%. ^ V V '.: ^ ^ oo tf*« ^ H -7* «* PRACTICAL TREATISE ON GEARING oJ^Ls s JUQstS^ • o 1 4* 3 I 3 SIXTH EDITION. PROVIDENCE, R. I. BROWN & SHARPE MANUFACTURING COMPANY. 1900. TWO COPIES RECEIVED, Library of C8Bgr§«% (Jfflee of tb® APR 1 1900 tegltter of Copyrights, . B3-7 )C[00 61040 Entered according to Act of Congress, in the year lyuO by BROWN & SHARPE MFG. CO., In the Office of the Librarian of Congress at Washington. Registered at Stationers' Hall, London, Eng. AH rights reserved. O0-2V9? P& SECOND GOPY, PREFACE. This Book is made for men in practical life ; for those that would like to know how to construct gear wheels, but whose duties do not afford them sufficient leisure to acquire a technical knowledge of the subject. <\\»\1 Af^.1^,1^ CONTENTS PAKT I . Chapter I. PAGE. Pitch Circle — Pitch — Tooth — Space — Addendum or Face — Flank — Clearance 1 Chapter II. Classification — Sizing Blanks and Tooth Parts from Circular Pitch — Ce n ter D istan ce 5 Chapter III. Single Curve Gears of 30 Teeth and more 9 Chapter IV. Rack to Mesh with Single Curve Gears having 30 Teeth and more 12 Chapter V. Diametral Pitch — Sizing Blanks and Teeth — Distance be- tween the Centers of Wheels 16 Chapter VI. Single-Curve Gears, having Less than 30 Teeth — Gears and Racks to Mesh with Gears having Less than 30 Teeth... 20 Chapter VII. Double-Curve Teeth— Gear of 15 Teeth— Rack 25 Chapter VIII. Double-Curve Gears, having More and Less than 15 Teeth — Annnlar Gears 30 VI CONTENTS. Chapter IX. PAGK. Bevel Gear Blanks 3-t Chapter X. Bevel Gears — Form and Size of Teeth — Cutting Teeth 41 Chapter XI. Worm Wheels — Sizing Blanks of 32 Teeth and more G3 Chapter XII. Sizing Gears when the Distance between the Centers and the Ratio of Speeds are fixed — General Remarks — Width of Face of Spur Gears — Speed of Gear Cutters — Table of Tooth Parts 79 PART II. Chapter I. Tangent of Arc and Angle 8? Chapter II. Sine, Cosine and Secant — Some of their Applications in Machine Construction 93 Chapter III. Application of Circular Functions — Whole Diameter of Bevel Gear Blanks — Angles of Bevel Gear Blanks 102 Chapter IV. Spiral Gears — Calculations for Pitch of Spirals 109 Chapter V. Examples in Calculations of Pitch of Spirals — Angle of Spiral — Circumference of Spiral Gears — A few Hints on Cutting 113 CONTENTS. VII Chapter VI. PAGE. Normal Pitch of Spiral Gears — Curvature of Pitch Surface — Formation of Cotters 116 Chapter VII. Cutting Spiral Gears in a Universal Milling Machine 122 Chapter VIII. Screw Gears and Spiral Gears — General Remarks 129 Chapter IX. Continued Fractions — Some Applications in Machine Con- struction 132 Chapter X. Angle of Pressure 137 Chapter XI. In fcernal Gears — Tables — Index 139 Chapter XII. Strength of Gears — Tables 142 PART I. CHAPTER I. PITCH CIRCLE, PITCH, TOOTH, SPACE, ADDENDUM OR FACE, FLANK, CLEARANCE. Let two cylinders, Fig. 1, touch each other, their . original Cyi axes be para" lei and the cylinders be on shalts, turning freely. If, now, we turn one cylinder, the adhesion of its surface to the surface of the other cylinder will make that turn also. The surfaces touching each other, without slipping one upon the other, will evi- dently move through, the same distance in a given . . Linear Veloci time. This surface speed is called linear velocity. ty. TANGENT CYLINDERS. Linear Velocity is the distance a point moves along a line in a unit of time. The line described by a point in the circumference of either of these- cylinders, as it, rotates, may be called an arc. The length of the arc (which may be greater or less than the circumference of cylinder), described in a unit of time, is the velocity. The length, expressed in lineal* units, as inches, feet, etc., is the linear velocity. EEOWS" t SHAP.PE MEG. CO. The length, expressed in angular units, as degrees, is the angular Telocity. If now, instead of l 3 we take 360 = , or one turn, as Angular "v e- t ] le ann^ar nnit, and 1 minute as the time unit, the angular velocity will be expressed in turns or revolu- tions per minute. If these two cylinders are of the same size, one will make the same number of turns in a minute that the Relative An- other makes. It cne cylinder is twice as large as the gular "\ elocitr. _ " ° other, the smaller will make two turns while the larger makes one. but the linear velocity of the surface of each cylinder remains the same. This combination would be very useful in mechan- ism if we could be sure that one cylinder would always turn the other without slipping. _-s- LancL Addendum. Tooth. Gear. Train. -:::•.:_>.' - ~z- LINE _~ : : t - =?:_£ IFi~. 3 In the periphery of these two cylinders, as in Fig. 2, cut equidistant grooves. In any grooved piece the places between grooves are called lands. Upon the lands add part-; ; these parts are called addenda. A land and its al.lendum is called a tooth. A toothed cylinder is called a gear. Two or more gears with teeth interlocking are called a train. A Hue. c c\ Fig. PROVIDENCE, R. I Addendum Circle. 2 or 3, between the centers of two wheels is called the Line of Cen line of centers. A circle just touching the addenda ters ' is called the addendum circle. The circumference of the cylinders without teeth is called the pitch circle. This circle exists geometri- Pltch Circle - cally in every gear and is still called the pitch circle pitcll circle or the primitive circle. In the study of gear wheels, it th e alS PrinStive is the problem to so shape the teeth that the pitch circle, circles will just touch each other without slipping. On two fixed centers there can turn only two circles, one circle on each center, in a given relative angular yelocity and touch each other without slipping. THICKNESS OF WIDTH OF SPACE j TOOTH AT PITCH LINE | AT PITCH LINE Fig. 4 4 BROWN & SHARPE MEG. CO. Space. The groove between two teeth is called a space. In cut gears the width of space at pitch line and thick- ness of tooth at pitch line are equal. The distance between the center of one tooth and the center of the circular Pitch, next tooth, measured along the pitch line, is the cir- cular pitch ; that is, the circular pitch is equal to a Tooth Thick- tooth and a space; hence, the thickness of a tooth at ness. x . the pitch line is equal to one-half the circular pitch. •^P^^V Let D = diameter of addendum circle. tions of Parts for Teeth and t< D — " " pitch " " P' = circular pitch. " t= thickness of tooth at pitch line. " s = addendum or face, also length of working part of tooth below pitch line or flank. " 2s=D" or twice the addendum, equals the working depth of teeth of two gears in mesh. " f— clearance or extra depth of space below working depth. " s +f= depth of space below pitch line. " "D" +f= whole depth of space. " N= number of teeth in one gear. " 7r=3.1416 or the circumference wlien diameter is 1. P' is read "P prime/' D" is read "D second." n is read, "pi." If we multiply the diameter of any circle by tt, the To find the product will bs the circumference of this circle. If and Diameter we divide the circumference of any circle by tt, the quotient will be the diameter of this circle. CHAPTER II. CLASSIFICATION -SIZING BUNKS AND TOOTH PARTS FROM CIRCULAR PITCE— CENTRE DISTANCE— PATTERN GEARS. If we conceive the pitch, of a pair of cears to be Elements of x * ° the Teeth. made the smallest possible, we ultimately come to the conception of teetn that are merely lines upon the original pitch surfaces. These lines are called ele- ments of the teeth. Gears may be classified with reference to the elements of their teeth, and also with reference to the relative position of their axes or -shafts. In most gears the elements of teeth are either straight lines or helices (screw-like lines). Part I. of this book, treats upon three kinds of gears. First — Spur Gears ; those connecting parallel shafts ^ tVX GearB - and whose tooth elements are straight. Second — Bevel Gears; those connecting shafts Bevel Gears, whose axes meet when sufficiently prolonged, and the elements of whose teeth are straight lines. In bevel gears the surfaces that touch each other, without slipping, are upon cones or parts of cones whose apexes are at the same point where axes of shafts meet. Third — Screw or Worm Gears; those connecting w |££ e Jlars. r shnfts that are not parallel and do not meet, and the elements of whose teeth are helical or screw-like. The circular }>itcfi and number of teeth in a wheel giz . n? beingf o-iven, the diameter of the wheel and size of Blanks, &<$. tooth parts are found as follows : Dividing by 3.1410 is the same as multiplying by Now - — J— T — 3183; hence, multiply the cir- 3-14 16- 3-14 16 cumference of a circle by .3183 and the product will be the diameter of the circle. Mult' ply the circular pitch by .3183 and the product will be the same part of the G BBOWM & SHARPE MFG. CO. diameter of pitch circle that the circular pitch is of the circumfeivnce of pitch circle. This part or module is a Diameter C; ,ii e d a diameter pitch. There are as manv diameter Pitch, or Mod- \ uie. pitches contained in the diameter of a pitch circle as there are teeth in the wheel. ai^the^ddeH 6 Most mechanics make the addendum of teeth equal ?ili^^ a ^f to the module. Hence we can designate the module or the >anie. raai- o an y- diameter pitch by the same letter as we do the adden- dum ; that is, let s=the module. .3183 V'=$, or circular pitch multiplied by .3183=5. or the module. X>=D', or number of teeth in a wheel, multiplied by Diameter of ' * J Pitch circle, the module, equals diameter of pitch circle. (N-j-2) s=D, or add 2 to the number of teeth, mul- wiioieDiam-tinv the sum by s and product will be the whole di- eter. L - - r a meter. -=/, or one tenth of thickness of tooth at pitch-line clearance. equals amount added to bottom of space for clearance. Some mechanics prefer to make/ 1 equal to T *g- of the working depth of teeth, or .0625 D'\ One-tenth of the thickness of tooth at pitch-line is more than one- sixteenth of working depth, being .07S54 D". Example. Example. — Wheel 30 teeth. U" circular pitch. c . _, . P'=1.5" : then t—~o" or thickness of tooth equals §■". Sizes ol Blank J- * and Tooth s = 1.5 " x .31 83=.47 • ^> " = moduie for 1^ " P'. (See - for Gear of so teeth. \% tables of tooth parts, pages 145 -14 s ). phcA; 11 D' = 30X.4775" = 14.325' = diaineter of pitch-circle. D=(30+2)X.4775"=15.280''=diameter of adden- dum circle. / — tV °f -75'' = .075" = clearance at bottom of space. 'D" = '2x. 4775" = . 9549" = working depth of teeth. D''-/-2x.4775'>.075" = 1.0299" = whole depth of space. s+/=. 4775" -.075" = . 5525" = depth of space inside of pitch-line. D"=-2s or the working depth of teeth is equal to two modules. In making calculations it is well to retain the fourth place in the decimals, but when drawings are passed into the workshop, three places of decimals are suffi- cient. PROVIDENCE, K. I. Fig. 5, Spur Gearing. 8 BROWN & SHARPE MFG-. CO. Distance be- The distance between' the centers of two wheels is Iween centers of two Gears, evidently equal to the radius of pitch-circle of one wheel added to that of the other. The r.idius of pitch-circle is equal to s multiplied by one-half the number of teeth in the wheel. Hence, if we know the number of teeth in two wheels, in mesh, and the circular pitch, to obtain the distance between centers we first find s ; then multiply s by one- half the sum of number of teeth in both wheels and the product will be distance between centers. Example — "What is the distance between the centers of two wheels 35 and 60 teeth, 1J" circular pitch. We first find s to be 11" x .3183 = . 3979". Multiplying by 47.5 (one-half the sum of 35 and 60 teeth) we obtain 18.899" as the distance between centers. Q^T™ f fJT Pattern Gears should be made large enough to cnnnKage in o o Gear Castings, allow for shrinkage in casting. In cast iron the shrinkage is about ijj inch in one foot. For gears one to two feet in diameter it is well enough to add simply -j-j-g- of diameter of finished gear to the pattern. In gears about six inches diameter or less, the moulder will generally rap the pattern in the sand enough to make any allowance for shrinkage unnecessary. In pattern gears the spaces between teeth should be cut wider than finished gear spaces to allow for rapping and to avoid having too much cleaning to do in order to have gears run freely. In cut patterns of iron it is generally Metal Pattern enough to make spaces .015" to .02 ' wider. This makes clearance .03" to .04" in the patterns. Some moulders might want .06" to .07" clearance. Metal patterns should be cut straight ; they work better with no draft. It is well to leave about .005" to be finished from side of patterns after teeth are cut ; this extra stock to be taken away from side where cutter comes through so as to take out places where stock is broken out. The finishing should be done with file or emery wheel, as turning in a lathe is likely to break out stock as badly as a cutter might do. If cutters are kept sharp and care is taken when coming through the allowance for finishing is not nec- essary and the blanks may be finished before they are cut. CHAPTER III. SINGLE-CURVE GEARS OF 30 TEETH AND OVER. Single-curve teeth are so called because they have T g^ le Curve but one curve by theory, this curve forming both face and flank of tooth sides. In any gear of thirty teeth and more, this curve can be a single arc of a circle whose radius is one-fourth the radius of the pitch circle. In gears of thirty teeth and more, a fillet is added at bottom of tooth, to make it stronger, equal in radius to one-seventh the widest part of tooth space. A cutter formed to leave this fillet has the advantage of wearing longer than it would if brought up to a corner. In gears less than thirty teeth this fillet is made the same as just given, and sides of teeth are formed with more than one arc, as will be shown in Chapter VI. Having calculated the data of a efear of 30 teeth, f Example of a , . , fo ' 4 Gear, N=30, P inch circular pitch (as we did in Chapter II. for lh" -%"• pitch), we proceed as follows : 1. Draw pitch circle and point it off into parts equal Geometrical , , ,„ ., . , .. , Construction. to one-hall the circular pitch. Fig. 6. 2. From one of these points, as at B, Fig. 6, draw radius to pitch circle, and upon this radius describe a semicircle; the diameter of this semicircle being equal to radius of pitch circle. Draw addendum, working depth and whole depth circles. 3. From the point B, Fig. 6, where semicircle, pitch circle and outer end of radius to pitch circle meet, lay off a distance upon semicircle equal to one-fourth the radius of pitch circle, shown in the figure at BA, and is laid off as a chord. 4. Through this new point at A, upon the semicircle, draw a circle concentric to pitch circle. This last is 10 BROWS k SHAEPE MEG. CO. Fig. 6 r^^SWM GEAR, 30 TEETH, "CIRCULAR PITCH, P'= = for .75" N = =30 P =4.1888" t = .375" S = .2387* D' = .4775" S+/ = .2762' D'+f = .5150" D' = 7.1610" D = 7.6384' SINGLE CURVE GEAR. PROVIDENCE, R. I. 11 called tlie base circle, and is the one for centers of tooth arcs. In the system of single cnrve gears, we have adopted the diameter of this circle is .968 of the diameter of pitch circle. Thus the base circle of any gear 1 inch pitch diameter by this system is .968". If the pitch circle is 2" the base circle will be 1.936." 5. With dividers set to one-quarter of the radius of pitch circle, draw arcs forming sides of teeth, placing one leg of the dividers in the base circle and letting the other leg describe an arc through a point in the pitch circle that was made in laying off the parts equal to one-half the circular pitch. Thus an arc is drawn about A as center through B. 6. With dividers set to one-seventh of the widest part of tooth space, draw the fillets for strengthening teeth at their roots. These fillet arcs should just touch the whole depth circle and the sides of teeth already described. Single curve or involute gears are the only gears invori^Gear- that can run at varying distance of axes and transmit lng- unvarying angular velocity. This peculiarity makes involute gears specially valuable for driving rolls or any rotating pieces, the distance of whose axes is likely to be changed. The assertion that gears crowd harder on bearings Pressure on ° ° bearings. when of involute than when of other forms of teeth, has not been proved in actual practice. Before taking next chapter, the learner should make Practice, De- several drawings of gears 30 teeth and more. Say next chapter. make 35 and 70 teeth 1|" P'. Then make 40 and 65 teeth !" F An excellent practice will be to make drawing on cardboard or Bristol-board and cut teeth to lines, thus making paper gears ; or, what is still better, make them of sheet metal. By placing these in mesh the learner can test the accuracy of his work. 12 BKOWN & SHAKPE MFG. CO. CHAPTER IV. RACK TO MESH WITH SINGLE-CURYE GEARS HAYING 30 TEETH AND OYER. made a g re ^ra' This gear (Fig. 7) is made precisely the same as gear tory to drawing i n Chapter III. It makes no difference in which direc- tion the construction radius is drawn, so far as obtain- ing form of teeth and making gear are concerned. Here the radius is drawn perpendicular to pitch line of rack and through one of the tooth sides, B. A semi- circle is drawn on each side of the radius of the pitch circle. The points A and A' are each distant from the point B, equal to one-fourth the radius of pitch circle and correspond to the point A in Fig. 6. In Fig. 7 add two lines, one passing through B and A and one through B and A'. These two lines form angles of 75- 10 (degrees) with radius BO. Lines BA and BA' are called lines of pressure. The sides of rack teeth are made perpendicular to these lines. Rack. A Rack is a straight piece, having teeth to mesh with a gear. A rack may be considered as a gear of infinitely long radius. The circumference of a circle approaches a straight line as the radius increases, and when the radius is infinitely long any finite part of the construction circumference is a straight line. The pitch line of a of Pitch Line of . .,-.'. Hack. rack, then, is merely a straight line just touching the pitch circle of a gear meshing with the rack. The thickness of teeth, addendum and depth of teeth below pitch line are calculated the same as for a wheel. (For pitches in common use, see table of tooth parts.) The term circular pitch when applied to racks can be more accurately replaced by the term linear pitch. Linear applies strictly to a line in general while circular pertains to a circle. Linear pitch means the distance between the centres of two teeth on the pitch line whether the line is straight or curved. PROVIDENCE, R. I. 13 A rack to mesh with a single-curve gear of 30 teeth or more is drawn as follows : 1. Draw straight pitch line of rack ; also draw ad- dendum line, working depth line and whole depth line, each parallel to the pitch line (see Fig. 7). Rack. Fig. 7. RACK TO MESH WITH SINGLE CURVE GEAR HAVING 30 TEETH AND OVER. 14 BROWN i: SHARPE MFG. CO. 2. Point off the pitch line into parts equal to one- half the circular pitch, or = t. 3. Through these points draw lines at an angle of 7-H with pitch lines, alternate lines slanting in oppo- site directions. The left-hand side of each rack tooth is perpendicular to the line BA. The right-hand side of each rack tooth is perpendicular to the line BA'. 4. Add fillets at bottom of teeth equal to 4- of the width of spaces between the rack teeth at the adden- dum line. sid\? g of e nick Tlie sketch, Fig. 8, will show how to obtain angle cf Teeth. sides of rack teeth, directly from pitch line of rack, without drawing a gear in mesh with the rack. Upon the pitch line b b', draw any semicircle — b a a' 1/ . From point b lay off upon the semicircle the distance b «, equal to one-quarter of the diameter of semicircle, and draw a straight line through b and a. This line, b a, makes an angle of 75.}° with pitch line bb', and can be one side of rack tooth. The same construction, b' a', will give the inclination 75|° in the opposite direction for the other side of tooth. The sketch, Fig. 9, gives the angle of sides of a tool for planing out spaces between rack teeth. Upon any line OB draw circle OABA'. From B lay off distance BA and BA'. each equal to one-quarter of diameter of the circle. Draw lines OA and OA . These two lines form an angle of 29°, and are right for inclination of sides of rack tool. PROVIDENCE, R. I. 15 Make end of rack tool .31 of circular pitch, and then ^Y 11 * 1 of ? ack ■t ' Tool at end. round the corners of the tool to leave fillets at the bottom of rack teeth. Thus, if the circular pitch of a rack is 1J" and we multiply by .31, the product .465" will be the width of tool at end for rack of this pitch before corners are taken off. This width is shown at x y. A Worm is a screw that meshes with the teeth of a gear. This sketch and the foregoing rule are also right for Worm Thread a worm-thread tool, but a worm-thread tool is not usually rounded for fillet. In cutting worms, leave width of top of thread .335 of the circular pitch. When this is done, the depth of thread will be right. SKETCH OF WORM THREAD 16 CHAPTER V. DIAMETRAL PITCH— SIZING BLANKS AND THE TEETH OF SPUR GEARS —DISTANCE BETWEEN THE CENTRES OF WHEELS. In making drawings of gears, and in cutting racks, When it is . . ° ° 7 - , necessary to it is necessary to know the circular pitch, both on know the Cir- . " . . . cuiar Pitch. account oi spacing teeth and calculating then* strength. It would be more convenient to express the circular pitch in whole inches, and the most natural divisions in a complete of an inch, as 1" P', f" P', \'' P', and so on. But as Pitch circum- the circumference of the pitch circle must contain the ference must • i ■. i i i -, m • ■ contain the cir- circular pitch soine whole number ot times, corre- some whole sponding to the number of teeth in the gear, the times ber ° diameter of the pitch circle will often be of a size not readily measured with a common rule. This is because the circumference of a circle is equal to 3 1416 times the diameter, or the diameter is equal to the circum- ference multiplied by .3183. In practice, it is better that the diameter should be Pitch, j .n f some size conveniently measured. The same applies Terms ot the ■ L L Diameter. fc the distance between centers. Hence it is generally more convenient to assume the pitch in terms of the diameter. In Chapter II. was given a definition of a diameter pitch, and also how to get a diameter pitch from the circular pitch. "We can also assume a diameter pitch and pass to its circular Pitch equivalent circular pitch. 11 the circumference of the and a JJiame- x x ter Pitch. pitch circle is divided by the number of teeth in the gear, the quotient will be the circular pitch. In the same manner, if the diameter of the pitch circle is divided by the number of teeth, the quotient will be a diameter pitch. Thus, if a gear is 12 inches pitch diameter and has 48 teeth, dividing 12" by 43, the quotient J" is a diameter pitch of this gear. In prac- PROVIDENCE, R. I. 17 tice, a diameter pitch is taken in some convenient part of an inch, as A-'' diameter pitch, and so on. It ^Abbreviation ' ^ t * ' of Diameter is convenient in calculation to designate one of these Pitch, diameter pitches by s, as in Chapter II. Thus, for \" diameter pitch, s is equal to J". Generally, in speak- ing of diameter pitch, the denominator of the fraction only is named. J" diameter pitch is then called 3 diametral pitch. That is, it has been found more con- venient to take the reciprocal of a diameter pitch in making calculation. The reciprocal of a number is 1, divided by that number. Thus the reciprocal of J is a NuE? 4, because \ goes into 1 four times. Hence, we come to the common definition : Diametral Pitch is the number of teeth to one inch pitch. of diameter of pitch circle. Let this be denoted by P. Thus, J" diameter pitch we would call 4 diametral pitch or 4 P, because there would be 4 teeth to every inch in the diameter of pitch circle. The circular pitch and the different parts of the teeth are derived from the diametral pitch as follows. ^iifi = P', or 3.1416 divided by the diametral pitch Given the Di- r i/ x ametral to find. is equal to the circular pitch. Thus to obtain the cir-*£ e Circular cular for 4 diametral pitch, we divide 3.1416 by 4 and get .7854 for the circular pitch, corresponding to 4 c £^ ta pitoh diametral pitch. traiPit?n. ame " In this case we would write P=4,P' = .7854", *=i". £-"— s, or one inch divided by the number of teeth to an inch, gives distance on diameter of pitch circle occupied by one tooth. The addendum or face of tooth is the same distance as s. i _ s =P, or one inch divided by the distance occupied by one tooth equals number of teeth to one inch. ±*l—t t or 1.57 divided by the diametral pitch gives amJSprtchto thickness of tooth at pitch line. Thus, thickness of JJes^of Tooth teeth along the pitch line for 4 diametral pitch is .392". L* n e he Pitch ^=D', or number of teeth in a gear divided by the N umber *o? diametral pitch equals diameter of the pitch circle, and the a Diam- Thus for a wheel, 60 teeth, 12 P, the diameter of^h^Diam- pitch circle will be 5 inches. gg^* Pitch — i^=D, or add 2 to the number of teeth in a wheelN umber of _. -1-1 -i Teeth ina wheel and divide the sum bv the diametral pitch, and the and the Diame- J r tral Pitch to find the Wholo Diameter. 18 BKOWN & SHARPE MFG. CO. quotient will be the whole diameter of the gear or the diameter of the addendum circle. Thus, for 60 teeth, 12 P, the diameter of gear blank will be 5fv inches. £,=P, or number of teeth divided by diameter of pitch circle in inches, gives the diametral pitch or number of teeth to one inch. Thus, in a wheel, 24 teeth, 3 inches pitch diameter, the diametral pitch is 8. — jp = P, or add 2 to the number of teeth; divide the sum by the whole diameter of gear, and the quotient will be the diametral pitch. Thus, for a wheel 3^'' diameter, 14 teeth, the diametral pitch is 5. D' P=N, or diameter of pitch circle, multiplied by diametral pitch equals number of teeth in the gear. Thus, in a gear, 5 pitch, 8" pitch diameter, the num- ber of teeth is 40. D P — 2=N or multiply the whole diameter of the gear by the diametral pitch,subtract 2, and the remain- der will be the number of teeth. j^p = s, or divide the whole diameter of a spur gear by the number of teeth plus two, and the quotient will be the addendum, or a diameter pitch. " Pitch >iameter ^ n f u ^ ure ? "vvhen we speak of a diameter pitch, we shall mean the addendum distance or s. If we speak of so many diameter pitches, we shall mean so many The Diame- times s, (-3= s). "When we say the diametral pitch we tral Pitch. F ' - J * shall mean the number of teeth to one inch of diameter of pitch circle, or P, (— =P). s To obtain Di- Pitc When the circular pitch is given, to find the corre- pitch Circular sponding diametral pitch, divide 3.1416 by the circular pitch. Thus 1.57 P is the diametral pitch correspond- ing to 2-inch circular pitch, (^l^i^=P). Example. What diametral pitch corresponds to J-" circular pitch ? Remembering that to divide by a fraction we multiply by the denominator and divide by the numer- ator, we obtain 6.28 as the quotient of 3.1416 divided by J . 6.28 P, then, is the diametral pitch corresponding to J circular pitch. This means that in a gear of \ inch circular pitch there are six and twenty-eight one hundredths teeth to every inch in the diameter of the pitch circle. In the table of tooth parts the diametral PROVIDENCE, R. I. 19 pitches corresponding to circular pitches are carried out to four places of decimals, but in practice three places of decimals are enough. When two gears are in mesh, so that their pitch circles just touch, the distance between their axes or centers is equal to the sum of the radii of the two gears. The number of the diameter pitches between centers is equal to half the sum of number of teeth in both gears. This principle is the same as given in Chapter II., page 6, but when the diametral pitch an d -o^nce ^ht numbers of teeth in two gears are given, add together tween centers. the numbers of teeth in the two wheels and divide half the sum by the diametral pitch. The quotient is the center distance. A gear of 20 teeth, 4 P, meshes with a gear of 50 Example, teeth : what is the distance between their axes or centers ? Adding 50 to 20 and dividing half the sum by 4, we obtain 8f" as the center distance. ' The term diametral pitch is also applied to a rack. Thus, a rack 3 P, means a rack that will mesh with a gear of 3 diametral pitch. It will be seen that if the expression for a diameter Fractional . -,, , \ Diametral pitch has any number except 1 for a numerator, we Pitch, cannot express the diametral pitch by naming the denominator only. Thus, if the addendum or a diam- eter pitch is f^, the diametral pitch will be 2^, because 1 divided by T \ equals 2£. In Chapter II, the term modtde is used in the same sense as the term a diameter pitch. Modtde is much used where gears are made to metric sizes, for the reason that, the millimeter being so short, the module is conveniently expressed in millimeters. If we know the module of a gear we can figure the other parts as easily as we can if we know either the circular pitch or the diametral pitch. The module is, in a sense, an actual distance, while the diametral pitch, or the num- ber of teeth to an inch, is a relation or merely a ratio. The meaning of the module is not easily mistaken. 20 EEOWU i SHiEPE 5IT(i. CO. CHAPTER VI. SUGLE-GURYE GEARS HAYING LESS THAN 30 TEETH— GEARS AND RACKS TO MESH WITH GEARS HAVING LESS THAN 30 TEETH. Construction, Jq Yig. 10. tlie construction of the rack is the same Fig. 10. ° as the construction of the rack in Chapter TV. Tne gear in Fig. 10 is drawn from base circle out to adden- dum circle, by the same method as the gear in Chapter III., but the spaces inside of base circle are drawn as follows : Flanks of T n gears, 12 to 19 t-eth, the sides of spaces inside Gears in low = ' ' " numbers of,,f the base circle are radial for a distance, a b, equal Teeth. «, - » ' l to ^, or 3.5 divided by the product of the pitch by the number of teeth. In gears with more than 10 teeth the radial construction is omitted. construction Then, with one leg- of dividers in pitch circle in of Fig. 10 con- ° * tinued. center of next tooth, e, and other leg just touching one of the radial lines at l>. continue the tooth side into c, until it will touch a fillet arc. whose radius is i the width of space at the addendum circle. The part, V c', is an arc from center of tooth f/, etc. The flanks of teeth or spaces in gear, Fig. 11, are made the same as those in Fig. 10. This rule is merely conventional or not founded upon any principle other than the judgment of the de- signer, to efTe:-t the object to have spaces as wide as practicable, just below or inside of base circle, and then strengthen flank with as large a fillet as will clear & addenda of any gear. If flanks in any gear will clear addenda of a rack, they will clear addenda of any Internal Gear, other gear, except internal gears. An internal gear is one having teeth upon the inner side of a rim or ring. Now, it will be seen that the gear, Fig. 10, has teeth PROVIDENCE, R. I. 21 Fig. 10 22 re. :w>- i snAzrz irre. ::. too much rounded at the points or at the addendum arcle. In gears cf pitch coarser than 10 to r_ch (10 AddTSof ^)» an( ^ having less than 30 teeth, this rounding 1 - — becomes obj ectionable. This rounding occur s, be a nse in these gears arcs of en : les depart too far from the true involute carve, being so much that points of teeth get no bearing on flanks of teeth in other wheels. In gear. Fir 11. r"_e teetii outside of base circle are made as nearly true involute as a workman will be able t:- ret vri:L: : =^ e: ;.l n; :_r_r.~. This is accomplished Ajproxtma-ag follows: draw three or four tan rents to the base Don : _rue jz.- o volute, circle, * t", jj\ k k\ 1 1', letting tie points of tangen sy on base circle i'.j\ k', V be about ^ or J the circular pitch apart : the first point, *', being distant from i, equal to | the radius of pitch circle. "With dividers set to -J- the radius of pitch circle, placing one leg in /', draw the arc, a' * j/ with one leg in j' 7 and radius j' j, dxawj U; with one leg in k . and radius I: k draw J: 7. Should the addendum circle be outside of 7. the I : oth side can be completed with the last radius, I' /. The arcs, a ij, j k and k l. together form a very close approximation to a true involute from the base circle, i' j' k' I'. The exact involute for gear teeth is the curve made by the end of a band when unwound from a cylinder of the same diameter as base circle. The foregoing operation of drawing tooth sides, although te 1: : - in les iription. is very easy of practical application. Bounding of It will also be seen that the addenda of rack teeth A i '- r z. _ i : : _ Rack. in Tig. 10. interfere with the gear-teeth nanks, as ;.: m n; to avoid this interference, the teeth of rack, Fig. 11. are rounded at points cr addenda- It is also necessary to round off the points of invo- lute teeth in high numbered gears, when they are to interchange with low nunir. e: ed _ e ~ In iniercshang - able sets of gears the lowest-numbered pinion is usual- Temple:? iv 12. Just how much to round off can be 1 earned by K::i::i. : _ - -- ■-.'-•- - ----_■- -~ -; a Zzii:'^ :n: :: :_:n met.-- cr Points of teeth. ° L ..dboard. for the gear and rack, or, two gears re- quired, and fitting addenda of teeth to clear flar k - However accurate we may make a diagram, it is qu mOVIDENCE, E. I. 23 Fig. 24 BROWN ft SHARPE MFG. CO. as well to make templets in order to shape cutters accurately. It is best to make cutters to corrected diagrams, as in Fig. 11. TVhen corrected diagrams are made, as in Fig. 11. take the following: For 12 and 13 teeth, diagram of 12 teeth. a Set of Cut- ters. 14 to 16 17 " 20 21 " 25 26 " 34 35 " 54 55 " 134 135 " rack, tc ; i 14 a ti a 17 ic tc i. i 21 it (< 26 i i It a 35 a hi t i 55 i . -135 i i Templets for large gears must be fitted to run with 12 teeth. PROVIDENCE, R. I. 25 CHAPTER VII. DOUBLE-CURYE TEETH— GEAR, 15 TEETH— RACK. In double-curve teeth the formation of tooth sides ah Double- curve Tooth changes at the pitch line. In all gears the part of Fa °es are con- teeth outside of pitch line is convex ; in some gears the sides of teeth inside pitch line are convex ; in some, radial; in others, concave. Convex faces and concave flanks are most familiar to mechanics. In interchange- able sets of gears, one gear in each set, or of each pitch, has radial flanks. In the bast practice, this gear has fifteen teeth. Gears with more than fifteen teeth, have concave flanks ; gears with less than fifteen teeth, have convex flanks. Fifteen teeth is called the Base of this system. We wid first draw a gear of fifteen teeth. This construction ° of Fig. 12. fifteen-tooth construction enters into gears of any number of teeth and also into racks. Let the gear be 3 P. Having obtained data, we proceed as follows : 1. Draw pitch circle and point it off into parts equal to one-thirtieth of the circumference, or equal to thick- ness of tooth =i^. 2. From the center, through one of these points, as at T, Fig. 12, draw line OTA. Draw addendum and whole-depth circles. 3. About this point, T, with same radius as 15-tooth pitch circle, describe arcs A K and k. For any other double-curve gear of 3 P., the radius of arcs, A K and O k, will be the same as in this 15-tooth gear=2J". In a 15-tooth gear, the arc, O k, passes through the center O, but for a gear having any other number of teeth, this construction arc does not pass through center of gear. Of course, the 15-tooth radius of arcs, A K and O k, is always taken from the pitch we are working with. 26 BROWN k SHAEPE MTG. CO. EiS- 12 DOUBLE CURVE GEAR. PROVIDENCE, R. I. 27 4. Upon these arcs on opposite sides of line OTA, lay off tooth thickness, A K and O k, and draw line KT&. 5. Perpendicular to K T k, draw line of pressure, L T P ; also through O and A, draw lines A R and O r, perpendicular to K T &. The line of pressure is at an angle of 78° with the radius of gear. 6. From 0, draw a line R to intersection of AR with KT^. Through point c, where R intersects L P, describe a circle about the center, 0. In this circle one leg of dividers is placed to describe tooth faces 7. The radius, c d, of arc of tooth faces is th*. straight distance from c to tooth-thickness point, b, on the other side of radius, O T. With this radius, c b, describe both sides of tooth faces. 8. Draw flanks of all teeth radial, as e and Of The base gear, 15 teeth only, has radial flanks. 9. With radius equal to one-seventh of the widest part of space, as g h, draw fillets at bottom of teeth. The foresroiner is a close approximation to epicy- . Approxima- ° *=> rr r J tion to Epicy- cloidal teeth. To get exact teeth, make two 15-tootheioidai Teeth. gears of thin metal. Make addenda long enough to come to a point, as at n and q. Make radial flanks, as at m and p, deep enough to clear addenda when gears are in mesh. First finish the flanks, then fit the long addenda to the flanks when gears are in mesh. When these two templet gears are alike, the centers standard are the right distance apart and the teeth interlock without backlash, they are exact. One of these tem- plet gears can now be used to test any other templet gear of the same pitch. Gears and racks will be right when they run cor- rectly with one of these 15-tooth templet gears. Five or six teeth are enough to make in a gear templet. Double- curve Rack. — Let us draw a rack 3 P. R^^ig^ST 9 Having obtained data of teeth we proceed as follows : 1. Draw pitch line and point it off in parts equal to one- half the circular pitch. Draw addendum and whole-depth lines. 2. Through one of the points, as at T, Fig. 13, draw line OTA perpendicular to pitch line of rack. 28 BBOWS s 8HABPE MFG. CO. ZFi s . 13 DOUBLE CURVE RACK. PROVIDENCE, R. I. 29 3. About T make precisely the same construction as was made about T in Fig. 12. That is, with radius of 15-tooth pitch circle and center T draw arcs Jc and A K ; make O k and A K equal to tooth thickness ; draw KT/;; draw r, A R, and line of pressure, each perpendicular to K T k. 4. Through R and r, draw lines parallel to A. Through intersections c and c' of these lines, with pressure line L P, draw lines parallel to pitch line. 5. In these last lines place leg of dividers, and draw faces and flanks of teeth as in sketch. 6. The radius c' cV of rack-tooth faces is the same length as radius c d of rack-tooth flanks, and is the straight distance from c to tooth-thickness point b on opposite side of line A. 7. The radius for fillet at bottom of rack teeth is equal to i- of the widest part of tooth space. This radius can be varied to suit the judgment of the designer, so long as a fillet does not interfere with teeth of engaging gear. Fig. 14 Racks of the same pitch, to mesh with interchange- able gears, should be alike when placed side by side, and fit each other when placed together as in Fig. 14. In Fig. 13, a few teeth of a 15-tooth wheel are shown in mesh with the rack. 30 CHAPTER VIII. DOUBLE-CURVE SPUR GEARS, HAYING MORE AND FEWER THAN 15 TEETH— ANNULAR GEARS, Consxracvion ^et us draw two gears, 12 and 24 teeth, 4 P, in Ol rig. Id. d mesh. In Fig. 15 the construction lines of the lower or 24-tooth gear are full. The upper or 12-tooth gear construction lines are dotted. The line of pressure, L P. and the line K T Jc answer for both gears. The arcs A K and O k are described about T. The radius of these ares is the radius of pitch circle of a gear 15 teeth 4 pitch. The length of arcs A K and O k is the tooth thickness for 4 P. The line K T k is obtained the same as in Chapter YH. for all double-curve gears, the distances only varying as the pitch. Having drawn the pitch circles, the line K T Jc. and, perpendicular to K T Jc, the lines A Pi. r and the line of pressure L T P. we proceed with the 24-tooth gear as follows : 1. From center C. through r, draw line intersecting line of pressure in m. Also draw line from center C to E. crossing the line of pressure L P at <:■. 2. Through m describe circle concentric with pitch circle about C. This is the circle in which to place one leg of dividers to describe flanks of teeth. 3. The radius, m n. of flanks is the straight distance from m to the first tooth-thickness point on other side of hue of centers. C C". at v. The arc is continued to n. to show how constructed. This method of obtain- ing radius of double-curve tooth flanks applies to all gear's having more than fifteen teeth. 4. The construction of tooth faces is similar to 15- tooth wheel in Chapter VII. That is : Draw a circle through c concentric to pitch circle : in this circle place one leg of dividers to draw tooth faces, the radius of tooth faces being c b. PROVIDENCE, R. I. 31 \\ \ \ \ \ \ /v, \ \ •^ A \\ A \ > — \\ A y\ / \ \ \ \ PINION, 12 TEETH, GEAR 24 TEETH, 4 P. P=4 N=12 and 24. P'= .7854" t = .3927" S = .2500" D 1 ^ .5000" s+ / = .2893" D"+/ = .5393" D'= 3" Id= ( D'= 6' C< DOUBLE CURVE GEARS IN MESH. 32 zzz^zs - t =ez_zzz zzzz ::. ~ "~ T . ". rhc : lira :z z_t:s it roots of ---~z is equal to no-seventh the width of space at addendum circle. The constructions for flanks of !■"-. !•: :-—'. 14 line :: pressure in u. Tizrzngh u draw circle about C. Iz :zis circle :ze leg :: .".-—_ lers :> zlzcei zdt iz \~ zzz z.zil_:5. '2 __e rzzizz :: zzzizis is :ze 5zz:z:_:e zzzzz z ::> :Lt ziz: : :zzz.- :z iczzzc s s r :izz --. ;:.:ir,- z :.'z o: CTC. Tzis zi _ r^ ::i _ r: zlzz/zs. Tiie zc :~ c:z- :i_.e . :: •. :: ~z:~ czzszrzctizz. 3. This arc f:r zzzzzis i= : miizzzei z :: z — zr '.. ~.z~ zzzter. :zlj a":: : z: :ze siszz ;: :ze — zrlczzz zerzz :: J s.) ; the lower part of Hank is similar to flanks of gear in Chapter VI. Ciizzter VII.. zzz ::■ :ze Ji-t::zz gear iz. zze fire- going, Hie I lira zemg to y ; the arc is continued to a^ ATJiz^r ; rirs. _Ajz-zz_z. Gziez Gears with teeth inside of a rim :r zizg zze cailez \~ r -zlar :r Iz.:einzl Grzrs. TJze construction of tooth outlines is similar to the f ore- g: iz.gr. c':: :ze z//::z5 :: z sjzzr ez:e:zizl zez: ::e::zze tze :-":"" :: :.z e_.r"rizr gezz. Prof. MacCord has shown that in the system just ze=::r*:e~. :ze z:zi:z zirsizzrg — _:z ez ar_zzziar .ez' zizz: :li±er zrzrz i: cv :' '. .-: :::tt1 Zr:'_. Tlzz a gear :: -- teeth zz:: work with an annular gear of 36 teeth, but it will work with annular gears of 39 tee:b and more. The n'.ers at the roots of the teeth mzs: re •: z I e s.s re'ivs :z..z iz :riiz:-.:v srzr g:ars. .An az:: z : ge.-.r : frrizz .zm :ts rz..:e cr i.-=s iz.z 15 tee: e z: r. This ^vill be shown in Part IL Azzzziaz-gezz y.atreizis rerzzie zz:re jzzziz :':: zzz'zizi^gtzazz ezrerzai or z z gears pzz:_=. In speaking :: linereni -siz ed gears, the smallest Tze zze of pressure iz. all z^:^^ except involute, eonstaiitlj :L i_ges. 78" is the pressure angle in double-curve, or epicrcloidal gears foe an instant PROVIDENCE. R. I. 33 only; in our example, it is 78° when one side of a fcooth reaches the line of centers, and the pressure against teeth is applied in the direction of the arrows. The pressure angle of involute gears does not change. An explanation of the term angle of pressure is given in Part II. We obtain the forms for ejncycloidal gear cutters by means of a machine called the Odontom Engine. This machine will cut original gears with theoretical accuracy. It has been thought best to make 24 gear cutters 24 Double- ° ° curve Gear for each pitch. This enables us to fill any require- Cu "era for ± j i. each Pitcb. ment of gear-cutting very closely, as the range covered by any one cutter is so small that it is exceedingly near to the exact shape of all gears so covered. Of course, a cutter can be exactly right for only one gear. Special cutters can be made, if desired. 1 PITCH TOOTH CURVES from the ODONTOM ENGINE. 34 CHAPTER IX. BEVEL-GEAR BLANKS, Bevel Gears connect shafts whose axes meet when Be ^ J« Qg ai ? f sufficiently prolonged. The teeth of bevel gears are cones. formed upon formed about the frustrums of cones whose apexes frustrums of l are at the same point where the shafts meet. In Fig. 16 we have the axes A O and B 0, meeting at O, and the apexes of the cones also at O. These cones are called the pitch cones, because they roll upon each other, and because upon them the teeth are pitched. If, in any bevel gear, the teeth were sufficiently pro- longed toward the apex, they would become infinitely small ; that is, the teeth would all end in a point, or vanish at 0. We can also consider a bevel gear as beginning at the apex and becoming larger and larger as we go away from the apex. Hence, as the bevel gear teeth are tapering from end to end, we may say BEVEL GEAR PITCH CONES. Fig. 16. that a bevel gear has a number of pitches and pitch circles, or diameters : in speaking of the pitch of a bevel gear, we mean always the pitch at the largest PROVIDENCE, R. I. 35 pitch circle, or at the largest pitch diameter, as at b d, Fig. 17. Fig. 17 is a section of three bevel gears, the gear o B q being twice as large as the two others. The outer surface of a tooth as m m' is called the face of Construction of Bevel Gear the tooth. The distance m m' is usually called the B1 anks. length of the face of the tooth, though the real length is the distance that it occupies upon the line O i. The outer part of a tooth at m n is called its large end, and the inner part m' n' the small end. Almost all bevel gears connect shafts that are at right angles with each other, and unless stated other- wise we always understand that they are so wanted. The directions given in connection with Fig. 17 apply to gears with axes at right angles. Having decided upon the pitch and the numbers of teeth : — 1. Draw centre lines of shafts, A O B and COD, at right angles. 2. Parallel to A O B, draw lines a b and c d, each distant from A B, equal to half the largest pitch diameter of one gear. For 24 teeth, 4 pitch, this half largest pitch diameter is 3" '. 3. Parallel to COD, draw lines e f and g h, dis- tant from COD, equal to half the largest pitch diameter of the other gear. For a gear, 12 teeth, 4 pitch, this half largest pitch diameter is 1 J ff . 4 At the intersection of these four lines, draw lines O i, O j, O k, and 1 ; these lines give the size and shape of pitch cones. We call them " Cone Pitch Lines." 5. Perpendicular to the cone- pitch lines and through the intersection of lines a b, c d, e f , and g h, draw lines m n, o p, q r. We have drawn also u v to show that another gear can be drawn from the same diagram. Four gears, two of each size, can be drawn from this diagram. 6. Upon the lines m n, o p, q r, the addenda and depth of the teeth are laid off, these lines passing 36 BROWN i- SHARPE MFG. CO. through the largest pitch, circle ;: the gears, Layoff the s id lentlnm. it beins; in these 2:ears 4- . This sives distance m n. o p, q r. and a v equal to the working depth of teeth, which in these gears is J . The addendum of course is measured perpendicu'.aily from the cone pitch lines as at k r. 7. Draw lines m, n, p, o. . r. These lines _ the height of teeth above the cone- :teh lines as they approach O, and would vanish entirel '. It is quite us well never to have the length of teeth, or face, m n longer than one-third th- apex distance m 0, nor more than two and one- half times the circular pitch 8. Having decided upon the length of face, draw limiting lines in n perpendicular :: i 0. q r perpen- dicular to k O. and so on. TL- listance between the cone-pitch lines at the inner ends of the teeth m n and q r is called the inner or smaller pitch diameter, and the circle at these points is called the smallest pitch circle. We now Lave the outline ;: a -rtion of the gears through their axes. The listance m i is the whole diameter of the pinion. ;f el ^ hol f The distance q o is the whole diameter of the gear. - In practice these diameters can be obtained bvmeasur- obtained by i n2 r th c Irawing. The diameter of pinion is 3.45 and Measuring c - x Drawings. " f - r g eai __ We can find the angles also by measuring the drawing with a protractor. In the absence of a protrack r, templetes can be cut to the drawing. The angle form . line m m with a b is the angle of face of pinion, in this pinion o'j c 11 . or 59^° nearly. The lines q and g h give us angle of face of gear, for this gear 22 z 19 . or '2'2^ = nea:~. The angle formed by m n with a b is called the angle of edge of pinion, in our sketch 2»3 : 34', or about 26J : The angle of edge of gear, line q r with g h, is or about 63J°. In turning blanks to these angles we place one arm of the protractor or templet against the end of the hub, when trying angles of a blank. Some designers give the ausrJes from the axes of gears, bat PROVIDENCE, E. I. 37 Fig. 17 38 BE0WX & SHARPE MFG. CO., it is not convenient to- try blanks in this way. The method that we have given comes right also for angles as figured in compound rests. When axes are at right angles, the sum of angles of edge in the two gears equals 90°, and the sums of angle of edge and face in each gear are alike. The angles of the axes remaining the same, all pairs of bevel gears of the same ratio have the same angle of edge ; all pairs of same ratio and of same numbers of teeth have the same angles of both edges and faces independent of the pitch. Thus, in all pairs of bevel gears having one gear twice as large as the other, with axes at right angles, the angle of edge of large gear is 63° 26 , and the angle of edge of small ffear is 26° 34'. > DO G In all pairs of bevel gears with axes at right angles, one gear having 24 teeth and the other gear having 12 teeth, the angle of face of small gear is 59° 11'. 4, n 9 t ] ie , r The following method of obtaining the whole diam- rnethod ot ob- c s taining Whole ^ er f bevel gears is sometimes preferred : Diameter ot c l Blanks. From k lay off : upon the cone-pitch line, a distance K w, equal to ten times the working depth of the teetk=10D". Xow add ts of the shortest distance of w from the line g h, which is the perpendicular dotted line w x. to the outside pitch diameter of gear, and the sum will be the whole diameter of gear. In the same manner fu of w y. added to the outside pitch diameter of pinion, gives the whole diameter of pinion. The part added to the pitch diameter is called the » diameter increment. Part II gives trigonometrical methods of figuring bevel gears : in our Formulas in Gearing there arc trigonometrical formulas for bevel gears, and also tables for angles and sizes. Construction ^ somewhat similar construction will do for bevel of Bevel-Gear Blanks whose gears whose axes are not at right angles. Axes are not ° ct ^ at Right An- J n Fig. 18 the axes are shown at O B and D, the angle BOD being less than a right angle. 1. Parallel to B. and at a distance from it equal to the radius of the gear, we draw the lines a b and c d. PROVIDENCE, R. I. 39 INSIDE BEVEL GEAR AND PINION. Fig.no 40 BROWN & SHARPE MFG. CO. 2. Parallel to O D, and at a distance from it equal to the radius of the pinion, we draw the lines e f and g h- 3. Now, through the point j at the intersection of c d and g h, we draw a line perpendicular to O B. This line k j, limited by a b and c d, represents the largest pitch diameter of the gear. Through j we draw a line perpendicular to D. This line j 1, limited by e f and g h, represents the largest pitch diameter of the pinion. 4. Through the point k at the intersection of a b with k j, we draw a line to O, a line from j to 0, and another from 1, at the intersection j 1 and e f to 0. These lines O k, O j, and O 1, represent the cone- pitch lines, as in Fig. 17. 5. Perpendicular to the cone-pitch lines we draw the lines u v, op, and q r. Upon these lines we h"ry off the addenda and working depth as in the previous figure, and then draw lines to the point O as before. By a similar construction Figs. 19 and 20 can be drawn. STOCKING CUTTER. PROVIDENCE, K. I. . 41 CHAPTER X. BEVEL GEARS. FORMS AND SIZES OF TEETH. CUTTING TEETH. To obtain the form of the teeth in a bevel gear we Form of bevel g e a r do not lay them out upon a pitch circle, as we do in a teeth, spur gear, because the rolling pitch surface of a bevel gear, at any point, is of a longer radius of curvature than the actual radius of a pitch circle that passes through that point. Thus in Fig. 21, let f g c be a cone about the axis O A, the diameter of the cone being f c, and its radius g c. Now the radius of curvature of the surface, at c, is evidently longer than g c, as can be seen in the other view at C ; the full line shows the curvature of the surface, and the dotted line shows the curvature of a circle of the radius g c. It is extremely difficult to represent the exact form of bevel gear teeth upon a flat surface, because a bevel gear is essentially spherical in its nature ; for practical purposes we draw a line c A perpendicular to O c, letting c A reach the centre line O A, and take c A as the radius of a circle upon which to lay out the teeth. This is shown at c n m, Fig. 22. For con- venience the line c A is sometimes called the back cone radius. Let us take, for an example, a bevel gear and a Fi Ex .4 mi>le ' pinion 24 and 18 teeth, 5 pitch, shafts at right angles. To obtain the forms of the teeth and the data for cutting, we need to draw a section of only a half of each gear, as in Fig. 22. 42 BROWN & SHAItPE MFG. CO., 1. Draw the centre lines A and B 0. then the lines g h and c d. and the gear blank lines as des- cribed in Chapter IX. Extend the lines op' and o p until they meet the centre lines at A B' and A B. 2. With the radius A c draw the arc c n m, which we take as the geometrical pitch circle upon which to lay out the teeth at the large end. The distance Ac is taken as the radius of the geometrical pitch circle at the small end : to avoid confusion an arc of this circle is drawn at c n m' about A. 3. For the pinion we have the radius B c for the geometrical pitch circle at the large end and Be for the small end: the distance B c is transferred to B c . 4. Upon the arc cum lay off spaces equal to the tooth thickness at the large pitch circle, which in our example is .314'. Draw the outlines of the teeth as in previous chapters : for single curve teeth we draw a semi-circle upon the radius A c. and proceed as des- cribed in chapter III. For all bevel gears that arc to be cut with a rotary disk cutter, or a common gear cutter, single curve teeth are chosen : and no attempt should be made to cut double curve teeth. Double curve teeth can be drawn by the directions given in chapters VII aud VIII. We now have the form of the teeth at the large end of the gear. Eepeat this operation with the radius B C about B, and we have the form of the teeth at the large end of the pinion. 5. The tooth parts at the small end are designated by the same letters as at the large, with the addition of an accent mark to each letter, as in the right hand column, Fig. 22, the clearance, f. however, is usually the same at the small end as at the large, for con- venience in cutting the teeth. Sizes of the The sizes of the tooth parts at the small end are in tooth parts. the same proportion to those at the large end as the line c is to c. In our example O c is _ . and O c is 3" : dividing c' by c we have -f. or .666, as the ratio of the sizes at the small end to those PROVIDENCE, R. J. 43 CO O u IE o JH h- e* Q. Lu fe( o UJ £E z> h- DC ID (J 44 BROWN A- SHARFE MFG. CO. at the large : t is .209 :>rf of .314". and so on. If the distance n m is equal to the outer tooth thickness, t, upon the arc c n ni, the lines n A and m A will be a distance apart equal to the inner tooth thickness t upon the arc c n m . The addendum, s . and the working depth. D . are at o c and o p . 6. Upon the arcs c n m and c '" we draw the forms of the teeth of the gear and pinion at the inside. Example of As an example of the cutting; of bevel sears with Cuttins. -,• i , rotary disk cutters, or common gear cutters, let us take a pair of 8 pitch. 1'2 and 24 teeth, shown in Fig. 23. Length of In making the drawing it is well to remember that nothing is gained by having the face F E longer than five times the thickness of the teeth at the large pitch circle, and that even this is too long when it is more than a third of the apex distance c. To cut a bevel gear with a rotary cutter, as in Fig. 24, is at best but a compromise, because the teeth change pitch from end to end. so that the cutter, being of the right form for the large ends of the teeth can not be right for the small ends, and the variation is too great when the length of face is greater than a third of the apex distance c. Fig. 23. In the example, one-third of the apex distance is T V . but F E is drawn only a half inch, which even though rather short, has changed the pitch from 8 at the outside to finer than 11 at the inside. Frequently the teeth have to be rounded over at the small ends by filing : the longer the teeth the more we have to file. If there is any doubt about the strength of the teeth, it is better to lengthen at the large end. and make the pitch coarser rather than to lengthen at the small end. Data : : These data are needed before beginning to cut : 1. The pitch and the numbers of the teeth the same as for spur gears. 2. The data for the cutter, as to its form : some- times two cutters are needed for a pah' of bevel gears. 3. The whole depth of the tooth spaces, both at :-:Ti^r. PEOVIDENCE, E. I. 45 S = .200" D"= .400" S+/= .231" D"+/ = .431" D'"= .266" S'+f =.165" D'"-h/ = .298" Fig. 22. BEVEL GEARS, FORM AND SIZE OF TEETH, 46 BROWN & SHARPE MFG. CO. the outside and inside ends ; D" + f at the outside, and D'" -J- f at the inside. 4. The thickness of the teeth at the outside and at the inside ; t and t'. 5. The height of the teeth above the pitch lines at the outside and inside ; s and s'. 6. The cutting angles, or the angles that the path of the cutter makes with the axes of the gears. In Fig. 23 the cutting angle for the gear c D is A Op, and the cutting angle for the pinion is B O o. selection of The form of the teeth in one of these gears differs cutters. so much from that in the other gear that two cutters are required. In determining these cutters' we do not have to develop the forms of the gear teeth as in Fig. 22 ; we need merely measure the lines A c and B c, Fig. 23, and calculate the cutter forms as if these distances were the radii of the pitch circles of the gears to be cut. Twice the length Ac, in inches, multiplied by the diametral pitch, equals the number of teeth for which to select a cutter tor the twenty- four-tooth gear : this number is about 54, which calls for a number three bevel gear cutter in the list of bevel gear cutters, page 61. Twice B c, multiplied by 8, equals about 13, which indicates a No. 8 bevel gear cutter for the pinion. This method of selecting cutters is based upon the idea of shaping the teeth as nearly right as practicable at the large end, and then filing the small ends where the cutter has not rounded them over enough. In Fig. 25 the tooth L has been cut to thickness at both the outer and inner pitch lines, but it must still be rounded at the inner end. The teeth M M have been filed. In thus rounding the teeth ihey should not be filed thinner at the pitch lines. There are several things that affect the shape of the teeth, so that the choice of cotters is not always so simple a matter as the taking of the lines A c and B c as radii. In cutting a bevel gear, in the ordinary gear cutting PROVIDENCE, R. I. 47 BEVEL GEAR DIAGRAM FOR DIMENSIONS. 4* BROWS i SHARPE 3JTG. CO. i_ii:_ines. the finished spaces are not always of the same form as the cutter might be expected to makr. because of the changes in the positions of the eutter and of the gear blank in order to cut the teeth of the right thickness at both ends. The cutter must of course be thin enough to pass through the small end of the spaces, so that the large end has to be cut to the right width by adjusting either the cutter or the blank side wise, then rotating the blank and cutting twice around. Widening T-T.15. Ill r \Z- '--^ ■ i 1'fiT :V_ 11 u C~I"~ _ ;._r 5-rt Z~. _ V r the Urge a space widened at the large end e . and the last chip to be cut off by the right side of the eutter, the cutter having been moved to the left, and the blank rotated in the direction of the arrow : in a Universal Milling _ Machine the same result would be attained by moving the blank to the right and rotating it in the direction of the arrow. It may be well to remember that in setting to finish the side of a tooth, the tooth and the cutter are first separated side wise, and the blank is then rotated by indexing the spindle to bring the large Tteeth Mr _ end of the tooth up against the cutter. This trnds 't^re "JJ^jj not only to cut the spaces wider at the large pitch at root. circle, but also to cut off still more at .the face of the tooth ; that is, the teeth may be cut rather thin at the face and left rather thick at the root. This tendency is greater as a cutting angle BOo. Fig. 23, is smaller, or as a bevel gear approaches a spur gear, because when the cutting angle is small the blank must be rotated through a greater arc in order to set to cut the right thickness at the outer pitch circle. This can be understood by Figs. 26 and 27. Fig. i-i is a radial- toothed clutch, which for our present purpose can be regarded as one extreme of a bevel gear in which the trrth are cut square with the axis: the dotted lines indicate the different positions of the cutter, the side of a tooth being finished by the side of the cutter that is on the centre line. In setting to cut these : there is the same side adjustment and rotation of the PROVIDENCE, R. I. Tig. 24 SETTING BEVEL GEAR CUTTER OUT OF CENTRE. 50 BROWN & SHAHl'K MFG. CO. spindle as in a bevel gear, but there is no tendency to make a tooth thinner at the face than at the root. On the other hand, if we apply these same adjustments to a spur gear and cutter, Fig. 27, we shall cut the face F much thinner without materially changing the thick- ness of the root R. Fig* 26 Almost all bevel gears are between the two extremes of Figs. 26 and 27, so that when the cuttiug angle B O o, Fig. 23, is smaller than about 30°, this change in the form of the spaces caused by the rotation of the blank maybe so great as to necessitate the substitution Fig 1 28 FINISHED GEAR. PROVIDENCE, R. I. 51 of a cutter that is narrower at e e', Fig. 24, than is called for by the way of figuring that we have just given : thus in our own gear cutting department we might cut the pinion with a No. G cutter, instead of a No. 8. The No. 6, being for 17 to 20 teeth, cuts the tooth sides with a longer radius of curvature than the No 8, which may necessitate considerable filing at the small ends of the teeth in order to round them over enough. Fig. 28 shows the same gear as Fig. 25, but in this case the teeth have all been filed similar to M M, Fig. 25. Different workmen prefer different ways to com- Filing the ,i ,,. £\-i Trri teeth at the promise m the cutting or a bevel gear. When a email end. blank is rotated in adjusting to finish the large end of the teeth there need not be much filing of the small end, if the cutter is right, for a pitch circle of the radius B c, Fig. 23, which for our example is a No. 8 cutter, but the tooth faces may be rather thin at the large ends. This compromise is preferred by nearly all workmen, because it does not require much filing of the teeth : it is the same as is in our catalogue by which we fill any order for bevel gear cutters, unless otherwise specified. This means that we should send c 2SS? ti whcn a No. 8, 8-pitch bevel gear cutter in reply to an order J ee j*j are to for a cutter to cut the 12-tooth pinion, Fig. 23 ; while in our own gear cutting department we might cut the same pinion with a No. 6, 8-pitch cutter, because we prefer to file the teeth at the small end after cutting them to the right thickness at the faces of the large end. We should take a No. G instead of a No. 8 only for a 12-tooth pinion that is to run with a gear two or three times as large. We generally step off to the next cutter for pinions fewer than twenty-five teeth, when the number for the teeth has a fraction nearly reaching the range of the next cutter : thus, if twice the line B c in inches, Fig. 23, multiplied by the diametral pitch, equals 20.9, we should use a No. 5 cutter, which is for 21 to 25 teeth inclusive. In filling an order for a gear cutter, we do not consider 52 x a- shap.fz :;: the fraction but send the cotter indicated by the whole number. Later on ^c- will refer to other compromises that are madv :: :lr sotting >f bevel gears. The sizes of the H-piteh tooth parts, Fig. 23. at the large end. are copied from the table of spar _ teeth, pages 145 to 148. The distance Oc is seven-tenths of the apex tance Oc. so that the sizes of the tooth y ts t the gear cutting small end, rxieptf, are seven-tenths the larse. The order for cutting these gears goes to the workmen in tV-is :o::n : Form of Senir _ : ~ ir. ;. ::.:re Lab 5b i.tear. P = 5 N = 24 y — f = .270' D — f = .195' t = .: t = .137' s = .12o 5 = . ^7 Cot: ins: Ansle = 59° 10' - r Small Gkar. X C:::::: Angle = l. : IS Fig. B2 is aside view of a Geai Cutting Machine. A evel gar blank A is held by the index spindle B. The cotter C is sanied y the cutter-slide D. The cutter-slide- carriag E n be set to the cutting angle, the degrees being indicated on the quadrant F . Fig l Z is a plan of the machine : in this view the : ::er-slide-carriage. in order to show the detaiis little p'ainer, is not set to an angle. Before bes^nnino; to cut the cutter is set central with the index spindle and the dial G is set :; zero, so that we can adjust the softer to any required distance out of centre, in either direction. Set the c itternBlide- carriage E. Fig. 32, to the cutting angle of the gear. which for 24-teeth is 59° 10': the quadrant beii _ divided to half -degrees, vre estimate thai l( or \ de- PROVIDKNCK, K. I 53 gree more than 59°. Mark the depth of the cut at the outside, as in Fig. 30 : it is also well enough to mark the depth at 'he inside as a check. The thickness of the teeth at the large end is conveniently deter- mined by the solid gauge, Fig. 29. The gear-tooth J GEAR TOOTH GAUGE. DEPTH GAUGE. _Fi#. 30 GEAR TOOTH CALIPER. JFig.31 vernier caliper, Fig. 31, will measure the thickness of teeth up to 2 diametral pitch. In the absence of the vernier caliper we can file a gauge, similar to Fig 29, to the thickness of the teeth at the small end. The index having been set to divide to the right 8 i^.°o? tooth number we cut two spaces central with the blank, being finished leaving a tooth between that is a little too thick, as in the upper part of Fig. 25. If the gear is of cast iron, and the pitch is not coarser than about 5 diametral, this is as far as we go with the central cuts, and we proceed to set the cutter and the blank to finish first one side of the teeth and then the other, going around only twice. The tooth has to be cut away more in proportion from the large than from the small end, which is the reason for setting the cutter out of centre, as in Fig. 24. 54 BllOWN & SHARPE MFG. CO. Fig. 32 AUTOMATIC GEAR CUTTING MACHINE. SIDE ELEVATION. rrsoviDENCE, r. i. 55 It is important to remember that the part of the cutter that is finishing one side of a tooth at the pitch line should be central with the gear blank, in order to know at once in which direction to set the cutter out of centre. We can not readily tell how much out of centre to set the cutter until we have cut and tried, because the same part of a cutter does not cut to the pitch line at both ends of a tooth. As a trial distance out of centre we can take about one-tenth to one- eighth of the thickness of the teeth at the large end. The actual distance out of centre for the 12-tooth pinion is .021" : for the 24-tooth gear, .030", when using cutters listed in our catalogue. After a little practice a workman can set his blank Necessity of 1 central cuts. the trial distance out of centre, and take his first cuts, without any central cuts at all ; but it is safer to take central cuts like the upper ones in Fig. 25. The depth of cut is partly controlled by the index-spindle raising-dial-shaft H, Fig. 33, which determines the height of the index spindle, and partly by the position of the cutter spindle. We now set the cutter out of centre the trial distance by means of the cutter-spindle dial-shaft, I, Fig. 33. The trial distance can be about one-tenth the thickness of the tooth at the large end in a 12-tooth pinion, and from that to one-eighth the thickness in a 24-tooth gear and larger. The principle of trimming the teeth more at the large end than at the small is illustrated in Fig. 24, which is to move the cutter away from the tooth to be trimmed, and then to bring the tooth up against the cutter by rotating the blank in the direction of the arrow. Adustments The rotative adjustment of the index spindle is accomplished by loosening the connection between the index worm and the index drive, and turning the worm,: the connection is then fastened again. The cutter is now set the same distance out of centre in the other direction, the index spindle is adjusted to trim the other side of the tooth until one end is down nearly to the right thickness. If now the thickness of the D6 BROWN & SHAKPE MFG. CO. small end is in the same proportion to the large end as Oc' is to Oc, Fig. 23, we can at once adjust to trim the tooth to the right thickness. But if we find that the large end is still going to be too thick when the small end is right, the out of centre must be increased. It is well to remember this : too much out of centre leaves the small end proportionally too thick, and too little out of centre leaves the small end too thin. After the proper distance out of centre has been learned the teeth can be finish-cut by going around out of centre first on one side and then on the other with- out cutting any central spaces at all. The cutter spindle stops, J J, can now be set to control the out of centre of the cutter, without having to adjust by the dial G. If, however, a cast iron gear is 5-pitch or coarser it is usually well to cut central spaces first and then take the two out-of -centre cuts, going around three times in all. Steel gears should be cut three times around. Blanks are not always turned nearly enough alike to be cut without a different setting for different blanks. If the hubs vary in length the position of the cutter spindle has to be varied. In thus varying, the same depth of cut or the exact J)" -f" f may not always be reached. A slight difference in the depth is not so objectionable as the incorrect tooth thickness i hat it may cause. Hence, it is well, after cutting once around and finishing one side of the teeth, to give careful attention to the rotative adjustment of the index spindle so as to cut the right thickness. After a gear is cut, and before the teeth are filed, it is not always a very satisfactory-looking piece of work. In Fig. 25 the tooth L is as the cutter left it, and is ready to be filed to the shipe of the teeth M M, which have been filed. Fig. 34 is the pair of gears that we have been cutting ; the teeth of the 12-tooth pinion have been filed. PROVIDENCE, K. I. 57 58 BROWN" & SHARPE MFG. CO. a second ^ second approximation in cutting with a rotary approxima- x l ^ tion- cutter is to widen the spaces atthe large end by swing- ing either the index spindle or the cutter-slide-carriage, so as to pass the cutter through on an angle with the blank side-ways, called the side-angle, and not rotate the blank at all to widen the spaces. This side-angle method is employed in our No. 11 Automatic Bevel Gear Cutting Machines : it is available in the manufac- ture of bevel gears in large quantities, because with the proper relative thickness of cutter, the tooth- thickness comes right by merely adjusting for the side-angle ; but for cutting a few gears it is not much liked by workmen, because, in adjusting for the side- angle, the central setting of the cutter is usually lost, and has to be found by guiding into the central slot already cut. if the side-angle mechanism pivots about a line that passes very near the small end of the tooth to be cut, the central setting of the cutter may not be lost. In widening the spaces at the large end, the teeth are narrowed practically the same amount at the root as at the face, so that this side-angle method requires a wider cutter at e e . Fig. 24. than the first, or rotative method. The amount of filing required to correct the form of the teeth at the small end is about the same as in the first method. a third ap- A third approximate method consists in cutting proximation. . . the teeth right at the large end by going around at least twice, and then to trim the teeth at the small end and toward the large with another cutter, going around at least four times in all. This method requires skill and is necessarily a little slow, but it contains possi- bilities for considerable accuracy. a fourth ap- A fourth method is to have a cutter fully as thick as proximation. , . the spaces at the small end, cut rather deeper than the regular depth at the large end, and go only once around. This is a quick method but more inaccurate than the three preceding : it is available in the manu- facture of large numbers of gears when the tooth-face PROVIDENCE, K. I. 59 Fig. 34 FINISHED GEAR AND PINION 60 BROWN & SIIARPE MFG. CO. is sho.rt compared with the apex distance. It is little liked, and seldom employed in cutting a few gears : it may require some experimenting to determine the form of cutter. Sometimes the teeth are not cut to the regular depth at the small end in order to have them thick enough, which may necessitate reducing the addendum of the teeth, s, at the small end by turning the blank down. This method is extensively employed by chuck manufacturers. A machine that cuts bevel gears with a reciprocating motion and using a tool similar to a planer tool is called a Gear Planer and the gears so cut are said to be planed. pianino- G f ^ ne f° rm of Gear Planer is that in which the prin- bevei gears. c [pi e embodied is theoretically correct ; this machine originates the tooth curves without a former. Another form of the same class of machines is that in which the tool is guided by a former. Usually the time consumed in planing a bevel gear is greater than the time necessary to cut the same gear with a rotary cutter, thus proportionately increasing the cost. Pitches coarser than 4 are more correct and some- times less expensive when planed ; it is hardly prac- ticable, and certainly not economical, to cut a bevel ge-ir as coarse as 3P. with a rotary cutter. In gears as fine as 1GP. planing affords no practical gain in quality. While planing is theoretically correct, yet the wear- ing of the tool may cause more variation in the thick- ness of the teeth than the wearing of a rotary cutter, and even a planed gear is sometimes improved by filing. Mounting of ^ gears are not correctly mounted in the place where gears. they are to run, they might as well not be planed. In fact, after taking pains in the cutting of any gear, when we come to the mounting of it we should keep right on taking pains. Angles and The method of obtaining the sizes and angles per- geais. taining to bevel gears by measuring a drawing is quite convenient, and with care is fairly accurate. Its PROVIDEXCK, K. I. 61 accuracy depends, of course, upon the careful measur- ing of a good drawing. We may say, in general, that in measuring a diagram, while we can hardly obtain data mathematically exact, Ave are not likely to make wild mistakes. Some years ago we depended almost entirely upon measuring, but siuce the publication of this "Treatise" and our " Formulas in Gearing" we calculate the data without any measuring of a drawing. In the ""Formulas in Gearing " there are also tables pertaining to bevel gears. Several of the cuts and some of the matter in this chapter are taken from an article by O. J. Beale, in the "American Machinist," June 20, 1^95. Cutters for Mitre and Bevel Gears. Diametral Diameter of Hole in Pitch. Cutter. Cutter. 4 8 3-8" 1 1-4" 5 3 1-16 1 1 6 2 3-4 1 1-16 8 2 1-2 1 1 10 2 1-8 7-8 12 2 ii 14 2 (. i 16 1 15-10 1 1 20 1 7-8 I c 24 1 3-4 ( . WORM WHEEL. Number of Teeth, 34. Circular Pitch, 2J/.j. Throat Diameter, 44.39". Outside Diameter, 46' C3 CHAPTER XI. WORM WHEELS— SIZING BLANKS OF 32 TEETH AND OVER. A worm is a screw made to mesli with the teeth of Worm. a wheel called a worm-wheel. As implied at the end of Chapter IV., a section of a worm through its axis is, in outline, the same as a rack of corresponding pitch. This outline can be made either to mesh with single or double curve gear teeth ; but worms ' are usually made for single curve, because, the sides of involute rack teeth being straight (see Chapter IV.), the tool for cutting worm-thread is more easily made. The thread- tool is not usually rounded for giving fillets at bottom of worm-thread. The axis of a worm is usually at rigl t singles to the axis of a worm wheel: no other angle of axis is treated of in this book. The rules for circular pitch apply in the size of tooth parts and diameter of pitch-circle of worm-wheel. The pitch of a worm or screw is sometimes given in Pitch of worm a way different from the pitch of a gear, viz. : in num- ber of threads to one inch of the length of the worm or screw. Thus, to say a worm is 2 pitch may mean 2 threads to the inch, or that the worm makes two turns to advance the thread one inch. But a worm may be double-threaded, triple-threaded, and so on; hence to avoid misunderstanding, it is better always to call the advance of the worm thread the lead. Thus, a worm-Thread worm-thread that advances one inch in one turn we call one-inch lead in one turn. A single-thread worm 4 to 1" is J" lead. We apply the term pitch, that is the circular pitch, to the actual distance between the threads or teeth, as in previous chapters. In single- thread worms the lead and the pitch are alike. If we have to make a worm and wheel so many threads to 04 BROWN & SHAEPE MFG. CO. FIG. 35— WORM AND WORM-WHEEL The thread of Worm is left-handed ; Worm is single-threaded PROVIDENCE, R. I. 05 : -~ : : ir inehj we irs: ziride 1" hw lie number of threads Mm ne imdk 9 and the quotient iM-ihe circular ptidL Hence, ~_r-"-f:-" - F .z. : is - ::.;-".:::.::_. X :'/'j:::. sxprcoBBa tI .:.::- — bat i3 mean: f : ar pitch. ~ : T. r:::l « :'_e :>. .: rf :: :^ nz . ei :: use. : : ri.-.:r ~:f: t " : m= i I : . :k=. T _ e _~m.ljer of threads to one iceh linear, is the reciprocal : i ~ . r . . t i - tie: . . Multiply 3.1416 bj the number of threads to one inch, and the product will be the diametral pitch of the — ::=:- irr. r_~j. -r=i:~"-i =.".".: ^ iir'.'.r-:*-: f..:r 3 worm advancing 1" in 1^ turns that: Dnvimg of Iiead=§" or .7-5 . Iane.xT ::::': ::P = t " r -i^^^-AT " - - ii-=-~-il; :: :_ :: F = S : "" ^rr :e.":*_-e- :::; : :_ • -- Z: _ :t "_E-~_zr :_ ~::z_ ezlI — _-ee. ~r . ': : . __ V ::■. . =: ._ :_ : :i '. :::n 1 iTi- :-:_:-: jzlt A J iz.1 u\:_ i: - - :± :_e- _:- t: ii_ :: ~_t ELLim-rT-EEr :: •'::;_-:_;_ - ' _ t. ._ -: ".t : : -L-e-^t i — : \ : :_;- Li; :z :_t ii=- - .-. _ _ t _ : : : _ t - - ~ L ;. :" ifzi 1 : i_ = : - 3. From lay :± the itmtwocr c O equal to the riiii= :: :_- — ::_ T_t iiizi-E-Tr^ :: i — :-_ is _ ri.- • — -i L^~ :z" :1t ii-:^z:»5-~ : ; evlI ~ - -e ~ . :_ e~: . : : _ - _ t - ! - t ;-. : _ : e- ;- : :::_::::::_-;■:: t :" I_ ::__ . .\-.~. - i_:,- eh:! 5 :: j Xlr-s* r-e~:'TETz: tie ~_:1- Iie^-tEe: :: ~::_ :-._.*. :_r : _- .:": ;: ': :::_ :: -:::_-:_. e '. : I : .■.- ! :_" . :.: :-_ .-.ei_ _t :::!"::- :" ' — :e_ A ! r_r-T __-- .-. r - : '.-'_ :: :i:- :: — :l_--_tt.. Z_e . :^_ ml : ~_e.^" ".. : - i : : i: '.'. rz.v~ -. _ : ; :_: ; ~_i5 : t.::::_ :- - -" : :_.-:t- :_r :^tln.e :: — ::n-— LettI. F:: :. :tt"_ _ ". : e :_- " t_ . _ _ _t:t: :: D. ~_:.^1 ::~ :..^>-z. i: :_t :_: : : :r 5i_:._tE-: L_e.l_t:t. :: — jl-e-t! - . - 7-_ - JS/f---* ; .«'f PROVIDENCE, R. I. 67 threaded with a tool of the same angle as the tool that threads the worm, the end of the tool beinme ri:ri:. The hob has cut away flanks of teeth still more than in Fig. 3S. The pitch circle in Kg. 39 is the same diameter as the pitch-circle in Fig. 38. The same hob was used for both wheels. The flanks in this wheel are so much undercut as to mate- rially lessen the bearing surface of teeth and worm- thread. interference In Chapter ~YX the interference of teeth in high- numbered g r ears and racks with flanks of 12 teeth vr remedied by rounding off the addenda. Although it w raid be move systematic to round off the threads ol a worm, making them, like rack-teeth, to mesh w::_ PROVIDENCE, R. I. 71 interchangeable gears, yet this has not generally been done, because it is easier to make a worm-thread tool with straight sides. Instead of cutting away the addenda of worm- thread, we can avoid the interference with flanks of wheels having less than 30 teeth by making wheel blanks larger. The flanks of wheel in Fig. 40 are not undercut, be- Fig. 40. cause the diameter of wheel is so large that there is hardly any tooth inside the pitch-circle. The pitch-circle in Fig. 40 is the same size as pitch- circles in Figs. 38 and 39. This wheel was sized by the following rule : Multiply the pitch diameter of Diameter at Throat to Avoid the wheel by .937, and add to the product four times interference, the addendum (4 s) ; the sum will be the diameter for the blank at the throat or small part. To get the whole diameter, make a sketch with diameter of throat to the foregoing rule and measure the sketch. It is impractical to hob a wheel of 12 to about 16 or 18 teeth when blank is sized by this rule, unless the wheel is driven by independent mechanism and not by the hob. The diameter across the outermost parts of teeth, as at A B, is considerably less than the largest diameter of wheel before it was bobbed. In general it is well to size all blanks, as by page 66 and Figs. 36 and 38, when the wheels are to be hobbed ; of course the cutter should be thin enough to leave stock for finishing. The spaces can be cut the full depth, the cutter being dropped in. When worm-wheels are not hobbed it is better to turn blanks like a spur-wheel. Little is gained by g ^J a n J £ h Li J e a having wheels curved to fit worm unless teeth are fin- ished with a hob. The teeth can be cut in a straight path diagonally across face of blank, to fit angle of worm-thread, as in Figs. 41 and 44. For dividing wheels to gear-cutting engines the Wheels for blanks are turned like a spur-wheel and a cutter about Machines. T *g-" larger diameter than the worm, is dropped in, as in Figs. 42 and 45. : : - 5: J _ v Fn 40 PROVIDENCE, K. I. 73 Some mechanics prefer to make dividing wheels in two parts, joined in a plane perpendicular to axis, hob teeth , then turn one part round upon the other, match teeth and fasten parts together in the new position, and hob again with a view to eliminate errors. With an accurate cutting engine we have found wheels like Figs. 42 and 45, not nobbed, every way satisfactory. As to the different wheels, Figs. 43, 44 and 45, when Figures 43, 44 ° , . and 45. worm is in right position at the start, the life-time of Fig. 43, under heavy and continuous work, will be the longest. Fig. 44 can be run in mesh with a gear or a rack as well as with a worm when made within the angular limits commonly required. Strictly, neither two gears made in this way, nor a gear and a rack would be mathematically exact, as they might bear at the sides of the gear or at the ends of the teeth only and not in the middle. At the start the contact of teeth in this wheel upon worm-thread is in points only; yet such wheels have been many years successfully used in ele- vators. Fig. 45 is a neat-looking wheel. In gear cutting engines where the workman has occasion to turn the work spindle by hand, it is not so rough to take hold of as Figs. 43 and 44. The teeth are less liable to in- jury than the teeth of Figs. 43 and 44. BROWN & SHAUPE MFG. CO. Fig. 41. "Worm-wheel with teeth cut in a straight path diagonally across face. Worm is double-threaded. PROVIDENCE, E. I. 7a Fig. 42. Worm and Worm- Wheel, for Gear-cutting Engine. 7G BROWN & SHAIirE MFG. CO. UaaMjamik--- Fig. 43. Fig. 44. Fig. 45. PROVIDENCE, R. I. 77 Some designers prefer to take off the outermost part of teeth in wheels (Figs. 35 and 43), as shown in these two figures, and not leave them sharp, as in Figs. 36 and 80. We do not know that this serves any purpose except a matter of looks. In ordering worms and worm wheels the centre dis- tances should be given. If there can be any limit allowed in the centre distance it should be so stated. For instance, the distance from the centre of a worm to the centre of a worm wheel might be calculated at 6" but 5 31-32" or 6 1-32" might answer. By stating all the limits that can be allowed , there may be a saving in the cost of work because time need not be wasted in trying to make work within narrower limits than are necessary. HOBS WITH RELIEVED TEETH. We are prepared to make hobs of any size with the teeth relieved the same as our gear cutters. The teeth can be ground on their faces without changing their form. The hobs are made with a precision screw so that the pitch of the thread is accurate before hard- ening. BIIOTVX & SHARFE MFG. CO. GASHING TEETH OF HOB, lO Inches Ou.tside Diameter. PROVIDENCE, E. I, CHAPTER XII. SIZING GEARS WHEN THE DISTANCE BETWEEN CENTRES AND THE RATIOS OF SPEEDS ARE FIXED— GENERAL REMARKS— WIDTH OF FACE OF SPOR GEARS— SPEED OF GEAR CUTTERS— TABLE OF TOOTH PARTS. Let us suppose that we have two shafts 14" apart, center to center, and wish to connect them by gears so, center dis- ' J ° tance and Ratio that they will have speed ratio 6 to 1. We add the 6 fixed - and 1 together, and divide 14" by the sum and get 2" for a quotient; this 2", multiplied by 6, gives us the radius of pitch circle of large wheel === 12". In the same manner we get 2" as radius of pitch circle of small wheel. Doubling the radius of each gear, we obtain 24" and 4" as the pitch diameters of the two wheels. The two num- bers that form a ratio are called the tsrms of the ratio. We have now the rule for obtaining pitch-circle diame- ter of two wheels of a given ratio to connect shafts a given distance apart: Divide the center distance bii the sum of the terms of Rule f °r r>u ° ^ J ameter of Pitch the ratio; find the product of tin ice the quotient by each circles. term sparately, and the two products ivill he the pitch diameters of the two ivheels. It is well to give special attention to learning the rules for sizing blanks and teeth ; these are much oftener needed than the method of forming tooth out- lines. A blank 1-J-" diameter is to have 16 teeth: what will the pitch be? What will be the diameter of the pitch circle ? See Chapter V. A good practice will be to compute a table of tooth parts. The work can be compared with the tables pages 86-89. Sj BBovrs- jc szat.iz mtg. In compute:, i: is well to take re to more than four rlaces. ,t to nine places = 3.141592653. •= to nine places = .318309886. There is no such thing as pure rolling contact in teeth of wheels: they always rub, and. in time, will wear themselves out of snape and niay become noisy. Bevel gears, vriier. correctly ft-raieiL ran smoother than spur gears of same diameter and pitch, because the teeth continue in contact longer than the teeth of spur gears. F r this reas : n annular gears run smoother than either bevel or spur gears. Sometimes gears have to be cut a little deeper than designed, in order to run easily on their shafts. If any departure is made in ratio of pitch diameters it :- better to have the driving gear the larger, that is, cut the follower smaller. For wheels coarser than eight diametral pitch (8 P'». it is generally better to cottwi x around, when accurate wxk is wanted, also for lar _ -e wheels, as the expansion of parts from heat often causes inaccurate work when cut but once around. There is not so much trouble from heat in plain or we _ e is as in arm gears. w-. .-_:-; „r Tiie -~: "t"_ c : cast-ii:n _ear :a:es ::•: r/ererai rtir- poses can be made to the following rule : It v:\J.-i S :y ' -to the quo* it : sum toiU be width of face for the pitch Example : "VThat width of face for gear 4 P Divid- ing 8 by -i and adding J" we obtain 2^", for width of face. For change gears on lathes, where it is desira- ble not to have face very wide, the following rule can be used: Divide -± ' -ch and add J Bv the latter rule a 4 P change gear would have but H race Speedof Gear The speed of gear cutters is subject t : s :> many con- ditions that dehnite rules cannot be given. TVe append table of average speeds. A coarse pitch cutter for pinion. 12 teeth, would usually be fed slower than a cutter for a large gear of same pitch. PROVIDENCE, E. I. 81 TABLE OF AVERAGE SPEEDS FOR GEAR-CUTTERS. A bfi tI) S ». u • c ; O «w ^.S ° . C3^ A S3 « ^"p o S| O 5S ftp g w ~ m ~ P.3 ^ o at § m d d Kg t- d *■" ■— c O.SM S3 &S3^£ Diametral of Cut u 5 aa °H P S3 -u JJ3 3 Er = ° ^ 3cm +» 7J CO r - 2<^ 9^ ^s5 r- "S — OQ 2 5 in. 24 18 .025 in .011 in. .60 in. . 20 in. 2* ±i " 30 24 .028 " .013 " .84 " .31 " 3 16 36 28 .031 " .015 " 1.12 " .42 " 4 3f " 42 32 .034 " .017 " 1.43 « .54 " 5 3 T V " 50 40 .037 " .019 " 1.85 " .76 " 6 2U- " 16 75 55 .030 " .016 " 2.25 " .88 " 7 2-9- '« "" 1 6 85 65 .032 " .018 " 2'. 72 « 1.17 " 8 9± " ~2 95 75 .034 " 020 " 3.23 " 1.50 " 10 01 « Z S 125 90 .026 " .014 " 3.25 " 1.26 " 12 2 " 135 100 .027 " .017 " 3.64 " 1.70 " 20 H " 145 115 .029 " .021 " 4,20 " 2.41 " 32 H « 160 135 .031 '• .025 " 4.96 " 3.37 " In brass the speed of gear-cutters can be twice as B ^ g e e d s ? fast as in cast iron. Clock-makers and those making a specialty of brass gears exceed this rate even. A 12 P cutter has been run 1,200 (twelve hundred) turns a minute in bronze. A 32 P cutter has been run 7,000 (seven thousand) turns a minute in soft brass. In cutting 5 P cast-iron gears, 75 teeth, a No. 1, BP^^g 9 cutter was run 136 (one hundred and thirty-six) turns a minute, roughing the spaces out the full 5 P depth ; the teeth were then finished with a 5 P cutter, running 208 (two hundred and eight) turns a minute, feeding by hand. The cutter stood well, but, of course, the cast iron was quite soft. A 4 P cutter has finished teeth at one cut, in cast-iron gears, 86 teeth, running 48 (forty-eight) turns a minute and feeding T y at one turn, or 3 in. in a minute. Hence, while it is generally safe to run cutters as in the table, yet when many gears are to be cut it is well to see if cutters will stand a higher speed and more feed. In gears coarser than 3 P it is more economical to cut first the full depth with a stocking cutter and then finish with a gear cutter. This stocking cutter is made 82 BROWN & SHAEPE MEG. CO. on the jDrinciple of a circular splitting saw for wood. The teeth, however, are not set ; but side relief is ob- tained by niakino- sides of cutter blank hollowing. The shape of stocking cutter can be same as bottom of spaces in a 1 --tooth gear, and the thickness of cutter can be J- of the circular pitch, see page 40. Keep cutters The matter of keeping cutters sharp is so important that it has sometimes been found best to have the work- man grind them at stated times, and not wait until he can see that the cutters are dull. Thus, have him m.ind everv two hours or after cutting a stated number of gears. Cutters of the style that can be ground upon their tooth faces without changing form are rap- idly destroyed if allowed to run after they are dull. Cutters are oftener wasted bv trying to cut with them Avhen they are dull than by too much grinding. Grind Ihe faces radial with a free cutting wheel. Do not let the wheel become glazed, as this will draw the temper of the cutter. In Chapter YI. was given a series of cutters for cut- ting gears having 12 teeth and more. Thus, it was there implied that any gear cf same pitch, having 135 teeth, 13G teeth, and so on up 1 :> the largest gears, and, also, a rack, could be cut with one cutter. If this cut- ter is 4 P, we would cut with it all 4 P gears, having 135 teeth or more, and we would also cut with it a 4P rack. Now. instead of alwavs ref erring to a cutter bv the number of teeth in gears it is designed to cut, it has been found convenient to designate it by a letter or by a number. Thus, we call a cutter of 4 P, made to cut gears 135 teeth to a rack, inclusive, Nj. 1, 4 P. We have adopted numbers for designating involute involute Gear orear-cutters a-s in the following table: Cutters. No. 1 will cut wheels from 135 teeth to a rack inclusive. 55 CI 134 teeth 35 " 54 " 26 u 34 - : 21 « 25 ki 17 .. 20 - 14 it 16 " 12 « 13 " PROVIDENCE, R. I. 83 By this plan it takes eight cutters to cut all gears having twelve teeth and over, of any one pitch. Thus it takes ei»ht cutters to cut all involute 4 P O gears having twelve teeth and more. It takes eight other cutters to cut all involute gears of 5 P, having 12 teeth and more. A No. 8, 5 P cutter cuts only 5 P gears having 12 and 13 teeth. A No. 6, 10 P cutter cuts only 10 P gears having 17, 18, 19 and 20 teeth. On each cutter is stamped the number of teeth at the limits of its range, as well as the number of the cutter. The number of the cutter relates only to the number of teeth in gears that the cutter is made for. In ordering cutters for involute spur-gears two things must be given : 1. Either the number of teeth to be cut in the qear How to order J J Involute Cut- or the number of the cutter, as given in the foregoing ters. table. 2. Either the pitch of the gear or the diameter and number of teeth to be cut in the gear. If 25 teeth are to be cut in a 6 P involute gear, the cutter will be No. 5, 6 P, which cuts all 6 P gears from 21 to 25 teeth inclusive. If it is desired to cut gears from 15 to 25 teeth, three cutters will be needed, No. 5, No. 6 and No. 7 of the pitch required. If the pitch is 8 and gears 15 to 25 teeth are to be cut, the cutters should be No. 5, 8 P, No. 6, 8 P, and No. 7, 8 P. For each pitch of epicycloidal, or double-curve gears, Epicycioidai 24 cutters are made. In coarse-pitch gears, the varia- cur ^ e cutters, tion in the shape of spaces between gears of consecu- tive-numbered teeth is greater than in fine-pitch gears. A set of cutters for each pitch, to consist of so large a number as 24, has been established because double curve teeth have generally been preferred in coarse-pitch gears, though the tendency of late years is toward the involute form. Our double curve cutters have a guide shoulder on each side for the depth to cut. When this shoulder just reaches the periphery of the blank the depth is right. The marks which these shoulders make on the blank, should be as nar- row as can be seen, when the blanks are sized right. 81 zz. : _ y h - terete z tee.-. :o. Double-curve gear-cutters are designate'.! bv letters instead of by numbers : this is to avoid confusion in :: ;~e:::i_ F allowing is the list of epicycloidal or double-c urve gear-cutters : — ^-^vf 1 ^ Cutter A :-tits 1*2 teeth. Cutter 31 :u: = 27 t: '2? teeth IknUe-curre .. -n .. -i •-> t; .. v" . ■">-,..- .. C .. U .. .. Q .. 0± .. ;- .. •• D ■■ 15 u • P •• 3S •• 42 • " E •• 16 •• • Q •■ 43 - 49 - u y ■• 17 " B " 50 < ; 59 " .. q - is •■ S " 60 "74 '■ « H - 19 " - T " 75 - 99 •• I 20 " u T • 100 ' : 149 " • J - 21 t: -_ V 150 " 24:9 " - K - 23 t: 24 ■■ TV •• 250 - Rack. L • 24 to 26 u X " Rack. A cutter that cuts more than one gear is made of proper form for the smallest gear in its range. Thus, :tttte: J ::t 21 t: 22 teeth is crrrect tor 21 teeth: cutter S for 60 to 74 teeth is correct for 60 teeth, and so on. E r . : ~_ :;-. :;y In ordering epicycloidal gear-cutters designate the Outers. letter of the cutter as in the foregoing table, also either give the pitch or give data that will enable us to determine the pitch, the same as directed for invo- lute cutters. AI ;.re care is required in making and adjusting epi- cycloidal gears than in making involute gears. ^ ~-~ -" In : lering bevel gear cutters three things must s Ee~el (.Tear ^ c o ./-"t:- 1 . h er of teeth in each gear. 2 Either the pitch of gears or the largest pitch dio gear; set Fig- -7 3. The length of tooth face. If the shafts are not to run at right angles, it should be so stated, and the angle given. Involute cutters onlv are used for cutting bevel gears. Nc tempt should be made to cut epicyclodial tooth bevel gears with rotarv dite — For any pitch not in the folio wing tables : find corresponding part : — multiply the tabular value for one inch by the circular pitch required, and the ;: will be the value for the pitch given. Exam- ple : What is the value of s for 4 inch circular pitch? .3183 = $ for 1 P and .3183 ■ 4= 1.2732=8 for 4" P. The expression "Addendum and — " (addendum and the modules mean- the distance of a tooth outside of pitch line and also the distance occupied for every tooth upon the diameter of pitch circle. ■■■ : = '•' ---i-z :i z - :- si' ~: :~ PART II. CHAPTER I. TANGENT OF ARC AND ANGLE. In Pakt II. we shall show how to calculate some e |p2fned t0 be of the functions of a right-angle triangle from a table of circular functions, the application of these calcula- tions in some chapters of Part I. and in sizing blanks and cutting teeth of spiral gears, the selection of cutters for spiral gears, the application of continued fractions to some problems in gear wheels and cutting odd screw-threads, etc., etc. A. Function is a quantity that depends upon another quantity for its value. Thus the amount a workman earns is a function of the time he has worked and of fi ^ nctl0u de ~ his wages per hour. In any right-angle triangle, O A B, we shall, for Right - angle t i o Triangle. convenience, call the two lines that form the right angle O A B the sides, instead of base and perpen- dicular. Thus O A B, being the right angle we call the line O A a side, and the line A B a side also. When we speak of the angle A O B, we call the line O A the side adjacent. When we are speaking of the Skle ad J acent - angle ABO we call the liue A B the side adjacent. The line opposite the right angle is the hypothenuse. Hypothenuse. In the following pages the definitions of circular functions are for angles smaller than 90°, and not strictly applicable to the reasoning employed in ana- lytical trigonometry, where we find expressions for angles of 270°, 760°, etc. 88 Tangent. BROWN & SHAEPE MFG. CO. The Tangent of an arc is the line that touches it at one extremity and is terminated by a line drawn from the center through the other extremity. The tangent is always outside the arc and is also perpendicular to the radius which meets it at the point of tangency. *iff ^7 Thus, in Fig. 46, the line A B is the tangent of the arc A C. The point of tangency is at A. An angle at the center of a circle is measured by the arc intercepted by the sides of the angle. Hence the tangent A B of the arc A C is also the tangent of the angle A O B. In the tables of circular functions the radius of the arc is unity, or. in common practice, we take it as one inch. The radius O A being 1", if we know the length of the line or tangent A B we can, by looking in a table of tangents, find the number of degrees in the angle A O B. To find the Thus, if A B is 2.25" long, we find the angle A O B Degrees in an . ° ° Angle. is 66° very nearly. That is, having found that 2.2460 is the nearest number to 2.25 in the table of tangents at the end of this volume, we find the corresponding degrees of the angle in the column at the left hand of the table and the minutes to be added at the top of the column containing the 2.2460. The table gives angles for every 10', which is suf- ficient for most purposes. PROVIDENCE, R. I- 89 Now, if we have a right-angle triangle with an angle the same as O A B, but with O A two inches long, the line A B will also be twice as long as the tangent of angle A O B, as found in a table of tangents. Let us take a triangle with the side OA=5" long, fln J xa ™£% £ and the side AB = 8" long ; what is the number of *™ e e f in au degrees in the angle A O B ? Dividing 8" by 5 we find what would be the length of A B if O A was only 1" long. The quotient then would be the length of tangent when the radius is 1" long, as in the table of tangents. 8 divided by 5 is 1.6. The nearest tangent in the table is 1.6003 and . the corresponding angle is 58°, which would be the angle A O B when A B is 8" and the radius O A is 5" very nearly. The difference in the angles for tangents 1.6003 and 1.6 could hardly be seen in practice. The side opposite the required acute angle corresponds to the tangent and the side adjacent corresponds to the radius. Hence the rule : To find the tangent of either acute angle in a right- T JJ e ^J d the angle triangle : Divide the side opposite the angle by the side adjacent the angle and the quotient will be the tangent of the angle. This rule should be com- mitted to memory. Having found the tangent of the angle, the angle can be taken from the table of tan- gents. The complement of an angle is the remainder after complement subtracting the angle from 90°. Thus 40° is the com- plement of 50°. The Cotangent of an angle is the tangent of the cotangent, complement of the angle. Thus, in Fig. 47, the line A B is the cotangent of A O E. In right-angle tri- angles either acute angle is the complement of the other acute angle. Hence, if we know one acute angle, by subtracting this angle from 90° we get the other acute angle. As the arc approaches 90° the tangent becomes longer, and at 90° it is infinitely long. The sign of infinity is oo. Tangent 90° = oo, AMi^by 1 ^? ~^7 a ta ^ e °^ tangents, angles can be laid out npon Tangent Ex- 5neet zinc. ere. T_:s is often an advantage, as i: is not =■ - e - - ' - convenient to lay protractor flat down so as to mark angles up to a sharp point. If we could lay off the length of a line ~ e could take tangents dire :■: from table and obtain angle at moe. It. however, is generally bet:-: : multiply the tangent by 5 or 10 and make an riilarged triangle. If, then, there is a slight error in laying off length of lines it will not make so much difference with the angle. Let it be required to lay off an angle of 14 r 30'. By the table we find the tangent to 1: e _ : B 1 Multiply- ing .25861 hj 5 we obtain, in the enlarged triangle. 1.29305 as the length of side opposite the angle l-±~ 30 . As we have made the side opposite five times as large, we must make the sid^ •int five times large, in order to keep angle the same. Hence. F:_ 48, draw* the line A B 5 long;; perpendicular to this line at A Iraw ._e:ieA J 1.203 long; now draw the line O B, and the angle A B O will be 14° 30'. If special accuracy is required, the tangent can be multiplied by 10 ; the line AO will then be 2.5S6' long and the line A B 10 long. Remembering that the acute angles of a right-angle triangle are the comple- ments of each other, we subtract 11 3U' from 90' and btain 75" 30 as the angle of A O B. The reader will remember these angles as occurring in Paj.i I.. Chapter IT., and obtained in a different way. A Srimcirele upon the line O B touching the extremitic ill just touch the right angle at A. and the line 3:-: :ur times as long as O A L: it be required to turn a piece 4 long. 1 diam- eter at small end. with a taper of 10 r one side with the fcher ; what will be the diameter of the piece at th^ large end ? A section. Fig. 49, through the axis of this piece is To .^lcuiate l ] le saine as if we added two riorht-anofle triangles. Diameter of a -_ Tapering^ B and A' B . to a straight piecr A A B B . 1 piece. Fig. 50. ,, =» * wide and 4" long,the a angles 1 an.i B bein_r 5 . thus making th: sides ' B and OBI with -ach _ ther. PROVIDENCE, R. I. 91 -1^293-+ Fig. 4S. Fig. 49 92 BROWN & SHARPE MFG. CO. The tangent of 5° is .08748, which, multiplied by 4 , gives . 34992 " as the length of each line, A O and A' O', to be added to 1" at the large end. Taking twice .34992" and adding to 1" we obtain 1.69984" as the diameter of large end. This chapter must " be thoroughly studied before taking up the next chapters. If once the memory becomes confused as to the tangent and sine of an angle, it will take much longer to get righted than it will to first carefully learn to recognize the tangent of an angle at once. If one knows what the tangent is, one can tell better the functions that are not tangents. 03 CHAPTER II. SINE— COSINE AND SECANT : SOME OF THEIR APPLICATIONS IN MACHINE CONSTRUCTION. Sine of Arc and Angle. The Sine of an arc is the line drawn from one extremity of the arc to the diameter passing through the other extremity, the line being perpendicular to the diameter. Another definition is : The sine of an arc is the dis- tance of one extremity of the arc from the diameter, through the other extremity. The sine of an angle is the sine of the arc that measures the angle. In Fig. 50 , A C is the sine of the arc B C, and of the angle BOC. It will be seen that the sine is always inside of the arc, and can never be longer than the radius. As the arc ap- proaches 90°, the sine comes nearer to the radius, and at 90° the sine is equal to 1, or is the radius itself. From the defini- tion of a sine, the side A C, opposite the angle A O C, in any right-angle triangle, is the sine of the angle A O C, when O C is the radius of the arc. Hence the rule : In any right-angle triangle, the side to find the opposite either acute angle, divided by the hypothe- nuse, is equal to the sine of the angle. The quotient thus obtained is the length of side opposite the angle when the hypothenuse or radius is unity. The rule should be carefully committed to memory. Fig. 50. Sine. 94 BROWN & SHARPE MFG. CO. Chord of an ^ Chord is a straight line joining the extremities of an arc, and is twice as long as the sine of half the angle measured by the arc. Thus, in Fig. 51, the chord B C is twice as long as the sine A C. Fig. r>l Let there be four holes equidistant about a circle 3" in diameter — Fig. 51 ; what is the shortest distance between two holes ? This shortest distance is the nnd X th™chord° chord A B, which is twice the sine of the angle COB. The angle A O B is one quarter of the circle, and C O B is one-eighth of the circle. 360°, divided by 8=45°, the angle COB. The sine of 45° is .70710, which multiplied by the radius 1.5", gives length C B in tho circle, 3" in diameter, as 1.0G0G5". Twice this length is the required distance A B=2.1213". "When a cylindrical piece is to be cut into any num- ber of sides, the foregoing operation can be applied to obtain the w T idth of one side. A plane figure bounded Polygon. by straight lines is called a polygon. PROVIDENCE, E. I. 95 When the outside diameter and the number of sides of a regular polygon are given, to find the length of one of the sides: Divide 3G0° by twice the number of To fi ^, tlie J J length of Side. sides ; midtiply the sine of the quotient by the outer diameter, and the product will be the length of one of the sides. Multiplying by the diameter is the same as multi- plying by the radius, and that product again by 2. The Cosine of an angle is the sine of the comple- Cosine - ment of the angle. In Fig. 50, C O D is the complement of the angle A O C ; the line C E is the sine of COD, and hence is the cosine of B O C. The line O A. is equal to C E. It is quite as well to remember the cosine as the part of the radius, from the center that is cut off by the sine. Thus the sine A C of the angle A O C cuts off the cosine O A. The line O A may be called the cosine because it is equal to the cosine C E. In any right-angle triangle, the side adjacent either acute angle corresponds to the cosine when the hypothenuse is the radius of the arc that measures the angle ; hence: Divide the side adjacent the acute To find the angle by the hypothenuse, and the quotient vnll be the cosine of the angle. When a cylindrical piece is cut into a polygon of any number of sides, a table of cosines can be used tOg.^iverth of obtain the diameter across the sides. g° n - 96 BROWN «s SHABPE MFG. CO. Let a cylinder, 2 " diameter. Fig. 53. be cut six-sided: what is the diameter across the sides ? The angle A O B. at the center, occupied by one of these sides, is one-sixth of the circle, =60 c . The cosine of one-half this angle, 30". is the line C O; twice this line is the diameter across the sides. The cosine of 30 : is .86602. which, multiplied by 2, gives 1.7320-4" as the diameter across the sides. Of course, if the radius is other than unity, the cosine should be multiplied by the radius, and the product again by 2. in order to get diameter across the sides ; or what is the same thing, multiply the cosine by the whole diameter or the diameter across the corners, ameter f aoroi ^e ru - e f or obtaining the diameter across sides of sides of a Po1 ^- regular polvgon. when the diameter across corners i> given, will then be: Multiply the cosine of 360 D divided l>y twice the number of sides, ly the diameter across corners, and the product will be the diameter across sides. Look at the right-hand column for degrees of the cosine, and at bottom of page for minutes to add to the degrees. The Secant of an arc is a straight line drawn from the center through one end of an arc, and terminated bv a tangent drawn from the other end of the arc. Thus, in Fig. 53, the line OB is the secant of the angle. COB. A C B Fig. J3. To cud the In a n v light -angle triangle, divide the hypothenuse Secant. c by the side adj icent either acute angle, and the quo- tient will le the secant of that angle. PROVIDENCE, R. I. 97 That is, if we divide the distance OB by O C, in the right-angle triangle COB, the quotient will be the secant of the angle COB. The secant cannot be less than the radius ; it in- creases as the angle increases, and at 90° the secant is infinity =co . A six-sided piece is to be H" across the sides ; towp^^^J large must a blank be turned before cutting the sides ? JF™ R)iygon! rS Dividing 360° by twice the number of sides, we have 30°, which is the angle COB. The secant of 30° is 1.1547. The radius of the six-sided piece is .75". Multiplying the secant 1.1547 by .75", we obtain the length of radius of the blank O B ; multiplying again by 2, we obtain the diameter 1.732" + . Hence, in a regular polygon, when the diameter across sides and the number of sides are given, to find diameter across corners : Multiply the secant of 360° divided by tvnce the number of sides, by the diameter across sides, and the product will be the diameter across corners. It ^Till be seen that the side taken as a divisor has been in each case the side corresponding to the radius of the arc that subtends the angle. The versed sine of an acute angle is the part of radius outside the sine, or it is the radius minus the cosine. Thus, in Fig. 50, the versed sine of the arc BC is AB. The versed sine is not given in the tables of circular functions : when it is wanted for any angle less than 90° we subtract the cosine of that angle from the radius 1. Having it for the radius 1, we can multiply by the radius of any other arc of which we may wish to know the versed sine. Fig. 54 is a sketch of a gear tooth of IP. In measuring gear teeth of coarse pitch it is sometimes a convenience to know the chordal thickness of the tooth, as at ATB, because it may be enough shorter than the regular tooth-thickness AHB, or t, to require attention. It may be also well to know the versed sine of the angle 1J, or the distance II, in order to tell where to measure the chordal thickness. 98 BliOWN & SHAKPK MI"G CO. PROVIDENCE, R. I. 99 On pages 104 and 105 are tables of data pertaining to chordal thickness of IP. teeth. For any other diametral pitch, divide the number in the tabic by that pitch. GEAR TOOTH CALIPER. 100 BROWN & SHAEPE MFG. CO. CHORDAL THICKNESS OF TEETH FOR GEARS AND CUTTERS, ON A BASTS OF 1 DIAMETRAL PITCH. T = D' sin. ff H = R (1— cos. £'). N = Number of teeth in gears. T = Chorclal thickness of Tooth. H = Height of Arc. D' = Pitch Diameter. R = Pitch Radius. yS f = 90° divided by the number of teeth. Note. — When the tooth of a gear is measured, add the height of arc to (S); and when gear cutter is measured subtract the height of arc from (S + f). Involute. Cutter. T H .0047 Corrected S+f forCutt. Corrected S for Gear. No.l — 135 T- -1P 1.5707 1.1524 1.0047 « 2 — 55 T- -IP 1.5706 .0112 1.1459 1.0112 u g__ 35T- -IP 1.5702 .0176 1.1395 1.0176 « 4— 26T- -IP 1.5698 .0237 1.1334 1.0237 «5— 21 T- -IP 1.5694 .0294 1.1277 1.0294 « 6— 17 T- -IP 1.5686 .0362 1.1209 1.0362 "'7— 14 T- -IP 1.5675 .0440 1.1131 1.0440 «8— 12 T- -IP 1.5663 .0514 1.1057 1.0514 11 T- -IP 1.5654 .0559 1.1011 1.0559 10T- -IP 1.5643 .0616 1.0955 1.0616 9T- -IP 1.5628 , .0684 1.0887 1.0684 8T- -IP 1.5607 .0769 1.0802 1.0769 providence, r. i. Epicycloidal. 101 Cutter. T H Corrected S + f f or Cutt. Corrected S for Gear. A — 12T- -1P 1.5663 .0514 1.1057 1.0514 B — 13 T- -IP 1.5670 .0474 1.1097 1.0474 C - — 14 T- -IP 1.5675 .0440 1.1131 1.0440 D — 15T- -IP 1.5679 .0411 1.1160 1.0411 E — 16 T- -IP 1.5683 .0385 1.1186 1.0385 F — 17 T- -IP 1.5686 .0362 1.1209 1.0362 G — 18 T- -IP 1.5688 .0342 1.1229 1.0342 H- — 19T- -IP 1.5690 .0324 1.1247 1.0324 I — 20 T- -IP 1.5692 .0308 1.1263 1.0308 J - — 21 T- -IP 1.5694 .0294 1.1277 1.0294 K — 23T- -IP 1.5696 .0268 1.1303 1.0268 L — 25 T- -IP 1.5698 .0247 1.1324 1.0247 M — 27T- -IP 1.5699 .0228 1.1343 1.0228 N- — 30 T- -IP 1.5701 .0208 1.1363 1.0208 — 34 T- -IP 1.5703 .0181 1.1390 1.0181 P - 38 T- -IP 1.5703 .0162 1.1409 1.0162 Q — 43 T- -IP 1.5705 .0143 1.1428 1.0143 R - — 50 T- -IP 1.5705 .0123 1.1448 1.0123 S - — 60T- -IP 1.5706 .0102 1.1469 1.0102 T - 75T- -IP 1.5707 .0083 1.1488 1.0083 U- — 100T- -IP 1.5707 .0060 1.1511 1.0060 V- — 150T- -IP 1.5707 .0045 1.1526 1.0045 w — 250T- -IP 1.5708 .0025 1.1546 1.0025 Special. No. Teeth. 9T — IP 10 T — IP 11T — IP 1.5628 1.5643 1.5654 H .0684 .0616 .0559 Corrected S + f for Cutt. 1.0887 1.0955 1.1012 Corrected S for Gear. 1.0684 1.0616 1.0559 103 CHAPTER III. APPLICATION OF CIRCULAR FUNCTIONS— WHOLE DIAMETER OF BEVEL GEAR BLANKS— ANGLES OF BEVEL GEAR BLANKS. The rules given in this chapter apply onl} T to bevel gears having the center angle c O i not greater than 90°. To avoid confusion we will illustrate one gear only. The same rules apply to all sizes of bevel gears. Fig. 55 is the outline of a pinion 4 P, 20 teeth, to mesh with a gear 28 teeth, shafts at right angles. For making sketch, of bevel gears see Chapter IX.. Part I. In Fig. 55, the line O m' m is continued to the line a b. The angle c O I that the cone pitch-line makes with the center line may be called the center angle. Angle of The center angle c O i is equal to the angle of edge ' c i c. c i is the side opposite the center angle c' O i, and c' O is the side adjacent the center angle, c i = 2.5"; & O : = 3.5". Dividing 2.5" by 3.5" we obtain .71128" + as the tangent of c' O i. In the table we find .71329 to be the nearest tangent, the corre- sponding angle being 35° 30'. 35 |°, then, is the center angle c O i and the angle of edge c i n, very nearly. When the axes of bevel gears are at right angles the angle of edge of one gear is the complement of angle of edge of the other gear-. Subtracting, then, 35^° from 90° we obtain 54^° as the angle of edge of gear 28 teeth, to mesh with gear 20 teeth, Fig. 55. from which we have the rule for obtaining centre angles when the axes of gears are at right angles. Divide the radius of the pinion by the radius of the gear and the quotient will be the tangent of centre angle of the pinion. Xow subtract this centre ande from 90 deer, and we have the centre angle of the gear. The same result is obtained by dividing the number of teeth in the pinion by the number of teeth in the gear ; the quotient is the tangent of the centre angle. PROVIDENCE, R. I. 103 Fig. 55. BEVEL GEAR DIAGRAM, 104 BROWN & SHARPE MFG. CO. Angle of Pace. To obtain angle of face O m" c\ the distance c O becomes the side opposite and the distance m" c is the side adjacent. The distance c O is 3.5", the radius of the 28 tooth bevel gear. The distance c m" is by measurement 2.82". Dividing 3.5 by 2.82 we obtain 1.2411 for tangent of angle of face O m" c. The nearest tangent in the table is 1.2422 and the corresponding angle is 51° 10'. To obtain cutting angle c O n" we divide the distance c n" by c O. By measurement c' n" is 2.2". Divid- ing 2.2 by 3.5 we obtain .62857 for tangent of cutting angle. The nearest corresponding angle in the table is 32°10'. The largest pitch diameter, kj, of a bevel gear, as in Fig. 56, is known the same as the pitch diameter of any spur gear. Now, if we know the distance b o or its equal a q, we can obtain the whole diameter of bevel gear blank by adding twice the distance b o to the largest pitch diameter. cr?ment ter Fig" Twice the distance b o, or what is the same thing, 5b - the sum of a q and b o is called the diameter incre- ment, because it is the amount by Avhich we increase the largest pitch diameter to obtain the whole or out- side diameter of bevel gear blanks. The distance b o can be calculated without measuring the diagram. The angle b o j is equal to the angle of edge. The angle of edge, it will be remembered, is the angle formed by outer edge of blank or ends of teeth with the end of hub or a plane perpendicular to the axis of gear. The distance b o is equal to the cosine of angle of edge, multiplied by the distance j o. The distance j o is the addendum, as in previous chapters ( = s). Hence the rule for obtaining the diameter increment of any bevel gear: Multiply the cosine of angle of edge by the working depth of teeth (D"), and the product will be the diameter increment. By the method given on page 102 we find the angle of edge of gear (Fig. 56) is 56° 20'. The cosine of 56° 20° is .55436, which, multiplied by §", or the e? e U r Side Diam " depth of the 3 P gear, gives the diameter increment of the bevel gear 18 teeth, 3 P meshing with pinion of 12 PROVIDENCE, R. I. 105 10G BROWN 6 SHABPE MFG. CO. teeth, j of .55436 = . 369."— I or .37". nearly). Adding the diameter increment. .37 . to the largest pitch diameter of gear, 6". we have 6.37" as the outside diameter. In the same manner, the distance c J is half the diameter increment of the pinion. The angle c J J: is equal to the center angle of pinion, and when axes are at right angles is the complement of center angle of gear. The center angle of pinion is 33° 40'. The cosine, multiplied by the working depth, gives .555" for diameter increment of pinion, and we have 4-555" for outside diameter of pinion. In turning bevel gear blanks, it is sufficiently accu- rate to make the diameter to the nearest hundredth of an inch. Angle in-re The small angle o / is called the angle increment. merit. ° f u When shafts are at right angles the face angle of one gear i 3 equal to the center angle of the other gear, minus the angle increment. Thus the angle of face of gear 'Fig. 56) is less than the center angle D k\ or its equal 0./ k by the angle o Oj. That is. subtracting o Oj from Oj A\ the re- mainder will be the angle of face of gear. Subtracting the angle increment from the center angle of gear, the remainder will be the cutting angle. The angle increment can be obtained by dividing oj. the side opposite, by Oj. the side adjacent, thus finding the tangent as usual. The length of cone-pitch line from the common center. t) j. can be found, without measuring dia- gram, by multiplying the secant of angle 0,/ A - , or the center angle of }:>inion, by the radius of largest pitch diameter of gear. The secant of angle Oj k, 33"" 40', is 1.2015, which. multiplied by 3". the radius of gear, gives 3.6045" as the length of line ,/. Dividing oj by Oj, we have for tangent .092-4. and for angle increment 5" 20'. The angle increment can also be obtained by the following rule : PROVIDENCE, E. I. 107 Divide the sine of center angle by half the nun* ber of teeth, and the quotient will be the tangent of increment angle. Subtracting the angle increment from, center angles of gear and pinion, we have respectively : Cutting angle of gear, 51°. Cutting angle of pinion, 28° 20'. .Remembering that when the shafts are at right angles, the face angle of a gear is equal to the cutting angle of its mate (Chapter X. part 1), we have: Face angle of gear, 28° 20'. Face angle of pinion, 51°. It will be seen that both the whole diameter and the angles of bevel gears can be obtained without making a diagram. Mr. George B. Grant has made a table of different pairs of gears from 1 to 1 up to 10 to 1, con- taining diameter increments, angle increments and centre angles, which is published in his " Treatise on Gears." " Formulas in Gearing," published by us, also contains extensive tables for bevel gearing. We have adopted the terms "diameter increment," ''angle incre- ment," and "centre angle" from him. lie uses the term (i back angle" for what we have called angle of edge, only he measures the angle from the axis of the gear, instead of from the side of the gear, or from the t Tol , a y° u *?: n ° ' & ' Angle oy the end of hub, as We have done ; that is, his "back angle "Sine. is the complement of our angle of edge. In laying out angles, the following method may be -t'i'j. 52. 108 BROWN & SHARPE MFG. CO. preferred, as it does awny with the necessity of making aright angle: Draw a circle, ABO (Pig. o't), ten inches in diameter. Set the dividers to ten times the sine of the required angle, and point off this distance in the circumference as at A B. From any point O in the circumference, draw the lines O A and O B. The angle AOB13 the angle required. Thus, let the re- quired angle be 12°. The sine of 12° is .20791, which, multiplied by 10, gives 2.0791", or Zj$q-" nearly, for the distance A B. Any diameter of circle can be taken if we multiply the sine by the diameter, but 10" is very convenient, as- all we have to do with the sine is to move the decimal point one place to the right. If either of the lines pass through the centre, then the two lines which clo not pass through the centre will form a right angle. Thus, if O B passes through the centre then the two lines A B and A O will form a right angle at A. 109 CHAPTER IV. SPIRAL GEARS — CALCULATIOHS FOR PITCH OF SPIRALS. When the teeth of a gear are cut, not in a straight s P iral Gear « path, like a spur gear, but in a helical or screw- like path, the gear is called, t actinically, a twisted or screw gear, but more generally among mechanics, a spiral gear. A distinction is sometimes made between a screw gear and a twisted gear. In twisted gears the pitch surfaces roll upon each other, exactly like spur gears, the axes being parallel, the same as in Fig. 1, Part I. In screw gears there is an end movement, or slipping of the pitch surfaces upon each other, the axes not being parallel. In screw gearing the action is analogous to a screw and nut, one gear driving another by the end movement of its tooth path. This is readily seen in the case of a worm and worm-wheel, when the axes are at right angles, as the movement of wheel is then wholly due to the end movement of worm thread. But, as we make the axes of gears more nearly parallel, they may still be screw gears, but the distinction is not so readily seen. We can have two gears that are alike run together, with their axes at right angles, as at A B, Fig. 59. The same gear may be used in a train of screw gears or in a train of twisted gears. Thus, B, as it relates to A, may be called a screw gear; but in connection with C, the same gear, B, may be called a twisted gear. These distinctions are not usually made, and we call all helical or screw-like gears made on the Universal Milling Machine spiral gears. When two extarnal spiral gears run together, with Direction of. their axes parallel, the teeth of the gears must have erence to Axes. opposite hand spirals. 110 BROWN & SHAKPE MFG. CO. Thus, in Fig. 59 the gear B has right hand spiral teeth, and the gear C has left hand spiral teeth. When the axes of two spiral gears are at right angles, both gears must have the same hand spiral teeth. A and B, Fig. 59. have right hand spiral teeth. If both gears A and B had left hand spiral teeth, the relative direc- tion in which they turn would be reversed, spiral Lead. ^he spiral lead or lead of spiral is the distance the spiial advances in one turn. A cylinder or gear cut with spiral grooves is morel)' a screw of coarse pitch or long lead ; that is, a spiral is a coarse lead screw, and a screw is a fine lead spiral. Since the introduction and extensive use of the Universal Milling Machine, it lias become customary to call any screw cut in the milling machine a spiral. The spiral lead is given as so many inches to one turn. Thus, a cylinder having a spiral groove that advances six inches to one turn, is said to have a six inch spiral. In screws the pitch is often given as so many turns to one inch. Thus, a screw of -J-" lead is said to be 2 turns to the inch. The reciprocal expression is not much used with spirals. For example, it would not be convenient to speak of a spiral of 6" lead, as -J- turns to one inch. The calculations for spirals are made from the func- tions of a right angle triangle. Example, Cut from paper a right angle triangle, one side of showing the r l ° ° . nature of a He- the right angle 6" long, and the other side of the lix or Spiral. c " c right angle 2 . Make a cylinder 6 in circumference. It will be remembered (Part I., Chapter II.) that the circumference of a cylinder, multiplied by .3183, equals the diameter — 6" X .3183=1.9098". Wrap the paper triangle around the cylinder, letting the 2" side be parallel to the axis, the 6" side perj)endicular to the axis and reaching around the cylinder. The hypoth- eneuse now forms a helix or screw-like line, called a spiral. Fasten the paper triangle thus wrapped around. See Fig. GO. PROVIDENCE, R. I. Ill FIG. 58 -RACKS AND GEARS. Fig. 59.-SPIRAL GEARING. 11*2 i ?. ; ~>~ : z vy- :: Wig SO If we dow turn this cylinder ABCD one lorn in the direction of the arrow, the spiral will adva nee from to E. This advai.ee is the lead of the sp The angle E F, which the spiral makes with the axis E 3 is the angle of the spiral. This ;mgle isfonnd as in Chapter I. The circumference of the cylinder :;s: :n::s ::■ t :- side opposite the angle. T . -:■ pilch of the spiral corresponds to the si seat the angle. Hence the role for angle of spiral: elating tie Divide tlte circumference of the parts of a spi -fry f] te numo er of inches of 5 the quotient will he the tangent of angle - When the angk >fspi and : '.inference are given, to find the lead : DivU/e the circrr/i/'trence by the tangent of angle, the quotient will be the lead of the spi When the angle of spiral and the lead or pitch of spiral are sriven, to find, the ci : rence : MuVij the t gent of angle by the lead, and the product will be t ce. When applying calculations to spiral gears the angle is reckoned at the pitch circumference and not at the outer or addendum circle. It will be seen that when two spirals of different diameters lave the b the spiral of less diame- ter will have the smaller angle. Thus in Fig. GO if ihe paper triangle bid teen 4" lonsr ii s: of C" the diam- eter of t e cylinder would have been 1.27" .d the angle of the spiral would have been _ legrees. 113 CHAPTER V. EXAMPLES IN CALCULATION OF THE LEAD OF SPIRAL— ANGLE OF SPIRAL— CIRCUMFERENCE OF SPIRAL GEARS— A FEW HINTS ON CUTTING. It will be seen that the rules for calculating the cir- cumference of spiral gears, angle and the lead of spiral are the same as in Chapter I., for the tangent and angle of a right angle triangle. In Chapter IV., the word "circumference" is substituted for "side opposite," and the words "lead of spiral" are substituted for "side adjacent." When two spiral gears are in mesh the angle of r ^ 1 ff^ ( ^ e f JjJ" spiral should be the same in one gear as in the other, *f C Q h l f t An s le in order to have the shafts parallel and the teeth work properly together. When two gears both have right hand spiral teeth, or both have left hand spiral teeth, the angle of their shafts will be equal to the sum of the angles of their spirals. But when two gears have different hand spirals the angle of then* shafts will be equal to the difference of their angles of spirals. Thus, in Fig. 59 the gears A and B both have right hand spirals. The angle of both spirals is 45°, their sum is 90°, or their axes are at right angles. But C has a left hand spiral of 45°. Hence, as the difference between angles of spirals of B and C is 0, their axes are parallel. If two 45° gears of the same diameter have the same number of teeth the lead of the spiral will be alike in both gears: if one gear has more teeth than the other the lead of spiral in the larger gear should be longer in the same ratio. Thus, if one of these gears has 50 teeth and the other has 25 teeth, the lead of spiral ra ^ s ea d f ^lil^ in the 50 tooth gear should be twice as long as that of ent diameters. the 25 tooth gear. Of course, the diameter of pitch 114: KBOWH £ SHARPE MFG. CD. circle should s a large in the 50 tooth as in the "25 tooth gear. In spirals where the angle is 45" the circumference is the same as the spiral lead, because the tangent of 45° is 1. oiS2S? on *** Sometimes the circumference is varied to suit a pitch v, 1 rPTl TO TPTPT1 Of* -I - "-" ■- Si nraL that can be cut on the machine and retain the angle required. This would apply to catting rolls for mak- ing diamond-shaped impressions where the diameter of the roll is not a matter of importance. When two gears are to run together in a given velocity ratio, it is well first to select spirals that the machine will cut of the same ratio, and calculate the numbers of teeth and angle to correspond. This will often save considerable time in fisfurinof. The calculations for spiral gears present no special difficulties, but sometimes a little ingenuity is required to make work conform n the machine and to such cutters as we may have in stock. Let it be required to make two spiral gears to run with a ratio of 4 to 1, the distance between centres be 3.125" m"), the axes to be parallel. By rule given in Chapter XII.. Part I., we fiud the diameters of tch : rclea will be 5" and 1\". Let us take a spiral of 4S" lead for the large gear, and a spiral of 12" lead for the small gear. The circumfer- ence of the 5" pitch circle is 15.70796". Dividing the circumference by the lend of the spiral, we have 1 5 4g 7 9 6 =-32724" for tangent of angle of spiral. In the ta 3 the :.■:. rest angle to tangent, .32724", is 1S° 1 1 ' . As before stated, the angle of the teeth in the small gear will be the same as the angle of teeth or spiral in the large gear. inAiigtesafctop Now, this rule gives the angle at the pitch surface spSjlGr^oave? on ^• I^P 0U looking at a small screw of coarse pitch, it will be seen that the angle at bottom of the thread is not so great as the angle at top of thread; that is, the thread at bottom is nearer parallel to the centre line than that at the top. This will be seen in Fig. 61, where A is the centre line; i f shows lirectiou of bottom of thread, and d g PROVIDENCE, K. I. 115 shows direction of top of thread. The angle Afb is less than the angle A g d. The difference of angle being dne to the warped nature of a screw thread. A cylinder 2" diameter is to have spiral grooves 20° Example in J ib calculation of with the centre line of cylinder; what will be the lead Leadof spiral. of spiral? The circumference is 6.2832". The tan- gent of 20° is .36397. Dividing the circumference by the tangent of angle, we obtain ^: § f § § 7 — 17.26"+ for lead of spiral. Fig. 01. 116 CHAPTER VI. NORMAL PITCH OF SPIRAL GEARS— CURVATURE OF PITCH SURFACE— FORM OF CUTTERS. Normal to a ^ Normal to a curve is a line perpendicular to the tangent at the point of tangency. In Fig. 62, the line B C is tangent to the arc DEF, and the line A E O, being perpendicular to the tan- gent at E the point of tangency, is a normal to the arc. Fig. 63 is a representation of the pitch surface of a spiral gear. A' D' C is the circular pitch, as in Part I. A D C is the same circular pitch seen upon the periphery of a wheel. Let A D be a tooth D and a space. Now, to cut this space D C, the path of cut- ting is along the dotted line a b. By mere inspection, we can see that the shortest distance between two teeth along the pitch surface is not the distance ADC. Let the line A E B be perpendicular to the sides of teeth. upon the pitch surface. A continuation of this line, perpendicular to all the teeth, is called the Normal Helix. The line A E B, reaching over a tooth and a space along the normal helix, is called the Normal Pitch. PUOVIDENCK, K. I. n Fig. 63. IIS " ~-'- -~>" : -:~z :■:?:- : Roma: pitch. Xhe Xormal Fitch of a spiral gear is then: _ shortest distance betwee the centers of ' :-:-;-\ ■ - :r...'.: ..' : : <- p'::\ >:. ;~h-\ In spur gears the normal pitch and circular pitch are alike. In the rack D D. Fig ' 8, the linear pitch and normal pitch are alike. ltter for From the foregoing it will be seen that, if we should at the Bp:v:e D C with :-. ::er 3 the thickness of which at the pitch line is equal to one-half the circular pitch. a? in spur ^h—is. the space would be too - ide, and the teeth would be too thin. Hence, spiral gears should be it with thinner cutters than spur gears of the same circular pitch. The aztgie CAB is equal to the angle of the spiral. The line AEBc rrresponds to the cosine of the angle C A B. Hence the rule : Mu It y lythe - gle mlTpPeL^ * °f *P* ra ? ty the t far pitch, t d he product will be at pitch. One-half the normal pitch is the proper thickness of cutter at the pitch line. If the normal pitch and the angle are kno~ L vide e normal pitch by the cost al pitch, and not to the circular tch. Thus, if the normal pitch is 12 di the addendum should be -jV', the t hkness .13 hi'". on. The di -. of pitch circle of a spiral gear is calculated fr m the etral pitch. Thus a gear of 30 teeth 10 P w raid .itch diametr . Normal Pitch varies. PROVIDENCE, R. I. 119 But il the normal pitch is 12 diametral pitch, the blank will be 3 T y diameter instead of 3 r y. It is evident that the normal pitch varies with the angle of spiral. The cutter should be for the normal pitch. In designing spiral gears, it is well first to look over list of cutters on hand, and see whether there are cutters to which the gears can be made to conform. This may avoid the necessity of getting a new cutter, or of changing both drawing and gears after they are under way. To do this, the problem is worked the reverse of the foregoing; that is: First calculate to the next finer pitch cutter than gl e ?sp e irai would be required for the diametral pitch. cutter^^lven Let us take, for example, two gears 10 pitch and 30 teeth, spiral and axes parallel. Let the next finer cut- ter be for 12 pitch gears. The first thing is to find the angle that will make the normal pitch .2G18", when the circular pitch is .3142". See table of tooth parts. This means (Fig. 63) that the line A D C will be .3142" when A E B is .2018". Dividing .2015" by .3142" (see Olnip. IV.), we obtain the cosine of the angle CAB, which is also the angle of the spiral, ;§f J§=.833. The same quotient comes by dividing 10 by 12, yf =.833 + ; that is, divide one pitch by the other, the larger number being the divisor. Looking in the table, we find the angle corresponding to the cosine .833 is 33° 30'. We now want to find the pitch of spiral that will give angle of 33 1-° on the pitch sui face of the wheel, 3" diameter. Dividing the circumference by the tan- gent of angle, we obtain the pitch of spiral (see Chap. V.) The circumference is 9.4248". The tangent of 33° 30' is .06188, £;f 1^= 14.23 ; and we have for our spiral 14.23" lead. When the machine is not arranged for the exact When exact • p •! ' ii Pitch cannot b© pitch of spn*al wanted, it is generally well enough to cut. take the next nearest spiral. A half of an inch more or less in a spiral 10" pitch or more would hardly be noticed in angle of teeth. It is generally better to take the next longer spiral and cut enough deeper to bring center distances right. When two gears of the same size are in mesh with their axes parallel, a change 120 BROWS .s SHAKPE MFG. CO. of angle of teeth or spiral makes no difference in the correct meshing < f the teeth. &&** Gears g nt when gears of different size are in mesh, clue of Different & sizes of Mesh. re g ar< j must be had to the spirals being in pitch, pro- portional to then angular velocities (see Chapter V. ) We come now to the curvature of cutters for spiral gears; that is, then shape as to whether a cutter is made to cut 12 teeth or 100 teeth. A cutter that is light, Shape of cut- to cut a spur gear 3 diameter, may not be right for a spiral gear 3" diameter. To find the curvature of cutter, fit a templet to the blank along the line of the normal helix, as A E B. letting the templet reach over about two or three normal pitches. The curvature of this templet will be nearer a straight line than an arc of the addendum circle. Xow find the diameter of a circle that will approximately fit this templet, and con- eider this circle as the addendum circle of a gear for which we are to s lect a cutter, reckoning the gear as of a pitch the fame as the normal pitch. Fig 64 Thus, in Fig. 64. suppose the templet fits a circle 3^-" diameter, if the normal pitch is 12 to inch, dia- metral, the cutter required is for 12 P and 40 teeth. The curvature of the templet will not be quite circular, but is sufficiently near for practical purposes. Strictly, • PROVIDENCE, R. I. 121 a, flat templet cannot be made to coincide with the normal helix for any distance whatever, but any greater refinement than we have suggested can hardly be car- ried out in a workshop. This applies more to an end cutter, for a disk cutter may have the right shape for a tooth space and still round off the teeth too much on account of the warped nature of the teeth. The difference between normal pitch and linear or circular pitch is plainly seen in Figs 58 and 59. The rack T> D, Fig. 58, is of regular form, the depth of teeth being -^ of the circular pitch, nearly (.6866 of the pitch, accurately). If a section of a tooth in either of the gears be made square across the tooth, that is a normal section , the depth of the tooth will have the same relation to the thickness of the tooth as in the rack just named. But the teeth of spiral gears, looking at them upon the side of the gears, are thicker in proportion to their depth, as in Fig. 59 This difference is seen between the teeth of the two racks D D and E E, Fig. 58. In the rack D D we have 20 teeth, w T hile in the rack E E we have but 14 teeth ; yet each rack will run with each of the spiral gears A, B or C, Fig. 59, but at different angles. The teeth of one rack will accurately fit the teeth of the other rack face to face, but the sides of one rack will then be at an angle of 45° with the sides of the other rack. At F is a guide for holding a rack m mesh with a gear. The reason the racks will each run with either of the three gears is because all the gears and racks have the same normal pitch. When the spiral gears are to run together they must both have the same normal pitch. Hence, two spiral gears may run correctly together though the circular pitch of one gear is not like the circular pitch of the other gear. If a rack is to run at any angle other than 90° with the axis of the gear it is well to determine the data from a diagram, as it is very difficult to figure the angles and sizes of the teeth without a sketch or diagram. 122 CHAPTER VII. CUTTING SPIRAL GEARS IN A UNIVERSAL MILLING MACHINE. A rotary disk cutter is generally preferable to a shank cotter or end mill on account of cutting faster and hold- ing its shape longer. In catting spiral grooves, it is sometimes necessary to use an end mill on account of the warped character of the grooves, but it is very Sel- dom necessary to use an end mill in cutting spiral gears. Proving the Before catting into a blank it is well to make a slight Setting "f the ° c Machine. trace of the spiral with the cutter, after the change sears are in place, to see whether the gears are correct. If the material of the gear blanks is quite expensive, it is a safe plan to make trial blanks of cast iron in order to prove the setting of the machine, before cutting into the expensive material. The cutting of spiral gears may develop some curi- ous facts to one that has not studied warped surfaces. The gears. Fig. 59, were cut with a planing tool in a shaper, the spiral gear mechanism of a Universal Mill- ing Machine having been faste. el upon the shaper. The tool was of the same form as the spaces in tl D D, Fig. 58. Ail spiral gears of the same pitch can be cut in this manner with one tool. The nature of tl is cutting operation can be understood from a considera- tion of the meshing of straight side rack teeth with a spiral gear, as in Fig. 58. Spiral gears that run cor- rectly with a rack, as in Fig. 5S, will run correctly with each other when tin. ir axes are parallel, as at B C, Fig. 59 j but it is not considered that they are quite correct, theoretically, to run together when the gears have the same hand spiral, and their axes are at right PKOVIDENCE, II. I. 12 > Q B Fig. 65 C k 'rrn- / tti ? \ « \ / \ _Fi<7. ec :u 31. ~> - szi_?.ji ::e angles, as A B . Fig. ' \ thoush tbev will ran we :::::■. .It. Tie c^.^::.. c: c:::::g s : : i- a planer tool is sometimes caHlodplanimg the tee: log is an accurate way of shaping teeth th with rack teeth and for gears on parallel si Lr:'.::lh:.si.7r , j ez; i : -■.:". :: c •:.: s"i: 1 ti c.'.re p'ner i./o-6=, :: . : his r.:: .eer. :::m: ::; geie :..". use. I: is c:Lve::t!i: :: live tie 5i:i :_ si . t .. : s :i :ne ::^:~.ii i..b.e : h. p:. _ __ j. _ __, -ion. No. of Teeth Pitch Diam Eft 1 . <~ iiside Lis i_e:e: . Circular Pit :ch Alible of Teeth with Axis N nnal Cin c-nlar Pitch P:::L :■: Ci truer . A lie- ivr ■ s . Tni?k-ess . : Tooth t Wh-> I'er :LI» -: >\ : er - act Lead of Spiral A] _ : rin :e Lei i : : Si :: - 1 - - \ rs n ~\T : lhng Machine to Cut Spiral Gear on W< 1st Gear on >r: :1 2nd Gear 01 *v - - - 't-s.: :i_ .^;r e" 7 ^" A spiral of any angle I : - : : can generally be cat in a Universal Milling Machine without special attach- ments, the cntter being at the top of the work. The cutter is placed on the arbor in such position tha: can reach the work centrally after the table is s : it :o the angle of the spiral. In order to cut central, it ia generally well enough to place the table, before setting it to the angle, so that the work centres will be central with the cutter, then swing the table and set it to the ui: :- : t'.e s ; ir;-/ . For very accurate work, i: is s\fer to test the posi-^^ 1 *** 1 Set * tion of the centres after the table has been set to the angle. PROVIDENCE, R. I. 125 Fig. 67. USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING SPIRAL GEARS. 120 BROWN & SHARPE MFG. CO. This can be done with a trial piece, Fig. 65 ? which is simply a round arbor with centre holes in the ends. It is mounted between the centres, and the knee is raised until the cutter sinks a small gash, as at A. This gash shows the position of the cutter; ami if the gash is central with the trial piece, the cutter will be central with the work. If preferred, the arbor can be dogged to the work spindle; and the line B C drawn on the side of the arbor at the same height as the cen- tres ; the work spindle should then be turned quarter way round in order to bring the line at the top. The gash A can now be cut and its position determined with the line. In cutting small gears the arbor can be dogged to the work spindle ; the distance between the gear blank and the dog should be enough to let the dog pass the cutter arbor without striking. A spiral gear is much more likely to slip in cutting than a spur gear. For gears more than three or four inches in diameter it is well to have a taper shank arbor held directlv in the work spindle, as shown in Figs. 67 and GS ; and for the heaviest work, the arbor can be drawn into the spin- dle with a screw in a threaded hole in the end of the shank. After cutting a space the work can be dropped away from the cutter, in order to avoid scratching it when coming back for another cut. Some workmen prefer not to drop the work away, but to stop the cutter and turn it to a position in which its teeth will not touch the work. To make sure of finding a place in the cut- ter that will not scratch, a tooth has sometimes been t iken out of the cutter, but this is not recommended. The safest plan is to drop the work away. Angie^reater In cutting spiral gears of greater angle than 45°, a vertical spindle milling attachment is available, as shown in Figs. G7 and 6$. In Fig. G7 the cutter is at 90° with the work spindle when the table is set to 0, so that the proper angle at which the table should be set, is the difference between the angle of the spiral and 90°. Thus, to cut a 70° PROVIDENCE, R. I. 127 Fig. 68. USE OF VERTICAL SPINDLE MILLING ATTACHMENT IN CUTTING SPIRAL GEARS. 128 bbow>" fc ^:-:aspe mt spiral, we subtract ?i : from £ . : . and the remainder, 20°, is the angle to se: the :. ... In cutting on the top, Fig. 67, the attachment is set to 0. In Fig. 69 the cutter is at the side of the work; the h ile is set to 0, and the attachment is set to the differ- e :>:: between 9 1 : an :I the r-: : .ire 1 ^n_'e ci spiral. In setting the cotter central it is convenient :: have a small knee as at K. Fig. 66. A line is drawn uponthc knee at Lhe same height as : the cent es. The cutter art 9i is . : glit to tl. : angle as just shown, and a gash is cut in the knee. Wh n the g ish is central with the lit r. the cutter wiD be central with the work. The cutter can be sir to act apon either side :: the gear to be : ". :::r ling as a right hand or a left hand spiral is wanted. The setting in Fig. 68 is for a right hand spiral. If the gear blank were bronght in front of the c ::- t:-r. and the reversing gen: s.t between two chance geai a, the machine w . al .1 be 5t: f« r a left hand spiral. For cc iraer pitches than abon: 12 IP diametral, it is well to cut more than once around, the finishing cut being quite light so as to en: sn;::th. 129 CHAPTER VIII. SCREW GEARS AND SPIRAL GEARS— GENERAL REMARKS. The working of spiral gears, when their axes are working of ., , . n iii * Spiral Gears. para. lei, is generally smoother than spur gears. A tooth does not strike along its whole face or length at once. Tooth contact first takes place at one side of the gear, passes across the face and ceases at the other side of the gear. This action tends to cover defects in shape of teeth and the adjustment of centres. Since the invention of machines for producing accu- rate epicyloidal and involute curves, it has not so often been found necessary to resort to spiral gears for smoothness of action. A greater range can be had in the adjustment of centers in spiral gears than in spur gears. The angle of the teeth should be enough, so that one pair of teeth will not part contact at one side of the gears until the next pah' of teeth have met on the other side of the gears. When this is done the gears will be in mesh so long as the circumferences of their addendum circles intersect each other. This is some- times necessary in roll gears. Relative to spur and bevel gears in Part I., Chapter XII., it was stated that all gears finally wore them- selves out of shape and might become noisy. Spiral gears may be worn out of shape, but the smoothness of action can hardly be impaired so long as there are any teeth left. For every quantity of wear, of course, there will be an equal quantity of backlash, so that if gears have to be reversed the lost motion in spiral gears will be as much as in any gears, and may be more if there is end play of the shafts. In spiral gears End Pressure there is end pressure upon the shafts, because of the Spiral Gears, screw-like action of the teeth. This end pressure is sometimes balanced by putting two gears upon each shaft, one of right and one of left hand spiral. 130 BROWN & SHARPE MFG. CO. The same result is obtained in solid cast gears by making the pattern in two parts — one right and one left-hand spiral. Such gears are colloquially called "herring-bone gears.'' In an internal spiral gear and its pinion, the spirals of both wheels are either right-handed or left-handed. Such a combination would hardly be a mercantile product, although interesting as mechanical feat. In screw or worm-gears the axes are generally at right angles, or nearly so. The distinctive features of screw gearing may be stated as follows : The relative angular velocities do not depend upon the diameters of pitch- cylinders, as in Chapter I., Distinctive Part I. Thus the worm in Chapter XL, Fig-. 35, can teatures of x ' ° ' Screw Gearing, be any diameter — one inch or ten inches — without affecting the velocity of the worm-wheel. Conversely if the axes are not parallel we can have a pair of spiral or screw gears of the same diameter, but of different numbers of teeth. The direction in which a worm-wheel turns depends upon whether the worm has a right-hand or left-hand thread. When angles of axes of worm and worm-wheel are oblique, there is a practical limit to the directional relation of the worm-wheel. The rotation of the worm-wheel is made by the end movement of the worm-thread. The term worm and worm-wheel, or worm -gearing, is applied to cases where the worms are cut in a lathe, and the shapes of the threads or teeth, in axial section, are like a rack. The shape usually selected is like the rack for a single curve or involute gear. See Chap. IV., Parti. Worms are sometimes cut in a milling machine. If the form of the teeth in a pair of screw gears is determined upon the normal helix, as in Chap. VI., the gears are usually called Spiral Gears. If we let two cylinders touch each other, their axes being at right angles, the rotation of one cylinder will have no tendency to turn the other cylinder, as in Chapter L, Part I. PROVIDENCE, R. I. 13J We can now see why worms and worm-wheels wear why Wor ™ J Wheels wear out faster than other gearing. The length of worm- sofast - thread, equal to more than the entire circumference of worm, comes in sliding contact with each tooth of the wheel during one turn of the wheel. The angle of a worm-thread can be calculated the same as the angle of teeth of spiral gear ; only, the angle of a worm thread is measured from a line or plane that is perpendicular to the axis of the worm. CHAPTER IX. CONTINUED FRACTIONS— SOME APPLICATIONS IN MACHINE CONSTRUCTION. Definition of _^_ continued fraction is one which has unitv for its a Continued Fraction. numerator, and for its denominator an entire number plus a fraction, which fraction has also unity for its numerator, and for its denominator an entire number plus a fraction, and thus in order. The expression. * 4 — i_ 3 — 1_ 5 is called a continued frac- tion. By the use of continued fractions, we are ena- Pracucai u=e D ; e d to nnd a fraction expressed in smaller numbers. of Continued *- Fractions. that, for practical purposes, may be sufficiently near in value to another fraction expressed in large numbers. If we were required to cut a worm that would mesh with a gear -4 diametral pitch -i P. . m a lathe having 3 to 1-inch linear leading screw, we might, without continued fractions, have trouble in finding- change gears, because the circular pitch corresponding to •4 diametral pitch is expressed in large numbers : This example will be considered farther on. For illustration, we will take a simpler example. What fraction expressed in smaller numbers is near- est in value to T : -- '■ Dividing the numerator and the denominator of a fraction by the same number does not change the value of the fraction. Dividing both Example in terms f s;_ bv 29. we have ^T~ or. what is the Continued u« .. " J T ~ ' same thing expressed as a continued fraction, s-kju The continued fraction s+j_ is exactly equal to -f£j- ^ now. we reject the v 7 , the fraction 4- will be larger than 5— :_. because the denominator has been diniin- 2 9' islied, 5 beii _ less than 5gV- i ' d something ^-_V expressed in smaller numbers than 29 for a PROVIDENCE, E. I. 133 numerator and 146 for a denominator. Eeducing \ and y 2 ^ to a common denominator, we have ^=i|f an d i 2_9_ = ±|i-. Subtracting one from the other, we have t Jq, which is the difference between \ and -f£-$. Thus, in thinking of T % as £, we have a pretty fair idea of its value. There are fourteen fractions with terms smaller than 29 and 146, which are nearer j 2 ¥ V than \ is, such as ■if, -if and so on to f 2 f T . In this case by continued frac- tions we obtain only one approximation, namely -J, and any other approximations, as T f, -J-f, &c, we find by trial. It will be noted that all these approximations are smaller in value than T 2 I %. There are cases, how- ever, in which we can, by continued fractions, obtain approximations both greater and less than the required fraction, and these will be the nearest possible approxi- mations that there can be in smaller terms than the given fraction. In the French metric system, a millimetre is equal to .03937 inch ; what fraction in smaller terms ex- presses .03937" nearly? .03937, in a vulgar fraction, is ToSood* Dividing both numerator and denominator by 3937, we have 25II1JL, Rejecting from the de- nominator of the new fraction, -Jf Jf , the fraction -fa gives us a pretty good idea of the value of .03937". If in the expression, Tz+TJJA, we divide both terms of the fraction |jj|j by 1575, the value will not be changed. Performing the division, we have 1 25 + 1 2 + 787 1575 • "We can now divide both terms of r \ 8 T 7 T by 787, without changing its value, and then substitute the new fraction for ^ 8 T \ * n ^ ne continued fraction. Dividing again, and substituting, we have : 1 25 + 1 2 + l_ 2+ 1 787 as the continued fraction that is exactly equal to .03937. 134 BROWN & SHAEPE MFG. CO. In performing the divisions, the work stands thus 3937) 100000 (25 7874 21260 19685 1575) 3937 (2 3150 787) 1575 (2 * 1574 1) 787 (787 787 •o- That is, dividing- the last divisor by the last remain- der, as in finding the greatest common divisor. The quotients become the denominators of the continued fraction, with unity for numerators. The denominators 25, 2, and so on, are called incomplete quotients, since they are only the entire parts of each quotient. The first expression in the continued fraction is ^ or .04 — a little larger than .03937. If, now, we take 25 + i , we shall come still nearer .03937. The expres- sion 25~+t is merely stating that 1 is to be divided by 25^. To divide, we first reduce 25J to an improper fraction, ^, and the expression becomes IT, or one divided by ^-. To divide by a fraction, "Invert the divisor, and proceed as in multiplication." We then have -f T as the next nearest fraction to .03937. -g2 T - 0392 + , which is smaller than .03937. To get still nearer, we take in the next part of the continued frac- tion, and have 1 25 + 1 2 + 1 2" We can bring the value of this expression into a fraction, with only one number for its numerator and one number for its denominator, by performing the operations indicated, step by step, commencing at the last part of the continued fraction. Thus, 2 + J, or 2-|, is equal to ■§-, Stopping here, the continued frac- tion would become } 25 -H_ 2- 1 1__ Now, 5 equals -|, and we have 25 +_»_. 25f equals 2 5 1 \ 3 -; substituting again, we have rhi. Dividing 1 by J-f- 1 , w^e have T | T . T | T is the nearest fraction to PROVIDENCE, E. I. 135 .03937, unless we reduce the whole continued fraction l 25+1 2 + 1 2 + L„, which would give us back the .03937 itself. tot T f T =. 03937007, which is only ^^^^ larger .03937. It is not often that an approximation will come so near as this. This ratio, 5 to 127, is used in cutting millimeter Practical use of the foregoing thread screws. If the leading screw of the lathe isExampie. 1 to one inch, the change gears will have the ratio of 5 to 127; if 8 to one inch, the ratio will be 8 times as large, or 40 to 127; so that with leading screw 8 to inch, and change gears 40 and 127, we can cut milli- meter threads near enough for practical purposes. The foregoing operations are more tedious in de- scription than in use. The steps have been carefully noted, so that the reason for each step can be seen from rules of common arithmetic, the operations being merely reducing complex fractions. The reductions, ■gig-, T 2 T , T ^j, etc., are called conver gents, because they come nearer and nearer to the required .03937. The operations can be shortened as follows: Let us find the fractions converging towards .7854", Example. the circular pitch of 4 diametral pitch, .7854= T \ 8 Tr 5 / (r ; reducing to lowest terms, we have -§-§-§-£. Applying the operation for the greatest common divisor: 3927) 5000 (1 3927 1073) 3927 (3 3219 708) 1073 (1 _708 365) 708 (1 365 843) 365 (1 343 22) 343 (15 22 123 110 13) 22 (1 13 9) 13 (1 9 4) 9 (2 8 1) 4 (4 4 Bringing the various incomplete quotients as de- nominators in a continued fraction as before, we have : 136 BROWN & SHARPE MFG. CO. 1 + 1 3 + 1 1 + 1 1 + 1 _ 1 + 1 _ 15 + 1 1 + 1_ i + i ; 2 + ^ Now arrange each partial quotient in a line, thus : 13111 15 1 1 2 4 1 3. 4 1 J_l ±T_2 183 3 5 5 8_93_ 3 9 27 x 4 1 9" 14 219 233 4~T2 T13 1 "5"0 Now place under the first incomplete quotient the first reduction or convergent -f, which, of course, is 1 ; put under the next partial quotient the next reduction or convergent \— 7 or ^ which becomes f . 1 is larger than .7854, and J is less than .7854. Having made two reductions, as previously shown, we can shorten the operations by the following rule for next eonvergents : Multiply the numerator of the convergent just found by the denominator of the next term of the con- tinued fraction, or the next incomplete quotient, and add to the product the numerator of the preceding convergent ; the sum will be the numerator of the next convergent. Proceed in the same way for the denominator, that is multiply the denominator of the convergent just found by the next incomplete quotient and add to the product the denominator of the preceding convergent ; the sum will be the denominator of the next convergent. Continue until the last convergent is the original frac- tion. Under each incomplete quotient or denominator from the continued fraction arranged in line, will be seen the corresponding convergent or reduction. The convergent \\ is the one commonly used in cutting racks 4 P. This is the same as calling the circumference of a circle 22-7 when the diameter is one (1) ; this is also the common ratio for cutting any rack. The equivalent decimal to \\ is .7S57 X , being about x g large. In three set- tings for rack teeth, this error would amount to about .001" For a worm, this corresponds to -if- threads to 1"; now, with a leading screw of lathe 3 to 1", we would want gears on the spindle and screw in a ratio of 33 to 14. Hence, a gear on the spindle with 66 teeth, and a gear on the 3 thread screw of 28 teeth, would enable us to cut a worm to fit a 4 P gear. CHAPTER X. ANGLE OF PRESSURE. 137 In Fig. 69, let A be any flat disk lying upon a hori- zontal plane. Take any piece, B, with a square end, a b. Press against A with the piece B in the direction of the arrow. Fig. 69. Fig. It is evident A will tend to move directly ahead of B in the normal line c d. Now (Fig. 70) let the piece B, at one corner f, touch the piece A. Move the piece B along the line d e, in the direction of the arrow. It is evident that A will not now tend to move in the line d e, but will tend to move in the direction of the normal c d. When one piece, not attached, presses against another, the tendency to move the second piece is in the direction of the normal, at the point of contact. This normal is called the line of pressure. ^ineofPreas- J L ure. The angle that this line makes with the path of the impelling piece, is called the angle of pressure. In Part I., Chapter IV., the lines B A and B A' are called lines of pressure. This means that if the gear 138 BE0WX k SHAEPE MFG. CO. drives the rack, the tendency to move the rack is not in the direction of pitch line of rack, but either in the direction B A or B A', as we turn the wheel to the left or to the right. The same law holds if the rack is moved in the direction of the pitch line; the tendency to move the wheel is not directly tangent to the pitch circle, as if driven by a belt, but in the direction of the line of pressure. Of course the rack and wheel do move in the paths prescribed by then connections with the framework, the wheel turning about its axis and the rack moving along its ways. This pressure, not in a direct path of the moving piece, causes extra friction in all toothed gearing that cannot well be avoided. Although this pressure works out by the diagram, as we have shown, yet, in the actual gears, it is not at all certain that they will follow the law as stated, because of the friction of teeth among themselves. If the driver in a train of gears has no bearing upon its tooth-flank, we apprehend there will be but little tendency to press the shafts apart. The arc through which a wheel passes while one of its teeth is in contact is called the arc of action. Base of Sys- Until within a few vears, the base of a svstem of tern of Inter- J ' change a bie c | iible-cmve interchangeable gears was 12 teeth. It Gears. o fc> is now 15 teeth in the best practice (see Chapter VII., Part I.) The reason for this change was : the base, 15 teeth, gives less angle of pressure and longer arc of contact, and hence longer lifetime of gears. Arc of Action. 139 CHAPTER XL INTERNAL GEARS. In Part L, Chapter VIII., it is stated that the space of an internal gear is the same as the tooth of a spur gear. This applies to involute or single-curve gears as well os to double-curve gears. The sides of teeth in involute internal gears are hollowing. It, however, has been customary to cut internal gears with spur gear-cutters, a No. 1 cutter generally being used. This makes the teeth sides convex. Special cutters should be made for coarse Special Cut, . , , , , -, t -i . . . , , ters for coarse pitch double-curve gears. In designing internal gears, Pitch. it is sometimes necessary to depart from the system with 15-tooth base, so as to have the pinion differ from the wheel by less than 15 teeth. The rules given in Part I., Chapters VII. and VIII., will apply in making gears on any base besides 15 teeth. If the base is low-numbered and the pinion is small, it may be neces- sary to resort to the method given at the end of Chap- ter VII., because the teeth may be too much rounded at the points by following the approximate rules. The base must be as small as the difference between Base f or in- t>6ril£Ll Ct P ft t* the internal gear and its pinion. The base can be Teeth, smaller if desired. Let it be required to make an internal gear, and pinion 24 and 18 teeth, 3 P. Here the base cannot be more than 6 teeth. In Fig. 71 the base is 6 teeth. The arcs A K and O k, drawn about T, have a radius equal to the radius of the pitch circle of a 6-tooth gear, 3 P, instead of a 15-tooth gear, as in Chapter VIII., Part I. The outline of teeth of both gears and pinion is pescription of made similar to the gear in Chapter VIII. The same 140 BROWN & SHARPE MFG. CO. GEAR, 24 TEETH. PINiON, 18 TEETH, 3P P = 3 N =24 and 18 P'= 1.0472" t— 5236" S= .3333" D"= .6666" }+/= -3857" P"+/= 7190" NTERNAL GEAR AND PINION IN MESH PROVIDENCE, R. I. 141 letters refer to similar parts. The clearance circle is, however, drawn on the outside for the internal gear. As before stated, the spaces of a spur wheel become the teeth of an internal wheel. The teeth of internal gears require but little for fillets at the roots ; they are generally strong enough without fillets. The teeth of the pinion are also similar to the gear in Chapter VIII., substituting 6-tooth for 15-tooth base. To avoid confusion, it is well to make a complete sketch of one gear before making the other. The arc of action is longer in internal gears than in external gears. This property sometimes makes it necessary to give less fillets than in external gears. In Fig. 71 the angle K T A is 30° instead of 12°, as in Fig. 12. This brings the line of pressure L P at an angle of 60° with the radius C T, instead of 78°. A system of spur gears could be made upon this 6-tooth base. These gears would interchange, but no gear of this 6-tooth system would mesh with a double- curve gear made upon the 15-tooth system in Part 1. c:-:.-.?7z?. :• :-Eii3': _ . . . _- _ '" -. _ : : '. : - _ _ _ : ".. '. -. : : \-\ r-_ : : ; ; : :k. 117 lri_ :-r : _'.t ; _ :5r ;i-r:, -1 :: • sZ- ■ - z -■■- 1 -. — ^ i - - - - ■ - .»__ . . . _ : _ -• : . : : : ; :: _ :: - . . . . - . . _ :-55~ 7 T i 7 _U.~ ^ JT = :: ~; . . . . — . T . Iirjr _' - : : - :.'. r :: ;iLS 4 "* ~ -• ~^~- 7Z T 7Z :: -tt. - i.— -'-^ - . — • : - 1 1-4 ~ - 40 1460 € 1 9-16 ' - -' 2220 5 --- ; - 24* — . . -,.._-_, . -\ 1 ;:r£ 1 : : t _ -. — - ; ~-n . — — : V^r 1 szir rrS5— r i: 1- 3 of the : :rr^o"_i* - ~ -. . li ■ T ... 10 Fitdb 353 1-3 Ubs. at the Pitch line. 8 - _ 6 "40 823 1-3 hen it is 2} times the cJrealar pitch. PROVIDENCE. II. I. 143 TABLE OF DECIMAL EQUIVALENTS of 8ths, 16ths, 32nds and 64ths of an inch. 8ths. i=.126 i=.250 #=.375 i = .500 f=.750 £=.875 16ths. T V=.0625 A=-1875 f V=3125 T V=.4375 T 9 g = .5625 U = .6875 |f =.8125 if =.9375 32nds. J^ = .03125 A=. 09375 A=. 15625 j - 32" 21875 .28125 .34375 .40625 .46875 .53125 .59375 .65625 .71875 .78125 .84375 .90625 .96875 64ths. ^ = . 015625 A=- 046875 5 — 6 4 — 7 — 64 — _9_ — 64 — 11 — 64 — 13 — 64 — 15. — 64 — ±1 — 64 — 9_ 32 11 32 1 Ji 32 15. 32 LI 32 19- "3 2 2 1_ 32 2.3. 32 25 32 21- 3 2 _2_iL 32 ill- 32 .078125 .109375 .140625 .171875 .203125 .234375 .265625 JL.9 64 2.1 64 2.3. 64 2.5. 64 2JT 64 29 64 _3_1 64 3.3. 64 3.5. "6 4 3.7 64 3.9. 64 41 64 43 64 15 64 11 64 19. 64 5_J_ 64' 5.3 64" A5_- 64' £1- 64" II- 64" 6.1- 64" 6.3- 64" . 296875 .328125 .359375 .390625 .421875 .453125 . 484375 .515625 . 546875 .578125 .609375 . 640625 .671875 . 703125 734375 . 765625 . 796875 .828125 . 859375 : 890625 . 921875 .953125 .984375 144 BBOWH fc SHABPE MFG. CO. GEAB WHEEL-. TABLE OF TOOTH PAP.TS nBCUTAP. PIT 3H IX TTRST C 1 LUiiy. - - - = :=- — n - — r 7 .- — - P : :- f YY'+f p .31 p ■-: r o 3 - 1.5708 i . o :• :■•:■ 0360 1.2732 .7360 1.3732 :_: 6700 1 = ? 1.6755 .9375 .596- L.1937 . ' : "90'i 1.2874 .5813 .6281 1-S- 4 1.7952 .8750 . 5570 1.1141 . 6445 1.201c .5425 .3-:: If 8 13 1.9333 .$125 .5173 1 . 0345 .5985 1.1158 5038 .5444 1: 2 3 - , D944 .7500 .4775 .9549 . 5525 1.0200 .4650 .5025 1-"- ± 1G 16 23 2.1855 .7187 .4576 . 9151 .5201 .9870 .4456 .4816 11 8 11 2.2848 875 .4377 -754 .5064 .9441 .4262 .41 1 'I 16 »1 2.31 .6562 .4178 .8 .4834 0- 4069 4397 1 n 4 a 2.5133 .6252 3979 .7958 .4604 .8583 3875 .4188 1A 19 2.645C .5937 .37-: .7" ; : .4374 .8156 .3681 .3978 H 9 2.7925 .5625 .35-1 .7162 .4143 .7724 .3488 37G9 1-- A 16 J_6 2.9568 .5312 .3382 .6764 .3. : '1; .7295 .3294 .3559 1 1 3.1416 . 5000 3183 .■ .3683 —:■: - : : 335C 15 16 ItV 3.3510 .4687 .2984 . 5968 .3453 .6437 _ : 6 . 3141 7 8 H 3.5904 .4375 .2785 .557 .3223 .6007 .2713 .2931 1 3 16 1* 3.866C .4: 52 .2586 .5173 .2993 .5579 .2519 .2722 3. i- 4.1858 .3750 .2387 .4775 .2762 . 5 1 5 _ .2325 .2513 1 1 16 x ll 4.5 .3437 .21 SO .4377 2532 .4"2: .2131 .2303 2 3 u 4.7124 .3333 . 2122 .4244 .2455 .4577 . .2233 PROVIDENCE, R. I. Ho TABLE OF TOOTH PARTS.— Continued. CIRCULAR PITCH IN FIRST COLUMN. e3 ■ 11 P' 5 8 Threads or Teeth' per inch Linear. Diametral Pitch. Thickness of Tooth on Pitch Line. Addendum and ~- ft hflO .g E-" G C D" Depth of Space below Pitch Line. ft ~ Width of Thread-Tool at End. Width of Thread at Top. 1 " w 1 3 P t s «+/ .4291 Px.31 .1938 P'x.S35i 5.0265 .3125 .1989 .3979 .2301 .2094 9 1 G IT L 9 5.5851 .2812 .1790 .3581 .2071 .3862 .1744 .1884 1 2 2 6.2832 .2500 .1592 .3183 .1842 .3433 .1550 .1675 7 16 2f 7.1808 .2187 .1393 .2785 .1611 .3003 .1356 .1466 2 "5 2i 7.8540 .2000 .1273 .2546 .1473 .2746 .1240 .1340 3 8 2| 8.3776 .1875 .1194 .2387 .1381 .2575 .1163 .1256 1 3 3 9.4248 .1666 .1061 .2122 .1228. 2289 .1033 .1117 5 1 G H 10.0531 .1562 .0995 .1989 .1151 .2146 .0969 .1047 2 1 3} 10.9956 .1429 .0909 .1819 .1052 .1962 .0886 .0957 1 4 4 12.5664 .1250 .0796 .1591 .0921 .1716 .0775 .0838 2 9~ 4-L *2 14.1372 .1111 .0707 .1415 .0818 .1526 .0689 .0744 1 "5 5 15.7080 1000 .0637 .1273 .0737 .1373 .0620 .0670 3 1 G °3 16.7552 .0937 .0597 .1194 .0690 .1287 .0581 .0628 I G 18.8496 .0833 .0531 .1061 .0614 .1144 .0517 .0558 1 1 7 21.9911 0714 .0455 0910 .0526 .0981 .0443 .0479 1 8 8 25.1327 .0625 .0398 .0796 .0460 .0858 .0388 .0419 1 9 9 28.2743 .0555 .0354 .0707 .0409 .0763 .0344 .0372 1 1 10 31.4159 .0500 .0318 .0637 .0368 .0687 .0310 .0335 1 1 6 1G 50.2655 .0312 .0199 .0398 .0230 .0429 .0194 .0209 i 14 'J BROWN & SHABTE ilFG. CO. GEAR WHEELS. TABLE OF TOOTH PART; -DIAMETRAL PITCH EN FIRST COLUMN. ■a — « T- Z — 71 — < 5 C p P t s D' »+/ D"+/ 1 2 6.2832 3.1416 2 . 0000 4.0000 2.3142 4.3142 a 4 4.1888 2 . 0944 1.3333 2.6666 1 . 5428 2.8761 1 3.1416 1 . 5708 1.0000 2 . 0000 1 .1571 2.1571 li 2.5133 1 , 2566 . 8000 1.6000 .9257 1.7257 H 2 . 0944 1.0472 .6666 1 . 3333 .7714 1.4381 If 1.7952 .8976 .5714 1 1429 .6612 1.2326 2 1.5708 . 7854 .5000 1 . 0000 . 5785 1 . 0785 01 1.3963 .6981 .4444 .8888 .5143 . 9587 2i 1.2566 .6283 .4000 .8000 .4628 .8628 -4 1.1424 .5712 .3636 .7273 .4208 .7844 3 1 . 0472 . 5236 . 3333 .6666 . 3857 .7190 31 .S976 .4488 .2857 .5714 .3306 .6163 4 . 7854 .3927 .2500 . 5000 .2893 . 5393 5 .6283 .3142 .2000 .4000 .2314 .4314 G .5236 .2618 .1666 .3333 .1928 .3595 i .4488 .2244 . 1429 .2857 . 1653 .3081 8 .3927 .1963 .1250 . 2500 .1446 .2696 9 .3491 . 1745 .1111 2222 .1286 .2397 10 .3142 .1571 .1000 .2000 .1157 .2157 11 .2856 . 142S . 0909 .1818 .1052 .1961 12 .2618 . 1309 0S33 .1666 . 0964 .1798 13 .2417 . 1208 .0769 .1538 .0890 .1659 14 .2244 .1122 .0714 . 1429 . 0S26 .1541 PROVIDENCE. R. I. 117 TABLE OF TOOTH PARTS— Continued. DIAMETRAL PITCH IN FIRST COLUMN. Diametral Pitch. Circular Pitch. Thickness of Tooth on Pitch Line. Addendum and J' Working Depth of Tooth. Depth of Space below Pitch Line. Whole Depth of Tooth. P. P'. t. s. D". .0771 D"+/. 15 .2094 .1047 .0666 .1333 .1438 16 .1963 .0982 .0625 .1250 .0723 .1348 17 .1848 .0924 .05 8 .1176 .0681 .1269 18 .1745 .0873 .0555 .1111 .0643 .1198 19 .1653 .0827 .0526 .1053 .0609 .1135 20 .1571 .0785 .0500 .1000 .0579 .1079 22 .1428 .0714 .0455 .0909 .0526 .0980 24 .1309 .0654 .0417 .0833 .0482 .0898 26 .1208 .0604 .0385 .0769 .0445 .0829 28 .1122 . 0561 .0357 .0714 .0413 .0770 30 .1047 .0524 .0333 .0666 .0386 .0719 32 .0982 .0491 .0312 .0625 .0362 .0674 34 .0924 .0462 .0294 .0588 .0340 .0634 36 .0873 .0436 .0278 .0555 .0321 .0599 38 .0827 .0413 .0263 .0526 .0304 .0568 40 .0785 .0393 .0250 . 0500 .0289 .0539 42 .0748 .0374 .0238 .0476 .0275 .0514 44 .0714 .0357 .0227 .0455 .0263 .0490 46 .0683 .0341 .0217 .0435 .0252 .0469 48 .0654 .0327 .0208 .0417 .0241 .0449 50 .0628 .0314 .0200 .0400 .0231 .0431 56 .0561 .0280 .0178 .0357 .0207 .0385 60 .0524 .0262 .0166 .0333 .0193 . 0360 us BROWN & SHARPE MFG. CO. NATURAL SINE. T>eg. 0' 10' 20' 33' 40' 50' 60' : .00000 .00291 .00581 .00872 .01163 .01454 .01745 89 1 .01745 .02036 .02326 .02617 .02908 .03199 .03489 : 88 2 .03489 .03780 . 04071 .04361 .04652 .04943 .05233 i 87 3 .052S3 .05524 .05814 .06104 .06395 .06685 .06975 1 80 4 .06975 .07265 .07555 .07845 .08135 .08425 .08715 85 5 .08715 .09005 .09295 .09584 .09874 .10163 . 10452 j 84 6 .10452 .10742 .11031 .11320 .11609 .11898 .12186 , 83 7 .12180 .12475 .12764 .13052 .13341 . 13629 .13917 i 82 8 .13917 .14205 .14493 . 14780 .15068 .15356 .15643 ' 81 9 .15643 .15980 .16217 .16504 .16791 .17078 .17364 i 80 10 . 17364 .17651 .17937 .13223 .18509 .18795 .19080 ■ 79 11 .19080 .19366 .19651 .19936 .20221 .20506 .20791 78 12 .20791 .21075 .21359 .21644 .21927 .22211 .22495 77 13 .22495 .22778 .23061 .23344 .23627 .23909 .24192 ! 7G 14 .24192 .24474 .24756 .25038 .25319 .25600 .25881 75 15 .25881 .26162 .26443 .26723 .27004 .27284 .27563 ■ 74 16 .27563 .27843 .28122 .28401 .28680 .28958 .29237 73 17 .29237 .29515 .29793 .30070 .30347 .30624 .30901 1 72 18 .30901 .31178 .31454 .31733 .32000 .32281 .32556 71 19 .32556 .3283L .3310G .33380 .33654 .83928 . 34202 70 20 .34202 .34475 .34748 .35020 .35293 .35565 .35836 69 21 .35836 .36103 .36379 .36650 .36920 .37190 .37460 68 22 .37460 .37780 .37999 .38268 .38536 .38805 .39073 ! 67 23 .39073 .39340 .39607 .39874 .40141 .40407 .40673 ; 66 24 .40673 .40989 .41204 .41469 .41733 .41998 .42261 65 25 .42261 .42525 .42788 .43051 .43313 .43575 .43887 64 26 .43837 .44098 .44359 .44619 .44879 .45139 .45399 63 27 .45399 .45658 .45916 .46174 .46482 .46690 .46947 62 28 .46947 .47203 .47460 .47715 .47971 .48226 .48481 61 29 .48481 .48735 .48989 .49242 .49495 .49747 .50000 60 30 .50000 .50251 .50503 .50753 .51004 .51254 .51503 59 31 .51503 . 51752 • .52001 .52249 .52497 .52745 .52991 58 32 .52991 .53288 .53484 .53730 .53975 .54219 .54463 57 33 .54463 .54707 .54950 .55193 .55436 .55677 .55919 56 34 .55919 .56160 .56400 .56640 .56880 .57119 .57357 55 35 .57357 . 57595 .57833 . 58070 .58306 .58542 .58778 54 36 .58778 .59013 .59243 . .59482 .59715 .59948 .60181 53 37 .60181 .60413 .60645 .60876 .61106 .61336 .61566 52 38 .61566 .61795 .62023 .62251 .62478 .62705 .62932 51 39 .62932 .63157 .63383 .63607 .63832 .64055 .64278 50 40 .64278 .64501 .64723 .64944 .65165 .65386 .65605 49 41 .65605 .65825 .66043 .66262 . 66479 .66696 .66913 48 42 .66913 .67128 .67344 .67559 .67773 .67986 .68199 47 43 .68199 .68412 .68624 . 68835 .69046 .69256 .69465 46 44 .69465 .69674 .69883 .70093 .70298 . 70504 . 70710 45 ■ 60' 50' 40' 30' 2Y 10' C i ~Deg. NATURAL COSINE. PROVIDENCE. R. I. 149 NATURAL SINE. Beg. 0' ! 10 ' 20' 30' 40' 50' 60' j 45 .70710 | .70916 .71120 .71325 . 71528 .71731 .71934 44 46 .71934 . 72185 .72336 .72537 . 72737 . 72936 .73135 43 47 . 73185 . 73333 . 73530 .73727 . 73923 .74119 .74314 42 48 74314 .74508 .74702 .74895 .75088 .75279 .75471 41 49 .75471 .75661 . 75851 . 76040 .76229 .76417 .76604 40 50 .76U04 .76791 .76977 .77162 .77347 .77531 .77714 39 51 .77714 .77897 .78079 .78260 . 78441 .78621 . 78801 38 53 .78801 .78979 .79157 .79385 .79512 .79688 . 79863 37 53 .79863 .80038 .80212 .80385 .80558 .80730 .80901 36 54 .80901 .81072 .81242 .81411 .81580 .81748 .81915 iiO 55 .81915 .82081 .82247 .82412 .82577 .82740 .82903 34 56 .82903 .83066 .83227 .83383 .83548 .83708 .83867 33 57 .83867 .84025 .84182 .84339 .84495 . 84650 .84804 32 53 .84804 .84958 .85111 .85264 .85415 .85566 .85716 31 59 .85716 .85866 .86014 .86162 .88310 .86456 . 86602 30 GO .86602 .86747 .86892 .87035 .87178 .87320 .87462 29 Gl .87462 .87602 . 87742 .87881 .88020 .88157 .88294 28 G2 .88294 .88430 .88566 .88701 . 88835 .88968 .89100 27 03 .89100 .89232 .89363 .89493 .89622 .89751 .89879 £6 64 .89879 .90006 .90132 .90258 .90383 .90507 . 90630 25 65 .90630 .90753 .90875 .90996 .91116 .91235 .91354 24 G6 .91854 .91472 .91589 .91706 .91821 .91936 .92050 23 67 .92050 .92163 .92276 .92388 .92498 .92609 .92718 22 G8 .92718 . 92827 .92934 .93041 .93148 .93253 . 93358 21 G9 .93358 .93461 .93565 .93667 .93768 .93869 . 93969 20 70 .93969 .94068 .94166 .94264 .94360 .94456 .94551 19 71 .94551 .94646 .94789 .94832 .94924 .95015 .95105 18 72 .95105 .95195 .95283 .95371 * .95458 .95545 .95630 17 73 .95630 .95715 .95799 .95882 .95964 .96045 .98126 16 74 .96126 .96205 .96284 . 96363 .96440 .96516 . 98592 15 75 .96592 .96667 .96741 .96814 .96887 .96958 .97029 14 76 .97029 .97099 .97168 .97237 .97304 .97371 .97437 13 77 .97437 .97502 .97566 .97623 .97692 .97753 .97814 12 78 .97814 .97874 .97934 .97992 .98050 .98106 .98162 11 79 .98162 .98217 .98272 .98325 .98378 .98429 .98480 10 80 .98480 .98530 .98580 .98628 .98676 . 98722 .98768 9 81 .98768 .98813 .98858 .98901 .98944 .98985 .99026 8 82 . 99026 .99066 .99106 .99144 .99182 .99218 .99254 7 83 .99254 .99289 .99323 .99357 .99889 .99421 .99452 6 84 .99452 .99482 .99511 .99539 .99567 .99593 .99619 5 85 .99619 .99644 .99668 .99691 .99714 . 99735 .99756 4 86 .99756 .99776 .99795 99813 .99880 .99847 .99863 3 87 .99863 .99877 .99891 .99904 .99917 .99928 . 99939 2 88 .99939 .99948 .99957 .99965 .99972 .99979 .99984 1 89 .99984 .99989 .99993 .99993 .99998 .99999 1.0000 "" 60' 50' 40' 30' £0' w 0' Deg. NATURAL COSINE. 150 BUOWN & SHARPK MFG. CO. NATURAL TANGENT. Dej;. ; 0' 10' 20' 80' 40' 50' 60' ———— .00000 .00290 .00581 . 00872 .01168 .01454 .01745 89 1 .01745 .02036 .02327 .02618 .02909 .03200 .03492 88 2 .03492 .03783 .04074 .04366 .04657 .04949 .05240 87 3 .05240 .05532 .05824 .06116 .06408 .06700 .06992 86 4 .06992 .07285 .07577 .07870 .08162 .08455 .08748 85 5 .08748 .09042 .09335 .09628 .09922 .10216 .10510 84 (5 .10510 .10804 .11099 .11393 .11688 .11983 . 12278 83 7 .12278 .12573 .12869 .13165 .13461 . 13757 .14054 82 8 .14054 .14350 .14647 . 14945 .15242 .15540 .15838 81 9 .15838 .16136 . 16435 . 16734 .17033 .17332 .17632 80 10 .17632 . 17932 . 18233 .18533 .18834 .19186 .19438 79 11 . 19438 . 19740 .20042 .20345 .20648 .20951 .21255 78 12 .21255 .21559 .21864 .22169 .22474 .22780 .23086 77 13 .23086 .23393 .23700 .24007 .24315 .24624 .24932 76 14 .24932 .25242 .25551 .25861 .26172 .26483 .26794 75 15 .26794 .27106 .27419 .27732 .28046 .28360 .28674 74 16 .28674 . 28989 .29305 .29621 .29938 .30255 .30578 73 17 .30573 .30891 .31210 .31529 .31850 .32170 .32492 72 18 .32492 .32813 .33136 .33459 .33783 .34107 .34432 71 19 .34482 . 34758 .35084 .35411 .85789 .36067 .36397 70 20 .36397 .3672(5 .37057 .37388 .37720 . 38053 .38386 69 21 .38386 .38720 .39055 .39391 .39727 .40064 .40402 68 22 .40402 .40741 .41080 .41421 .41762 .42104 .42447 67 23 .42447 .42791 .43135 .43481 .43827 .44174 .44522 60 24 .44522 .44871 .45221 . 45572 .45924 .46277 .46630 65 25 .46630 . 46985 .47341 .47697 .48055 .48413 .48773 64 26 .48773 .49133 .49495 .49858 .50221 .50586 .50952 63 27 .50952 .51319 .51687 . 52056 .52427 . 52798 .53170 62 28 .53170 .53544 .53919 . 54295 . 54672 .55051 .55430 61 29 .55480 .55811 .56195 .56577 .56961 . 57847 .57735 60 30 .57785 .58123 .58513 .58904 .59297 . 59690 . 60086 59 31 .60086 .60482 .60880 .61280 .61680 .62083 . 62486 58 32 . 62486 .62892 .63298 .63707 .64116 .64528 .64940 57 33 .64940 .65355 .65771 .66188 .66607 .67028 .67450 5,5 84 .67450 .67874 .68300 .68728 .69157 . 69588 .70020 55 35 .70020 .70455 .70891 . 71329 .71769 .72210 .72654 5-1 36 .72654 .73099 .73546 .73996 .74447 .74900 .75355 55 37 .75355 .75812 .76271 . 76732 . 77195 . 77661 .78128 52 38 .78128 .78598 .79069 .79543 .80019 .80497 . 80978 51 31) .80978 .81461 .81946 . 82433 .82923 .83415 .83910 50 40 .83910 .84406 .84906 . 85408 .85912 .86419 . 86928 40 41 .86928 .87440 . 87955 .88472 . 88992 .89515 .90040 48 42 .90040 .90568 .91099 .91633 .921(59 .92709 .93251 47 4:5 .93251 .93796 .94845 .94896 .95450 .96008 .96568 46 44 .96568 ! .97132 .97699 .982(59 .118843 .99419 1.0000 45 * CO' 50' 40' 30' 20' 10' 0' Deg. NATURAL COTANGENT. PROVIDENCE, R. I. 151 NATURAL TANGENT. Beg. o' 10' 20' 30' 40' 50' 60 45 1.0000 1.0058 1.0117 1.0176 1.0235 1.0295 1.0355 44 46 1.0355 1.0415 1.0476 1.0537 1.0599 1.0661 1.0723 43 47 1.0723 1.0786 1.0849 1.0913 1.0977 1.1041 1.1106 42 48 1.1106 i.-;m 1 . 1236 1 . 1302 1.1369 1.1436 1.1503 41 49 1.1503 1.1571 1 1639 1.1708 1.1777 1.1847 1.1917 40 50 1.1917 1.1988 1.2059 1.2131 1.2203 1.2275 1 2349 39 51 1.2349 1.2422 1.2496 1.2571 1.2647 1.2723 1.2799 38 52 1.2799 1.2876 1.2954 1.3032 1.3111 1.3190 1.3270 37 53 1.3270 1.3351 1.3432 1.3514 1.3596 1.3680 1.3763 36 54 1.3763 1.3848 1.3933 1.4019 1.4106 1.4193 1.4281 35 55 1.4281 1.4370 1.4459 1.4550 1.4641 1.4733 1.4825 34 56 1 4825 1.4919 1.5013 1.5108 1.5204 1.5301 1.5398 33 57 1.5398 1.5497 1.5596 1.5696 1.5798 1.5900 1.6003 32 58 1.6003 1.6107 1.6212 1.6318 1.6425 1.6533 1.6642 31 59 1.6642 1.6753 1.6864 1.6976 1 . 7090 1.7204 1.7320 30 60 1.7320 1.7437 1.7555 1 . 7674 1.7795 1.7917 1.8040 29 61 1.8040 1.8164 1.8290 1.8417 1.8546 1.8676 1.8807 28 62 1.8807 .1.8940 1.9074 1.9209 1.9347 1.9485 1.9626 27 63 1.9626 1.9768 1.9911 2.0056 2.0203 2.0352 2.0503 26 64 2.0503 2.0655 2.0809 2.0965 2.1123 2.1283 2.1445 25 65 2.1445 2.1609 2.1774 2.1943 2.2113 2.2285 2.2460 24 66 2.2460 2.2637 2.2816 2.2998 2.3182 2.3369 2.3558 23 67 2.3558 2.3750 2.3944 2.4142 2.4342 2.4545 2.4750 22 68 2.4750 2.4959 2.5171 2.5386 2.5604 2.5826 2.6050 21 69 2.6050 2.6279 2.6510 2.6746 2.6985 2.7228 2.7474 20 70 2.7474 2.7725 2.7980 2.8239 2.8502 2.8770 2.9042 19 71 2.9042 2.9318 2.9600 2.9886 3.0178 3.0474 3.0776 18 72 3.0776 3.1084 3.1397 3.1715 3.2040 3.2371 3.2708 17 73 3.2708 3.3052 3.3402 3.3759 3.4123 3.4495 3.4874 16 74 3.4874 3.5260 3.5655 3.6058 3.6470 3.6890 3.7320 15 75 3.7320 3.7759 3.8208 3.8607 3.9136 3.9616 4.0107 14 76 4.0107 4.0610 4.1125 4.1653 4.2193 4.2747 4.3314 13 77 4.3B14 4.3896 4.4494 4.5107 4.5736 4.6382 4.7046 12 78 4.7046 4.7728 4.8430 4.9151 4.9894 5.0653 5.1445 11 79 5.1445 5.2256 5.3092 5.3955 5.4845 5.5763 5.6712 10 80 5.6712 5.7693 5.8708 5.9757 6.0844 6.1970 6.3137 9 81 6.3137 6.4348 6.5605 6.6911 6.8269 6.9682 7.1153 8 82 7.1153 7.2687 7.4287 7.5957 7.7703 7.9530 8.1443 7 83 S.1443 8.3449 8.5555 8.7768 9.0098 9.2553 9.5143 6 84 9.5143 9.7881 10.078 10.385 10.711 11.059 11.430 5 85 11.430 11.826 12.250 12.706 13.196 13.726 14.300 4 86 14.300 14.924 15.604 16.349 17.169 18.075 19.081 3 87 19.081 20.205 21.470 22.904 24.541 26.431 28.636 2 88 28.636 31.241 34.367 38.188 42.964 49.103 57.290 1 89 57.290 68.750 85.939 114.58 171.88 343 . 77 00 60' 50' l 40' 30' 20' 10' 0' Deg. NATURAL COTANGENT. 152 BROWN J: SHARPE MFG. CO. NATURAL SECANT. 1 Beg. 0' 10' 20' 30' 4/ 50 60' 1.0000 1 . 0000 1.0000 1 . 0000 1 . 0000 1.0001 1 . 0001 89 1 1.000L 1 . 0002 1 . 0002 1.0003 1.0001 1 . 0005 l.OOOo 88 | 2 1 . 0006 1.0007 1 0008 1 . 0009 1.0010 1.0012 1.0013 87 3 1.0013 1 . 0015 1.0016 1.0018 1.0020 1 . 0022 1.0024 86 i 4 1 . 0024 1 . 002a 1.0028 1.0030 1.0033 1.0035 1.0033 85 5 1.0038 1 . 0040 1 . 0043 1.0016 1.0049 1.0052 1 . 0055 84 | 6 ; 1.0055 1 . 0058 1.0031 1.0064 1.0068 1.0071 1.0075 83 7 i 1.0075 1.0078 1 . 0082 1 . 0086 1.0090 1 . 0094 1 . 0098 , 82 8 1.0098 1.0102 1.0103 1.0111 1.0115 1.0120 1.0124 1 81 9 1.0124 1.0129 1.0134 1 . 0139 1.0144 1 . 0149 1.0154 i 80 10 1.0154 1.0159 1.0164 1.0170 1.0175 1.0181 1.0187 ; 79 11 1.0187 1 . 0192 1.0198 1.0204 1.0210 1.02L7 1 . 0223 1 78 12 1 . 0223 1.0229 1 . 02:36 1.0242 1 . 0249 1 . 0256 1.0263 77 u 1 . 0263 1.0269 1.0277 1 . 0284 1.0291 1 . 0298 1.0303 1 76 14 1.0303 1.0313 1.0321 1 . 0329 1.0336 1 . 0344 1.0352 75 15 1.0352 1.0380 1.0369 1.0377 1 . 0385 1 . 0394 1.0402 74 1(3 1.0402 1 . 0411 1 . 0420 1.0429 1.0438 1.0147 1 . 0456 73 17 1 . 0456 1.0436 1 . 0475 1.0485 1 . 0494 1.0504 1.0514 1 72 18 1.0514 1.0524 1 . 0534 1 . 0544 1 . 0555 1.0565 1.0576 71 19 1.0576 1.0586 1.0597 1.0:08 1.0319 1 0630 1.0641 70 20 1.0641 1 . 0853 1.06C4 1.0.'i76 i.o::87 1 . 0699 1.0711 C9 21 : i.07ii 1.0723 1 . 0735 1 . 0747 1.0760 1.0772 1.0785 68 23 1.0785 1.0798 1.0810 1.0823 1.0837 1 . 0850 1.0833 67 23 1 . 0863 1.0877 1 . 0890 1.0904 1.0918 1.0932 1.0946 66 1 24 1.0946 1.0960 1 . 0974 1.0989 1.1004 1.1018 1.1033 65 25 1 . 1033 1.1048 1.1063 1.1079 1.1094 1.1110 1.1126 64 26 1.1126 1.1141 1.1157 1.1174 1.1190 1.1206 1.1223 63 27 1.1223 1.1239 1.1256 1.1273 1.1290 1.1308 1.1125 62 28 1.1325 1 . 1343 1.1361 1.1378 1.1396 1.1415 1.1433 61 29 1.1433 1.1452 1.1470 ' 1 . 1489 1.1508 1 . 1527 1 . 1547 60 , 30 1.1547 1 . 1566 1.1583 1.1605 1 . 1625 1.1646 1.1665 59 1 3L 1.1666 1.1686 1.1707 1.1723 1.1749 1.1770 1.1791 58 32 1.1791 1.1813 1.1835 1.1856 1.1878 1.1901 1.1923 57 33 1.1923 1.1946 1 . 1969 1.1992 1.2015 1.20-8 1 . 2062 56 34 1.2032 1.2085 1.2109 1.2134 1.2158 1.2182 1.2207 55 35 1.2207 1.2232 1.2257 1.2283 1.2308 1.2334 1.2360 54 | 3) 1.2360 1.2386 1.2413 1.2440 1 . 24 56 1.2494 1.2521 53 37 1.2521 i 1.2548 1.2576 1.2304 1.2632 1.2661 1.2690 52 | 33 1.26v 9 1 1.2719 1.2748 1.2777 1.2807 1.2837 1.2887 : 51 39 1.2867 1.2898 1.2328 1.2959 1.2990 1.3022 1.3054 50 49 1.3054 1.3086 1.3118 1.3150 1=3183 1.3216 1.3250 49 41 1.32-30 1.3283 1.3317 i 1.3351 : 1.3336 1.3421 1.3456 48 42 1 . 3456 1.3491 1.3527 1.3563 1.3599 1.3636 1.3673 47 43 1 3673 1.3710 1.3748 ; 1.3785 ; 1.3824 1.3862 1.3901 46 44 1.3901 1.3940 1.39S0 1.4020 1.40J0 1.4101 1.4142 45 1 GO' 50' 10' 30' 20' 10' Deg. ' NATURAL COSECANT. PROVIDENCE, R. I. 153 NATURAL SECANT. 1 Beg. 0' 10' 20' 30' 40' 50' 60' : 45 1.4142 1.4183 1.422j 1.4267 1.4309 1.4352 1.4395 44 46 1.4395 1.4439 1.4483 1.4527 1.4572 1.4617 1.4662 43 47 1.4662 1.4708 1.4755 1.4801 1.4849 1.4896 1.4944 42 48 1.4944 1.4993 1 . 5042 1.5091 1.5141 1.5191 1.5242 41 49 1.5242 1.5293 1.5345 1.5397 1.5450 1 . 5503 1.5557 40 50 1.5557 1.5611 1.5666 1.5721 1.5777 1.5833 1.5890 39 51 1.5890 1.5947 1.6005 1.(3063 1.6122 1.6182 1.6242 38 52 1.6242 1.6303 1.6364 1.6426 1.6489 1.6552 1.6616 37 53 1.6616 1.6680 1 . 6745 1.6811 1.6878 1.6945 1.7013 36 54 1.7013 1.7081 1.7150 1.7220 1.7291 1.7362 1.7434 | 35 55 1.7434 1.7507 1.7580 1.7655 1.7730 1.7806 1.7882 ! 34 56 1.7882 1.7960 1.8038 1.8118 1.8198 1.8278 1.8360 1 33 57 1.8360 1.8443 1.8527 1.8611 1.8697 1.8783 1.8870 1 32 58 1.8870 1.8959 1.9048 1.9138 1.9230 1.9322 1.9416 31 59 1.9416 1.9510 1.9603 1.9702 1.9800 1.9899 2.0000 30 60 2.0000 2.0101 2.0203 2 0307 2.0412 2.0519 2.0826 29 61 ! 2.0623 2.0735 2.0845 2.0957 2.1070 2.1184 2.1300 ; 28 62 ! 2.1300 2.1417 2.1536 2.1656 2.1778 2.1901 2.2026 27 63 2.2026 2.2153 2.2281 2.2411 2.2543 2.2376 2.2811 26 64 2.2811 2.2948 2.3087 2.3228 2.3370 2.3515 2.3662 25 65 2.3662 2.3810 2.3961 2.4114 2.4239 2.4426 2.4585 24 66 2.4585 2.4747 2.4911 2.5078 2.5247 2.5418 2.5593 ! 23 67 2.5593 2.5769 2.5949 2.6131 2.6316 2.6503 2.6694 22 68 2.6694 2.6883 2.7085 2.7285 2.7488 2.7694 2.7904 '. 21 69 2.7904 2.8117 2.8334 2.8554 2.8778 2.9006 2.9238 20 70 2.9238 2.9473 2.9713 2.9957 3.0205 3.0458 3.0715 19 71 3.0715 3.0977 3.1243 3.1515 3.1791 3.2073 3.2360 18 i 72 3.2360 3.2853 3.2951 3.3255 3.3564 3.3880 3.4203 17 1 73 3.4203 3.4531 3.4867 3.5209 3.5558 3.5915 3.6279 16 74 3.6279 3.6651 3.7031 3.7419 3.7816 3 8222 3.8637 15 75 3.8337 3.9061 3.9495 3.9939 4.0393 4.0859 4.1335 14 76 4.1335 4.1823 4.2323 4.2836 4.3362 4.3901 4.4454 13 77 4.4454 4.5021 4.5604 4.6202 4.6816 4.7448 4.8097 12 78 '4.8097 4.8764 4.9451 5 0158 5.0886 5.1635 5.2408 11 79 5.2408 5.3204 5.4026 5.4874 5.5749 5.6653 5.7587 10 80 5.7587 5.8553 5.9553 6.0588 6.16C0 6.2771 6.3924 9 81 6.3924 6.5120 6.6363 6.7654 6.8997 7.0396 7.1852 8 82 7.1852 7.3371 7.4957 7.6612 7.8344 8.0156 8.2055 17 i 83 8.2055 8.4046 8.6137 8.8336 9.0651 9.3091 9.5667 6 84 9.5667 9.8391 10.127 10.433 10.758 11.104 11.473 5 85 11.473 11.868 12.291 12.745 13.234 13.763 14.335 4 86 14.335 14.957 15.633 16.380 17.198 18.102 19.107 3 87 19.107 20.230 21.493 22.925 24.562 26.450 28 653 2 88 28.653 31.257 34.382 38.201 42 975 49.114 57.298 1 89 57.298 68.757 85.945 114.59 171.88 343.77 00 GO' 50 43' 80' 20' 10' 0' Deg. NATURAL COSECANT. :.:.4 SHABPE MFG. :o. TABLE OF DECIMAL EQIIVALEXTS : IHILIMZTEES AXI' FRACTIONS OF MILLrAEETEP.-. inm. Inehi s. 1 . 5 0" _a_. •f 3 - 5 ' 50" 5 50 :_ « ■ ' ' 9 50 1 ' ' 1 1 ■' ' ■ : . ' i 1 3_- * Ji I - 1 5 - ' 1 6- | " - LA 50 1 S S " go " ' 'f " : - ' ■: t. • r |S 5 .0007! .00157 236 .00315 .00394 .('0551 .001 .007 .--7-7 . 00945 .01024 .01102 .01181 .012 01339 .01417 014 . 01575 . -1654 .017;: 2 .01811 .01890 . 01 ■ mm. Inches. 2047 02126 02205 02283 02362 02441 02520 02598 02677 02756 02835 02913 02992 03071 03150 03228 03307 033-6 03465 03543 03622 03701 03780 03858 03937 •: : ■ a : ' Li- : I s ■: 30 3J_ 3A 3 5 - 6- n a _ I 8.- " " 50 40- " " 1 2- 50" 43- ." 50" . S ti- 5 4 S- " 4 ■ 1 = mrn. Inches. 2 = .07874 3 = .11811 4 = .15748 5= .19685 6= .2362- i = .2755V' 8 = .31496 ,' = . 35433 10= .39370 11 = 433-7 12= .47244 13= .51181 14= .55118 15= .59055 16= .62992 17= I 18= ">•:' 19= .74- 20= .7874C 21 = .-2677 22= 34 S14 23 = .90551 24= .94488 25= -425 26 = 1 .02:; . 10 mm. = l Centimeter=0.3937 in<: es ... =1 Decimeter =c 2 10 dm. =1 Meter =39.37 '2~>A mm.=l Enirli>h Inch. INDEX. A. PACK Abbreviations of Parts of Teeth and Gears ... 4 Addendum 2 Angle, How to Lay OS. an SB, 107 Angle Increment 106 Angle of Edge 102 Angle of Face 104 Angle of Pressure 137 Angle of Spiral 113 Angular Velocity 2 Annular Gears - 32, 139 Arc of Action 138 B. • Base Circle 11 Base of Epicycloidal System 25 Base of Internal Gears 130 Bevel Gear Blanks 34 Bevel Gear Cutting on B. & S. Automatic Gear Cutter 52 Bevel Gear Angles by Diagram 36 Bevel Gear Angles by Calculation 102, 106 Bevel Gear, Form of Teeth of 41 Bevel Gear, Whole Diameter of 36, 104 C. Centers, Line of 2 Circular Pitch 4 Classification of Gearing 5 Clearance at Bottom of Space G Clearance in Pattern Gears : 8 Condition of Constant Velocity Katio 2 Contact, Arc of '. 138 Continued Fractions 132 Coppering Solution 85 Cutters, How to Order 83 Cutters, Table of Epicycloidal 8t 15G INDEX. PAGE. Cutters, Table of Involute 82 Cutters, Table of Speeds for 81 Cutting Bevel Gears on B. & S. Automatic Gear Cutter 52 Cutting Spiral Gears on a Universal Milling Machine 122 D. Decimal Equivalents, Tables of 143, 154 Diameter Increment 101 Diameter of Pitch Circle G Diameter Pitch 5 Diametral Pitch 17 Distance between Centers 8 E. Elements of Gear Teeth 5 Epicycloidal Gears, with more and. less than 15 Teeth 30 Epicycloidal Gears, with 15 Teeth 25 Epi cycl oid al Rack 27 F. Face, Width of Spur Gear 80 Flanks of Teeth in Low-numbered. Pinions 20 G. Gear Cutters, How to Order 83 Gear Patterns ." 8 Gearing Classified 5 Gears, Bevel , 31, 41, 102 Gears, Epicycloidal 25 Gears, Involute 9 Gears, Spiral 109, 122 Gears, Worm G3 II. Herring-bone Gears 130 I. Increment, Angle 100 Increment, Diameter 104 Interchangeable Gears 24 Internal or Annular Gears 32, 139 Involute Gears, 30 Teeth and over 9 Involute Gears, with Less than 30 Teeth 20 Involute Rack 12 INDEX. 157 L PAGE. Lead of a Worm 62 Limiting Numbers of Teeth in Internal Gears 32 Line of Centers 2 Line of Pressure 12, 137 Linear Velocity 1 M. Machine, B. & S., for Cutting Bevel Gears 52 N. Normal 11G Normal Helix 116 Normal Pitch 116 0. Original Cylinders 1 P. Pattern Gears 8 Pitch Circle 3 Pitch, Circular or Linear -4 Pitch, a Diameter 6 Pitch, Diametral 17 Pitch, Normal 116 Pitch of Spirals 112 Polygons, Calculations for Diameters of ,. 05 11 Rack 12 Pack for Epicycloidal Gears •. 27 Pack for Involute Gears 12 Pack for Spiral Gears 121 Relative Angular Velocity 2 Rolling Contact of Pitch Circle 3 S. Screw Gearing 109, 130 Single-Curve Teeth 9 Speed of Gear Cutters 81 158 INDEX. PAGE. Spiral Gearing 109. 122 Standard Templets ". 27 Strength of Gears 142 Table of Decimal Equivalents 143, 154 Table of Sines, etc 148, 153 Table of Speeds for Gear Cutters 81 Table of Tooth Parts 144, 147 Velocity, Angular 2 Velocity, Linear 1 Velocity. Relative 2 W. Wear of Teeth SO, 129 Worm Gears 63 PRACTICAL TREATISE ON GEARING BROWN &- SHARPE MFG. CO. Providence, R. I., U. S. A. 'WT&&* / •»* -V V '>> O. OCT V > V ' o o. A - *>/ \ ^, ■■' - O ;** ^ . ^ . ^ V*. \ -*■ V- yj- V *> V q> ^ $ ."fe, * 8 « '■ *P o .