cabinets or supports which are broad at the base and intended to be fastened to the foundation, and legs similar to those of a table or chair. The latter are intended to simply rest on the floor, should be firmly fastened to the machine and should be larger at the upper end where the greatest bending moment will come. The use of legs instead of cabinets is an assumption that the frame is stiff enough to withstand all stresses that come upon it, unaided by the foundation, and if that is the case intermediate supports are unne- cessary. Whether legs or cabinets are best adapted to a cer- tain machine the designer must determine for himself. Where two supports or pairs of legs are necessary under a frame, it is best to have them set a certain distance from the ends, and make the overhanging part of the frame of a parabolic form, as this divides up the bending moment and allows less deflection at the center. Trussing a long cast-iron frame with iron or steel rods is objectionable on account of the differ- ence in expansion of the two metals and the liability of the tension nuts being tampered with by work- men. The sprawling double curved leg which originated in the time of Louis XIV and which has served in turn for chairs, pianos, stoves and finally for engine lathes is wrong both from a practical and aesthetic stand- point. It is incorrect in principle and is therefore ugly. 20 MACHINE DESIGN. EXERCISE. 1. Apply the foregoing principles in making a written criticism of some engine or machine frame and its supports. (a) Girder frame of engine. (b) Tangye bed of air compresser. (c) Bed, uprights and supports of iron planing machine. (d) Bed and supports of engine lathe. (e) Cabinet of shaping or milling machine. (f) Frame of upright drill. 10. Machine Frames. For general principles of frame design the reader is referred to Chapter 2. Cast iron is the material most used but steel castings are now becoming common in situations where the stresses are unusually great, as in the frames of presses, shears and rolls for shaping steel. Cored vs. Rib Sections. Formerly the flanged or rib section was used almost exclusively, as but a few castings were made from each pattern and the cost of the latter was a considerable item. Of late years the use of hollow sections has become more common ; the patterns are more durable and more easily molded than those having many projections and the frames when finished are more pleasing in appearance. The first cost of a pattern for hollow work, including the cost of the core-box, is sometimes considerably more but the pattern is less likely to change its shape and in these days of many castings from one pattern, this latter point is of more importance. Finally it may be said that hollow sections are usually stronger for the same weight of metal than any that can be shaped from webs and flanges. MACHINE FRAMES. 21 Resistance to Bending. Most machine frames are exposed to bending in one or two directions. If the section is to be ribbed it should be of the form shown in Fig. 3. The metal being of nearly uniform thickness and the flange which is in tension having an area three or four times that of the compression flange. In a steel casting these may be more nearly equal. The hollow Fig. 3. 0w/y//MMW//t/ \ % ; ; p In B Fig. 4. section may be of the shape shown in Fig. 4, a hollow rectangle with the tension side re-enforced and slightly thicker than the other three sides. The re-enforcing flanges at A and B may often be utilized for the attaching of other mem- bers to the frame as in shapers or drill presses. The box section has one great advantage over the I section in that its moment of resistance to side bending or to twisting is usually much greater. The double I or the TJ section is common where it is necessary to have two parallel ways for sliding pieces as in lathes and planers. As is shown in Fig. 5 the two Is are usually connected at intervals by cross girts. Besides making the cross-section of the most economical form, it is often desirable to have such a longitudinal profile as shall give a Fig. 5. uniform fiber stress from end to 22 MACHINE DESIGN. end. This necessitates a parabolic or elliptic outline of which the best instance is the housing or upright of a modern iron planer. A series of experiments made in 1902 under the direction of the author, on the modulus of rupture of cast iron beams of the same weight but different cross- sections gave interesting results. Beginning with the solid circular section, which failed under a transverse load of 7,500 lb., square, rectangular, hollow and / shaped sections were tested until a maximum was reached in the / section with heavy tension flange which broke under a load of 38,000 lb. Channel and T shaped sections such as are appropriate for fly-wheel rims were also tested with the ribs in tension and in compression. The strength of such sections was found to be from two to three times as great when the ribs were in com- pression as when they were in tension. Resistance to Twisting. The hollow circular section is the ideal form for all frames or machine members which are subjected to torsion. If subjected also to bending the section may be made elliptical or, as is more common, thickened on two sides by making the core oval. See Fig. 6. As Fig. 6. has already been pointed out the box sections are in general better adapted to resist twisting than the ribbed or / sec- tions. Frames of Machine Tools. The beds of lathes are subjected to bending on account of their own weight and that of the saddle and on account of the downward MACHINE FRAMES. 23 pressure on the tool when work is being turned. They are usually subjected to torsion on account of the un- even pressure of the supports. The box section is then the best ; the double I commonly used is very weak against twisting. The same principle would apply in designing the beds of planers but the usual method of driving the table by means of a gear and rack prevents the use of the box section. The uprights of planers and the cross rail are subjected to severe bending moments and should have profiles of uniform strength. The uprights are also subject to side bending when the tool is taking a heavy side cut near the top. To pro- vide for this the uprights may be of a box section or may be reinforced by outside ribs. The upright of a drill press or vertical shaper is exposed to a constant bending moment equal to the upward pressure on the cutter multiplied by the distance from center of cutter to center of up- right. It should then be of constant cross-section from the bottom to the top of the straight part. The curved or goose- ^ necked portion should Fig 7 then taper gradually. The frame of a shear press or punch is usually of the Gr shape in profile with the inner fibers in tension and the outer in compression. The cross-section should be as in Fig. 3 or Fig. 4, preferably the latter, and should be graduated to the magnitude of the bending moment at each point. (See Fig. 7.) 24 MACHINE DESIGN. EXERCISES. 1. Discuss the stresses and the arrangement of material in the girder frame of a Corliss engine. 2. Ditto in the G frame of a band saw. PROBLEM. Design a G frame similar to that shown in Fig. 7, for a shear press capable of shearing a bar of mild steel 1| by 1 inches and having a gap four inches high and twenty-six inches deep. CHAPTEE III. Fig. 8. CYLINDER AND PIPES. ii. Thin Shells. Let Fig. 8 represent a section of a thin shell, like a boiler shell, exposed to an inter- nal pressure of p pounds per sq. inch. Then, if we consider any diameter AB, the total upward pressure A\ on upper half of the shell will balance the total down- ward pressure on the lower half and tend to separate the shell at A and B by tension. Let d= diameter of shell in inches. r= radius of shell in inches. 1= length of shell in inches. t= thickness of shell in inches. S= tensile strength of material. Draw the radial line CP to represent the pressure on the element P of the surface. Area of element at P=lrdO. Total pressure on element =plrd&. Vertical pressure on element =plr sin OdO. Total vertical pressure on APB= ^ C plr sin OdO=%plr. 25 26 MACHINE DESIGN. The area to resist tension at A and B=2tl and its total strength =2tlS. Equating the pressure and the resistance pr pd 2 S 4:S . . . . (13) The total pressure on the end of a closed cylindrical shell =Trr*p and the resistance of the circular ring of metal which resists this pressure =2irrtS. Equating : t Therefore a shell is twice as strong in this direction as in the other. Notice that this same formula would apply to spherical shells. In calculating the pressure due to a head of water equals h, the following formula is useful : p=OAMh. . . . (15) In this formula h is in feet and p in pounds per square inch. PROBLEMS. 1. A cast-iron water pipe is 12 inches in internal diameter and the metal is .45 inches thick. What would be the factor of safety, with an internal pressure due to a head of water of 250 feet ? 2. What would be the stress caused by bending due to weight, if the pipe in Ex. 1 were full of water and 24 feet long, the ends being merely supported ? 3. A standard lap- welded steam pipe, 8 inches in nominal diameter is 0.32 inches thick and is tested with an internal pressure of 500 pounds per sq. inch. What is the bursting pressure and what is the factor of safety above the test pres- sure, assuming S= 40000 ? THICK SHELLS. -ft 12. Thick Shells. There are several formulas for thick cylinders and no one of them is entirely satis- factory. It is however generally admitted that the tensile stress caused by internal pressure in such a cylinder is greatest at the inner circumference and diminishes according to some law from there to the exterior of the shell. This law of variation is expressed differently in the different formulas. Barlow's Formulas. Here the cylinder diameters are assumed to increase under the pressure, but in such a way that the volume of metal remains constant. Experiment has proved that in extreme cases this last assumption is incorrect. Within the limits of ordinary practice it is, however, approximately true. Let di and d 2 be the interior and exterior diameters j _ ,7 in inches and let t= 2 * be the thickness of metal. 2i Let I be the length of cylinder in inches. Let Si and S* be the tensile stresses in Ibs. per sq. inch at inner and outer circumferences. The volume of the ring of metal before the pressure is applied will be : and if the two diameters are assumed to increase th amounts x l and x z under pressure the final volume will be : Assuming the volume to remain the same : df - d? = ( 28 MACHINE DESIGN. Neglecting the squares of Xi and x z this reduces to or the distortions are inversely as the diameters. The unit deformations will be proportional to and the stresses Si and S 2 will be in the same ratio : j^f / y* C$ /~J ^ or the stresses vary inversely as the squares of the diameters. Let S be the stress at any. diameter d, then : S=^=~~ (where r is radius) and the total stress on an element of the area l.dr is : Integrating this expression between the limits -^ and -~ for r and multiplying by 2 we have : ~ Eq uating this to the pressure which tends to produce rupture, pdl, where p is the internal unit pressure, there results : The formula (13) for thin shells gives P= By comparing this with formula (16) it will be seen that in designing thick shells the external diameter determines the working pressure or : THICK SHELLS. Lame's Formula. In this discussion each particle of the metal is supposed to be subjected to radial com- pression and to tangential and longitudinal tension and to be in equilibrium under these stresses. Using the same notation as in previous formula : for the maximum stress at the interior, ', (18) for the stress at the outer surface. Fig. 9 illustrates the variation in S from inner to outer surface. Solving for cZ a in (17) we have Fig. 9. *-*^Ps .(19) A discussion of Lame's formula may be found in most works on strength of materials. PROBLEMS. 1. A hydraulic cylinder has an inner diameter of 8 inches, a thickness of four inches and an internal pressure of 1500 Ibs. per sq. in. Determine the maximum stress on the metal by Barlow's and Lame's formulas. 2. Design a cast-iron cylinder 6 inches internal diameter to carry a working pressure of 1200 Ibs. per sq. in. with a factor of safety of 10. 3. A cast-iron water pipe is 1 inch thick and 12 inches in- ternal diameter. Required head of water which it will carry with a factor of safety of 6. 30 MACHINE DESIGN. 13. Steel and Wrought Iron Pipe. Pipe for the transmission of steam, gas or water may be made of wrought iron or steel. Cast-iron is used for water mains to a certain extent, but its use for either steam or gas has been mostly abandoned. The weight of cast-iron pipe and its unreliability forbid its use for high pressure work. Wrought iron pipe up to and including one inch in diameter is usually butt-welded, and above that is lap- welded. Steel pipes may be either welded or may be drawn without any seam. Electric welding has been successfully applied to all kinds of steel tubing, both for transmitting fluids and for boiler tubes. The following tables are taken by permission from the catalogue of the Crane Company and show the stand- ard dimensions for steam pipe and for boiler tubes. Ordinary standard pipe is used for pressures not exceeding 100 Ib. per sq. in., extra strong pipe for the pressures prevailing in steam plants where compound and triple expansion engines are used, while the double extra is employed in hydraulic work under the heavy pressures peculiar to that sort of transmission. STEEL AND WROUGHT IRON PIPE. 31 hi * .s ! Ii fa II g s oo I 10 MACHINE DESIGN. ^ 02 Q O I f Si 1 1 Q 52; <3 Number of Threads per Inch of Screw. 00 00 00 00 oo 00 00 oo 00 00 00 3& 9 Pounds. i 05 TH O5 oi TH JO TH 1 00 TH S t- T 1 oo 00 JO oo' I ' Pipe Containing One Cubic Foot. 1 1 TH CO TH TH o O5 S 3 co" oo 00 oi JO h 11 1 | 1 1 g CO co JO GO ^ O3 t- co C5 CO s i Son TH o il 1 1 00 i 1 JO o JO CO 1 CO CO 1 J 0i 3 1 05 TH CO CO 1 1 CO o I o 05 10 s I 03 CO CO ^ JO co 00 I i-H TH CO 2 1 "ej g i OO CO CO O5 05 00 88 "oo CO CO OS i i I 'a cc 05 TH JO TH 05 T-H % % S S O5 CO 1 a g -d a 1 g 1 i 1 CO CO 1 co i i co i i TH JO O5 TH 0$ 3 3 ^ g 00 o o? 1 1 S 0. TH 00 1 i CO o g o JO J^ 1 g TH 0> T* 10 05 SI JO % TH co 1 g S a g i 1 TH I 5 CO i-H 00 i i o? 05 JO 8 o i - 2 3 JO TH 0^ CO o? t- 8 CO CO co |i 1 Oi t- co OJ OJ O7 oo o? I co CO i 1 Wl 1 to o 1 1 i 1 O5 i 05 3; a * 1 CO TjJ "* 10 CO *" *" oo o 3^ TH 1 ll 1 JO CO JO co JO 1 JO o JO to "1 a ^ ^ JO 10 CO l ~ oo O5 ^ T- 11 II j CO -f< JO CO - 00 OS TH TH O7 STEEL AND WROUGHT IRON PIPE. 33 w ft ( ( PH g H ^ m gth o are Sq o5 si |*I 1! 3? ^ ^N MACHINE DESIGN. 02 =1 Ift Pounds. s - 9) rr e> IO 50 8 OS I X i : 3 i Pipe per Foot of u a> 1 s 1 Oft CD i- -t 1 I g 04 1 1 t- ~ i s 5 5 II | s I i g 1 S i X | g C rf\ .!_> t-l r^ ^ CO oi 09 ,_^ rH ^ J W M 1 .J I g g OS S i i I S IE 1 X a i i a.| ^ ^^ -r-1 Tf lO o X - s s "3 a | 1 I g 13 g 5 2 1 s s 1 i | a a i i IH s "* 10 '~ K H 1 J s i | 1 i S? i i 1 | g "M a: " oi e - CO OS Z- a " cS W 6 1 1 g 1 7 g 00 i re S g ? | g co 1 0) c a re * ta ** x; O> s S 3 s w i i r? i- 1 r 1 1 | re f: CO 30 "K o a w 90 * tQ <- ~ N -r ^ w Icfi o ! 1 x IN fc - - 8 S 8 = 5 S 1*1 DC OJ g s I I i | | S i i 8 H &i|s w' a> ^3 I i i 'g I = 1 i ~ i 1 1 |e|5 a M ^ ~ " N 04 re * "* u 0) "5 a n .- ifi _ .0 0> Jfi g s ' = : --- OS re X Q ia i cS Si "Q at - O5 9 S . e o S S i g . -r t- S'5 ** I rH rH " CO CO CO + lO co : 1* 'cS o5 O 9$ 0> i> X C5 rH J CO ^ o M I-H 1-1 i i B 1 1 (1 r -fj i 5 o X s Oil 1 gj i g t- OJ H S CO CO -h IO i> 6- 00 C5 N Tjj Q w ' ' 1 1 1 3 CO CO CO 00 CO CO I I CO T-l U^ CO CO JO O 8 IS p 5 1 s! ^ ,< i ; I II! .a l-l O. p& I.I i 5* I 6 i oo ioo ioo fe c3 i ^ , I I rr I I a L g 3 in lp ^ 1 1> I < CO CO - g blag SI l co ^ Sg.J ^7.^ ?^^ i*S ill O e O S 55 H S^ 3 $ 5. 2 a W _' ^ ^ ga ssi M M C3 4> o 1*8 5 -3 M a a 2 * O> -i w m '.! u = i .3.2 S 11| 11-3 1~J s 1 1 lit * o a" Ml -tJ 3 s-s H M "II 38 MACHINE DESIGN. 14. Strength of Boiler Tubes. When tubes are used in a so-called fire-tube boiler with the gas inside and the water outside, they are exposed to a collapsing pres- sure. The same is true of the furnace flues of internally fired boilers. Such a member is in unstable equilibrium and it is difficult to predict just when failure will occur. Experiments on small wrought-iron tubes have shown the collapsing pressure to be about 80 per cent. of the bursting pressure. With short tubes set in tube sheets the length would have considerable in- fluence on the strength, but ordinary boiler tubes collapsing at the middle of the length would not be influenced by the setting. The strength of such tubes is probably proportional -j ) where t is the thickness and d the diameter. dj D. K. Clark gives for large iron flues the following formula : ....... (20) where P is the collapsing pressure in Ib. per sq. in. These flues had diameters varying from 30 in. to 50 in. and thickness of metal from f in. to T 7 T in. Prof. K. T. Stewart has recently made some in- teresting experiments on the collapsing pressure of lap-welded steel tubes and reported the results to the American Society of Mechanical Engineers. (See Transactions, Vol. XXVTI.) It will only be possible here to give some of the gen- eral conclusions, as stated by the author in his paper : 1. The length of tube, between transverse joints tending to hold it to a circular form, has no practical STRENGTH OF BOILER TUBES. 39 influence upon the collapsing pressure of a commercial lap-welded steel tube so long as this length is not less than about six diameters of tube. 2. The formulae, as based upon the present research, for the collapsing pressure of modern lap- welded Bes- semer steel tubes, are as follows : P=1000(l- J 1-1600 ~.) . . .(A) P=86670-~-1386 (B) Where P= collapsing pressure, pounds per sq. inch. d= outside diameter of tube in inches = thickness of wall in inches Formula (A) is for values of P less than 581 pounds, or for values of -r- less than 0.023, while formula (B) is for values greater than these. These formulae, while strictly correct for tubes that are 20 feet in length between transverse joints tending to hold them to a circular form, are, at the same time, substantially correct for all lengths greater than about six diameters. They have been tested for seven diameters, ranging from 3 to 10 inches, in all obtainable thicknesses of wall, and are known to be correct for this range. 3. The apparent fiber stress under which the different tubes failed varied from about 7000 pounds for the relatively thinnest to 35000 pounds per square inch for the relatively thickest walls. Since the average yield point of the material was 37000 and the tensile strength 58000 pounds per square inch, it would appear that the strength of a tube subjected to a collapsing fluid pressure is not dependent 40 MACHINE DESIGN. alone upon either the elastic limit or ultimate strength of the material constituting it. 15. Pipe Fittings. Steam pipe up to and including pipe two inches in diameter is usually equipped with screwed fittings, including ells, tees, couplings, valves, etc. Pipe of a larger size, if used for high pressures, should be put together with flanged fittings and bolts. One great advantage of the latter system is the fact that a section of pipe can easily be removed for repairs or alterations. Small connections are usually made of cast-iron or malleable iron. While the latter are neater in ap- pearance they are more apt to stretch and cause leaky joints. The larger fittings are made of cast-iron or cast-steel. Such fittings can be obtained in various weights and thicknesses, to correspond to those grades of pipe listed in the tables. The designer should have at hand catalogues of pipe fittings from the various manufacturers, as these will give in detail the proportions of all the different con- nections. For pressures not exceeding 100 Ibs. per sq. in. rubber and asbestos gaskets can be used between the flanges, but for higher pressures or for superheated steam corrugated metallic gaskets are necessary. In 1905 some very interesting experiments on the strength of standard screwed elbows and tees were made by Mr. S. M. Chandler, a graduate of the Case School, and published by him in " Power" for October, 1905. The fittings were taken at random from the stock of the Pittsburg Valve and Fittings Co., and three of PIPE FITTINGS. 41 each size were tested to destruction by hydraulic pres- sure. The following table gives a summary of the results obtained. The values which are starred in the table were obtained from fittings which had purposely been cast with the core out of center so as to make one wall thinner than the other. These values are not included in the averages. TABLE X. BURSTING STRENGTH OF STANDARD SCREWED FITTINGS, PRESSURES IN POUNDS PER SQUARE INCH. SIZE. ELBOWS. AVERAGE. 2* 3500 3300 3400 3400 3 2400 2600 2100* 2500 8* 2100 1700* 2400 2250 4 2800 2500 2500 2600 4i 2000* 2600 2600 2600 5 2600 2500 2500 2538 . 6 2600 2200 2300 2367 7 1800 2100 1900* 1950 8 1700 1600 1700 1667 9 1800 1800 1900 1833 10 1800 1700 1600 1700 12 1100 1200 900* 1150 SIZE. TEES. AVERAGE. li 3400 3300 3300 3333 H 3400 3200 2800* 3300 2 2500 2800 2500 2600 2i 2400 2100* 2500 2450 3 1400* 1900 1800 1850 3* 1200* 1500 1800 1650 4 1800 2100 1700 1867 4* 1100* 1400 1400 1400 5 1700 1300* 1500 1600 6 1400 1500 1100* 1450 7 1400 1400 1500 1433 8 1200* 1400 1300 1350 9 1300 1400 1200 1300 10 1100 1300 1200 1200 12 1100 1000 1100 1067 * Made with eccentric core. 42 MACHINE DESIGN. These tests show a large apparent factor of safety for any pressures to which screwed fittings are usually subjected. The failure of such fittings in practice must be at- tributed to faulty workmanship in erection, such as screwing too tight, lack of allowance for expansion and poor drainage. The average tensile strength of the cast-iron used in the above fittings was 20000 Ibs. per sq. in. PROBLEMS. 1. Determine the bursting pressure of a wrought iron steam pipe 6 inches nominal diameter. (a) If of standard dimensions. (b) If extra strong. (c) If double extra strong. 2. Compare the above with the strength of standard screwed elbows and tees of the same size. 3. Determine the probable collapsing pressure of a charcoal iron boiler-tube of two inches nominal diameter. 16. Steam Cylinders. Cylinders of steam engines can hardly be considered as coming under either of the preceeding heads. On the one hand the thickness of metal is not enough to insure rigidity as in hydraulic cylinders, and on the other the nature of the metal used, cast-iron, is not such as to warrant the assump- tion of flexibility, as in a thin shell. Most of the for- mulas used for this class of cylinder are empirical and founded on modern practice. Van Bur en's formula for steam cylinders is : * t=.0001pd+.l5\/d (21) * See Whitham's "Steam Engine Design," p. 27. STEAM CYLINDERS. 43 A formula which the writer has developed is some- what similar to Van Buren's. Let s' = tangential stress due to internal pressure. Then by equation for thin shells Let s" be an additional tensile stress due to distor- tion of the circular section at any weak point. Then if we regard one-half of the circular section as a beam fixed at A and B (Fig. 11) and assume the maximum bending moment as at C some weak point, the tensile stress on the outer fibres at C due to the bending will be proportional to =2* i by the laws of flexure, or where c is some unknown constant. The total tensile stress at C will then be t 2 sr j_ pd* 2d .(a) *&+&, (22) Solving for c Solving for t a form which reduces to that of equation (13) when c=0. An examination of several engine cylinders of 44 MACHINE DESIGN. standard manufacture shows values of c ranging from .03 to .10, with an average value : e=.06. The formula proposed by Professor Barr, in his paper on* " Current Practice in Engine Proportions," as representing the average practice among builders of low speed engines is : =.05 d+.3 inch (23) In Kent's Mechanical Engineer's Pocket Book, the following formula is given as representing closely existing practice : t=. 0004 dp + 0.3 inch (24) This corresponds to Barr's formula if we take p= 125 pounds per square inch. Experiments f made at the Case School of Applied Science in 1896-97 throw some light on this subject. Cast iron cylinders similar to those used on engines were tested to failure by water pressure. The cylinders varied in diameter from six to twelve inches and in thickness from one-half to three-quarters inches. Contrary to expectations most of the cylinders failed by tearing around a circumference just inside the flange. (See Fig. 12). * Transactions A. S. M. E., vol. xviii, p. 741. t Transactions A. S. M. E. vol. xix. STEAM CYLINDERS. TABLE XI. 45 No. Diam. d Pres- sure. P Thick- ness. t Line of Failure. Formulas Used. Strength of Test-bar. 18 s =^ 14 s=% a c= a 12.16 800 .70 Circum. 6940 3470 .046 18000 Ibs. d 12.45 700 .56 Longi. 7780 .047 24000 Ibs. e 9.12 1325 .61 Circum. 9900 4950 .048 24000 Ibs. f 6.12 2500 .65 Circum. 11800 5900 .055 24000 Ibs. 1 9.58 600 .402 Longi. 7150 .049 24000 Ibs. 2 9.375 1050 .573 Circum. 8590 4300 .055 24000 Ibs. 3 9.13 975 .596 Circum. 7470 3740 .072 24000 Ibs. 4 12.53 700 .571 Longi. 7680 .048 24000 Ibs. 5 12.56 875 .531 Circum. 10350 5180 .028 24000 Ibs. Average of c=.05 Table XI gives a summary of the results. Out of nine cylinders so tested, only three failed by splitting longitudinally. This appears to be due to two causes. In the first place, the flanges caused a bending moment at the junction with the shell due to the pull of the bolts. In the second place, the fact that the flanges were thicker than the shell caused a zone of weakness near the flange due to shrinkage in cooling, and the pres- ence of what founders call " a hot spot." The stresses figured from formula (14) in the cases where the failure was on a circumference, are from one-fifth to one-sixth the tensile strength of the test bar. 4:6 MACHINE DESIGN. The strength of a chain is the strength of the weakest link, and when the tensile stress exceeded the strength of the metal near some blow hole or "hot spot," tearing began there and gradually extended around the circumference. Values of c as given by equation (a) have been cal- culated for each cylinder, and agree fairly well, the average value being c=.05. To the criticism that most of the cylinders did not fail by splitting, and that therefore formulas (a) and (22) are not applicable, the answer would be that the chances of failure in the two directions seem about equal, and consequently we may regard each cylinder as about to fail by splitting under the final pressure. If we substitute the average value of c=.05 and a safe value of s=2000, formula (21) reduces to : d d I p* * Subsequent experiments made at the Case School in 1904 show the effect of stiffening the flanges by brackets. Four cylinders were tested, each being 10 inches internal diameter by 20 inches long and having a thickness of about f inches. The flanges were of the same thickness as the shell and were re-enforced by sixteen triangular brackets as shown in Fig. 13. The fractures were all longitudinal there being but little of the tearing around the shell which was so marked a feature of the former experiments. This shows that the brackets served their purpose. Table XII gives the results of the tests and the calculated values of c. * Machinery, N. Y., Nov. 1905. FIG. 12. FRACTURED CYLINDER. FIG. 13. FRACTURED CYLINDER. STEAM CYLINDERS. TABLE XII. BURSTING PRESSURE OF CAST-IRON CYLINDERS. Internal Average Bursting Value pd Diameter. Thickness. Pressure. of c. 9f 10.125 0.766 1350 .0213 9040 10.125 0.740 1400 .0152 10200 10.125 0.721 1350 .0126 9735 10.125 0.720 1200 .0177 9080 Average value of c=.0167. Comparing the values in the above table with those in Table XI we find c to be only one- third as large. The tensile strength of the metal in the last four cylinders, as determined from test bars, was only 14000 Ibs. per sq. in. Comparison with the values of due to direct tension as given by the formula shows that in a cylinder of this type about one-third of the stress is " accidental" and due to lack of uni- formity in the conditions. In. Table XI about two- thirds must be thus accounted for. PROBLEMS. 1. Referring to Table XI, verify in at least three experiments the values of 8 and c as there given. Do the same in Table XII. 2. The steam cylinder of a Baldwin locomotive is 22 ins. in diameter and 1.25 ins, thick, Assuming 125 Ibs. gauge pres- 48 MACHINE DESIGN. sure, find the value of c. Calculate thickness by Van Buren's and Barr's formulas. 3. Determine proper thickness for cylinder of cast-iron, if the diameter is 38 inches and the steam pressure 100 Ibs. by formulas 13, 21, 23, 24 and 25. 4. The cylinder of a stationary engine has internal diameter =12 in. and thickness of shell =1 in. Find the value of c for jp=120 Ibs. per sq. in. 17. Thickness of Flat Plates. An approximate formula for the thickness of flat cast-iron plates may be derived as follows : Let Z= length of plate in inches. b= breadth of plate in inches. t= thickness of plate in inches. p= intensity of pressure in pounds. S = modulus of rupture Ibs. per sq. in. A plate which is supported or fastened at all four edges is constrained so as to bend in two directions at right angles. Now if we suppose the plate to be represented by a piece of basket work with strips crossing each other at right angles we may consider one set of strips as resisting one species of bending and the other set as resisting the other bending. We may also consider each set of strips as carrying a fraction of the total load. The equation of condition is that each pair of strips must have a common deflec- tion at the crossing. Suppose the plate to be divided lengthwise into flat strips an inch wide I inches long, and suppose that a fraction p' of the whole pressure causes the bending of these strips. THICKNESS OF FLAT PLATES. 49 Regarding the strips as beams with fixed ends and uniformly loaded : bh* and the thickness necessary to resist bending is : In a similar manner, if we suppose the plate to be divided into transverse strips an inch wide and b inches long, and suppose the remainder of the pressure^ p' equals p" to cause the bending in this direction, we shall have : t I But as all these strips form one and the same plate the ratio of p' to p" must be such that the deflection at the center of the plate may be the same on either supposition. The general formula for deflection in this case is WT ZSIEI and I = Tn for each set of strips. Therefore the deflec- tion is proportional to ^- and ^-jr- m the two cases. But p' + p" = p Solving in these equations for p f and p" ,_ pb* V 50 MACHINE DESIGN. Substituting these values in (a) and (b) : t= !., P. As l>b usually, equation (27) is the one to be used. If the plate is square l=b and .(88) If the plate is merely supported at the edges then formulas (26) and (2Y) become : For rectangular plate : / 9Q N ZM%- For square plate : A round plate may be treated as square, with side = diameter, without sensible error. The preceding formulas can only be regarded as approximate. G-rashof has investigated this subject and developed rational formulas but his work is too long and complicated for introduction here. His for- mulas for round plates are as follows : Round plates : Supported at edges : Fixed at edges : '^1. . .(32) THICKNESS OF FLAT PLATES. 51 where t and p are the same as before, d is the diameter in inches and S is the safe tensile strength of the material. Comparing these formulas with (28) and (30) for square plates, they are seen to be nearly identical if allowance is made for the difference in the value of S. Experiments made at the Case School of Applied Science in 1896-97 on rectangular cast iron plates with load concentrated at the center gave results as follows : Twelve rectangular plates planed on one side and each having an unsupported area of 10 by 15 inches were broken by the application of a circular steel plunger one inch in diameter at the geometrical center of each plate. The plates varied in thickness from one-half inch to one and one-eighth inches. Numbers 1 to 6 were merely supported at the edges, while the remain- ing six were clamped rigidly at regular intervals around the edge. To determine the value of S, the modulus of rup- ture of the material, pieces were cut from the edge of the plates and tested by cross-breaking. The average value of S from seven experiments was found to be 33000 Ibs. per sq. in. In Table XIII are given the values obtained for the breaking load W under the different conditions. If we assume an empirical formula : and substitute given values of S, I and b we have nearly : W = 100kt 2 ......... (b) Substituting values of W and t from the Table XIII we have the values of k as given in the last column, 52 MACHINE DESIGN. If we average the values for the two classes of plates and substitute in (a) we get the following empirical formulas : For breaking load on plates supported at the edges and loaded at the center : (31) and for similar plates with edges fixed : TT=442^ (32) in both formulas is the modulus of rupture. TABLE XIII. CAST IRON PLATES 10x15 INS. Thickness Breaking Constant. No. Load. t W k 1 .562 7500 237 2 .641 11840 288 3 .745 14800 267 4 .828 21900 320 5 1.040 31200 289 6 1.120 31800 254 7 .481 9800 424 8 .646 17650 422 9 .769 26400 446 10 .881 33400 430 11 1.020 47200 454 12 1.123 59600 477 Those plates which were merely supported at the edges broke in three or four straight lines radiating from the center. Those fixed at the edges broke in four or five radial lines meeting an irregular oval inscribed in the rectangle. Number 12 however failed by shearing, the circular plunger making a circular hole in the plate with several radial cracks. THICKNESS OF FLAT PLATES. 53 Some tests were made in the spring of 1906 at the Case School laboratories by Messrs. Hill and Nadig on the strength of flat cast-iron plates under uniform hydraulic pressure. The plates tested were of soft gray iron, having a low tensile strength of about 12000 Ibs. per square inch, and were of the following sizes : 12 by 12 by f inches. 12 by 12 by 1 inches. 12 by 18 by 1.25 inches. 12 by 18 by |f inches. These burst at the following pressures respectively : 375 Ibs. 675 Ibs. 650 Ibs. 450 Ibs. The fractures started at the center of the plates and ran to the sides in irregular lines. The square plates were somewhat weaker than would have been expected from the formula and the rectangular plates somewhat stronger. PROBLEMS. 1. Calculate the thickness of a steam-chest cover 8 X 12 inches to sustain a pressure of 90 Ibs. per sq. inch with a factor of safety =10. 2. Calculate the thickness of a circular manhole cover of cast-iron 18 inches in diameter to sustain a pressure of 150 Ibs. per sq. inch with a factor of safety =8, regarding the edges as merely supported. 3. Determine the probable breaking load for a plate 18 by 24 in. loaded at the center, (a) when edges are fixed, (b) When edges are supported. 4. In experiments on steam cylinders, a head 12 inches in diameter and 1.18 inches thick failed under a pressure of 900 Ibs. per sq. in. Determine the value of S by formula (28). CHAPTER IV. FASTENINGS. 18. Bolts and Nuts. Tables of dimensions for U. S. standard bolt heads and nuts are to be found in most engineering hand-books and will not be repeated here. These proportions have not been generally adopted on account of the odd sizes of bar required. The standard screw-thread has been quite generally ac- cepted as superior to the old V- thread. Roughly the diameter at root of thread is 0. 83 of the outer diameter in this system, and the pitch in inches is given by the formula p = .24i/d+.625-.lT5. .... .(33) where d= outer diameter. TABLE XIV. SAFE WORKING STRENGTH OF IRON OR STEEL BOLTS. Diam. of Bolt. Inch. Thr'ds per Inch. No. Diam. at Root of Thread. Inches. Area at Root of Thread. Sq. In. Safe Load in Tension. Lb. Safe Load in Shear. Lb. 5000 Ib. per sq. in. 7500 Ib. per sq. in. 4000 Ib. per sq. in. 6000 Ib. per sq. in * 20 .185 .0269 135 202 196 294 A 18 .240 .0452 226 340 306 460 I 16 .294 .0679 340 510 440 660 T 7 * 14 .344 .0930 465 695 600 900 i 13 .400 .1257 628 940 785 1175 BOLTS AND NUTS. 55 TABLE XIV (Continued). SAFE WORKING STRENGTH OF IRON OR STEEL BOLTS. Diam. of Bolt. Inch. Thr'ds per Inch. No. Diam. at Root of Thread. Inches. Area at Root of Thread. Sq. In. Safe Load in Tension. Lb. Safe Load in Shear. Lt. 5000 Ib. per sq. in. 7500 Ib. per sq. in. 4000 Ib, per sq. in. 6000 Ib. per sq. in. & 12 .454 .162 810 1210 990 1485 i 11 .507 .202 1010 1510 1230 1845 i 10 .620 .302 1510 2260 1770 2650 t 9 .731 .420 2100 3150 2400 3600 i 8 .837 .550 2750 4120 3140 4700 H 7 .940 .694 3470 5200 3990 6000 1* 7 1.065 .891 4450 6680 4910 7360 if 6 1.160 1.057 5280 7920 5920 7880 H 6 1.284 1.295 6475 9710 7070 10600 tt 51 1.389 1.515 7575 11350 8250 12375 If 5 1.490 1.744 8720 13100 9630 14400 U 5 1.615 2.049 10250 15400 11000 16500 2 4| 1.712 2.302 11510 17250 12550 18800 The shearing load is calculated from the area of the body of the bolt. Bolts may be divided into three classes which are given in the order of their merit. 1. Through bolts, having a head on one end and a nut on the other. 2. Stud bolts, having a nut on one end and the other screwed into the casting. 3. Tap bolts or screws having a head at one end and the other screwed into the casting. 56 MACHINE DESIGN. The principal objection to the last two forms and especially to (3) is the liability of sticking or breaking off in the casting. Any irregularity in the bearing surfaces of head or nut where they come in contact with the casting, causes a bending action and consequent danger of rupture. This is best avoided by having a slight annular projection on the casting concentric with the bolt hole and finishing the flat surface by planing or counter- boring. Counter-boring without the projection is a rather slovenly way of overcoming the difficulty. When bolts or studs are subjected to severe stress and vibration, it is well to turn down the body of the bolt to the diameter at root of thread, as the whole bolt will then stretch slightly under the load. A check nut is a thin nut screwed firmly against the main nut to prevent its working loose, and is commonly placed outside. As the whole load is liable to come on the outer nut, it would be more correct to put the main nut outside. (Prove this by figure.) After both nuts are firmly screwed down, the outer one should be held stationary and the inner one reversed against it with what force is deemed safe, that the greater reaction may be between the nuts. Numerous devices have been invented for the purpose of holding nuts from working loose under vibration but none of them are entirely satisfactory. Probably the best method for large nuts is to drive a pin or cotter entirely through nut and bolt. A flat plate, cut out to embrace the nut and fastened to the casting by a machine screw, is often used. MACHINE SCREWS. 57 19. Machine Screws. A screw is distinguished from a bolt by having a slotted, round head instead of a square or hexagon head. The head may have any one of four shapes, the round, fillister, oval fillister and flat as shown in Fig. 14. A committee of the American Society of Mechanical Engineers has recently recommended cer- tain standards for machine screws.* The form of thread recommended is the U. S. Stand- ard or Sellers type with provision for clearance at top and bottom to insure bearing on the body of the thread. The sizes and pitches recommended are as follows : TABLE XV. MACHINE SCREWS. Standard Diameter. .070 .085 .100 .110 .125 .140 .165 .190 .215 28 .240 .250 .270 .320 .375 Threads per inch. 72 64 56 48 44 40 36 32 24 24 22 20 16 Reference is made to the report itself for further details of heads, taps, etc. 20. Eye Bolts and Hooks. In designing eye bolts it is customary to make the combined sectional area of the sides of the eye one and one half-times that of the bolt to allow for obliquity and an uneven distribution of stress. Large hooks should be designed to resist combined * Trans. A. S. M. E., Vol. xxvii. 58 MACHINE DESIGN. bending and tension ; the bending moment is equal to the load multiplied by the longest perpendicular from the center line of hook to line of load. The tension due to this bending must be added to the direct tension and the body of the hook designed ac- cordingly. PROBLEMS. 1. Discuss the effect of the initial tension caused by the screwing up of the nut on the bolt, in the case of steam fittings, etc. ; i. e. should this tension be added to the tension due to the steam pressure, in determining the proper size of bolt ? 2. Determine the number of f inch steel bolts necessary to hold on the head of a steam cylinder 15 inches diameter, with the internal pressure 90 pounds per square inch, and factor of safety =12. 3. Show what is the proper angle between the handle and the jaws of a fork wrench. (1) If used for square nuts ; (2) If used for hexagon nuts ; illustrate by figure. 4. Determine the length of nut theoretically necessary to prevent stripping of the thread, in terms of the outer diameter of the bolt. (1) With U. S. standard thread. . (2) With square thread of same depth. 5. Design a hook with a swivel and eye at the top to hold a load of one ton with a factor of safety 5, the center line of hook being three inches from line of load, and the material wrought iron. 21. Riveted Joints. Eiveted joints may be divided into two general classes : lap joints where the two plates lap over each other, and butt joints where the edges of the plates butt together and are joined by over-lapping straps or welts. If the strap is on one RIVETED JOINTS. 59 D c side only, the joint is known as a butt joint with one strap ; if straps are used inside and out the joint is called a butt joint with two straps. Butt joints are generally used when the material is more than one half inch thick. Any joint may have one, two or more rows of rivets and hence be known as a single riveted joint, a double riveted joint, etc. Any riveted joint is weaker than the origi- nal plate, simply CP^^ A ~- C\~ O -/Z? oecause holes cannot ^^^ be punched or drilled C-^'N /^v s~\ r in the plate for the ^sL/ i i introduction of rivets without removing some of the metal. Fig. 15 shows a double riveted lap joint and Fig. 16 a single riveted butt joint with two straps. Eiveted joints may fail in any one of four ways : 1. By tearing of the plate along a line of rivet holes, as at AB, Fig. 15. 2. By shearing the rivets. 3. By crushing and [ wrinkling of the plate in front of each rivet as at C, Fig. 15, thus causing leakage. 4. By splitting of the plate opposite each rivet as at D, Fig. 15. The last manner of failure may be pre- 15. of o O =1 o O ==1 Fig. 16. 60 MACHINE DESIGN. vented by having a sufficient distance from the rivet to the edge of the plate. Practice has shown that this distance should be at least equal to the diameter of a rivet. Experience has shown that lap joints in plates of even moderate thickness are dangerous on account of the liability of hidden cracks. Several disastrous boiler explosions have resulted from the presence of cracks inside the joint which could not be detected by inspection. The fact that one or both plates are out of the line of pull brings a bending moment on both plates and rivets. Some boiler inspectors have gone so far as to condemn lap-joints altogether. Let = thickness of plate. d= diameter of rivet-hole. p= pitch of rivets. n= number of rows of rivets. T= tensile strength of plate. C = crushing strength of plate or rivet. S= Shearing strength of rivet. Average values of the constants are as follows : Material. T C 8 Wrought Iron 50 000 80 000 40 000 Soft Steel.., 56 000 90 000 45 000 The values of the constants given above are only average values and are liable to be modified by the exact grade of material used and the manner in which it is used. LAP JOINTS. 61 The tensile strength of soft steel is reduced by punching from three to twelve per cent according to the kind of punch used and the width of pitch. The shearing strength of the rivets is diminished by their tendency to tip over or bend if they do not fill the holes, while the bearing or compression is doubtless relieved to some extent by the friction of the joint. The values given allow roughly for these modifica- tions. 22. Lap Joints. This division also includes butt joints which have but one strap. Let us consider the shell as divided into strips at right angles to the seam and each of a width =p. Then the forces acting on each strip are the same and we need to consider but one strip. The resistance to tearing across of the strip between rivet holes is (pd)tT. ...... (a) and this is independent of the number of rows of rivets. The resistance to compression in front of rivets is ndtC .......... (b) and the resistance to shearing of the rivets is (c) If we call the tensile strength T= unity then the relative values of C and $ are 1.6 and 0.8 respectively. Substituting these relative values of T, C and S in our equations, by equating (b) and (c) and reducing we have d=2.55t ........ (34) Equating (a) and (c) and reducing we have (35) 02 MACHINE DESIGN. Or by equating (a) and (b) p = d+1.6nd ....... (36) These proportions will give a joint of equal strength throughout, for the values of constants assumed. 23. Butt Joints with two Straps. In this case the resistance to shearing is increased by the fact that the rivets must be sheared at both ends before the joint can give away. Experiment has shown this increase of shearing strength to be about 85 per cent and we can therefore take the relative value of S as 1.5 for butt joints. This gives the following values for d and p dUl.36* ........ (37) p=d+l.lf ...... (38) p=d+l.Qnd ....... (39) In the preceding formulas the diameter of hole and rivet have been assumed to be the same. The diameter of the cold rivet before insertion will be T V inches less than the diameter given by the formulas. Experiments made in England by Prof. Kennedy give the following as the proportions of maximum strength : Lap joints cZ=2.33 Butt joints d=l.St 24. Efficiency of Joints. The efficiency of joints designed like the preceding is simply the ratio of the section of plate left between the rivets to the section BETT JOINTS WITH UNEQUAL STRAPS. 63 of solid plate, or the ratio of the clear distance between two adjacent rivet holes to the pitch. From formula (35) we thus have. ........ (40) This gives the efficiency of single, double and triple riveted seams as .615, .762 and .828 respectively. Notice that the advantage of a double or triple riveted seam over the single is in the fact that the pitch bears a greater ratio to the diameter of a rivet, and therefore the proportion of metal removed is less. 25. Butt Joints with unequal Straps. One joint in common use requires special treatment. It is a double-riveted butt joint in which the inner strap is made wider than the outer and an extra row of rivets added, whose pitch is double that of the original seam ; this is sometimes diamond See Fig. 17. This outer row of rivets is then exposed to single shear and the original rows to double shear. Consider a strip of plate of a width = 2p. c a 1 led riveting. i_ Fig. 17. Then the resistance to tearing along the outer row of rivets is (2p-d)tT 64 MACHINE DESIGN. As there are five rivets to compress in this strip the bearing resistance is As there is one rivet in single shear and four in double shear the resistance to shearing is + (4 X 1.85) d*S = 6. Solving these equations as in previous cases, we have for this particular joint d=l.&2t ....... (41) (42) (43) 26. Practical Rules. The formulas given above show the proportions of the usual forms of joints for uniform strength. In practice certain modifications are made for economic reasons. To avoid great variation in the sizes of rivets the latter are graded by sixteenths of an inch, making those for the thicker plates considerably smaller than the formula would allow, and the pitch is then calculated to give equal tearing and shearing strength. Table XVI shows what may be considered average practice in this country for lap-joints with steel plates and rivets. PRACTICAL RULES. 65 TABLE XVI. RIVETED LAP JOINTS. Thick- ness of Plate. Diam. of Rivet. Diam. of Hole. Pitch. Efficiency of Plate. Single. Double. Single. Double. i * A If If .59 .68 T 5 6 f H If 2i .58 .68 1 f if U 2* .57 .67 I 7 . It 1 2 2f .56 .68 1 1 H 2 21 .53 .67 The efficiencies are calculated from the strength of plate between rivet holes and the efficiencies of the rivets rnay be even lower. Comparing these values with the ones given in Art. 24 we find them low. This is due to the fact that the pitches assumed are too small. The only excuse for this is the possibility of getting a tighter joint. TABLE XVII. RIVETED BUTT JOINTS. Thickness Diam. Diam. Pitch. of of of Plate. Rivet. Hole. Single. Double. Triple. i t if 3f 4 5* f it 1 2f 3f 5i * if 2f 3| 5i * if 1 2f 8| 5 1 1 h* 2f 3| 5 06 MACHINE DESIGN. Table XVII has been calculated for butt joints with two straps. As in the preceding table the values of the pitch are too small for the best efficiency. The tables are only intended to illustrate common practice and not to serve as standards. There is such a diversity of practice among manufacturers that it is advisable for the designer to proportion each joint according to his own judgment, using the rules of Arts. 22-25 and having regard to the practical considerations which have been mentioned. A committee of the Master Steam Boiler Makers' Association has made a number of tests on riveted joints and reported its conclusions* The specimens were prepared according to generally accepted practice, but on subjecting them to tension many of them failed by tearing through from hole to edge of plate. The committee recommends making this distance greater, so that from the center of hole to edge of plate shall be perhaps 2d instead of 1.5~ 2.55.R when S is the safe shearing strength. Substituting this value of P in (32) we have for the safe deflection : 31. Square Wire. The value of the stress for a square section is : where d is the side of square. 76 MACHINE DESIGN. The distortion at the corners caused by twisting through an angle is : = Od ll/2 Equation (c) then becomes : SP'Dl The three principal equations (46), (47) and (48) then reduce to : X== GkP * * * * (49) (51) The square section is not so economical of material as the round* 32. Experiments. Tests made on about 1700 tem- pered steel springs at the French Spring Works in Pittsburg were reported in 1901 by Mr. R. A. French.* These were all compression springs of round steel and were given a permanent set before testing by being closed coil to coil several times. Table XVIII gives results of these experiments. * Trans. A. S. M. E., Vol. XXIII. SSOJ1S 3UUB811S EXPERIMENTS. '888888888888! jmtoisaoj, >|||||||gg|gS888888888Sl [S S 1 8 1 8 8 8 S88 8 8 8 8 88 | 88 8| 8 8 < Snijds esoio ocj 111 ,,H ,H TH T-KN inch Ibs. The work per square inch of projected area is then : ld Solving in (a) for Z Z = !lLJLl!lir. (b) w Let 2 = C a co-efficient whose value is to be ob- tained by experiment ; then C= j and /= ~ -. . . (58) Crank pins of steam engines have perhaps caused more trouble by heating than any other form of jour- nal. A comparison of eight different classes of propel- lers in the old U. S. Navy showed an average value for C of 350,000. 106 MACHINE DESIGN. A similar average for the crank pins of thirteen screw steamers in the French Navy gave (7=400,000. Locomotive crank pins which are in rapid motion through the cool outside air allow a much larger value of (7, sometimes more than a million. Examination of ten modern stationary engines shows an average value of (7=200,000 and an average pressure per square inch of projected area =300 Ib. The investigations of Professor Barr above referred to show a wide variation in the constants for the length of crank pins in stationary engines. He prefers to use TJ"D the formula : l=K- T -+B where K and B are con- JL/ stants and L= length of stroke of engine in inches. We may put this in another form since : HP WN - T -= - where Wis the total mean pressure. JL/ lyoUuu The formula then becomes : The value of B was found to be 2.5 in. for high- speed and 2 in. for low-speed engines, while K fluc- tuated from .13 to .46 with an average of .30 in the the former class, and from .40 to .80 with an average of .60 in the low-speed engines. If we adopt average values we have the following formulas for the crank-pins of modern stationary engines : WN High-speed engines Z=-+ 2. 5 in. . . WN Low-speed engines 1= +2 in ..... (61) EXPERIMENTS. 107 Compare these formulas with (58) when values of C are introduced. In a discussion on the subject of journal bearings in 1885,* Mr. Geo. H. Babcock said that he had found it practicable to allow as high as 1200 Ib. per sq. in. on crank pins while the main journal could not carry over 300 Ib. per sq. in. without heating. One rule for speed and pressure of journal bearings used by a well-known designer of Corliss engines is to multiply the square root of the speed in feet per second by the pressure per square inch of projected area and limit this product to 350 for horizontal engines and 500 in vertical engines. 48. Experiments. Some tests made on a steel journal 3^ inches in diameter and 8 inches long running in a cast-iron bearing and lubricated by a sight-feed oiler, will serve to illustrate the friction and heating of such journals. The two halves of the bearing were forced together by helical springs with a total force of 1400 pounds, so that there was a pressure of 54 Ib. per sq. in. on each half. The surface speed was 430 ft. per min. and the oil was fed at the rate of about 12 drops per minute. The lubricant used was a rather heavy automobile oil having a specific gravity of 0.925 and a viscosity of 174 when compared with water at 20 deg. Cent. The length of the run was two hours and the tem- perature of the room 70 deg. Fahr. (See Table XXII.) * Trans. A. S. M. E., Vol. VI. 108 MACHINE DESIGN. TABLE XXII. FRICTION OF JOURNAL BEARING. Time. Rev. per min. Temp. Fahr. Coeff. of friction. 10:03 500 69 .024 10:15 482 82 .0175 10:30 506 100 .013 10:45 506 115 .010 11:00 516 125 .010 11 :15 135 .004 11 :30 145 .004 11 :45 512 147 .004 12:00 151 .007 49- Strength and Stiffness of Journals. A journal is usually in the condition of a bracket with a uniform load, and the bending moment M =&- Therefore by formula (6) \.\Wl The maximum deflection of such a bracket is A W? ~SE I 64 WV If as is usual A is allowed to be stiffness d- or approximately d=4.J E *IWF E ' inches, then for (63) .(64) CAPS AND BOLTS. 109 The designer must be guided by circumstances in determining whether the journal shall be calculated for wear, for strength or for stiffness. A safe way is to design the journal by the formulas for heating and wear and then to test for strength and deflection. Kemember that no factor of safety is needed in formula for stiffness. Note that W in formulas for strength and stiffness is not the average but the maximum load. 50. Caps and Bolts. The cap of a journal bearing exposed to upward pressure is in the condition of a beam supported by the holding down bolts and loaded at the center, and may be designed either for strength or for stiffness. Let : P=max. upward pressure on cap. L= distance between bolts. b= breadth of cap at center. h= depth of cap at center. A = greatest allowable deflection. Strength: M-* Stiffness: 25ZT (65) T _bh'_WL* 12 ~ .(66) If A is allowed to be T J T inches and E for cast iron is taken =18000000 then: &.01115Zr~f-7- (67) 110 MACHINE DESIGN. The holding down bolts should be so designed that the bolts on one side of the cap may be capable of carrying safely two thirds of the total pressure. PROBLEMS. 1 . A flat car weighs 10 tons, is designed to carry a load of 20 tons more and is supported by two four-wheeled trucks, the axle journals being of wrought iron and the wheels 33 inches in diameter. Design the journals, considering heating, wear, strength and stiffness, assuming a maximum speed of 30 miles an hour, factor of safety =10 and C= 300000. 2. The following dimensions are those generally used for the journals of freight cars having nominal capacities as indicated : CAPACITY. DIMENSIONS OF JOURNAL. 100000 lb. 4. 5 by 9 in. 60000 lb. 4.25 by 8 in. 40000 lb. 3.75 by 7 in. Assuming the weight of the car to be 40 per cent of its carrying capacity in each instance, determine the pressure per square inch of projected area and the value of the constant C {Formula (58)} . 3. Measure the crank pin of any modern engine which is accessible, calculate the various constants and compare them with those given in this chapter. 4. Design a crank pin for an engine under the following con- ditions : Diameter of piston =28 inches. Maximum steam pressui-e=90 lb. per sq. in. Mean steam pressure =40 lb. per sq. in. Revolutions per minute =75 Determine dimensions necessary to prevent wear and heating and then test for strength and stiffness. 5. Design a crank pin for a high speed engine having the following dimensions and conditions : Diameter of piston =14 inches STEP-BEARINGS. HI Maximum steam pressure =100 Ib. per sq. in. Mean steam pressure =50 Ib. per sq. in. Revolutions per minute=250. 6. Make a careful study and sketch of journals and journal bearings on each of the following machines and analyze as to (a) Materials, (b) Adjustment, (c) Lubrication. a. An engine lathe. b. A milling machine. c. A steam engine. d. An electric generator or motor. 7. Sketch at least two forms of oil cup used in the labora- tories and explain their working. 8. The shaft journal of a vertical engine is 4 in. in diameter by 6 in. long. The cap is of cast iron, held down by 4 bolts of wrought iron, each 5 in. from center of shaft, and the greatest vertical pressure is 12000 Ib. Calculate depth of cap at center for both strength and stiff- nesSi and also the diameter of bolts. 9. Investigate the strength of the cap and bolts of some pillow block whose dimensions are known, under a pressure of 500 Ib. per sq. in. of projected area. 10. The total weight on the drivers of a locomotive is 64000 Ib. The drivers are four in number, 5 ft. 2 in. in diameter, and have journals 7 in. in diameter. Determine horse power consumed in friction when the locomotive is running 50 miles an hour, assuming tan=.05. 51. Step-Bearings. Any bearing which is designed to resist end thrust of the shaft rather than lateral pressure is denominated a step or thrust bearing. These are naturally most used on vertical shafts, but may be frequently seen on horizontal ones as for ex- ample on the spindles of engine lathes, boring machines and milling machines. Step-bearings may be classified according to the shape of the rubbing surface, as flat pivots and collars, conical pivots, and conoidal pivots of which the Schiele 112 MACHINE DESIGN. pivot is the best known. When a step-bearing on a vertical shaft is exposed to great pressure or speed it is sometimes lubricated by an oil tube coming up from below to the center of the bearing and connecting with a stand pipe or force-pump. The oil entering at the center is distributed by centrifugal force. 52. Friction of Pivots or Step-bearings. Flat Pivots. Let W-- weight on pivot di=-.outer diameter of pivot p= intensity of vertical pressure T= moment of friction /= co-efficient of friction tan We will assume p to be a constant which is no doubt approximately true. area Let r=the radius of any elementary ring of a width = c?r, then area of element =2irrdr Friction of element = fp X^ Moment of friction of element = 2fpirr*dr and T=2fp7r i ^-r 2 dr (a) or - ,,. (68) The great objection to this form of pivot is the un- even wear due to the difference in velocity between and circumference f CONICAL PIVOT. 113 53. Flat Collar. Let d 2 = inside diameter Integrating as in equation (a) above, but using limits l and 2 we have 2i 2i In this case P= and T=lWJ%=%j. . . (69) 54. Conical Pivot. Let a = angle of inclination to the vertical. VJ P= 7 dW dP As in the case of a flat ring the intensity of the vertical pressure is 4:W \ /and the vertical pressure on an elementary ring of * the bearing surface is Fig. 47. \ dW=-^^rdr=^ -C*J Cti 1*2 As seen in Fig. 4T the normal pressure on the elementary ring is flp_dW SWrdr 3 MACHINE DESIGN. The friction on the ring is fdP and the moment of this friction is (dl J - (70) As a approaches | the value of T approaches that of a flat ring, and as a approaches the value of T ap- proaches QO . If d 2 =0 we have T = The conical pivot also wears unevenly, usually as- suming a concave shape as seen in profile. 55. Schiele's Pivot. By experimenting with a pivot and bearing made of some friable material, it was shown that the outline tended to become curved as shown in Fig. 49. This led to a mathematical investi- gation which showed that the curve would be a trac- trix under certain conditions. This curve may be traced me- chanically as shown in Fig. 48. Let the weight W be free to move on a plane. Let the string SW be kept taut and the end S moved along the straight line SL. Then will a pencil attached to the F . 4g center of W trace on the plane a tractrix whose axis is SL, SCHIELE'S PIVOT. 115 In Fig. 49 let SW= length of string =r, and let P be any point in the curve. Then it is evident that the tangent PQ to the curve is a constant and =r l Also -t-=i\ sinO Let a pivot be generated by revolving the curve around its axis SL. As in the case of the conical pivot it can be proved that the normal pressure on an element of convex surface is SWrdr Fig. 49. dP = .(a) Let the normal wear of the pivot be assumed to be proportional to this normal pressure and to the velocity of the rubbing surfaces, i. e. normal wear proportional to pr, then is the vertical wear proportional to pr sinff But sinO is a constant, therefore the vertical wear will be the same at all points. This is the characteristic feature and advantage of this form of pivot. As shown in equation (a) dp SWr.dr =~~ SWfr.rdr and d\ .(72) T is thus shown to be independent of d 9 or of the length of pivot used. 116 MACHINE DESIGN. This pivot is sometimes wrongly called antifriction. As will be seen by comparing equations (68) and (72) the moment of friction is fifty per cent, greater than that of the common flat pivot. The distinct advantage of the Schiele pivot is in the fact that it maintains its shape as it wears and is self- adjusting. It is an expensive bearing to manufacture and is seldom used on that account. It is not suitable for a bearing where most of the pressure is side ways. 56. Multiple Bearings. To guard against abrasion in flat pivots a series of rubbing surfaces which divide the wear is sometimes provided. Several flat discs placed beneath the pivot and turning indifferently, may be used. Sometimes the discs are made alter- nately of a hard and a soft material. Bronze, steel and raw hide are the more common materials. Notice in this connection the button or washer at the outer end of the head spindle of an engine lathe and the loose collar on the main journal of a milling machine. See Figs. 41 and 42. Pivots are usually lubricated through a hole at the center of the bearing and it is desirable to have a pressure head on the oil to force it in. The compound thrust bearing generally used for propeller shafts consists of a number of collars of the same size forged on the shafts at regular intervals and dividing the end thrust between them, thus reducing the intensity of pressure to a safe limit without making the collars unreasonably large. A safe value for p the intensity of pressure is, ac- cording to Whitham, 60 Ib. per sq. in. for high speed engines. MULTIPLE BEARINGS. A table given by Prof. Jones in his book on Machine Design shows the practice at the Newport News ship- yards 011 marine engines of from 250 to 5000 H. P. The outer diameter of collars is about one and one-half times the diameter of the shafts in each case and the number of collars used varies from 6 in the smallest engine to 11 in the largest. The pressure per sq. in. of bearing surface varies from 18 to 46 Ib. with an average value of about 32 Ib. The hydraulic foot step sometimes used for the vertical shafts of turbines is in effect a rotating plunger supported by water pressure underneath and so packed in its bearing as to allow a slight leakage of water for cooling and lubricating the bearing surfaces. PROBLEMS. 1. Design and draw to full size a Schiele pivot for a water wheel shaft 4 inches in diameter, the total length of the bear- ing being 3 inches. Calculate the horse-power expended in friction if the total vertical pressure on the pivot is two tons and the wheel makes 150 revs, per min. and assuming /=. 25 for metal on wet wood. 2. Compare the friction of the pivot in Prob. 1, with that of a flat collar of the same projected area and also with that of a conical pivot having =30 deg. 3. Design a compound thrust bearing for a propeller shaft the diameters being 14 and 21 inches, the total thrust being 80,000 Ibs. and the pressure 40 Ib. per sq. in. Calculate the horse-power consumed in friction and compare with that developed if a single collar of same area had been used. Assume /=. 05 and rev. per min. =120. CHAPTER VIII. BALL AND ROLLER BEARINGS. 57. General Principles. The object of interposing a ball or roller between a journal and its bearing, is to substitute rolling for sliding friction and thus to re- duce the resistance. This can be done only partially and by the observance of certain principles. In the first place it must be remembered that each ball can roll about but one axis at a time ; that axis must be determined and the points of contact located accord- ingly. Secondly, the pressure should be approximately normal to the surfaces at the points of contact Finally it must be understood, that on account of the contact surfaces being so minute, a comparatively slight pressure will cause distortion of the balls and an entire change in the conditions. 58. Journal Bearings. These may be either two, three or four point, so named from the number of points of contact of each ball. The axis of the ball may be assumed as parallel or inclined to the axis of the journal and the points of contact arranged accordingly. The simplest form con- sists of a plain cylindrical journal running in a bearing of the same shape and having rings of balls interposed. The successive rings of balls should be separated \)y thin loose collars to keep them in place. These collars are a source of rubbing friction, and to do away with them the balls are sometimes run in grooves either in journal, bearing or both. 118 JOURNAL BEARINGS. 119 Fig. 50 shows a bearing of this type, there being three points of contact and the axis of ball being parallel to that of journal. The bearings so far mentioned have no means of adjustment y for wear. Conical -J bearings, or those in which the axes of the balls meet in a com- mon point, supply Fig. 50. this deficiency. In designing this class of bearings, either for side or end thrust, the inclination of the axis is assumed according to the obliquity desired and the points of contact are then so located that there shall be no slipping. Tig. 51 illustrates a common form of adjustable or cone bearing and shows the method of designing a three point contact. A C is the axis of the cone, while the shaded area is a section of the cup, so called. Let a and b be two points of contact be- tween ball and cup. Draw the line a b and produce to cut axis in A. Through the center of ball draw the line A B ; then will f 4/^m (T u-c Fig. 51. this be the axis of rotation of the ball and a c, b d will be the projections of two circles of rotation. As the radii of these circles have the same ratio as the radii of revolution an, b m, there will be no slipping and the ball will roll as a cone inside another cone. The 120 MACHINE DESIGN. exact location of the third point of contact is not material. If it were at c, too much pressure would come on the cup at b ; if at d there would be an excess of pressure at a, but the rolling would be correct in either case. A convenient method is to locate p by drawing A D tangent to ball circle as shown. It is recommended however that the two opposing surfaces at p and b or a should make with each other an angle of not less than 25 deg. to avoid sticking of the ball. To convert the bearing just shown to four point contact, it would only be necessary to change the one cone into two cones tangent to the ball at c and d. To reduce it to two point contact the points a and b are brought together to a point opposite p. As in this last case the ball would not be confined to a definite path it is customary to make one or both surfaces con- cave conoids with a radius about three fourths the diameter of the ball. See Fig. 52. 59. Step-Bearings. The same principles apply as in the preceding article and the axis and points of contact may be varied in the same way. The most common form of step-bearing consists of two flat circular plates Fig. 52. separated by one or more rings of balls. Each ring must be kept in place by one or more loose retaining collars, and these in turn are the cause, of some sliding friction. This is a bear- ing with two point contact and the balls turning on horizontal axes. If the space between the plates is filled with loose balls, as is sometimes done, the rubbing B STEP-BEARINGS. 121 of the balls against each other will cause considerable friction. To guide the balls without rubbing friction three point contact is generally used. Fig. 58 illustrates a bearing of this character. The Fig. 53. method of design is shown in the figure, the principle being the same as in Fig. 51. By comparing the letter- ing of the two figures the similarity will be readily seen. This last bearing may be converted to four point contact by making the upper collar of the same shape as the lower. To guide the balls in two point contact use is sometimes made of a cage ring, a flat collar drilled with holes just a trifle larger than the balls and disposing them either in spirals or in irregular order. See Fig. 54. This method has the advantage of making each Fig. 54. MACHINE DESIGN. ball move in a path of different radius thus securing more even wear for the plates. 60. Materials and Wear. The balls themselves are always made of steel, hardened in oil, tempered and ground. They are usually accurate to within one ten thousandth of an inch. The plates, rings and journals must be hardened and ground in the same way and perhaps are more likely to wear out or fail than the balls. A long series of experiments made at the Case School of Applied Science on the friction and endur- ance of ball step-bearings showed some interesting peculiarities. Using flat plates with one circle of quarter inch balls it was found that the balls pressed outward on the retaining ring with such force as to cut and indent it seriously. This was probably due to the fact that the pressure slightly distorted the balls and changed each sphere into a partial cylinder at the touching points. While of this shape it would tend to roll in a straight line or a tangent to the circle. Grinding the plates slightly convex at an angle of one to one and-a-half degrees obviated the difficulty to a certain extent. Under even moderately heavy loads the continued rolling of the ring of balls in one path soon damaged the plates to such an extent as to ruin the bearing. A flat bearing filled with loose balls developed three or four times the friction of the single ring and a three point bearing similar to that in Fig. 53 showed more than twice the friction of the two point. A flat ring cage such as has already been described was the most satisfactory as regards friction and en- durance. The general conclusions derived from the experi- ROLLER BEARINGS. 123 ments were that under comparatively light pressures the balls are distorted sufficiently to seriously disturb the manner of rolling and that it is the elasticity and not the compressive strength of the balls which must be considered in designing bearings. 61. Design of Bearings. Figures on the direct crushing strength of steel balls have little value for the designer. For instance it has been proved by numer- ous tests that the average crushing strengths of J inch and | inch balls are about Y500 Ib. and 15000 Ib. respectively. Experiments made by the writer show that a \ inch ball loses all value as a transmission element on account of distortion, at any load of more than 100 Ib. Prof. Gray states, as a conclusion from some ex- periments made by him, that not more than 40 Ib. per ball should be allowed for f inch balls. This distortion doubtless accounts for the failure of theoretically correct bearings to behave as was ex- pected of them. Ball bearings should be designed as has been explained in the preceding articles and then only used for light loads. 62. Roller Bearings. The principal disadvanta^ of ball bearings lies in the fact that contact is only , a point and that even moderate pressure causes exc< . sive distortion and wear. The substitution of cylinders or cones for the balls is intended to overcome this difficulty. The simplest form of roller bearing consists of a plain cylindrical journal and bearing with small cylin- drical rollers interposed instead of balls. There are two difficulties here to be overcome. The rollers tend 124 MACHINE DESIGN. JLLi Lc to work endways and rub or score whatever retains them. They also tend to twist around and become unevenly worn or even bent and broken, unless held in place by some sort of cage. In short they will not work properly F j g> 55< unless guided and any form of guide entails sliding friction. The cage generally used is a cylin- drical sleeve having longitudinal slots which hold the rollers loosely and prevent their getting out of place either sideways or endways. The use of balls or convex washers at the ends of the rollers has been tried with some degree of success. See Fig. 55. Large rollers have been turned smaller at the ends and the bearings then formed allowed to turn in holes bored in revolving collars. These collars must be so fastened or geared together as to turn in unison. 63. Grant Roller Bearing. The Grant roller is conical and forms an intermediate between the ball and the cylindrical roller having some of the advantages of each. The principle is much the same as in the adjustable ball bearing, Fig. 52, rolling cones being substituted for balls, Fig. 56. The inner cone turns loose on the spindle. The conical rollers are held in posi- tion by rings at each end, while the outer or hollow cone ring is adjustable along the axis. Fig. 56. HYATT ROLLERS. 125 Two sets of cones are used on a bearing, one at each end to neutralize the end thrust, the same as with ball bearings. 64. Hyatt Rollers. The tendency of the rollers to get out of alignment has been already noticed. The Hyatt roller is intended by its flexibility to secure uniform pressure and wear under such conditions. It consists of a flat strip of steel wound spirally about a mandrel so as to form a continuous hollow cylinder. It is true inform and comparatively rigid against com- pression, but possesses sufficient flexibility to adapt itself to slight changes of bearing surface. Experiments made by the Franklin Institute show that the Hyatt roller possesses a great advantage in efficiency over the solid roller. Testing f inch rollers between flat plates under loads increasing to 550 Ib. per linear inch of roller developed co-efficients of friction for the Hyatt roller from 23 to 51 per cent, less than for the solid roller. Subsequent examination of the plates showed also a much more even distribution of pressure for the former. A series of tests were conducted by the writer in 1904-05 to determine the relative efficiency of roller bearings, as compared with plain cast iron and bab- bitted bearings under similar conditions. 1 The bear- ings tested had diameters of 1||, 2 T \, 2 T 7 -g-, and 2{| inches and lengths approximately four times the diameters. In the first set of experiments Hyatt roller bearings were compared with plain cast iron sleeves, at a uniform speed of 480 rev. per min. and under loads varying from 64 to 264 pounds. The cast iron bearings were copiously oiled. 1 Machinery, N. Y., Oct. 1905. 126 MACHINE DESIGN. As the load was gradually increased, the value of / the coefficient of friction remained nearly constant with the plain bearings, but gradually decreased in the case of the roller bearings. Table XXIII. gives a summary of this series of tests. TABLE XXIII. COEFFICIENTS OF FRICTION FOR ROLLER AND PLAIN BEARINGS. Diameter of Journal. Hyatt Bearing Plain Bearing. Max. Min. Ave. Max. Min. Ave. 1H .036 .019 .026 .160 .099 .117 2& .052 .034 .040 .129 .071 .094 2 T V .041 .025 .030 .143 .076 .104 2}| .053 .049 .051 .138 .091 .104 The relatively high value of / in the 2 T 3 g- and 2 roller bearings were due to the snugness of the fit be- tween the journal and the bearing, and show the ad- visability of an easy fit as in ordinary bearings. The same Hyatt bearings were used in the second set of experiments, but were compared with the McKeel solid roller bearings and with plain babbited bearings freely oiled. The McKeel bearings contained rolls turned from solid steel and guided by spherical ends fitting recesses in cage rings at each end. The cage rings were joined to each other by steel rods par- allel to the rolls. The journals were run at a speed of 560 rev. per min. and under loads varying from 113 to 456 pounds. Table XXIV. gives a summary of the second series of tests. ROLLER STEP-BEARINGS. 127 TABLE XXIV. COEFFICIENTS OF FRICTION FOR ROLLER, AND PLAIN BEARINGS. Diam. of Jo'rnal . Hyatt Bearing. McKeel Bearing. Babbitt bearing. Max. Min. Ave. Max. Min. Ave. Max. Min. Ave. 1A 2& 2i 7 i .032 .019 .042 .012 .011 .025 .018 .014 .032 .033 .017 .022 .074 .088 .114 .029 .078 .083 .043 .082 .096 .028 .015 .021 2*i .029 .022 .025 .039 .019 .027 .125 .089 .107 The variation in the values for the babbited bearing is due to the changes in the quantity and temperature of the oil. For heavy, pressures it is probable that the plain bearing might be more serviceable than the others. Notice the low values for /in Table XXII. Under a load of 470 pounds the Hyatt bearing de- veloped an end thrust of 13.5 pounds and the McKeel one of 11 pounds. This is due to a slight skewing of the rolls and varies, sometimes reversing in direction. If roller bearings are properly adjusted and not overloaded a saving of from two-thirds to three-fourths of the friction may be reasonably expected. 65. Roller Step-Bearings/ In article 60 attention was called to the fact that the balls in a step-bearing under moderately heavy pressures tend to become cylinders or cones and to roll accordingly. This has suggested the use of small cones in place of the balls, rolling between plates one or both of which are also con- ical. A successful bearing of this kind with short 128 MACHINE DESIGN. cylinders in place of cones is used by the Sprague-Pratt Elevator Co., and is described in the American Machin- ist for June 27, 1901. The rollers are arranged in two spiral rows so as to distribute the wear evenly over the plates and are held loosely in a flat ring cage. This bearing has run well in practice under loads double those allowable for ball bearings, or over 100 Ib. per roll for rolls one-half inch in diameter and one-quarter inch long. Fig. 57 illustrates a bearing of this character. Col- lars similar to this have been used in thrust bearings for propeller shafts. The discussion referred to in Art. 46 also included ball and roller bearings and should be read by the de- signer. Mr. Mossberg, de- signer of the roller bear- ings of that name, recom- mends rollers of spring tempered tool steel, cages of tough bronze and boxes of high carbon steel with a hard temper. Mr. Charles R. Pratt reports the limit of work for ^ inch balls in thrust bearings to be 100 pounds per ball at 700 revolu- tions per minute and 6 inches diameter circle of rotation. Mr. W. S. Eogers gives the maximum load for a 1 inch ball as 1000 pounds and for a inch ball as 200 pounds. Mr. Henry Hess states that in a roller bear- ing one fifth of the number of rollers multiplied by the length and diameter of one roller may be considered as the projected area of the journal. For ball bearings one fifth the total number of balls multiplied by the F< ROLLER STEP-BEARINGS. 129 square of the ball diameter may be used in the same way. Space forbids reference to all of the many varieties of ball and roller bearings shown in manufacturers' catalogues. These are all subject to the laws and limit- ations mentioned in this chapter, While such bearings will be used more and more in the future, it must be understood that extremely high speeds or heavy pressures are unfavorable and in most cases prohibitive. Furthermore, unless a bearing of this character is carefully designed and well constructed it will prove to be worse than useless. CHAPTER IX. SHAFTING, COUPLINGS AND HANGERS. 66. Strength of Shafting. Let D= diameter of the driving pulley or gear. N= number rev, per minute. P= force applied at rim. T= twisting moment* The distance through which P acts in one minute is irDN inches and wor]s.= PirDN in. Ib. per min. PD But =T the moment, and 2-n-N = the angular velocity. .-. work = moment X angular velocity. One horse power = 33000 ft. Ib. per min. = 396000 in. Ib. per min. 396000 ~ 396000 HP =^ ^ il*r T 63Q25 HP ~N~ p 126050 HP ,^-N D^ The general formula for a circular shaft exposed to torsion alone is 130 STRENGTH OF SHAFTING. But r= 63025 where N=no. rev. per min. Substituting in formula for d ^3210001 - nearly . . . (76) S may be given the following values : 45000 for common turned shafting. 50000 for cold rolled iron or soft steel. 65000 for machinery steel. It is customary to use factors of safety for shafting as follows : Headshafts or prime movers 15 Line shafting 10 Short counters 6 The large factor of safety for head shafts is used not only on account of the severe service to which such shafts are exposed, but also on account of the incon- venience and expense attendant on failure of so im- portant a part of the machinery. The factor of safety for line shafting is supposed to be large enough to allow for the transverse stresses produced by weight of pulleys, pull of belts, etc., since it is impracticable to calculate these accurately in most cases. Substituting the values of S and introducing factors of safety, we have the following formulas for the safe diameters of the various kinds of shafts. 132 MACHINE DESIGN. TABLE XXV. DIAMETERS OF SHAFTING. KIND OF SHAFT. Head Shaft. Line Shaft. Counter Shaft. MATERIAL. Com'n Iron A Y 415 5 3.50 HP Soft Steel Mach'y Steel 4.00 3.38 3.10 The Allis-Chalmers Co. base their tables for the horse power of wrought iron or mild steel shafting on the formula HP=cd*N where c has the following values : c Heavy or main shafting .008 Shaft carrying gears .010 Light shafting with pulleys .013 This is equivalent to using values of S as 2570 Ib. , 3200 Ib. and 41TO Ib. per sq. in. in the respective classes and would give for co-efficients in Table XXV. the numbers 5, 4.64 and 4.25 which are somewhat larger than those given for similar cases in the table. A table published by Wm. Sellers & Co. in their shafting catalogue gives the horse powers of iron and steel shafts for given diameters and speeds. An invest!- COUPLINGS. 133 gation of the table shows it to be based upon a value of about 4000 Ib. for S or a co-efficient of 4.31 in Table XXV. In case there is a known bending moment M, com- bined with a known twisting moment T, then a re- sultant twisting moment is to be substituted for Tin the formulas (73) to (75). Mr. J. B. Francis has published a table in the Journal of the Franklin Institute which gives the greatest admissible distance between bearings for line shafts of different diameters, when subject to no transverse forces except from their own weight. This distance varies from 16 feet for 2 inch shafts up to 26 feet for 9 inch shafts, the span being proportional to the cube root of the diameter. The distance should be much less when the shaft carries numerous pulleys with their belts. 67. Couplings. The flange or plate coupling is most commonly used for fastening together adjacent lengths of shafting. Fig. 58 shows the pro- portions of such a coup- ling. The flanges are turned accurately on all r- sides, are keyed to the Q. shafts and the two are centered by the projec- tion of the shaft from one part into the other Fig 58 as shown at A. The bolts are turned to fit the holes loosely so as not to in- terfere with the alignment, 134 MACHINE DESIGN. The projecting rim as at B prevents danger from belts catching on the heads and nuts of the holts. The faces of this coupling should he trued up in a lathe after being keyed to the shaft. Jones and Laughlins in their shafting catalogue give the following proportions for flange couplings. Diam. of Shaft. Diani. of Hub. Length of Hub. Diam. of Coupling. 2 4i 3 8 21 5| 4 10 3 6* 5, 12 3* 8 6i 14 4 9 7 16 5 HI 8f 20 There are five bolts in each coupling. The sleeve coupling is neater in appearance than the flange coupling but is more complicated and expensive. In Fig. 59 is illustrated a neat and effective coupling B Fig. 59. of this type. It consists of the sleeve S bored with two tapers and two threaded ends as shown. The two conical, split bushings BB are prevented from turning by the feather key K and are forced into the conical recesses by the two threaded collars C C and thereby CLUTCHES. 135 clamped firmly to the shaft. The key K also nicks slightly the center of the main sleeve S, thus locking the whole combination. Couplings similar to this have been in use in the Union Steel Screw Works, Cleveland, Ohio, for many years and have given good satisfaction. The Sellers coupling is of the type illustrated in Fig. 59, but is tightened by three bolts running parallel to the shaft and taking the place of the collars C C. In another form of sleeve coupling the sleeve is split and clamped to the shaft by bolts passing through the two halves as illustrated in Fig. 60. The "muff" coupling, as [ i I L J its name implies is a plain pj g> sleeve slipped over the shafts at the point of junction, accurately fitted and held by a key running from end to end. It may be regarded as a permanent coupling since it is not readily removed. 68. Clutches. By the term clutch, is meant a coup- ling which may be readily disengaged so as to stop the following shaft or pulley. Clutch couplings are of two kinds, positive or jaw clutches and friction clutches. The jaw clutch consists of two hubs having sector shaped projections on the adjacent faces which may interlock. One of the couplings can be slid on its shaft to and from the other by means of a loose collar and yoke, so as to engage or disengage with its mate. This clutch has the serious disadvantage of not being readily engaged when either shaft is in motion. 136 MACHINE DESIGN. Friction clutches are not so positive in action but can be engaged without difficulty and without stopping the driver. Three different classes of friction clutches may be distinguished according as the engaging members are flat rings, cones or cylinders. The Weston clutch, Fig. 61, belongs to the first named class. A series of rings inside a > sleeve on the follower B interlocks with a similar series outside a smaller sleeve on the driver A somewhat as Fig. 61. in a thrust bearing (Art. 56). Each ring can slide on its sleeve but must rotate with it. When the parts A and B are forced together the rings close up and engage by pairs, producing a considerable turning moment with a moderate end pressure. Let : P= pressure along axis. n= number of pairs of surfaces in contact. f = coefficient of friction. r=mean radius of ring. T turning moment. Then will : T=Pfnr (77) If the rings are alternately wood and iron, as is usually the case, / will have values ranging from 0.25 to 0.50. The cone clutch consists of two conical frustra, one external and one internal, engaging one another and CLUTCHES. 137 driving by friction. Using the same notation as before, and letting a= angle between element of cone and axis, the normal pressure between the two surfaces JL J J.'K ^'^2-l^^ -,411 "U^ -*-.! will be : ^ and the friction will be : sin a. sin a Therefore : T = .(T8) sin a a should slightly exceed 5 deg. to prevent sticking and/ will be at least 0.10 for dry iron on iron. Substituting / = 0.10 and sin a =0.125 we have T = O.SPr as a convenient rule in designing. Fig. 62 illustrates the type of clutch more generally used on shafting for transmitting mod- erate quantities of power. As shown in the figure one member is attached to a loose pulley on the shaft, but this same type can be used for con- necting two in- dependent shafts. The ring or hoop H, finished inside and out, is gripped at intervals by pairs of jaws JJ having wooden faces. These jaws are actuated as shown by toggles and levers connected with the slip ring R. The toggles are so adjusted as to pass by the center and lock in the gripping position. These clutches are convenient and durable but Fig. 62. 138 MACHINE DESIGN. occupy considerable room in proportion to their trans- mitting power. The Weston clutch is preferable for heavy loads. The roller clutch is much used on automatic ma- chinery as it combines the advantages of positive driving and friction engagement. A cylinder on the follower is embraced by a rotating ring carried by the driver. The ring has a number of recesses on its inner surface which hold hardened steel rollers. These recesses being deeper at one end allow the rollers to turn freely as long as they remain in the deep portions. The bottom of the recess is inclined to the tangent of the circle at an angle of from 9 to 14 deg. When by suitable mechanism the rollers are shifted to the shallow portions of the recesses they are im- mediately gripped between the ring and the cylinder and set the latter in motion. A clutch of this type is almost instantaneous in its action and is very powerful, being limited only by the strength of the materials of which it is composed. Several small rolls of different materials and diam- eters were tested by the writer in 1905 with the following results : Material. Diameter. Length. Set load. Ultimate load. Cast Iron Soft Steel Hard Steel 0.375 0.75 1.125 0.4375 0.4375 0.4375 1.5 1.5 1.5 1.5 1.5 1.5 5500 6800 7800 8800 11100 35000 12400 19500 29700 20000 69. Coupling Bolts. The bolts used in the ordinary flange couplings are exposed to shearing, and their SHAFTING KEYS. 139 combined shearing moment should equal the twisting moment on the shaft. Let n number of bolts. d t = diameter of bolt. D diameter of bolt circle. We will assume that the bolt has the same shearing strength as the shaft. The combined shearing strengll of the bolts is .IStedinS and their moment of resistance to shearing is This last should equal the torsion moment of the shaft or Solving for d l and assuming D=3d as an average value, we have d^ , ................. (79) v 1 1/ In practice rather larger values are used than would be given by the formula. 70. Shafting Keys. The moment of the shearing stress on a key must also equal the twisting moment of the shaft. Let b= breadth of a key. Z= length of key. h= total depth of key. S' = shearing strength of key. The moment of shearing stress on key is and this must equal -gy Usually &=j- For shafts of machine steel S=S f , and for iron shafts $=f$' nearly, as keys should always be of steel. 140 MACHINE DESIGN. Substituting these values and reducing : For iron shafting l=l.2d nearly. For steel shafting 1=1. Qd nearly, as the least lengths of key to prevent its failing by shear. If the key way is to be designed for uniform strength, the shearing area of the shaft on the line A B, Fig. 63, should equal the shearing area of the key, if shaft and key are of the same material and AB= Fig. 63. These proportions will make the depth of key way in shaft about =|6 and would be appropriate for a square key. To avoid such a depth of key way which might weaken the shaft, it is better to use keys longer than required by preceding formulas. In American practice the total depth of key rarely exceeds f & and one-half of this depth is in shaft. To prevent crushing of the key the moment of the compressive strength of half the depth of key must equal T. dlh^ Sd 3 f . or 2 X -2 X&= -0 ........ ,..-(a) where S e is the compressive strength of the key. For iron shafts S e =2S g and for steel shafts $ c =n$ a Substituting values of S c and assuming h=^b=-f^d we have Iron shafts l=2.5d nearly. Steel shafts l=3\d nearly, as the least length for flat keys to prevent lateral crushing. HANGERS AND BOXES. 141 The above refers to parallel keys. Taper keys have parallel sides, but taper slightly between top and bot- tom. When driven home they have a tendency to tip the wheel or coupling on the shaft. This may be par- tially obviated by using two keys 90 deg. apart so as to give three points of contact between hub and shaft. The taper of the keys is usually about \ inch to one foot. The Woodruff key is sometimes used on shafting. As may be seen in Fig. 64 this key is semi-circular in shape and fits a recess sunk in the shaft by a milling cutter. Fig. 64. 71. Hangers and Boxes. Since shafting is usually hung to the ceiling and walls of buildings it is necessary to provide means for adjusting and aligning the bearings as the movement of the building disturbs them. Furthermore as line shafting is continuous and is not per- fectly true and straight, the bearings should be to a certain extent self-adjusting. Eeliable experiments have shown that usually one-half of the power developed by an engine is lost in the friction of shafting and belts. It is important that this loss be prevented as far as possible. The boxes are in two parts and may be of bored cast- iron or lined with Babbitt metal. They are usually about four diameters of the shaft in length and are oiled by means of a well and rings or wicks. (See Art. 142 MACHINE DESIGN. 44.) The best method of supporting the box in the hanger is by the ball and socket joint ; all other con- trivances such as set screws are but poor substitutes. Fig. 65 shows the usual arrangement of the ball and socket. A and B are the two parts of the box. The center is cast in the shape of a partial sphere with C as a center as shown by the dotted lines. The two sockets S S can be adjusted vertically in Fig. 65. the hanger by means of screws and lock nuts. The horizontal adjustment of the hanger is usually effected by moving it bodily on the support, the bolt holes being slotted for this purpose. Counter shafts are short and light and are not subject to much bending. Con- sequently there is not the same need of adjustment as in line shafting. In Fig. 66 is illustrated a simple bearing for counters. The solid cast iron box B with a spheric- al center is fitted directly in a socket in the hanger Fig. 66. H and held in position by the cap C and a set screw. There is not space here to show all the various forms HANGERS AND BOXES. 143 of hangers and floor stands arid reference is made to the catalogues of manufacturers. Hangers should be symmetrical, i. e., the center of the box should be in a vertical line with center of base. They should have relatively broad bases and should have the metal dis- posed to secure the greatest rigidity possible. Cored sections are to be preferred. Fig. 67. e Fig. 68. Fig. 67 illustrates the proportions of a Sellers line- shaft hanger. This type is also made with the lower half removable so as to facilitate taking down the shaft. Fig. 68 shows the outlines of a hanger for heavy shafting as manu- factured by the Jones & Laughlins Com- pany while Fig. 69 il- lustrates the design of Fig. 69. the box with oil wells and rings. The open side hanger is sometimes adopted on ao MACHINE DESIGN. count of the ease with which the shaft can be removed, but it is much less rigid than the closed hanger and is suitable only for light shafting. The countershaft hanger shown in Fig. 70 is simple, strong and sym- metrical and is a great improve- ment over those using pointed set screws for pivots. Hangers similar to this are used by the Brown & Sharpe Mfg. Co. with some of their machines. PROBLEMS. 1. Calculate the safe diameters of head shaft and three line shafts for a factory, the material to be rolled iron and the speeds and horse-powers as follows : 200 rev. per min. 120 rev. per min. 250 rev. per min. 200 rev. per min. least two lines of Head shaft 100 H P Machine shop 30 H P Pattern shop 50 H P Forge shop 20 H P 2. Determine the horse-power of at shafting whose speeds and diameters are known. 3. Design and sketch to scale a flange coupling for a three inch line shaft including bolts and keys. 4. Design a sleeve coupling for the foregoing, different in principle from the ones shown in the text. 5. A four-inch steel head shaft makes 100 rev. per min. Find the horse-power which it will safely transmit, and design a Weston ring clutch capable of carrying the load, HANGERS AND BOXES. 145 There are to be six wooden rings and five iron rings of 12 in. mean diameter. Find the moment carried by each pair of surfaces in contact and the end pressure required. 6. Find mean diameter of a single cone clutch for same shaft with same end pressure. 7. Find radial pressure required for a clutch like that shown in Fig. 62, the ring being 24 in. in mean diameter and there being four pairs of grips. Other conditions as in pre- ceding problems. 8. Select the line shaft hanger which you prefer among those in the laboratories and make sketch and description of the same. 9. Do. for a countershaft hanger. 10. Explain in what way a floor-stand differs from a hanger. CHAPTER X. GEARS, PULLEYS AND CRANKS. 72. Gear Teeth. The teeth of gears may be either cast or cut, but the latter method prevails, since cut gears are more accurate and run more smoothly and quietly? The proportions of the teeth are essentially the same for the two classes, save that more back lash must be allowed for the cast teeth. The circular pitch is obtained by dividing the circumference of the pitch circle by the number of teeth. The diametral pitch is obtained by dividing the number of teeth by the diam- eter of the pitch circle and equals the number of teeth per inch of diameter. The reciprocal of the diametral pitch is sometimes called the module. The addendum is the radial projection of the tooth beyond the pitch circle, the dedendum the corresponding distance inside the pitch circle. The clearance is the difference be- tween the dedendum and addendum ; the back lash the difference between the widths of space and tooth on the pitch circle. Let circular pitch = p. module =^=m. 7T diametral pitch = = p m addendum =a. dedendum or flank =/. clearance =fa=c. height =a+f=h. width =w. (See Fig. 72.) 146 GEAR TEETH. The usual rule for standard cut teeth is to make w = & allowing 110 calculable back-lash, to make a=m 2i and f=-^- or h=^m and clearance =~- 8 o There is however a marked tendency at the present time towards the use of shorter teeth. The reasons urged for their adoption are : first, greater strength and less obliquity of action ; second, less expense in cutting. * Several systems have been proposed in which the height of tooth h varies from 0.425p to 0.55p. According to the latter system a=0.25p, /=0.3p, and c=.05p. In modern practice the diametral pitch is a whole number or a common fraction and is used in describing the gear. For instance a 3 pitch gear is one having 3 teeth per inch of diameter. The following table gives the pitches in common use and the proportions of long and short teeth. If the gears are cut, w=^ ; if cast gears are used, 2t to 0.48p. * See American Machinist, Jan. 7, 1897, p. 6. 148 MACHINE DESIGN. TABLE XXVI. PROPORTIONS OF GEAR TEETH. PITCH. STANDARD TEETH. SHORT TEETH. Diame- tral. Circular. Addend. a Height. Clear- ance. c Addend. a Height. n Clear- ance. c * 6.283 2. 4.25 0.25 1.571 3.456 . 0.314 f 4.189 1.33 2.82 0.167 1.047 2.303 0.209 1 3.142 1. 2.125 0.125 0.785 1.728 0.157 1 2.513 0.8 1.7 0.1 0.628 1.383 0.125 H 2.094 0.667 1.415 0.083 0.524 1.152 0.105 if 1.795 0.571 1.212 0.071 0.449 0.988 0.09 2 1.571 0.5 1-062 0.062 0.392 0.863 0.078 8* 1.396 . 445 0.945 0.056 0.349 0.768 0.070 24 1.257 0.4 85 0.05 0.314 0.691 0.063 2f 1.142 0.364 0.775 0.045 0.286 0.629 0.057 3 1.047 0.333 0.708 0.042 0.262 0.576 0.052 3| 0.898 0.286 0.608 0.036 0.224 0.494 0.045 4 0.785 25 0.531 0.031 0.196 0.432 0.039 5 0.628 0.2 0.425 0.025 0.157 0.345 0.031 6 0.524 0.167 0.354 0.021 0.131 0.288 0.026 7 0.449 0.143 0.304 0.018 0.112 0.246 0.022 8 0.393 0.125 0.266 0.016 0.098 0.216 0.020 9 0.349 0.111 0.236 0.014 0.087 0.191 0.017 10 0.314 0.1 0.212 0.012 0.079 0.174 0.016 11 0.286 0.091 0.193 0.011 0.071 0.156 0.014 12 0.262 0.0834 0.177 0.010 0.065 0.143 0.013 13 0.242 0.077 0.164 0.010 0.060 0.132 0.012 14 0.224 0.0715 0.152 0.009 0.056 0.123 0.011 15 0.209 0.0667 0.142 0.008 0.052 0.114 0.010 16 0.196 0.0625 0.133 0.008 0.049 0.108 0.010 73. Strength of Teeth. Let P = total driving pressure on wheel at pitch cir- cle. This may be distributed over two or more teeth, but the chances are against an even distribution. Again, in designing a set of gears the contact is likely to be confined to one pair of teeth in the smaller pinions, STRENGTH OF TEETH. 149 Each tooth should therefore be made strong enough to sustain the whole pressure. Rough Teeth. The teeth of pattern molded gears are apt to be more or less irregular in shape, and are especially liable to be thicker at one end on account of the draft of the pattern. In this case the entire pressure may come on the outer corner of a tooth and tend to cause a diagonal fracture. Let O in Fig. 71 be the point of application of the pressure P, and AB the line of probable fracture. Drop the 1 CD on AB Let AB = x and angle CAD = a t Fig. 71. The bending moment at section AB is M=Py, and the moment of resistance is Fig. 72. were S = safe transverse strength of material. and (a) 150 MACHINE DESIGN. If P and w are constant, then S is a maximum when -2- is a maximum. # But 11 = h sina and x= - COSa cosa which is a X maximum when a = 45 and -^- = t/ o r> Substituting this value in (a) we 3 P But in this case w =A7p and therefore $= 2 . ip and p=3.684\S ...... < diametral pitch, ^ = ' 853 \/p ...... ( 81 ) Unless machine molded teeth are very carefully made, it may be necessary to apply this rule to them as well. Cut Gears. With careful workmanship machine molded and machine cut teeth should touch along the whole breadth. In such cases we may assume a line of contact at crest of tooth and a maximum bending moment. M=Ph The moment of resistance at base of tooth is M l = ^Sbw 2 when 5 is the breadth of tooth. In most teeth the thickness at base is greater than w, but in radial teeth it is less. Assuming standard proportions for cut gears : LEWIS' FORMULAS. 151 h = w = .p and substituting above : . 6T6 5 p ....... (82) For short teeth having h = .55p formula (82) reduces to: P = .0758&Sp . . ' ....... (83) The above formulas are general whatever the ratio of breadth to pitch. The general practice in this coun- try is to make b= 3p Substituting this value of b in (82) and (83) and reducing : Long teeth : p = 2.326 Short teeth :p = 2.098 J| ......... (85) The corresponding formulas for the diametral pitch are : Long teeth :L= 1.35 ^p' ' ' ..... (86) Short teeth: =1.49 j ........ ( 8Y ) m \p 74. Lewis' Formulas. The foregoing formulas can only be regarded as approximate, since the strength of gear teeth depends upon the number of teeth in the wheel ; the teeth of a rack are broader at the base and consequently stronger than those of a pinion. This is more particularly true of epicycloidal teeth. Mr. Wilfred Lewis has deduced formulas which take into 152 MACHINE DESIGN. account this variation. For cut spur gears of standard dimensions the Lewis formula is as follows : .(88) where n= number of teeth. This formula reduces to the same as (82), for %=14 nearly. Formula (82) would then properly apply only to small pinions, but as it would err on the safe side for larger wheels, it can be used where great accuracy is not needed. The same criticism applies to the other formulas in Art. 73. The value of S used should depend on the material and on the speed. The following safe values are recommended for cast iron and cast steel. Linear velocity ft. per min. 100 300 300 600 900 1200 1800 2400 Cast Cast Iron Steel. . . . 8000 24000 6000 15000 4800 12000 4000 10000 3000 7500 2400 6000 2000 5000 1700 4250 For gears used in hoisting machinery where there is slow speed and liability of shocks a writer in the American Machinist recommends smaller values of S than those given above * and proposes the following for four different metals : * American Machinist, Feb. 16, 1905. EXPERIMENTAL DATA. 153 Linear velocity ft. per min. Gray Iron Gun Metal Cast Steel. Mild Steel, 100 200 300 600 900 1200 1800 2400 4800 4200 3840 3200 2400 1920 1600 1360 7200 6300 5760 4800 3600 2880 2400 2040 9600 8400 7680 6400 4800 3840 3200 2720 12000 10500 9600 8000 6000 4800 4000 3400 The experiments described in the next article show that the ultimate values of S are much less than the transverse strength of the material and point to the need of large factors of safety. 75. Experimental Data. In the American Machinist for Jan. 14, 1897, are given the actual breaking loads of 'gear teeth which failed in service. The teeth had an average pitch of about 5 inches, a breadth of about 18 inches and the rather unusual velocity of over 2000 ft. per minute. The average breaking load was about 15000 Ib. there being an average of about 50 teeth on the pinions. Substituting these values in (88) and solving we get =1575 Ib. This very low value is to be attributed to the con- dition of pressure on one corner noted in Art. 73. Substituting in formula for such a case. This all goes to show that it is well to allow large factors of safety for rough gears, especially when the speed is high. Experiments have been made on the static strength 154 MACHINE DESIGN. of rough cast iron gear teeth at the Case School of Applied Science by breaking them in a testing machine. The teeth were cast singly from patterns, were two pitch and about 6 inches broad. The patterns were constructed accurately from templates representing 15 deg. involute teeth andcycloidal teeth drawn with a describing circle one-half the pitch circle of 15 teeth ; the proportions used were those given for standard cut gears. There were in all 41 cycloidal teeth of shapes cor- responding to wheels of 15-24-36-48-72-120 teeth and a rack. There were 28 involute teeth corresponding to numbers above given omitting the pinion of 15 teeth. The pressure was applied by a steel plunger tangent to the surface of tooth and so pivoted as to bear evenly across the whole breadth. The teeth were inclined at various angles so as to vary the obliquity from to 25 deg. for the cycloidal and from 15 deg. to 25 deg. for the involute. The point of application changed accord- ingly from the pitch line to the crest of the tooth. From these experiments the following conclusions are drawn : 1. The plane of fracture is approximately parallel to line of pressure and not necessarily at right angles to radial line through center of tooth. 2. Corner breaks are likely to occur even when the pressure is apparently uniform along the tooth. There were fourteen such breaks in all. 3. With teeth of dimensions given, the breaking pressure per tooth varies from 25000 Ib. to 50000 Ib. for cycloids as the number of teeth increases from 15 to infinity ; the breaking pressure for involutes of the same pitch varies from 34000 Ib. to 80000 Ib. as the number increases from 24 to infinity. TEETH OF BEVEL GEARS. 155 4. With teeth as above the average breaking pres- sure varies from 50000 Ib. to 26000 Ib. in the cycloids as the angle changes from deg. to 25 deg. and the tangent point moves from pitch line to crest ; with involute teeth the range is between 64000 and 39000 Ib. 5. Reasoning from the figures just given, rack teeth are about twice as strong as pinion teeth and involute teeth have an advantage in strength over cycloidal of from forty to fifty per cent. The advantage of short teeth in point of strength can also be seen. The modulus of rupture of the material used was about 36000 Ib. Values of S calculated from Lewis' formula for the various tooth numbers are quite uniform and average about 40000 Ib. for cycloidal teeth. Involute teeth are to-day generally preferred by manufacturers. William Sellers & Co. use an obliquity of 20 deg. in- stead of 14 J or 15 deg. the usual angle. 76. Teeth of Bevel Gears. There have been many formulas and diagrams proposed for determining the strength of bevel gear teeth, some of them being very complicated and inconvenient. It will usually answer every purpose from a practical standpoint, if we treat the section at the middle of the breadth of such a tooth as a spur wheel tooth and design it by the foregoing formulas. The breadth of the teeth of a bevel gear should be about one-third of the distance from the base of the cone to the apex. One point needs to be noted ; the teeth of bevel gears are stronger than those of spur gears of the same pitch and number of teeth since they are developed from a pitch circle having an element of the normal cone as a radius. To illustrate, we will suppose that we are designing the teeth of a miter gear and that 156 MACHINE DESIGN. the number of teeth is 32. In such a gear the element of normal cone is j/ 2 times the radius. The actual shape of the teeth will then correspond to those of a spur gear having 32 j/ 2=45 teeth nearly. NOTE. In designing the teeth of gears where the number is unknown, the approximate dimensions may first be obtained by formula (84) or (85) and then these values corrected by using Lewis' formula. PROBLEMS. 1. The drum of a hoist is 8 in. in diameter and makes 5 rev. per minute. The diameter of gear on the drum is 36 inches and of its pinion 6 in. The gear on the counter shaft is 24 in. in diameter and its pinion is 6 in. in diameter. The gears are all cut. Calculate the pitch and number of teeth of each gear, as- suming a load of one ton on drum chain and $=6000. Also determine the horse-power of the machine. 2. Calculate the pitch and number of teeth of a cut cast steel gear 10 in. in diameter, running at 250 rev. per min. and transmitting 20 HP. 3. A cast-iron gear wheel is 30 ft. 6| in. in pitch diameter and has 192 teeth, which are machine-cut and 30 in. broad. Determine the circular and diameter pitches of the teeth and the horse-power which the gear will transmit safely when making 12 rev. per min. 4. A two pitcli cycloidal tooth, 6 in. broad, 72 teeth to the wheel, failed under a load of 38000 Ib. Find value of S by Lewis' formula. 5. A vertical water-wheel shaft is connected to horizontal head shaft by cast iron gears and transmits 150 HP. The water-wheel makes 200 rev. per min. and the head shaft 100. Determine the dimensions of the gears and teeth if the latter are approximately two pitch. 6. Work Problem 1, using short teeth instead of standard. HIM AND ARMS. 157 77. Rim and Arms. The rim of a gear, especially if the teeth are cast, should have nearly the same thickness as the base of tooth, to avoid cooling strains. It is difficult to calculate exactly the stresses on the arms of the gear, since we know so little of the initial stress present, due to cooling and contraction. A hub of unusual weight is liable to contract in cooling after the arms have become rigid and cause severe tension or even fracture at the junction of arm and hub. A heavy rim on the contrary may compress the arms so as actually to spring them out of shape. Of course both of these errors should be avoided, and the pattern be so designed that cooling shall be simultaneous in all parts of the casting. The arms of spur gears are usually made straight without curves or taper, and of a flat, elliptical cross- section, which offers little resistance to the air. To support the wide rims of bevel gears and to facilitate drawing the pattern from the sand, the arms are some- times of a rectangular or T section, having the greatest depth in the direction of the axis of the gear. For pulleys which are to run at a high speed it is important that there should be no ribs or projections on arms or rim which will offer resistance to the air. Experiments by the writer have shown this resistance to be serious at speeds frequently used in practice. A series of experiments conducted by the author are reported in the American Machinist for Sept. 22, 1898, to which paper reference is here made. Twenty-four pulleys having 3^ inches face and diameters of 16, 20 and 24 inches were broken in a testing machine by the pull of a steel belt, the ratio of the belt tensions being adjusted by levers so as to be 15$ MACHINE DESIGN. two to one. Twelve of the pulleys were of the ordi- nary cast-iron type having each six arms tapering and of an elliptic section. The other twelve were Medart pulleys with steel rims riveted to arms and having some six and some eight arms. Test pieces cast from the same iron as the pulleys showed an average modu- lus of rupture of 35800 for the cast-iron and 50800 for the Medart. In every case the arm or the two arms nearest the side of the belt having the greatest tension, broke first, showing that the torque was not evenly distributed by the rim. Measurements of the deflection of the arms showed it to be from two to six times as great on this side as on the other. The buckling and springing of the rim was very noticeable especially in the Medart pulleys. The arms of all the pulleys broke at the hub showing the greatest bending moment there, as the strength of the arms at the hub was about double that at the rim. On the other hand some of the cast iron arms broke simultaneously at hub and rim, showing a negative bending moment at the rim about one-half that at the hub. The following general conclusions are justified by these experiments : (a) The bending moments on pulley arms are not evenly distributed by the rim, but are greatest next the tight side of belt. (b) There are bending moments at both ends of arm, that at the hub being much the greater, the ratio depending on the relative stiffness of rim and arms. The following rules may be adopted for designing the arms of cast iron pulleys and gears : 1. Multiply the net turning pressure, whether caused SPROCKET WHEELS AND CHAINS. 159 by belt or tooth, by a suitable factor of safety and by the length of the arm in inches. Divide this product by one-half the number of arms and use the quotient for a bending moment. Design the hub end of arm to resist this moment. 2. Make the rim ends of arms one-half as strong as the hub ends. 78. Sprocket Wheels and Chains. Steel chains con- necting toothed wheels afford a convenient means of getting a positive speed ratio when the axes are some distance apart. There are three classes in common use, the block chain, the roller chain and the so-called " silent " chain. Mr. A. Eugene Michel publishes quite a complete discussion of the design of the first two classes in Machinery for February, 1905, and reference is here made to that journal. Block chain is that commonly used on bicycles and small motor cars, so named from the blocks with round ends which are used to fill in between the links. The sprocket teeth are spaced to a pitch greater than that of the chain links and the blocks rest on flat beds between the teeth, Fig. 73. Eoller chains have rollers on every pin and have inside and outside links. The sprocket teeth have the same pitch as the chain links, the rollers fitting circular recesses between the sprockets, Fig. Y4. 160 MACHINE DESIGN. The most serious failing of the chain is its tendency to stretch with use so that the pitch becomes greater than that of the sprocket teeth. To obviate this difficulty in a measure considerable clearance should be given to the sprocket teeth as indicated in Fig. T4. As the pitch of the chain increases it will then ride higher upon the sprockets until the end of the tooth is reached. The teeth are rounded on their side faces, that they may easily enter the gaps in the chain and have side clearance. Mr. Michel gives the following values for the tensile strength of chains as determined by actual tests. EOLLER CHAIN. Fig. 74. Pitch inches i 5 "8 i 1 H H If 2 Tensile Strength Ib. 1200 1200 4000 6000 9000 12000 19000 25000 BLOCK CHAIN. 1 inch pitch 1200 to 2500 Ib. li " ' " 5000 " Mr. Michel further recommends a factor of safety of from 5 to 40 according to the severity of the condi- tions as to speed and shocks. The tendency is to use short links and double or triple width chains to increase the rivet bearing sur- SILENT CHAINS. 161 face, as it is this latter factor which really determines the life of a chain. Boiler chains may be used up to speeds of 1000 to 1200 feet per minute. The sprocket should be so designed that one tooth will carry the load safely with the pressure near the crest since these conditions obtain as the chain stretches. Use values of S as in Art. 74. 79. Silent Chains, The weak points in the ordinary chain, whether it be made with blocks or rollers, are the rivet bearings. It is the continual wear of these, due to insufficient area and lack of proper lubrication, that shortens the life of a chain. The so-called " silent- chain " with rocker bear- ings,- is comparatively free from this defect. Fig. 75 illustrates the shapes of links, rivets and sprockets for this >. kind of chain as man- ufactured by the Morse Chain Company. The chain proper is entirely outside of the sprocket teeth so that the latter may be continuous across the face of the wheel, save for a single guiding groove in the center. Projections on the under side of the links engage with the teeth of the sprocket, E being the point of contact for the driver and / a similar point for the follower when the rotation is as indicated. Each rivet consists practically of two pins called by the makers the rocker pin and the seat pin. Each pin is fastened in its particular gang of links and the n 162 MACHINE DESIGN. relative motion is merely a rocking of one pin on the other without appreciable friction. The pins are of hardened tool steel with softened ends. The combination of this freedom from rubbing contact with the adaptation of the engaging tooth profiles, gives a chain which can be safely run at high speeds without objectionable vibration or appreciable wear. The chains can be made of almost any width from one-half inch up to eighteen inches, the width de- pending upon the pitch of the chain and the power to be transmitted. The following are the working loads (and limiting speeds) of chains two inches in width and of different pitches, taken from a table published by the makers : Pitch in inches 1 1 3 .9 1.2 1.5 Working load in pounds 130 190 236 380 520 760 Limiting Speed 2000 1600 1200 1100 800 600 Eev. per min. The number of teeth in the small sprocket may vary from 15 to 30 according to the conditions. Assuming 17 teeth and the number of revolutions given in the above table the speed of chain would be 1420 feet per minute for the |- inch pitch and 1275 feet per minute for the 1.5 inch. Chains of this character have been run successfully at 2000 feet per minute. CRANKS AND LEVERS. PROBLEMS. 163 1. Design eight arms of elliptic section for a gear 48 inches pitch diameter, to transmit a pressure on tooth of 800 pounds. Material, cast iron having a working transverse strength of 6000 pounds per square inch. 2. Two sprocket wheels of 75 and 17 teeth respectively are to transmit twenty horse-power at a chain speed of about 800 feet per minute, with a factor of safety of 12 Determine the proper pitch of roller chain, the pitch diam- eters of the sprockets, and the numbers of revolutions. 3. Suppose that in Problem 2, a "silent" chain is to be used and the chain speed increased to 1200 feet per minute. Determine the proper pitch of chain to be used if the width of chain is 3 inches. Determine diameters and revolutions of sprockets as before. Cranks and Levers. A crank or rocker arm which is used to transmit a continuous or reciprocating rotary motion is in the condition of a cantilever or bracket with a load at the outer end. If the web of the crank is of uniform thickness theory requires that its profile should be parabolic for uniform strength, the vertex of the parabola being at the load point. A convenient approximation to this shape can be attained by using the tangents to the parabola at Fig. 76. ' points midway between the hub and the load point. See Fig. 76, The crank web is designed of the right 164: MACHINE DESIGN. thickness and breadth to resist the moment at AB, and the center line is produced to Q, making PQ = \ PO. Straight lines drawn from Q to A and B will he tangent to the parabola at the latter points and will serve as contour lines for the web. Assume the following dimensions in inches : Z = length of crank == OP. t thickness of web. h = breadth " " = AB. d = diameter of eye = cd. d 1= " " pin. b = breadth of eye. D diameter of hub = CD. A= " " shaft. B = breadth of hub. If the pressure on the crank pin is denoted by P PI then will the moment at AB be and the equa- A tions of moments for the cross-section will be : Pl_Sth* 2 " 6 [Bee Formula (3)] and from this the dimensions at AB maybe calculated. The moment at the hub will be PI and will tend to break the iron on the dotted lines CD. The equation of moments for the hub is therefore : Pl=~- (ZP-IV) From this equation the dimensions of the hub may be calculated when Z) x is known. The eye of a crank is most likely to break when the pressure on the pin is along the line OP, and the fracture will be along the dotted lines cd. The bending moment will be P mul- CRANKS AND LEVERS, 165 tiplied by the distance from center of pin to center of eye measured along axis of pin. If we call this dis- tance Xj then will the equation of moments be : It is considered good practice among engine builders to make the values of x, b and B as small as practicable, in order to reduce the twisting moment on the web of the crank and the bending moment on the shaft. In designing the hub, allowance must be made for the metal removed at the key- way. PROBLEM. Design a cast steel crank for a steam engine having a cylin- der 12 by 30 inches and an initial steam pressure of 120 Ib. per sq. in. of piston. The shaft is 6 inches and the crank pin 3 inches in diameter. The distance x may be assumed as 4 inches. Calculate, 1. Dimensions of web at AB. 2. Dimensions of hub allowing for a key 1 xf inches. 3. Dimensions of eye for pin, make a scale drawing in ink showing profile of crank complete, S may be assumed as 6,000 Ib. per sq. in. CHAPTEK XI. FLY-WHEELS. 81. In General. The hub and arms of a fly-wheel are designed in much the same way as those of pulleys and gears, the straight arm with elliptic section being the favorite. The rims of such wheels are of two classes, the wide, thin rim used for belt transmission and the narrow solid rim of the generator or blowing engine wheel. Fly-wheels up to eight or ten feet in diameter are usually cast in one piece ; those from ten to sixteen feet in diameter may be cast in halves, while wheels larger than the last mentioned should be cast in sections, one arm to each section. This is a matter, not of use, but of convenience in transportation. The joints between hub and arms and between arms and rim need not be specially considered here, since wheels rarely fail at these points. The rim and the joints in the rim cannot be too carefully designed. The smaller wheel cast in one piece is more or less sub- ject to stresses caused by shrinkage. The sectional Fig. 77. ' wheel is generally free from such stresses but is weakened by the numerous joints. Kim joints are of two general classes according as bolts or links are used for fastenings. 166 SAFE SPEED FOR WHEELS. 167 "Wide, thin rims are usually fastened together by internal flanges and bolts as shown in Fig. 77, while the stocky rims of the fly-wheels proper are joined directly by links or T head " prisoners " as in Fig. 78. As will be shown later, the former is a weak and unreliable joint, especially when located mid- way between the arms. The principal stresses in fly-wheel rims are caused by centrifugal force. 82. Safe Speed for Wheels. The centrifugal force developed in a rapidly revolving pulley or gear pro- duces a certain tension on the rim, and also a bending of the rim between the arms. We will first investigate the case of a pulley having a rim of uniform cross section. It is safe to assume that the rim should be capable of bearing its own centrifugal tension without assist- ance from the arms. Let _D=mean diameter of pulley rim. t= thickness of rim. b= breadth of rim. w= weight of material per cu. in. = .26 Ib. for cast-iron. = .28 Ib. for wrought iron or steel. n= number of arms. N= number rev. per min. v= velocity of rim in ft. per sec. 108 MACHINE DESIGN. First let us consider the centrifugal tension alone. The centrifugal pressure per square inch of concave surface is ( \ p= - - .... (a) gr where W is the weight of rim per square inch of con- cave surface =wt, and r= radius in feet = OTC The centrifugal tension produced in the rim by this force is by formula (13) Substituting the values of p, W and r and reducing : 8 =a>* .... (89) and v= >fef (90) For an average value of w=.27, (89) reduces to a convenient form to remember. The corresponding values of S for dry wood and for leather would be nearly : Wood S Leather S=~ oO If we assume S as the ultimate tensile strength, 16500 Ibs. for cast-iron in large castings and 60000 Ibs. for soft steel, then the bursting speed of rim is : for a cast-iron wheel ^=406 ft. per sec. . (91) and for steel rim v=7T5 ft. per sec. . (92) and these values may be used in roughly calculating the safe speed of pulleys. SAFE SPEED FOR WHEELS. 169 It has been shown by Mr. James B. Stanwood, in a paper read before the American Society of Mechanical Engineers,* that each section of the rim between the arms is moreover in the condition of a beam fixed at the ends and uniformly loaded. This condition will produce an additional tension on the outside of rim. The formula for such a beam when of rectangular cross-section is Wl_ Sbd* ,,. "12"" 6 W in this case is the centrifugal force of the fraction of rim included between two arms. The weight of this fraction is * Dbtw and its cen- trifugal force W=X or n g gn Also i= n Substituting these values in (b) and solving for S : o o r^oDwv 2 / x ~w ....... ' ' If w is given an average value of .27 then 8jg- nearly ...... (d) and the total value of the tensile stress on outer sur- face of rim is Solving for v : T~jT~ v=^l~p T . (94) In a pulley with a thin rim and small number of * See Trans. A. S. M. E. Vol. XIV. 170 MACHINE DESIGN, arms, the stress due to this bending is seen to be con- siderable. It must, however, be remembered that the stretching of the arms due to their own centrifugal force and that of the rim will diminish this bending. Mr. Stan wood recommends a deduction of one-half from the value of S in (d) on this account. Prof. Gaetano Lanza has published quite an elab- orate mathematical discussion of this subject. (See Vol. XVI. Trans. Am. Soc. Mech. Engineers.) He shows that in ordinary cases the stretch of the arms will relieve more than one-half of the stress due to bending, perhaps three-quarters. 83. Experiments on Fly- Wheels. In order to de- termine experimentally the centrifugal tension and bending in rapidly revolving rims, a large number of small fly-wheels have been tested to destruction at the Case School laboratories. In all ten wheels, fifteen inches in diameter and twenty-three wheels two feet in diameter have been so tested. An account of some of these experiments may be found in Trans. Am. Soc. Mech. Eng. Vol. XX. The wheels were all of cast- iron and modeled after actual fly-wheels. Some had solid rims, some jointed rims and some steel spokes. To give to the wheels the speed necessary for de- struction, use was made of a Dow steam turbine capa- ble, of being run at any speed up to 10000 revolutions per minute. The turbine shaft was connected to the shaft carrying the fly-wheels by a brass sleeve coup- ling loosely pinned to the shafts at each end in such a way as to form a universal joint, and so proportioned as to break or slip without injuring the turbine incase of sudden stoppage of the fly-wheel shaft. EXPERIMENTS ON FLY-WHEELS. 171 One experiment with a shield made of two-inch plank proved that safety did not lie in that direction, and in succeeding experiments with the fifteen inch wheels a "bomb-proof constructed of 6X12 inch white oak was used. The first experiment with a twenty- four inch wheel showed even this to be a flimsy contri- vance. In subsequent experiments a shield made of 12x12 inch oak was used. This shield was split re- peatedly and had to be re-enforced by bolts. A cast steel ring about four inches thick lined, with wooden blocks and covered with three inch oak plank- ing, was finally adopted. The wheels were usually demolished by the ex- plosion. No crashing or rending noise was heard, only one quick, sharp report, like a musket shot. The following tables give a summary of a number of the experiments. TABLE XXVI. FIFTEEN INCH WHEELS. Bursting Speed. Centrifugal No. Tension v* Remarks. Rev. Feet per i~n per Minute. Second =v. 1U 1 6,525 430 18,500 Six arms. -...-2 6,525 430 18,500 Six arms. 3 6,035 395 15,600 Thin rim. 4 5,872 380 14,400 Thin rim. 5 2,925 192 3,700 Joint in rim. 6 5,600* 368 13,600 Three arms. 7 6,198 406 16,500 Three arms. 8 5,709 368 13,600 Three arms. 9 5,709 365 13,300 Thin rim. 10 5,709 361 13,000 Thin rim. Doubtful. MACHINE DESIGN. TABLE XXVII. TWENTY-FOUR INCH WHEELS. Shape and Size of Rim. Weight f\f d fc Diam- Breadth Depth Area Wheel, eter Inches. Inches. Inches. Sq. Inches. Style of Joint. Pounds 11 24 3i 1.5 3.18 Solid rim. 75.25 12 24 *rV .75 3.85 Internal flanges,bolted 93. 13 24 4 .75 3.85 <( u u 91.75 14 24 4 .75 3.85 95. 15 16 24 24 a .75 2.1 3.85 2.45 Three lugs and links. 94.75 65.1 17 24 1.2 2.1 2.45 Two lugs and links. 65. TABLE XXVIII. FLANGES AND BOLTS. FLANGES. BOLTS. No. Thickness. Effective Breadth. Effective Area. No. to each Diameter. Total Tensile Inches. Inches. Inches. Joint. Inches. Strength. Pounds. 12 f 2.8 1.92 4 T'V 16,000 13 2.75 1.89 4 A 16,000 14 15 ij 2.75 2.5 2.58 2.34 4 4 t 16,000 20,000 BY TESTING MACHINE. Tensile strength of cast-iron =19,600 pounds per square in. Trans verse strength of cast-iron =46, 600 pounds per square in. Tensile strength of T 5 ^ bolts =4,000 pounds. Tensile strength of f bolts =5,000 pounds. EXPERIMENTS ON FLY-WHEELS. TABLE XXVIX. FAILURE OP FLANGED JOINTS. 173 Bursting Cent. rfi 5 w Speed. Tension. s s j-j 02 * o No. "^ i PH Rev. Ft. per Per REMARKS. 2 1 i i 3 .2 per Sec. Sq. In. Total bursting speed of iron in feet per second, for different materials and different shapes. TABLE XXXII. BURSTING SPEEDS IN FEET PER SECOND. Metal. S-g *& .2 Tensile Strength. Values of v. Thin Ring. Perforated Disc (Stodola). Flat Disc. Taper Disc. Logar- ithmic Disc. w *=! ,=9 fc=4 fc=a i=w Cast Iron. .026 .0315 .028 18000 60000 60000 430 715 760 500 825 880 745 1240 1315 1050 1750 1860 1215 2050 2140 Manganese Bronze Soft Steel PROBLEMS. 1. Determine bursting speed in revolutions per minute, of a gear 48 inches in diameter with six arms, if the thickness of rim is .75 inch. (1) Considering centrifugal tension alone. (2) Including bending of rim due to centrifugal force as- suming that f the stress due to bending is relieved by the stretching of the arms. 2. Design a link joint for the rim of a fly-wheel, the rim being 8 in. wide, 12 in. deep and 18 ft. mean diameter, the links to have a tensile strength of 65000 Ib. per sq. in. Determine the relative strength of joint and the probable bursting speed. BURSTING SPEEDS. 183 3. Discuss the proportions of one of the following wheels in the laboratory and criticise dimensions. (a) Fly-wheel. Allis engine. (b) Fly-wheel, Fairbanks gas engine. (c) Fly-wheel, air compressor. (a) Fly-wheel, Ball engine. (e) Fly-wheel, ammonia compressor. 4. Determine the value of C in formula (103) by calculation. 5. A Delaval turbine disc is made of soft steel in the shape of the logarithmic curve without any hole at the center. Determine the probable bursting speed if the disc is 8 inches in diameter. 6. A wheel rim is made of cast iron in the shape of a ring having diameters of 4 feet and 6 feet, inside and outside. Determine probable bursting speed. 7. Substitute the value for centrifugal force in place of internal pressure in Barlow's formula (b) Art. 12, and derive a value for Sin a rotating ring. Test this for d=~ and compare A with formulas in preceding article. CHAFTER XII. TRANSMISSION BY BELTS AND ROPES. 91. Friction of Belting. The transmitting power of a belt is due to its friction on the pulley, and this friction is equal to the difference between the tensions of the driving and slack sides of the belt. Let w = width of belt. Ji= tension of driving side. T 2 = tension of slack side. R = friction of belt. /= coefficient of friction be- tween belt and pulley. = arc of contact in circu- r. 82. lar measure. The tension T at any part of the arc of contact is in- termediate between Ti and T 2 . Let AB Fig. 82 be an indefinitely short element of the arc of contact, so that the tensions at A and B differ only by the amount d T. dT will then equal the friction on AB which we may call dR. Draw the intersecting tangents OTand OT to rep- resent the tensions and find their radial resultant OP. Then will OP represent the normal pressure on the arc AB which we will call P. being changed. The ultimate tensile strength of manila and hemp rope is about 10000 Ib. per sq. in. To insure durability and efficiency it has been found best in practice to use a large factor of safety. Prof. Forrest E. Jones in his book on Machine Design recommends a maximum tension of 200 d z pounds where d is the diameter of rope in inches. This cor- responds to a tensile stress of 255 Ib. per sq. in. or a factor of safety of about 40. The value of / for manila on metal is about 0.12, but as the normal pressure between the two surfaces 192 MACHINE DESIGN. is increased by the wedge action of the rope in the groove we shall have the apparent value of/: / 1 = /-s-sin-^- where A a. = angle of groove, For a = 4r5 to 30 f l varies from 0.3 to 0.5 and these values should IK used in formula (110). (1 e~* ) in this formula, for an arc of contact of 150, becomes either .54 or .73 according as/ 1 is taken 0.3 or 0.5. If T l is assumed as 250 Ib. per sq. in., the force R transmitted by the rope varies from 135 Ib. to 185 Ib. per sq. in. area of rope section. The following table gives the horse-power of manila ropes based on a maximum tension of 255 Ib. per sq. in. TABLE XXXIII. Table of the horse-power of transmission rope, reprinted from the transactions of the American Society of Mechanical Engineers, Vol. 12, page 230, Article on "Rope Driving" by C. W. Hunt. The working strain is 800 Ib. for a 2-inch diameter rope and is the same at all speeds, due allowance having been made for loss by centrifugal force. WIRE ROPE TRANSMISSION. 193 Diameter- II Rope, Inches. || SPEED OF THE ROPE IN FEET PER MINUTE. 1st 1500 2000 2500 3000 3500 4000 4500 5000 6000 7000 1 3.3 4.3 5.2 5.8 6.7 7.2 7.7 7.7 7.1 4.9 30 7 4.5 5.9 7.0 8.2 9.1 9.8 10.8 10.8 9.3 6.9 36 1 5.8 7.7 9.2 10.7 11.9 12.8 13.6 13.7 12.5 8.8 42 It 9.2 12.1 14.3 16.8 18.6 20.0 21.2 21.4 19.5 13.8 54 1* 13.1 17.4 20.7 23.1 26.8 28.8 30.6 30.8 28.2 19.8 60 If 18.0 23.7 28.2 32.8 36.4 39.2 41.5 41.8 37.4 27.6 72 2 23.1 30.8 36.8 42.8 47.6 51.2 54.4 54.8 50.0 35.2 84 98. Wire Rope Transmission. Wire ropes have been- used to transmit power where the distances were too great for belting or hemp rope transmission. The increased use of electrical transmission is gradually crowding out this latter form of rope driving. For comparatively short distances of from 100 to 500 yards wire rope still offers a cheap and simple means of carrying power. The pulleys or wheels are entirely different from those used with manila ropes. Fig. 84 shows a section of the rim of such a pulley. The rope does not touch the sides of the groove but rests on a shallow depression in a Fig. 84. wooden, leather or rubber filling at the bottom. The high side flanges prevent the rope from leaving the pulley when swaying on account of the high speed. The pulleys must be large, usually about 100 times '3 MACHINE DESIGN. the diameter of rope used, and run at comparatively high speeds. The ropes should not be less than 200 feet long unless some form of tightening pulley is used. Table XXXIV. is taken from Roebling. Long ropes should be supported by idle pulleys every 400 feet. TABLE XXXIV. TRANSMISSION OF POWER BY WIRE ROPE. Showing necessary size and speed of wheels and rope to obtain any desired amount of power. M a ~ t| ~ l -iJ 03 ; !_, ^ LJ i _ ;_, > Diamete] Wheel in Number Revoluti Diamete of Rope. f Diamete Wheel in Number Revoluti -2 13 I 1 S "8 o w 4 80 5-8 3.3 10 80 11-16 58.4 100 5-8 4.1 100 11-16 73. 120 5-8 5. 120 11-16 87.6 140 5.8 5.8 140 11-16 102.2 5 80 7-16 6.9 11 80 11-16 75.5 100 7-16 8.6 100 11-16 94.4 120 7-16 10.3 120 11-16 113.3 140 7-16 12.1 140 11-16 132.1 6 80 12 10.7 .12 80 3-4 99.3 100 1-2 13.4 100 3-4 124.1 120 1-2 16.1 120 3-4 148.9 140 1-2 18.7 140 3-4 173.7 7 80 9-16 16.9 13 80 3-4 122.6 100 9-16 21.1 100 3-4 153.2 120 9-16 25.3 120 3-4 183.9 8 80 5-8 22. 14 80 7-8 148. 100 5-8 27.5 100 7-8 185. 120 5-8 33.0 120 7-8 222. 9 80 5-8 41.5 15 80 7-8 217. 100 5-8 51.9 100 7-8 259. 120 5-8 62.2 120 7-8 300. WIRE ROPE TRANSMISSION. 195 PKOBLEMS. 1. Design a main driving belt to transmit 150 HP. from a belt wheel 18 ft. in diameter and making 80 rev. per min. The belt to be double leather without rivets. 2. Investigate driving belt on Allis engine and calculate the horse-power it is capable of transmitting economically. 3. Calculate the total maximum tension per inch of width due to load and to centrifugal force of the driving belt on the motor used for driving machine shop, assuming the maximum load to be 10 HP. 4. Design a manila rope drive, English system, to transmit 500 HP. , the wheel on the engine being 20 feet, in diameter and making 60 rev. per min. Use Hunt's table and then check by calculating the centrifugal tension and the total u 2 maximum tension on each rope. Assume S= ^ where v= feet per second. 5. Design a wire rope transmission to carry 120 HP. a dis- tance of one-quarter mile using two ropes. Determine working and maximum tension on rope, length of rope, diameter and speed of pulleys and number of supporting pulleys. INDEX. ART. PAGE. ABBREVIATIONS 2 1 ADJUSTMENT OF BEARINGS 43 96 ALLOYS 3 4 ARMS OF PULLEYS 77 157 BALL BEARINGS, In general 57 118 Conical 58 119 Cylindrical 58 118 Design of 61 123 Materials of 60 122 Step or thrust 59 120 BEAMS, formulas for 5 12 Of uniform strength 6 12 BEARINGS, Adjustment of 43 96 Ball 57 118 Cylindrical... 42 96 Engine... , 43 97 Lathe ......:... 43 98 Lubrication of 44 99 Roller , 62 123 Sliding 36 86 Steporthrust 51 111 Thrust 56 116 BELTING, Centrifugal tension of ...:. 95 189 Friction of .V 91 184 Speed of 95 188 Strength of 92 186 Taylor's experiments on. 93 186 Width of 94 187 BOILER SHELLS ... 11 25 Tubes 14 38 197 198 MACHINE DESIGN. ART. PAGE. BOLTS, Coupling 69 138 Dimensions of 18 54 Eyeorhook 20 57 BRONZES 3 5 BUTT JOINTS 23 62 CABINET SUPPORTS 9 19 CAPS AND BOLTS 50 109 CAST IRON 3 3 CENTRIFUGAL OILERS 44 101 CHAIN DRIVING, Block 78 159 Roller 78 160 Silent 79 161 CLUTCHES, Conical 68 137 Roller ' 68 138 Weston 68 136 COLLAR BEARING 53 113 Compound 56 116 COLUMN FORMULAS 5 9 CONSTANTS, Columns 5 10 Cross-sections ,.. 5 11 COTTERS 29 69 COUPLINGS, Bolts for 69 138 Clutch 68 135 Flange 67 133 . Muff 67 135 Sleeve 67 134 CRANKS 80 163 CRANK PINS, Heatingof 47 104 Pressure on 46 103 CROSS-SECTIONS 10 21 CYLINDERS, Hydraulic 12 27 Steam 16 42 DESIGN, GENERAL PRINCIPLES OP 8 15 Discs, ROTATING , . 86 177 INDEX. 199 ART. PAGE. Disc, Continued. Conical 88 180 Logarithmic 89 181 Plain 87 178 Speeds of 90 181 FACTORS OF SAFETY 7 13 FLAT PLATES, Formulas for 17 48 Tests of 17 51 FLYWHEELS 81 166 Experiments on 83 170 Rirn Joints of 83 175 Safe Speed of 82 167 FLUES, STRENGTH OF. 14 38 FORMULAS, GENERAL 5 8 FRAME DESIGN 10 20 FRICTION, Belts 91 184 Journals 45 102 Experiments on 48 107 Pivots 52 112 Schiele pivot 55 114 GEARS, Arms of 77 157 Rims of , 77 157 Teethof 72 146 GIBS 28 72 " 38 89 GUIDES, CIRCULAR 40 91 HANGERS 71 141 HEATING OF JOURNALS 47 107 HOOKS, DESIGN OF 20 57 IRON, Cast t 3 3 Malleable 3 4 Wrought 3 2 JOINTS, Butt 23 62 Lap 22 61 Riveted. 21 58 JOINT PINS. . 28 69 200 MACHINE DESIGN. ART. PAGE. JOURNALS. 42 96 Experiments on 48 107 Friction of 45 102 Heating of 47 104 Pressure on 46 103 Strength of , 49 108 KEYS, Cotter 28 69 Shafting 70 139 Woodruff 70 141 LAP JOINTS 22 61 LEGS OF MACHINES 9 19 LEVERS 80 163 LUBRICATION OF BEARINGS 44 99 MACHINE FRAMES 10 20 MALLEABLE IRON 3 4 MATERIALS OF CONSTRUCTION 3 2 NOTATION 4 8 OIL CUPS , 44 100 PACKINGS FOR GLANDS 41 92 PIPE, Sizes 13 30 Fittings ,..15 40 PIVOTS, Conical .... 54 113 Flat ... 52 112. Schiele 55 114 PLATES, Flat 17 48 Narrow 27 68 PRESSURE ON JOURNALS 46 103 PULLEYS, Arms of 77 157 Cast Iron ,, 83 175 Wooden 84 176 RING OILER. . ... 44 101 RIVETED JOINTS 21 58 Efficiency of 24 63 Narrow Plates 27 68 Practical rules for 26 fit Special forms of 25 63 INDEX. 201 ART. PAGE. ROLLER BEARINGS, Conical 63 124 Cylindrical 62 123 Hyatt.... 64 125 Step.... 65 127 Tests of 64 126 ROPE, TRANSMISSION, Manila...:..... 96 190 Strength of 97 191 Wire....:. 98 193 SCHIELE PIVOT 55 114 SCREWS, MACHINE 19 57 SHAFTING, Diameter of 66 132 Keys for 70 139 Span of 66 133 Strength of 66 130 SHELLS, STRENGTH OF, Thick 12 27 Thin 11 25 SLIDES, Angular 37 87 Flat 39 89 Gibbed 38 88 SPRINGS, Elliptic 35 83 Flat 34 81 Helical 30 73 Square wire 31 75 Testsof f 32 76 Torsion 33 79 STEAM CYLINDERS, Strength of 16 42 Testsof 16 45 STEEL 3 3 STRENGTH OF METALS, Cast 3 7 Wrought t 3 6 STUFFING BOXES 41 91 SUPPORTS} MACHINE 9 is 202 MACHINE DESIGN. ART. PAftE. TEETH OF GEARS, Bevel 76 155 Cut 73 150 Experiments on 75 153 Lewis' formula for 74 151 Proportions of 72 143 Strength of 73 148 Velocity of 74 153 THRUST BEARING 56 116 TUBES, BOILER 14 33 UNITS AND DEFINITIONS 1 i WROUGHT IRON 3 g Hall's College Laboratory Manual of Physics By EDWIN H. 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