G2I.& L57n 4 *- ~ v ;VA . \D* "" f •••''■ 1 'Pr-: : >■ m- v -,4 1 . r :> ■ - -■■■<■ , / a. A-*S T N v. s ■■■inuik ‘•r-i* 'V -j^V '* \ : '* CV > / . r **• ^ :.V fc .i y • a • • r* • ir\ \&t y ' %yPi ■< ■'-- ••' »■* . o* ~V . > •— • >*-tv •■ • >• V ,. i . jfcs. : < / .- v -#• "Y . * -v : . ' v :^r • ■ ■ ■** . W .V i . ■* > / /Y v r \ ‘ / V. '-H - /;• ^ ■ -V y -7. ; s L <* 4.. • ■' 7 S t ■—* " < 7 -: V t A yL: *7 V ‘ ;'( ' < 7 ' / . • •**'">■*. / * ' ^;. 7 .V W* & THE UNIVERSITY OF ILLINOIS LIBRARY 62 . 1.8 L57n Return this book on or before the Latest Date stamped below. University of Illinois Library THE UNIVERSITY OF ILLINOIS LIBRARY &2.I.8 L57n Mimeographed by Edwards Brosc Ann Arbor, Mich, II 0 T E S on ELEMENTARY M ' A 0 H I IT E DESIGN Frepared for Student in the Mechanical Engineering Department 0. A. Leutwiler and W . V . Dunk in „ University of Illinois September , 1911 . Published by The U 3 of Ic Supply Store Oharapa i rn , 111. Digitized by the Internet Archive in 2017 with funding from University of Illinois Urbana-Champaign Alternates https://archive.org/details/notesonelementarOOIeut CHAPTER I Materials U s ed in the Co n struc t i on of Machine Parts c The principal materials used in the construction of machine parts are cast iron, steel, wrought iron, copper, trass, bronze, babbitt metal, wood and leather* 1. Cast iron is more commonly used than any other material in machine parts* This is because (a) of its high compressive strength and (t) because it can be given easily any desired form* A wood pattern (sometimes metal) of the piece desired is made; and from this a mold is made in the sand* The pattern is next, removed from the mold and the liquid metal poured in, which on cooling assumes the form of the pattern* Cast iron is obtained directly from the melting of the iron ore in the blast furnace. This product from the blast furnace is known commercially as pig iron* It fuses easily (melting point PP00° F. ) but it cannot be tempered, or welded under ordinary conditions. The composition of cast iron varies considerably but in general is as follows’ Metallic iron *.90*0 to 95.0$ Garb cn . 1.5 " 4 . 5$ Till icon . 0.5 " 4.0 $ less than 0.15$. Sulphur 192 - ' ' ’ ■ • • Phosphorus Manganese a e . 0 9 06 to I o 5$ . trace to 5,0^, Carbon may be united either chemically with the iron, in which case the product is known as white iron, or it may exist in the free state when the product is known as grey iron. The white iron is very brittle and hard, and is therefore but little used in machine parts . In the free state the carbon is known as graphite . Silicon is an important constituent of cast iron because of the influence it exerts on the condition of the carbon present in the iron, From 0,25$ to of silicon tends to wake the iron soft and strong* but beyond 2,0 ' silicon the iron becomes weak and hard. An increase of silicon causes less shrinkage in the castings, but a further increase (above 5 %) may cause an increase in the shrinkage, kith about l t 0';4 silicon the tendency to pro- duce blowholes in the castings is reduced to a minimum. Sulphur in cast iron causes the carbon to unite chemically with the iron, thus producing hard, white iron, which is hard and brittle. For good castings, the sulphur content should not exceed 0.15//. Phosphorus in cast iron tends to produce weak and. brittle castings. It also causes the metal to bo very fluid when melted, thus causing the metal to take an excellent impression of the mold. Phosphorus is, therefore, a desirable constituent in pig iron for the production of fine thin castings where no great strength is required. For this purpose, from 5 to 5 " of the ol em- inent may be used. The absence of phosphorus from the iron give-’ rise to soft and malleable castings, which are also lacking 192 - I f ' V- 3 in strength and soundness . For strong castings of good quality, the amount of phosphorus rarely exceeds n.55$, hut when fluidity and softness is more important than strength, from ifo to 1,5$ may he used. Manganese when present in cast, iron up to about 1.5'* tends to make the castings harder to machine; but renders them more suitable for smooth or polished surfaces. It also causes a fine granular structure in the castings and prevents the absorption of the sulphur during melting. Manganese may also be added to cast iron to soften the metal. This is due to the fact that the manganese counteracts the effects of the sulphur and silicon by eliminating the former amd counteracting the latter. However, when the iron is remelted, its hardness returns since the manganese is oxidised and more sulphur is absorbed. The transverse strength of oast iron nay be increased about 30.0$, and the shrinkage and depth of chill decreased 25 o 0$, while the combined carbon is diminished one-half by adding to the molten metal, powdered ferro- manganese in the proportion of one pound of the latter to about 300 pounds of the former. » Pig inon is the basis for the manufacture of all iron products. It is, not only used practically unchanged to produce castings of a great variety of form and quality, but it is also used in the manufacture of wrought iron and steel. For each special purpose, the iron must have composition within certain limits. It follows, therefore, that pig iron offers a considerable variety of composition. The practice of purchasing pig iron by analysis is generally followed at the present time. The following table 192 . . . . ■ ■■ gives the specifications as required by one large manufacturing concern. (I. A p u % l — 1 o > • p 829). Table I 0 Class Silicon Phosphorus Mangane s Sulphur Total Carbon * % Hot over /° Hot over i Hot under <4 1 1.5 to 8.0 0 e 2 to 0.75 1.0 0.040 5 .0 2 2.0 to 0(5 0,2 to 0.75 1.0 0 .035 r *5 S 2.5 to 3,0 0.2 to 0.75 1 .0 0.050 O c c 4 2.0 to 2.5 1.0 to 1.50 1.0 0 . 040 37 K 5 4.0 to 5,0 0,2 to 0.80 1.0 0 . 040 3v0 Analysis i s made from drillings from a pig selected at random from each four tons of every carload as unloaded. The right is reserved to reject a portion or all of the material which. does not conform to above specifications in every particular. In a general way, the specified limits for the composition of the chief grades of pig iron are as follows: Grade of Iron Silicon Table II. Sulphur Phosphorus Manganese Ho. 1 Foundry 7® / 3 2.5 to 3.0 Under .055 c o r~* o SSL-P LO o ! Under 1 .0 Ho . 2 " 2.0 to 2.5 ti .045 0.5 to 1 ,00 it 1 .0 No . 3 ,r 1.5 to 2.0 (t .055 0.5 to 1.00 it 1 .0 Malleable 0.7 to 1.5 ft ,050 Under 0,20 it 1,0 Gray Forge Und_er 1.5^ II .100 " 1,00 ii 1 ,0 Bessemer 1.0 to 2.0 II ,050 0.10 it 1 .0 Low Phosphorus Under 2.0 II j 0 5 0 ” 0 , 30 ii 1 .0 Basic M 1.0 tl *■050 n 1.00 ii 1.0 Basic Bessemer ” 1.0 If .050 2,0 to 5,00 1.0 to 2.0 According to U30, pig iron may be separated roughly into tvr o great clan ses. The first cla ss includes those grades used 192 r * . 4 I l ' 1 5 the production of foundry and malleable irons, while the second includes those used in the manufacture of wrought iron and steel. In the process of remelting or manufacturing, the first class undergo little if any chemical change, while the second class undergo a complete chemical changer 2 . Malleable Castings are made by thoroughly cleaning and heating foundry castings (preferably with the sulphur content low) in an annealing furnace in connection with some substance that will absorb the carbon from the cast iron. Hematite or brown iron ore in pulverised form is used extensively. The in- tensity of heat required is, on the average, about 1650° F. The length of time the castings remain in the furnace depends upon the degree of maleability required and upon the size. Usually the light castings require a minamum of cO hours while the heavier ones may require 72 hours or longer. 3. Chilled castings are those which have a hard and. durable surface. The iron used is, generally, close grained gray iron, low in silicon. A chilled casting is formed by making that part of the mold in contact with the surface of the casting to be chilled, of a construction, ouch that the heat will be rapidly withdrawn. The mold for causing the chill usually consists of iron bars or plates, placed such that their surfaces will be in contact with the molten iron. These plates abstract heat rapid- ly from the iron, with the result that the part of the casting in contact with the cold surface assumes a state similar to white iron, while the rest of the casting remains in tho for?" of gray 1 Oo . ' . 6 iron. The withdrawal of heat is hastened by the circulation of cold water through pipes, circular or rectangular in cross section, placed near the surface to he chilled. Chilled castings offer great resistance to crushing forces. The outsid.e, or "shin" of the ordi- nary casting is in fact a chilled surface, hut by the arrangement mentioned above the depth of the "shin" is greatly increased with a corresponding increase in strength and wearing qualities. Car wheels and chilled rolls are familiar examples of chilled castings. Car wheels require great strength combinedrwith a hard durable tread. The depth of chill varies from 3/8" to 1", (I.A.Vol.76, p 162). Chilled rolls are used in rolling steel and iron sheets, and also tin plates, because their hard smooth surfa.ce gives to the sheets and plates a smooth surface, {I. A., 1903, Apr .23, p 2.) 4. Wrought iron is formed from pig iron by melting the latter in a "puddling furnace" t During the process of melting, the impurities in the pig iron are removed by oxidation leaving the pure iron and"slag", both is a pasty condition. In this condition the iron and slag is formed into "muck balls", weighing about 130 pounds, and removed from the furnace. These balls are put into a "squeezer" and compressed, thereby removing a large amount of the slag, and then rolled into bars. The bars, known as "muck bars", are cut into smaller bars or strips and arranged in piles, the con- secutive layers being at right angles to each other, These piles are raised to a welding heat and rolled into "merchant bars" . These bars are the ordinary wrought iron of commerce. Wrought iron is not so extensively used now as formerly, steel, to a great extent having taken its place. Wrought iron, however, still finds 192 ' « _ 7 extensive use in the manufacture of pine, formings, parts of elec- trical machinery, small structural shape"’ arv.i crucible steel. 5. Stee l is a compound in which iron and carbon are the principal parts. It is made from pig iron by burning out the carbon, silicon, manganese and. other impurities, and recarbonising to any degree desired. The principal processes or methods of manufacturing steel are (l) the Bessemer, (2) the Open Hearth and ( o ) the Cement at i on * Bessemer Process. - In the Bessemer process, several tons (usually about ten) of molten pig iron are poured into a pear shaped vessel called a converter. Through this mass of iron large quantities of cold air are passed. In about four minutes, all the silicon and manganese of the pig iron has combined with the oxygen of the air. The carbon in the pig iron no i;r begins to unite with the oxygen forming carbon-monoxide, which, burns out through the mouth of the converter in a long brilliant flame. The burning, of the carbon-monoxide continues for about six minuted, when the flame shortens, thus indicating that nearly all the carbon has boon burned out of the iron, and that the air supply should ho shut off. The burning out of these impurities has raised the temperature of the iron to white heat and left a relatively pure mass of iron. To this mass is added a certain amount of carbon in the form of a very pure iron high in carbon and manganese. The metal in then poured into molds forming ingots, which aro tolled while hot into the desired shapes. The characteristics of the Bessemer process are: 192 8 (l) great rapidity of reduction, about ton minutes perjb.eat; (8) no extra fuel required; (f>) metal is not molted in the furnace where the reduction takes place, Bessemer steel was formerly used almost entirely in the manufature of wire, skelps for tubing, wire nails, shafting, machine steel, tank plates, and structural shapes. Open hearth steel has, however, very largely superseded the Bessemer products in the manufacture of these articles. Open Hearth Process . In the manufacture of open hearth steol, the molten pig iron direct from the reducing furnace, is poured into a long hearth, the top of which has a fire brick lin- ing. The impurities in the iron are burned by heat reflected from this refractory lining, and obtained from burning gas and air. The slag is first burned, and the slag in turn oxidises the im- purities. The time required for purifying is from 6 to 10 hours, after which the metal is recarbonized, cast into ingots and rolled as in the Bessemer process. The characteristics of the open hearth process are: (1) relatively long time to oxidize impurities; (8) large quantities (85 to 70 tons) purified and recarbonized in one charge; (5) ex- tra fuel (gas) required; (4) a part of the charge (steel scrap and iron ore, added to the charge at the beginning of the process) are melted in the furnace. Open hearth steel in used in the manufacture of cutlery, files, shovels, picks, boiler plato, and armor plate, in addition to the articles mentioned above. Cementation Process, In this process of manufacturing 192 . / o steel, bars of wrought iron, imbedded in charcoal are hostel for several clays. The wrought iron absorbs carbon fron th s ?•••» srcooJ. and is thus transformed into steel. Y ? hen the bars of iron c,r ' re- moved they aro found to be covered with scales or "blisters'. Tho name given to this product is "blister steel" , By removing the sclaes and blisters and subjecting the bars to a cherry red heat for a few days, a more uniform distribution of the carbon is obtained. Blister steel, when heated and rolled directly into the finished bars, is known as German Steel. Bars of blister steel may be cut up and forged together under the hammer, forming a product called "s' oar steel". By re- peating the process with the shear steel, we obtain "double s? oar" steel . Crucible Steel. Crucible or cast steel is very uniform and homogenous in structure. It in made by molting blister s.t' / ul in a crucible, casting it in ingots and rolling into bars. By this method is produced the finest crucible or* cast steels. Another method of producing crucible-cast steel is to melt Swedish iron, (wrought iron obtained from the reduction of a very pure iron ore in the blast furnace, and in using charcoal instead of coke in the puddling flame) and charcoal in a sealed vessel, tho contents of which are poured into a larger vessel or ladle, containing a simi- lar product from other sealed vessels. The metal in this larger vessel is cast into ingots, which are tubsequontly forged or roll- ed. into bars. By far the greater portion of cast stel is produced by this method. 192 . • ' * i • ' 1 10 6. S teel C asting s, Castings, similar to cast iron cast- ings, may be formed in almost any desired stare from too molten steel. The open hearth steel is considered surer i or to Bessemer steel for steel castings. In texture, these castings arc course and crystalline, since the steel has been allowed to cool without drawing or rolling. Formerly, trouble was experienced in obtaining good, sound castings; but by great care and irriproved methods in the production of molds, first class castings may now be obtained. Steel castings are used for those machine parts, requiring greater strength than is obtained by using pig iron castings, 7. Cold Rolled Steel, This steel is rolled hot to approximately the required dimensions. The surface is then care- fully cleaned, usually by chemical means, and rolled cold to a very accurately gauged thickness between smooth rollers. A very smooth and hard surface, with greatly increased strength, is thus given to the steel. Cold rolled steel is use’ chiefly for shafting. 8. Spe c ial Steels . A special steel is one in vrhicli some other element rather than carbon distinguishes the metal. The principal elements used to form these special steels are nickel, tungsten, chromium, manganese, vanadium, aluminium and silicon,. ITickel steel contained 0.0$ to 1.0;! carbon, and up to >5$ nickel. It possesses great tensile strength and ductility. Tests show that its tensile strength varies from 100,000 to 275,000 pounds per square inch, with an elastic limit of 40,000 to 75,000 pounds per square inch. The effect of the nickel is not always uniform; thus a nickel coontent up to 8$ increases the tensile strength and elastic limit, while between 8$ and 15$ brittleness is 192 ■ * . . ■ ' ' 11 produced; but above 15$ the strength and elastic Unit return. Nickel in used in ordnance work and in making am. or "slate. It does not, in general, crack when pierced by a projectile. R .its made from this steel, show better wearing qualities than tbos \*de from Bessemer steel. On account of its ability to withstand heavy shocks and torsional stresses, nicket steel is used for crank shafts shafting, connecting rods, explosive engines, and in automobile work . Tungsten steel is an alloy of iron, carbon, tungsten, and manganese, and sometimes chromium. The element which gives this steel its peculiar: property - self or air hardening - is not tungsten but manganese combined with carbon. The s tungsten, however, is an important element, since it enables the alloy to contain a larger combined carbon content. On account of its hard- ness, this steel can not be easily machined, but -■■■ust be forged to the desired shape. Its great use is for cutting tools -- roiwMrg tools . Chromium steel is formed by adding to carbon steel, from 1 e n$ to 2.0$ chromium. The steel thus produced is extremely hard and also self or air hardening. It is homogeneous and very fine grained. Because of the rapid oxidation of the chromium, the steel deteriorates rapidly when redressed. 'This stool is used, for armor plate, shells and metal cutting tools. M The boat high speed tool steels now contain from 5.0 $ to 6.0$ chromium." Manganese steel is formed by adding f erro-manganoso to iron or steel. The proportion of carbon present is high, on ac- count of the process of manufacturing. The influence of the manganese is not always the sane, thus with the manganese 192 ' ■ ■ . varying from a mere trace to 6*0$ and with carbon not exceeding 0.4$ the strength and ductility of the compound diminished while the hardness and brittleness increase. From 8$ to 10$ manganese the strength and ductility increase rapidly. With 12$ manganese, the steel becomes we air but on increasing the manganese to about 14.0 $ the strength of the metal becomes a maximum. Above 14.0$ manganese the strength of the steel remains fairly constant until about 17*0$ manganese when the strength diminishes rapidly. Manganese steel is in general free from blow holes, but is difficult to cast on account of its high shrinkage which is about two and one-half times as great as cast iron. The thermal conductivity of this steel is low, and its electric resistance is practically constant with '.varying temperatures. It possesses great hardness which is not diminished by annealing, and also a high tensile strength, combined with great toughness and ductility These qualities would make this steel the ideal metal for machine construction, were it not for the fact that its great hardness pro vents it from being machined in any way but by abrasive processes. It in the most durable metal known to resist wear, and therefore is used extensively for steam shovel teeth, dredge pins, plow points, frogs, switches, crossings, crusMng*rolls for ore, rook screens, gear sprockets, or any machine mart which must resist a grinding wear in dust. The best composition in Mn 1^-15$: 0 not over 0.5$: 3 not over 0.4$. The castings should always be anneal- ed* Vanadium steel is formed by adding a small amount of vanadium 192 . . ' 13 usually loss than 0.5$ — to carbon stsel. Other elements ouch as chromium and manganese may also be present. The vi' radium; tends to produce brittleness; this steel should, therefore, be annealed, The principal use of vanadium steel is for metal cutting tools. Aluminum steel contains only a small per cent of that element, since it easily united with the oxygen and passes off with the slag in the process of manufacturing. Aluminum increases the strength of the steel, but decreases its ductility. This steel is not in extensive use, since the same grade of steel may be ob- tained by employing less expensive elements, as for instance, silicon and manganese. Silicon steel, as now made, contains about 5.0$ Silicon and shows a very fine grained structure. It nay bo made very hard hut it is more liable to crack in hardening than ordinary steel. Silicon steel is more expensive and possesses less strength than carbon steel. 9. Cooper . Copper in a pure state is a soft motal and is extremely ductile. Good castings can not be made from the pure metal because of blow holes and great shrinkage in cooling. Copoer is used extensively with other metals to form alloys. Other uses are for certain parts of electrical machinery, wire, tubing, expansion joints, etc. 10. Alloys . Alloys may be made of two or more metals that have an affinity for each other. The compound or alloy thus 192 . ' ' » f |K> !;r '*l * ' <■ 14 produced has properties and characteristics which none of the metals possess. The principal alloy use:, in machine construction may bo obtained by combining two or more oC the foil owing mot. Is; copper, zinc, tin, lead, antimony, bismuth and aluminum. Brass is an alloy of copper and zinc. Its composition may vary from 06 parts cooper and 24 parts zinc to 70 parts co^mr and 30 parts zinc. Lead is sometimes added* especially in f-e cheaper grades of brasses. Brass is easily worked and may be made to take a fine finish. For this reason it finds extensive use in machine parts, especially for those parts not requiring great strength. Bronze or gun metal is an alloy of copper and tin. Zinc is some tines added to cheapen the alloy or to change its color and to increase its malleability. Soft bronze contains about 90 parts copper and 10 parts 'tin, while hard bronze contains 02 parts cop- per and 18 parts tin. Castings stronger than those made from pig iron may be made from bronze. The softer grades are m' 1 for cocks and fittings, and the harder grades for bearings r ' bushings The strength and toughness of bronze are greatly increased b^ rapid cooling. Phosphor bronze varies somewhat in composition* but in general is abCut as follows: copper 80 parts, tin 10 parts, lead 9 parts and phosphorus 1 part. It is easily cast and is stronger than cast iron castings. Phosphor bronze is used for bushings, bearings, gear wheels, and nay be drawn into wire for the -manu- facture of springs. Manganese bronze has about the following composition: 192 ■ • ' . ■ • ' ■ * 15 copper 88 parts, tin 8 parts, and manganese d parts. Several qual- ities are made by varying the percent of these metals. It is stronger than phosphor bronze and for t; is reason is much usod for propel lor blade castings. It is also usO'' 1 for boa, rings and bush- ings, and for mining screens on account of its non-corrosive qualities . Babbitt mot a 1 L s om etim e s calls d v h i t e o t a 1 or rrh j t o bra •••s, is an alloy of copper, tin and antimony in varying pro- portions . Usually these proportions are as follows * tin C A to 80 parts, cooper 8 to 3 parts, and antimony 8 to 7 parts. It gives rise to less friction than either bronze or brass, but crushes more easily. It is not, therefore, suitable for use in severe and heavy work. Its only use is for hearings. 1VS . ■ I 3YSICAL CONSTANT; CO' I 1 CQ p bb •rH © r s- • O 4^ P-i pH . . a ft a o O G O O C C O O O 10 O O CO !' IP \J' H ! LO <© C; «•*’ Si 1 P‘ V LO p p p i-"* “ *v *\ K ^ K •H ft ft © ft o c c CO o ft P p pp C C C CO c ft ft O o o c c c ft 1 O'©© • • n *» •» *\ «\ P o P P eft CO c: O to eo CO p w t~i eo l — 1 1 — i 1 — I Pi ft p coo coo c c C; O *H c c c. c co c c. c. P o c c o o c c cco CO O *H ft • *\ *\ •» f\ V« A *\ *\ ft ft © ft o q o o c o o c c i — 1 • P pH P CO c. o c c c oo c ftp© C O C C O LO G O C © © © • • •N A »l iv k k ^ OOP P eft C O LO LO LO P L O LO CO a © h .ft © CO CO Cft P P © p p p © p ft • o o o o c o ft ft bO o ft O O O’ C C C s ? c Ph-H C O C i> C CO •H X © •s #\ A *\ P © ft • • O b> 0; (ft i — 1 i — 1 4-3 P Cft P to p ft © P © p « « (—• e— o b r-> o ft • l •rH • © ft C C O C C. C C C C p p Pft *H O C C CO c c c o © p c c c c c c c c o ft E w • r» a a a *\ *\ r, «\ *\ i — 1 *H P eft O ’• 1 lb C. t*/ rH CV- rH H P P © LO to Cft Cft 1 — © top p c p c5 -H bO E ft ft •H c3 © ft • © C Ph-h o c p c c c o o o V r c r O c o o o o c c. c c c LO C CQ P o Eh O ■p &H fH Eh &q <3 CO © p © o © o s r— H •H eft P o ft ft o a (ft ' — " © cq © ft p Cft © © bO bO ft P •rH *H ft ft (X: c3 **- > sc, o p © © o ft P CQ CO to ft {>5 *H r d 'u c5 i — l © *H •P ft CO PQ G <0 LC C c CO o : — ! LO CO i AVERAGE PHYSICAL CONSTANTS , 1C Ultima te t ensile strength Ultimate compressive strength Ultimate shearing strength ! Elastic {Ultimate limit. {Flexural j Strength Modulus or coef. of Eleotricity Shearing Modulus of Electricity Height LI . per sq . in. Lb . per sq. in. Lb « per sq . in . T Lb . per 1 sq. in. Lb . per sq. in. Lb . per sq. in. Lb . per oq. in. Lb . per cu . ft . Hard Stool 100,000 180, ono ro , ooo 50,000 110,000 50,000,000 18,000,000 •190 Structural Steel 50,000 00,000 50 f nor c 5,o no 50,000,000 18,000,000 490 V r ought iron 50,000 50,000 / n ooo 2 5 , ooo 85,000 ,000 10,000,000 -IPO 5 5,000 Ton. Cast iron P.0,000 90.000 20 ,ooo )£0 ,onr Oom. 35,000 15,000,000 0 ,.000,000 4 50 Copper 30 , 000 15 r 0 00 ,.000 6 ,-000,-900 55 0 Timber Y'ith Grain 10,090 0,000 coo 3 ,0°0 9,000 1 r 50 0., 090 40 Across Grain 3,000 400,000 •V) Concrete 300 5 , 000 1,000 1,000 70Q 5 ,-000 r 000 150 Stone 6 , 000 1,500 2 ,nno P ,000 6,000,000 ICO Brick . 3,000 . - 1,000 1,000 BOO 8,000,000 185 SAFETY FACTORS. Material For steady Stress For Varying Stress For Shocks (Buildings) ^ Bridges) (Machines) Hard Steel 5 6 15 Structural Steel 4 A C 10 Wrought Iron 4 6 10 Ca3t Iron c 10 20 Timber 6 TO ’fj CHAPTER II 17 Strength of Mater ial s . The object of a machine is to transmit motion through its various links, to some particular part where useful vr ork is to be done* The transmitting of this motion gives rise to forces which must, be resisted by the parts of the machine through which the force is acting. In order to safely withstand these forces, each machine part must be constructed in accordance with certain laws . These laws may bo the result of theoretical investigation or they may be obtained from existing conditions of design. For- mulas for the size of the machine parts obtained from this latter source are called empirical - In the design of machine parts many of the formulas used are more or less empirical. This is especially true of the greater number of the formulas usod in this elementary design in machine design. The student is not far enough advanced in his applied mathematics to enable him to derive but few of the formulas. He should, however, familiar- ize himself with the meaning of the various symbols occurring in the formulas, and should also bo able to use readily any formula occurring in these notes. 1 1 . External For c es, Stress ^ a rd d t ra ins. - C on s i de r an iron rod whose cross section is A square inches, su on ended from so^'6 fixed point, and sup nr o ting at its lower end a load or weight of P pounds. Evidently the pull downwards at- the end of the rod is P pounds, and if equilibrium is to be maintained , the rod, in any section near the bottom, must exert a pull eoual 102 i ' ' ■ ■ (Hi. 1G to P and in the opposite direction. This force, which arises in the rod due to the pull of the weight P, is called a stress, and map be defined as the internal resistance which the mole- cules of the rod offer to the force P. In other words , the internal resistance offered by a body to any force tending to overcome the force of cohesion is called a stress. In the above rod if S represents the stress per square inch of cross section, then evidently; Area of cross section of rod multiplied by unit stress = total stress induced in section, but the total stress induced in section is due to the pull of the external force, hence AS = P (1) From (l) if we know the magnitude of the external force and the value of the stress per square inch we nay easily find the area of the section. Stresses are usually measured in pounds or tons per unit area. In this course the unit of area will be the square inch and the unit stress or intensity of stress will be given in pounds per square inch. The values of the unit stresses, are obtained by experiment, that is, by testing pieces to destruction in testing machines. For values of these stresses for various materials see Table I. Strain , - A body subjected to an external force no matter how small, undergoes a deformation or change of form, tiie amount of which is called a strain . Thus a rod 10 inches \ long has suspended from- it a weight such that its length while supporting the ’weight is 10.01 inches. 192 The total * . . - strain is therefore *01 inches and the unit strain or strain per unit of length is .001. Within certain limits strains are directly proportional to the stresses which produce them. Let 1 = length of body e = total deformation or elongation of body s = unit of strain then e = Is . ' (2) There are three kinds of simple stresses induced in a machine part by the external forces acting upon the part. They are as follows: (1) Tensi le, in which the externa]- force s are tec ling to pull the body apart. As an illustration of this stress con- sider the rod mentioned above. The rod may fail by being nulled apart. Here the external forces are acting away from each oth'.T. (2) Com p ressiv e , in which the external forces are ten- tending to crowd the parts of the body together. The piston rod of an engine on the forward stroke, legs for the support of lathes, columns, are examples of machine parts subjected to compressive stress. (3) Shearing , in which the external forces are act- ing on the body in parallel lines very near each other, in the opposite sense. A good illustration is the punching of a hole in a plate, or the shearing of bars or plate : in a shearing machine. In addition to the simple stres -es given, above, may also be mentioned bearing stress. This is a form, of compres- sive stress, and is caused by two surfaces pressing or bearing 192 ■ . . - 2.0 against each other « Plato edges on rivets or nine, cotter edges and keys in key ways are all illustrations of machine warts in which bearing stress is induced. There are two kinds of stresses, viz: torsion and bond- ing. A brief discussion of these will be given later. Experiments have shown that when a body in subjected to a small stress a small strain is produced, and, upon removal of the external force causing the stress, the body returns to its original shape. This property, belonging to the body, i3 called elastic ity . All bodies are more or less elastic, and within certain limits the strain is directly proportional to the stress. That is, if a force produces a stress of 10000 pounds and a corresponding strain of . O'*! inches, thin doubling the force would produce twice the stress wit! twice the strain, or taking one-half the external force, the corresponding stress would he 5000 pounds and the strain <-0015 inch. The external force acting upon the body may, however, bo of such a magnitude that the resulting stress produces a strain which is part ly per- manent. Under these conditions the strain is called a set . The strain is no longer proportional, to the stress, which produced it. If tho external force be increased then the stress will be increased and the strain will rapidly increase until the body is broken. The stress at the point where the strain becomes perma- nent is called, the elastic limit . Evidently no machine part should be loaded so a-: to produce a stress near the elstic limit. The coefficient of elasticity of a body is the ratio 192 . I ' ! 21 of the unit stress to the unit strain. Let S = the unit stress s = the unit strain E = the coefficient of elasticity. Then by the definition and combination of (l) and (2) we have E (3) If in (3) E be regarded as a force, also that a = 1 and A » unit area, we should have E = P. The interpretation of this is left as an exercise for the student. The ultimate strength of a body is that unit stress which is just sufficient to break it. 12. Stresses due _t o_ Bon ding, .-If w e c on s i d e r t h e forces acting in any machine we shall find thi t some of them act without leverage, thus producing direct stress in the machine part. As an example, consider the piston rod of an engine. The steam pressure is the external force transmitted through the rod, thus producing a stress within the rod, the magnitude of which is equal to the external force. The stress in uniformly distri- buted over the cross section since the resultant of the steam pressure acts through the geometric axis of the rod. This stress is called a direct stress . Again, if wo continue our analysis we shall find that some of the forces act with a leverage, thus producing bending in the machine part. As an example, consider the crank of the steam engine. The force transmitted through the connecting rod to the crank through the crank pin produced a 192 moment about the ' op center of the crank shaft® The stresses thus induced in the shaft are not direct stresses, and the determination of "d>ich - require some knowledge of the theory of beams. 15 . Beams , - Consider the bar in Pig. 1 resting on the support R^ and Rg and supporting a load W at a distance x from the left support. Without the load W the bar will be in the position indicated by the full liners, but supporting the load the bar will tend to assume the p o s i t i on indicated by the dotted lines. If we assume 9 as we may, that the bar is made up of an infinite number of fibers running the long way of the bar, then those fibers lying in the top surface containing the line ran, will be shortened or compressed, and those fibers in the surface containing the line op will be lengthened or stretched. In other words, the upper fibers will be subjected to a compressive stress and the lower fibers to a tensile stress. Evidently somow K -vro between the upper and lower fibers is a surface the fibers in which are neither shortened or lengthened. Such a surface is called the neutral surface and always contains the center of gravity, provided the loading is as shown. Referring to Pig. 1, the lino qr represents the neutral surface. Let us consider the forces acting on the bar in Pig. 1. The weight W acting downward causes the supports at R-j and R 0 to exert an upward pressure on the bar. To find the r-Z> value of these upward pressures or reactions, we employ the principle that the sum of the moments of r. 11 the forces witj respect to any point is equal to zero, nrovided the forces are in equilibrium. Since this is true, with resnect to any noint, 19? . . . ■ • • , ■ . let up. take moments , about action of either or R 0 . right reaction Rr> , then we of the bar a point containing tho lino of Suppose we take moments about the shall have, if 1 equals the length. Ri 1 or W(l-x) = 0 W(l-x) 1 also if we take moments about the left support, we shall have Here we have considered forces acting upward as positive, and those acting downward as negative. Let us next determine the bending moment in the bar due to the load W at any section as ab. The bonding moron t is measured by the resultant moment of the external force on either side of the section ah . We shall consider tho forces to the left of the section ah. The only force on the left of the sec- tion ab_ is the reaction and tho tendency of this force to cause dending in tho section depends upon its magnitude and its distance from the section. If wo consider the section cd, then we shall have two forces to the left of the section - tho reaction R]_ and tho weight W, The tendency of 1?^ is to cause rotation of the bar in a clockwise direction about a pdint in the section cd and the magnitude of this tendency to rotate the bar is measured by the moment of tho force R-j with respect to the point in the section. The tendency of W is to cause rotation of the bar in a counter clockwise direction, 192 and the magnitude . - n A of this tendency to rotate is also measured by the moment of W with respect to the point in the section. The algebraic sum. of these tendencies to produce rotation about a point in the section od is called the bending m oment , or the algebraic sum of the moments of the forces on the left of the section with respect to a point in that section called the b endi ng moment . Expressing this bending moment by means of a formula, we shall have Bending moment = &i x i ~ Wxg (6) 14 o B ee j sting m om en t, - Let us consider that part of the bar to the right of the section cd. to be removed as shown in Fig. 2 < The fibers in the upper part of the bar an wo have seen are in compression while those fibers in the lower part are in tension. Evidently the compressive stresses tend to produce a rotation of the upper part of the bar in a counter clockwise direction, while the tensile stresses tend also to produce a rotation of the lower part of the bar in a counter clockwise direction. The tension below nd the compression above have resultants R and R whose moment is Em, The moment of one of these forceps with respect to the line of action of the other is known as the mome nt of resistance of the section. Evidently, for equilibrium, the bonding moment must equal the resisting moment, or R^xq - Wxp = Em ( 7 } 15. - A bar subject to the loading shown in Fig. 1 is called a simple bea m. A cantilever beam is one having one end fixed and the other free. A restrained beam is one which has both ends fixed. Each of these beams with the loading is 192 . ' r . I represented in Fig, 3. This figure shows the bea T ”is as all sup- porting loads concentrated at a point. Instead of this arrange*- ment , the beam may have the loads distributed over its length, either uniformly or varying according to some given law, or several concentrated loads c Exercise: Visit the shops and laboratories and write down the name of at least one machine part loaded as a beam; that is, (a) One piece acting as a simple beam with a concentrated load . ti ti " load uniforml cantilever bean with a concen- f! II restrained (b) " " " distributed . ( c ) One " 11 trated load. ( d ) One " " formly distributed. (e) One piece " trated load. ( f } One " *' formly distributed. The student in his shop work should observe the charac ter of the loading of the parts of the machines with which he works. The beam is of frequent occurrence in machine construc- tion. and one should be able at once to know the character of the loads in order that the bending moments may bo accurately determined and the part thus properly proportioned. 192 if il II If If II It II load uni concen- " load uni- . . ; ' . . ' | The bending moment; M may bo calculated from formulas given in table in Kent, p. 268. In the first column is given the character of the beam and the nature of the load. The column headed "Maximum moment of Stress" gives the value of the bending moment M while the column headed "Moment of Rupture gives the resisting moment,. 16 . - We have seen that for equilibrium the bending moment must equal the resisting moment (Lq. 7), Let us next- find an expression for its value in terms of the dimension of the section. Let S = tensile or compressive fiber stress upon fiber farthest away from the gravity axis, c = distance from gravity axis to most remote fiber, in inches . 5 = distance from gravity axis to any fiber whose area equals a sq. in. Then by the following law, sometimes called Hookes Law which is "the amount of elongation or compression in any fiber is directly proportional to its distance from the gravity axis" , we have §. = stress per sq. in. at a distance of 1" from gravity axis, c c _ O s C It It it if ii it ll if :j ii it It I! o 2 (— Ba)r = a = moment of this stress r* ' v-' ^ is "a" about the gravity axis. in the fiber who s ■_ area How we consider the piece as being made up of an infinite number of fibers whose area is "a" : hence the sum of 192 ■ ' . .. - ' ■ . °7 all the moments of the stresses induced in. each fiber would be the sum of all the expressions: ba or Sa 3 s > a5 since S and c are con- C \ 0 c ^ p stants. But the expression > a Z 0 being the sum of the products formed by multiplying each of the elementary strips by the square of its distance from the gravity axis is called the moment of inertia of that section with t.respect to the gravity axis. The moment of inertia in usually denoted by I. We have therefore q p SI resisting moment Em = _ > aZ = — 7 or denoting the bending c 0 moment by M we have M = 31 ( 8 ) In ( 8 ) the ratio =*— • is called the section modulus, c values of which, for various sections may be found from tables in Kent, p. 249. 192 . ■ .. . . ■ _ • • CHAPTER III 28 FASTENINGS 0 Ri veted Joi n ts , 17. R ivets. - The most common means of uniting plates as used in boilers, tanks and structural work is by means of ri- vets. A rivet is a round bar consisting of an upset end called, the head and a long part called the shank. It is a permanent fastening, removable only by chipping off the head. Rivots should in general be placed at right angles to the forces which cause them to fail. The greatest stress thus induced in then is that of shearing. If rivets are to resist a tensile stress a greater number should be used than when they are resisting a shearing stress. Rivets are made of wrought iron and soft steel and are formed in suitable dies while hot from round bars out to length. The shank usually has parallel sides for about one-half its length, the remaining length tapering every slightly. When used the rivets are brought up to a red heat, placed in the holes of the plates to be connected and a second head formed, oith/ r by hand or machine work. In the former hammers are used; tf e head of the rivet being held firmly against the under side of the plate. In the latter, the rivet is pressed between two dies. Generally speaking, machine riveting is better than hand work, as the hole in the plates in nearly always filled with the rivet body, while in hand work the effect of the blow does not appear to reach the interior of the rivet, thus producing n) 192 . movement of the metal in the rivet hole. 18- Rivet Ho les. For the sake of economy rivet hoi -s are usually punched- There are two serious objections to thus forming the hole. The metal around the holes is injured by the lateral flow of the metal under the punch. This may, however, be remedied by punching smaller holes and then reaming them to size.- Next, the sjmcing of the holes in the two parts is not accurately done in the case of punching so that it becomes neces- sary to either ream out the holes (in which case the rivets may not completely fill the hole thus enlarged) or use a drift pin. The drift pin should be used only with light weight harpers * The size of the rivet hole is about l/l6 inch larger than the rivet. This is subject, however, to some variation, depending upon the class and character of the work. This clearance space allows for some inaccuracy in punching the plate and also permits clearance space for driving the rivet when hot » Drilling the holes is the best method of perforation of plates. The late improvements in drilling machinery has made it possible to accomplish this work with almost the same economy as in punching. The metal is not injured by the drillin of holes; indeed there are tests which show an increase in unit- strength of the metal between the rivet holes . 19- Forms of Rivets. - Rivets are made from a very tough and ductile quality of iron and steel, They are formed in dies from the round bars while hot and in this 192 c ond 3 1 i on . . 4 ■ - ■ ■ . . ' . . ■ ' • ■ . . r 50 are called rivet blanks,, The rivet blank is composed of two parts, viz., the head and the shank* For convenience the head, which is formed during the process of driving is called the point. The amount of shank necessary to form the point depends upon the diameter of the rivet . Since the length of rivet is measured under the head, the length required is equal to the length of shank necessary to form the point plus the thickness of the plates. The thickness of the plates, or the distances between the head and point after the rivet is driven is called the grip of the rivet . To find the length of rivet required for uniting plates, add to the combined thickness of plates a length equal to 1 1/2& for steeple points (see Fig. 4 button points (Fig. 4, b) add to combined thic^nsr A - ) pj 1 r\ 7 1 -* c r Cl of pi at os a . .p a { '"'t C\ Uj Fig. 4, c). Lengths of rivets should always he taken in quarter inch lengths on account- of stock sizes. Any length up to five or six inches, however, may be obtained, but the odd sizes will cost more than the standard sizes. TO. Forms of Heads . - Rivets with many different forms of heads may be found in mechanical work, but the ones in general use in boiler work are only three viz . , cone head , button head , and countersunk head . These are shown in Fig. 4, (a), (b), and (c) respectively. The proportions advocated by different manu- facturers vary somewhat. The proportions given in Fig. 4 are those used by the Champion Rivet Company. The steeplo poirr , Fig. 4 (a) is one easily made by hand driving and is, therefore , rauoh used. This form, however, on account of the thinness of the 192 ' ' ■ 1 pn edges, is weak to resist tension and should not therefore be used on important work. The cone head, Fig. 4 (&), is one of tgreat strength and is used a great deal in boiler work. It is not generally used aa a form for the point on account of difficulty in driving. The button head type, Fig. 4 (b) is widely used for points and may be easily formed in hand work by the aid of a sna p-,. A snap is much used in forming points. It is a piece of steel with a forming die in one end. The die is placed over the rivet and the snap struck with a heavy hammer, in this manner an almost perfect point may be formed. The countersunk point weakens the plate so much that it is used only when, projecting heads would be objectionable, as under flanges of fittings, in the direct line of the play of the flames. Its use is sometimes imperative for both heads and points, but it should be avoided whenever possible. The counter- sink in the plate should never exceed 3/4 of the thickness of the plate, and for that reason the height of the rivet point is generally from l/l6 to l/8 of an inch greater than the depth of the countersink. The point then projects by that amount or if the plate is required to be perfectly smooth, the point chipped off level with the surface. SI. Rivete d Jo i nts . - There are two methods of con- necting bars abd plates by means of rivets. The arrangement by which the edge of one plate overlaps the edges of the o\ her is called the lap joint . Such a joint may ho riveted up as shown in Fig. 5 with only one row of rivets, in which 192 0 O r* ,-TI a, o . 15; " ' jj , . ■ ■ cal led single riveted lap joint „ With the arrangement is two rows of rivets the joint is .known as a double riveted lap joint , and with three rows a triple rive ted lap join t. When more than one row of rivfets is used they may be arranged as shown in Pig. 6, (b) or (a). In the former the joint is chain riveted and in the latter z ig-zag ri v eted c When the plates or bars butt against each other and are joined by overlapping plates or straps the connection is called a butt joint . Such a joint may have one plate on the outside or one plate on the outside and one on the inside; may be single, double or triple riveted and may have chain or zig-zag riveting. Fig. 7 shows a single riveted butt joint with two cover plates . 22 . Failure of Riveted Jo ints . - Joints may fail in one of several ways, as shown by the following examples. (a) Shearing of the Rivet. - In the case of all lap joints and butt joints with one strap, the rivets tend to fail along one section, while in butt joints with two straps failure tonds to take place along two sections. Thus in Pig. P (a) the tendency would be for the rivets to fail along the section mm and after failure the conditions would be represented by Fig. S (b). Such a rivet is said to be in single shear. If P is the force tending to pull the plates past each other, S„ the o ultimate shearing strength of a square inch of section of the rivet and A the cross sectional area of the rivet, then from (l) 192 . ' • l p (9) '7ta 2 s s 4 Exerc ise , Sketch a butt joint with two straps and deduce formula for shearing resistance of one rivet. (b) Crushing the P late or Rivet. - If the rivet should be strong enough to resist the shearing force then the plate may fail bj‘ crushing and wrinkling; as shown at A in Fig. 9. The resistance to crushing offered by any small portion MIT , Fig. 10 on the circumference, equals the projection of MI! on the diameter perpendicular to the line of action of the force. The total resistance, therefore, would be the sun of all such projections times tho thickness of the plate. Evidently the sum of all such projections would be equal to the diameter of if e / rivet and the total resistance, therefore, would be dtS c in which d = diameter of the river, t ■ = thickness of plate, and S 0 = the ultimate crushing strength of plate. Since this must equal the force P tending to cause crushing, we have P = dtS (10) c (c) Bursting of Plate. - A joint may fail by split- ting of the plate opposite each rivet, as shown at B in Fig, 9. This manner of failure may be prevented by having a suf- ficient distance from the rivet to the edge of the plate. It has been found, experimentally, that if this distance is at least equal to the diameter of the rivet then failure will not, in general, take place in this way. (d) Tearing of the Plate. - The nlateo may bo torn 195 , ■ • . * . - or pulled apart along the line of rivets as shown at CD, Fig. 9. Evidently the least resisting area offered by the plate to tearing would be the net area of the plate along the lino of rivets. If p = pitch of rivets or distance from center to center of rivet holes, t = the thickness of plate, and St = ultimate tensile strength of plate, then the resistance of the plate to tearing would be (p~d)tS+, or equating to exter- nal force, P - ( p-d ) tS' t (11) fe) Shearing of th e Plate . - A plate night fail by shearing along the linos in front of the rivet, as shown at E in Fig. 9. Failure in this way is not likely to occur. To find the resistance of the plate to shearing, if a equals the margin or distance from center of rivet hole to the edge of plate, then the shearing resistance offered by the plate would be 2atS 0 , and the force required to cause failure would be P = 2at8 (12) 8 23 * Regarding the use of the lap joint in the boiler construction, experience has shown that such joints are dan- gerous on account of the liability of hidden cracks. These defects are due chiefly to the fact that the lines of action of the forces in the two plates do not coincide. A couple is thun set up which causes the plate to herd, and nay therefore cause the plate to crack. Some boiler Inspectors condemn + ho use of lap joints in boiler construction. Theoretical formulas have been deduced which, give proportions of boiler joints for uniform strength. These 192 - . ' . ■ . ’ ■ 35 formulas, however, are not adhered to in practice for economic reasons. . The usual average practice for joints perhaps in this country are those proportions advocated by the Hartford Steam Boiler Inspection and Insurance Co. These designs may be found in the bach of Scully's Steel and Iron Co's. Stock List. 108 • 1 :,NMi IMFf v,v Ifj gjgjtv;?; . Ippi ► / ■ i“ • • ■ CHAPTER IV 56 Fa sjb e Ti irigs . Bolts and Hut s « 24. B olts, - A bolt is a round bar on which lias been formed a helical projection or thread. Usually, only one end of the rod is fitted with a thread while the other is upset to form the head. When used as a fastening a hollow cylindrical part which has threads formed on its inner surface in used. This part is called the nut. Sometimes one or two of the parts to ho united may have the threads in the hole through which the bolt passes. In such cases the nut as an extra part is not required. The object of bolts in to fasten machine parts firmly, and also allow the part3 to be easily separated, or disconnected. The bolt passes through the parts to be connected and when the nut is screwed down, surface compression is caused in the parts thus united, while the parts themselves react on the head and nut, producing tension in the bolt. Ordinarily bolts must resist a tensile force, although they may in some cases be required to resist a shearing f oroe . To find the bolt diameter for a given load use formula ( 1 ) . In this case should be taken rather low especially in bolts less than one inch in daimeter. A equal's the net area at the base of thread. For United States Standard Threads the diameter of bolt for a given area at base of thread is given in table in Kent, p, 205. For other threads calculations must be made for the diameter of bolt when area at the base of thread is known. 192 i . ' ' • 37 The commercial forme in which bolts aro made are as follows: ( a ) Thr o u g h B o 1 1 s , usually rough with square lioads and nuts (Pig. 11, a) or hexagonal heads and nuts (Pig. 11, b). The standard lengths of "through bolts" as givsn by the manu- facturers T catalogue are as follows: between 1" and 5" length's vary by one quarter inch; between 5" and IS" lengths vary by one- half inch; above IS" lengths vary one inch. Any ©length of bolt, however, may be obtained, but odd lengths cost more than the atan dard lengths. By length of bolt is meant the distance from point to the inner side of head. The length of the threaded part is from three to four times the height of nut. Any desired length of thread may be obtained, however. The usual forms of through bolts are machine bolts, rough or finished, and. carriage bo lts, rough (see Fig. 11, c). For special forms see catalogues. In general, the use of machine bolts is to connect iron parts, while carriage bolts are used in wood construction. The proportions for heads and nuts, U.-.d, unfinished, are as follows: Diameter of hexagonal or square head or nut across plate = 1 1/2 d + 1/8” Height of nut = d Height of head = 3/4 d + l/l6" For finished heads and nuts subtract l/lG" from above formulas. The chamfer on heads and nuts is generally drawn as shown in Fig. 11 (a) and. (b), and if the curves are drawn with 192 ■i . ■ the radii given t^ey will he in good proportion. 38 (b) Tap Bolts, more frequently called cap screws , do nor require a nut, but screw directly into one of the pieces to be fastened, the head pressing against the other piece. They may be obtained both rough and finished, and with hexagonal, square, round or filister, flat, or button heads . Cap screws are threaded either U.S.S.or V thread. In Fig. 12 are illus- trated the various forms of heads for cap screws. In this Fig. (a) represents the square head, (b) the hexagonal head, (c) and (d) the filister head, (e) the flat head and (f) the button head. The filister head may be obtained either flat or oval on top. All cap screws , except those with filister heads are thread- ed three-fourths of the length for one inch or less in diameter lengths less than four inches. Beyond these dimensions threads are cut one-half the length. For proportions of cap screws, see Kent, p. 208. Lengths vary by one-quarter inch between the limits given, and height of head except in the flat head is equal to the diameter of the screw e The angle between the sides of the flat head is 7 6° . Radius of round head 3/4 d, 25,. Set Screws . - These a. re screws which press against a piece and by friction prevent relative motion between the two parts. They are usually made with square heads and case hardened points, and may be obtained with either U.3.G. or V threads. Usually the short diameter of the head is equal to the diameter of the body of the screw. The hight of head is always equ 1 to diameter of body. Lengths vary from three-fourths to five inches by quarter inches. The headless set screw shown in Fig. 13 ( g) 192 ■ : /,* t ; . . . ‘ . 3/8" is made only in the following sizes; x «? . 5 1/2" x 9/lo" • 5/8" x ll/l6" r 3/4" x 7/8" „ The principal distinguishing feature of set screws is in the form of the point. Fig. 13 illustrates the various forms of points , Only cup and round point set screws are regular, ail other being special. In dimensioning set screws give the diameter first and length last, thus, 3/4" x 8" i.G. 26, Stud Bo lts . - A stud, is a bolt in which the head is replaced by a threaded end and having a small plain port.? on in the middle as shown in Fig, 14. It passes through one of the parts to be connected and is screwed into the other parts, thus remaining always in position when the parts are disconnected . with this construction the wear and crumbling of threads in a weak material such as cast iron is avoided < Studs are generally used to secure the heads of cylinders in engine design. There is no standard for lengths of threaded ends.,, hence this length must, always be specified. They may be obtained mil- led at B, Fig, 14, or rough and the ends threaded either or V thread. Lengths vary from 1 l/4" to 8" by 1-/4" for milled studs. For rough studs lengths vary from 1* l/2" to 4" by l/4" and from 4" to 6" by l/2" . Usually one end is made a tight f it T'hile the other is standard. 27. Machine Screw s., Machine screw is a term used, in / l its broad sense, to Include all screws that fasten into iron and metal as distinguished from those screws that fasten into ood . We shall, however, use the term to designate those screws which go by gage number rather than by diameter of the body. The table Kent p. 209 gives all the proportions which are required in 192 • ' ■ • fc ’ " ■' ' ■i. *i mmM ■ 40 drawing such screws* It will be observed that machine screws have no standard number of threads per inch, hence in dimensioning these screws, give the number of the screw, the number of threads and the length, thus, No. 30 - 16 x 1 l/f 1 ' hach. Sc. 08 «■ Forms of Sc r ew 'Threads , - The different forms given to screw threads aro shown in Fig. 15. Of these, the United states Standard and the V threads are used for fastenings, while the Square, Trapezoidal and Acme threads are used principally for transmitting motion. By pitch of screw is meant the distance from a point in one thread to the corresponding point in the next thread. Evident- ly the number of threads per inch of length is equal to the reciprocal of the pitch for a single threaded screw. When a sciew is used to transmit motion it is often desirable to have the nut advance a considerable distance per revolution of the screw. It is evident that the advance of the nut along the axis of the screw, per revolution, is equal to the pitch. If a considerable advance per revolution is desired it may happen that the pitch necessary will be too great for tli •: diameter of the screw. This difficulty is obviated by cutting two or more parallel threads, each having the same pitch or lead. Such screws are known ao multiple threaded screw s; when a screw has two threads it is called a double thread e d screw; three threads a triple threaded screw , etc. The distance between two consecutive thro ads of a multiple threaded screw in called the divided p itch and is equal to tho lead divided by the number of threads, 192 . ' • " 41 Tlie proportions of the various threads shown in Fig. are given by the following formulas: d = diameter of bolt body d]_= diameter at root of thread n = number of threads per inch p = pitch of screw t = depth of thread (a) V Thread d^ = d - 1 .733 n tt = 0 . 8f 6p , . . < - p and n same as (18) and (20). (b) United States Standard Thread. p = 0 c 24-\/d + 0.685 1.299 0.175 dl -d - 1 n= i t = 0 . 65p n \ t t (17) ( 18 ) (19) ( 20 ) ( 21 ) (c) Square Thread for transmitting motion. (Wm. Sellers 5 Co. ID ia . of | Screw 1 4 5 16 rr o 8 7 16 1 2' 5 8 id 4 7 8 1 1 18 1 1Z td 1q 1 In Threads per In 10 9 8 7 6 1 & 2 5 4 1 o A ' O 3 n Dia. of Screw - - - -- r— 1— x 8 i! if 2 n 2— cl Cj o ry 3 : .l *1 r~r y ■’z 4 i Threads per in. 2l o <5 pi o k 2* o Cj 2 r* ILL 4 1.2 4 i.2 8 i r > 8 li 2 li r> 1192 ■ (d) Trapezoidal Thread for t rails mi t ting mo t i on . * P t ( e ) Acme Thread . 2d 15 d io ( 22 ) (23) n - 1 1 ll • 2 2 4 3 1 ^ 1 1 5 — 6 7 8 9 j 10 Si .3655 2914 2419 .1801 1451 .1183 ! .087-5 ! r\ n n 4 J w o .05 66 .0470: .0411 .0361 .0319;- b .5707 .2906 2471 1853 .1483 .1235 .0927 .0741 .0618 .0529 .0463 ,041 5 .0371 t .5100 .4100 .3433 .2600 .2100 .1767 .1350 .1100 .0955 .0814 .07° 5 .0855 ,n goo In using the above formulas, after having found the pitch, use (20) in every case tofind the number of threads per inch. If this should give a number other than some convenient alch-quot part, the pitch should be changed slightly in order to give the convenient number of threads. A problem will illustrate this: Suppose it is desired to find the pitch of threads on a 3" bolt, threads to be U.S.Std. Substituting in (18) we have p = 0.28" and from (20) n = 3.57. Evidently this would be an inconvenient number of threads to cut on the lathe, hence suppose we sfiy n = 3.5, from which p = 0.288". (f) Gas Pipe Threads, - The rules for the depth and pitch of screw threads given above do not ap^ly to pipe threads, since the calculated depth would in every case bo greater th n the 192 * . • ' ... . 45 thickness of the pipe. A section of a standard pipe thread is shown in Fig. 15 ( g.) . It will bo noticed that the total length of thread is made up of three parts : full thread over a tapered rj p ?) 4. 4. p length of — — — ----- 9 when D represents t3=e outside diameter 11 of pipe and n the number of threads per inch; two threads ful'i at the root but incomplete at the top and not on a taper; and four imperfect threads. The roots of these last four threads lie on a straight line, which passes from full depth to no depth. The total ta'per is 3/4" per foot. For number of threads per inch, outside diameter, inside diameter, weight per foot, etc., see table in Kent, pp. 194-5. It should be remembered that gas pipe goes only by inside measurement, i.e. by the nominal diameter. The actual inside diameter varies somewhat from the nominal, but only the latter is used in speaking of commercial sizes. 59. Hut Locks. Since nuts must have a small clearance in order 'to allow them to turn freely on the bolt, the tendency is for them to unscrew or slack back. This tendency is especially true in the case of nuts subjected to vibration. In order to prevent this unscrewing a great many lifferont arrange- ments have been devised. (a) The cheapest and most common device is the loch or jam nut shown in Fig. 16 (a). Two nuts are used, ono of which is about half as thick as the standard nut. Since the load is thrown on the outer nut (why 1 /), that nut should be the thicker. The jam but is not always to be depended upon as a locking device. For proportions, see Fig. 1C (a). Very often the size 195 - ■ : ' . ■ , S, of the loci: nut is the same as the standard. (b) Another effective way of locking nuts is by means of sot screws, as shown in Fig. 16 (b). This arrangement is called a collar nut . The lower portion of the nut is a cylinder on the surface of which is cut a groove. This cylindrical part of the nut fits in a collar, in which is fastened a dowel pin, as shown. A set screw prevents relative motion between the collar and. tho The following proportions have proven satisfaoto Height of nut above ring 5 * = Z d (24) Height of ring = 0.55d (25) Ins ide diameter of ring = 1.5 d rr If (26) Outside diameter of ring = 2.25d — (27) 16 Diameter of cylindrical parts of nuts = 1.5d and 1.45d (28) »» *1 Diameter of sot screw = 0.2d + — — (29) 16 Diameter of. dowel-pin = Q.ld + 0.1" ( '30 ) (c) Fig. 17 (a) shows a device for locking a nut by means of a stop plate fastened at one side of it. Tho plate is bolted to one of the pieces to be secured and will hold the nut in either of two positions, or in other words , the nut nay be locked at intervals of one-twelfth of a revolution. Frequently double lock plates are used as in the case for the nuts on the studs which secure the propoller blades to the hub. The foil ow- ing proportions may be used: 192 ' .. . . . Thickness of plato = —• A 1 Diameter of cap screw = - _ . 4G (? 1 ) (30) Distance A = 1 l/8 times the short diameter of nut (33). (d) A nut lock used considerably on track work .1.3 shown in Fig, 17 (b) and consists essentially of one complete turn of a helical spring placed, between tJ e nut and the piece to bo fastened o When the nut is screwed down tightly the wash ?r is flattened out and its >. lasticity produces a pressure upon the nut thereby preventing it from s ladling off. (e) Very frequently split p ins are used to prevent nuts from backing off. ^ (f ) When using fine pitch screws nut locks arc not always necessary, as the thread angle is less than the angle .of friction. A common example of the use of fine threads is on a bicycle . 190 A * CHAPTER V 46 Keys a nd Cotters . 50 - Keys , - The principal function of keys and pins is to prevent relative rotary motion between two parts of a mach- ine, as of a pulley about a shaft on which it fits. In general keys are made either straight or slightly tapering. The straight keys are to be preferred since they will not disturb the alignment of the parts to be keyed, but have the disadvantage that they re- quire accurate fitting between the hub and shaft. The taper keys by taking up the slight play between hub and shaft are very apt to throw them out of alignment, but they have the advantage tret any axial motion between the parts is prevented, due to the wedging action. Keys may be divided into three classes, as follows; (l) sunk keys; (3) keys on flats; (3) friction keys. 51. Sunk Keys. - The types of sunk keys used chiefly are those having rectangular cross sections, though occasionally round or pin keys are used. (a) Square Key . - The so-called square key is only ap- proximately square in cross section and has its opposite sides parallel. As shown in Pig. 18 (a) this type of key bears only on the sides of the key seats, and being provided with a slight clearance at the top and bottom, it has no tendency to exert a bursting pressure upon the hub. To prevent axial movement of the hub, set screws bearing upon the key, or other mean/ 192 . must be provided. The square key la used where accurate con- centricity of the keyed parts is required, also when they may have to be disconnected frequently, as in machine tools. It is suitable for heavy loads provided set screws or other means are used to prevent tipping in its seat. (b2 Fla t Key . - The flat key has parallel sides, but its top and bottom taper. As shown in Fig. 18 (b) its thickness is considerably loss than its width b, and that it fits on all sides thus tending to spring the connected parts atid at the same time introducing a bursting pressure upon the hub. This type of key is used for both heavy or light service in which the objections just mentioned are permissible. (c) Feather or spline. - The feather key or spline is a aquare key fitted only on the sides and permits free axial movement of the hub along its shaft. Its thickness is usually greater than its width, thereby increasing the contact surface and at the same time decreasing the wear. The feather key is fastened to either the hub or shaft, while the key-way in the other part is made a working fit. The key is secured to the shaft by a countersunk machine screw, or when it is required to fasten it to the hub, dovetailing or riveting may be resorted to. Quite frequently two feather keys sot 180° apart are used, thereby equalizing the strain. ( d ) 17 o odruff Key, - The Woodruff key shown i: 18 (a) is a modified form .of the sunk key. It is patented and is manufactured by the Whitney llfg. Co. of Hartford, Conn, 188 The key-seat in the hub is of the usual form, but that in the shaft has a circular outline and is considerably deeper than the ordinary key-way. The extra depth of course weakens the shaft, but the deep base of the key precludes all possibility of tip- ping; and the freedom of the key to adjust itself to the key seat in the hub makes an imperfect fit almost impossible, while with the ordinary taper key a perfect fit is very difficult to obtain. In securing long hubs the depth of the key-way may bo diminished by using two or more Woodruff keys at intervals in the same key-seat. This form of key may be obtained in both the square and flat types . (e) Lev/ is Key. - Some twenty years ago Mr. Wilford Lewis invented the type of key shown in Fig. 19 (b). This key is subjected practically to a pure compression, and is used ex- tensively by one manufacturer on large engine shafts. Its main disadvantage lies in the fact that it is expensive to fit. (f) Barth Key. - Several years ago Mr. C. G. Barth invented a key shown in Fig. SO (a). It consists of an ordinary restangular key with one half of both sides beveled off at 45° .• It is not necessary that the key forms a tight fit since the pressure tends to force it better into its seat. This key has no tendency whatever to turn in its seat, and is subjected to a pure compression. In that respect it is similar to the Lewis key and has the advantage that it costs less to fit. This v ey may also be used as a feather key, and in a great many cases has replaced troublesome rectangular feather keys and has always proved satisfactory. 192 ' • • • - * ■ AO (g) Roun d or Pin Ke y. ~ A round or pin key ferns a cheap and accurate means of securing a hub to the end of a shaft. This form of fastening is used only for light and si^all work. - The pin, either cylindrical or tapering, is fitted half way into the shaft and hub, as shown in Fig. 20 (b). Sometimes a screw is employed in place of the pin. When using a taper pin it is advisable to use "standard taper pins", as those may be purchased at less cost than they can be made in small lots. Reamers to fit these various sizes may be obtained from any machinist suppy com- pany, 52, Keys o n Flats . - A key on the flat has par. ilel sides with its top and bottom slightly tapering and • is used for transmitting light powers . Fig. 21 (a) shows this form or fasten- ing . 55. Friction Keys. - The most common .form of friction key is the saddle key shown in Fig. 21 fb) . The sides are paral- lel arid its top and bottom are slightly tapering. The bottom fits the shaft and the holding power of the key is that due to friction alone. This form of key is intended for very light work, or in some cases for temporary service as in setting an eccentric, 54 . Strength of Keys . - Keys are generally proportioned by empirical formulas and tables, and in almost all cases these are based upon the diameter of the shaft, neither the twisting moment on the shaft nor the length of the key are consider mi. in arriving at the. cross section. Since a key is used for tori iof* alone the twisting moment to be transmitted should fix its 102 * . ' • ■ 50 dimensions, and not the diameter of the shaft, as in most cases the shaft must also re ist a bonding moment in addition to the twisting moment, thus requiring a shaft of larger diameter than is necessary for simple twisting. This means that the empirical formulas give a larger hey than is really needed, thereby in- creasing its cost and at the same time decreasing the effective strength of the shaft. The length of thbjcey should bo considered to determine its crushing and shearing resistance. Generally the twisting moment to be transmitted by the hey is determined by some machine element other than the shaft, as, for example, a gear or pulley on the shaft. In calculating the dimensions of the hey, the size of shaft hould not be disregarded altogether or the result might be a key too small to be fitted properly, or one that is too large. Prom this we arrive at the following method of procedure | calculate the required dimensions and modify these to suit practical considerations , It is generally supposed that hoys fail by ’hearing across, but this is. seldom the case. A large number of f llsr's arc due to the crushing of the side of the hey or key-seat, and it is always well to determine the crushing stress. (a) Crushing Stren gth, - To determine the crushing stress in the side of any key-seat, the following method may be used (See Pig. 22 ). Lot P = driving force exerted by the hub S- D = crushing fiber stress 192 I . . ' • v . * 51 S s = shearing fiber stress T = twisting moment to be transmitted 1 = length of key. The crushing resistance of the key is tlS ™ and its moment about the center of the shaft is approximately tld£ A Equating this moment to T, we have O, = 4T_ " b t Id (34) Assuming S-^ and having given values for T, t and d, (34) may be used for calculating the required length of the key. Occasionally a key is required to transmit the full power of the shaft; hence making its strength equal to that of the shaft, we get 5 ( 1X d S B t • 16 = — from which t = ' > S s b 4t S|~j ( 35 ) (b ) fi h o a r i n g S t r e n g t h , - To determine the shearing stress equate the moment T to the product of the radius of the shaft and the stress over the area exposed to shear, whence (36) s bid Equating the value of T from (34) to that obtained from (36) we get 2 S„b t = (37) 5 b 192 ' • . If S-., = 2S , as is generally assumed, (57) calls for a 3 square key. To facilitate fitting, the width of the key is very often made greater than its depth, which has the effect of de- creasing S relative to S, . From this it follows that invest! ga~ tions for the crushing stress are more essential than those for the shearing stress, as the latter in an actual key ta'-'os care of itself . 55. Gib-head key . - The gib-head key. Fig. .35. (b) is exactly the same as the flat key shown in Fig. 18 (b) with the head added, and is used when it cannot be driven out from the small end conveniently. It is variously called gib-head key, hoo head key, nose key and draw key. Proportions of the head are shown in the figure. 56 o Goiters. -- A cotter is a cross key used for jo ini rods and hubs that are subjected to a tension or compression in the direction of their axis, as in a piston rod and cross head; a strap and connecting rod, valve rod and sterna Fig. 03 shows one method of joining two parts by means of a cotter. Strength and Proportions. - (1) Assuming the joint in Fig. 23 to be loaded axially as shown, the following relations between the external force and the internal strength at the va- rious sections may readily be obtained. Let S>b = allowable bearing or crushing stress fl shearing stress " tensile stress. S B = S t = (a) For tension in rods 0 _ TTd 2 St (58) 192 end across the slot ^ <9 (b) For tension in the rod p = d x t s t L 4 (o) For tension in the s ocliet across the slot 2 P = | — ( D L 4 (0..; For shearing the cotter dr) - (D - )t J O JL. L ( 59 ) ( 40 ) = 2btS, (4-1) / \ <, e / For shearing the rod end P = 2 ad^Sg (42; (f) For shearing the socket end P = 2c (D - d 1 ) S s (41) (g) For crushing along the surface AB P = ditSfc (A4) (h) For crushing along the surface CE and FGf P = (D - dp) tS b (45) (2) Assuming the load on the joint to be reversed in direction so that the rods are in compression instead of tension, '.76 then have (i) For shearing the collar off the rod F = ^d ie S s (4?) (j) For crushing the collar ■Q yf 2 P = — ( 9-0 - 4 ■ d x Sx * ) -g ( 47 ) The taper on the cotter should not be j: lade ex c e o s i v e so as to permit loosening due to the load t ran e 3 ■;i ittod. 0 "c a s i on ally set screws are used to guard against this loosening of the 192 54 cotter, A practical taper is 1/2 inch per foot, but thi" m ay be increased to 1 l/2 inch per foot if some locking device is applied to the cotter. In the figure the cotter is shown as being square ended, but more often it is made semi-circular, which has the following advantages: (l) avoids sharp corners that arc liable to start cracks | (2) gives more shearing area oh the sides of the slots; (3) cheaper to make the slots. Exercise. - Deduce empirical formula for a, b, c, d n , d n D, e, and t in terms of d, assuming S g = 0,8S 192 t ' i .. kD ^ b s CHAPTER VI o a Shafting* Shafting is of three kinds, namely, rough, turned and cold-rolled, and is made almost exclusively of steel, wrought iron having gone out of use to a large extent. The rough shafting is turned only at journals and couplings, and is very little iced. The t urne d. s h af t in g is used extensively for accurate work , and the standard sizes run l/lo of an inch under the quarter inch- sizes: this is due to the fact that the regular sizes of rough shafting are turned down that amount. Cold rolled shafting has a very tough crust on its surface, acquired by its treat -nt in manufacture, and for that reason from 20 y. to 50 y stronger than turned shafting of the same size. It may he obtained in almost any size and in cheaper than turned shafting, hut the variation in diameter, which is one of its characteristics , prohibits its 57 o Shaft Calculations M 1 the following two important points must be considered: (1 ) for short or very large shafts the strength determines their size * (?) for long shafts stiffness and rigidity generally fix the diameter to be used* The general principles governing straining actions to rtinj ay be subjected, are as follows: (a) Simple twist- ing: (b) Simple bending : (c) C or 'bine d twisting and bending: (d) Combined twisting and compression. 19? Strength - Shafting is very 58 , S imple Twistin g. - rarely subjected to simple twisting, since the weights of pulleys and gears, belt pulls and gear tooth pressure cause bending stresses. These stresses are quite frequently difficult to determine before- hand, and as they are apt to complicate calculations they are omitted in many cases * Whenever they are omitted a large factor of safetj - is used in order to tal:e care of the bending. Let d = diameter of shaft P = load acting at the end of lever arm S = allowable shearing stress*; Equating the moment of the load to the moment of resistance, Pa = V- d ° S s = ? Tra- il O] 16 (d?) S s TC d° Taole I gives the values of for different- diameter 16 of shafts, and — = constant in the table. Problem. - Find the diameter of a shaft which is to sustain a twisting moment of 80,000 inch pounds. Fiber stre^ ' nc to exceed 18000 pounds per square inch. E =6.67. The nearest constant in Table I above 3-* 37 C* 8 is 6.75. The corresponding diameter is 3 \/'- y which is the diameter of the required shaft. Angular Deflection The relation between the twistir CD moment 7 and the angular deflection x may be derived as follow; 108 . ■ - . > 57 Lot = torsional modulus of elasticity. 1 = length of shaft in inches. x = deflection in inches measured on the -urfe.oe of the shaft shown in Fig a 25 „ TABLi] I . d 0 4 r- 1 "i ± 3 1 o n / n 1 16 8 16 4 16 8 16 o (-U 1 ±198 , 235 ,279 . . 32S .363 .443 ,510 r, c : R 0 ... I s— *_/ f ; c 6C1 j 9 1 o 57 1 72 1 o 881 2 c 05 2,23 2 .44 2, 63 2 . 34 3,05 3 5 0 29 5 „ 63 5 c, 98 6 c 35 6 a 73 7.13 7,54 7.937 8.41 4 12 o 55 13 o 15 13,77 14 . 40 15.- 05 15,73 16.42 17. .14 17.87 R 24 e 51 25 c 44 26 0 40 27 « 36 28,37 29,41 30 - 45 31,53 32.62 ' 6 4 2 o 3 5 43 * 69 45 r 06 43 , 95 47 c 87 49./S3 50.80 52.32 R r Or ' c Jo 7 67 o 25 69 3 08 70 o 92 72 c 81 . . 74,72 , , - - 76,68 78 o 65 80 . 60 82.72 8 100.39 102 o 77 105.17 107 e 62 110,10 i 112,63 115,16 1 ■ 1 117,73 120*41 ±563 ' > - . . o 1C CD|Ol 11 '16' ic. 1 ! Si 5 | l _ i 7 O o 15 16 j d i ! 1 i • i .84" . 945 1.05 1,17 1,29 1 . 43 1 . . 4 . 3.30 rr r- O o oO 3.81 • 4.08 4.37 4.66 4.97 1 2 i P Q7 O • o / Q rr a a „ - _ , 9.84 , ... . - . , 10.34- IQ. 87 T 11.41 11.98 I ' 3 i 1 L 18. S3 19,40 20.20 21,01 . ... 21,86 22.72 O ^ /3 wO « *w ; 4 ! ■'.'3.76 34.90 ' 36.09 37.28 38.52 39.76 41.05 . ! 5 j i 55.48 57.01 - 58.65 60 , 30 62.06 63, 7g 65.48 5 i 84.82 86.95 89,10 91 o 27 93 51 op, na '.J Si mple Bend ing - St rength - 'In designing machinery we frequent ly use stationary shafts, upon which certain heavy parts revolve, these revolving pieces being brass bushed. Such shafts are sometimes called pins, and may be proportioned for a simple bending moment, since it is not required to transmit any twisting moment. Equating the bending moment M to the moment of resis- tance, we have M = 7Td‘ 32 or M = 0.1 d° (nearly) ! n;o N i V Od ) Table II gives the values of 0.1 for different M diameters of shafts, and 3 = the constant in the table. 192 TABLE II 60 |d 0 1 16 I s rr _£ 16 1 4 -4 l lo ' 1 to |(D 7 16 1 2 |l 0.1 ■ • 1 .119 • .142 .169 0.195 - . Q 3 .260 * ' .30 ,338 g .8 . 88 » 960 1 <>03 1.14 1.24 1.54 1.4-6 1.56 5 2.7 2.88 5 c 05 r* o sx » O v> 3.45 3.64 3.04 4.07 4,29 4 c A o .A 6.72 7.02 7.3-5 7. 67 “ - 8.03 8 . 37 8.75 9.11 K i 12.5 12.99 1 3 . 4-6 15.97 14.47 15.01 V> 5" 1c . 09 16 . 34 6. 21.6 22 . 30 22.98 23,70 24.41 @5.17 9^ O'] 26.09 27.46 !a 34 . 3 n cl at\ 50 •- ffcO 36,17 37 . 84 ' 38.10 39,-11 • 40 . 11 ' 41 . 15 ' 42,1 9 1 a nr i o W ±- 9 Cj ' 53.64 r 54.90 i 5G.15 57.45 58 , 74 60.08 61,41 S 16 5 8 11 16 5 A 13 16 7 8 LC 16 d .383 f .. .429 .484 . 800 ,598 — p nr r* . / o 1 1.69 / , 1.81 1 . 95 — 2.08 i 2.24 ! 2.38 — , — ; 2.53 o 4.53/ ^ 4.76 5.02 — 5,27 5.55 5 . 82 6,12 3 Q Ak-j 9.89 10.31 10 c 72 11.10 11.59 12,05 4 j 17. #2 17.79 18.41 £9,01 19.65 20 . 28 20 . 95 5 23a 2 8 29.08 29 o 92 30.75 31.63 32 . 50 r— r-r A - 1 4' O c 4*1 ,, I 6 43.27 A/. 1 45.45 7 ' 46.55 47.70 48 . 84 50.05 r. 1 7 I 102 ,70 64.16 T~ 65 o 5Qf 66 . 99 68.46 69.91 — 71.41 3 1 192 ■ ■ . ■ . 61 Problem* - Find, the diameter of a shaft vfhich is to take a bending moment of 45*000 inch pounds. Fiber stress not to exceed 15*000 pounds per square inch* M = Z a The nearest constant in Table II above 5 is S 5.05, corresponding to a diameter of 5 1/8", which is the required diameter . Tr a nsverse Seflection . - Transverse deflection is that due to the bending of the shaft and may occur alone or in con- nection with the angular deflection discussed in Art. (58). For line and counter shafts a deflection of 0.01 of an in c h per foot of length is considered good practice, while for certain kinds of machinery this would be excessive. Transverse deflection depends upon the method of loading the shaft, and formulas for these deflec- tions may be obtained from the corresponding beam formulas given in Kent „ 40. Combined Twis t ing and Be nding * - A rotating shaft carrying gears, pulleys, drums, etc., is subjected to both bending and twisting, when used for transmitting power. Calculating the diameter of the shaft by either of the above tables, ignoring the other, would result in a weak shaft* We may, however, substitute for the simple twisting moment T a -greater twisting moment T 0 , which is the equivalent of the combined twisting moment T and the bending moment M. Thi3 equivalent twisting moment is derived direc ly from the expression for combined stress, as follows: Merriman in his Mechanics of Materials, p. 265, gives the following formula for combined stress: 192 ■ 62 3 ‘ / 2 -g 2 Max*- '"ten • . or~T*-omp 0 S"t:r&-su S e - -g* +y S s ^ + '•% ( 53 ) Substituting in (53) the values of S a and . S. from (48) and (52) above ^0 srrd 3 is 16 T d l \ \L2 M + /T" + M' 2 (54) n» - m\/ t 2 + K-ow since — — equals the equivalent twisting moment T e M 2 ( 55 ) To find the diameter of the shaft suitable for the combined moments T and M, substitute the value of T e for T in (48) and proceed as outlined in Art. 38. Equation (55) is the one used almost altogether by American designers. Guests * L aw - In 1800 Prof. J. J. Guest published an article in the Philosophical Magazine giving results of a series of experimental researches on the strength of ductile materials under combined stress. He found that the shearing stress is the factor which governs the yielding of the material when subjected to combined stress. From this theory he deduced the following formula for the equivalent bending moment i M e =\/~m2 + T 2 (56) In England (56)' is now adopted, as the correct formula to be used in the design of shafting, since it' is based upon ac- tual experiments. It will, also be considered as the standard formula to be used in the solution of problems pertaining to this course. To find the diameter of a shaft subjected to comb'ned- bending and torsion, calculate the equivalent bending moment 192 •• \ • ' i! L ■ ' ■ ’ by means of (56) and substitute it for M in (52) and proceed as outlined in Art. 39. Problem,,- Find the diameter of a shaft subjected to a twisting moment of 80,050 inch pounds and a bending moment of 4-5000 inch pounds. Fiber stress not to exceed 12,000 pounds per square inch. From (56) M g = 91780, and from (52) —A? — = 7.65. The nearest constant in Table II above 7„35 is 7.67, corres- ponding to a diameter of 4- l/4” , which is the required diameter. To show the difference in the results obtained by unin (55) we find the diameter of shaft to be 3 7/8” , thus showing that it is better to use (53). By comparing the three problems just shown, one readil sees the importance of considering both the bending and the twin- ing moments upon any shaft that is subject to these actions. 4:1. Combined. Twisting and Compression . - Propeller shafts of steamers and vertical shafts carrying considerable weights are subjected to these straining actions. Two cases arise in practice, as follows: (a) Then the span, or distance, between bearings is so smal tfcat the shaft may be considered as subjected to simple compres- sion, so far as the action of the thrust is concerned. (b) '.Then the span is so great that the shaft must be consi- dered as a column liable to buckle. Case ( a ) - The intensity of compressive stress fci a solid shaft is AP Q — 0 7rn ’ in which 192 p equals the thrust. From . , (48) the intensity of shearing stress due to the twisting moment 16 T on the shaft is S. ■ 7T d' : ’he resultant maximum stress due to the combined action of S c and S g may be found from the expres’ sion for combined stress, see (58) „ / Max, Compressive Stress is \L 2 , + v o s s 2 b c (t7) Substituting values of S Q and S s in (57), we have o j j ^4- 1 Max o Comp 0 Stress = ^ ^ 2 j P * \/ P * up j ( 58) To find d for given values of F, f and maximum compres- sive stress, assume a trial value of d (somewhat larger than required for twisting moment alone), and then check for maximum- stress „ Case (b) - The value of mean intensity of compressive stress in a long column is given by Hitter’s formula (p. 711, Meriman’s Mechanics of ‘Materials), that is H TC"~hr in which P = the thrust or load L = unbraced length A = area of cross section E = coefficient of elasticity S = unit stress at the elastic limit S’= the greatest compressive unit stress on the concave side, m = varies from .25 to 4. = least radius of gyration of cross section. 172 r / Gb Since the stress found by (59) is the re an intensity of compress j ve stress in the long column which corresponds to a maximum compressive stress S 0 , a short compression member of the same cross section A would be capable of supporting a load which is greater than P in the ratio of to 3’ or the new load c c C p pt = u g (do) a i C Plow to find the diameter of shaft, necessary to sup- port a load P and twisting moment T, use (58) as before, but substitute P' for P» 42 Hollow Shafts c - In any shaft the outer fibers are mere useful in resisting bending or twisting than the fibers near the center, from which it follows that the weight of a ho 1 1 ow shaft is diminished in greater proportion than its strength. Let d = the diameter of a solid shaft having the sane strength as the hollow shaft. d-j_ = outer diameter of hollow shaft, dp = inner " " " ” For a hollow shaft subjected to simple twisting m _ £ s ^fdf - af) X ’ ... ■ - ■ ■■ - M - - — ■ - IS d x The value of T for a solid shaft is given by (48) Henoe to make the two shafts equally strong, we must have (51) 13 d : 1 99 s / IS •r i ee or d 1 " = d- d 0 a i - { » ) Letting d- = m d i = \ l m*' V ~ 1 d \ /— For average value of m = 2 (62) ( ) d 1 = 1.02d. ( 64 )_ 45. Bending Moments , - In calculating the bending moment that a shaft is subjected to, the various authorities differ in the method used for determining the length of the mo- ment arms. It is generally assumed, by all, that a shaft running freely in its bearings is so loose that it could easily deflect to the center of the bearing. Therefore for a shaft of this kind the moment arms are measured to the centers of the bearings, and a shaft designed on this assumption is generally on the safe side as far as strength is concerned. Whenever a machine part is driven or forced tightly upon a shaft, it is practically impossible for the shaft to break at the center of the hub, but may fail near either end. This is due to the fact that the hub may become loose since the bending of the shaft would tend to localize the crushing at those points. According to Mr. C. L. Griffin this distance may be assumed to lie between l/2 inch and 1 inch. The method of assuming the moment arms as extenGin * to the center of the hubs and bearing is the one coma 'only used, and 192 , 67 -.Till be adopted in all problems in this course. It is simpler and saves considerable time, 44., Problems. - The following problems are given to illustrate the general method of procedure in obtaining the bending moment diagrams and finally determining the size of shaft required in any given case. (a) Two Gea rs be twe en Bearing s. - Let Fig. 23 show the method of loading. 1-Tow calculating the reactions at A and B due to the load on the gear e, we can readily find the bending moments at the points G. In like manner we can determine the bending moments at D due to the load on the pinion f . TTow con- struct the bending moment diagrams ASB and AFB as shown, thus showing at a glance where the maximum moment occurs, and at the same time permitting of a graphical addition. Besides sustaining the bending moments the shaft also transmits a twisting moment T = PR. To obtain the required diameter find the value of M and proceed as explained in Art. 40, ( b) Gear and Two P ini ons, - The loading of the gear and pinions is shown in Fig. 27 (a). Proceeding as outlined above lay off to some convenient scale the bending moment dia- grams for each load, and since P and W act in the same direction, the bending moments due to these loads will oppose each other. Thus the shaded portion of the diagram represents the algebraic sum of the two moment diagrams. How combining the twisting- pis moment T = — — with the maximum bending moment as outlined in •Zj Art. 40, the required diameter of the shaft may be obtained. ' , 68 Assuming the loads P and U acting at so s angle to each other. In order to find the bending moment at any point on the shaft as at D, lay off separately in the direction of each force, the bending moment ordinate for that force. For example, at the point C the moment M-j due to the loads W is BF in Fig. 87 (a), and the moment Mg due to P is CG« Assuming the loads acting at right angles to each other, lay off in Fig. 87 (b) Mn parallel to W and Mg parallel to P, their resultant M gives the bonding moment at the point C. Having found the maximum moment by this method, and knowing the twisting moment, find the necessary dia- meter of the shaft as outlined above. Note. - The methods described in the above problems may be applied to ail forms -of leading. (o) Hoisting Dr um. - Fig. 88 shows a common method of supporting the hoisting drum of a crane trolley. The drum is hushed and runs loose on the shaft, hence no twisting moment is transmitted through the shaft. Let G - the weight of the gear. D = " ,f " 11 drum. . ? = " lohd oh the gear teeth acting horizontally, * W = " ?f " each rope . Reducing all load.s to the more heavy load support we have for the horizontal lead at A H = J (1 - t) (87) and for the vortical load at A 192 > V = Gr( 1 t) + W(b + 2c 0 + Dd (3?) . £ Combining H and V in a manner -similar to that used in Prob. (h) for moments, the resultant pressure at A is obtained* By similar reasoning the reaction and resultant pres- sure at B may be obtained* Knowing the resultant pressures at A and B calculate the bending moments at C and S, and whichever is the maximum must be used in finding the required size of the shaft, using the method given in Art. 39. (d) In Fig. 29 is shown diagrammatic ally a hoisting drum, gear and pinion, in which E is the center of gravity of the rope loads W. Whenever there are two ropes on the drum, the position of E is constant while for one rope E moves along the drum, and for the latter case several solutions should be made with varying positions of E, The load W is supported upon throe points, namely, the journals A and B and the gear tooth at D„ The load W taken separately, causes an upward reaction on each journal, and. Is divided proportionally between them. The tooth load P Is also divided, proportionally between A and E, and causes a downward reaction on each journal. The algebraic sum of the two loads upon the journals gives the amount and. direction of the resultant load on the journal. In Fig. 29 draw BD and DE produced to cut AB at C. We thus represent the rope load W as supported eccentrically upon a beam BD with an arm EH and prevented from rotating about 3D by 1 *T OO 1 * 7*3 I ■ ’ 70 the reaction of the bearing upon the journal A acting HG. From this we derive the following relations: A downward pressure of the bearing upon the journal A An upward " " " " " " " E with an arm EF ■W=W ? FA FD BD (W + TV ) m f? BF on the gear teeth at D = grj (W + W f ) . The condition of loading at the journal A is seen from, the position of the point 0, which lying beyond B as in Fig. 2 r - , indicates a downward pressure upon A; a position at B would indi cate no pressure upon A, and a position between A and B would indicate an upward pressure on the journal. In the above dis- cussion the weight of the drum and gear was neglected. 192 1 CHAPTER VII* 71 Bearings and Journale j 45. Bearings . - Bearings may be divider?, into two general classes: (l) Sliding; (2) Rolling* Sliding Bea’rin-g . - There are three types of sliding bearings : (a) Right line bearings in which the motion is parallel to the elements of the sliding surfaces: This type may again be subdivided into three hinds as follows: (1) Ordinary flat guides a,s used on engine crossheads and punching machine rams. (2) Angular guides as used on planers and lathes. (5) Circular guides as used on some engine crossheads, piston and stuffing boxes. (b) Journal bearings. - When two machine parts rotate relatively to each other, the part which is enclosed by and rubs against the other is called the journal , while the part which en- closes the journal is called the box or less specifically the bearing . Journal bearings may be divided into three classes: (i) Journal rotating inside a fixed bearing, for example a shaft in its bearing. (2) Journal fixed and bearing rotating as a loose « pulley on its shaft. (3) Journal and shaft both having a definite motion as in the crank end of a connecting rod, (c) Thrust bearings. - A thrust bearing is one designed for taking the end thrust of a shaft. There are two kinds: (l) Step or pivot bearing is used on vertical shafts and supports the weight of the entire shaft together with its attached parts, and any other force acting vertically downwards . The shaft terminates at the bearing. 192 ' •in- ■ >1 j ;• 72 (2) Collar or thrust bearing as used on a propeller shaft or on a spindle of a drill press. In this case the shaft extends through and beyond the bearing. Rolling Bearings. - Rolling bearings may be divided into two types as follows: (a) Ball bearings, in which hardened steel balls are placed between the journal and its box, thereby reducing frictional resistance, since sliding friction is changed to rolling friction. (b) Roller bearings using either cylindrical or conical rollers in place of the balls. This arrangement distributes the load over a larger surface. Each one of the above types maybe subdivided into the following: (l) Right line bearings; (2) Journal bearings: (?) Thrust bearings. 46. Bearing Surfaces for Journals . - Bearing surfaces are made of many different substances, depending largely on the class of work in which the bearing is used, and the conditions under which it must run. The following is a list of some of the materials that are used for bearing surfaces: phosphor bronze, ordinary bronze, lumen bronze, babbit, cast iron, chilled cast iron, tempered steel, case-hardened steel, mild steel, also the self-lubricating materials, fiber-graphite stone and wood. The materials phosphor bronze, ordinary bronze, lumen bronze, and babbitt may be classed as anti-friction metals, and together with other alloys of a similar character, are U3ed for lining bearings on machinery of rather high grade. One of the objects of these linings jj_?s to prevent wear on the journal, by 192 running it in contact with a metal softer than itself. This result is very desirable because the bee ring metal cen be much- more eaoi iy and cheaply replaced than the journal. The main requirements for a good bearing metal are the following: (&) Least liability to heating. (b) Sufficient strength # *r to prevent squeezing out under the load. (c) High melting point. (d) Least abrasion in service. (o) Least possible abrasion of j ournal . Experimental Conclusions. Some years ago the Pennsyl- vania Railroad made 3one exhaustive service teste with various combinations of cop.-ar, tin and lead, in order to determine the composition which would be best suited to thsir requirements, and the following conclusions were drawn from the experiments: (a) Ordinary bronze shows 50% more wear then phosrher bronze . (b) The phosphorus plays mo part in preventing wear, save by producing sound castings. (c) Wear diminishes with tne increase of lead. (d) Wear diminishes with d^mi’nution of tin. (e) Alloys containing more than 15-/? of lead or loss than 8;": of tin could not be produced because of segregation: but it is believed that if the lead coulch be still farther increased- and the tin decreased and still ha>ve the resultant alloy homogeneous, a, better metal would result. Since that time allojye have been successfully forced containing a much larger percentage of lead, 192 and tests b av e I '*3 * 74 born© out th© conclusion of the Pennsylvania Railroad. How- ever, the gain in ivearing qualities is somewhat off? et by the decreased compressive strength of the metal. Leadj - Of all bearing metals lead is by f tr the first in anti-friction qualities, but it has so little compressive strength that many alloys containing large percentag s of it will yield under ordinary pressure, with the result hat the metal is squeezed into the oil inlets and stops lubricatio: .. Bronzes . - The composition of phosphor br nze and ordi- nary bronze ae used by the Penn. R. R. , is as folio s: Phosphor bronze contains 79.7 y'o copper, 10 fi tin, 9.5^ lead and .8$ phos- phorus; Ordinary bronze 87.5^ copper and 12.5^ tin. Lumen bronze, invented by Prof. R. C. Carp nter of Cornell Univ . and manufactured by the Bierbaum and Merrick Metal Co. of Buffalo, is claimed to be an excellent substi ute for phosphor bronze. It has a high compressive strength and pos- sesses the peculiar property when heated of incr9asi: g in strength until a temperature of 350® F. is reached. Babbitt Metal . - Babbitt metal is probably the ?->ost extensively used of all the bearing metals, one roan n for this being that the term is applied to an innumerable numb < r of dif- ferent white metal alloys. A good composition is as follows: lead 70^, antimony 20;'?, and tin 10fe. For further information regarding bearing rr stal alloys see Kent pp. 319 to 358. The physical condition and structure often ias as 192 , . .. i. *-;*•$ *n " I ■ - . ♦ • ■ • . ‘ J) 75 much to do with the running qualities of a bearing motrl as the chemical composition. This may be determined by a micro- scopic examination of a fractured section which has been polished and etched with acid. The microstructure test should be included in specifications for bearing metals in important work. The conditions which cause heating of the bearing and should be particularly avoided are : (a) Segregation of the metal: (b) Coars'e crystal lino structure; (c) Dross or oxidation products and an excessive amount of enclosed gas in the metal. The materials cast iron, chilled cast iron and mild steel ar-- often used in cheap work. Cast iron bearings with steel journals have met with considerable success when the bearings were long, and some eminent engineers have advocated their exten- sive use, claiming that the surface will in a short time wear to a glassy finish and run with very little friction. However, if for any reason lubrication fails and heating begins, the result is liable to be either serious injury or total destruction to both bearing and shaft, and for that reason the cast iron bearing has fallen into disfavor. Tempered steel and case-hardened steel have been used rather extensively for bearings where the pressures are great. Self-lubricating materials are used in places whore i lubrication would be very difficult and liable to be neglected or where oil or grease would be harmful. Fiber-graphite ie a patent substance, and is composed of pulverized graphite mixed with wood pulp in a bath, of water. The mixture is 192 then moulded . .. ' , ' * ■ '' . . • . 76 to the desired shape, and the water is squeezed out by heavy -pres- sure, after which it is treated with oil and baked. It is said to work very well under ordinary pressures. Stone is very little used for bearings in this coun- try, except for light work when minimum fric'tion'is desir- ed, in which case the precious stones are used extensively. The jewels of watches, of electrical instruments, furnish excellent examples of this class of bearings. Wooden bearings are frequently used when the bearing is submerged in water, and are also used occasionally in cheap machinery. . The stern tube bearing for the propeller shaft of a ship furnishes a good example of a bearing submerged in water, and is made of lignum vitae, a wood which is well adapted for service of this kind. 47. Lubrication . -,The function of the lubricating device is to keep up a continuous film of oil between the journal and its bearing, so that the metals will never be in actual contact. When this is done the journal practice. lly floats in a bath of oil, and runs with very little friction. However, when lubrication fails the coefficient of friction is at once, increased, the bearing begins to heat and the journal either seizes or cuts out the bearing metal very rapidly. The lubricants used for different kinds of bearings range from the thinnest oils to heavy grease, and in some cases even to graphite, a solid. For slow speeds, with heavy contin- uous pressures grease i3 better than oil, because it is not so 192 77 4 easily squeezed out from between the surface of contact: while with faster speeds the oil is continuously carried to the point of’ maximum pressure so rapidly that there is not time enough for the oil film to be squeezed out. In general it may bo said that the viscosity or "body" of the lubricant varies inversely as the speed of the journal, and directly as the pressure upon it. Usually the oil reservoir is a part of the bearing itself, but when the bearing is not a casting it is more conven- • ient to have the oil cups separate. It is now quite common to oil the bearings of engines in large power plants by forced lubrication. The oil is forced * j by means of a pump through various bearings, after which it is returned to the reservoir, filtered, and again forced through the system. In this way the bearings are oiled continuously, and also cheaply, because the oil is used over and over again and needs very little replenishing, 46. Design of Journals . - In designing a journal there are usually three things which must be considered; first, the journal must be strong enough as a beam to support the load which comes upon it, and if it is also subjected to a twisting moment this must be combined with the flexure in calculating the strength second, it must be sufficiently rigid to obviate springing and consequent heating of the bearing; and third, the pressure per square inch of projected area must be Iovt enough to prevent squeezing the lubricant from between the bearing surfaces. 192 . * ' 78 (a) Strength . - End journals are generally considered cantilever beams loaded uniformly, (soe Fig. 30). Equating the bending moment to the moment of resistance, PI _ 7? d 3 S d 32 from which d = 1.7 v ; p 1 si (77) (b) Stiffness. - From the table of beams in Kent’s p. 238,. the deflection A = P1 ' 3 (en) p-pT W-h -A- In good practice the valu ) of A is generally limited to — - — of an inch. 100 Substituting in ( ■38) this value of A and for I its equivalent in terms of the diameter we have approximately 4 AS d = 4 \ i 1 1 (69) ’ E Allowable Bearing Presaun e. ~ The following table, taken from Unwins, and other authori ties, shows the allowable pressure per square inch of projects l area for different types of machines. (a) Bearings for slow speed.e an L intermittent load.s Main journal on punching m .chines 2000 - 3000 Grant: pins on punching mac! ines 5000 - 80 n 0 (b) Engine crank pins High speed engine 250 - 600 192 Low speed engine 859 — 1300 Locomotive 1500 - 1700 Marine 400 - 000 Petrol 350 - 400 (c) Engine cross head pins High speed .engine soo - 1700 Low ,f H 1000 - 1800 Marine 1000 ~ 1300 Petrol soo -!■ in no (a) Engine main journals Center crank high speed engine 100 - 240 Side u low " f( 100 ?G0 Petrol 550 - 400 Marine 150 400 (e) Locomotive driving journals Passenger 190 Fr e i ght GOO Switching G20 (?) Car journals 300 600 (g) Motors and generators 50 - 90 (h) Thrust blocks for propeller shafts 40 - 80 (i) Engine slides Marine 50 - ICO Stationary o r fj s..* - 40 (3) Eccentric 3heaves so _ 100 (k) Hoisting machinery shafting 192 GO . so The length of a journalln proportion to the diameter is a ratio which the designer must first choose according to his own judgment, and afterwards adjust to the three conditions men- tioned above. The ratio varies from .5 to 5 or 6, usually ranging from 2 to 3 for ordinary rigid bearings when there is no limitation on space, from 3 to 4 for rigid shaft bearings, and from 4 to 5 for self-aligning shaft bearings. In general, the ratio decreases as the diameter increases and increases as the speed increases. 49. Design of Bearing . - Having determined the dimensions of the journal, the next step is to design the bearing. This is an excellent example of the kind of machine design, where calcu- lations are of little value. Should an exact analysis of the stresses be made, it would be found that they arc very complicated, and in many cases so small that the proportions of the part would be determined by other considerations. For this reason it is impracticable to calculate the thickness of walls and gen- eral proportions of the bearing, and they are usually taken from parts which have been in actual service and have been found successful under conditions similar to those of the new design, scale drawings are often contained in manufacturers T catalogs with a table of the principal dimensions for different sizes, and these are a great help to inexperienced designers, A designer, however , should make himself independent of those aids bv stud v- ing standard designs, and becoming so familiar with t-'em that ho is able to proportion parts of this kind simply from judgment. In addition to these pi pfybrtions there are a few impor- tant points of design with which he should be thoroughly familiar. 192 . I . y- u ■ «r • (1) Some means of adjustment for wear should be provided. This adjustment may be made in various ways but the most com- mon methoe is to use a split bearing bolted together, in which case the wear may be taken up by simply tightening the bolts as shown in Fig. 31. (2) The line of division of the bearing should bo perpenuicU' lar to the line of pressure so that the surface having maximum pressure on it will be continuous and not have any sharp edges as they would tend to scrape off the oil film. The division should be made as shown in Fig. 31 for two reasons: first, be- cause when made in that way it helps to prevent the box under pressure from springing together at the 3id.es and gripping the shaft: second, because it prevents the lubricant from escaping at the split. (3) Eearings should always fit loosely at the points where they are divided, since the sharp edge would tend to scrape the oil film off the shaft if it were tight, and also because it is the natural tendency of a box which has been bored absolutely true to fit tightly at the 3ides, and more especially so if it begins to beat. This action is illustrated by Fig. 32. (4) The method of applying the lining to a bearing is some- what different for the different metals. The brasses and hard alloy linings are cast separate, and when used in split bearings are aquare or some irregular shape to keep them from turning in the bearing. Boxes made in this manner are hand fitted, but when they are used in a solid box it is customary to turn the 192 ■ * 4 . * . ’ ■ * bushings on the outside, bore the bearing and then drive the bushing to place. The babbitt linings are formed by pouring the molten metal around a mandrel the same size as the journal, into a recess in the bearing which was provided, f'cr that purpose. After the babbitt cools the shaft may be readily withdrawn, In better work the babbitt is poured around a mandrel. of smaller diameter, and then hammered or peened in and boned out to size. (5) In a bearing of any considerable length the pressure between the journal and the box is not uniformly distributed along the entire length and the maximum pressure naturally occurs at the point where the box is rigidly held. This point is usually located in the center of the bearing, so for this reason the oiling device on a cylindrical journal should be at that point if possible. The lubricant will naturally move in the direction of least resistance and then find, its way to the ends of the bear- ing; 1 at ?v r hich points, either wipers or drip pans should, be Pro- vided so that the oil cannot run out on the shaft and. drip to the floor. (6) Then it is impracticable on account of the length of a bearing, or its construction, to place the oiling device at the center, the proper distribution may be accomplished by the use of oil grooves in the bearing metal. Oil grooves may bo divided into two classes: first, the grooves proper which distri- bute the oil in the bearing: and second, the -feeders which carry the oil to the grooves. The oil grooves should always be cut 192 ■* ■ . ■ . ■ RPW ' . ■ - . $pi parallel to the journal, and be placed at a point where the prespure is slight, so that they wi' 1 serve to fcr^ a uniform and continuous film of oil on the 3 urnal , and not act as a wiper to scrape off what oil film ha 3 air )ady been formed. The feeders should always be placed at right ar ^les to the journal. Although the groove is often cut obliquely : t is bad practice as the oblique edges are more injurious t< the film than straight ones, and they cannot be placed a,t point of least pressure. (7) ho escape should be provided for the oil at points of maximum pressure. 50. Design of Ca;^ and I sits . ~ The oar of a bearing subjected to an upward pressure if generally regarded as a beam supported by the holding down bol s or screws and loaded at the center. Both its strength and st .ffness should be considered. Let b = distance between bo? is e = depth of cap as she n in Fig. 33 1 = length of bearing P = maximum upward pres jure £ = maximum, allowable 6 ^flection = ” r gf“ of an inch. Strength - Equating the ber ting moment to the moment of resistance Pb _ Sle^ A from which o V 84 Stiffness -From the table of beams in Kent * 3 p. 808 , the deflection 4S - Pb' 48 El Substituting in (71), le' 100 (71) of an inch for & f 18,0PQ,P00 for E (cast iron), and for I, then 18 e = . 0112 b (72) Bolts or Screws . - The holding down bolts, screws or studs should bo designed for tension, assuming two thirds of the max- imum upward pressure comes upon one bolt. 192 t - -■ iig ' CHAPTER VI 1 I Sp ur Gearing. Spur gearing may be of tvo forms , friction gearing and toothed gearing . 51. Friction Gearing . - The simplest form of gearing: is the plain friction, consisting of a pair of cylinders or cones hold in contact with sufficient pressure to produce rotation of the driven gear simply by the friction between the two surfaces. (See the bevel frictions on the shaper in the University Wood Shop.) These gears aro used when the speed of rotation is high, and when an absolutely positive drive is either non-essential or un- desirable. In friction drives the driving wheel is often made- of paper of come sort of fiber, while the driven wheel is made of metal. However, in many cases both gears ar ? made of metal. The velocity ratio of a pair of frictions, provided there is no slip- ping between the surfaces of contact, is D, Ho — = — (73) Jo in ~*hich D and IT are the diameters ond revolutions per minute res- pectively. The subscript 1 refers to the driver and 2 to the driven, V.'ith this type of gearing there is always a certain amount of slip- page which must be taken into consideration, and which with properly designed gears ^ hould be about 2 : / , 5P. Toothed Gearing. - Then it is desired to transmit an absolutely positive velocity ratio, or when the surface speed is not 19£ very great, it becomes necessary to provide the surface with projections and grooves as shown in Fig. f>4. The original sur- faces of the frictions then become the pitch surfaces of the toothed gears, and the projections together with the grooves form the teeth. These teeth must be of such a f err. as to satisfy the following conditions: (a) They must transmit a uniform velocity ratio. In order to do this the common normal at the point of contact of the tooth profiles must always pass through the pitch point, i.e. the point of tangency of the two pitch lines. (b) The relative motion of one tooth upon the other should be as much a rolling motion as possible, on account of the greater friction and wear attendant to sliding. With toothed gearing, however, it is impossible to have pure rolling contact and still maintain a constant velocity ratio. (c) The tooth should conform as nearly as possible to c cantilever beam of uniform strength, and should be symmetrical on both sides so that the gear may run in either direction, (d) The arc of action should be rather long, so that more than one pair of teeth may be in mesh at the same time. Tooth Curves . - There are a great many different curves that would serve as profiles for teeth and satisfy the above con- ditions with sufficient accuracy for all practical purposes, but the ones in actual general use are only two, namely, the involute and the cycloidal . As regards strength and efficiency, the two forms aro practically on a par. However, the involute tooth has 192 I . orr one decided advantage over the cycloidal; namely, the distance "between centers may be slightly greater or less then the theoreti- cal distance, without affecting the velocity ratio. The cycloi- dal tooth, also, has one important advantage over the involute; namely, a convex surface surface in always in contact with o, concave and although the contact is theoretically a line, practically it is net, and consequently the wear is not so rapid as ’■•ith involute teeth when the surfaces are all convex. 55. Methods of Manufacture. - Gear teeth are formed in practice by two distinct processes: moulding acid m achine cut ■: ing . Originally all gears were cast and the mould was formed from a complete pattern of the gear. Of late years, however, gear mould- ing machines have been used to a considerable extent end the re- sults obtained are far superior to the pattern moulded gear. Even with machine moulding , however, the teeth are somewhat rough and warped out of shape, so that the gears always run with con- siderable friction and are not suited to high speeds. In this country the gears of ordinary size s/re almost always cut, except in cheap machinery. The method which is most commonly used is to cut them with a milling cutter, which has been formed to the ex- act shape of the tooth. There are also two stylos of gear planers, one of which generates mathematically correct profiles by virtue of the motion given to the cutter and blank, and the other forms the outlines by following a previously shaped templet. 54. Materials of Gearing. - The material used for gear teeth are machine steel, steel casting, cast iron, bronze, rawhide, 192 - - ' ■ I . . . . . I 86 fiber and wood. Machine steel pinions are often used with large cast iron gears in order to he up for the weehre 00 of the teeth on the pinion, due to their decreased section at the root, by using a strong material. Steel castings are used when the gears are of large size and are subjected to violent shocks and heavy loads. Bronze is frequently used for srur uinions meshing with steel or iron gears, and when properly cut ^ay be run at very high speeds. In worm gearing the gear is often made of bronze and the worm of steel. Good bronze is considerably stronger than cast iron. Cast iron is the material which is more frequently used by far than any other. Rawhide and fiber gears are used when quiet and smooth running free from vibrations is desirable. Rawhide gears are stronger and preferable to fiber. The Hew Process Rawhide Co. of Syracuse, N.Y. claim their gears to be equally as strong as cast iron gear of the same dimensions. They are furnished with or ’without metal flanges and bushings, and the teeth are cut just the same as the metal gear. They are usually of smell size, although larger ones are sometimes made in which the teeth only are of rawhide, the center being made of cast iron. They are often used as the driving pinions on motors, and the fact that rawhide is a non-conductor is a marked advantage in this instance. Wooden teeth are sometimes used for the large gear of a pair, in which case the teeth of the mating gear a.r^ usually cast. The wooden teeth or cogs are morticed into a cast iron rim. and 1S2 their purpose is the same as in the c&ae of the rev hide and liber gears, to reduce the noise and vibration by their greater elac- f ticity . 55. Involute System, - In the involute system ef veer- ing the outline of the tooth in an involute of a circle called the base circle. However, when the tooth extends b'dow the bc.se circle that portion of the profile is made radial. The simplest conception of an involute is as follows: if a cord, which ha" been previously wound around any given plane curve, and has a pencil attached to its free end, is unwound, keeping the cord perfectly tight, the pencil will trace the involute of the given curve. The base circle may easily be obtained by 'bring through the pitch point a line making an angle with the tangent to 1:' e pitch circle at this point, equal to the angle of obliquity of action; then the circle drawn tangent to this line will be the required base circle. In order to manufacture .gears economically it is very essential that any gear of a given pitch should work correctly with any other gear of the same pitch, thus making an interchange' able set. To accomplish this end standard proportions have b ?sn adopted for the teeth. The angle of obliquity o f action which is generally accepted as standard in this country is 15°, although in oases of special design this angle is often made greater, and is some- times as large as 30°. When the angle of obliquity is increased, the component of the pressure tending to force the gears awart and producing friction in the bearings is of course increased, 192 ■ . . v , ^ H «< . but on the other hand the profile of the tooth *Hd?r at the base and consequently the strength is correspondingly greater. These special gears are used when' the conditions are unusual and' the standard tooth form is not suitable. In England teeth of greater obliquity of action and less depth then -:he standard are quite common, and are now being used to sene extent in this country. In designing teeth of this kind care must be taken to make the arc of action at least as great as the circula pitch,* otherwise the teeth would not be continuous ly in mesh and would probably come together in such a 7 ; ay as to lock and prevent further rotation. The standard angle of obliquity of action adopted by manufacturers of gear cutters iv slightly at variance with the usual standard for cast teeth, being 14° 28’ 4-0", the sine of which is .25. The smallest involute gear of standard proportion that will mesh correctly with a rack of thes&me pitch, contains 30 teeth; however, this difficulty is remedied by slightly cor- recting the points of all the teeth in the set, so th t a gear of twelve teeth may mesh with any of the other gears of the same pitch. The profilea of these teeth may be drawn with almost exact accuracy by circular arcs with their centers on the base circle, and the values of these radii for a 15. involute have been carefully worked out by Mr. G. B. Grant of the Grant Goer Works. These values arc given in the following table. 192 . i i 1 4 . * ■ r 91 table gf radii for 15 ° INVOLUTE ! TT XI » Centers on Base Line. Divide by the Multiply by the T* 0 ,ri -f Vs Diametral Pitch Circular Pitch. Face Flank Face Flank Radius Radius Radius Radius 10 2.28 .69 .73 .22 11 2 .40 .83 .76 .77 10 2.51 ;96 .80 .31 13 2.62 1.09 .83 . 34 14 2.72 1.22 .87 .39 15 2 .82 1.34 .90 .43 10 O CO *~J 9 -u 1.46 . 9*^ .47 17 5.02 1.58 .96 .50 18 3.12 1.69 .99 .54 IS 3.7,7 1 .79 1.03 .57 20 3.32 1.99 1.06 .60 21 3 .41 1.98 1.09 • c* rr • 0 22 3.49 2.06 1.11 4* • CO O " A L/ 5.71 2.15 1.13 .39 04 /! • ‘"A" 2.24 1.16 .71 05 5.71 2.33 1.18 .74 03 3.78 O * O 1.20 .77 27 3.85 2.50 1.23 .80 28 P T QO -• # Cj 2.59 1.25 • 87 29 3 • ss 2.69 1.27 .05 30 4.06 2.76 1.29 .08 51 4.13 2.85 1.51 .91 VO 4.20 2.93 J. • w — * • «-* v> O 4) /. O Q • .O *s 3.01 “I *2 A J. • O c ■ oe 34 4.33 3.09 "1 rr r> 1 » -JO .99 bb . .39 3.16 i;39 1.01 36 4.45 3.23 1.41 1.03 37-4-0 4.20 1.34 41-4-5 4.63 1.48 43-51 5.06 1.61 37-60 5.74 1.83 61-70 6.52 2.07 • 71-90 7.72 2.46 91-120 9.78 3.11 121-180 13.38 4.26 181-360 21.62 6.88 92 It will be noticed that this table 1 3 for 13° involute end therefore does not note the standard form for cut gears. The forms given, however, may be used on the drawing, because in cutting a gear the workman needs only to know the number of the cutter, and all that is required on a drawing is an approximate representation of the tooth profile. The table also gives values down to a ten tooth gear, while the standard cut gear sets only rim down to twelve teeth. This is theoretically the smallest standard involute gear that will have an arc of action equal to the circular pitch, however, and in the ten and eleven tooth gears the error is so slight that it is practically unnoticeable. It was found necessary to devise a separate means of drafting the rack. The tooth is drawn in the usual way, the sides of the tooth making angles of 15° with the lines of centers from the root line to a point midway between the pitch and the addendum lines. The outer half of the face is formed by a circular arc, with its center on the pitch line and its radius equal to 2. 10" divided by the diametral pitch or 0.67" multiplied by the cir- cular pitch. The radius of the fillet at the root of the tooth is taken as l/4 of the widest part of the tooth space. Standard Involute Cutters . - Brown and Sharpe, the leading manufacturers of formed gear cutters in this country, furnished involute cutters in sets of 8 for each pitch as follows: No. 1 will cut gears from 135 teeth to a rack 2 " " " " 55 " " 134 teeth - Joa t o Lvp: i * . f si! t ,• f « " ot two I <■ ' ' ' £ • | - . 4 wi 1 1 cut gears from 26 teeth to 54 teeth 5 t» K » it 21 u I! 25 IT 6 it 11 r? ft 17 it n 20 11 7 ft It u tt 14 ti tt IS ft 8 ii 11 n it 12 ii it 13 ft When more accurate tooth forms are desired they also furnish cutters to order of the half sizes making a set of 15 cutters instead of 8. These cutters are commonly based on diametral pitch and are made in the follovjlng sizes: from 1 to 4 by quarters: from 4 to 6 by halves: from 6 to 16 by whole numbers; from 13 to 52 by even numbers only; then 36, 58, 40, 44, 48, 50, 56, SO, 64, 70, 80, and 120. However, they also furnish cutters at a slightly greater cost based on circular pitch, and the sizes vary as follows; from l/8 M to 1” by sixteenths; from 1" to 1 l/2" by eighths, and from 1 l/2 n to 3 n by quarters. Action of Involute Teeth. - Fig. 35 illustrates very clearly the action of a pair of involute teeth. Let the circles A and B represent the base circles of a pair of involute gears, the pitch circles of which would be the circles described about A and B with radii of AC and BC respectively. Imagine a cord attached to A at L extending around the circumference to the point D, from there direatly ©.cross to E, and abound the circum- ference of B to Me Let the central point of the string be per- manently marked in some manner and be denoted by C. Now rotate A in the direction of the arrow and trace the path of the point 192 ' ' *m -m# ■ ' . ■ . *4 $ K fr ■ ' ' C on the surface of A extended, on the surface of B extended, and also its actual path in space. It is evident that these throe curves will be CG, OH, and CJ, and that CG and CH will be por- tions of the involutes of the two base circles A and B. ¥ ow reverse the rotation of B and rewind the string on E until C reaches the point K. During this motion it will complete the tooth forms OF and Cl. Bearing in mind that C is always the point of contact of the teeth, its path is evidently JE and coincides exactly with the line of pressure between the teeth, since the line CD is always normal to the involute curve it is generating. If the centers A and B should be misplaced slightly on account of wear in the journals, a uniform velocity ratio would still be transmitted because the no male would still pass through the point C. The only result of this shifting of centers would be to change the obliquity of pressure of the teeth and the length of the arc of contact. The outlines of the teeth would not be changed a particle. 56 . Cycloidal System. - The cycloidal sy tw although the oldest is not so popular as the involute system, an' 7 seems to be gradually going out of use. Mr. Grant in his "Treatise or Gear Wheels" says: "There is no core need for two different kinds of tooth curves for gears of the same pitch than there is need for different kinds of threads for standard screws, or of two different kinds cf coins of the same value, and the cycloi dal tooth would never be missed if it were dropped altogether. But it was the first in the field, is simple in theory, 192 is / ~ r 95 easily dram, has the recommendation of many Fell m oaring teachers, and holds its position by means of human inertia,* or the natural reluctance of the average human mind to adopt a change particu- larly a change for the better" „ This view is probably a little biased, but nevertheless there is a great deal of sound truth in it. The proportion of machine-cut cycloidal teeth to machine-cut involute teeth is very small, but in some classes of Fork, and especially when the loads are heavy, they are still used exten- sively . Form of Tooth. - The outline of a cycloidal tooth is made up of two curves. The faces of the teeth are epi cy cloids and the flanks are h ypocycloids , with two exceptions, namely, internal gearing and racks. In the former case the faces are hypocycloids and the flanks are epicycloids, while in the latter both curves are plain cycloids. (When a circle rolls on a fixed straight line the path generated by any assumed, point of the circle is a cycloid: should the circle roll on the outside of another circle, the path of this point would be an epicycloid, and should roll on the inside of another circle, it would be a hypo- cycloid . ) These rolling circles are generally spoken of as de~ scribing circles, and their size determines the for'*' of the tooth, the arc of contact, and the angle of obliquity of action. The angle of obliquity in the system is constantly changing, but it« average value when the proportions of the teeth ahe standard, is about 15°, the same as in involute gearing, The circle upon ’■’■•h'h the describing circles are rolled is the pitch circle. 192 ' ' V. : - ' " Then the diameter of the rolling circle is equal to the radius of the pitch circle, the flanks of the teeth, are radial, and when it is smaller than the radius of the pitch circle, the flanks of the teeth are undercut. In addition to the objection that undercut teeth are weak, the amount of undercut must be very slight if the teeth are to be cut with a rotating cutter. Fig. 33 shows the limit of undercut possible with a ro- tating cutter; the width of the space must either remain constant or decrease as it approaches the -root line. The sane describing circle must always be used for these parts of the teeth which work together, i.e. the faces of a tooth on the one gear must be formed by the same describing circle as the flanks of the tooth it meshes with. In interchange- able sets it is desirable to use the same size describing circle for both the faces and the flanks of all the gears of the same pitch, and the size of the describing circle which j- generally accepted as standard is one whose diameter is equaljtc the radius of a twelve tooth gear of the same pitch. Here s,gain, however, the manufacturers of gear cutters are at variance, and use 15 teeth as the base of the system. This does not mein that the 15 tooth gear is the smallest gear in the set, but •imply means that smaller gears will have undercut flanks. The profiles of these teeth, as in the case of involute teeth, may be very accurately represented by circular arcs, and the following table gives values for these radii, with radial distances from their centers to the witch circle as determined by Mr. Grant. The centers of faces lie inside of the pitch 192 * • • 1 ' ' * ■ ■ t, . ' 97 circle, while the centers of flanks lie outside of it. The smallest gear in the set is again one having ten teeth, while the smallest one for which standard cutters are manufactured is one having twelve teeth. The form, given by the table is also slightly different from, the form of standard cut- ters on account of the difference in describing circles, but os in the case of involute profiles nay be used on the drawing. 1C 1 08 TABLE OF RADII FOR CYCLOIDAL TEETH • Number of For One Diametral TP ni i.' V^yJ r One Jn. Circular Teeth in Pitch. For any Pitch. For any other the Gear other pitch divide pitch multiply b y by that pitch. that pitch. Faces Flanks Faces TP lanks E^IcbC "fc i I nt ’ v 1 s Radr Dis „ ! I Rad. I ->-j 1 s # Rad . jJiS Rad . Dis . i 10 10 1 O Q -L 0 0 .02 8 c 00 A . no * 7 £* .01 r> - r~ — • / J) 1.87 j 1 11 1 11 ! 2.00 .04 | 11,05 n rzr\ • O A- ; .33 .01 . rr rr A 0 ,07 ! 12 12 2 o01 i .03 | _ , 34 .02 ; \i4 — 13-14 2.04 .07 i 15.10 9.45 n rr .00 • Oy? 4.00 15l Q 15-16 ■ 2.10 .09 j 7,86 7 .46 .37 » 0 0 0 po 1 ~i ir s ] 17l 2 17-18 2 . 14 .11 1 6.13 2.20 ,68 • 0*4: 1.95 7- 1 .... 20 19-21 2.20 .13 : 5 , 12 1.57 .70 . 04 i.:o b i .30 25 22-24 2.23 .15 4.50 1.13 .72 .08 ■” n j * l 27 25-29 2.33 ,16 4.10 on • «-/ sj .74 .05 1.30 .89 I ry r-r OO 50-3 S 2.40 .19 5.80 .72 n c • ' V_ - •° ? 1.20 1 .-3 j f Ao 37-48 2.48 .22 3.52 .S3 * ■ .07 1.12 .80 j l 58 49-72 1 2.60 I .25 ^ cr* O .54 .83 .08 • 1.06 • .17 i 97 i 73-14-4 | 2.85 ,28 3 . 14 .44- .90 • .09 1.00 . 1 290 145-300 2.92 • 0 X 3. CO * 08 ,93 .10 .05 i 0 . 1,. Rack | pop | Cj • w j rzA i 2.96 n£. • - OA © - — .11 0 /. » ~ - 192 ' . . * ' Standard Ov&loidal Cutters Broun and Shame furnish sets of cycloidal cutters based on diametral pitch only, and the sizes vary as follows: from 2 tv 3 by quarters; from 5 to 1 by by halves; from 4 to 10 by whole numbers, and from 10 to 13 by t even numbers only. Each set consists of 24 cutters as follows: Cutter A cuts 12 teeth Cutter M cuts 27-29 teeth B C D " 15 14 15 E " 16 P " 17 G 11 IS H 19 I " 20 J f f q n no ft & JL o K ” 23-24 " L n 25-26 " jf f! 30-73 ” 0 ,J 34-37 n P " 38-42 " Q, " 43-49 " R " 50-59 » S n 30-74 t: rn T? r? r; nr\ f? X t <_.'•* j . U " 100-149 V n 150-249 ,f W ” 250 or more teeth X racks. Action of Cycloidal Teeth . - Fig. 37 illustrates the action of a pair of cycloidal teeth. Let the circles A and B represent the pitch circles of a pair of cycloidal gears, and the circles D and E represent their rolling circles. Let C be the pitch point, and let C^ and C e be the points on the circles D and E which coincide with C when the teeth are in the position shown. How let the centers of the circles A, B, D and E be fined and rotate A in the direction indicated by the arrow. Let ' . ■ . ■ I f the contact at C be so arranged that the circles B, D, and E are driven with the sane peripheral speed as A. Then trace the oath of the point on the surface of A extended, on the surface of E extended, and also its actual path in spao 3 . These paths will evidently b9 respectively the hypocycloidal flank of a tooth CF, the epicycloidal face of the meshing tooth OH, and the path of the point of contact 0 J . Nov: replace the mexhanism in its original position, rotate A in the 0 "> -osite direction and trs.ee the path of the point C a in the sane '-am? or . This forms the curves CG, Cl, and CK, and completes both tooth. forms and the path of contact. As the line of pressure between the teeth which of course coincides with the common norms. 1 at the point of contact must always pass through the point C in order to transmit a uniform velocity, the angle of obliquity varies from PP' JCL to zero during the arc of aorr oach, and. from zero to the KCM, which equals - > \/~ JCL, during the angle of recess. In order to show that with this form of tooth, the normal to the tooth profile at the point of contact always passes through the pitch point C let us observe Fig. IS. It is evident that the generating point C 0 as well as every other point in the rolling circle, is at any given instant rotating about the point of contact C of the rolling circle ^ith the pitch circle. Therefore on the instant in question the line CC g is a radius for the point C Q and. is consequently normal at that point to the curve which C @ is generating. Now referring again to Fig. 57, the point at which the rolling circle is 192 * ; ■ ' ■ - 101 always in contact with pitch circle is evidently the pitch point, and therefore the common normal at the point of contact always passes through it, 57 » Strength of Toot h. - Having determine the proper form of a gear tooth the next step is to determine its propor- tions for strength. Owing to the inaccuracy of forming and spacing the teeth, it is customary to provide sufficient strength for transmitting the entire load by one tooth, rather than con- sidering the load distributed over the whole number of teeth in theoretical contact. The load on a single tooth, when the gears are cast, from w ood patterns, is often concentrated at some one point, usually an outer corner, on account of the draft on the teeth and the natural warp of the castings. The same result is liable 1 to be produced when the shaft is weak or when the gears are not supported on a rigid foundation. However, in the case of well supported machine-moulded or cut gears the load may be consider d uniformly distributed along the tooth. ?or the reasons just staged the subject of strength of teeth will be discussed under tvo heads, as follows: (a) Strength of cast teeth: (b) Strength of cut teeth. (a) Cast Teeth. - In deriving the formula for this cla.so of gearing, it will be sufficiently accurate to consider the shape of the tooth as rectangular, and the load as acting at the outer end. The load may be concentrated at one corner, or it may be uniformly distributed along the length of the tooth. 192 ■ ■ ’ ■ - ■ 102 (l) When the load is concentrated at as shown in Fig. 39, we have the following: The bending moment due to the load W and equating thi3 to the moment of resistance, Wh cos *, = --- -- - 6 sin cc where S is the allowable stress of the material S = 5W sin 2 t 2 ~ The stress S is maximum when 2 « is a when a = 45°, therefore, 3W max . S = ~ t 2 an outer corner, = Wh* = Wh cos we get , . From this (74) maximum or (75) or. (2) When the load is the length of the tooth, we have moment and the resisting moment, t 2 fs Wh = from which S = 6Wh t 2 f distributed uniformly along by equating the bending (76) Equating the stresses given by (73) and (76), we got f = 2h = 1.4p’ (7-) when h = 0.7p* and p’ = the circular pitch. Although as shown by (77), the theoretical length of face at which the teeth will be of equal strength for both cases of loading is 1.4p', a well known American engineer, C. V. Hunt, taking his data from actual failures in his own work work, states that the face should be about 2p’ in order to 192 . ■ ■ ■ v . . " • . • ' 105 satisfy this condition. The seeming discrepancy between theory and actual results may be easily explained, when one takes into consideration the fact that even though the load may be entirely concentrated at t ■- e corner at the beginning of application of pressure, it is very probable that before the full pressure is brought to bear, a slight deflection of the outer corner will cause the load to be disturbed along a sconsiderable length of the face. Another condition also which adds to the length of the face is that of proper proportions for wearing qualities, and in some cases the faces are made extra long for that purpose alone. It is customary in American practice to make the face 2 to 3 times the circular pitch, the length of the face increasing as the quality of the work improves 3 Common proportions for cast teeth may be found in Kent p. 889. The values given in columns two and five are used quite frequently. The thickness of the tooth is given in column five as 0.475p’: now substituting the values for h and t given in Kent's in (78) we get W = O.O 54 Sp’f (78) This formula has the same form as the well known Lewis formula derived below. It is a rather difficult matter to give ■ proper values for 3 in (78) for different conditions, but the • table in Kent p. 901 or Fig. 42 may be used. (b) Cut Teeth. - In 1893 Mr. Wilford Lewis of Wm. Sellers & Co. presented at a meeting of the Engineers’ Club of Philadel- phia a very excellent method of calculating the strength of gear * 192 ■ • i ■ ‘ >• " • • ■ ■ . • mi ' 'r f ' * " i> f 104 teeth* His investigation was the first one to take into consi- deration the form, of tooth profile, and the fact that the direction of pressure is always normal to this outline. It has since that time been almost universally adopted for calculating the strength of teeth, when the workmanship is of high grade as in cut gears. In his investigation, Mr. Lewis assumed that at the beginning of contact the load was concentrated at the end of the tooth, with its line of action normal to the tooth profile in the direction A B, Fig. 40. The actual thrust P was then resolved at the point B into two components, one acting radially producing pure compression, and the other W acting tangent ly, When the material of which the gears are made is stronger in compression than in tension this radial component adds to the strength of the tooth, and when the tensile and compressive strength are approximately equal, it is a source of weakness. However, in either case the effect is slight not exceeding 10 ^, and in the original investigation was neglected altogether. The strength of the tooth may now be determined by drawing through B, Fig. 40, a parabola which is tangent to the toothprofile at some point D. This parabola then encloses a cantilever beam of uniform strength, the weakest section of the tooth then lies along the line DE, and whose strength is evidently a measure of the strength of the tooth. The problem now is to find an expression for W in known terms* By similar triafigles „ t2 h = ( 79 ) 4x 192 ' ■ ... ■ ... * * m ' 105 Equating the bending moment to the moment of resistance ^2 ~ Wh = or W = ^ fSx Dividing and multiplying by p’ W = Sp’fy (80) O V when v = . -—h - is a factor depending upon the pitch and the form 3p' of the tooth profile. Values of y and S are tabulated in Kent, p. 901, also in Pig. 41 and 42. For cut gears f varies from 2 l/ to 4 times the circular pitch. In some cases when the speed and the conditions otherwise are not favorable in regard to wear, f i3 made even greater than 4 times the circular pitch. For pro- portions of standard cut teeth see Kent, p. 890 and column ten of the table on p. £89. 5 8. Methods of strengthening Teeth . When it is desir- able to have the teeth of a gear extra strong, any one of the five following methods of strengthening may be used: (a) shroud- ing; (b) using short teeth; (c) increasing the angle of obli- quity; (d) stub tooth; (e) using helical teeth; (f) using a butressed tooth. (a) Shrouding. - The gain in strength due to shrouding depends upon the face of the gear, the effect being more marked in the case of a narrow face than in a wider one. Wilford Lewis considers shrouding bad practice, However, when the face equals 2 l/2 times the circular pitch, he has demonstrated by a crude theoretical investigation that single shrouding (Fig. 192 ■■ . ■' . . . . ■ ■ 103 43, a) will increase the strength at least 10/, and. double at least 30/« Single shrouding is illustrated by Fig. 43 (a), double by Fig. 43 (b)* and half by Fig. 43 (c)« (b) Sh o rt Tooth, - Gear teeth whose heights are less than that given by common proportions are considerably stronger, and furthermore, they run with less noise. In this country C. *7. ( Hunt advocates this type of tooth, and the following proportions for involute teeth are the ones he has successfully used on gears for coal ho i/s ting engines and similar machinery. Addendum = 0.2 p' (SI) of ge-ar = 2p’ 1 (82) ( Clea^ance= .05 (p’ * 1") (83) They' following table used by Hunt gives working and maximum loacws on a cast iron spur gear of °0 teeth, which is the smallest he uses. i 7- 'Circular / _ . , t Load in Pounds Circular Pitch Load in P ound 3 i i i Pitch , . ! Working | l Maximum Working Maximum l / ’ i i/' i 1320 1650 2 1/4 6700 p. 2Q c 1 1/4 2300 2600 2 l/2 8300 1050Q 1 1/2 3000 3700 2 3/4 10000 12500 j 1 3/4 4100 5000 3 12000 j 14800 (c) Increasing Angle of Obliquity. - The gain in strength due to increasing the angle of obliquity is shown in Fig. 44 which figure consists of the left half of a tooth of 22 l/%° ob- liquity and the right half of a tooth having the same witch but 192 ■ . . " - * ■ • • - • - ' . • • • « . • • •; ' ' ' ' 107 having an angle of obliquity equal to 15°. 1 Toy; the factor y which 2x appears in Lewis* formula is equal to wp , from which it is evident that an increase of x when p ? remains constant will result in an increase of y and consequently an increase in the strength of the tooth. This increase of x i m - shown in the figure » A further advantage aside from the increase of strength lies in the fact that the size of the smallest pinion which will mesh with a rack without correction for interference diminishes rapidly as the angle of obliquity increases. Thus with an angle of obliquity of 15° the 30 toothed pinion is the smallest one which can be used without correction, while with an obliquity of 22 l/g° the smallest gear in an uncor- rected set is theoretically 14, but practically it may be re- duced to 12 * (d) Stub Tooth . - Another method of strengthening gear teeth, which is now being introduced quite extensively in automobile transmission gears consists of a combination of (b) and (c), and is known as stub tooth. The angle of obliquity used is 20o, and the following table gives tooth dimensions as recommended by the "Fellows Gear Shaper Co." of Springfield, Vermont . i oo jl ■ ' i - i • ini..** ' 107 TABLE OF STUB TOOTH DIMENSIONS 1 Pitch Thickness on Pitch Line Addendum Dedendum 4/5 e 3925 .200 .250 ' 5/7 .SIS . 142 S .1785 3/8 .2617 .125 1 1 r.,-2 0 j 1 7/9 .2243 olll 1 .1389 8/10 .1962 .100 .125 jo/n . 1744 . 1909 .1137 10/12 .157 uOSoo ,1042 112/14 ! .1308 0.0714 0007 a \j ■ ^ o (e) Helical Teeth . - Gears having accurately made helical teeth and if properly supported, will run much smoother than ordinary gears. In the latter form of gearing there is a time in each contact when the whole load is concentrated on the edge of the tooth, thus having a leverage equal to the height of the tooth. With helical gearing, however, the points of contact at any instant are distributed over the entire working surface of the tooth or such parts of two teeth in contact at the same time. Therefore the mean lever arm with which the load may act in order to break the tooth cannot be more than half the weight of the tooth. Prom the above remarks it follows that the helical teeth are considerably stronger than the straight ones, (f) Buttressed Tooth . - The buttress or ho ok- tooth gear can be used in cases where the power is always transmitted in the same direction. The loaded side of the tooth has the 1S2 . . •• ■ . ' U ■: k ,1 , ■ . IOC usual standard profile, while the back side has a greater obliquity as shown in Fig. 45. To compare its strength to that of the standard tooth, use the following method: make a drawing of the two teeth and measure their thickness at the top of fillet; then the strength of the hook tooth is to the standard as the square of its thickness is to the square of the thickness of t’-'e stan- dard tooth. no ^ «_/ the arm of a flexure, and is uniformly Letting then Gear W heel Proportions . - Arms. - In proportioning gear it may be considered a cantilever beam under the usual assumption regarding the load ic that it distributed over the arm. — = section modulus of the arm at the center. c S = allowable fiber stress. T = twisting moment transmitted, by the gear n = number of arms T _ SI (94) n c From (84) a value of — may be found from which the 0 dimensions of the adopted sections may readily be determined. Fig. 46 shows four types of arms used in gear construction, of which (a) (b) and (d) are intended for large gears and (c) for lighter gears, though quite frequently it is used for heavy gears. The method of laying out is plainly shown in Fig. 46 (e). Assuming the proportions as shown in (e), (84) reduces to h = AjZP-l ( 0" ) nS 192 ‘ 110 For web centers make the thickness of the web l/? T p*. Rimo* - Calculations for rim dimer * 1 ions are of little value, and in actual designing empirical formula, give the best results. Fig. 46 shows the proportions used for the various rim sections commonly met with in gear construction. Hubs. ~ The common solid hub should have a reinforce- ment of metal over the key as shown in Fig. 4-6. In olace of a solid, hub one that is split may be used, thereby reducing the cool ing strains* in the wheel and at the same time permitting an easy adjustment on the shaft. Keys should always be placed under an arm in the case of a solid hub, and in a split hub amrom.imately at right angles to the center split. The following formulas by Herman Johnson published in the American Machinist, Jan. 14, 1904, are applicable to gear wheels, pulleys and other machine parts. They represent the actual practice of four large manufacturing concerns. Diameter of Hub in Terms of the Bore. Mature of Power Tran s mitte d. Cast Iron Stee l Castings . Heavy, very great shock 2d 1 3/4 »• l/s T! 1 3/4 + i/8 " 1 5/3 + 3/le" 1 5/8 1/8" 1 1/2 d + 1/4" Standard, medium shock Light, no shook 19 ? ' ' Ill Length of Hub in Terms of the Bore. Bearings 3d to 4d Levers 1 1/2 d Gear Wheels 1 5/4d to 2 l/4d Pulleys 2/3 Pace Hand Wheels 1 1/5? d to 2 d Truck Wheels 2d to ? l/4d 60. Definitions,, - (l) A spur gear is a toothed gear the teeth of which are parallel to its axis. (?) When a large and a small gear mesh together the large one is called the gear and the small one the minion, (3) A rack is a gear wheel with an infinite radius, or, in other words, a straight bar with gear teeth formed on it. (<_•) An internal ge ar is one having its teeth on the inside of the rim and is also called an annular gear n (5) A gear blank is the solid wheel, from which the gear is formed before the teeth are cut e (c) The face of a gear is the wiCth of the gear, i.e. the length of the tooth. (7) The pitch circles of a pair of gears are imaginary circles, the diameters of which are the same as the diameters of a pair of friction gears that would replace the spur gears. (8) The base circle is an imaginary circle used in involute gearing to generate the involutes which form the tooth- profiles . (?) The describing circle i ' an imaginary circle use' 1 in cycloidal gearing to generate the epicycloidal and hypo cy- cloidal curves which form the tooth profiles. There are two de- scribing circles used, one inside and ere outside of the -itch 192 ' ' ■ .... ■ ■ , ;.iij . . . . 112 circle, and are usually the sane size. (10) Angle of o bliquity of ac tion is the inclination of the pressure between a pair of mating teeth to a line drawn tan- gent to the pitch circle at the pitch point, i.e, the angle DCF in Fig. 47. (11) Arc of approach is the arc measured on the ^itch circle of 3, gear from the position of the tooth at the beginning of the contact to the central position, i.e. the arc EC in Fig. 47. (12) Arc of recess is the arc measured on the pitch circle from the central position of the tooth to its position where contact ends, i.e. the arc Cl in Fig. 47, (13) Arc of action is the sum of the arcs cf approach and recess, (14) D iametral pitch is the number of teeth divided by the pitch diameter. It is not any diameter on the gear, but is simply a convenient ratio to use, (15) Circular pitch is the circumference of the pitch circle divided by the number of teeth, and is equal to the distance from one tooth to a corresponding point on the next, measured on the pitch circle. (16) Chordal pitch is the distance from, one tooth to a corresponding point on the next measured on a chord of the pitch circle instead of the circumf erenae . This is only used for "'aking the drawing, or if the teeth are t.to be formed on a wood pattern, by the pattern maker, ’ (17) Thickness of the too(th always means the thickness l i9te on the pitch line • ' ■ . ■ . ■ • ■ • ■ TO' 113 (18) Tooth spac e means the width of the rmace on the pitch line. (IS) Backlash is the difference be twee - tooth srr-c- and thickness of tooth. (80) C lea rance is the difference between addendum and dedendum, or, in other words, ’the amount of space between the root of a tooth and the point of the tooth which meshes with it. (21 ) A ddendu m is the radial distance from the witch circle to the ends of the teeth. (22) Dedendum is the radial, distance from the pitch circle to the roots of the teeth. (23) The face of a tooth is that portion of its profile which lies outside of the pitch circle. (24) The flank of a tooth is that portion of the Pro - file which lies inside the pitch circle. (25) Line of centers is the line pa'" sinr £ erectly through both centers of a pair of gears. (26) Pitch point is the point at which the pitch circles of the tvro gears are tangent to each other. (27) Velocity ratio always ^eans t] o ratio of "we revo- lutions of the driver to the revoltuions of t’ o driven. t tepftrtant • Points , - (l) The word w diavoter f: when used in connecton with gears is always understood to ween witch dia- meter. (2) Circular pitch is an actual dimension and is always in inches; the diametral pitch, however, is only a ratio. (3) The circumference of the pitch circle cf a gear 192 114 must always be a multiple of the circular pitch or number of teeth in the gear will not co”-e out even. (4) The relation between diam.etra 1 a.v. A circular pitch is expressed by the formula pn 1 = Tf (5) The relation between the diametral pitch, pitch dia- F meter &,nd number of teeth is p = — t . (S) For Brown and Sharpe proportions the relation be- tween diametral pitch, outside diameter, and number of teeth is N + 3 p = — _ — (7) In designing out gears care should always be taken to malie the diametral pitch 3ome regular standard pitch. Although gear cutters can be obtained, which are haded on circular pitch, the-; are not usually kept in stock, and gears designed to bo cut with then will almost always cause vexatious delays in the shop 0 (S) In designing cast gears the proportion of the teeth are usual 1;^ given in terms of the circular pitch, and if the teeth are to be formed by the pattern maker, it is just as well to ignore the diametral pitch altogether. However, if the pattern is to be of metal its teeth will probably be cut and they should be designed with that end in view. When two mating gears of such size that the circum- ference of one is an even multiple of the circumference of the other, it is evident that any tooth on the large one will always mesh with the same tooth on the small one. This condition ayr vat os inequalities of wear, and when the distances between costers and the velocity ratio will permit of a slight variation, an extra or hunting tooth is sometimes added to one of the gears. 192 11 " CHAPTER IX. Bevel Rearin g r When two shafts which intersect each other are to be connected by gearing, the result is a po.ir cf bevel gears. The origin of the name is obvious as the faces of the gears must be beveled toward the axis m order to fit each other. Occasionally, however, the shafts are inclined at an angle to each other, but do not intersect, in which case the gears have teeth which are not in the same plane with the axis, and are called shew bevels « Friction bevels are quite common, and what has been said in the previous chapter regarding spur frictions will apply in general to bevels also. The form, of tooth which is almost universally used for tooth bevel gears is the involute. This is probably due to the fact that slight errors in its form are not nearly so disastrous to the running of the gears as when the tooth curves are cycloidal. 61 „ Processes of Manufacture . - These gears may be either cast or cut, and are made in both ways , The process of cast- ing is not materially different from that used in spur gearing, but the process of cutting is much more difficult on account of the continuously changing form and size of the tooth from one end to the other. As in the case of spur gearing, there are several dif- ferent methods of cutting the teeth, some of which form the teeth with theoretical accuracy, while others produce only approximately 192 . . . ' ■ n. i ■ 116 correct forms. Throe of these methods give very accurate results, hut they require expensive special machines and are used only when very high grade work is desired. The three methods are: the templet-planing process represented by the Gleason gear planer: the temp let -grinding proces s, represented by a machine manufactured by the Leland and Paulcover Company and the moulding-p l aning pro - cess , represented by the Bilgram bevel gear planer 0 In all these processes the path of the cutting tool always passes through the apex of the cone, i.e« the point of intersection, of the two shafts, and consequently the grower con- vergence is given to the tooth. With a formed rotating cutter, however, it is impossible to produce the groper convergence and in many cases the teeth have to be filed after they are cut before they will mesh properly. Nevertheless, the milling machine is very commonly used for cutting bevel gears, for the simple reason that the equipment of most shops includes a milling machine, while comparatively few do enough bevel gear cutting to justify the purchase of an expensive special machine for that purpose, 62, Form of Teeth , - When the gears are plain frictions \ it is evident that the face of the gears must be frustrums of a pair of cones whose vertices are at the point of intersection of the axes. These cones may now he considered the pitch cones of a pair of tooth gears, and the teeth may be generated in a manner analagous to the methods used for spur gearing. In discussing the method of forming the teeth, the involute system only wil" 1 be considered, as the cycloidal form is very little used. 192 4 ■ . ' c . 117 In Fig, 48 (this discussion refer:' - to the dotted lines only), let the cone OHI represent the base cone of a bevel gear from which the involute tooth surfaces are to be wrapped, In order to simplify the conception of the process of unwrapping, imagine the cone to be enclosed in a very thin flexible covering, vdiich is cut along the line OE 0 Now unwrap the covering, tabing care to keep it perfectly tight, and the surface generated by the edge or ele- ment OF is the desired involute surface. The -point E, while it evi- dently generates an involute r . of the circle HI, is also constrained to remain at a constant distance from 0 equal to GE, or in other words, it travels on the surface of a sphere HAI , From that re. son the curve FE is called a spherical involute. Fig. 49 illustrates the method of finding the base cone. In this figure the cones GC1 and OFG represent the pitch cones of a pair of bevel gears whose axes intersect at the point 0 and whose element of contact is the line OE, The axis of the cone OFG happens by mere accident to coincide with the extreme right hand edge of the cone OCD 0 Now on account of the fact that the ends of the teeth as generated above, lie on the surface of the sphere £EPS , let us consider the surfaces contained Inside the sphere only. Pass a plane LM tangent to the cone OOD/ at the element OJ, Then pass a second piano HP through / the el/ement OJ making an angle with the plane LSI equal to the J angLjfV of obliquity, and construct a cone OHI tangent to the plane W o Thir. cone is the base cone for the gear, and it evidently corres- ponds exactly to the base cylinder of a spur gear. It is pi inly ^ - een that as the point of intersection of the axe - or apex of the 192 ' ' ■ ; • ; ’ • V . ■ '• ' lie pitch cones move farther away the gears approach the spn.rv.goar form, an" actually become spur gears when the apex recedes to in- finity. The spherical surfaces which should theoretically form the tooth profile is a very hard surface to deal with in pr• we have tSsdl t = dW = 6 h h- From the geometry of the figure, h= _L J 1 it- and (8c) Substituting these values in (86) 2 / dW = St j jdl 6h r 1 3? ow the moment of the force dY.' equals /dW, or dM = St ! 2 * * S / S dJ Integrating M St 6h x Tl 2 IP h^l 2 , ° = Stl > ( 1 16h 1 D 1 ? / - i - " 1 ->2 v - v •'87 > V ^ ' i (88) M represents the' total turning moment around the apex of the pitch cone, therefore, to find, the force acting 192 • • • 121 at any point as at the greatest addendum, divide M by the dis- tance of the point from apex. Let W* represent a force which if applied at the large end of the tooth would produce a turning moment equivalent to M, then ^ 8- , _ M Sti^li W' = 1- 18 hj F *7 *7 V V' D- ( 39 ) F Do From Fig. 52 ~ = 1 - _£ substituting the value h Di of 1]_ from this equation in (89) , St-, 2 f W’ = ± 18h, V - ® 8 3 O \ D 1 < =(D 1 -D2; (9.0) Assuming the same proportions for the teeth as in spur gears (h^ = 0.7 p’ and t^ = 0.475p’) and substituting these values in (90) (Do)' w = .oiesp'f From Fig. 52 1 1 + D, D- D- (91) D, 1 - f X 1 : substituting this in (91) we have W* = Sp’frc 1 in which rn = 0.018 3-3 f (f) S In good practice £ should never exceed l/c . For values of S for various mater- ials consult Fig. 42, and the following table, gives values of f m for different values of — • 192 , •• ■ ' ■ .... ■ 122 F 7i cl .2 • £ .4 . 5 .6 .7 .8 .9 o • i — 1 n .049 .044 .039 .035 n*c c 028 .025 <•028 .02 .018 (b) Out Teeth . - The formula generally adopted by de- signers for calculating the strength of cut bevel teeth, is the cne proposed by Wilfred Lewis in a paper before the Engineers' Club of Philadelphia in 1893. For the assumption made in his investigation consult Art. 57 (b). The formula as derived by i Hr. Lefais is given in Kent p. 902, or may be found as follows: (see Pig. 53). In this proof the same assumption regarding the distribution of the pressure on the teeth will be made as in (a) above. Furthermore, equations (86) to (90) inclusive hold in the present case. From (79) x = - 1 - , so substituting this value 4h , in (90) and multiplying through by p we get Sp’f W ? = r” _ 1 ^ r — _ 5 P _ 3 Dl - 1)2 2z Dp 2 (Dl -Dp) i 3p 5 - J (S3) 2x Letting y = p p t as in Art. 57 (b) V- 1 _ O y, T = Sp’fyn i f where n = 1 - — + f 1 1 I£LL f J (£:■!) , values of which for various ratios of -j are given in the following table: J i . f 7i ,1 .2 rr » o .4 . 5 . 6 .7 .8 — .9 i i o n .903 00 i — 1 .75 .655 .583 • A. .465 .415 .57 ^rr r- «* ».AO f ’J 192 . • . . - ■ ■ • ■ ■ As stated before 12 ? should never exceed l/o and t^e value cf f which is commonly used is about 2 1/2 p*. For values of’ S for different materials at various speeds see Fig. 42. In obtaining a value for y from Fig. 41 use the formative number of teeth. In order to save time the following method may be used in finding the strength cf bevel gear teeth: (a) Multiply the number of teeth in the bevel gear by ^1 (see Fig. 52) which gives the formative number of teeth. r (b) Find the strength of a spur gear having this num- ber of teeth and of the same face and pitch as the bevel gear. (c) Multiply the strength of this spur gear by the f value r taken from the table for assumed value of — r . 1 1 64. Gear Y/heel Proportions . - Arms , - In bevel gears the T arm is remarkably well adapted for resisting the strains that cone upon it and for that reason is quite extensively used in gear wheels of large size. In small gears, however, the extra expense of construction more than offsets the saving of material, therefore the web and solid centers are in common use. Fig. 54 shows a T arm. as applied to bevel gears. The rib or feather at the back of the arm is added to give lateral stiffness, and in the derivation of the formula •for its strength their effect will be neglected, since they add very little to the resistance of the arms to bonding in the plane of the wheel. As in the case of spur gears the arm is treated as a cantilever beam under flexure and that each arm will carry — part of the load transmitted, by the gear. n ’ 192 ■ ■ ■ . ■ • V ■ ? ■ 124 Let b _ thickness of the arm h - width of arm at the center D]_= pitch diameter at large end of tooth S = allowable fiber stress W ' =equivalent load at large end of tooth n = number of arms W’D then 2n Sbh 2 e~ from which h = 35{ 1 D . bSn 1 f or Exercise. - Assuming b = 7T p ’ and S = C00' n for cast iron, establish the formula for h. Rim and Feathers . - Fig. 54 gives the proportions of the rim and feathers commonly used. Hub. ~ For hub dimensions consult the table given in 'i Art . 59 „ 65. Definitions . - (l) The back cone radius is the length of an element of the back cone. See Fig. 50. (2) The formative number of teeth is the number of teeth, of the given pitch which would be contained in a complete spur gear of a radius equal to the back cone radius. (5) The edge angle is the angle between o plane which is tangent to the back cone, and the plane containing the pitch circle. See Fig. 50. (4) The center angle is the angle between a plane, tangent to the pitch cone and the axis of the gear. See Fig. 50. 192. . . . . . _ 1 or (5) The out t. l nr~, angle ip the angle between a plant tangent to the root cone, and the axis of the gear. See Fig. f 0 , { 3 ) Backing is the distance from the addendum at the large end of the teeth to the end of the hub which is on the large end of the gear, measured parallel to the axis of the gear. See Fig. 50. (7) A miter ge ar is a bevel gear whose center angle is 4-5°, and a pair of miter gears arc always exact duplicates of one another . Important points. - (a) As bevel gear teeth taper from end to end they actually have a number of pitches and pitch diameters, but in speaking of the pitch or the diameter, the values at the large end are always meant. (b) The formulas given in Kent p. POO for spur gears may all be used for bevel gears except the ones in which b the outside diameter appears. This diameter for a bevel gear may be calculated by the following formula; N -4- 2 cos ... . D = (96) P when u is the center angle. (Derive this formula.) (c) The back cone radius is used for determining the formative number of teeth. (d) The formative number of teeth is used for all pur- poses which pertain to the shape of the teeth; e.g. in selecting the proper cutter for cutting the gear, and in obtaining the factor y for calculating the strength. 197 ' ' . . in diameter breaks under a tensile 1. A round red 2 n lead of 190000 pounds. Find the unit breaking stress. 2 . Using same unit stress as in problem ( 1 } That load will cause a 1 l/4" square red to fail by pulling apart? c. Assuming a working stress equal to one-half the elastic limit, what must be the width of a piece of bar steel (structural) to carry a load cf 50000 pounds, if the thickness of bar is ?>/■'" ~ 4. A bar of structural steel 1 c/4 ,! in diameter fails under a tensile load cf 150000 pounds, what was the ultimate tensile strength of bar? 5. Using same tensile strength as in problem {-'■) what will be the unit breaking stress in a round bar of structural steel 1 7/b n diameter? 6. A steel bar 1 ,; in diameter and 10 ,f long has a total value for the elastic limit cf 30000 pounds, and a tot"! ultimate tensile strength of 50000 pounds. The length of the bar at the elastic limit was 10.0180" and at failure was 15. "8". Find elastic limit, ultimate tensile strength, and the unit elongation for the elastic limit and for failure. 7. The maximum pressure in a steam engine cylinder, l° ff diameter, is 1 20 pounds per square inch. Using a factor of safety of 10 find diameter of piston rod. 8. A short (structural) steel bar is 3" in diameter, -; r hat cempressiwe load will it take using a factor of safety cf 3? cf ic ISC ' 2h * • * I . . ■ ■ •j t! - < ? : , • . . . 1 on -L / 9, Find the load which a short, hollow coot iron coliwn IF" external dia: eter , 9 1/2" internal diameter, wil.l take with factor of safety of 6. 10. A simple beam is 58" long and carries a concentrated load of 6500 pounds at a distance 17" toward the right from, the left support . Find both reactions, How many forces acting on the beams. Make a sketch indicating direction and magnitude of each force. 11. A simple beam OF" long carries two cone n t rated lea's of 5350 and 1975 pounds. The two loads are 1C" apart and the former is 15" from the left support. Find both reactions. How many forces acting on the beam? Mak' sketch indicating direction and. magnitude of each force. IF. A simple beam 26" between, sun -orts earnin'- three con- centrated loads of 1800 , 5000 and lF0 n pounds. The first is 7" to the left of the left support , the second to the right of the left supnort , and the third is 9" to the left of the right support. Find both reactions. Make sketch indicating direction and magnitude of each force. 15. A simple beam 2F" between sup sorts carries a load of 3-050 pounds at a distance of 5" to the left of the left sun ert. Find reactions. Make sketch indicating direction an-’ magnitude of eash force. 14. In protier’s ( 10) , (ll), (IF) and (13) find (a ) Bending moment in the plane of each fo^ce. (b) Draw diagram of bending moment due to each load, (a) " " " " " " " total loads, 192 - -V ' (d) Locate section in bean of maximum bonding moment . (e) Find size of bean assuming circular section and St = 9000 pounds per square inch. 15 o A gear wheel SO” in diameter has 5 arms and tr. nsraito 18 E.P. at 800 R.P.M. (a) What kind of beams are the arms? i (b) Find bending moment which each arm must resist. (c) Find size of arms at center of gear assuming elliptical section with major axis equal to twice the minor axis. Take St = 2000 pounds per square inch. I for ellipse 16. In problem, (ll) assume rectangular section with depth- equal to three tim.es the breadth. Find size of beam. Take St = 800 pound::- per square inch. 17 . In a single riveted lap joint the thickness of the plate is 3/3" , diameter of rivets p/d" , pitch of rivets 2 P/l6 t? , margin 1 l/d" , Sketch joint (two views) and find the fo 1.1 owing*. (a ) Resistance of the. joint to she ring of rivets. (b \ fr / it 11 ir crushing the plate and rivets. (0 ) 11 11 11 it bursting of plate opposite the rivet . (d } Resistance 11 u ii tt tearing apart of plate (as shown at CL Fig. 9 ) , (e ) Resistance ii 11 11 11 s h e a r i n g of > 1 a t e , . . r • , . . ] °9 18. A double riveted lap joint has t’ o following dimen- sions : Thickness of plo.ro 1 /?" , diameter of rivets 7 j 8” , pitch of rivets 1 13/l6” , margin 1 7 /.1G” . Sketch joint and investigate as in problem (17) using the following working stresses. - St - -8000, S s = G300 and = 130 n 0 pounds per sq. in, 19. A triple riveted butt joint has the following dimen- sions: Thickness of plates 5/3” , diameter of rivets 1”, thick- ness of strops l/s" , width of outer plate 11 5/8” , width cf inner plate 18”, pitch of rivets 5 7/8”, and 7 3/4” ( -outside rovrs), distance apart of rivet rows 3 3/lo” and 2 5 / 8 ” (two rows on each side of joint). Sketch joint and investigate as in problem (17) using the following - St = 75 nr ', S 3 = 5000 and S = 10000 pounds per square inch. 20 n The outside diameter of a bolt threaded. U.S.Std. is 1 3/d” . Find pitch, number of threads per inch and. tensile strength (St = 4000 pounds per square inch). 81. Find diameter of wrought iron holt that is to sustain a ste:dy load of 5 tons. 22, VJliat steady load v.ill 7 studs bolts (U.S.Std.) S/'” in diameter safely resist? 33. Calculate diameter of wrought iron bo], t that is to sustain a varying load of 1800 pounds. 84. Sketch and. dimension fully a finished bolt 7/8" diameter with hexagonal head and. nut. 193 * ■ i : • . • : ■ . \J ' “| rv -r 1 4 0 64. In problem 62, find size of arms of gear, (assume elliptical cross section with major axis = twice minor axis also 6 arms). Take = 2500. 65 e In a pair of spur gears with cut teeth the pinion is made of machinery steel and the gear of cast iron. The diameter of driver is 5" and the velocity ratio is 8.2 to 1. find diame- tral pitch, and number of cutter required in each case. 66 . In problem 65 find ratio of strength of pinion to strength of gear. 67. A cast iron bevel gear with cast teeth, 1 1 nitch, 60T will transmit what H.P. at 150 R.P.M. ? 68. In problem (67) find formative number of teeth, (center angle = 32°). 69. Show that formative, number of teeth is equal to actual number of teeth in bevel gear multiplied by the ratio — , (See Fig. 52 ) , r 70. In a pair of cast iron bevel gears with cast teeth; the diameter of pinion' is 7.55' 1 and the diameter of gear is 25,79". What is velocity ratio if pinion is driver? 71 , In problem 70 if the geah is driver and its R.P.M. is 200 find H.P. transmitted. Take pitch ~ 1 l/p " . 72, Find circular pitch of bevel gear °9,61 lf diameter, which must transmit 30 H.P . at 150 R.P.M. 73. Design a pair of cast iron bevel gears to trasnnit 55 E.P.. R.P.M. of driver = 60; velocity ratio = 3/7; diameter f pinion = 15" . Teeth to be standard involute cut on a milling machine « 192 « ' ■ ■ . ■ • ms < HIE UBMW of tv:-. rm- - ‘ • utt * fc '• r *;■«: H *’ '.«Pf . |-K It! § ss - ij-\ •Si- 5^3 s o. |S .sJ&jT^ >>,v| §■§•! £ V'CtR ^•5^-S so 'c-Si jj> 1 ' ^ ^55 £g§V§^- *■ - ^ i v ^4i-5 'Srs'g^-s ‘Vi •A) Q.'g. ?s-ssH:Sf -kS^H sf- CQ bo • N bo • ^4 li* Bill'S to rs r- ^3 ^ s.' §^~§ o ^r.Q' -J •§ !S ^ Ji> ^-1^1 < 6 ^ r* Q .^•ssll ■S li ft's ^ ", ■^t: 3^ S s lov&’it; 1 1) ^^41 rl^ N ' Ip.telP Pi 8J«P|. t-Sis v-s gl-t^ ,\ Q \ ^ .3 ■s v * 5 h 5 s iO ^ l« Is S-® S-I-? ^ fc S ST^t § § 4 | r- 5 ■P; in to Eju be • 9*4 b. £; o c; CL^> | c; c: 0 ) ^ ri O ^ V ^‘ ^ Q X. £ *l«» £ >• .a Cb ^. s *£ a-S^t-S* |;§^-| cn-c? S-> ^ ^ ^ S-K ^ O ^ § j| IK' >SW 3 . v$ * 'U ^ <0 ^ V i ^ <0 CN £- QJ §^.S 8 ^<3^ S> «jl3 x^.N K “> ^ ^ ^ Qj K Q -fK^IIK 5;^ J QJ vi 0 Is-sb-m •Si^SWvL ^")|<0 Fig G X T Q ki N VO W) 3 3z.g iwi is? siby r w Fig 15 la) (b) Fig 18 Fig 20 "ij? I*~ Fig 22 1 — 1 -1 — 1 1 1 • ( r~r I i U) (U Fig 19 n? Fig 26 Fig 33 THE U* SARY Number or Teeth. 0 10 EO 30 -40 SO &0 70 80 90 IOO II O IZO \ 30 140 Number of Te®t-h Fig 41 Fig 42 Values of Strength Foic.+or p Fig 27 and 28 Fig 29 Fig 45 THE UBKAWY OF THE UMYE&Slflf GF IU-UI01S Fig 50 THE LIBRARY OF THE UNIVERSITY Cr ILLINOIS IKE U2» r< APR 2 4 1930 UNIVERSITY of *