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 COPYRIGHT DEPOSm 
 
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NOTE S 
 
 ON 
 
 MACHINE DESIGM 
 
 BY 
 
 • / 
 
 CHARLES H. BENJAMIN, M. E, 
 
 PROFESSOR OF MECHANICAL ENGINEERING, 
 
 CASE SCHOOL. OF APPLIED SCIENCE. 
 
 }^-. I 
 
 SECOND EDITION. 
 
 * • • • < 
 • • • • • 
 
 COPYRIGHTED 
 
 • • • 
 
 
 • ' • -» • 
 
 CLEVELAND : 
 ChARI^ES H. HOI^MES, PUBI.ISHER, 2303 EUCI.ID AvBw 
 
 1902. 
 
T3" zi o 
 
 THE LIBRARY OF 
 
 CONGRESS, 
 Oni Oopy Reccivco 
 
 FEB. 24 1902 
 
 CofrrmoHT rwtry 
 
 C1LAS$ ^ XXc No. 
 
 OOPY B. 
 
 m « 
 • « « 
 
©xjrnt^nt^* 
 
 OHARTER 7. 
 
 Page. 
 
 Units used. Materials, properties and strength. Nota- 
 tion. Formulas. Constants of cross-sections. Formulas for 
 loaded beams 3 
 
 CHAPTER 2, 
 
 General principles governing the design of frames and 
 supports 12 
 
 CHAPTER 3. 
 
 Stationary Machine Members: 
 
 Thin and thick shells. Steam, gas and water pipe. 
 Cast-iron steam cylinders. Flat plates. Machine frames. i6 
 
 CHAPTER 4. 
 
 Springs : 
 
 Tension and compression. Torsion. Flat or leaf 
 springs 31 
 
 CHAPTER a. 
 
 Fastenings: 
 
 Bolts and nuts. Riveted joints. Joint pins and cotters. 39 
 
 CHAPTER e, 
 
 S1.1DING Bearings: 
 
 General rules. Angular slides. Gibbed slides. Flat 
 slides. Circular guides. Stuffing-boxes .... . r^*-, . 53 
 
 CHAPTER 7. 
 
 jouRNAi^, Pivots and Bearings : 
 
 Adjustment, Lubrication. Friction. I/imits of pressure. 
 Heating. Strength and stiffness. Caps and bolts. Friction 
 of pivots. Conical pivot. Schiele's pivot. Collar bearings. 61 
 
CHARTER 8. 
 
 Bai.1. and R01.1.ER Bearings: 
 
 General principles. Journal and step ball bearings. 
 Materials and wear. Design. Roller bearings. Hyatt 
 rollers. Roller steps 76 
 
 CHARTER a. 
 
 Shafting, Couplings and Hangers : 
 
 Strength of shafting. Couplings. Coupling bolts and 
 keys. Hangers and boxes 83 
 
 CHARTER JO. 
 
 Gears, Pui.i.evs and Fi.y-WheeivS. 
 
 Gear teeth. Proportions and strength. Experimental 
 data. Teeth of bevel gears. Rim and arms. Safe speed for 
 wheels. Bursting of fly-wheels. Rims of gears 91 
 
 CHARTER n. 
 
 Transmission by Bei.ts and Ropes. 
 
 Friction of belting. Strength of belting. Rules for 
 horse -power. Centrifugal tension. Manila rope trans- 
 mission. Rules and tables. Wire rope transmission. . . 112 
 
preface to ^jeconif @Mtf xtn. 
 
 In presenting this book no claim is made of 
 ^^, originality of subject matter, as nearly every 
 ^ thing in it can be found elsewhere. The object 
 in preparing the book was to gather together in small 
 compass the more simple formulas for the strength and 
 stiffness of machine parts, with an explanation of the 
 principles involved, and with such tables and general 
 information as the designer of machinery might find 
 useful. 
 
 The book pre-supposes an acquaintance with math- 
 ematics and the laws of the strength of materials. 
 
 In short, the aim has been to put the mathematical 
 principles of macnine design in a compact form at a 
 moderate price for the use of the student and the young 
 engineer. 
 
 In revising the text for a second edition some 
 additions have been made to the physical constants in 
 tables I and II as the result of recent experiments. 
 Experimental data obtained by the author in the labor- 
 atories of the school have also been added, notably 
 those in regard to iron and steel pulleys, belts, fly 
 wheels, gear teeth, ball bearings, and the friction of 
 steam packings. 
 
 The author wishes to acknowledge the great assist- 
 ance given him by Mr. J. Verne Stanford in the 
 preparation of drawings for the cuts in this edition. 
 
(S^haptev 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 resist- 
 ance of material to distortion per unit of sectional area. 
 The word strain will be used to denote the distortion 
 of a piece per unit of length. The word set will be 
 used to denote total permament 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. Materials. The principal materials used in 
 machine construction are given in the following tables 
 with the physical characteristics of each. 
 
 By wrougt 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. 
 
 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 throu g h 
 
4 MACHINE DESIGNo 
 
 a series of rolls, structural steel and rods or rails result. 
 Steel castings are poured directly from the open 
 iiearth furnace and allowed to cool without any draw- 
 ing or rolling. They are coarser and more crystalline 
 than the rolled steel. 
 
 Open hearth steel is generally used for boiler plates 
 and of these, two grades are commonly known as 
 marine steel and flange steel. 
 
 Bessemer steel is largely used in the manufacture 
 of rails for steam and electric roads. 
 
 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 ma^e directly from the pig by remelt- 
 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. 
 
 Malleable iron is cast iron annealed and partially de- 
 carbonized by being heated in an annealing oven in con- 
 tact with some oxidising 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 constants for strength and elasticity are only 
 fair average values, and should be determined for an^ 
 special material by direct experiment when it is prac- 
 ticable. Many of the constants are not given in the 
 table on account of the lack of reliable data for their 
 determination. 
 
MACHINE DESIGN. 
 
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MACHINE DESIGN. 
 
 
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MACHINE DESIGN. 7 
 
 3. Notation. 
 
 Let S = Stress per square inch, 
 
 W== Total load applied in pounds. 
 M = Bending moment in pounds inches. 
 T = Twisting moment in pounds inches. 
 b = Breadth of cross -section in inches. 
 h=: Depth of cross-section in inches. 
 d = Diameter of circular section in inches. 
 A=Area of cross-section in square inches. 
 l=Ivength of piece in inches. 
 I = Rectangular moment of inertia. 
 J = Polar moment of inertia. 
 c = Half depth of beam or shaft in inches. 
 r= Radius of gyration of section in inches. 
 
 — = Section modulus for bending. 
 
 —= Section modulus for twisting. 
 
 4. Formulas. 
 
 Simple Stress. 
 
 Tension, compression or shear, S=-t- (i) 
 
 Be7iding under Transverse Load, 
 
 SI 
 General equation, M= — (2) 
 
 Rectangular section, M= — - — (3) 
 
 Rectangular section, bh^= — - — (4) 
 
 Sd^ 
 
 Circular section, M= (s) 
 
 ' 10^2 ^^^ 
 
 Circular section, ' d=3 r^-^^ (5) 
 
 Torsion or Twisting, 
 
 S T 
 General equation, T=— ^ (7) 
 
8 
 
 MACHINE DESIGN. 
 
 Sd^ 
 
 5-1 
 
 -n 
 
 S.I 
 
 s 
 d^-d,* 
 
 ..(8) 
 
 .(9) 
 .(id) 
 
 Circular section, T 
 
 Circular section, d 
 
 Hollow circular section, 
 
 Other values of — and — may be taken from Table 4. 
 c c 
 
 Combined Bending and Twisting, 
 
 Calculate shaft for a twisting moment, 
 
 T^=M + n/M' + T' (11) 
 
 Column subject to Bending, 
 W 
 
 Use Rankine's formula, -^= 
 
 i+q 
 
 r 
 
 (12) 
 
 The values of r^ may be taken from Table IV. The 
 subjoined table gives the average values of q, while 
 S is the compressive strength of the material. 
 
 TABLE III,— Values of q in formula 12. 
 
 Material. 
 
 Both 
 ends 
 fixed. 
 
 Fixed 
 
 and 
 
 round. 
 
 Both 
 ends 
 round. 
 
 Fixed 
 and 
 
 free. 
 
 Timber 
 
 Cast Iron 
 
 Wrought Iron... 
 Steel 
 
 I 
 
 1.78 
 3000 
 
 1.78 
 5000 
 
 1.78 
 36000 
 
 1.78 
 
 25000 
 
 4 
 
 16 
 3000 
 
 t6 
 5000 
 
 16 
 36000 
 
 16 
 25000 
 
 3000 
 
 I 
 
 3000 
 4 
 
 5000 
 
 5000 
 
 4 
 36000 
 
 4 
 25000 
 
 36000 
 
 I 
 
 25000 
 
MACHINE DESIGN. 9 
 
 In this formula, as in all such, the values of the 
 
 constants should be determined for the material used 
 
 by direct experiment if possible. 
 
 W 1 
 
 Or use straight line formula, -r- =S-— k — (12a) 
 
 A r 
 
 TABLE Ilia. — Values of S and k in fornnula (12a). 
 
 (Memman's Mechanics of Materials.) 
 
 Kind of Column. 
 
 S 
 
 k 
 
 Limit — 
 r 
 
 Wrought Iron : 
 
 Flat ends 
 
 42000 
 42000 
 42000 
 
 52500 
 52500 
 52500 
 
 80000 
 80000 
 80000 
 
 5400 
 
 128 
 
 157 
 203 
 
 179 
 
 220 
 284 
 
 438 
 537 
 693 
 
 28 
 
 218 
 178 
 138 
 
 195 
 159 
 123 
 
 122 
 99 
 
 77 
 
 128 
 
 Hinp'ed ends 
 
 Round ends 
 
 Mild Steel : 
 
 Flat ends 
 
 Hinged ends 
 
 Round ends 
 
 Cast Iron : 
 
 Flat ends 
 
 Hinged ends , 
 
 Round ends 
 
 Oah : 
 
 Flat ends 
 
 
 See also Carnegie's Pocket Companion (pp. 129, 
 147 and 152) for applications of these formulas. 
 
 For values of — less than 90 mild steel columns 
 
 are calculated for direct compression. 
 
lO 
 
 MACHINE DESIGN. 
 
 TABLE 
 
 IV.— CONSTANTS 
 
 OF CROSS-SECTION. 
 
 Form of 
 
 Square of 
 
 Moment 
 
 Section 
 
 Polar 
 
 Tortion 
 
 Section 
 
 Radius of 
 
 of 
 
 Mod'lus 
 
 Moment 
 
 ModUm 
 
 and Area 
 
 Gyration. 
 
 Inertia 
 
 I 
 
 of Iner- 
 
 J 
 
 A 
 
 r' 
 
 I=Ar^ 
 
 c 
 
 tia. J 
 
 c 
 
 Rect'ngle 
 bh 
 
 12 
 
 bh2 
 
 12 
 
 bh2 
 
 6 
 
 bh»H- bh3 
 
 bh3 + b^h 
 
 12 
 
 ^v/ba2th2 
 
 Square 
 d^ 
 
 d2 
 12 
 
 d* 
 
 12 
 
 d3 
 
 6 
 
 d^ 
 
 6 
 
 d3 
 4.24 
 
 Hollow 
 
 
 
 
 
 
 Rect'ngle 
 or I-beam 
 
 bh3-b,h? 
 
 bh3-bihf 
 
 bh3-blhf 
 
 
 
 (12bh-bihi) 
 
 12 
 
 6h 
 
 bh— b,h, 
 
 
 
 
 
 
 Circle 
 
 d2 
 
 Trd^ 
 
 d3 
 
 7rd4 
 
 d3 
 
 'd. 
 
 16 
 
 64 
 
 10.2 
 
 32 
 
 5.1 
 
 4 ' 
 
 
 
 
 
 
 Hollow 
 
 
 
 
 
 
 Circle 
 -^(d.-dD 
 
 d2+ df 
 i6 
 
 7r(d^-dt) 
 
 d^-df 
 10.2d 
 
 7r(d4-df) 
 
 d^-df 
 5.1 d 
 
 64 
 
 32 
 
 Ellipse 
 
 %b 
 
 4 
 
 a2 
 16 
 
 7rba3 
 64 
 
 ba2 
 10.2 
 
 TTba' + ab^ 
 
 ba3- ab3 
 
 64 
 
 10.2 a 
 
 Values of I and J for more complicated sections 
 can be worked out from those in table, 
 
MACHINE DESIGN. 
 
 I*: 
 
 TABLE V.-FORMULAS FOR LOADED BEAMS. 
 
 Beams of Uniform Cross-section. 
 
 Maxi- 
 mum 
 Moment 
 
 M 
 
 Maxi- 
 mum 
 
 Deflec- 
 tion 
 
 A 
 
 Cantilever, load at end 
 
 Cantilever uniform load 
 
 Wl 
 Wl 
 
 2 
 Wl 
 
 4 
 
 Wl 
 
 8 
 
 3W1 
 i6 
 
 Wl 
 8 
 
 Wl 
 8 
 
 Wl 
 
 12 
 
 Wl 
 2 
 
 WV 
 3E1 
 
 Wl' 
 
 8EI 
 
 WV 
 
 48Ei 
 
 5WI' 
 384 E I 
 
 .0182 WP 
 Ei 
 
 .0054Wr 
 
 Simple beam, load at middle 
 
 Simple beam, uniform load 
 
 Beam fixed at one end, supported 
 at other, load at middle 
 
 Beam fixed at one end, supported 
 at other, uniform load 
 
 Ei 
 
 WV 
 192 Ei 
 
 WV 
 384 Ei 
 
 WV 
 
 "l"2El 
 
 Beam fixed at both ends, load at 
 middle , 
 
 Beam fixed at both ends, uniform 
 load 
 
 Beam fixed at both ends, load at 
 one end, (pulley arm) 
 
 The maximum deflection of cantilevers and beams 
 of uniform strength is greater than when the cross- 
 section is uniform, fifty per cent, greater if the breadth 
 varies, and one hundred per cent greater if the depth 
 varies. 
 
FRAME DESIGNS. 
 
 5. General Principles. 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 tortion, and curves of uniform 
 stress for pieces subjected to bending action. 
 
 Superfluous metal must be avoided and this ex- 
 cludes all ornamentation as such. There should be a 
 good practical reason for every pound of metal in the 
 machine. It may be sometimes necessary to waste 
 metal in order to save labor in finishing, and in general 
 the aim should be to save labor at the expense of the 
 stock. 
 
 It is thus necessary for the designer to be familiar 
 with all the shop processes as well as the principles of 
 
MACHINE DESIGN. 1 3 
 
 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 appearance 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 
 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- 
 priate 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 through- 
 out, never combining cored or box sections with ribbed 
 sections. In cast pieces the thickness of metal should 
 be uniform to avoid cooling strains, and for the same 
 reason sharp corners should be absent. When aper- 
 tures are cut in a frame either for core-prints or for 
 lightness, the hole or aperture should be the symetrical 
 figure, and not the metal that surrounds it, to make 
 the design pleasing to the eye. 
 
 Machine design has been a process of evolution. 
 The earher 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 
 
14 MACHINE DESIGN. 
 
 construction, so that in the earher 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 pos- 
 terity as actually advertising an upright drill designed 
 in pure Tuscan. 
 
 6. Machine Supports. The fewer the number of 
 supports the better. Heavy frames, as of large engines, 
 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 
 foimdation. The use of intermediate supports is 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 principle' ' . 
 
 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 great- 
 est 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 unnecessary. 
 
 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 
 
MACHINE DESIGN. 1 5 
 
 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 difference in expan- 
 sion of the two metals and the liabihty of the tension 
 nuts being tampered with by workmen. 
 
 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 standpoint. 
 It is incorrect in principle and is therefore ugly. 
 
 Exercise i. — Apply the foregoing principles in 
 making a written criticism of some engine or machine 
 frame and its supports. 
 
(S^lfaptev 3* 
 
 STATIONARY MACHINE MEMBERS, 
 
 Thin Shells 
 
 Fig. I 
 
 Let Fig. I represent a section of a 
 thin shell, like a boiler 
 shell, exposed to an inter- 
 nal pressxH-e of p pounds 
 per' sq. inch. Then, if 
 we consider any diameter 
 B AB, will the total upward 
 pressure on upper half of 
 the shell balance the total 
 downward pressure on 
 the lower half and tend 
 to separate the shell at A 
 and B by tension. 
 
 I^t d= diameter of shell in inclies. 
 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=lrd(? 
 Total pressure on element =plrd<^. 
 Vertical pressm-e on element =plr sin 6d0. 
 
 Total vertical pressure on APB=^ I plr sin 6d6--—2plT 
 
 The area to resist tension at A and B=2tl and its 
 total strength = 2 tlS. 
 
 Equating the pressure and the resistance 
 2tlS = 2plr 
 pr_pd 
 
 t = 
 
 2S 
 
 (13) 
 
MACHINE DESIGN. 1 7 
 
 The total pressure on the end of cyUnder==:rr^p 
 and the resistance of a circular ring of metal to this 
 pressiu:e=27rrtS 
 
 2:rrSt=:7rr'p 
 
 t- PL-Pi fTA^ 
 
 Therefore a shell is twice as strong in this direc- 
 tion 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=o.434h (15) 
 
 exampi.es. 
 
 1 . A cast-iron water pipe is 1 2 inches in 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. i 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 presstue of 500 pounds per sq. inch. What 
 is the bursting pressure and what is the factor of safety 
 above the test pressure, assuming 8=40000? 
 
 7. 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 in such a cyhnder caused by internal pres- 
 sure is greatest at the inner circumference and dimin- 
 ishes according to some law from there to the exterior 
 of the shell. This law of variation is expressed differ- 
 ently 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 
 
18 
 
 MACHINE DESIGN. 
 
 TABLE VI.— WROUGHT IRON WELDED TUBES, 
 
 For Steam, Gas or Water. 
 
 y^ to 14 inclusive, butt-welded, tested to 300 lbs. 
 per sq. inch hyraulic pressure. 
 
 I ^/i inch and upwards, lap-welded, tested to 500 
 lbs. per sq. inch hydraulic pressure. 
 
 o 
 S 
 
 n' 
 
 0) 
 
 W- O 
 
 p 
 
 Pa 
 
 0) 
 
 
 g' 
 
 ^ 
 
 P 
 
 tf> 
 
 
 
 (V 
 
 P- 
 P 
 
 
 
 
 P- 
 
 0) 
 
 
 
 ?- 
 
 2 
 p 
 
 s 
 
 ^ 
 
 0) 
 
 •-t 
 
 
 * 
 
 w 
 
 
 o 
 
 W 
 n 
 
 O) 
 P 
 p. 
 
 w 
 
 
 w 
 
 P-* 
 
 (T) 
 P 
 
 74. 
 3/8 
 
 1/ 
 
 I 
 
 2 
 
 2^ 
 
 3 
 
 3'A 
 
 4 
 
 4^ 
 
 4 
 6 
 
 7 
 8 
 
 9 
 10 
 
 .40 
 
 •27 
 
 .24 
 
 27 
 
 ■54 
 
 •36 
 
 •42 
 
 18 
 
 .67 
 
 •49 
 
 •56 
 
 18 
 
 .84 
 
 .62 
 
 •85 
 
 14 
 
 1,05 
 
 .82 
 
 1. 12 
 
 14 
 
 I-3I 
 
 I 04 
 
 1.67 
 
 11^ 
 
 1.66 
 
 138 
 
 2.25 
 
 11% 
 
 1.90 
 
 1. 61 
 
 2.69 
 
 11^ 
 
 2-37 
 
 2 06 
 
 3.66 
 
 ii'A 
 
 2.87 
 
 2,46 
 
 577 
 
 8 
 
 3-50 
 
 3.06 
 
 7^54 
 
 8 
 
 400 
 
 3-54 
 
 9-05 
 
 8 
 
 450 
 
 4.02 
 
 10 72 
 
 8 
 
 5.00 
 
 450 
 
 12.49 
 
 8 
 
 5-56 
 
 504 
 
 1456 
 
 8 
 
 6.62 
 
 606 
 
 18.77 
 
 8 
 
 7.62 
 
 7.02 
 
 2341 
 
 8 
 
 8.62 
 
 7.98 
 
 28.35 
 
 8 
 
 9.68 
 
 9.00 
 
 3407 
 
 8 
 
 10.75 
 
 10.01 
 
 40.64 
 
 8 
 
 .0572 
 .1018 
 .1886 
 .3019 
 .5281 
 
 .8495 
 1.4956 
 
 2.0358 
 
 33329 
 47329 
 7-3529 
 9.8423 
 
 12.6924 
 
 15-9043 
 199504 
 
 28.8426 
 
 387048 
 
 50.0146 
 63.6174 
 
 80.1186 
 
MACHINE DESIGN. I9 
 
 practice it is, however, approximately true. 
 
 Let d^ and d^ be the interior and exterior diam- 
 eters in inches and let t=-^ be the thickness. 
 
 2 
 
 lyCt 1 be the length of cylinder in inches. 
 Let Si and Sg be the stresses in lbs. per sq. inch 
 at inner and outer circumferences. 
 Then it may be proved that 
 
 or the stresses vary inversely as the squares of the cor- 
 responding diameters. 
 
 Integrating, the total stress on the area 2tl is found to be 
 
 ^='^^'^i (^) 
 
 Equating this to the pressure which tends to produce 
 
 rupture, pdl, where p is the internal unit pressure, 
 
 2S t 
 
 there results : * P=^i — r — (16) 
 
 di+ 2t ^ 
 
 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 : 
 
 d/-dx^ 
 
 Pi 
 
 (17) 
 
 Fig. 2. 
 
 for the maximum stress 
 Qat the interior. 
 
 andS,=-pf-f^p,..(i8) 
 
 for the stress at the outer 
 surface, f 
 
 Fig. 2 illustrates the 
 variation in S from inner 
 to outer surface. 
 
 For discussion see Merriman's Mechanics of Materials: * p. 26; t pp. 310-14. 
 
20 MACHINE DESIGN. 
 
 Solving for d, in (i8) we have 
 
 ^=-■^IPI ^-> 
 
 exampi.es. 
 
 1. A hydraulic cylinder has an inner diameter of 
 8 inches, a thickness of four inches and an internal 
 pressure of 1500 lbs. per sq. in. Determine the maxi- 
 mum stress on the metal by Barlov^'s and Lame"s 
 formulas. 
 
 2. Design a cast iron cylinder 6 inches internal 
 diameter to carry a working pressure of 1 200 lbs. per 
 sq. in. with a factor of safety of 10. 
 
 3. A cast iron water pipe is i inch thick and 12 
 inches internal diameter. Required head of water 
 which it will carry with a factor of safety of 6. 
 
 8. 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 assumption 
 of flexibility, as in a thin shell. Most of the formulas 
 used for this class of cylinder are empirical and founded 
 on modern practice. 
 
 Va7i Bureri's formula for steam cylinders is : 
 
 ^ .0001 pd + .i5^/~d~ (20) 
 
 A formula which the writer has developed is 
 somewhat similar to Van Buren's. 
 
 lyct s'=tangential stress due to internal pressure. 
 
 Then by equation for thin shells 
 
 2t 
 
 Let s" be an additional tensne 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. 3) and assume the 
 
 * See Whitham's "Steam Engine Design", p. 27. 
 
MACHINE DESIGN. 
 
 21 
 
 maximum bending moment as at C some weak point, 
 
 the tensile stress on the 
 
 outer fibres at C due to 
 
 the bending will be pro- 
 
 pd^ 
 portional to-^^^ by the 
 
 Q laws of flexure, or 
 cpd^ 
 
 s"= 
 
 un- 
 
 where c is some 
 known constant. 
 
 The total tensile stress 
 at C will then be 
 , cpd^ 
 
 Solving for c c=— r^ 
 
 (a) 
 
 Solving for t 
 
 cpd' pM^ 
 S i6S' 
 
 (21) 
 
 a form which reduces to that of equatiop (13) when c=o. 
 
 An examination of several engine cylinders of 
 standard manufacture shows values of c ranging from 
 .03 to .10, with an average value : 
 
 c = .o6 
 
 The formula proposed by Professor Barr, in his 
 recent paper on * ''Current Practice in Engine Propor- 
 tions'', as representing the average practice among 
 builders of low speed engines is : 
 
 t=.05 d-+-.3 inch ,.,...(22) 
 
 Experiments 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 
 
 * Transactions. A. S. M. E,, vol. xviii, p. 741. 
 
22 
 
 MACHINE DESIGN. 
 
 failed by tearing around a circumference just inside 
 the flange. 
 
 In Table VII are assembled the results of the 
 various experiments for comparison. The values of S 
 by formula (13) are calculated for each cylinder, and by 
 formula (14) for all those which failed on a circum- 
 ference. It will be noticed that six out of nine faiied 
 in the latter way. 
 
 
 t+H 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 m t/5 
 
 'v/i 
 
 'Ji tn 
 
 t/3 
 
 (/3 
 
 7) 
 
 C/3 
 
 
 
 r^ U 
 
 Xi ^ 
 
 ^ 
 
 rd ^ 
 
 ^ 
 
 rO 
 
 rQ 
 
 ^ 
 
 
 
 f oJ 
 
 ^-H T- 
 
 ^ 
 
 '"' 
 
 »— 1 y- 
 
 ^ 
 
 •"^ 
 
 '""' 
 
 '■"' 
 
 '-"' 
 
 
 
 bJO^ 
 
 
 
 Q 
 
 8 8 
 
 
 
 
 
 
 
 
 
 
 
 
 8 8 
 
 g 
 
 g 
 
 g 
 
 g 
 
 g 
 
 
 
 00 ^ 
 
 ^ 
 
 ^ ^ 
 
 ^ 
 
 ^ 
 
 '^ 
 
 ^ 
 
 
 
 CO^ 
 
 M CN 
 
 c< 
 
 (N (N 
 
 <N 
 
 <N 
 
 (N 
 
 CN 
 
 
 
 oj II 
 
 VsO t^ 
 
 00 
 
 10 o\ 
 
 10 
 
 01 
 
 00 
 
 00 
 
 
 
 ^ -^ 
 
 ^ 
 
 10 ^ 
 
 LO 
 
 t^ 
 
 ^ 
 
 CN 
 
 
 
 o 
 
 q 
 
 q 
 
 q q 
 
 q 
 
 q 
 
 q 
 
 q 
 
 
 
 C/3 
 
 
 
 
 
 
 
 
 
 
 
 
 'O ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 - II 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 t/1 
 
 t^ 
 
 
 10 
 
 
 
 
 
 
 ''^t- 
 
 
 
 00 
 
 
 
 Cj 
 
 ^ 
 
 
 On 
 
 On 
 
 
 CO 
 
 t^ 
 
 
 
 1— ( 
 
 
 
 r— ^ 
 
 3 
 
 tn 
 
 ro 
 
 
 -^ 
 
 UO 
 
 
 -^ 
 
 CO 
 
 
 
 10 
 
 LO 
 
 
 s 
 
 
 
 
 
 
 
 
 
 
 
 
 
 'O +^ 
 
 
 
 
 
 
 
 
 
 
 • 
 
 
 
 Cu ^ 
 
 C 
 
 ^ 
 
 
 
 c 
 
 :> 
 
 
 
 
 
 
 
 
 
 11 
 
 __ 
 
 ^ w 
 
 '^ 
 
 3 
 
 
 
 L 
 
 
 
 On 
 
 t^ 
 
 00 
 
 10 
 
 > 
 
 OS t 
 
 ~^ 
 
 0^ 
 
 00 H 
 
 ■H 
 
 10 
 
 ^ 
 
 VO 
 
 SP 
 
 (J 
 
 
 
 m 
 
 VO t 
 
 ■^ 
 
 C^ 
 
 M t 
 
 ■^ 
 
 00 
 
 t^ 
 
 t^ 
 
 
 
 
 UJ 
 
 
 
 
 
 1— ( 
 
 
 
 
 
 M 
 
 <-t-l 
 
 
 
 
 
 
 
 
 
 
 
 
 
 CO 
 
 ^2i 
 
 S ., 
 
 ■H 
 
 S 
 
 a ., 
 
 _; 
 
 3 
 
 a 
 
 • tH 
 
 a" 
 
 (U 
 
 <r 
 
 ine 
 ailu 
 
 a \ 
 
 5J0 
 
 T^ 
 
 3 t 
 
 ji) 
 
 ^ 
 
 ^u 
 
 biO 
 
 ^ 
 
 bX) 
 
 h- 
 
 .^ ^ 
 
 3 
 D 
 
 g 
 
 ^ \ 
 
 -1 
 3 
 
 u 
 
 •T-t 
 
 a 
 
 • 1— t 
 
 a 
 
 
 
 
 
 h4P^ 
 
 h 
 
 :r 
 
 
 
 K 
 
 :j 
 
 
 
 
 
 .-r 
 
 a 
 
 > 
 
 < 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 -tI ^ 
 
 
 
 
 c 
 
 ^j 
 
 CO 
 
 VO 
 
 M 
 
 M 
 
 
 
 .Si ^ ^ 
 
 vi 
 
 D 
 
 M 
 
 10 ( 
 
 D 
 
 r^ 
 
 On 
 
 t>» 
 
 CO 
 
 
 
 
 !>- I 
 
 
 
 ^ 
 
 VO . 
 
 ^ 
 
 LO 
 
 10 
 
 10 
 
 10 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Pres- 
 sure. 
 
 P 
 
 ( 
 
 3 
 
 10 
 
 ( 
 
 D 
 
 
 
 LO 
 
 (3 
 
 LO 
 
 
 ( 
 
 3 
 
 CN 
 
 ( 
 
 D 
 
 UO 
 
 r^ 
 
 
 
 t^ 
 
 
 
 00 i 
 
 ^>» 
 
 CO 
 
 to vj 
 
 D 
 
 
 
 ON 
 
 t^ 
 
 00 
 
 
 
 
 
 H-1 
 
 cs 
 
 
 M 
 
 
 
 
 
 
 
 
 
 
 
 10 
 
 
 
 
 
 B 
 
 vO 1 
 
 LO 
 
 M 
 
 <N C 
 
 
 
 t^ 
 
 CO 
 
 CO 
 
 VO 
 
 
 
 rt nj 
 
 1— 1 
 
 ^ 
 
 l-H 
 
 HH \ 
 
 
 
 CO 
 
 h-l 
 
 10 
 
 LO 
 
 
 
 S 
 
 O) ( 
 
 N 
 
 Cn 
 
 VO ( 
 
 :> 
 
 On 
 
 On 
 
 (N 
 
 oi 
 
 
 
 t— t 
 
 — 1 
 
 
 
 
 
 
 M 
 
 M 
 
 
 6 
 
 cJ -1 
 
 d 
 
 CD 
 
 «-^-( 
 
 - 
 
 M 
 
 CO 
 
 ^ 
 
 •o 
 
MACHINE DESIGN. 23 
 
 This appears to be due to two causes. In the first 
 place, the influence of the flanges extended to the 
 center of the cylinder, stiffening the shell and prevent- 
 ing the splitting which would otherwise have occurred. 
 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. 
 
 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 
 calculated for each cylinder, and agree very well except 
 in numbers 3 and 5. 
 
 To the criticism that most of the cylinders did not 
 fail by splitting, and that therefore formulas (a) and 
 (21) are not applicable, the answers 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 = 20oo, formula (21) reduces to: 
 
 t=:-Pl-+ JL lp + _P^ (2^) 
 
 8000 200\ ^ 1600 ^ ^^ 
 
 In Kent's Mechanical Engineer's Pocket Book 
 p. 794, the following formula is given as representing 
 closely existing practice : 
 
 t=.ooo4dp+o.3 inch (24) 
 
 This corresponds to Ban's formula if we take p=i25 
 pounds per square inch. 
 
24 MACHINE DESIGN. 
 
 exampi.es. 
 
 1. Referring to Table VII, verify in at least three 
 experiments the values of S and c as there given. 
 
 2. The steam cylinder of a Baldwin locomotive 
 is 22 ins. in diameter and 1.25 ins. thick, Assuming 
 125 lbs. gauge pressure, 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 pres- 
 sure 100 lbs. by formulas 13, 20, 21, 23 and 24. 
 
 9. Thickness 6f Flat Plates. An approximate 
 formula for the thickness of flat cast-iron plates may 
 be derived as follows : 
 
 Let l=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 lbs. per sq. in. 
 
 Suppose the plate to be divided lengthwise into flat 
 strips an inch wide 1 inches long, and suppose that 
 a fraction p' of the whole pressure causes the bending 
 of these strips. 
 
 Regarding the strips as beams with fixed ends and 
 uniformly loaded : 
 
 6M _ 6W1 ^ pT^ 
 
 bh^ ~ i2bh^~ 2t^ 
 and the thickness necessary to resist bending is : 
 
 =g^ 
 
 s « 
 
 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— p' 
 equals p" to cause the bending in this direction, we 
 shall have : 
 
 h *' 
 
 But as all these strips form one and the same plate 
 
 -J 
 
MACHINK DESIGN. 25 
 
 the ratio of p' to p'' must be such that the deflection at 
 the center of the plate may be the same on either sup- 
 position. The general formula for deflection in this 
 case is 
 
 t' 
 and 1= — for each set of strips. Therefore the deflec- 
 12 
 
 tion IS proportional to -^y~ ^^^ ^~r~ '^ ^^^ ^^^ cases. 
 
 .-. pl^=p''b^ 
 
 But '+p''=P 
 
 Solving in these equations for p^ and p^^ 
 
 p i*+b* 
 
 Substituting these values in (a) and (b) : 
 
 ^=^^^>l.-s(iw (^^) 
 
 As 1 > b usually, equation (26) is the one to be 
 used. If the plate is square 1 ^^^ b and 
 
 2 \ s 
 
 (27) 
 
 If the plate is merely supported at the edges then 
 formulas (25) and (26) become : 
 For rectangular plate : 
 
 *-T\i S(P+b*) ^"^> 
 
 For square plate : 
 
 =A |3P 
 
 ■ 2\2S 
 
 * — ^J:;o (29) 
 
25 MACHINE DESIGN. 
 
 Formulas for the thickness of a fiat plate under a 
 concentrated load at the center, can be derived in a 
 similar manner. A round plate may be treated as 
 square, with side = diameter, without sensible error. 
 
 The preceding formulas can only be reparded as 
 approximate. Grashof 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 : 
 
 '=Tji (3°) 
 
 Fixed at edges : 
 
 '=tJ •■•■ •■■■(-) 
 
 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 (27) and (29) for 
 square plates, they are seen to be nearly identical. 
 
 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 ten by 1 5 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 i 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 lbs. per sq. in. 
 
MACHINE DESIGN. 
 
 27 
 
 In Table VIII are given the values obtained for 
 the breaking load W under the different conditions. 
 
 Those plates which 
 were merely supported at 
 the edges broke in three 
 or four straight lines rad- 
 iating from the center. 
 Those fixed at the edges 
 broke in four or five rad- 
 ial lines meeting an irreg- 
 ular oval inscribed in the 
 rectangle . Number 1 2 
 however failed by shear- 
 ing, the circular plunger 
 making a circular hole 
 in the plate with several 
 radial cracks. 
 
 EXAMPLES. 
 
 1 . Calculate the thick- 
 ness of a steam-chest 
 cover 8X12 inches to 
 sustain a pressure of 90 
 lbs. per sq. inch with a 
 factor of safety = 10. 
 
 2. Calculate the thick- 
 ness of a circular man- 
 hole cover of cast-iron 18 
 inches in diameter to sus- 
 tain a pressure of 150 lbs. 
 per sq. inch with a factor 
 of safety =8, regarding 
 the edges as merely sup- 
 ported. 
 
 3. Work out formulas for a rectangular plate 
 having a concentrated load= W at the center, and with 
 edges either supported or fixed. 
 
 4. Test values for W given in Table VIII by 
 formulas obtained in example 3. 
 
 TABLE VIII. 
 
 Cast iron plates : 
 
 [OX15 ins. 
 
 No. 
 
 Thick- 
 ness. 
 t 
 
 Breaking 
 
 Load. 
 
 W 
 
 I 
 
 .562 
 
 7500 
 
 2 
 
 .641 
 
 1 1840 
 
 3 
 
 .745 
 
 14800 
 
 4 
 
 .828 
 
 21900 
 
 5 
 
 1.040 
 
 31200 
 
 6 
 
 I. 120 
 
 31800 
 
 7 
 
 .481 
 
 9800 
 
 8 
 
 .646 
 
 17650 
 
 9 
 
 .769 
 
 26400 
 
 ID 
 
 .881 
 
 33400 
 
 II 
 
 1.020 
 
 47200 
 
 12 
 
 1. 123 
 
 59600 
 
28 MACHINK DESIGN. 
 
 5. In experiments on steam cylinders, a head 12 
 inches in diameter and 1.18 inches thick failed under a 
 pressure of 900 lbs. per sq. in. Determine the value 
 of S by formula (31). 
 
 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 pattern for hollow work, includ- 
 ing 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. 
 
 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 y/ 
 
 shown in Fig. 4. The metal ^ 
 
 being of nearly uniform thickness ^ 
 
 and the flange which is in tension /X 
 
 having an area three or four times 
 that of the compression flange. 
 In a steel casting these may be 
 
 i 
 
 in a sreei casting tnese may oe (7 //nJ// //? y///y?/ % 
 more nearly equal. The hollow V////^<////u//////> 
 
 section may be of the shape shown . 
 
 in Fig. 5, a hollow rectangle with ^^* 4« 
 
MACHINE DESIGN. 
 
 29 
 
 VZY////////?m 
 
 the tension side re-enforced and slightly thicker than 
 the other three sides. The re-enforcing flanges at A 
 
 and B may often be utiUzed 
 for the attaching of other 
 members 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 
 B much greater. The double 
 Fig. 5. I or the U section is com- 
 
 mon where it is necessary to 
 have two parallel ways for sliding pieces as in lathes 
 and planers. As is shown in Fig. 6 the two Is are 
 i^yyyy J uyy jjy i usually conucctcd at intervals bv 
 
 ^T ^ , P^ 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 uniform fibre stress from 
 end to end. This necessitates a 
 parabolic or elUptic outhne of 
 which the best instance is the 
 housing or upright of a modern 
 iron planer. 
 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 I 
 bending the section may be made 
 elliptical or, as is more common, 
 thickened on two sides by making 
 the core oval. See Fig. 7. As 
 has already been pointed out the ^ 
 
 box sections are in general better 
 adapted to resist twisting than the ribbed or I sections 
 
 , 
 
 ! 
 
 Fig. 6. 
 
30 MACHINE DESIGN. 
 
 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 
 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 up- 
 rights are also subject to side bending when the tool 
 is taking a heavy side cut near the top. To provide 
 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 X the distance from cen- 
 ter of cutter to center of upright. 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 then taper gradually. 
 
 The frame of a shear press or punch is usually of 
 the G shape in profile with the inner fibers in tension 
 and the outer in compression. The cross-section should 
 be as in Fig. 4 or Fig. 5, preferably the latter and should 
 be graduated to the magnitude of the bending moment 
 at each point. 
 
 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. 
 
II. spiral Springs. The most common form of 
 spring used in machinery is the spiral or hehcal spring 
 made of round brass or steel wire. Such springs may 
 be used to resist extension or compression or they may 
 be used to resist a twisting moment. 
 
 Tension and Compression, 
 lyCt Iy=length of axis of spring. 
 D=mean diameter of spring. 
 1= developed length of wire. 
 d= diameter of wire. 
 n= number of coils. 
 P= tensile or compressive force. 
 x= corresponding extension or compression. 
 S=safe torsional or shearing strength of wire. 
 =2500 for spring brass wire. 
 = 75000 to 1 15000 for cast steel tempered. 
 G= modulus of torsional elasticity. 
 = 6000000 for spring brass wire. 
 = 12000000 to 15000000 for cast steel, tempered. 
 
 Then l=\/^^^D^n^+I?" 
 
 If the spring were extended until the wire became 
 straight it would then be twisted n times, or through an 
 angle=27rn and the stretch would be 1 — L. 
 
 The angle of torsion for a stretch =x is then 
 
 ^ 27rnx , . 
 
 ''=1=17 (^^ 
 
 Suppose that a force P' acting at a radius will 
 
 twist this same piece of wire through an angle d caus- 
 ing a stress S at the surface of the wire. Then will 
 
 the distortion of the wire per inch of length be s=— i- 
 
 5.1T 5iP^D .K^ 
 
32 MACHINE DESIGN. 
 
 ^ S I0.2 P^Dl . . 
 
 •'•^=T=-7d^ ^"^ 
 
 In thus twisting the wire the force required will 
 vary uniformly from o at the beginning to P^ at the 
 end provided the elastic limit is not passed, and the 
 average force will be 
 
 P^ P^D<9 
 
 = — The work done is therefore 
 
 2 4 
 
 If the wire is twisted through the same angle by 
 the gradual application of the direct pressure P, com- 
 pressing or extending the spring the amount x , the 
 work done will be 
 
 Px _ P^D^ Px 
 — But = — 
 
 2 42 
 
 2Px 
 .•.P^= - (d) 
 
 Substituting this value of PMn (c) and solving 
 for x: 
 
 _Gd^ 
 ^~io.2Pl 
 Substituting the value of from (a) and again 
 solving for x : 
 
 10, 
 
 X 
 
 If we neglect the original obliquity of the wire 
 then 1 = -Dn and L=o and equation (e) reduces to 
 
 2,55P1D^ 
 "" Gd* ^^^ 
 
 Making the same approximation in equation (d) 
 we have P^ = P 
 
 i, e, — a force P will twist the wire through approxi- 
 mately the same angle when applied to extend or 
 compress the spring, as if applied directly to twist a 
 piece of straight wire of the same material with a 
 
 lever arm = — 
 2 
 
 This may be easily shown by a model. 
 
MACHINE DESIGN. 33 
 
 The safe working load may be found by solving- 
 for P^ in (b) and remembering that P = P^ 
 
 P=^4^ (33) 
 
 2.55 D ^^^^ 
 
 when S is the safe shearing strength. 
 
 Substituting this value of P in (24) we have for 
 
 the safe deflection : 
 
 IDS , , 
 
 13. Square Wire. The value of the stress for a 
 square section is: 
 
 where d is the side of square. 
 
 The distortion at the corners caused by twisting 
 through an angle 6 is: 
 
 ^ ^d 
 
 Equation (c) then becomes: 
 ^ 6P'D1 
 
 The three principal equations (32), (33) and (34) 
 then reduce to: 
 
 1.5PID' , . 
 
 "=-5d^ (35) 
 
 T3 Sd' , .. 
 
 P=T7^ (36) 
 
 "=05^ (37) 
 
 The square section is not so economical of material 
 as the round. 
 
 13. Experiments. Tests made on about 1700 
 tempered 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 
 
34 
 
 MACHINE DESIGN. 
 
 closed coil to coil several times. Mr. French as a result 
 of the experiments arrives at the following conclusions: 
 
 1. The average value of G is 14,500,000. 
 
 2. The safe working stress S depends upon the 
 proportions of the spring and varies from 75000 to 
 112,000 lbs. per sq. inch for a good grade of steel 
 properly tempered. 
 
 3. If R=--T- the ratio of spring diameter to wire 
 
 diameter, the following values of S may be safely 
 assumed. 
 
 VALUES OF S. 
 
 
 R=3 
 
 R=8 
 
 d = s/s inch or less 
 
 d=-Y^ inch to ^/i inch.. 
 d=-Y|-inch to i^ inch... 
 
 112,000 
 110,000 
 105,000 
 
 85,000 
 80,000 
 75,000 
 
 4. When a spring is subjected to sudden shocks 
 a smaller value of S must be used. 
 
 5. In designing close coil extension springs the 
 value of G will be as above but the values of S should 
 not be over two-thirds the corresponding values for 
 compression springs. 
 
 14. Spring in Torsion. If a spiral spring is used 
 to resist torsion instead of tension or compression, the 
 wire itself is subjected to a bending moment. We will 
 use the same notation as in the last article, only that 
 P will be taken as a force acting tangentially to the 
 
 circumference of the spring at a distance — from the 
 
 axis, and S will now be the safe transverse strength 
 of the wire, having the following values: 
 S=3ooo for vSpring brass wire. 
 = 90,000 to 125000 for cast steel tempered. 
 
MACHINE DKSIGN. 35 
 
 E = 9000000 for Spring brass wire. 
 
 = 30000000 for cast steel tempered. 
 Let ^= angle through which the spring is turned 
 by P. 
 
 The bending moment on the wire will be the 
 
 PD 
 same throughout and = This is best illustrated 
 
 by a model. 
 
 To entirely straighten the wire by unwinding the 
 spring would require the same force as to bend straight 
 wire to the curvature of the helix. 
 
 To simplify the equations we will disregard the 
 obliquity of the helix, then will l=7rDn and the radius 
 of curvature _ D 
 
 2 
 Let M ^= bending moment caused by entirely 
 5>i:raightening the wire ; then by mechanics 
 
 ^^ EI 2EI ' 
 
 and the corresponding angle through which spring is 
 turned is 27rn. 
 
 But it is assumed that a force P with a radius — 
 turns the spring through an angle 0, ^ 
 
 PD 2EI d 
 
 2 D 2-n 
 
 _EI^_EI^ 
 ■~-Dn~ 1 
 Solving for ^ : 
 
 -li <>) 
 
 and if wire is round 
 
 ^ 10.2PDI 
 
 '=-^d^ ^38) 
 
 The bending moment for round wire will be 
 
 PD Sd^ . , 
 
 — -irrz (39) 
 
 2 10,2 
 
3b MACHINK DESIGN. 
 
 and this will also be the safe twisting moment that 
 can be applied to the spring when vS = working 
 strength of wire. The safe angle of deflection is founa 
 
 PD 
 
 by substituting this value of in (38): 
 
 2IS 
 Reducing: ^^^d~ **" ^^^^ 
 
 15. Flat Springs. Ordinary flat springs of uniform 
 rectangular cross-section can be treated as beams and 
 their strength and deflection calculated by the usual 
 formulas. 
 
 In such a spring the bending and the stress are 
 greatest at some one point and the curvature is not 
 uniform. 
 
 To correct this fault the spring is made of a con- 
 stant depth but varying width. 
 
 If the spring is fixed at one end and loaded at the 
 other the plan should be a triangle with the apex at 
 the loaded end. If it is supported at the two ends and 
 loaded at the center, the plan should be two triangles 
 with their bases together under the load forming a 
 rhombus. The deflection of such a spring is one and 
 a half times that of a rectangular spring. 
 
 As such a spring might be of an inconvenient 
 
 width, a com- 
 pound or leaf- 
 spring is made 
 by cutting the 
 triangular 
 I spr i ng into 
 
 I ^ I -| strips parallel 
 
 I [ to the axis, and 
 
 I piling one a- 
 bove another as 
 Fig. 8. in Fig. 8. 
 
 f 
 
 This arrangement does not change the principle, 
 save that the friction between the leaves may increase 
 the resistance somewhat. 
 
MACHINE DESIGN. 37 
 
 Let l=length of Span. 
 b=breadth of leaves, 
 t^^thickness of leaves. 
 n=: number of leaves. 
 W=load at center. 
 A=deflection at center. 
 
 S and E may be taken as 80000 and 30000000 
 respectively. 
 
 Strength : 
 
 ^^ Wl Snbt: 
 
 M= — = 
 
 Elasticity : 
 
 4 6 
 w=— -^ — (41) 
 
 WP ^ , nbt' 
 
 A= — =rrwnere I = 
 
 32EI 12 
 
 3WI' , , 
 
 •■• ^=8E^' <42) 
 
 For the benefit of those who wish to design 
 
 springs in quantity, reference is made to Trans. Am. 
 
 Soc. Mech. Eng. Vol. XVII. p. 340, where will be 
 
 found very complete tables for both helical and flat 
 
 springs. 
 
 EXAMPLES. 
 
 1. A spring balance is to weigh 25 pounds with 
 an extension of 2 inches, the diameter of spring being 
 y^ inches and the material, tempered steel. 
 
 Determine the diameter and length of wire, and 
 number of coils. 
 
 2. Determine the safe twisting moment and angle 
 of torsion for the spring in example i, if used for a 
 torsional spring. 
 
 3. Design a compound flat spring for a locomotive 
 to sustain a load of 16000 lbs. at the center, the span 
 being 40 inches, the number of leaves 1 2 and the ma- 
 terial steel. 
 
 4. Determine the maximum deflection of the 
 above spring, under the working load. 
 
38 MACHINE DESIGN. 
 
 5. Measure various indicator springs and deter- 
 mine value of G from rating of springs. 
 
 6. Measure various brass extension springs cal- 
 culate safe static load and safe stretch. 
 
 7. Make an experiment on torsion spring to de- 
 termine distortion under a given load and calculate 
 value of E. 
 
FASTENINGS. 
 
 i6. Bolts and Nuts. Tables of dimensions for 
 U. S. standard bolts and nuts are to be found in all 
 hand books and will not be repeated here. 
 
 Roughly the diameter at root of thread is .83 Ox 
 the outer diameter in this system, and the pitch in 
 inches is given by the formula 
 
 p=.24\/d+.625 — .175 (43> 
 
 where d::= outer diameter. 
 
 In designing bolts to resist simple tension, calcu- 
 late the area needed to resist the given tension, divide 
 this by the number of bolts to be used and the quotient 
 will be the area of one bolt at the root of thread. 
 
 From the tables the corresponding diameter and 
 the diameter of body of bolt can be determined. 
 
 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. 
 
 The principal objection to the last two forms and 
 especially to (3) is the liability of sticking or break- 
 ing 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. 
 
 17. 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 
 
4° 
 
 MACHINE DESIGN. 
 
 TABLE IV.-SAFE WORKING STRENGTH 
 OF WROUGHT IRON BOLTS. 
 
 Diam 
 
 of 
 Bolt. 
 
 Inch. 
 
 1 6 
 
 3/8 
 
 7 
 
 16 
 
 V2 
 
 9 
 16 
 
 5/8 
 3/ 
 /8 
 I 
 
 iM 
 
 I 3/^ 
 
 l5/i 
 13/ 
 I^ 
 2 
 
 Diam. 
 
 at 
 Root of 
 Thread. 
 Inches. 
 
 344 
 400 
 
 454 
 507 
 620 
 
 731 
 
 837 
 940 
 
 065 
 
 160 
 
 284 
 
 389 
 490 
 
 615 
 712 
 
 Area 
 at 
 Root of 
 Thread. 
 Sq. Ins. 
 
 .0269 
 
 .0452 
 
 .0679 
 
 .0930 
 
 .1257 
 
 .162 
 
 .202 
 
 .302 
 
 .420 
 
 .550 
 .694 
 
 .891 
 
 1.057 
 1.295 
 1. 515 
 1.744 
 2.049 
 2.302 
 
 Safe 
 lyoad in 
 Tension 
 
 Lbs. 
 
 150 
 250 
 
 375 
 
 510 
 
 690 
 
 890 
 
 mo 
 
 1660 
 
 2310 
 
 3025 
 
 3815 
 4900 
 
 5815 
 7^25 
 
 8335 
 
 9590 
 
 11270 
 
 12660 
 
 Safe 
 Load in 
 Shear, 
 
 Lbs. 
 
 220 
 
 350 
 500 
 
 675 
 880 
 
 1120 
 
 1380 
 
 2000 
 
 2700 
 
 3535 
 
 4475 
 
 5520 
 
 6680 
 
 7950 
 
 9330 
 
 10825 
 
 12425 
 
 14130 
 
 Thr'ds 
 per 
 Inch. 
 
 No. 
 
 20 
 18 
 16 
 
 14 
 
 13 
 12 
 
 II 
 
 10 
 
 9 
 
 8 
 
 7 
 
 7 
 6 
 
 6 
 
 5K2 
 
 5 
 5 
 4/2 
 
MACHINE DKSIGN. 41 
 
 the bolt to allow for obliquity and an uneven distri- 
 bution of stress. 
 
 Large hooks should be designed to resist com- 
 bined bending and tension ; the bending moment is 
 equal to the load X the longest perpendicular from the 
 center line of hook to line of load. 
 
 Check Nuts: A check 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 out- 
 side. 
 
 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. 
 
 The foregoing table is convenient for determining 
 the size of bolt needed to resist tension or shear and is 
 based on the U. S. standard form of thread using a 
 factor of safety =10. 
 
 For steel bolts, increase the loads given in the 
 table 20 per cent. The loads given are correct within 
 10 pounds. The shearing area used is that of the 
 body of the bolt. 
 
 EXAMPLES. 
 
 1. Discuss the effect of the initial tension caused 
 by the screwing up of the nut on tb e bolt, in the case 
 of steam fittings, etc.; i, e, should this tension be added 
 to the tension due to the steam pressure, in determin- 
 ing the proper size of bolt ? 
 
 2. Determine the number of Y^ inch bolts neces- 
 sary 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 
 
 (i) If used for square nuts: 
 
 (2) If used for hexagon nuts; illustrate by figure. 
 
42 
 
 MACHINE DESIGN. 
 
 4. Determine the length of nnt theoretically 
 necessary to prevent stripping of the thread, in terms 
 of the outer diameter of the bolt. 
 
 (i) 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. 
 
 18. Riveted Joints. No attempt will be made to 
 go into the details of this subject, but only to state 
 the general principles involved in designing joints. 
 
 Riveted 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 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 
 
 original plate, 
 
 nsimply because 
 
 holes cannot be 
 
 punched or drilled 
 
 in the plate for 
 the introduction of 
 rivets without re- 
 moving some of 
 the metal. 
 
 Fig. 9 shows a double riveted lap joint and Fig. 
 10 a single riveted butt joint with two straps. 
 
 Riveted joints may fail in any one of four Vays : 
 
c 
 
 c 
 
 i 
 i 
 
 ) 
 
 ) 
 
 MACHINE DESIGN. 43 
 
 1. By tearing of 
 the plate along a 
 line of rivet holes, 
 as at AB, Fig. 9. 
 
 2. By shearing of 
 the rivets. 
 
 3. By crushing and 
 wrinkling of the 
 plate in front of each 
 rivet as at C, Fig. 
 9, thus causing 
 leakage. 
 
 4. By splitting of the plate opposite each rivet as 
 at D, Fig. 9. The last manner of failure may be pre- 
 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. 
 
 Let : t= thickness of plate. 
 
 d= diameter of rivet-hole 
 
 p= pitch of rivets. 
 
 n=number of rows of rivets. 
 
 T= tensile strength of plate. 
 
 C= bearing or crushing strength of plate. 
 
 S= Shearing strength of rivet. 
 
 Average values of the constants are as 
 
 follows : 
 
 Material. 
 
 T 
 
 C 
 
 S 
 
 Wrought Iron. 
 Soft Steel. 
 
 50 000 
 56 000 
 
 80 000 
 90 000 
 
 40 000 
 45 000 
 
 19. 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. 
 
44 MACHINE DKSIGN. 
 
 The resistance to tearing across of the strip be- 
 tween rivet holes is (p— d)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 
 
 ^nd^S (c) 
 
 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. 
 
 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 average values given allow roughly for these 
 modifications. 
 
 If we call the tensile strength T=unity then the 
 relative values of C and S 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 (44) 
 
 Equating (a) and (c) and reducing we have 
 
 p=d+.628-^ (45) 
 
 Or by equating (a) and (b) 
 
 p=d+i.6nd (46) 
 
 These proportions will give a joint of equal 
 strength throughout, for the values of constants as- 
 sumed. 
 
 20. Butt Joints with two Straps. In this case the 
 resistance to shearing is increased by the fact that the 
 
MACHINE DESIGN. 45 
 
 rivets must be sheared at both ends before the joint 
 can give way. 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 
 
 d=i.36t .^ (47) 
 
 p=d+i.i8^ (48) 
 
 p=d+i.6nd (49) 
 
 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 -^\ 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 : 
 
 lyap joints d = 2.33t 
 
 p=d+i.375nd 
 
 Butt joints d=i.8t 
 
 p=d+r.55nd 
 
 21. Efficiency of Joints. The efl&ciency of joints 
 designed like the preceding is simply the ratio of the 
 section of plate left between the rivets to the section 
 of solid plate, or the ratio of the clear distance between 
 two adjacent rivet holes to the pitch. From formula 
 (35) we thus have 
 
 Eflficiency= — -^— ^ — (50) 
 
 i + i.6n 
 
 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. 
 
4^ machine: design. 
 
 22. Butt Joints with unequal Straps. One joint 
 in common use requires special treatment. 
 
 It is 
 the inner 
 
 ?? 
 
 fi 
 
 c 
 
 1 
 
 c 
 
 i 
 
 ) 
 ) 
 
 a double - riveted butt joint in which 
 strap is made wider than the outer 
 
 and an 
 
 extra row 
 
 of rivets 
 
 added, 
 
 whose 
 
 pitch is 
 
 double 
 
 that of 
 the origi- 
 nal seam ; 
 
 this is 
 
 sometimes 
 
 called 
 
 diamond 
 
 riveting. 
 
 See Fig. 
 II. 
 
 Fig. II. 
 
 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 widths 2 p. Then 
 the resistance to tearing along the outer row of rivets 
 is (2p— d)tT 
 
 As there are five rivets to compress in this strip 
 the bearing resistance is 
 
 5dtC 
 
 As there is one rivet in single shear and four in 
 double shear the resistance to shearing is 
 
 I i + (4X 1.85) J — d^S=6.6d^S 
 
MACHINE DESIGN. 47 
 
 Solving these equations as in previous cases, we 
 have for this particular joint 
 
 d=i.52t (51) 
 
 2p=9d 
 P=4-5d .-(52) 
 
 Efficiency=-^ = — (53) 
 
 2p 9 
 
 23. 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. 
 
 The following table gives the proportions gener- 
 ally used in this country for lap joints, as given by 
 '^Locomotive'' 1882. 
 
 TABLE X.-RIVETED LAP JOINTS. 
 
 Thick- 
 ness of 
 
 Diam. 
 of 
 
 Diam. 
 of 
 
 Pitch. 
 
 Efficiency. 
 
 
 
 
 
 Plate. 
 
 Rivet. 
 
 Hole. 
 
 Single. 
 
 Double 
 
 Single. 
 
 Double 
 
 % 
 
 ^8 
 
 1 1 
 16 
 
 2 
 
 3 
 
 .66 
 
 •77 
 
 5 
 1 6 
 
 1 1 
 
 16 
 
 ^ 
 
 2tV 
 
 zyi 
 
 .64 
 
 .76 
 
 3/8 
 
 Va 
 
 1 3 
 
 1 6 
 
 2yi 
 
 z% 
 
 .62 
 
 •75 
 
 16 
 
 1 3 
 
 16 
 
 ^8 
 
 2-A- 
 
 33/8 
 
 .60 
 
 •74 
 
 y- 
 
 ^8 
 
 15 
 16 
 
 ^% 
 
 1% 
 
 •58 
 
 •73 
 
 This table is for iron plates and iron rivets. For 
 steel plates with iron or steel rivets increase the diam- 
 eter of rivets -^ inch, the pitch remaining the same. 
 
48 
 
 MACHINE DESIGN. 
 
 A similar table has been calculated for butt joints. 
 Table XI is for iron plates with iron rivets. For steel 
 plates increase the diameter of rivet -jj- inch, the pitch 
 remaining the same. 
 
 TABLE XI.- 
 
 -RIVETED BUTT JOINTS. 
 
 Thick- 
 ness of 
 
 Diam. 
 of 
 
 Diam. 
 of 
 
 Pitch. 
 
 
 
 
 Plate. 
 
 Rivet. 
 
 Hole. 
 
 Single. 
 
 Double 
 
 Triple. 
 
 V2 
 
 Va 
 
 13 
 
 16 
 
 2Y& 
 
 4 
 
 s'A 
 
 Vz 
 
 13 
 
 16 
 
 /8 
 
 2 3/^ 
 
 zV. 
 
 sY. 
 
 Va 
 
 n 
 
 15 
 16 
 
 2 3/^ 
 
 2,Va 
 
 sH 
 
 % 
 
 15 
 16 
 
 I 
 
 2 Yd 
 
 2>Va 
 
 5 
 
 I 
 
 I 
 
 i-iV 
 
 2 3/^ 
 
 3K 
 
 5 
 
 EXAMPIvES. 
 
 1. Investigate proportions of joints for half -inch 
 plate in Table X and criticise. 
 
 2. Criticise in same way the proportions of joints 
 for one inch plate in Table XI. 
 
 3. Show the effect of increasing the diameter of 
 rivets -jj- inch for steel plates and prove by example. 
 
 4. A cylindrical boiler 5X 16 ft. is to have long 
 seams double-riveted laps and ring seams single riveted 
 laps. If the internal pressure is 90 lbs. gauge pres- 
 sure and the material soft steel, determine thickness 
 of plate and proportions of joints. Factor of safety to 
 be five and efficiency of joints to be allowed for. 
 
 5. A marine boiler is 11 ft. 6 ins. in diameter and 
 14 ft. long. The long seams are to be diamond riveted 
 butt joints and the ring seams ordinary double riveted 
 butt joints. The internal pressure is to be 175 lbs. 
 gauge and the material is to be steel of 60,000 lbs. 
 
MACHINE DESIGN. 
 
 49 
 
 tensile strength. Determine thickness of shell and 
 proportions of joints. Net factor of safety to be five 
 allowing for efiiciency of joints. 
 
 6. Design a diamond riveted joint such as shown 
 in Fig. iia for a steel plate f^ inches thick. Outer 
 cover plate is 5^ inches and inner cover plate tV inches 
 thick. Determine efficiency of joints. 
 
 Fig. 1 1 a. 
 
 7. The single lap joint with cover plate, as shown 
 in Fig. 12, is to have pitch of outer rivets double that 
 of inner row. Determine diameter and pitch of rivets 
 for fi inch plate and the efficiency of joint. 
 
 \_7 
 
 ^ 
 
 Fig. 12 
 
 14. Joint Pins. A joint pin is a bolt exposed to 
 double shear. If the pin is loose in its bearings it 
 should be designed with allowance for bending, by 
 adding from 30 to 50 per cent to the area of cross- 
 section needed to resist shearing alone. Bending of the 
 pin also tends to spread apart the bearings and this 
 should be prevented by having a head and nut or cot- 
 ter on the pin. 
 
 If the pin is used to connect a knuckle joint as in 
 boiler stays, the eyes forming the joint should have a 
 a net area 50 per cent in excess of the body of the 
 stay, to allow for bending and uneven tension. 
 
50 MACHINE DESIGN. 
 
 25. Cotters. A cotter is a key which passes dia- 
 metrically through a- hub and its rod or shaft, to fasten 
 them together, and is so called to distinguish it from 
 shafting keys which lie parallel to axis of shaft. 
 
 Its taper should not be more than 4 degrees or 
 about I in 15, unless it is secured by a screw or check 
 nut. 
 
 The rod is sometimes enlarged where it goes in 
 the hub, so that the effective area of cross-section 
 where the cotter goes through may be the same as in 
 the body of the rod 
 
 Let: d=diameter of body of rod. 
 
 d^^^diameter of enlarged portion. 
 
 t=thickness of cotter, usually = — - 
 
 4 
 b= breadth of cotter. 
 
 1 = length of rod beyond cotter. 
 Suppose that the applied force is a pull on the rod 
 — causing tension on the rod and shearing stress on 
 the cotter. 
 
 The effective area of cross section of rod at cotter 
 
 is ^Ail^^ ^._.dy 
 
 4 4 ^^ 4 
 
 and this should equal the area of cross-section of the 
 body of rod. 
 
 4 4 
 
 ^^=^Jr^, = i.2id (54) 
 
 Let P=pull on rod. 
 
 S=shearing strength of material. 
 The area to resist shearing of cotter is 
 
 2 S 
 . K 2P 
 •• ''^^ • (^> 
 
MACHINE DKSIGN. 
 
 The area to resist shearing of rod is 
 
 51 
 
 and 1= 
 
 2d,S 
 
 (b) 
 
 If the metal of rod and cotter are the same 
 
 2d,l = ^A 
 
 1 
 
 (55) 
 
 Great care should be taken in fitting cotters that 
 they may not bear on corners of hole and thus tear the 
 rod in two. 
 
 A cotter or pin subjected to alternate stresses in 
 opposite directions should have a factor of safety 
 double that otherwise allowed. 
 
 Adjustable cotters, used 
 for tightening joints or bear- 
 ings are usually accompanied 
 by a gib having a taper equal 
 and opposite to that of the 
 cotter. (Fig. 13) In design- 
 ing these for strength the 
 two can be regarded as re- 
 sisting shear together. 
 
 For shafting keys see 
 chapter on shafting. 
 
 EXAMPI^ES. 
 
 1. Design a knuckle joint for a soft steel boiler 
 stay, the pull on stay being 12000 lbs. and the factor 
 of safety, six. 
 
 2. Determine the diameter of a round cotter pin 
 for equal strength of rod and pin 
 
 3. A rod of wrought iron has keyed to it a piston 
 18 inches in diameter, by a cotter of machinery steel. 
 
52 MACHINE DESIGN. 
 
 Required the two diameters of rod and dimensions 
 of cotter to sustain a pressure of 1 50 pounds per square 
 inch on the piston. Factor of safety = 8. 
 
 4. Design a cotter and gib for connecting rod of 
 engine mentioned in Ex. 3, both to be of machinery 
 steel and ,75 inches thick. 
 
SLIDING BEARINGS^ 
 
 26. Slides in General. The surfaces of all slides 
 should have sufficient area to limit the intensity of 
 pressure and prevent forcing out of the lubricant. No 
 general rule can be given for the Hmit of pressure. 
 Tool marks parallel to the sliding motion should not 
 be allowed, as they tend to start grooving. The 
 sliding piece should be as long as practicable to avoid 
 local wear on stationary piece and for the same reason 
 should have sufficient stiffness to prevent springing. 
 A slide which is in continuous motion should lap over 
 the guides at the ends of stroke, to prevent the wear- 
 ing of shoulders on the latter and the finished surfaces 
 of all slides should have exactly the same width as the 
 surfaces on which they move for a similar reason. 
 
 Where there are two parallel guides to motion as 
 in a lathe or planer it is better to have but one of these 
 depended upon as an accurate guide and to use the 
 other merely as a support. It must be remembered that 
 any sliding bearing is but a copy of the ways of the 
 machine on which it was planed or ground and in turn 
 may reproduce these same errors in other machines. 
 The interposition of handscraping is the only cure for 
 these hereditary complaints. 
 
 In designing a slide one must consider whether it 
 is accuracy of motion that is sought, as in the ways of 
 a planer or lathe, or accuracy of position as in the 
 head of a milling machine. Slides may be divided ac- 
 cording to their shapes into angular, flat and circular 
 slides. 
 
 27. Angular Slides, An angular slide is one in 
 which the guiding surface is not normal to the direc- 
 tion of pressure. There is a tendency to displacement 
 sideways, which necessitates a second guiding surface 
 inclined to the first. This oblique pressure constitutes 
 
54 
 
 MACHINK DKSIGN. 
 
 the principal disadvantage of angular slides. Their 
 principal advantage is the fact that they are either self 
 adjusting for wear, as in the ways of lathes and 
 planers, or require at most but one adjustment. 
 
 Fig. 14 shows one of the V^s of an ordinary planing 
 machine. The platen is held in place by gravity. 
 The angle between the two surfaces is usually 90° but 
 may be more in heavy machines. The grooves g, g are 
 intended to hold the oil in 
 place ; oiling is sometimes 
 effected by small rolls re- 
 cessed into the lower piece 
 and held against the platen 
 by springs. 
 
 The principal advantage of 
 this form of way is its abil- 
 ity to hold oil and the great 
 disadvantage its faculty for 
 catching chips and dirt. 
 
 /Fig. 15 shows an inverted V such as is com- 
 mon on the ways of engine lathes. The angle is about 
 the same as in the preceding form but the top of the V 
 should be rounded as a precaution against nicks and 
 bruises. 
 
 The inverted V is pre- 
 ferred for lathes since it 
 will not catch dirf and 
 chips. It needs frequent 
 lubrication as the oil runs 
 off rapidly. Some lathe 
 carriages are provided 
 with extensions filled 
 with oily felt or waste to 
 protect the ways from 
 dirt and keep them wiped and oiled. Side pressure 
 tends to throw the carriage from the ways; this action 
 may be prevented by a heavy weight hung on the car- 
 riage or by gibbing the carriage at the back (See Fig. 20) . 
 
 Fig. 15 
 
MACHINE DESIGN. 55 
 
 The objection to this latter form of construction is the 
 fact that it is practically impossible to make and k^p 
 the two V' s and the gibbed slide all parallel. 
 
 28. Gibbed Slides. All shdes which are not 
 self-adjusting for wear must be provided with gibs 
 and adjusting screws. Fig. 16 shows the most com- 
 mon form as used in tool shdes for lathes and planing 
 machines. 
 
 The angle employed is 
 
 usually 60°; notice that 
 
 ^ Ss -pL the corners c c are clip- 
 
 \ /A'^^"^- 3 ped for strength and to 
 
 g" 
 
 avoid a corner bearing; 
 notice also the shape of 
 gib. It is better to have 
 p. j^ the points of screws 
 
 ^' ' coned to fit gib and nol 
 
 to have flat points fitting recesses in gib. The latter 
 form tends to spread joint apart by forcing gib down. 
 If the gib is too thin it will spring under the screws and 
 cause uneven wear. 
 
 The cast iron gib, Fig. 
 17, is free from this latter 
 defect but makes the slide 
 rather clumsy. The screws 
 however are more accessi- 
 ble in this form. Gibs are 
 sometimes made slightly ta 
 pering and adjusted by a screw Fig. 17. 
 
 and nut giving endwise motion. 
 
 29. Flat Slides. This type of slide requires ad- 
 justment in two directions and is usually provided 
 with gibs and adjusting screws. Flat ways on ma- 
 chine tools are the rule in English practice and are 
 gradually coming into use in this country. Although 
 more expensive at first and not so simple they are 
 more durable and usually more accurate than the an- 
 gular ways. 
 
 
 — ?ri 
 
 1 t 
 
 \ 
 
 M 
 
MACHINK DESIGN. 
 
 a flat way for a planing ma- 
 
 _5 
 
 S6 
 
 Fig. 1 8 illustrates 
 chine. The other 
 way would be simi- 
 lar to this but with- 
 out adjustment. 
 The normal pres- 
 sure and the fric- 
 tion are less than 
 with angular 
 ways and no 
 amount of side 
 pressure will lift 
 the platen from 
 its position. 
 
 Fig. 19 shows a portion of the ram of a shaping 
 machine and illustrates the use of an L gib for adjust- 
 
 ^_^ . ,, — . ment in two di- 
 
 llP , I rections. Fig. 20 
 
 shows a g i b b e d 
 slide for holding 
 down the back of a 
 lathe carriage with 
 two adjustments. 
 
 The gib g is 
 tapered and adjust- 
 ed by a screw and 
 nuts. The saddle of 
 a planing machine 
 the table of a shaper usually has a rectangu- 
 lar gibbed slide above and a 
 taper slide below, this form 
 of the upper slide being ne- 
 cessary to hold the weight 
 of the overhanging metal. 
 Some lathes and planers are 
 built with one V or angular 
 way for guiding the carriage j 
 or platen and one flat wa}^ ^ 
 acting merely as a support. 
 
 or 
 
 u 
 
 9) 
 
 9 
 
 TXJ 
 
 Fig. 20. 
 
MACHINE DKSIGN. 
 
 57 
 
 30. Circular Guides. Examples of this form may 
 be found in the column of the drill press and the over- 
 hanging arm of the milling machine. The cross heads 
 of steam engines are sometimes fitted with circular 
 guides; they are more frequently flat or angular. One 
 advantage of the circular form is the fact that the cross 
 head can adjust itself to bring the wrist pin parallel to 
 the crank pin. The guides can be bored at the same 
 setting as the cylinder in small engines and thus se- 
 cure good alignment. 
 
 31, Stuffing Boxes. In steam engines and pumps 
 the glands for holding the steam and water packing 
 are the sliding bearings which cause the greatest fric- 
 tion and the most trouble. Fig. 2 1 shows the general 
 
 arrangement. 
 B is the stufl&ng 
 box attached to 
 
 the cylinder 
 head; R is the 
 piston rod ; G ^ 
 
 the gland ad- 4 
 justed by nuts 
 
 on the studs 
 shown ; P the 
 
 packing con- 
 tained in a re- 
 cess in the box 
 and consisting 
 of rings, either of some elastic fibrous material like 
 hemp and woven rubber cloth or of some soft metal 
 like babbit. The pressure between the packing and 
 the rod, necessary to prevent leakage of steam or wa- 
 ter, is the cause of considerable friction and lost work. 
 Experiments made from time to time in the labora- 
 tories of the Case School of Applied Science have 
 shown the extent and manner of variation of this fric- 
 tion. The results for steam packings may be sum- 
 marized as follows : 
 
 I. That the softer rubber and graphite packings, 
 
 Fig. 21 
 
58 MACHINE DESIGN. 
 
 which are self-adjusting and self-lubricating, as inNos. 
 2, 3, 7, 8, and 1 1, consume less power than the harder 
 varieties. No. 17, the old braided flax style, gave very 
 good results, 
 
 2. That oiling the rod will reduce the friction with 
 any packing. 
 
 3. That there is almost no limit to the loss caused 
 by the injudicious use of the monkey-wrench. 
 
 4. That the power loss varies almost directly with 
 the steam pressure in the harder varieties, while it is 
 approximately constant with the softer kinds. 
 
 The diameter of rod used — two inches— would be 
 appropriate for engines from 50 to 100 horse-power. 
 The piston speed was about 140 feet per minute in the 
 experiments, and the horse-power varied from .036 to 
 .400 at 50 pounds steam pressure, with a safe average 
 for the softer class of packings of .07 horse-power. 
 
 At a piston speed of 600 feet per minute, the same 
 friction would give a loss of from .154 to 1.71 with a 
 working average of .30 horse-power, at a mean steam 
 pressure of 50 pounds. 
 
 In Table 12 Nos. 6, 14, 15 and 16 are square, 
 hard rubber packings without lubricants. 
 
 Similar experiments on hydraulic packings under 
 a water pressure varying from ten to eighty pounds 
 per square inch gave results as shown in Table 14. 
 
 The figures given are for a two inch rod running 
 at an overage piston speed of 140 feet per minute. 
 
machine; design. 
 
 59 
 
 TABLE XII. 
 
 ^ be 
 
 1—i 
 
 
 ^g^g 
 
 , , 4> 
 
 
 o q 
 
 'u 
 
 H S-^ 
 
 oUr^pq 
 
 ^^^ 
 
 
 
 6 
 
 
 
 (A en . 
 
 Remarks on Leakage, etc. 
 
 I 
 
 5 
 
 22 
 
 .091 
 
 .085 
 
 Moderate leakage. 
 
 2 
 
 8 
 
 40 
 
 .049 
 
 .048 
 
 Easily adjusted; slight leakage. 
 
 3 
 
 5 
 
 25 
 
 •037 
 
 .036 
 
 Considerable leakage. 
 
 4 
 
 5 
 
 25 
 
 •159 
 
 .176 
 
 lycaked badly. 
 
 5 
 
 5 
 
 25 
 
 .095 
 
 .081 
 
 Oiling necessary; leaked badly. 
 
 6 
 
 5 
 
 25 
 
 .368 
 
 .400 
 
 Moderate leakage. 
 
 7 
 
 5 
 
 25 
 
 .067 
 
 .067 
 
 Easily adjusted and no Tkage. 
 
 8 
 
 5 
 
 25 
 
 .082 
 
 .082 
 
 Very satisfactory; slight I'kage. 
 
 9 
 
 3 
 
 15 
 
 .200 
 
 .182 
 
 Moderate leakage. 
 
 lO 
 
 3 
 
 
 .275 
 
 . 
 
 Excessive leakage. 
 
 II 
 
 5 
 
 25 
 
 .157 
 
 .172 
 
 Moderate leakage. 
 
 12 
 
 5 
 
 25 
 
 .266 
 
 •330 
 
 Moderate leakage. 
 
 13 
 
 5 
 
 25 
 
 .162 
 
 .230 
 
 No leakage; oiling necessary. 
 
 14 
 
 5 
 
 25 
 
 .176 
 
 .276 
 
 Moderate I'kage; oiling neces. 
 
 15 
 
 5 
 
 25 
 
 •233 
 
 .255 
 
 Difficult to adjust; no leakage. 
 
 i6 
 
 5 
 
 25 
 
 .292 
 
 .210 
 
 Oiling necessary; no leakage. 
 
 17 
 
 5 
 
 25 
 
 .128 
 
 .084 
 
 No leakage. 
 
 TABLE XIII. 
 
 Kind 
 of 
 
 Pack- 
 ing. 
 
 Horse Power consumed by each Box, 
 when Pressure was applied to Gland 
 Nuts by a Seven - Inch Wrench. 
 
 Horse Power 
 
 before and 
 
 after oiling 
 
 Rod. 
 
 
 5 
 
 Pounds 
 
 8 
 
 Pounds 
 
 10 
 
 Pounds 
 
 12 
 
 Pounds 
 
 14 
 
 Pounds 
 
 16 
 Pounds 
 
 Dry. 
 
 Oiled. 
 
 I 
 
 3 
 4 
 5 
 6 
 
 9 
 
 II 
 12 
 
 13 
 15 
 16 
 
 17 
 
 .120 
 
 .248 
 .220 
 .348 
 .126 
 
 .363 
 .6b6 
 
 .405 
 .161 
 
 .317 
 .526 
 
 .327 
 
 .198 
 
 .136 
 
 •430 
 .228 
 .500 
 
 •454 
 .242 
 
 •394 
 
 *.*86o 
 
 .277 
 
 •303 
 
 .260 
 .535 
 
 •359 
 .582 
 
 .380 
 
 •330 
 .520 
 
 .454 
 
 •390 
 
 .340 
 •533 
 
 .055 
 .154 
 
 .323 
 .067 
 
 •533 
 .666 
 
 .454 
 .454 
 
 .021 
 
 .123 
 
 .194 
 
 •053 
 ,236 
 .636 
 .176 
 .122 
 
6o 
 
 MACHINE 
 
 DESIGN 
 
 • 
 
 
 
 TABLE XiV. 
 
 
 
 No. of 
 Packing. 
 
 Av. H. P. 
 
 at 
 2o Lbs. 
 
 Av. H. P. 
 
 at 
 
 70 Lbs. 
 
 Max. 
 H. P. 
 
 Min. 
 H. P. 
 
 Av. H. P. 
 
 for entire 
 
 Test. 
 
 I 
 
 .077 
 
 .351 
 
 •452 
 
 .024 
 
 ■259 
 
 2 
 
 .422 
 
 .500 
 
 
 512 
 
 .167 
 
 .410 
 
 3 
 
 .130 
 
 .178 
 
 
 276 
 
 •035 
 
 .120 
 
 4 
 
 .184 
 
 .195 
 
 
 230 
 
 .142 
 
 .188 
 
 5 
 
 .146 
 
 .162 
 
 
 285 
 
 .069 
 
 .158 
 
 6 
 
 .240 
 
 .200 
 
 
 255 
 
 .071 
 
 .186 
 
 7 
 
 .127 
 
 .192 
 
 
 213 
 
 •095 
 
 •154 
 
 8 
 
 .153 
 
 .174 
 
 
 238 
 
 .112 
 
 .165 
 
 9 
 
 .287 
 
 .469 
 
 
 535 
 
 •159 
 
 •389 
 
 lO 
 
 .151 
 
 .160 
 
 
 226 
 
 •035 
 
 .103 
 
 ^ II 
 
 .141 
 
 156 
 
 
 380 
 
 .064 
 
 .177 
 
 12 
 
 .053 
 
 •095 
 
 
 •143 
 
 •035 
 
 .090 
 
 Packings Nos. 5, 6, 10 and 12 are braided flax 
 with graphite lubrication and are best adapted for low 
 pressures. Packings Nos. 3, 4 and 7 are similar to the 
 above but have parafine lubrication. Packings Nos. 
 2 and 9 are square duck without lubricant and are on- 
 ly suitable for very high pressures, the friction loss 
 being approximately constant. 
 
 EXAMPLES. 
 Make a careful study and sketch of the sliding 
 bearings on each of the following machines and ana- 
 lyze as to: (a) Purpose (b) Character, (c) Adjust- 
 ment, (d) Lubrication . 
 
 1 . One of the engine lathes in the shop. 
 
 2. One of the planing machines. 
 
 3. One of the shaping machines. 
 
 4. One of the milling machines. 
 
 5. One of the upright drills. 
 
 6. One of the engines. 
 
®Jtai:tt^r 7* 
 
 JOURNALS, PIVOTS AND BEARINGS. 
 
 32. Journals. A journal is that part of a rotating 
 shaft which rests in the bearings and is of necessity a 
 surface of revolution, usually cylindrical or conical. 
 The material of the journal is generally steel, some- 
 times soft and sometimes hardened and ground. 
 
 The material of the bearing should be softer than 
 the journal and of such a quality as to hold oil readily. 
 The cast metals such as cast iron, bronze and babbitt 
 metal are suitable on account of their porous, granu- 
 lar character. Wood, having the grain normal to the 
 bearing surface, is used where water is the lubricant, 
 as in water wheel steps and stern bearings of propel- 
 lers, 
 
 33. Adjustment. Bearings wear more or less rap- 
 idly with use and need to be adjusted to compensate 
 for the wear. The adjustment must be of such a char« 
 acter and in such a direction as to take up the wear 
 and at the same time maintain as far as possible the 
 correct shape of the bearing. The adjustment should 
 then be in the line of the greatest pressure. 
 
 Fig. 22 illustrates 
 some of the more com- 
 mon ways of adjusting 
 a bearing, the arrows 
 showing the direction 
 of adjustment and pre- 
 sumably the direction 
 of pressure, (a) is the 
 most usual where the 
 principal wear is ver- 
 tical, (d) is a form 
 frequently used on 
 the main journals of 
 
62 
 
 MACHINE DKSIGisr. 
 
 engines when the wear is in two directions, horizontal 
 on account of the steam pressure and vertical on ac- 
 count of the weight of shaft and fly wheel. All of 
 these are more or less imperfect since the bearing, af- 
 ter wear and adjustment, is no longer cylindrical but 
 is made up of two or more approximately cylindrical 
 surfaces. 
 
 A bearing slightly conical and adjusted endwise 
 as it wears, is probably the closest approximation to 
 correct practice. 
 
 Fig. 23 shows the 
 main bearing of the 
 Porter - Allen en- 
 gine, one of the 
 best examples of a 
 four part adjust- 
 ment. The cap, is 
 adjusted in the nor- 
 mal way with bolts 
 and nuts; the 
 bottom, can be 
 raised and lowered 
 the cheeks can be 
 
 Fig. 
 
 23. 
 underneath 
 
 by liners placed 
 moved in or out by means of the wedges shown. 
 Thus it is possible, not only to adjust the bearing for 
 wear, but to align the shaft perfectly. 
 
 The main bearing of the spindle in a lathe, as 
 shown in Fig. 24, offers a good example of symmetri- 
 cal adjustment. The headstock A has a conical hole 
 to receive the bearing B, 
 which latter can be moved 
 lengthwise by the nuts F G. 
 The bearing may be split 
 into two, three or four seg- 
 ments or it may be cut as 
 shown in (e) Fig. 22 and 
 sprung into adjustment. A 
 careful distinction must be 
 made between this class of 
 bearing and that before 
 
MACHINE DESIGN, 63 
 
 mentioned, where the journal itself is conical and ad- 
 justed endwise. A good example of the latter form is 
 seen in the spindles of many milling machines. 
 
 34 Lubrication. The bearings of machines which 
 run intermittently, like most machine tools, are oiled 
 by means of simple oil holes, but machinery which is 
 in continuous motion as is the case with line shafting 
 and engines requires some automatic system of lu- 
 brication. There is not space in these notes for a de- 
 tailed description of all the various types of oiling de- 
 vices and only a general classification will be at- 
 tempted. 
 
 lyubrication is effected in the following ways: 
 
 1 . By grease cups. 
 
 2. By oil cups. 
 
 3. By oily pads of felt or waste. 
 
 4. By oil wells with rings or chains for lifting 
 the oil. 
 
 5. By centrifugal force through a hole in the 
 journal itself. 
 
 Grease cups have little to recommend them except 
 as auxiliary safety devices. Oil cups are various in 
 their shapes and methods of operation and constitute 
 the chief class of lubricating devices. They may be 
 divided according to their operation into wick oilers, 
 needle feed, and sight feed. The two first mentioned 
 are nearly obsolete and the sight feed oil cup, which 
 drops the oil at regular intervals through a glass tube 
 in plain sight, is in common use. The best sight feed 
 oiler is that which can be readily adjusted as to time 
 intervals, which can be turned on or off without dis- 
 turbing the adjustment and which shows clearly by its 
 appearance whether it is turned on. On engines and 
 electric machinery which is in continuous use day and 
 night, it is very important that the oiler itself should 
 be stationary, so that it may be filled without stopping 
 the machinery. 
 
 For continuous oiling of stationary bearings as in 
 line shafting and electric machinery, an oil well below 
 
64 
 
 MACHINE DESIG 
 
 the bearing is preferred, with some automatic means 
 of pumping the oil over the bearing, when it runs 
 back by gravity into the well. Porous wicks and pads 
 acting by capillary attraction are un- 
 certain in their action and liable to 
 become clogged. For bearings of 
 medium size, one or more light steel 
 rings running loose on the shaft 
 and dipping into the oil, as shown 
 in Fig. 25, are the best. For large 
 bearings flexible chains are employed 
 which take up less room than the ring. 
 Centrifugal oilers are most used on 
 parts which cannot readily be oiled 
 when in motion, such as loose pulleys and the crank 
 pins of engines. 
 
 Fig. 26 shows two such devices as applied to an 
 engine. In A the oil is supplied by the waste from the 
 main journal; in B an external sight feed oil cup is 
 used which supplies oil to the central revolving cup C. 
 
 ^ 
 
 Fig. 25 
 
 
 
 ;L... . 
 
 B 
 
 
 
 
 
 
 c 
 
 Fig. 26. 
 
 Ivoose pulleys or pulleys running on stationary 
 studs are best oiled from a hole running along the axis 
 of the shaft and thence out radially to the surface of 
 the bearing. A loose bushing of some soft metal per- 
 forated with holes is a good safety device for loose 
 pulleys. 
 
 Note: For adjustable pedestal and hanger bearings 
 see the chapter on shafting. 
 
MACHINK DESIGN. 65 
 
 35. Friction of Journals: 
 
 Let W=the total load on a journal in lbs. 
 l=the length of journal in inches. 
 d=the diameter of journal in inches. 
 N= number of revolutions per minute. 
 v= velocity of rubbing in feet per minute. 
 F= friction at surface of journal in lbs. 
 = W tan (p nearly. 
 If a journal is properly fitted in its bearing and 
 does not bind, the value of F will not exceed W tan <p 
 and may be slightly less. The value of tan ^ varies ac- 
 cording to the materials used and the kind of lubrica- 
 tion, from .05 to .01 or even less. The work absorbed 
 in friction may be thus expressed : 
 
 ^ _^^^ TrdN TrdNWtans^ , ^. 
 
 Fv=Wtan^x = (56) 
 
 12 12 
 
 36. Limits of Pressure. Too great an intensity of 
 pressure between the surface of a journal and its bear- 
 ing will force out the lubricant and cause heating and 
 possibly ''siezing". The safe limit of pressure de- 
 pends on the kind of lubricant, the manner of its ap- 
 plication and upon whether the pressure is continuous 
 or intermittent. The projected area of a journal, or 
 the product of its length by its diameter, is used as a 
 divisor. 
 
 The journals of railway cars offer a good example 
 of continuous pressure and severe service. A limit of 
 300 lbs. per square inch of projected area has been 
 generally adopted in such cases. 
 
 In the crank and wrist pins of engines, the rever- 
 sal of pressure diminishes the chances of the lubricant 
 being squeezed out, and a pressure of 500 lbs. per sq. 
 in. is generally allowed. 
 
 The use of heavy oils or of an oil bath, and the 
 employment of harder materials for the journal and 
 its bearing allow of even greater pressures. 
 
 37. Heating of Journals. The proper length of 
 journals depends on the liability of heating. 
 
66 MACHINE DESIGN. 
 
 The energy or work expended in overcoming fric- 
 tion is converted into heat and must be conveyed away 
 by the material of the rubbing surfaces. If the ratio 
 of this energy to the area of the surface exceeds a cer- 
 tain Hmit, depending on circumstances, the heat will 
 not be conveyed away with sufficient rapidity and the 
 bearing will heat. 
 
 The area of the rubbing surface is proportional to 
 the projected area or product of the length and diame- 
 ter of the journal, and it is this latter area which is 
 used in calculation. 
 
 Adopting the same notation as is used in Art. 35, 
 we have from equation (56) 
 
 the work of friction = . ft. lbs, 
 
 12 
 
 or = :7rdNWtan ^ inch lbs. 
 The work per square inch of projected area is 
 then: TidNWtanc^' TrNWtan^ , . 
 
 ^= — id — = — I — ^^> 
 
 Solving in (a) for 1 
 
 TcNWtan^ ... 
 
 1= (b) 
 
 w 
 
 w 
 
 Let— =:C a co-efficient whose value is to be ob- 
 
 TTtan^ 
 
 tained by experiment; then 
 
 C = -^andl-=-^ (57) 
 
 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 350000. 
 
 A similar average for the crank pins of thirteen 
 screw steamers in the French Navy gave C= 400000. 
 
 Locomotive crank pins which are in rapid motion 
 through the cool outside air allow a much larger value 
 of C, sometimes more than a million. 
 
MACHINE DKSIGN. 67 
 
 Examination of ten modern stationary engines 
 shows an average value of C= 200000 and an average 
 pressure per square inch of projected area =300 lbs. 
 
 In general we may use these formulas for station- 
 ary practice : 
 
 To prevent heating: 1= (s8) 
 
 ^ ^ 200000 ^^ ^ 
 
 W 
 To prevent wear ld= (59) 
 
 38. Strength and Stiffness of Journals. A jour- 
 nal is usually in the condition of a bracket with a uni- 
 
 Wl 
 form load, and the bending moment M = — 
 
 Therefore by formula (6) 
 
 Wl 
 
 " --^T- w^ 
 
 A 3 fWl ,^ , 
 
 or d=i.72i' 1-^ (60) 
 
 The maximum deflection of such a bracket is 
 
 /\ = 
 
 8EI 
 
 ;rd' WV 
 
 64 8EA 
 64WP ^ 2.547WP 
 
 sttEa" ea 
 
 If as is usual A is allowed to be ih inches, then 
 
 for stiffness d=^ J?^^^^ (6i) 
 
 or approximately d=4 ^=:- (6.?) 
 
68 MACHINE DESIGN. 
 
 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 use all three of the formulas and take the largest 
 result. 
 
 Remember 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. 
 
 39. Caps and Bolts. The cap of a journal bearing 
 is in the condition of a beam supported by the holding 
 down bolts and loaded at the center, and may be de- 
 signed 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. 
 
 Qf .;. AT Sbh^ PL 
 Strength: M = — - — == — 
 
 Stiffness : A = 
 
 -J 
 
 2bS 
 WL' 
 
 (63) 
 
 48EI 
 _ bh^_ WL^ 
 
 12 48EA 
 
 '--'^^^r^ (64) 
 
 WL' 
 
 4bEA 
 
 If A is allowed to be xio inches and E for cast 
 iron is taken = 18000000- 
 
 then: h = .oiii5L'J^ (65) 
 
 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. 
 
MACHINE DESIGN. 69 
 
 exampi.es. 
 
 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=io and 0=^300000. 
 
 2. Measure the crank pin of any modern engine 
 which is accessible, calculate the various constants and 
 compare them with those given in this section. 
 
 3. Design a crank pin for an engine under the 
 following conditions : 
 
 Diameter of piston = 28 inches. 
 
 Maximum steam pressure = 90 lbs. per sq. in. 
 Mean steam pressure = 40 lbs. per sq. in. 
 Revolutions per minute = 75 
 
 Determine dimensions necessary to prevent wear 
 and heating and then test for strength and stiffness. 
 
 4. Make a careful study and sketch of journals 
 and journal bearings on each of the following 
 machines and analyze as to (a) Materials, (b) Ad- 
 justment, (c) lyubrication. 
 
 (i) One of the engine lathes in the shop. 
 
 (2) One of the milling machines. 
 
 (3) One of the steam engines. 
 
 (4) One of the electric generators. 
 
 5. Sketch at least one form of oil cup used in the 
 laboratories and explain its working. 
 
 6. The shaft journal of a vertical engine is 4 ins. 
 in diameter by 6 ins. long. The cap is of cast iron, 
 held down by 4 bolts of wTought iron, each 5 ins. 
 from center of shaft, and the greatest vertical pressure 
 is 12000 lbs. 
 
 Calculate depth of cap at center for both strength 
 and stiffness, and also the diameter of bolts. 
 
 7. Investigate the strength of the cap and bolts of 
 
70 MACHINE DESIGN. 
 
 some pillow block whose dimensions are known, under 
 a pressure of 500 lbs. per sq. in. of projected area. 
 
 8. The total weight on the drivers of a locomotive 
 is 64000 lbs. The drivers are four in number, 5 ft. 2 
 in. in diameter, and have journals 7}^ in. in diameter. 
 
 Determine the horse power consumed in friction 
 under each of the three above named conditions, when 
 the locomotive is running 50 miles an hour, assuming 
 tan^=:i.o5 . 
 
 40. Friction of Pivots or Step -Bearings.— Flat 
 Pivots. 
 
 Let W=weight on pivot 
 
 dj=outer diameter of pivot 
 p= intensity of vertical pressure 
 M = moment of friction 
 f= co-efficient of friction = tan ^ 
 
 We will assume p to be a constant which is no 
 doubt approximately true. 
 
 W 4W 
 
 Then p= 
 
 area TrdJ 
 
 Let r= the radius of any elementary ring of a 
 width = dr , then area of element=27rrdr 
 
 Friction on element = fp X 27rrdr 
 Moment of friction of element =2fp7rrMr 
 
 -.1 
 
 and M = 2fp7r f— ^'^^ ^^) 
 
 o 
 
 r^ d' 
 
 or M = 2fp7r =2fp7r — ^ 
 
 3 24 
 
 2f^d? 4W I .^^r. .^^. 
 
 ' X ^=--Wfd, (66) 
 
 24 ^d: 
 
 The great objection to this form of pivot is the 
 uneven wear due to the difference in velocity between 
 center and circumference. 
 
MACHINE DESIGN. 7 1 
 
 /It. Flat Collar. 
 
 lyet d2= inside diameter 
 Integrating as in equation (a) above, but using 
 
 limits 
 
 and 
 
 
 
 — ^and — ^ we have 
 
 2 2 
 
 
 
 M= 
 
 , dl-dl 
 =2fp" 2 4 
 
 this 
 
 case 
 
 
 
 
 P = - 
 
 4W 
 
 
 '(d^-d|) 
 
 
 M= 
 
 = ^Wf 
 
 d^-d^ 
 
 d?-d^ 
 
 (67) 
 
 42. Conical Pivot. 
 
 Let a = angle of inclination to the vertical. 
 
 M-p ' rA 
 
 
 X 
 
 \ \ I: 
 
 \ 
 
 ^ — 
 
 ■v 
 
 -^ 
 
 dW= 
 
 Fig. 27. 
 
 4W 
 -(d?-d^) 
 
 As in the case of a 
 flat ring the intensity 
 of the vertical pres- 
 sure is 
 
 4W 
 
 Kd?-dD 
 
 and the vertical pres- 
 sure on an elementary 
 ring of the bearing 
 surface is 
 
 X 2 TT rdrizz 
 
 8Wrdr 
 
 dj~d 
 
 As seen in Fig. 27 the normal pressure on the 
 elementary ring is 
 
 dP= -^ _ 8Wrdr 
 
 sina (dj— d2)sina 
 
72 MACHINE DESIGN. 
 
 The friction on the ring is fdP and the moment of 
 this friction is 
 
 ,-- , ,_ SWf rMr 
 dM=frdP^ ^^,_^.^^.^^ 
 
 M= ,./^!. PrMr 
 
 (df-d^)sinaJ d 
 
 2 
 
 sma dj— A\ 
 
 TT 
 
 As a approaches — the value of M approaches that of 
 
 a flat ring, and as a approaches o the value of M ap- 
 proaches 00 . 
 
 If d2 =o we have 
 
 Ayr T/ Wfd ,^ , 
 
 M=^-^ (69) 
 
 sina 
 
 The conical pivot also wears unevenly, ucually 
 assuming a concave shape as seen in profile. 
 
 43. 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. 29, 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- ,5 
 
 chanically as shown in Fig. 28. I 
 
 Let the weight W be free to j 
 
 move on a plane. Let the string ; 
 
 SW be kept taut and the end S • 
 
 moved along the straight line SL. j 
 
 Then will a pencil attached to the 1 
 
 center of W trace on the plane a ^ L 
 tractrix whose axis is SL. Fig- ^8. 
 
 o 
 
 w 
 
MACHINE DESIGN, 
 
 73 
 
 
 
 r, - 
 
 
 — >1 
 
 s 
 
 *r_.~j/p 
 
 /^ 
 
 --w 
 
 
 / d 
 
 w 
 
 > 
 
 "~d 
 
 Q 
 
 
 
 
 L 
 
 
 
 
 
 In Fig. 29 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 
 
 = 1*1 
 
 r 
 
 Also 
 
 sin ^' 
 
 Let a pivot be generated 
 by revolving the curve around 
 its axis SL. As in the case ol 
 the conical pivot it can be 
 ^^S' 29. proved that the normal pres- 
 
 sure on an element of convex surface is 
 
 dP= 
 
 8Wrdr 
 
 (d?-d^)sin^ 
 
 (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, /. e, normal wear 
 proportional to pr, then is the vertical wear 
 
 proportional to '^ 
 
 But- 
 
 is a constant, there- 
 
 sin<? * sin<? 
 
 fore 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) 
 
 8Wr,dr 
 
 dP: 
 
 dM: 
 
 8Wfr,rdr 
 
 and 
 
 M = 
 
 8Wfr, rl-vl _ Wfd, 
 
 d^-d 
 
 (70) 
 
 M is thus shown to be independent of d^ or of the 
 
 length of pivot used. 
 
74 MACHINE DESIGN. 
 
 This pivot is sometimes wrongly called anti- 
 friction. As will be seen by comparing equations (66) 
 and (70) 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 manu- 
 facture and is seldom used on that account. 
 
 It is not suitable for a bearing where most of the 
 pressure is side ways. 
 
 44. 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. 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 mak- 
 ing the collars unreasonably large. 
 
 A safe value for p the intensity of pressure is, ac- 
 cording to Whitham, 60 lbs. per sq. in. for high speed 
 engines. 
 
 A table given by Prof. Jones in his book on 
 Machine Design shows the practice at the Newport 
 News ship-yards on 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 
 
MACHINE DESIGN. 75 
 
 the smallest engine to ii in the largest. The pressure 
 per sq. in. of bearing surface varies from 1 8 to 46 lbs. 
 with an average value of about 32 lbs. 
 
 EXAMPLES. 
 
 1 . Design and draw to full size a Schiele pivot for 
 a water wheel shaft 4 inches in diameter, the total 
 length of the bearing 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 f = 
 .25 for metal on wet vvood. 
 
 2 . Design a compound thrust bearing for a pro- 
 peller shaft the diameters being 14 and 21 inches, the 
 total thrust being 80000 lbs. and the pressure 40 lbs. 
 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 f=.o5 and revs, 
 per min. = 120. 
 
BALL AND ROLLER BEARINGS. 
 
 45. General Principles. The object of interpos- 
 ing a ball or roller between a journal and its bearing, 
 is to substitute rolling for sliding friction and thus to 
 reduce 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. 
 
 46. 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 by 
 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. 
 
 Fig. 30 shows a bearing of this type, there being 
 three points of contact and the axis of ball being par- 
 
 allel to that of journal 
 
MACHINE DESIGN. 
 
 77 
 
 The bearings so 
 far mentioned have 
 no means of ad- 
 justment for wear, 
 onical bearings, 
 or those in which 
 the axes of the 
 ^. balls meet in a 
 
 ^^S' 30. common point, 
 
 supply this deficiency. In designing this class of bear- 
 ings, either for side or end thrust, the inclination of 
 the axis is assumed according to the obUquity desired 
 and the points of contact are then so located that there 
 shall be no slipping. 
 
 Fig. 31 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 between 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 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 
 theae circles have the same ratio as the radii of revo- 
 lution an, b m , there will be no slipping and the ball 
 
 will roll as a cone inside another cone. The exact 
 cation of the third point of contact is not material. 
 it were at c, too much 
 pressure would come on 
 the cup at b ; if at d 
 there would be an ex- 
 cess of pressure at a but 
 the rolling would be cor- 
 rect in either case. A 
 convenient method is to 
 locate p by drawing A D 
 tangent to ball circle as 
 shown. It is recom- 
 mended however that 
 the two opposing sur- 
 
 lo- 
 If 
 
 t--€^ -^ 
 
 3, 
 
78 MACHINE DESIGN. 
 
 faces at p and b or a should make with each other an 
 angle of not less than 25"^ 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 
 concave conoids with a radius about three fourths the 
 diameter of the ball. See Fig. 32. 
 
 47. Step -Bearings. 
 
 The same principles ap- 
 ply as in the preceding 
 article and the axis and 
 points of contact may 
 be varied in the same^- 
 way. The most com- m 
 mon form of step- i 
 bearing consists of two 
 flat circular plates sep- 
 arated 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 sHding friction. This is a bearing 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 of the balls 
 against each other will cause considerable friction. 
 
 To guide the balls without rubbing friction three 
 point contact is generally used. 
 
 Fig. 33 illustrates a bearing of this character. 
 The method of design is shown in the figure the prin- 
 ciple being the same as in Fig. 31. By comparing the 
 lettering of the two figures the similarity will be read- 
 ily seen. 
 
MACHINE DESIGN 
 
 79 
 
 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. 
 
 This method has 
 the advantage of 
 making each ball 
 move in a path of 
 different radius 
 thus securing more 
 even wear for the 
 plates. 
 
 48. 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 pe- 
 culiarities. 
 
 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 in- 
 dent 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 
 
8o MACHINE DESIGN. 
 
 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 beaKing similar to that in Fig. 33 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- 
 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. 
 
 49. 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 }^ inch 
 and ^ inch balls are about 7500 lbs. and 15000 lbs. 
 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 lbs. 
 
 Prof. Gray states, as a conclusion from some ex- 
 periments made by him, that not more than 40 lbs. per 
 ball should be allowed for Y^ 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. 
 
 50. Roller Bearings. The principal disadvantage 
 of ball bearings lies in the fact that contact is only at 
 a point and that even moderate pressure causes exces- 
 sive distortion and wear. The substitution of cylin- 
 ders or cones for the balls is intended to overcome this 
 difficulty. 
 
MACHINE DESIGN. 8 1 
 
 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 
 difl&culties here to be overcome. The rollers tend 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 prop- 
 erly unless guided and any form of guide entails sliding 
 friction. The cage generally used is a cylindrical 
 sleeve having longitudinal slots which hold the rollers 
 loosely and prevent their getting out of place either 
 sideways or endways. 
 
 The use of balls between the rollers at the ends 
 has been tried with some degree of success. Large rol- 
 lers 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. 
 
 SI. 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 in form and comparatively rigid against com- 
 pression, but possesses sufficient flexibility to adapt 
 itself to slight changes of bearing surface. This bear- 
 ing is readily caged by running rods through the rol- 
 lers and riveting them to collars at the ends. 
 
 Experiments made by the Franklin Institute show 
 that the Hyatt roller possesses a great advantage in 
 eflSciency over the solid roller. 
 
 Testing ^ inch rollers between flat plates under 
 loads increasing to 550 lbs. 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. 
 
82 MACHINE DESIGN. 
 
 Subsequent examination of the plates showed also a 
 much more even distribution of pressure for the for- 
 mer. 
 
 52. Roller Step Bearings. In article 48 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 is also 
 conical. A successful bearing of this kind with short 
 cylinders in place of cones is used by the Sprague- 
 Pratt Elevator Co., and is described in the American 
 Machinist 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 
 lbs. per roll for rolls one-half inch in diameter and 
 one-quarter inch long. 
 
SHAFTING, COUPLINGS AND HANGERS. 
 
 53. Strength of Shafting. 
 
 I^et D = diameter of the driving pulley or gear. 
 N= number revs, per minute. 
 P= force applied at rim. 
 T=^ twisting moment. 
 
 The distance through which P acts in one minute 
 is ttDN inches and work— Pt^DN in. lbs. per min. 
 
 PD 
 
 But = T the moment, and 27rN= the angular 
 
 2 
 
 velocity. 
 
 . • . work = moment X angular velocity 
 One horse power = 33000 ft. lbs. per min. 
 = 396000 in. lbs. per min. 
 
 HP= ^^^^ _ 27rTN 
 
 396000 396000 
 TN 
 
 ^^=^5 ^''' 
 
 also T=^3-^ (^^^ 
 
 126050 HP 
 
 DN ^'■^^ 
 
 The general formula for a circular shaft exposed 
 to torsion alone is 
 
 where N = no. revs, per min. 
 
84 
 
 MACHINE DESIGN, 
 
 Substituting in formula for d 
 
 d = ^ I321000HP 
 
 SN 
 
 nearly (74) 
 
 It 
 ing as 
 
 S may be given the following values : 
 45000 for common turned shafting. 
 50000 for rolled iron or soft steel. (0.15 C) 
 65000 for machinery steel. (0.55 C) 
 
 is customary to use factors of safety for shaft- 
 follows: f 
 Headshafts or prime movers 15 
 lyine 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 in- 
 convenience and expense attendant on failure of so 
 important 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 valuoe of S and introducing fac- 
 tors of safety, we have the following formulas for the 
 safe diameters of the various kinds of shafts. 
 
 TABLE XV. 
 
 Kind of 
 
 Material. 
 
 Shaft. 
 
 Common Iron 
 
 Soft Steel 
 
 Machy. Steel 
 
 Head Shaft. 
 
 Line Shaft. 
 
 Counter 
 
 Shaft. 
 
 , (HP 
 3 HP 
 3 fHP 
 
 , HP 
 
 3 HP 
 
 3 fHP 
 
 ^ » HP 
 
 3 HP 
 
MACHINE DESIGN. 85 
 
 In case there is a known bending moment M, com- 
 bined with a known twisting moment T, then a re- 
 sultant twisting moment 
 
 T=M-f x/m^ + T^ 
 
 is to be substituted for T in the above formulas. 
 
 54. Couplings. The flange or plate coupling is 
 most commonly used for fastening together adjacent 
 lengths of shafting. 
 
 Fig. 34 shows the proportions of such a coupling. 
 The flanges are turned accurately on all sides, are 
 keyed to the shafts and the two are centered by the 
 projection of the shaft from one part into the other as 
 
 shown at A. The 
 bolts are turned 
 to fit the holes 
 loosely so as not 
 to interfere with 
 the alignment. 
 
 The projecting 
 rim as at B pre- 
 vents danger 
 from belts catch- 
 ing on the heads 
 and nuts of the 
 bolts. 
 
 ^sssij 
 
 Fig- 34" 
 
 The faces of this coupling should be trued up 
 in a lathe after being keyed to the shaft. 
 
 The sleeve coupling is less clumsy than the fore- 
 going but is rather more complicated and expensive. 
 
86 
 
 MACHINE DESIGN. 
 
 In Fig. 35 is illustrated a neat and efifective coup- 
 lino^ of this type. It consists of the sleeve S bored 
 with two tapers and two threaded ends as shown. 
 The two conical, split bushings B B 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 clamped firmly to the shaft. The key K also 
 nicks slightly the center of the main sleeve S, thus 
 locking the whole combination. 
 
 Fig. 35. 
 
 Couplings similar to this have been in use in the 
 Jnion Steel Screw Works, Cleveland, Ohio, for many 
 years and have given good satisfaction. 
 
 In another form of sleeve coupling the sleeve is 
 split and clamped to the shaft by bolts passing through 
 the two halves. 
 
 55. Coupling Bolts. The bolts used in the ordi- 
 nary flange couplings are exposed to shearing, and 
 their combined shearing moment should equal the 
 twisting moment on the shaft. 
 
 Let n=number of bolts. 
 di=diameter of bolt. 
 D= diameter ot bolt circle. 
 We will assume that the bolt has the same shear- 
 ing strength as the shaft. The combined shearing 
 strength of the bolt is .7854dinS and their moment of 
 resistance to shearing is 
 
 .7854d^nSx — =.3927Dd,'nS 
 
MACHINK DESIGN. 87 
 
 This last should equal the torsion moment of the 
 
 shaft or .3927Dd?nS= 
 
 5.1 
 
 Solving for d^ and assuming D = 3d as an average 
 
 value we have d^= — — •••(75) 
 
 V 6n 
 
 In practice rather larger values are used than 
 
 would be given by the formula. 
 
 56. 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. 
 i=length of key. 
 h= total depth of key. 
 S'= shearing strength of key. 
 
 The moment of shearing stress on key is 
 
 bis'xA=_Mis: 
 
 2 2 
 
 Q 1 3 J 
 
 and this must equal — Usually b= — 
 
 5-1 4 
 
 For shafts of machine steel S=S', and for iron 
 shafts S=^S' nearly as keys should always be of 
 steel. 
 
 Substituting these values and reducing : 
 
 For iron shafting l=i. 2d nearly. 
 
 For steel shafting l=i.6d nearly, as the least 
 lengths of key to prevent its fail- 
 ing by shear. 
 
 If the key way is to be de- 
 signed for uniform strength, the 
 shearing area of the shaft on the 
 line AB Fig. 35a should equal 
 the shearing area of the key, if 
 shaft and key are of the same 
 material and AB=CD=b. 
 
S8 MACHINE DESIGN. 
 
 These proportions will make the depth of key way 
 in shaft about = ^b and would be appropriate for a 
 square key. 
 
 To avoid sucii 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 ^b 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. 
 
 d Ih ^ Sd^ 
 or — X - — X Sc= (a) 
 
 2 2 ' 5.1 
 
 where Sc is the compressive strength of the key. 
 
 For iron shafts Sc =28 
 
 and for steel shafts Sc =— S 
 
 2 
 
 Substituting values of S© and assuming h=^b 
 = J 2 d we have 
 
 Iron shafts l=2.5d nearly 
 
 Steel shafts 1= 3 j^d nearly, as the least 
 
 length for flat keys to prevent lateral crushing. 
 
 57. Hangers and Boxes. As shafting is usually 
 hung to the ceiling and walls of buildings it is neces- 
 sary 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 perfectly true and straight, the bearings should 
 be to a certain extent self-adjusting. Reliable experi- 
 ments 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. 
 
MACHINE DESIGN, 
 
 89 
 
 (See Chapter 7.) The best method of supporting the 
 box in the hanger is by the ball and socket joint ; all 
 other contrivances such as set screws are but poor 
 substitutes. 
 
 Fig. 36 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 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. 
 
 Fig. 36. 
 
 Counter shafts are short and light and are not 
 subject to much bending. Consequently there is not 
 the same need of ad- 
 justment as in line 
 shafting. 
 
 In Fig, 37 is il- 
 lustrated a simple 
 bearing for counters. 
 The solid cast iron 
 box B with a spherical 
 center is fitted directly 
 in a socket in the 
 hanger H and held in 
 position by the cap C 
 and a set screw. Fig. 37. 
 
90 MACHINE DESIGNc 
 
 There is not space here to show the various forms 
 of hangers and floor stands and 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. 
 
 EXAMPLES. 
 
 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 : 
 Head shaft loo HP 200 revs, per min. 
 Machine shop 30 HP 120 revs, per min. 
 Pattern shop 50 HP 250 revs, per min. 
 Forge shop 20 HP 200 revs, per min. 
 
 2. Determine the horse power of at least two lines 
 of shafting whose speed and diameter 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 one shown in the text. 
 
 5. Select the line shaft hanger which you prefer 
 among those in the laboratories and make sketch and 
 description of the same. 
 
 6. Do. for a countershaft hanger. 
 
 7. Explain in what way a floor stand differs from 
 a hanger. 
 
GEARS, PULLEYS AND FLY WHEELS. 
 
 58. Gear Teeth. The teeth of gears are made in 
 three ways and are accordingly known as pattern 
 molded, machine molded and machine cut, the first 
 being the least accurate in form and the last the most 
 so. 
 
 Let circular pitch 
 diameter pitch 
 
 pitch number 
 
 addendum 
 
 flank 
 
 clearance 
 
 height 
 
 width 
 
 space 
 
 P 
 P-^d 
 
 TT 
 
 TT _ I 
 
 a 
 
 f 
 
 f— a==c 
 
 f+a==h 
 
 w 
 
 p— w=s 
 
 The following table gives the ordinary proportions 
 in use for the three kinds of teeth. 
 
 TABLE XVL — PROPORTIONS OF GEAR TEETH. 
 
 Kind of 
 
 Adden- 
 dum. 
 
 a 
 
 Flank. 
 
 Clearance 
 
 Height . 
 
 Width. 
 
 *paoe. 
 
 Tooth. 
 
 f 
 
 C 
 
 h 
 
 W 
 
 S 
 
 Pattern 
 Molded 
 
 •32 P 
 
 •38 P 
 
 .06 p 
 
 •7P 
 
 •47 P 
 
 •53 P 
 
 Machine 
 Molded 
 
 •32 P 
 
 •38 P 
 
 .06 p 
 
 •7P 
 
 .48 p 
 
 •52 p 
 
 Machine 
 Cut 
 
 d 
 
 ij^d 
 
 m 
 
 2Hd 
 
 •5P 
 
 •5P 
 
92 
 
 MACHINE DESIGN. 
 
 Ill making machine cut teeth some mechanics 
 prefer to make the clearance = 2V p which is slightly 
 more than }id. 
 
 The diameter pitch bears the same relation to the 
 diameter as the circular pitch does to the circumfer- 
 ence, and may be obtained by dividing the diameter 
 by the number of teeth. 
 
 Its value is evidently d= 
 
 •3183 p. 
 
 The pitch number is simply the reciprocal of d 
 and shows the number of teeth in the wheel per inch 
 of diameter. 
 
 In modern practice the pitch number is either a 
 whole number or a common fraction. 
 
 The following table gives values of the diameter 
 and circular pitches for all the pitch numbers in use. 
 
 TABLE XVII.— PITCH OF GEAR TEETH. 
 
 Pitch 
 
 Diam - 
 
 Circu- 
 
 Pitch 
 
 Diam- 
 
 Circu - 
 
 Pitch 
 
 Diam- 
 
 Circu - 
 
 Num- 
 
 eter 
 
 lar 
 
 Num- 
 
 eter 
 
 lar 
 
 Num- 
 
 eter 
 
 lar 
 
 ber. 
 
 Pitch 
 
 Pitch 
 
 ber. 
 
 Pitch 
 
 Pitch 
 
 ber. 
 
 Pitch 
 
 Pitch 
 
 I 
 
 d 
 
 P 
 
 I 
 
 d 
 
 P 
 
 I 
 
 d 
 
 P 
 
 d 
 
 
 1 
 
 d" 
 
 
 
 d 
 
 
 
 K 
 
 2. 
 
 6.2832 
 
 7 
 
 .1429 
 
 .4488 
 
 24 
 
 .0417 
 
 .1309 
 
 u 
 
 1-3333 
 
 4.1888 
 
 8 
 
 -125 
 
 •3927 
 
 26 
 
 .0385 
 
 .1208 
 
 1 
 
 I. 
 
 3.1416 
 
 9 
 
 .1111 
 
 •3491 
 
 28 
 
 .0357 
 
 .1122 
 
 IX 
 
 .8 
 
 2.5133 
 
 10 
 
 .1 
 
 .3142 
 
 30 
 
 .0333 
 
 .1047 
 
 ^'A 
 
 .6667 
 
 2.0944 
 
 II 
 
 .0909 
 
 .2856 
 
 32 
 
 .0312 
 
 .0982 
 
 i^ 
 
 •5714 
 
 1.7952 
 
 12 
 
 -0833 
 
 .2618 
 
 34 
 
 .0294 
 
 .0924 
 
 2 
 
 .5 
 
 1.5708 
 
 13 
 
 .0769 
 
 .2417 
 
 36 
 
 .0278 
 
 .0873 
 
 2H 
 
 .4444 
 
 1-3963 
 
 14 
 
 .0714 
 
 .2244 
 
 38 
 
 .0263 
 
 .0827 
 
 2^ 
 
 .4 
 
 1.2566 
 
 15 
 
 .0667 
 
 .2094 
 
 40 
 
 .025 
 
 .0785 
 
 2H 
 
 •3366 
 
 I. 1424 
 
 16 
 
 .0625 
 
 .1963 
 
 42 
 
 .0238 
 
 .0748 
 
 3 
 
 •3333 
 
 1.0472 
 
 17 
 
 .0588 
 
 .1848 
 
 44 
 
 .0227 
 
 .0714 
 
 3>^ 
 
 .2857 
 
 .8976 
 
 18 
 
 .0555 
 
 -1745 
 
 46 
 
 .0217 
 
 .0683 
 
 4 
 
 .25 
 
 .7854 
 
 19 
 
 .0526 
 
 .1653 
 
 48 
 
 .0208 
 
 .0654 
 
 5 
 
 .2 
 
 .6283 
 
 20 
 
 .05 
 
 .1571 
 
 50 
 
 .02 
 
 .0628 
 
 6 
 
 .1667 
 
 .5236 
 
 22 
 
 .0455 
 
 .1428 
 
 56 
 
 .0179 
 
 .0561 
 
MACHINE DESIGN. 
 
 93 
 
 The proportions given in Table XVI. are those 
 most usual in practice, but many good authorities 
 recommend a shorter tooth as giving less obHquity and 
 sliding friction and as being much stronger. 
 
 The length generally recommend-^d in such cases is 
 equal to one-half the circular pitch plus the clearance. 
 Short teeth are being more used every year. 
 
 Strength of Teeth. 
 
 P = total driving pressure on wheel at pitch 
 
 59. 
 
 Let 
 
 circle. 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. 
 
 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 C in Fig. 39 be the point of application of the 
 pressure P, and AB the line of probable fracture. 
 
 B 
 
94 
 
 MACHINE DESIGN 
 
 Use the notation of Fig. 38 and the proportions 
 for pattern molded teeth in Table XVI. 
 
 The bending moment at 
 section AB is M=Py, and 
 the moment of resistance 
 
 is M' = -i-Sxw' 
 
 where S = safe transverse 
 strength of material. 
 
 Py =4-Sxw^ 
 6 
 
 and 
 
 Fig. 38. 
 
 S = 
 
 6Py 
 
 (a) 
 
 when 
 
 If P and w are constant, then S is a maximum 
 
 y 
 
 IS a maximum. 
 
 But 
 
 y = h sin a and x=- 
 
 cosa 
 
 y 
 
 — = sin« cosa which is a 
 
 maximum when «= 45"* and — = J^ 
 
 X ^ 
 
 ^P 
 
 Substituting this value in (a) we have S = -^ 
 
 7 p 
 But in this case w=.47p and therefore S = -^ — i 
 
 ^'^ .22Ip* 
 
 and p=3.684 f— 
 
 (~p 
 Diameter pitch, d= 1.173 — 
 
 S^ 
 
 P 
 
 Unless machine molded teeth are very carefully- 
 made, it may be necessary to apply this rule to them 
 as well. 
 
 Pitch number 
 
 ' "^ = -853 J 
 
 (76) 
 (77) 
 (78) 
 
MACHINK DESIGN. 95 
 
 Cut Gears, With careful workmanstiip 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 mo- 
 ment 
 
 M==Ph 
 
 The moment of resistance at base of tooth is 
 M^ = >^Sbw^ 
 when b is ^he 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 : 
 
 h — 2}id =.6765p 
 W--.5P 
 and substituting above : 
 
 A A T. Sbp^ 
 .6765 Pp=-^ 
 
 P = .o6i6bSp .(79) 
 
 The above formula is general whatever the ratio 
 of breath to pitch. The general practice in this country 
 is to make b=3p 
 
 Substituting this value of b in (79) and reducing: 
 p=2.326 I— (8o> 
 
 or about two thirds the value obtained in Case I. 
 
 |P" 
 Diameter pitch d=.74 I— - ,. (81) 
 
 I /"s~ 
 Pitch number ~d'=^-35j-^ (82) 
 
 60. 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. 
 
96 MACHINE DESIGN. 
 
 Mr. Wilfred Lewis has deduced formulas which take into 
 account this variation. For cut spur gears of standard 
 dimensions the Lewis formula is as follows : 
 
 P=bSp (.124-^^51) (83) 
 
 n 
 
 where n = number of teeth. 
 
 This formula reduces to the same as (79) for 
 n=i4 nearly. 
 
 Formula (79) would then properly apply only to 
 small pinions, but as it would err on the safe side for 
 (arger wheels, it can be used where great accuracy is 
 not needed. The same criticism applies to formulas 
 (80) (81) and (82). 
 
 The value of S used should depend on the ma- 
 terial and on the speed. 
 
 The following values are recommended for cast 
 iron and cast steel. 
 
 Ivinear Velocity o 500 1000 1500 2000 
 
 ft. per minute 
 
 Cast Iron 6000 4500 2500 2000 1800 
 Cast Steel 15000 loooo 7000 5000 4500 
 
 Good bronze will have about the same strength 
 as the steel. The smaller values of S at the higher 
 speeds are to allow for the blows and shocks which 
 always occur in quick running gears. 
 
 61 Experimental Data. Inth^ American Mac hinisf 
 for Jan. 14, 1897 ^^^ 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 lbs. there being an average of about 50 teeth on 
 the pinions. Substituting these values (83) and solv- 
 ing we get 
 
 8=1575 lbs. 
 
MACHINE DESIGN. 97 
 
 This very low value is to be attributed to the con- 
 dition of pressure on one corner noted in Art. 59. 
 Substituting in formula for such a case. 
 
 S=7^^7^ -8150 
 
 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 ou the static strength 
 of rough cast iron gear teeth at the Case School of 
 AppUed 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^ involute teeth and cycloidal teeth constructed with 
 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 correspond- 
 ing to numbers above given omitting the pinion of 15 
 teeth. 
 
 The pressure was applied by a steel plunger tan- 
 gent to the surface of tooth and so pivoted as to bear 
 evenly across the whole breadth. The teeth were in- 
 clined at various angles so as to vary the obliquity 
 from o to 25° for the cycloidal and from 15° to 25^ for 
 the involute. The point of application changing accord- 
 ingly from the pitch line to the crest of the tooth. 
 From these experiments the following conclusions were 
 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. 
 
98 MACHINE DESIGN. 
 
 3. With teeth of dimensions given, the breaking 
 pressure per tooth varies from 25000 lbs. to 50000 lbs. 
 for cycloids as the number of teeth increases from 15 
 to infinity; the breaking pressure for involutes of the 
 same pitch varies from 34000 lbs to 80000 lbs. as the 
 tooth number increases from 24 to infinity. 
 
 4. With teeth as above the average breaking pres- 
 sure varies from 50000 lbs. to 26000 lbs. in the cycloids 
 as the angle changes from 0° to 25^ and the tangent 
 point moves from pitch line to crest, with involute 
 teeth the range is between 64000 and 39000 lbs. 
 
 5. Reasoning from the figures just given, rack 
 teeth are about twice as strong as pinion teeth and in- 
 volute teeth have an advantage in strength over cy- 
 cloidal 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 lbs. Values of S calculated from Lewis' formula 
 for the various tooth numbers are quite uniform and 
 average about 40000 lbs. forcycloidal teeth. Involute 
 teeth are to-day generally preferred by manufacturers. 
 William Sellers & Co. use an obliquity of 20^ instead 
 of 14^ or 15^ the usual angle. 
 
 62. 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 de- 
 veloped 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 
 
MACHINE DESIGN. 99 
 
 that the number of teeth is 32. In such a gear the 
 element of normal cone is \/~2~times the radius. The 
 actual shape of the teeth will then correspond to those 
 of a spur gear having 32^/T"== 45 teeth nearly. 
 
 Note.— In designing the teeth of gears where the 
 number is unknown, the approximate dimensions may- 
 first be obtained by formula (80) and then these values 
 corrected by using Lewis' formula. 
 
 EXAMPLES. 
 
 1 . The drum of a hoist is 8 ins. in diameter and 
 makes 5 revs, per minute. The diameter of gear on 
 the drum is 36 inches and of its pinion 6 ins. The 
 gear on the counter shaft is 24 ins. in diameter and its 
 pinion is 6 ins. in diameter. The gears are all rough. 
 
 Calculate the pitch and number of teeth of each 
 gear, assuming a load of one ton on drum chain and 
 S = 6000, Also determine the horse-power of the 
 machine. 
 
 2. Calculate the pitch and number of teeth of 
 a cut cast steel gear 10 ins. in diameter, running at 
 250 revs, per min. and transmitting 20 HP. 
 
 3. A cast iron gear wheel is 30 ft. 6^ ins. in 
 pitch diameter and has 192 teeth, which are machine 
 cut and 30 ins. broad. 
 
 Determine the circular and diameter pitches of the 
 teeth and the horse-power which the gear will trans- 
 mit safety when making 12 revs, per min. 
 
 4. A two pitch cycloidal tooth, 6 ins. broad, 72 
 teeth to the wheel, failed under a load of 38000 lbs. 
 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 H P. The water wheel makes 200 revs, per min. 
 and the head shaft 100. 
 
 Determine the dimensions of the gears and teeth 
 if the latter are approximately two pitch. 
 
 LofC. 
 
lOO MACHINE DESIGN. 
 
 63. 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 simul- 
 taneous 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 import- 
 ant that there should be no ribs or projections on arms 
 or rim which will offer resistance to the air. Experi- 
 ments 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 Anierican Machinist for Sep. 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 tension being adjusted by levers so as to be 
 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 
 
MACHINE DESIGN. lOI 
 
 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 show- 
 ing 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 nega- 
 tive bending moment at the rim about one-halt 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 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. 
 
I02 MACHINE DESIGN. 
 
 64. Sate 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 lbs. for cast iron. 
 
 = .28 lbs. for wrought iron or steel. 
 n=number of arms. 
 N= number revs, per min. 
 v=r velocity of rim in ft. per sec. 
 
 First let us consider the centrifugal tension alone. 
 The centrifugal pressure per square inch of concave 
 surface is 
 
 Wv^ 
 
 P = "^ — ^") 
 
 where W is the weight of rim per square inch of con- 
 
 D 
 cave surface = wt, and r= radius in feet = ~^T~ 
 
 The centrifugal tension produced in the rim hj 
 this force is by formula (13) 
 
 ^— 2t 
 
 Substituting the values of p, W and r and reducing: 
 
 I2WV^ 
 
 
 and v=^^- (85) 
 
MACHINE DESIGN. IO3 
 
 For an average value of w=.27, (86) reduces to 
 
 v^ 
 S = — nearly. 
 
 a convenient form to remember. 
 
 If v/e assume S as the ultimate tensile strength, 
 16500 lbs. for cast iron in large castings and 60000 lbs. 
 for soft steel, then the bursting speed of rim is ; 
 
 for a cast iron wheel 
 
 v=4o6 ft. per sec (86) 
 
 and for steel rim ^=775 ft. per sec (87) 
 
 and these values may be used in roughly calculating 
 the safe speed of pulleys. 
 
 It has been shown by Mr. James B. Stanwood, 
 in a paper read before the American Society of Me- 
 chanical Engineers,^ that each section of the rim be- 
 tween 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^ 
 
 17=-^ (^) 
 
 W in this case is the centrifugal force of the frac- 
 tion of rim included between two arms. 
 
 TrDbtW 
 
 The weight of this fraction is ~ and its cen- 
 
 TrDbtw 24V^ 247rbtwv^ 
 
 trifugal force W = - X -^^orW=— ^^^^ 
 
 ttD 
 
 Also 1 — ~~z~~ and d=t 
 
 * See Trans. A. S. M. B. Vol. XIV. 
 
I04 MACHINE DESIGN. 
 
 Substituting these values in (b) and solving 
 for S : 
 
 Dwv" 
 8=3.678-^^ (c) 
 
 If w is given an average value of .27 then 
 
 Dv' 
 S = -^^ nearly (d) 
 
 and the total value of the tensile stress on outer sur- 
 face of rim is 
 
 Dv' v' 
 S'=^^ + j^nearly (88) 
 
 Solving for v : 
 
 V =.U . _i_ (89) 
 
 X tn2^ 10 
 
 In a pulley with a thin rim and small number of 
 arms, the stress due to this bending is seen to be con- 
 siderable. 
 
 It must however be remembered that the stretch- 
 ing of the arms' due to their own centrifugal force and 
 that of the rim will to some extent diminish this bend- 
 ing. Mr. Stanwood 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 re- 
 lieve more than one half of the stress due to bending, 
 perhaps three-quarters. 
 
MACHINE DESIGN. I05 
 
 65 Experiment^ 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 dameter 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 w^as made of a Dow steam ttu-bine capa- 
 ble of being run at any speed up to loooo 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 in case 
 of sudden stoppage of the fly wheel shaft. 
 
 One experiment with a shield made of two-inch 
 plank convinced us that safety did not lie in that di- 
 rection, 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 
 contrivance. In subsequent experiments a shield made 
 of 12x12 inch oak was used. Even this shield was split 
 repeatedly and had to be re-enforced by bolts. 
 
 A cast steel ring about four inches thick Uned with 
 wooden blocks and covered wiih 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. 
 
lo6 
 
 MACHINE DESIGN, 
 
 TABLE XVilL — FIFTEEN INCH WHEELS. 
 
 
 Bursting Speed. 
 
 Centrifugal 
 
 
 
 
 
 Tension 
 
 
 No. 
 
 Revs. 
 
 Feet per 
 
 v2 
 10 
 
 Remarks. 
 
 
 per Minute. 
 
 Second =v. 
 
 
 I 
 
 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. 
 
 lO 
 
 5.709 
 
 361 
 
 13,000 
 
 Thin rim. 
 
 * Doubtful. 
 
 TABLE XIX.— TWENTY- FOUR INCH WHEELS. 
 
 
 Shape and Size of Rim. 
 
 Weight 
 
 ^0. 
 
 
 
 
 
 
 of^ 
 
 
 Diam- 
 
 Breadth 
 
 Depth 
 
 Area 
 
 
 Wheel, 
 
 
 eter 
 
 
 
 Sq. 
 
 Style of Joint. 
 
 
 
 Inches 
 
 Inches 
 
 Inches 
 
 Inches 
 
 
 Pounds. 
 
 II 
 
 24 
 
 2H 
 
 1-5 
 
 3.18 
 
 Solid rim. 
 
 75.25 
 
 12 
 
 24 
 
 4A 
 
 .75 
 
 3.85 
 
 Internal flanges, bolted 
 
 93. 
 
 13 
 
 24 
 
 4 
 
 .75 
 
 3.85 
 
 (( (( (( 
 
 91-75 
 
 14 
 
 24 
 
 4 
 
 .75 
 
 3.85 
 
 
 95. 
 
 15 
 
 24 
 
 4A 
 
 .75 
 
 3.85 
 
 
 94.75 
 
 16 
 
 24 ^ 
 
 1.2 
 
 2.1 
 
 2.45 
 
 Three lugs and links 
 
 65.1 
 
 17 
 
 24 
 
 1.2 
 
 2.1 
 
 2.45 
 
 Two lugs and links. 
 
 65. 
 
MACHINE DESIGN. 
 
 TABLE XX — FLANGES AND BOLTS. 
 
 107 
 
 
 Flanges. 
 
 Bolts. 
 
 Ko. 
 
 Thick- 
 ness, 
 
 Inches. 
 
 Effective 
 Breadth, 
 
 Inches. 
 
 Effective 
 Area, 
 
 Inches. 
 
 No. to 
 each 
 Joint. 
 
 Diameter 
 Inches. 
 
 Total 
 
 Tensile 
 
 Strength, 
 
 Pounds. 
 
 12 
 
 13 
 14 
 15 
 
 if 
 
 2.8 
 2.75 
 2.75 
 2.5 
 
 1.92 
 2.34 
 
 4 
 4 
 4 
 4 
 
 
 16,000 
 16,000 
 16,000 
 20,000 
 
 BY TESTING MACHINE. 
 
 Tensile strength of cast iron = 19,600 pounds per square in. 
 Transverse strength of cast iron = 46,600 pounds per square in. 
 Tensile strength of -^^ bolts = 4,000 pounds. 
 Tensile strength of | bolts = 5,000 pounds. 
 
 TABLE XXI.— FAILURE OF FLANGED JOINTS. 
 
 
 Area of Rim, 
 
 Square Inches. 
 
 Effect. Area 
 flanges. Sq. Ins. 
 
 Total Strength 
 Bolts, Pounds. 
 
 Bursting 
 Speed. 
 
 Cent. 
 Tension. 
 
 
 No. 
 
 Rev. 
 
 per 
 Min. 
 
 Ft.per 
 Sec. 
 
 = V. 
 
 Per 
 Sq.In. 
 
 10 
 
 Total 
 Lbs. 
 
 Remarks, 
 
 II 
 
 3.18 
 3.85 
 
 3.85 
 3.85 
 
 
 
 3,672 
 
 385 
 
 lA 800 
 
 /It' rtnr\ 
 
 Solid riin. 
 
 12 
 
 1.92 
 
 ,2:58 
 2.34 
 
 16,000 
 16,000 
 16,000 
 20,000 
 
 
 Flange broke. 
 Flange broke. 
 Bolts broke. 
 Flange broke. 
 
 13 
 
 14 
 15 
 
 1,760 
 
 1,875 
 1,810 
 
 184 
 196 
 190 
 
 3,40013,100 
 3,85014,800 
 3,61013,900 
 
io8 
 
 MACHINE DESIGN. 
 
 TABLE XXII.— LINKED JOINTS. 
 
 
 
 Lugs. 
 
 
 
 Links. 
 
 
 No. 
 
 
 
 Area 
 Sq. In. 
 
 
 Effect 
 
 Breadth. 
 Inches. 
 
 d CO 
 
 Effective 
 Area, 
 Sq. Ids. 
 
 i6 
 
 .45 
 
 I.O 
 
 .45 
 
 3 
 
 •57 
 
 .327 
 
 .186 
 
 17 
 
 .44 
 
 .98 
 
 •43 
 
 2 
 
 .54 
 
 .380 
 
 .205 
 
 Rim. 
 
 Max. 
 
 Area, 
 
 Sq. Ins. 
 
 2.45 
 2.45 
 
 Net 
 
 Area, 
 
 Sq. Ins. 
 
 1.98 
 1.98 
 
 BY TESTING MACHINE. 
 
 Tensile strength of cast iron = 19,600. 
 Transverse strength of cast iron = 40,400. 
 Av. tensile strength of each link = 10, 180. 
 
 TABLE XXIII, — FAILURE OF LINKED JOINTS. 
 
 
 
 
 Bursting 
 
 Cent. 
 
 
 
 i 
 
 i 
 
 Speed. 
 
 Tension. 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 Ft. 
 
 Per 
 
 
 
 No. 
 
 
 0^ 
 
 Rev. 
 per 
 
 per 
 Sec. 
 
 Sq.In. 
 v2 
 
 Total. 
 
 Remarks. 
 
 
 % 
 
 ^ 
 
 -*-» 
 
 Min. 
 
 = v 
 
 10 
 
 
 
 
 tn 
 
 t/5 
 
 
 
 
 
 
 16 
 
 30,540 
 
 38,800 
 
 3,060 
 
 320 
 
 10,240 
 
 25,100 
 
 Rim broke. 
 
 17 
 
 20,360 
 
 38,800 
 
 2,750 
 
 290 
 
 8,410 
 
 20,600 
 
 Lugs and Rim 
 broke. 
 
MACHINE DESIGN. I09 
 
 The flanged joirrts mentioned had the internal 
 flanges and bolts common in large belt wheel rims 
 while the linked joints were such as are commonly 
 used in fly wheels not used for belts. 
 
 Subsequent experiments have given approximately 
 the same results as those just detailed. The highest 
 velocity yet attained has been 424 feet per second; 
 this is in a solid cast iron rim with numerous steel 
 spokes. The average bursting velocity for solid cast 
 rims with cast spokes is 400 feet per second. 
 
 Wheels with jointed rims burst at speeds varying 
 from 190 to 250 feet per second, according to the style 
 of joint and its location. The following general con- 
 clusions seem justified by these tests. 
 
 1. Fly-wheels with solid rims, of the proportions 
 usual among engine builders and having the usual 
 number of arms, have a sufficient factor of safety at a 
 rim speed of 100 feet per second if the iron is of good 
 quality and there are no serious cooling strains. 
 
 In such wheels the bending du*e to centrifugal 
 force is slight, and may safely be disregarded. 
 
 2. Rimjoints midway between the arms are a 
 serious defect and reduce the factor of safety very ma- 
 terially. Such joints are as serious mistakes in design 
 as would be a joint in the middle of a girder under a 
 heavy load. 
 
 3. Joints made in the ordinary manner, with in- 
 ternal flanges and bolts, are probably the worst that 
 could be devised for this purpose. Under the most 
 favorable circumstances they have only about one- 
 fourth the strength of the solid rim and are particularly 
 weak against bending. 
 
 In several joints of this character, on large fly- 
 wheels, calculation has shown a strength less than 
 one-fifth that of the rim. 
 
 4. The type of joint known as the link or prisoner 
 joint is probably the best that could be devised for 
 
no MACHINE DESIGN. 
 
 narrow rimmed wheels not intended to carry belts, 
 and possesses when properly designed a strength about 
 two-thirds that of the solid rim. 
 
 66. Rims of Cast Iron Gears. A toothed wheel 
 will burst at a less speed than a pulley because the 
 teeth increase the weight and therefore the centrifugal 
 force without adding to the strength. 
 
 The centrifugal force and therefore the stresses 
 due to the force will be increased nearly in the ratio 
 that the weight of rim and teeth is greater than the 
 weight of rim alone. 
 
 This ratio in ordinary gearing varies from 1.5 to 
 1.7. We will assume 1.6 as an average value. Neglect- 
 ing bending we now have from equation (86) 
 
 I2WV^ I9.2WV^ 
 
 S=i.6x— ^= (90) 
 
 and ^=S^ 
 
 \19.2w 
 
 = 326.2 ft. per second (91). 
 
 Including bending 
 
 «'=^-^^\^^+7^] (9^> 
 
 As the transverse strength of cast iron by experi- 
 ment is about double the tensile strength, a larger 
 value of S may be allowed in formulas (88), (89) and 
 
 (92). 
 
 In built up wheels it is better to have tlie joints 
 come near the arms to prevent the tendency of the 
 bending to open the joints, and the fastenings should 
 have the same tensile strength as the rim of the 
 wheel. 
 
MACHINE DESIGN. Ill 
 
 exampi.es. 
 
 1 . Design eight arms of elliptic section for a gear 
 48 ins. pitch diameter to transmit a pressure on tooth 
 of 800 lbs. 
 
 2. Determine bursting speed of the gear in pre- 
 vious example in revolutions per minute if the thick- 
 ness of rim is .75 inch. 
 
 (i) Considering centrifugal tension alone. 
 
 (2) Including bending of rim due to centrifugal 
 force, assuming that one-half the stress due to bending 
 is relieved by the stretching of the arms. 
 
 3. Design a link joint for the rim of a fly-wheel, 
 the rim being 8 ins. wide, 12 ins. deep and 18 ft. mean 
 diameter, the links to have a tensile strength of 65000 
 lbs. per sq. in. Determine the relative strength of joint 
 and the probable bursting speed. 
 
 4. Discuss the proportions of one of the following 
 wheels in the laboratory and criticise dimensions. 
 
 (a) Fly-wheel, AUis engine. 
 
 (b) Fly - wheel, Fairbanks gas engine. 
 
 (c) I^arge Medart pulleys, Electrical laboratory. 
 
 (d) Belt - wheel, AUis engine. 
 
TRANSMISSION BY BELTS AND ROPES. 
 
 67. 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. 
 
 Ti=tension of driving 
 
 side. 
 T2= tension of slack side. 
 R= friction of belt 
 
 f = co-efficient of friction " ^> 
 
 between belt and 
 
 pulley. 
 = arc of contact in cir- I 
 
 cular measure. T 
 
 Fig. 40. 
 
 The tension T at any part of the arc of contact is 
 intermediate between T^ and Tg. 
 
 Let AB Fig. 40, be an indefinitely short element 
 of the arc of contact, so that the tensions at A and B 
 differ only by the amount dT. 
 
 dT will then equal the friction on AB which we 
 may call dR. 
 
 Draw the intersecting tangents OT and OT' to 
 represent the tensions and find their radial resultant 
 OP. Then will OP represent the normal pressure on 
 the arc AB which we will call P. 
 
 <OTP= <ACB = d^ 
 .-. P=Td^ 
 
 B 
 
MACHINE DESIGN. 
 
 The friction on AB is 
 
 fP=:fTd^ 
 
 or dT=dR=fTd<? 
 
 dT 
 and fd^ ~"T^ 
 
 Integrating for the whole arc : 
 
 /.T^dT T, 
 
 T, id 
 T"=e 
 
 T,=^,=T,e- 
 R=T,-T,=T,(i-e-f^) 
 
 "3 
 
 ie 
 
 (93) 
 
 The average value of f for leather belts on iron 
 pulleys as determined by experiment is f=0.27 
 
 If we denote the expression (i — e — ^^) by C, then 
 for different arcs of contact C has the following values: 
 
 Arc of 
 Contact. 
 
 go^' 
 
 IIO° 
 
 130° 
 
 150'' 
 
 180° 
 
 210^ 
 
 240° 
 
 C 
 
 •345 
 
 .404 
 
 •458 
 
 .506 
 
 •571 
 
 .627 
 
 .676 
 
 The friction or force transmitted by a belt per 
 inch of width is then 
 
 R=CT, (94) 
 
 and T^ must not exceed the safe working tensile 
 strength of the material. 
 
 A handy rule for calculating belts assumes C=.5 
 which means that the force which a belt will transmit 
 under ordinary conditions is one -half its tensile 
 strength. 
 
114 MACHINE DESIGN . 
 
 68 Strength of Belting. The strength of belting 
 varies widely and only average values can be given. 
 According to experiments made by the author good oak 
 tanned belting has a breaking strength per inch of 
 width as follows: 
 
 Single. Double. 
 
 Solid leather 900 lbs. 1400 lbs. 
 
 Where riveted 600 lbs. 1200 lbs. 
 
 Where laced 350 lbs. 
 
 Canvas belting has approximately the same 
 strength as leather. Tests of rubber coated canvas 
 belts 4-ply, 8 inches wide, show a tensile strength of 
 from 840 lbs to 930 lbs. per inch of width. 
 
 69. Taylor's Experiments. The experiments of 
 Mr. F. W. Taylor, as reported by him in Trans. Am. 
 Soc. Mech. Eng. Vol. XV. afFord the most valuable 
 data now available on the performance of belts in actual 
 service. 
 
 These experiments were carried on during a period 
 of nine years at the Midvale Steel Works. Mr. Tay- 
 lor's conclusions may be epitomized as follows : 
 
 1 . Narrow double belts are more economical than 
 single ones of a greater width. 
 
 2. All joints should be spliced and cemented. 
 
 3. The most economical belt speed is from 4000 to 
 4500 ft. per min. 
 
 4. The working tension of a double belt should 
 not exceed 35 lbs. per inch of width, but the belt may 
 be first tightened to about double this. 
 
 5. Belts should be cleaned and greased every six 
 months. 
 
 6. The best length is from 20 to 25 feet between 
 centers. 
 
MACHINE DKSIGN. II5 
 
 70. Rules for Width of Belts It will be noticed 
 that Mr. Taylor recommends a working tension only to 
 to A- the breaking strength of the belt. He justifies 
 this by saying that belts so designed gave much less 
 trouble from stoppage and repairs and were conse- 
 quently more economical than those designed by the 
 ordinary rules. 
 
 In the following formulas 50 lbs. per inch of width 
 is allowed for double belts and 30 lbs. for single belts. 
 These are suitable values for belts which are not run- 
 ning continuously. The formulas may be easily changed 
 for other thicknesses and for other values of CT^. 
 
 Let HP=horse power transmitted. 
 
 D=diameter of driving pulley in inches. 
 N=no. revs, per min. of pulley. 
 
 The moment of force transmitted by belt is 
 
 RD _ CT.wD _ 
 2 2 ~ 
 
 TN CT.wDN 
 ^^^ ^^=6^S^ 126050 (95) 
 
 Substituting the values assumed for CT, and solving 
 
 for w : 
 
 HP 
 Single belts w = 4200 ^-^ (96) 
 
 HP 
 Doublebelts w= 2500 ^^ (97) 
 
 The most convenient rules for belting are those 
 which give the horse-power of a belt in terms of the 
 surface passing a fixed point per minute. 
 
 In formula (95) CT.wDN 
 
 ^^== 126050 
 
 we will substitute the following : 
 
Il6 MACHINE DESIGN. 
 
 w 
 W= width of belt in feet = — 
 
 12 
 
 V= velocity in ft. per mm. = 
 
 or HP: 
 
 i44CT,WV 
 
 1260507: 
 
 Substituting values of C and 1\ as before and 
 solving for WV=square feet per minute we have ap- 
 proximately : 
 
 Single belts WV= 90 HP (98) 
 
 Double belts WV= 55 HP (99) 
 
 71. Speed of Belting. As in the case of pulley 
 rims, so in that of belts a certain amount of tension is 
 caused by the centrifugal force of the belt as it passes 
 around the pulley. 
 
 From equation (84) i2wv^ 
 
 where v= velocity in ft. per sec. 
 
 \v= weight of material per cu. in. 
 S = tensile stress per sq. in. 
 
 To make this formula more convenient for use we 
 will make the following changes in the constants: 
 
 Let V = velocity of belt in ft. per minute = 6ov. 
 w= weight of ordinary belting. 
 
 = .032 per cu. in. 
 Si= tensile stress per inch width, caused by 
 
 centrifugal force. 
 =about TE S for single belts. 
 
 V 
 
 Then v=^ 
 
 16S, 
 
MACHINE DESIGN. 1 1 7 
 
 Substituting these values in (86) and solving for S^ 
 
 Si= I 610000 (^o^) 
 
 The speed usually given as a safe limit for ordi- 
 nary belts is 3000 ft. permin., but belts are sometimes 
 run at a speed exceeding 6000 ft. per min. 
 
 Substituting different values of V in the formula 
 we have : 
 
 V=30oo Si= 5.59 lbs. 
 
 V=4ooo Si= 9.94 lbs. 
 
 V=5ooo Si = 15 -53 lbs. 
 
 V=6ooo 81=22.36 lbs. 
 
 The values of Si for double belts will be nearly 
 twice those given above. At a speed of 5000 ft. per 
 minute the maximum tension per inch of width on a 
 single belt designed by formula ( 96 ) , if we call 
 C = .5, will be : 
 
 (3oX2) + i5.= 75 lbs. 
 
 giving a factor of safety of eight or ten at the splices. 
 
 In a similar manner we find the maximum tension 
 per inch of width of a double belt to be : 
 
 (50x2) +30= 130 lbs. 
 
 and the margin of safety about the same as in single 
 belting. 
 
 72. Manila Rope Transmission. Ropes are some- 
 times used instead of fiat belts for transmitting power 
 short distances. They possess the following advan- 
 tages: they are cheaper than belts in first cost ; chey 
 are flexible in every direction and can De carried 
 around corners readily. They have however the dis- 
 advantage of being less efiicient in transmission than 
 leather belts and less durable; they are also somewhat 
 diflEicult to splice or repair. 
 
Il8 MACHINE DESIGN. 
 
 There are two systems of rope driving in common 
 use : the English and the American. In the former 
 there are as many separate ropes as there are grooves 
 in one pulley, each rope being an endless loop always 
 running in one groove. 
 
 In the American system one continuous rope is 
 used passing back and forth from one groove to an- 
 other and finally returning to the starting point. 
 
 The advantage of the English system consists in 
 the fact that one of the ropes may fail without causing 
 a breakdown of the entire drive, there usually being 
 two or three ropes in excess of the number actually 
 necessary. On the other hand the American system 
 has the advantage of a uniform regulation of the ten- 
 sion on all the plies of rope. The guide pulley, which 
 guides the last slack turn of rope back to the starting 
 point, is usually also a tension pulley and can be 
 weighted to secure any desired tension. The English 
 method is most used for heavy drives from engines to 
 head shafts ; the American for lighter work in dis- 
 tributing power to the different rooms of a factory. 
 The grooves in the pulle^^s for manila or cotton ropes 
 usually have their sides inclined at an angle of about 
 45°, thus wedging the rope in the groove. 
 
 The Walker groove has curved sides as shown 
 in Fig. 41, the curvature increasing towards the 
 bottom. As the rope 
 wears and stretches it 
 becomes smaller and 
 sinks deeper in the 
 groove; the sides of the 
 groove being more ob- 
 lique near the bottom, 
 the older rope is not 
 pinched so hard as the 
 newer and this tends 
 to throw more of the 
 work on the latter. Fig. 41 
 
MACHINE DESIGN. II 9 
 
 73. Strength of Manila Ropes. The formulas for 
 transmission by ropes are similar to those for belts the 
 values for S and ^ being changed. The ultimate tensile 
 strength of manila and hemp rope is about loooo lbs. 
 per sq. in. 
 
 To insure durability and efficiency it has been 
 found best in practice to use a large factor of safety. 
 Prof. Forrest R. Jones in his book on Machine Design 
 recommends a maximum tension of 200 d^ pounds 
 where d is the diameter of rope in inches. This cor- 
 responds to a tensile stress of 255 lbs. per sq. in. or a 
 factor of safety of about 40. 
 
 The value of f for manila on metal is about 0.12, 
 but as the normal pressure between the two surfaces 
 is increased by the wedge action of the rope in the 
 groove we shall have the apparent value of f : 
 
 a 
 r = f-r-sin"^ where 
 
 a = angle of groove, 
 For a = 45° to 30'' 
 
 f^ varies from 0,3 to 0.5 and these values should be 
 used in formula (93). 
 
 (i — e"~f^) in this formula for an arc of contact of 
 150°, becomes either .54 or .73 according as T is taken 
 0.3 or 0.5. 
 
 If Tj is assumed as 250 lbs. per sq. in., the force 
 R transmitted b^^ the rope varies from 135 lbs to 185 
 lbs. per sq. in. area of rope section. 
 
 The following table gives the horse -power of 
 manila ropes based on a maximum tension of 255 
 lbs. per sq. in. 
 
I20 
 
 MACHINE DESIGN. 
 
 TABLE XXIV. 
 
 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 lbs. for a 2 inch diameter rope 
 and is the same at all speeds, due allowance having been made 
 for loss by centrifugal force. 
 
 n i 
 
 SPEED OF THE ROPE IN FEET PER MINUTE. 
 
 1 
 
 
 1500 
 
 2000 
 
 2500 
 
 3000 
 
 3500 
 
 4000 
 
 4500 
 
 5000 
 
 6000 
 
 7000 
 
 Smalle 
 Diam. 
 leys, I 
 
 fittl 
 
 3-3 
 
 4.3 
 
 5.2 
 
 5.8 
 
 6.7 
 
 7.2 
 
 7.7 
 
 7.7 
 
 7-1 
 
 
 ^ 
 
 4.9 
 
 30 
 
 H 
 
 4.5 
 
 5.9 
 
 7.0 
 
 8.2 
 
 9.1 
 
 9.8 
 
 10.8 
 
 10.8 
 
 9-3 
 
 6.9 
 
 36 
 
 I. 
 
 5.8 
 
 7.7 
 
 9.2 
 
 10.7 
 
 11.9 
 
 12.8 
 
 13.6 
 
 13-7 
 
 12.5 
 
 8.8 
 
 42 
 
 iX 
 
 9.2 
 
 12. 1 
 
 14.3 
 
 16.8 
 
 18.6 
 
 20.0 
 
 21.2 
 
 21.4 
 
 19.5 
 
 13.8 
 
 54 
 
 I'A 
 
 I3-I 
 
 17.4 
 
 20.7 
 
 23.1 
 
 26.8 
 
 28.8 
 
 30.6 
 
 30.8 
 
 28.2 
 
 19.8 
 
 60 
 
 iX 
 
 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 
 
 74. 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. 
 
MACHINE DESIGN, 
 
 121 
 
 Fig. 42 shows a section of the rim of such a pul- 
 ley. The rope does not touch the sides of the groove 
 but rests on a shallow depression in a wooden, leather 
 or rubber filling at the bottom. The 
 high side flanges prevent the rope 
 from leaving the pulley when sway- 
 ing on account of the high speed. 
 
 The pulleys must be large, usu- 
 ally about 100 times 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 XXV is taken from 
 Roebling. 
 
 Fig. 42. 
 
 lyong ropes should be supported by idle pulleys 
 every 400 feet. 
 
 EXAMPLES. 
 
 1. Design a main driving belt to transmit 150 HP 
 from a belt wheel 18 ft. in diameter and making 80 
 revs, 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 transmit- 
 ting economically. 
 
 3. Calculate the total maximum tension per inch 
 of width due to load and to centrifugal force of the 
 driving belt on the generator used for lighting, assum- 
 ing the maximum load to be 50 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 revs, per min. Use 
 
122 
 
 MACHINE DESIGN. 
 
 TABLE XXV. 
 TRANSMISSION OF POWER BY WIRE ROPES. 
 
 Showing necessary size and speed of wheels and rope to 
 obtain any desired amount of power. 
 
 Diameter of 
 Wheel in ft. 
 
 Number of 
 Revolutions. 
 
 Diameter 
 of Rope. 
 
 Horse - 
 Power. 
 
 «4-l 
 
 on. 
 
 0^ 
 
 Number of 
 Revolutions. 
 
 Diameter 
 of Rope. 
 
 Horse - 
 Power. 
 
 4 
 
 8o 
 
 3/8 
 
 3-3 
 
 10 
 
 80 
 
 II-16 
 
 58.4 
 
 
 lOO 
 
 3/8 
 
 4.1 
 
 
 100 
 
 II-16 
 
 73- 
 
 
 I20 
 
 3/8 
 
 5- 
 
 
 120 
 
 II-16 
 
 87.6 
 
 
 140 
 
 3/8 
 
 5-8 
 
 
 140 
 
 II-16 
 
 102.2 
 
 5 
 
 80 
 
 7-16 
 
 6.9 
 
 II 
 
 80 
 
 II-16 
 
 75-5 
 
 
 100 
 
 7-16 
 
 8.6 
 
 
 100 
 
 II-16 
 
 94-4 
 
 
 120 
 
 7-16 
 
 10.3 
 
 
 120 
 
 II-16 
 
 II3-3 
 
 
 140 
 
 7-16 
 
 12. 1 
 
 
 140 
 
 II-16 
 
 132. 1 
 
 6 
 
 80 
 
 V2 
 
 10.7 
 
 12 
 
 80 
 
 V\ 
 
 99-3 
 
 
 100 
 
 Vz 
 
 13-4 
 
 
 100 
 
 V\ 
 
 1 24. 1 
 
 
 i.?o 
 
 % 
 
 16.1 
 
 
 120 
 
 % 
 
 148.9 
 
 
 140 
 
 V2 
 
 18.7 
 
 
 140 
 
 Ya 
 
 173-7 
 
 7 
 
 80 
 
 9-16 
 
 16.9 
 
 13 
 
 80 
 
 Vx 
 
 122.6 
 
 
 100 
 
 9-t6 
 
 21. 1 
 
 
 100 
 
 Ya 
 
 153-2 
 
 
 120 
 
 9-16 
 
 25-3 
 
 
 120 
 
 Vx 
 
 183-9 
 
 8 
 
 80 
 
 /8 
 
 22. 
 
 14 
 
 80 
 
 Vz 
 
 148. 
 
 
 ICX> 
 
 ^8 
 
 27-5 
 
 
 100 
 
 /s 
 
 185. 
 
 
 120 
 
 5/8 
 
 33 -o 
 
 
 120 
 
 /8 
 
 222. 
 
 9 
 
 80 
 
 Vz 
 
 41-5 
 
 15 
 
 80 
 
 ^ 
 
 217. 
 
 
 100 
 
 Vz 
 
 51-9 
 
 
 100 
 
 Vz 
 
 259- 
 
 
 120 
 
 Vz 
 
 62.2 
 
 
 120 
 
 n 
 
 300- 
 
MACHINE DESIGN. 1 23 
 
 Hunt's table and then check by calculating the centri- 
 fugal tension and the total maximum tension on each 
 rope. 
 
 5. Design a wire rope transmission to carry 
 1 20 H P a distance 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. 
 
 PAGE. 
 
 Adjustment of Bearings 61-62 
 
 Ball Bearings 76-80 
 
 Barlow's Formula 17 
 
 Beams, Bending 7-8 
 
 Deflection 11 
 
 Bearings, Adjustment 61-62 
 
 Ball 76-80 
 
 Ivubrication 63-64 
 
 Roller 80-82 
 
 Rotating , 61-67 
 
 Sliding 53-57 
 
 Thrust 74 
 
 Belting, Friction of 112-113 
 
 Experiments 114 
 
 Speeds 116 
 
 Strength 114 
 
 Width 115-116 
 
 Boiler Shells 16 
 
 Bolts and Nuts 39 
 
 Coupling 86 
 
 Strength, Table 40 
 
 Bursting Fly-Wheels 105-109 
 
 Caps and Bolts 68 
 
 Columns, Strength 8-9 
 
 Cotters 50-51 
 
 Couplings, Shaft 85-86 
 
 Cylinders, Steam 20-24 
 
 Table 22 
 
 Deflection, Formulas 11 
 
 Design, Principles of . 12-13 
 
Fly- Wheels, Experiments 105-109 
 
 Formulas, General 7~9 
 
 Frame Design 12-13 
 
 Frames, Machine 28-30 
 
 Friction of Belts 112-113 
 
 Journals 65 
 
 Pivots 70-73 
 
 Gearing 100 
 
 Arms and Rim loo-ioi 
 
 Bevel 98 
 
 Safe Speed no 
 
 Spur 91-98 
 
 Guides 57 
 
 Hangers 88 
 
 Heating of Journals 66 
 
 Hooks 41 
 
 Hyatt Rollers 81 
 
 Iron, Cast 4 
 
 Malleable 4 
 
 Wrought 3 
 
 Joints, Riveted 42-48 
 
 Butt 45-46 
 
 I/ap 43-44 
 
 Tables 47-48 
 
 Joint Pins 49 
 
 Journals, Friction . 65 
 
 Heating 66 
 
 Pressure 65 
 
 Strength 67 
 
 Keys, Cotter 50-51 
 
 Shafting 87 
 
 Ivamc's Formulas 19 
 
 Lubrication 63-64 
 
 Materials of Construction 3 
 
 Notation Used 7 
 
 Nuts, Check 41 
 
 Pipe, Table 18 
 
 Pivots, Conical 71 
 
 Flat • 70 
 
 Schiele 72-73 
 
Plates, Flat 24-27 
 
 Experiments 26-27 
 
 Pulleys, Arms of loo-ioi 
 
 Safe Speed 102-104 
 
 Riveted Joints 42-49 
 
 Roller Bearings 80-82 
 
 Rope, Manila 117-120 
 
 Horse - Power 120 
 
 Strength 119 
 
 Rope, Wire 120-122 
 
 Tables 122 
 
 Sections, Cored 28-29 
 
 Section Moduli 10 
 
 Shells, Thin 16 
 
 Thick 17-20 
 
 Shafting 83-85 
 
 Slides, General 53 
 
 Angular 54 
 
 Flat 56 
 
 Gibbed 55 
 
 Speed, Safe 102-104 
 
 Springs, Experiments 33-34 
 
 I^lat 36-37 
 
 Tension and Compression 31-34 
 
 Torsion 34-35 
 
 Steel 3-4 
 
 Stress and Strain 2; 
 
 Strength of Metals 5-^' 
 
 Stuffing Boxes 57-58 
 
 Experiments 59-^' 
 
 Supports, Machine 14 
 
 Teeth of Gears 91-99 
 
 Experiments 96-9S 
 
 Thrust Bearings 74 
 
 Units Used •• 3 
 
TEXT BOOKS 
 
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 Professor of Mechanical Engineering, 
 CASE SCHOOL OF APPLIED SCIENCE. 
 
 HEGHANIGAL LABORATORY PRACTICE 
 
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