TJ im .L8 flu nil UillnR I It I i itiUi(liiliiuiiiiit!i)ii]i!iiiiiiiiitil Class Book IS.'^:C lA Copyilghtlj^. COPYRIGHT DEPOSIT. AMERICAN MACHINIST GEAR BOOK American Machinist Gear Book Simplified Tables and Formulas for Designing, and Pra6lical Points in Cutting All Commercial Types of Gears By Charles H. Logue Associate Editor American Machinist Formerly Mechanical Engineer R. D. Nuttall Co. I9IO Published by the American Machinist McGraw-Hill Book Company, Sole Selling Agents 239 West Thirty-ninth Street, New York 6 Bouverie Street, London, E. C. Berlin, N. W. 7, Unter Den Linden 71 Copyright, 1910, by the American Machinist -^ ^^ \- Afnerican Machinist, New York, U. S. A. CI. A 3685;;? PREFACE This book has been written to fill a pressing want; to give practical data for cutting, molding, and designing all commercial types, and to present these subjects in the plainest possible manner by the use of simple rules, diagrams, and tables arranged for ready reference. In other words, to make it a book for "the man behind the machine," who, when he desires information on a subject, wants it accurate and wants it quick, without dropping his work to make a general study of the subject. At the same time a general outline of the underlying principles is given for the student, who desires to know not only how it is made, but what is made. Controversies and doubtful theories are avoided. Tables and formulas commonly accepted are given without com- ment. A great deal of this matter has previously been published in the col- umns of the American Machinist, but is revised to make the subject more complete. Credit is given in all cases when the author is known; there may be cases however, where record of the original source of information has been lost, as is often the case when data are in daily use and the authority is ob- scure. Obviously in such cases the author's name cannot be given. Charles H. Logue. May I, 1910. CONTENTS SKCTION PAGE I. Tooth Parts i II. Spur, Gear Calculations . . . ./ 45 III. Speeds and Powers .*.... 49 IV. Gear Proportions and Details of Design no V. Bevel Gears 139 VI. Worm Gears 15^ VII. Helical and Herringbone Gears 205 VIII. Spiral Gears 221 IX. Skew Bevel Gears 241 X. Intermittent Gears . 246 XI. Elliptical Gears 257 XII. Epicyclic Gear Trains 264 XIII. Friction Gears 280 XIV. Odd Gearing 300 XV. Pattern Work and Molding 309 XVI. Suggestions for Ordering Gears 329 XVII. Practical Points in Gear Cutting 342 Index . 345 SECTION I Tooth gearing furnishes an efficient and simple means of transmitting power at a constant speed ratio, making it possible to time the movements of machine parts positively. Owing to refinements in the tooth form, the introduction of generating machines and facilities to cut gears of the largest size accurately, loads may now be transmitted at speeds which a comparatively short time ago were considered prohibitive. The need for gears that would answer the exacting requirements of automobile construction has done much to bring this about. Designing automobile gears, however, is a case of fitting the gears to the machine; it is a question of securing material that will stand the strain; the gear dimensions are practically self determined ; however, this is not the only kind of gearing that has been designed after this fashion. We have an excellent formula for the strength of gear teeth, but it contains a variable factor — the allowance to be made on account of impact — concerning which very little is known. The most important question of all, that of wear^ has heretofore been left practically untouched. The best data obtainable has been given. Few records have been kept of actual performances, and nothing whatever has been found relative to the abrasion of different materials in tooth contact. The various ways in which gears are mounted is responsible for the apparent contradiction of what few data are at hand, as a gear driver which is entirely satisfactory on one machine will be worthless on another at the same load and the same speed. The circumferential speed that may be allowed for gears of different types is another neglected factor, and last, but not least, what do we know of gear efficiency? In fact the most important information relative to gear trans- missions has been entirely a matter of guesswork. It is hardly to be assumed that this will ever be reduced to an exact equation, but there should be some basis from which to form our conclusions. Gears may be roughly divided into three general classes: Gears connecting parallel shafts; gears connecting shafts at any angle in the same plane; gears connecting shafts at any angle not in the same plane. In the first class are included spur, helical, herringbone, and internal gears. The second class covers bevel gears only. The third class includes worm, spiral and skew bevel gears. 2 AMERICAN MACHINIST GEAR BOOK Gears connecting parallel shafts are the most efficient, and from a point of efficiency may be graded into herringbone, internal, spur, and, lastly, helical gears. The efficiency of the second class, bevel gears, varies with the shaft angle, increasing as the angle approaches zero. As a general thing the third class should be avoided wherever possible, although worm gears have their peculiar uses; for instance, where a quiet, self-locking drive is required without reference to the loss of power. Spiral gears are employed where the load is light and the gear ratio is low, say under lo to i; worm gears are often emplgyed for low ratios down to I to I, but are extremely difficult to cut and therefore expensive. When the worm is made much coarser than quadruple thread there is generally trouble. Skew bevel gears are used where the distance between the shafts is not great enough to employ worm or spiral gears. Skew bevel gears are simpler, and easier to cut than has been generally supposed, but are still things to avoid. These three general classifications are commercially subdivided as follows: KIND RELATION OF AXES PITCH SURFACES NOTES Spur Parallel Cylinders Bevel Intersecting at any angle in the same plane Cones Helical Parallel Cylinders Herringbone Parallel Cylinders Double Helical Spiral At any angle not in same plane Cylinders For small ratios Worm At any angle not in same plane Cylinders For large ratios Skew Bevel At any angle not in same plane Hyperboloids Where shaft centers Internal Parallel or at any angle in are close same plane Cylinders or cones With teeth cut on in- Elliptical Parallel or at any angle in ner surface same plane Elliptical cylinders or elliptical cones Irregular Parallel or at right angles in same plane Any Irregular pitch lines Intermittent Parallel or at right angles in To give driven gear a same plane Cylinders period, or periods of rest during one rev- olution of driver Friction Parallel or at any angle in Contact surfaces rep- same plane Cylinders or cones resenting the pitch surfaces of a toothed gear Commercial Classification of Gears TOOTH PARTS TOOTH PARTS I Lxed axes are connected by imaginary pitch surfaces, which roll upon each other and transmit uniform motion without slipping. The object in toothed gearing is to provide these imaginary surfaces with teeth, the action of which will make the uniform motion of the pitch surfaces positive; not depending upon friction produced by direct pressure as in friction gears, which are an ex- cellent representation of pitdi surfaces. If the teeth are not so formed that this condition is fulfilled the movement of the driven gear will be made up of accelerations and retardations which will not only absorb a large percentage of the power but disintegrate the material of which the gear is constructed and seriously affect the operation of the machine. Tool marks on planer and bor- ing mill work corresponding to the teeth in the driven gear may be traced directly to this. There is but one form of tooth in common use — the involute; the cycloidal form has practically disappeared. For a thorough understanding of tooth con- tact, however, it must be included. FIG. GENERATING THE CYCLOIDAL TOOTH. CYCLOIDAL Generated by rolling a circle above and below the pitch circle of gear; a point on its circumference describing the tooth outline. See Fig. i. INVOLUTE Generated by rolling a straight line on the base circle of gear, any point on this line describing the involute curve. See Fig. 2. The same result is obtained by unwinding a string from the base circle.- See Fig. 3. OCTOID Conjugated by a tool representing a flat sided crown gear tooth; a modi- fication of the involute. Used only on bevel gear generating machines. See Fig. 33- THE CYCLOID An illustration of the manner in which the cycloidal tooth is generated is illustrated by Fig. i; the wheel A being the pitch circle and B and B' the describing circles which are of the same diameter. The point C will describe the face of the tooth as the circle B is rolled on the pitch circle, and the. flank of the tooth as the circle B' is rolled inside the pitch circle. In other words, AMERICAN MACHINIST GEAR BOOK the exterior cycloid is formed by rolling the describing circle on the outside of the pitch circle, this exterior cycloid engaging the interior cycloid, which is formed by rolling the describing circle on the inside of the pitch circle. The describing circle is commonly made equal to the pitch radius of a 15- tooth pinion of the same pitch as the gear being drawn. According to J. Howard Cromwell: "Roomer, a celebrated Danish astrono- mer, is said to have been the first to demonstrate the value of these curves for tooth profiles." But De la Hire is credited with demonstrating that it was Base Circle FIG. 2. THE INVOLUTE GEN- ERATED BY A STRAIGHT LINE. FIG. 3. THE INVOLUTE GEN- ERATED BY A STRING. possible to form both the face and flanks of any number of gears with the same describing circle. The pressure angle of the teeth is not constant in one direction, but varies from zero at the pitch point to about 22 degrees at the end of the contact with a rack tooth. The contact points of all the teeth engaged intersect the line of action, which is a segment of the describing circle drawn from the line of centers. See Fig. 34. Wilfred Lewis has said: "The practical consideration of cost demands the formation of gear teeth upon some interchangeable system. "The cycloidal system cannot compete with the involute, because its cutters are formed with greater difficulty and less accuracy, and a further expense is entailed by the necessity for more accurate center distances. Cycloidal teeth must not only be accurately spaced and shaped but their wheel centers must be fixed with equal care to obtain satisfactory results. Cut gears are not only more expensive in this system, but also when patterns are made for castings TOOTH PARTS 5 the double curved faces require far more time and care in chiseling. An involute tooth can be shaped with a straight-edged tool, such as a chisel or a plane, while the flanks of cycloidal teeth require special tools, approximating in cur- vature the outline desired. It is, therefore, hardly necessary to argue any further against the cycloidal gear teeth, which have been declining in popu- larity for many years, and the question now to be considered is the angle of obliquity most desirable for interchangeable involute teeth." In this same connection George B. Grant, of the Philadelphia Gear Works, wrote: "There is no more need of two difTerent kinds of tooth curves for gears of the same pitch than there is need of two different threads for standard screws, or of two different coins of the same value, and the cycloidal tooth would never be missed if it were dropped altogether. But it was first in the field, is simple in theory, is easily drawn, has the recommendation of many well-meaning teachers and holds its position by means of ' human inertia,' or the natural reluctance of the average human mind to adopt a change, partic- ularly a change for the better. " THE INVOLUTE The pressure on the teeth of involute gears is constantly in the direction of the line of action. The line of action is drawn through the pitch point at an angle from the horizontal equal to the angle of obliquity. All contact between the teeth is along this line. The base circle is drawn inside the pitch circle and tangent to the line of action. The action of a pair of involute gears is the same as if their base circles were connected by a cross belt; the point at which the belt crosses being the pitch point P; the straight portion of the belt not touching the base circles represent- ing the lines of action. See Fig. 4. At the pitch point the velocities of both gears are equal. To show^ that the involute is but a limiting case of the cy- cloidal system, consider the describing line as a curtT of infinite radius, which is rolled upon the pitch circle. As this describing line cannot be rolled inside the pitch circle to form the interior cycloid that wdll engage the exterior cycloid formed by rolling the describing line outside the pitch circle of the mating gear, the pitch circles upon which the cycloids are formed must be separated so as to allow the exterior cycloids to engage each other. The original pitch circles becoming the base circles. See Fig. 5. The distance between the pitch circle and the base circle, and therefore, the angle of obliquity, depends upon the proportionate length of tooth to be used and the smallest number of teeth in the system. To obtain contact for the full length of the tooth, the base circle must fall below the lowest point reached AMERICAN MACHINIST GEAR BOOK by the teeth of the mating gear. Below the base line there can be no contact of any value. There is such a difference between the largest possible gear and the rack that it is at first a little difficult to see the application of the methods used to describe the involute to the rack tooth. As the diameter of the gear is in- creased, the radii used to draw the Pitch Circle FIG. 4. THE ACTION OF INVOLUTE TEETH ILLUSTRATED BY A CROSSED BELT CON- NECTING THE BASE CIRCLES. FIG. 5. SEPARATING THE PITCH CIRCLES TO ALLOW THE EXTERIOR CYCLOIDS TO ENGAGE. involute curve are lengthened, and the teeth have less curvature. Until finally, when the radius of the pitch circle is of infinite length, the tooth radii are also infinite, and the involute is a straight line, drawn at right angles to the line of action. The theoretical rack tooth, therefore, has perfectly flat sides, each side being inclined toward the center of the tooth to an angle equaling the angle of ob- liquity. See Fig. 6. TOOTH PARTS ORIGIN OF THE INVOLUTE TOOTH The origin of the involute curve as applied to the teeth of gears is credited to De la Hire, a French scientist, a complete description and explanation of its use being published about 1694 in Paris. The first English translation of this work was published in London in 1696 by Mandy.* Professor Robinson, of Edinburgh, later describes this theory, references being made to his work in ''An Essay on Teeth of Wheels," by Robertson Buchannan, edited by Peter Nicholson and published in 1808. In this essay the involute as applied to the teeth of gears is fully described. Fig. 7 being a copy of a cut used therein for illustration. That the principal ad- vantage of the involute system was then well understood will be showTi in the following paragraph, referring to Fig. 7 : THE INVOLUTE RACK TOOTH. FIG. 7. ACTION OF THE INVOLUTE TOOTH. "It is obvious that these teeth will work both before and after passing the line of centers, they will work with equal truth, whether pitched deep or shal- low, a quality peculiar to them and of very great importance." The theory of the involute gear tooth is also described by Sir. David Brew- ster, Dr. Thomas Young, Mr. Thomas Reid and others. Professor Robert Willis, gives a very complete description of this form of tooth in his "Principles of Mechanism," 1841. Up to this period the involute tooth was not seriously considered, the cyclodial being the favorite. The in- * However, the origin of the involute gear tooth is surrounded by mystery, no two authorities agreeing upon the subject. According to Robert Willis, in his "Principles of Mechanism," the involute was first suggested for this purpose by Euler, in his second paper on the Teeth of Wheels. N.C. Petr XI. 209. 8 AMERICAN MACHINIST GEAR BOOK /» volute tooth was objected to on account of the great thrust supposed to be put on the bearings by the oblique action of the teeth. In an 1842 edition of M. Camus' work, "A treatise on the Teeth of Wheels,'* edited by John I. Hawkins, a series of experiments with wooden models was made to demonstrate the actual thrust occasioned by different angles of ob- liquity. The result of these experiments is given as follows: ''These experiments, tried with the most scrupulous attention to every cir- cumstance that might affect their result, elicit this important fact — that the teeth of wheels in which the tangent of the surfaces in contact makes a less angle than 20 degrees with the line of centers, possess no tendency to cause a separation of their axes: consequently, there can be no strain thrown upon the bearings by such an obliquity of tooth. " J. Howard Cromwell, in his treatise on Tooth Gearing, 1901, says: "Such an obliquity as 20 degrees must, unless counteracted by an opposing force, tend to separate the axes; and, as suggested FIG. 8. THE MOLDING PROCESS. by Mr. Hawklus, this opposing force is most probably the friction between the teeth, 'which would tend to drag the axes together with as much force as that tending to separate them." That the involute system is closely connected to the cycloidal system is shown by Dr. Brewster in his reference to De la Hire's work. "De la Hire considered the involute of a circle as the last of the exterior epicycloids; which it may be proved to be, if we consider the generating straight line (see Fig. 2) as a curve of infinite radius." The 14}^ degree angle of obliquity, as proposed by Professor Robert Willis in his " Principles of Mechanics," was adopted by the Brown & Sharpe Company some forty years ago. Since that time this system has come into general use. THE MOLDING PROCESS If a gear blank made of some pliable material is forced into contact with a rack, as shown in Fig. 8, the rack tooth would conjugate teeth in the blank. It does not matter what form is given the conjugating tooth, as long as it has a regular line of action; all gears formed by it will interchange. The Bilgram spur and spiral gear generating machine operates upon this principle. See Fig. 9. The cutter A , which is a reciprocating or planing tool having the profile of a correct rack tooth — namely, a truncated, straight-sided TOOTH PARTS wedge. While this tool reciprocates, it also travels slowly to the right, the blank meanwhile turning under it, the motion being that which would exist were the tool a rack tooth and the blank a gear. During this combined move- Emery Wheel FIG. 9. ACTION OF THE TOOL IN GENERATING A TOOTH. ment the tool cuts the tooth space in the manner indicated. In the Bilgram bevel-gear machine the tool does not move sidewise, the blank being rolled upon it as a complete gear might be rolled on a stationary rack, but in the spur-gear machine this action is reversed— the blank turning on a fixed center, while the tool moves over it, as it would be turned by a moving rack. The Fellows' gear shaper is designed on the same principle, but instead of a rack tooth as a planing tool, a gear of from 12 to 60 teeth is used, the motion of cutter and blank being the same as between gears in mesh. See Imaginary Rack J \ Cutter , 1 FIG. 10. ACTION OF THE FELLOWS' GEAR CUTTER. Cutter FIG. II. GENERATION OF THE FELLOWS' GEAR-CUTTER TEETH. Fig. 10. These cutters are ground to shape after being hardened as shown in Fig. II, in which the emery wheel is shaped as the planing tool in Fig. 9. The cutter being ground taking the place of the gear. lO AMERICAN MACHINIST GEAR BOOK TO DRAW THE INVOLUTE CURVE The involute curve is constructed on the base circle as follows: Draw the pitch circle and through pitch point P, Fig. 12, draw the line of action at the required angle of obhquity. Tangent to this line draw the base circle. Divide the base circle into any number of equal spaces, i', 2', 3', 4', 5', 6', as shown in Fig. 13. From each of these points draw lines intersecting at center 0. Draw lines I'-i, 2'-2, ^'-t,, etc., tangent to base circle and at right angles with lines extending to center. Make the length of line I'-i equal to one of the divisions of base circle: line 2 '-2 equal to two divisions, line T,'-2, equal to three divisions, and so on. Then through points i, 2, 3, 4, FIG. 12. LOCATING THE BASE CIRCLE. FIG. 13. DRAWING THE INVOLUTE. 5, 6, etc., trace the involute curve. Find a convenient radius, not necessarily on base circle, from which to draw the balance of the teeth, several radii sometimes being necessary to get the ' proper curve, especially for a small number of teeth. The involute curve does not extend below the base circle. Below the base circle drawing the teeth is simply a matter of obtaining suf- ficient clearance to avoid interference with the teeth of the mating gear. SINGLE CURVE TEETH This method of drawing gear teeth should be used only when the gear is to be pictured, not for templets. It is approximately correct only for 143^ degree teeth and for 30 teeth and over, although it may always be used for the curve between the base circle and the pitch circle. Referring to Fig. 14, draw the pitch diameter and locate addendum, deden- dum, and tooth spaces. With a radius of one half the radius of the pitch circle draw semicircle A from the center to the pitch line with the point of dividers TOOTH PARTS II located on the center line midway between these points. Take one half of this radius or one quarter the radius of pitch circle and, with point of Tooth Curve Radius One Quarter of Pitch Radius -Pitch Diameter "Addendum or Outside Diameter FIG. 14. LAYING OUT A SINGLE CURVE TOOTH. dividers at B on pitch circle draw an arc cutting semicircle A at point C. This is the center for the first tooth curve and locates the base circle for all tooth arcs. DEMONSTRATION OF INVOLUTE PRINCIPLE BY A MODEL An excellent Reuleaux model fgr demonstrating the principle of the involute system was loaned by Cornell University and is shown in Figs. 15 to 22. The segments in the model represent gears cf 21 and 17 teeth, about 1% inch circular pitch. The angle of obliquity is 30 degrees, which is sufficiently great to drop the base circle slightly below the bottom of the teeth in ths smallest gear of the pair, thereby securing a theoretical tooth free from under- cut or correction for interference. The teeth are carried to a point to show all the tooth action possible. The base circles upon which the involute curve is constructed are represented in this model by rims E and F, upon which is tightly wrapped the band //, which, when wound from the base circle of the gear to the base circle of the pinion, represents the line of action, also the angle of obliquity, or pressure, the 12 AMERICAN MACHINIST GEAR BOOK Tension Sprin PifcK Lffie fch Line FIGS. 15 AND 17. FIGS. 16 AND 18. INVOLUTE GEAR TOOTH MODEL. TOOTH PARTS 13 thrust of the teeth in contact being constantly in the direction of this band, which intersects all points of contact between the teeth. Referring to Fig. 2, this band represents a line rolled on the base circle, also, as is self-evident, a string unwound from its base circle, as in Fig. 3, any given point on which will describe the involute curve. The points describing teeth in the gear segments are shown on the model by the lines a, b, c and d on the band H connecting the base circles, any of which will follow the contour of both teeth engaged from top to bottom as the gears are rotated, as well as those not yet in action. In fact, the generating of the involute curve is begun just as soon as any point on the band is raised from the rim representing the base circles and continues until the movement of the gear is stopped, the tooth outhne being described by one of the points crossing the pitch line. The amount of this curve that is used above and below the pitch line depends upon the proportionate length ^of the tooth. The location of points a, b, c, d on band H have no significance in the model; they are placed to correspond with the location of the teeth, being projected on a radial line drawn from the bottom of the tooth curve. If the pitch of gear segments had been coarser these points would simply have been farther apart. In the model, the length of the tooth is restricted only by the meeting of the curves describing the opposite sides of the tooth. The tooth is carried below the pitch line the same distance as above, plus a sufhcient distance for clearance. In case the gear segments were taken off, the model would simply represent the base circles of two gears, connected by a band, the angle of which, from the horizontal, would indicate the angle of obliquity. The driven shaft is pro- pelled by the band, acting as a belt ; any point upon it will describe the proper tooth outline from the base line up, the pitch point being at the intersection of this band and the line of centers. See Fig. 4. Another important point is shown. The contact point of any two teeth en- gaged is followed by one of the points a, b, c, etc., as the gears are rotated and the band or line upon which these points are marked is always tangent to the two base circles. This illustrates the law of contact for involute teeth defined in counection with Fig. 35. The action between two involute teeth is that of two cylinders rolling and slipping upon each other. The diameter of these cylinders is constantly changing, one becoming larger and the other be- coming smaller as the teeth enter and leave contact. The impulse given the driven gear will be variable if these conditions are not fulfdled during the en- tire action. This illustrates the importance of having the tooth curves theo- retically correct. 14 AMERICAN MACHINIST GEAR BOOK FIGS. 19 AND 21. FIGS. 20 AND 22. INVOLUTE GEAR TOOTH MODEL. TOOTH PARTS 15 If the base circles E and F in the model were brought closer together it would reduce the pitch circles of both gears proportionately, also reduce the angle of obliquity, as the band or line representing the angle of obliquity must always be tangent to both base circles and pass through the pitch point, where the velocities of gears are equal. Drawing the base circles apart increases the pitch circles, also the obUquity, although the action of the teeth remains cor- rect as long as they are engaged. See Figs. 20, 21, and 22. In Fig. 15, the point b on band H is just touching the point of the tooth A as it enters into contact with tooth B. In Fig. 16, it has followed the contact, and therefore the outline of both engaging teeth to the pitch point P, and in Fig. 17, it is just leaving the point of the tooth B at the end of its contact with the tooth A. The point b will continue toward the upper tooth A until it com- pletes the involute and comes to rest on the base circle E. Figs. 18 and 19 show relation of points c and d With other teeth in the seg- ments as the gears are revolving to the left, and affords a better opporJ:unity to study the entire action of the model. In Figs. 20, 21, and 22 the centers have been widened M inch. Fig. 20 shows tooth A just entering contact with tooth B. In Fig. 21 the point b and band has followed the contact to the pitch point, w^hich is now midway between the two pitch circles as marked on segments. Fig. 22 shows the tooth A leaving contact, giving the entire range of action. This illustrates a peculiarity of the involute system, and explains how it is pos- sible to obtain correct tooth action if a pair of gears are moved from their proper centers. It will be noticed that the points a, b, c, and d follow the tooth outlines and points of contact just as accurately as when on proper centers, although the angle of obliquity is changed. The involute curve is always the same for a given base diameter and, as the pitch diameter, and not the base diameter is altered, a change in the center distance will make no change in the action of the teeth. This is illustrated by Fig. 23. There will be new pitch diameters automatically established at the pitch point as the centers are moved, simply a different portion of the involute curve is used for the tooth. It is apparent that the farther out the tooth is placed the greater will be the distance between the pitch and the base circles and the greater the angle of obHquity. With the model in this position, if the lines drawn on gear segments to represent the pitch diameter were moved out until they w^ere rolled together at the pitch point and the teeth made heavier at the pitch line to take up backlash, we would have gears of increased obliquity, which in turn, could be still farther apart as long as there were any teeth left, for with any increase of angle the length of tooth is necessarily shorter. i6 AMERICAN MACHINIST GEAR BOOK Involute Curve FIG. 23. DIAGRAM SHOWING HOW PROPER ACTION IS MAINTAINED AS THE GEAR AXES ARE SEPARATED. FIG. 24. GRAPHICAL DEMONSTRATION FOR INTERFERENCE OF SPUR GEARS. TOOTH PARTS 17 INTERFERENCE IN INVOLUTE GEARS The limitations and inaccuracies of the involute system are well explained in the following paragraphs by C. C. Stutz: While the general principles governing the interference of involute gears are well known, the following graphical demonstrations, formulas, and plotted dia- grams may place this general information in more efficient form for the use of many. Fig. 24 shows a graphical demonstra- tion of the interference of a 5-pitch, 15- tooth true involute form spur pinion and a 5-pitch, 48-tooth mating gear. The point F is the right-angled intersection of a line drawn from the center of the pinion, and at an angle of 143^ degrees with the common center line of the pinion and gear, with the line of pressure which is drawn through the point of tangency of the two pitch circles and at an angle of 143^ degrees to the common tangent at that point. If this point falls within the addendum circle of the meshing gear, the tooth of the meshing gear will interfere from this point up to its addendum circle. Therefore the tooth from this point on the curve must be corrected to overcome it. If the point F falls on or outside of the addendum circle of the meshing gear no interference will result. The point F' for fig. 25. an angle of obHquity of 20 degrees falls on the addendum circle and thus the gear and pinion indicated in the illustration would mesh without interference for this angle. FORMULA FOR LOCATING THE POINT OF INTERFERENCE OF SPUR GEARS Referring to Fig. 25: Let^F = c. AB = r,. A D = d. B E=r,. D E=f. a = the angle of tooth pressure. y = the distance from the center of the gear to the point at which in- terference begins. INTERFERENCE OF SPUR GEARS. l8 AMERICAN MACHINIST GEAR BOOK x=ihe distance from the point at which interference begins to the addendum circle of the gear measured along a radius. 0=the perpendicular distance from the point at which interference begins to the center line of the pinion and gear. Then /-, = the pitch radius of the gear. r, = the pitch radius of the pinion. D' = ihe pitch diameter of the gear. D =the outside diameter of the gear. Then c = r.. cos a, and d = c cos a = r^ cos^ a. Now J=r^-\-r,, — d. = r^-\-r., {i-cos' a), and = c sin a = r^ sin a cos a. Now y^ = f -\- O^ and y = V p -{- 0\ Then by substituting 3; = V [r, + r^ (i — cos^ a)Y + (r^ sin a cos a)^ For a pressure angle of 143^ degrees y = V (r, + 0.0627 r,y + (0.2424 rj-, and D X = y. 2 For a pressure angle of 20 degrees y = \''^ {r, + 0.1169 rj- + (0.3214 r^y, and D X = 2 Solving for x and y will give the point of interference for any particular case. DIAGRAM FOR LOCATION OF INTERFERENCE Fig. 26 shows a diagram giving the location of the point of the beginning of interference for one diametral pitch involute gears from 10 to 135 teeth mesh- ing with a i2-tooth pinion. The ordinates are the distances from the point where interference commences to the addendum circle of the gear measured along the radius. They correspond to the quantity x in the preceding equa- TOOTH PARTS 19 — — — — — ' — — — — _ — - b' — — — — — ' — — — _J _ s 3 -o fc S 73" 3 -0 ■tJ § iTi T3 -A < 1 -o T1 -^ 3 1 (0 6 in f 60 To c 6 lo f < r. p, 0) u 0) ai 1 3 i ;^ > p 4 3 Pi p . pi « P — . C I — — — — — u — ' _c J \ — — ~ — — _CJ 1 i \ \ — — — — .1 \ \ V — \ ^ ^ \ V — r \ s \ \ \ \ s, \ \ ^ \ \ \ .^ ^ a to s u ^ •y, s :2; < s IS H w n w u y, CO W - H 12; CM 1-1 a < o o • ooooooooooooooooooooc rroJOOO'ii'^.CMO OOtO -^fN OOOCD rfiCMOOOtO-^MO OOtO'^ O«5®l0ini0l0inr(<-^'^^T^rlJC0C0C0C0C0CV)ClCM(NCooooooooooooo Interference in Inches 20 AMERICAN MACHINIST GEAR BOOK tions. From this point to the addendum circle the tooth outUne must be cor- rected. The upper curve A is for a pressure angle of 14}^ degrees and an addendum of 0.3183 X circular pitch. The second, B, is for the same pressure angle and a shorter addendum, 0.25 X circular pitch. This addendum factor is for what is known as the stubbed tooth standard, as proposed by the author on page 23. The third curve, C, is for a pressure angle of 20 degrees and an addendum of 0.3183 X circular pitch, while the lowest one, D, is for the 20-degree angle and the stubbed tooth addendum. The diagram as plotted is for one diametral pitch. To find the correspond- ing ordinate for any other pitch divide the value given in the diagram by the required pitch. The quotient will be the distance desired. INTERFERENCE OF RACK AND PINION Interference will occur between the teeth of a rack and pinion when the point B, Fig. 27, which is the intersection of a perpendicular from the point to the line of pressure A L falls inside of the rack addendum line E E. In the figure FIG. 27. INTERFERENCE OF GEAR AND RACK. the distance over which interference takes place is C D. It is usual practice to shorten the rack teeth by the amount of this interference and the following equations give an easy method of computing this distance. Let Let Then TOOTH PARTS 21 A^=the number of teeth in the pinion. p = ihe diametral pitch. r=the pitch radius. 6 = the radius of the base circle. • a = the pressure angle. .r=the distance necessary to shorten the addendum of the rack tooth and 5= the normal addendum of the rack tooth. I 5 = > P 'AN r = J P b = r cos a, D = b cos a, O D = r COS' a, OC =r - s, X =0 D — C, and substituting = r COS' a — {r — s) Whence = —--{cos- a -i) + — : P P I — j/^'iV (l — COS^ a) X = P For a pressure angle of 141^ degrees I - 0.03135 .¥ X = P For a pressure angle of 20 degrees I — 0.05849 N X = Solving these equations we find that for the true involute form of tooth and a pressure angle of 14^^ degrees interference between the teeth of rack and pin- ion begins with a pinion of 31 teeth. Similarly for a 20-degree pressure angle the interference begins with a pinion of 1 7 teeth. 22 AMERICAN MACHINIST GEAR BOOK INTERFERENCE OF INTERNAL GEAR AND PINION The following method of correction and equations are true for all combina- tions when the pinion has less than 55 teeth. Referring to Fig. 28: Let FIG. 28. INTERFERENCE OF INTERNAL GEAR AND PINION. r, = the pitch radius of gear. r.^= the pitch radius of pinion. b = the radius of the base circle. c = the radius of the correction circle. d = the radius of the rounding off circle. e = the radius of the interference circle. = the radius of the tooth cutting. a = the pressure angle. Then b = r^ cos a, e = 1^ (^, + rX c = ^^ cos a, =-M '',, and TOOTH PARTS 23 EXISTING TOOTH STANDARDS -BROWN & SHARPENS The Brown & Sharpe system is perhaps the best known; the angle of obhquity being 143^ degrees. Addendum, = 0.3183 />' or Dedendum, = 0.^83 p^ or — f^ 2 Working depth, = 0.6366 p"" or — - Whole depth, = 0.6866 p' or -^^^ Clearance, = 0.05 p^ or — ' In which p^ = circular pitch, and p = diametral pitch. GRANT'S The Grant system has an angle of obliquity of 1 5 degrees, otherwise it is the same as Brown & Sharpe's. This system is used on the Bilgram generator. SELLERS' Wm. Sellers & Co. adopted a form of tooth some 32 years ago in which the angle of obliquity was 20 degrees, otherwise the same as Brown & Sharpe's. HUNT'S The C. W. Hunt Co. have a standard in which the angle of obliquity is i4>^ degrees; the tooth parts being as follows: Addendum, = 0.25 />\ or P Dedendum, = 0.30 p\ or 0.9424 P Working depth, = 0.50 p\ or 1.5708 P Whole depth, = 0.55 p\ or 1.7278 P. Clearance, ■ = 0.05 p\ or 0-157 P THE AUTHOR'S This system, presented in connection with a discussion of an interchangeable involute gear- tooth system at the December, 1908, meeting of the American Society of Mechanical Engineers, was originally published in American 24 AMERICAN MACHINIST GEAR BOOK Machinist, June 6, 1907. Angle of obliquity is 20 degrees. Balance of the tooth parts being the same as the Hunt system described above. FELLOWS' The stubbed tooth adopted by the Fellows Gear Shaper Company has an angle of obliquity of 20 degrees. The tooth parts, however, do. not bear a definite relation to the pitch; the addendum being made to correspond to a diametral pitch one or two sizes finer, as: Actual pitch _ 2 23^ 3457 ^ 10 12 14 10 12 Pitch depth 23^ 3 4579 10 12 14 The upper figures indicate the diametral pitch for tooth spacing and ths lower figures indicate the diametral pitch from which the depth is taken. In this system the addendum varies from 0.264 to 0.226 of the circular pitch; Fig. 30 Proposed Stubbed Involute Tooth Shape 12 Teeth, 20 Degrees. Addendum =0.25 X Circular Pitchy COMPARATIVE FORMS OF I4ix^-DEGREE AND 20-DEGREE STANDARDS. 0.25, which is the addendum for the Hunt and the author's standard, is a rough mean. The author's standard tooth is shown in Fig. 30 for comparison with the 143^-degree standard in Fig. 29. Wilfred Lewis discussed tooth standards before the American Society of Mechanical Engineers, 1900, as follows: "About 30 years ago, when I first began to study the subject, the only system TOOTH PARTS 25 of gearing that stood in much favor with machine-tool builders was the cy- cloidal. " For some time thereafter William Sellers & Co., with whom I was connected, continued to use cylindrical gearing made by cutters of the true cycloidal shape, but the well-known objection to this form of tooth began to be felt, and pos- sibly 25 years ago my attention was turned to the advantages of an involute system. The involute systems in use at that time were the ones here de- scribed as standard, having 143 2 degrees' obliquity, and another recommended by Willis having an obliquity of 15 degrees'. Neither of these satisfied the re- quirements of an interchangeable system, and with some hesitation I recom- mended a 20-degree system, which was adopted by William Sellers & Co., and has worked to their satisfaction ever since. I did not at that time have quite the courage of my convictions that the obliquity should be 22)^ degrees or one-fourth of a right angle. Possibly, however, the obliquity of 20 degrees may still be justified by reducing the addendum from a value of one to some fraction thereof, but I would not undertake at this time to say which of the two methods I would prefer. "I brought up the same question nine years ago before the Engineers' Club of Philadelphia, and said at that time that a committee ought to be appointed to investigate and report on an interchangeable system of gearing. We have an interchangeable system of screw threads, of which everybody knows the advantage, and there is no reason why we should not have a standard system of gearing, so that any gear of a given pitch will run with any other gear of the same pitch. " A UNIVERSAL STANDARD The question of recommending a standard for gear teeth is now in the hands of a special committee appointed by the American Society of Mechanical En- gineers. This is a matter of the utmost importance. There are now many different standards in this country alone. Also, owing to the inaccuracies of the present form of teeth, gears for heavy work are generally designed to meet the requirements of that particular pair, changing either the depth or tooth, angle, or both, just enough to avoid any doctoring of the involute; this is ex- pensive. I do not know of any gears heavier than 43^ inch circular pitch that of late years have been cut 143^ degrees. It is common practice to use a modi- fied tooth of some sort for all gears over one diametral pitch. If this is neces- sary for heavy gears, why not for smaller ones? To do away with this multiplicity of standards, and bring a universally accepted standard gear-tooth system out of the present chaos, is the work be- fore this committee. 26 AMERICAN MACHINIST GEAR BOOK Of course, there will be many natural objections raised to any such change, as there always is, but the amount of expense and trouble entailed in the adoption of such a standard is a small measure of the benefit that will be ultimately derived. Everybody who has studied this matter must admit that a standard tooth of some kind is desired. Putting it off will do no good, unless perchance, some genius discovers a better form than the involute, which is not likely. Therefore, let us hope for a speedy conclusion of the work of this com- mittee and a standard that is a standard. MODIFIED TEETH A common method of modifying the involute tooth to avoid either inter- ference, undercutting, or the necessity of departing from the true outline is by shortening the dedendum and lengthening the addendum of the pinion tooth. The opposite treatment is given the gear tooth, the dedendum being made deeper to accommodate the added addendum of the pinion and the addendum of the gear correspondingly short. This method is employed on all bevel-gear generating machines for angles less than 20 degrees to avoid interference, the amount of correction depending, of course, upon both the number of teeth being cut and the number of teeth in the engag- ing gear, or, in other words, depending upon the position of the base line. On bevel-gear. generating machines it is the practice to make no modification in the angle for a 20-degree tooth when cut- ting a depth equal to o.6866p\ For this depth of tooth and a pressure angle of 20 degrees interference beginning at 17 teeth, enough roll, however, can be given the blank to allow the generating tool to un- dercut the flank of the tooth, and avoid interference without any correction of the tooth parts. This is not the case, however, when cutting the standard 143^ or 15 degrees on account of limitation in the movements of the machine. This modification in the tooth parts for bevel gears is accompHshed by shift- ing the face angles and outside diameters, the pinion being enlarged and the gear reduced. The dedendum of the pinion is sometimes shortened for another reason: Often the bore is so large as to leave insufficient stock between the bottom of the teeth and the keyseat. See Fig. 31. When the pinion cannot be enlarged or the bore reduced the only possible recourse is to shorten the dedendum, taking the amount shortened from the point of the ^ear tooth. This practice FIG. 12 Teeth 3 Diamfetral Pitch 2V2 Inch Bore 31. SHORTENING THE DEDENDUM TO STRENGTHEN KEYWAY. TOOTH PARTS 27 is not to be recommended although extensively used; it would be much better to apply the short tooth of increased obliquity to such cases. THE OCTOID All bevel-gear generating machines operate on the octoid system, and not the involute, as is generally supposed. An involute spur gear may be generated by the action of a tool representing the rack tooth, as illustrated by Fig. 9. In generating a bevel gear, however, the tool representing the engaging rack tooth must always travel tow^ard the apex of the gear being cut, swinging in a partial circle instead of travelling on Involute Tooth Octoid Tooth Fig. 33. American Machinist FIG. 32. GENERATING THE OCTOID TOOTH. a straight line in the direction of the rotation of the gear, as is the case when cutting a spur gear. The base of the bevel-gear tooth is, therefore, a crown gear instead of a rack. An involute crown gear theoretically correct will have curved instead of straight sides as shown in Fig. 7,2. As it is not practical to make the generating tools this peculiar shape, they are made straight sided and the octoid tooth is the result. THE LINE OF ACTION There is a definite relation between the circle or line which will describe the tooth outline and the Kne of action. Thus, if the line of action is in the form of a circle, as shown in Fig. 34, that circle of which this line is a segment will de- scribe the tooth outline if rolled upon the pitch circle. The difficulties encoun- 28 AMERICAN MACHINIST GEAR BOOK tered in the general application of this law are well illustrated by George B. Grant in section 32 of his " Treatise on Gear Wheels," as follows: ''This ac- cidental and occasionally useful feature of the rolled curve has generally been made to serve as a basis for the general theory of the tooth curve, and it is re- sponsible for the usually clumsy and limited treatment of that theory. The general law is simple enough to define, but it is so difficult to apply, that but one tooth curve, the cycloidal, w^hich happens to have the circle for a roller, can be intelligently handled by it, and the natural result is, that that curve has received the bulk of the attention. For example, the simplest and best of all the odontoids (pure form of tooth curve), the involute, is entirely beyond its reach, because its roller is the loga- rithmic spiral, a transcendental curve that can be reached only by the higher mathematics. No tooth curve, which, like the in- volute, crosses the pitch line at any angle but a right angle, can be traced by a point in a simple curve. The trac- ing point must be the pole of a spiral, and therefore a mechanical impossibility. A practicable rolled odontoid must cross the pitch line at right angles. To use the rolled curve theory as a base of operations w411 confine the dis- cussion to the cycloidal tooth, for the involute can only be reached by abandon- ing its true logarithmk roller, and taking advantage of one of its peculiar prop- erties, and the segmental, sinusoidal, parabolic, and pin tooth, as well as most other available odontoids, cannot be discussed at all." riG. 34. RELATIOX OF THE LINE OF ACTION TO THE DESCRIBING CIRCLE. THE LAW OF TOOTH CONTACT To transmit uniform motion, any form of tooth curve is subject to this gener- al law: "The common normal to the tooth must pass through the pitch point. " That is, a line drawn from the pitch point P through the contact point of any pair of teeth, as at h, must be at right angles or normal to the engaging portions of both teeth. See Fig. 35. In the involute system the line of action always passes through the pitch point P, and the engaging teeth take their base from -the points / and y, where the line of action intersects the base circles. Conversely, a line drawn from the instantaneous radii of any two teeth engaged will pass through their TOOTH PARTS 29 point of contact if the teeth are correctly formed. For example: In Fig. 35, the point of contact between the teeth C and D is at b, on the line of action, the FIG. 35. THE ARC OF ACTION. radius of the engaged portion of the tooth C is at /, and the radius of the tooth D is at y, fulfilling the required conditions. THE ARC OF ACTION The tooth action between two gears is between the points a and b, at which points the line of action intersects the addendum circles of the two gears. The actual length of contact is along the pitch lines occupied by the teeth whose addendum circles intersect the line of action, or between the points c and d. See Fig. 35. The distance F — d passed over while the point of contact approaches the pitch point is the arc of approach, the distance P — c passed over as the point of contact leaves the pitch point is the arc of recess. By increasing the addendum of the driving gear the arc of approach is re- duced and the arc of recess is increased. The frictiort of the arc of approach is greater than in the arc of recess. 30 AMERICAN MACHINIST GEAR BOOK THE BUTTRESSED TOOTH The buttressed tooth shown in Fig. 36 is described by Professor Robert Willis in a paper published in the Transactions of the Institute of Civil Engi- neers, London, 1838. It is apparent that the object is to obtain a strong tooth FIG. 36. THE BUTTRESSED TOOTH. for a pair of gears operating continuously in one direction. This is accompHshed by increasing the angle of obliquity of the back of the tooth, the face of the tooth being any angle desired. If the back of the tooth is correctly formed the PIG. 37. BUTTRESSED TEETH IN CONTACT gears will operate satisfactorily in either direction although with an increased pressure on their bearings when using the back face of the teeth owing to the TOOTH PARTS 31 increased obliquity of action. For many purposes there is no objection to this, and it is a great wonder that this tooth is not more extensively used. Of course, there must be a limit to the angle of the back of the tooth. For practical purposes the curve at the top of tooth at the back should not extend further than the center line of the tooth; for an addendum of — or o.6866/>,, P this will occur at an angle of about 32 degrees. A greater angle than this will subject the tooth to breakage at the point. In Fig. 37 is shown a pair of but- tressed tooth gears in contact. STEPPED GEARS A stepped spur gear consists of two or more gears keyed to the same shaft, the teeth on each gear being slightly advanced. If mated with a similar gear the tooth contact will be increased, which increases the smoothness of action. A common form of this type of gear is that of two gears cast in one piece with a separating shroud. For a cut gear there must be a groove turned between the faces of sufficient width to allow the planing tool or cutter to clear. A tooth is placed opposite a space, when the gear is made in two sections. HUNTING TOOTH It has been customary to make a pair of cast tooth gears with a hunting tooth, in order that each tooth would engage all of the teeth in the mating gear, the idea being that they would eventually be worn into some indefinite but true shape. Some designers have even gone so far as to specify a pair of " hunt- ing-tooth miter gears." That is, one ''miter" gear would have, say, 24 teeth and its mate 25 teeth. There never was any call for the introduction of the hunting tooth even in cast gears, but in properly cut gears any excuse for its use has certainly ceased to exist. TEMPLET MAKING In making the templets for gear teeth there are several points of importance to be kept in mind, namely: Templets should be made of light sheet steel instead of zinc which is often employed; the surface of steel should be coppered by the application of blue vitriol. For spiral or worm gears, templets should always be made for the normal pitch. For spacing and tooth thickness, always use chordal measurements. Check the chordal distance over the end teeth of templet. This is of the utmost im- portance. 32 AMERICAN MACHINIST GEAR BOOK Put enough teeth in the templet to show the entire tooth action, and try the templets on centers before making up the cutters or formers. It is a good idea to make a standard templet of each pitch as they are required, to try out other templets that must be made later on. When a templet is required for a fine pitch gear it is good practice to lay out the teeth on white paper lo or even 20 times the actual size and reduce by photography. On this drawing the center should be plainly marked and inclosed in a heavy circle, also a short section of the pitch line should be made heavy with a connecting radial line indicat- ing the radius of pitch circle. If the pitch radius required is i}4 inches, it should be made, say, 15 inches on the drawing. The drawing is then photo- graphed, the camera being set until the radial line, which was drawn 15 inches, measures i}4 inches on the ground glass. See Fig. 38. FIG. 38. TOOTH OUTLINE AS PHOTO- GRAPHED FROM LARGE SCALE DRAWING. DEFINITION OF PITCHES Diametral pitch is the number of teeth to each inch of the pitch diameter. Circular pitch is the distance from the edge of one tooth to the corresponding edge of the next tooth measured along the pitch circle. Addendum FIG. 39. TOOTH PARTS. TOOTH PARTS 33 DIAMETRAL PITCH CIRCULAR PITCH THICKNESS OF TOOTH OF PITCH. LINE WHOLE DEPTH DEDENDUM ADDENDUM K 6.2832" 3.1416" 4.3142" 2.3142" 2.0000" M • 4.1888 2.0944 2.8761 1.5728 T--?>?>ii I 3.1416 1.5708 2.1571 I.1571 1 .0000 iM 2.5133 1.2566 1-7257 0.9257 0.8000 i>^ 2.0944 1.0472 I.4381 0.7714 0.6666 iM 1-7952 0.8976 1.2326 0.6612 0.5714 2 1.5708 0.7854 "1.0785 0.5785 0.5000 2M 1-3963 0.6981 0.9587 0.5143 0.4444 21/2 1.2566 0.6283 0.8628 0.4628 0.4000 2% 1. 1424 0.5712 0.7844 0.4208 0.3636 3 1.0472 0.5236 0.7190 0.3857 0.3333, 33-^ 0.8976 0.4488 0.6163 0.3306 0.2857 4 0.7854 0.3927 0.5393 0.2893 0.2500 5 • 0.6283 0.3142 0.4314 0.2314 0.2000 6 0.5236 0.2618 0.3595 0.1928 0.1666 7 0.4488 0.2244 0.3081 0.1653 0.1429 8. 0.3927 0.1963 0.2696 0.1446 0.1250 9 0.3491 0.1745 0.2397 0.1286 O.IIII 10 0.3142 O.1571 0.2157 O.II57 O.IOOO II 0.2856 0.1428 O.I961 0.1052 0.0909 12 0.2618 0.1309 0.1798 0.0964 0.0833 13 0.2417 0.1208 0.1659 0.0890 0.0769 14 0.2244 O.II22 O.I541 0.0826 0.0714 15 0.2094 0.1047 0.1438 0.0771 0.0666 16 0.1963 0.0982 0.1348 0.0723 0.0625 17 0.1848 0.0924 0.1269 0.0681 0.0588 18 0.1754 0.0873 O.II98 0.0643 0.0555 19 0.1653 0.0827 O.II35 0.0609 0.0526 . 20 0.1571 0.0785 0.1079 0.0579 0.0500 22 0.1428 0.0714 0.0980 0.0526 . 0.0455 24 0.1309 0.0654 0.0898 0.0482 0.0417 26 0.1208 0.0604 0.0829 0.0445 0.0385 28 0.1122 0.0561 0.0770 0.0413 0.0357 30 0.1047 0.0524 0.0719 0.0386 0.0333 32 0.0982 0.0491 0.0674 0.0362 0.0312 34 0.0924 0.0462 0.0634 0.0340 0.0294 36 0.0873 0.0436 0.0599 0.0321 0.0278 38 0.0827 0.0413 0.0568 0.0304 0.0263 40 0.0785 0.0393 0.0539 0.0289 0.0250 ,42 0.0748 0.0374 0.0514 0.0275 0.0238 44 0.0714 0.0357 0.0490 0.0263 OX)227 46 0.0683 0.0341 0.0469 0.0252 0.0217 48 0.0654 0.0327 0.0449 0.0241 0.0208 50 0.0628 0.0314 0.0431 0.0231 0.0200 56 0.0561 0.0280 0.9385 0.0207 0.0178 60 0.0524 0.0262 0.0360 0.0193 0.0166 Table i — Diametral Pitch Relation between Diametral Pitch and Circular Pitch, with corresponding Tooth Dimensions 34 AMERICAN MACHINIST GEAR BOOK CIRCULAR PITCH DIAMETRAL PITCH THICKNESS OF TOOTH OF PITCH LINE WHOLE DEPTH DEDENDUM ADDENDUM • 6 " 0.5236 3.0000" 4.II96" 2.2098" 1.9098" 5 " 0.6283 2.5000 3-4330 I.8415 1-5915 4 " 0.7854 2.0000 2.7464 1-4732 1.2732 3^" 0.8976 • I-7500 2.4031 1.2890 I.II40 3 " 1.0472 1.5000 2.0598 1. 1049 0.9550 2%" I. 1424 I-3750 1.8882 1.0028 0.8754 ^y^' 1.2566 1.2500 1-7165 0.9207 0.7958 2^" 1-3963 1. 1250 1-5449 0.8287 0.7162 2 " 1.5708 1 .0000 ^•372>2 0.7366 0.6366 ■ I%" 1-6755 0.9375 1.2874 0.6906 0.5968- iM" 1-7952 0.8750 1. 2016 0.6445 O.55/O I^" 1-9333 0.8125 1-1158 0.5985 0.5173 i3^" 2.0944 0.7500 1.0299 0.5525 0.4775 iJTe" 2.1855 0.7187 0.9870 0.5294 0.4576 i^/^" 2.2848 0.6875 0.9441 0.5064 0.4377 iV' 2.3936 0.6562 0.9012 0.-4837 0.4178 iM" 2.5133 0.6250 0.8583 0.4604 0.3979 i^e" 2.6465 0.5937 0.8156 0.4374 0.3780 i,^" 2.7925 0.5625 0.7724 0.4143 0.3581 lYe" 2.9568 0.5312 0.7295 0.3913 0.3382 I " 3-1416 0.5000 0.6866 0.3683 0.3183 - 'X^' 3-35IO 0.4687 0.6437 0.3453 0.2984 Vs" 3-5904 0.4375 0.6007 0.3223 0.2785 %" 3.8666 0.4062 0.5579 0.2993 0.2586 %" 4.1888 0.3750 0.5150 0.2762 0.2387 %" 4.5696 0.3437 0.4720 0.2532 0.2189 y%" 5.0265 0.3125 0.4291 0.2301 0.1989 ' %" 5-5851 0.2812 0.3862 0.2071 0.1790 y^' 6.2832 0.2500 0.3433 0.1842 0.1592 %" 7.1808 0.2187 0.3003 O.1611 0.1393 2 // 5 7.8540 0.2000 0.2746 0.1473 0.1273 %" 8.3776 0.1875 0.2575 O.1381 O.II94 . Vz" 9.4248 0.1666 0.2287 0.1228 O.I061 ^e" • 10.0531 0.1562 0.2146 O.II51 0.0995 2 /' 7 10.9956 0.1429 0.1962 0.1052 0.0909 Ji" 12.5664 0.1250 0.1716 0.0921 0.0796 2 /' 14.1372 O.IIII 0.1526 0.0818 0.0707 1 // ft 15.7080 O.IOOO 0.1373 0.0737 0.0637 %" 16.7552 0.0937 0.1287 0.0690 0.0592 H" 18.8496 0.0833 0.1144 0.0614 0.0531 1" 21.9911 0.0714 0.0981 0.0526 0.0455 3^" 25.1327 0.0625 0.0858 0.0460 0.0398 9 28.2743 0.0555 0.0763 0.0409 0.0354 i^o" 31-4159 0.0500 0.0687 0.0368 0.0318 :^6" 50.2655 0.0312 0.0429 0.0230 0.0199 Table 2 — Circular Pitch Relation between Circular Pitch and Diametral Pitch, with corresponding Tooth Dimensions TOOTH PARTS 35 AAA 20 D. P. 0.1571 Inch C. P. JVAA 18 D. P. 0.1745 Inch C. P. 16 D. P. 0.1963 Inch C. P. 14 D. P. 0.2244 Inch C. P. 12 D. P. 0.2618 Inch C. P. 10 D. P. 0.3142 Inch C. P. 9 D. P. 0.3491 Inch C. P. 8 D. P. 0.3927 Inch C. P. 7 D. P. 0.4488 Inch C. P. 6 D. P. 0.5236 Inch C. P. 5 D. P. 0.6283 Inch C. P. 4 D. P. 0.7854 Inch C. P. 3D. P. 1.0472 Inch C. P. COMPARATIVE SIZES OF GEAR TEETH — INVOLUTE FORM. 36 AMERICAN MACHINIST GEAR BOOK 2^2 D. P. 1.2566 In. C.P. 2 D.P. 1.5708 In. C.P. P4 D.P. 1.7952 In. C.P. 2.0944 In. C.P. COMPARATIVE SIZES OF GEAR TEETH—INVOLUTE FORMS. TOOTH PART^ 37 lU D. P. 2.5133 Inch C. P. 1 D. P. 3.1416 Inch C. P. COMPARATIVE SIZES OF GEAR TEETH — INVOLUTE FORM. 38 AMERICAN MACHINIST GEAR BOOK NO. PITCH NO. PITCH NO. PITCH NO. PITCH TEETH DIAMETER TEETH DIAMETER TEETH DIAMETER TEETH DIAMETER 8 2.550 43 13.687 78 24.828 113 35-968 9 2.870 44 14.006 79 25.146 114 36.286 lO 3-183 45 14.324 80 25-465 115 36-605 II 3-501 46 14.642 81 25-783 116 36-923 12 3.820 47 14.961 82 26.101 117 37-241 13 4.138 48 15-279 83 26.420 118 37-560 14 4-456 49 15-597 84 26.738 119 37-878 15 4-775 50 15.915 85 27.056 120 38-196 i6 5-093 51 16.234 86 27-375 i 121 38.514 17 5-4II 52 16.552 87 27.693 122 38.833 i8 5-730 S3 16.870 88 28.011 123 39.151 19 6.048 54 17.189 89 28.330 124 39.469 20 6.366 55 17.507 90 28.648 125 39-788 21 6.684 56 17.825 91 28.966 126 40.106 22 7.003 57 18.144 92 29.284 127 40.424 ^3, 7-321 58 18.462 93 29.603 128 40-743 24 7-639 59 18.780 94 29.921 129 41.061 25 7-958 60 19.099 95 30.239 130 41-379 26 8.276 61 19.417 96 30-558 131 41.697 27 8.594 62 19-735 97 30.876 132 42.016 28 8.913 63 20.053 98 31-194 133 42.334 29 9.231 64 20.372 99 31-513 134 42.652 30 9-549 65 20.690 100 31-831 135 42.971 31 9.868 66 21.008 lOI 32.148 136 43.289 32 10.186 67 21.327 102 32.468 137 43-607 d>3> 10.504 68 21.645 103 32.785 i 138 43.926 34 10.822 69 21.963 104 33.103 j 139 44-243 35 II. 141 70 22.282 105 33-421 140 44.562 36 11-459 71 22.600 106 33-740 141 44.881 3,7 11.777 72 22.918 107 34-058 j 142 45-199 38 12.096 73 23.237 108 34-376 1 143 45-517 39 12.414 74 23.555 109 34-695 144 45-835 40 12.732 75 23-873 no 35-013 145 46.154 41 13-051 76 24.192 '' III 35-331 146 46.472 42 13-369 77 24.510 112 35-650 147 46.790 Table 3 — Pitch Diameters for One-Inch Circular Pitch Teeth from 8 to 147 FOR ANY OTHER PITCH MULTIPLY BY THAT PITCH METRIC PITCH The module is the addendum, or the pitch diameter in milUmeters divided by the number of teeth in the gear. TOOTH PARTS 39 The pitch diameter in millimeters is the module multii)lied by the number of teeth in the gear. All calculations are in millimeters. M = module (addendum) D' = pitch diameter D = outside diameter N = number of teeth W = working depth of tooth W' = whole depth of tooth / = thickness of tooth at pitch line / = clearness r = root M = N = N D or or M t = M 1.5708, iV+ 2' M — 2, / D' = NM, D = {N -\r 2) M W = 2M W = PF + / M 1.5708 ,_ 1^ , r — — , or M 0.157, r = if + / 10 ^' -^ ENGLISH ENGLISH . ENGLISH MODULE DIAMETRAL MODULE DIAMETRAL MODULE DIAMETRAL PITCH PITCH PITCH 0.5 50.800 7 3.628 I 25.400 3 8.466 8 3-175 1-25 20.320 3-5 7-257 9 2.822 1-5 •16.933 4 6-350 10 2.540 1-75 14-514 4.5 5-644 II 2.309 2 12.700 S 5.080 12 • 2. 117 2.25 11.288 5-5 4.618 14 1.814 2-5 10.160 6 4-233 16 1.587 2.75 9.236 t Module in Millimeters Table 4 — Pitches Commonly Used CHORDAL PITCH The chordal pitch is the shortest distance between two teeth measured on the pitch line, in other words, the distance to which the dividers would be set to space around the gear on the pitch line. This pitch is not used except for laying out large gears and segments that cannot be cut on a gear cutter. For such cases, also for laying out templets, it is absolutely necessary to use the chordal pitch, as the chordal pitch of the pinion is different from the chordal pitch of , the gear, the circular pitch of each being equal. N = number of teeth, C = chordal pitch. 40 AMERICAN MACHINIST GEAR BOOK D' = pitch diameter, e = sine of one half of angle subtended by side at center. 1 80° e = sine — ,-^ — . N C D' = C' = D' e, or D' sine 180^ N Table 5, diameters for chordal pitch, will be found useful for sprocket gears. NO. PITCH NO. PITCH NO. PITCH NO. PITCH TEETH DIAMETER TEETH DIAMETER TEETH DIAMETER TEETH DIAMETER 4 1-414 39 12.427 74 23.562 1 109 34-701 5 1. 701 40 12.746 75 23.880 IIO 35-019 6 2.000 41 13.064 76 24.198 III 35-337 7 2-305 42 13-382 77 24-517 112 35-655 8 2.613 43 13.699 78 24-835 113 35-974 9 2.924 44 14.018 79 25-153 114 36.292 10 3-236 45 14-335 80 25-471 115 36.610 II 3-549 46 14-653 81 25.790 116 36.929 12 •3-864 47 14.972 82 26.108 117 37-247 13 4.179 48 15.291 83 26.426 118 37-565 14 4.494 49 15.608 84 26.744 119 37-883 15 4.810 50 15-927 85 27.063 120 38.202 16 5.126 51 16.244 86 27.381 121 38.520 17 5-442 52 16.562 87 27.699 122 38.838 18 5-759 53 16.880 88 28.017 123 39-156 19 6.076 54 17.200 89 28.335 124 39-475 20 6-392 55 17.516 90 28.654 125 39-793 21 6.710 56 17-835 91 28.972 126 40.111 22 7.027 57 18.152 92 29.290 127 40.429 23 7-344 58 18.471 93 29.608 128 40.748 24 7.661 59 18.789 94 29.927 129 41.066 25 7-979 60 19.107 95 30.245 130 41.384 26 8.297 61 19-425 96 30.563 131 41-703 27 8.614 62 19.744 97 30.881 132 42.021 28 8.931 63 20.062 98 31.200 133 42.339 29 9-249 64 20.380 99 ^ 31-518 134 42.657 30 9-567 65 20.698 100 31-836 135 42.976 31 9-834 1 66 21.016 lOI 32-154 136 43.294 32 10.202 67 21-335 102 32.473 137 43.612 33 10.520 68 21.653 103 32.791 138 43-931 34 10.838 : 69 21.971 104 33-109 139 44.249 35 II. 156 70 22.289 105 33-428 140 44-567 36 11.474 71 22.607 106 33-740 141 44.890 37 II. 791 72 22.926 107 34-058 142 45.204 38 12. no 73 23.244 108 34-376 , 143 45-522 Table 5 — Pitch Diameters for One-Inch Chordal Pitch Teeth from 4 to 14 j FOR any other pitch MULTIPLY BY THAT PITCH TOOTH PARTS 41 CHORDAL THICKNESS OF TEETH In order to correctly measure the teeth, the chordal thickness must be used, as illustrated by Fig. 40. Also as the location of the pitch hne on the sides of —j-f FIG. 40. CHORDAL TOOTH THICKNESS. the teeth falls below the pitch line at the center of tooth. The measurement for the addendum must also be corrected, if any degree of accuracy is expected. Table 6, gives these corrected dimensions for various standard pitches. Number of I D P. iK D. P. 2 D . p: 23^ D. P. Number of Teeth. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Teeth. 8 1.5607 1.0769 1.0405 0.7179 0.7804 0.5385 0.6243 0.4308 8 9 1.5628 1.0684 1. 0419 0.7123 0.7814 0.5342 0.6251 0.4273 9 10 1.5643 I.0616 1.0429 0.7077 0.7821 0.5308 0.6257 0.4246 10 II 1-5654 1.0559 1.0436 0.7039 0.7827 0.5279 0.6261 0.4224 II 12 1.5663 I.0514 1.0442 0.7009 0.7831 0.5257 0.6265 0.4206 12 14 1.5675 1 .0440 1.0450 0.6960 0.7837 0.5220 0.6270 0.4176 14 17 1.5686 1.0362 1.0457 0.6908 0.7843 O.5181 0.6274 0.4145 17 21 1.5694 1.0294 1.0463 0.6863 0.7847 0.5147 0.6277 0.4118 21 26 1.5698 1.0237 1.0465 0.6825 0.7849 O.5118 0.6279 0.4095 26 35 1.5702 1. 0176 1.0468 0.6784 0.7851 0.5088 0.6281 0.4070 35 55 1.5706 I.OII2 I.0471 0.6741 0.7853 0.5056 0.6282 0.4045 55 135 1.5707 1.0047 1. 047 1 0.6698 0.7853 0.5023 0.6283 0.4019 135 Number of 3 D . P. s'A D. P. 4 D . P. 5 E >. P. Number of Teeth. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Teeth. 8 0.5202 0.3589 0.4459 0.3077 0.3902 0.2692 O.3121 0.2154 8 9 0.5209 0.3561 0.4465 0.3052 0.3907 0.2671 0.3126 0.2137 9 10 0.5214 0.3538 0.4469 0.3033 0.391 1 0.2654 0.3129 0.2123 10 II 0.5218 0.3519 0.4473 0.3017 0.3913 0.2640 O.3131 0.2 II 2 II 12 0.5221 0.3505 0.4475 0.3004 0.3916 0.2628 0.3133 0.2103 12 14 0.5225 0.3480 0.4479 0.2983 0.3919 0.2610 0.3135 0.2088 14 17 0.5228 0.3454 0.4482 0.2961 0.3921 0.2590 0.3137 0.2072 17 21 0.5231 0.3431 0.4485 0.2941 0.3923 0.2573 0.3139 0.2059 21 26 0.5233 0.3412 0.4485 0.2925 0.3925 0.2559 0.3140 0.2047 26 35 0.5234 0.3392 0.4486 0.2907 0.3926 0.2544 0.3140 0.2035 35 55 0.5235 0.3371 0.4487 0.2889 0.3927 0.2528 O.3141 0.2022 55 135 0.5236 0.3349 0.4488 0.2871 0.3927 0.2512 O.3141 0.2009 135 Table 6 — Chordal Thicknesses and Addenda of Gear Teeth of Diametral Pitch Boston Gear Works 42 AMERICAN MACHINIST GEAR BOOK Number of 6 D. P. 1 7 D. P. 8 D. P. 9 D. P. Number of Teeth. Thickness Addendum Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Teeth. 8 0.2601 0.1795 0.2230 0.1538 0.1951 0.1346 0.1734 0.1197 8 9 0.2605 0.1781 0.2233 0.1526 0.1954 0.1336 0.1736 0.1187 9 lO 0.2607 0.1769 0.2235 0.1517 0.1955 0.1327 0.1738 0.1 180 10 , II 0.2609 0.1760 0.2236 0.1508 1 0.1957 0.1320 0.1739 0.1173 II 12 0.2610 0.1752 0.2238 0.1502 j 0.1958 0.1314 0.1740 O.1168 12 14 0.2612 0.1740 0.2239 0.1491 0.1959 0.1305 0.1742 0.1 160 14 17 0.2614 0.1727 0.2241 0.1480 0.1961 0.1295 0.1743 0.1151 17 21 0.2616 0.1716 0.2242 0.1471 0.1962 0.1287 0.1744 0.1144 21 26 0.2616 0.1706 0.2243 0.1462 0.1962 0.1280 0.1744 0.1137 26 35 0.2617 0.1696 0.2243 0.1454 0.1963 0.1272 0.1745 O.1131 35 55 0.2618 0.1685 0.2244 0.1445 0.1963 0.1264 0.1745 0.1 1 24 55 135 0.2618 0.1675 0.2244 0.1435 0.1963 0.1256 0.1745 O.1116 135 Number of 10 D. P. II D. P. 12 D. P. 13 D. P. Number of Teeth. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Teeth. 8 O.1561 0.1077 0.1419 0.0979 0.1301 0.0897 0.1201 0.0828 8 9 0.1563 0.1068 0.1421 0.0971 0.1302 0.0890 0.1202 0.0822 9 lO 0.1564 0.1061 0.1422 0.0965 0.1304 0.0885 0.1203 0.0816 10 II 0.1565 0.1056 0.1423 0.0960 0.1305 0.0880 0.1204 0.0812 II 12 0.1566 0.1051 0.1424 0.0956 0.1305 0.0876 0.1205 0.0809 12 14 0.1567 0.1044 0.1425 0.0949 0.1306 0.0870 0.1206 0.0803 14 17 0.1569 0.1036 0.1426 0.0942 0.1307 0.0863 0.1207 0.0797 17 21 0.1569 0.1029 0.1427 0.0936 0.1308 0.0858 0.1207 0.0792 21 26 0.1570 0.1024 0.1427 0.093 1 0.1308 0.0853 0.1207 0.0787 26 35 0.1570 0.1018 0.1427 0.0925 0.1309 0.0848 0.1208 0.0782 35 55 O.1571 O.IOII 0.1428 0.0919 0.1309 0.0843 0.1208 0.0777 55 135 O.1571 0.1005 0.1428 0.0913 0.1309 0.0837 0.1208 0.0772 135 Number of 14 D. P. 15 D. P. 1 16 D. P. 17 D. P. Number of Teeth. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Thickness. Addendum. Teeth. 8 O.I 1 15 0.0769 0.1040 0.0718 0.0975 0.0673 0.0918 0.0633 8 9 0.1116 0.0763 0.1042 0.0712 0.0977 0.0669 0.0919 0.0628 9 lO 0.1117 0.0758 0.1043 0.0709 0.0978 0.0664 0.0920 0.0624 10 II 0.1118 0.0754 0.1044 0.0704 0.0978 0.0659 0.0921 0.0621 II 12 O.1119 0.0751 0.1044 0.0701 0.0979 0.0657 0.0921 0.0618 12 14 ! 0.1119 0.0746 0.1045 0.0696 0.0980 0.0652 0.0922 0.0614 i 14 17 0.1120 0.0740 0.1046 0.0691 0.0980 0.0648 0.0923 0.0609 17 21 0.1 1 21 0.0735 0.1046 0-0686 0.0981 0.0643 0.0923 0.0605 21 26 0.1 1 21 0.0731 0.1046 0.0682 0.0981 0.0640 0.0923 0.0602 26 35 O.1122 0.0727 0.1047 0.0678 0.0981 0.0636 0.0924 0.0598 35 55 0.1 122 0.0722 0.1047 0.0674 0.0981 0.0632 0.0924 0.0595 55 135 O.1122 0.0718 0.1047 0.0670 0.0981 0.0628 0.0924 0.0591 135 Chordal Thicknesses and Addenda of Gear Teeth of Diametral Pitch — Continued TOOTH PARTS 43 Number of i8 D. P. 19 D. P. 20 D. P. -M ). P. Number of Teeth. Thickness. Addendum. Thickness Addendum. Thickness. Addendum. Thickness. Addendum. Teeth. 8 0.0867 0.0598 0.0821 0.0567 ■ 0.0780 0.0538 0.0650 0.0448 8 9 0.0868 0-05Q3 0.0822 0.0562 0.0781 0.0534 0.065 1 0.0445 9 lO O.086Q 0.0589 0.0823 0.0558 0.0782 C.0530 0.0651 0.0443 ■10 II 0.0869 0.0586 0.0824 0-0555 0.0783 0.0528 0.0652 0.9439 II 12 0.0870 0.0584 0.0824 0.0553 0.0784 0.0525 0.0653 0.0437 12 14 0.0871 0.0580 0.0825 0.0549 0.0784 0.0522 0.0653 0.0435 14 I? 0.0871 0-0575 0.0826 0.0545 0.0784 0.0518 0.0653 0.0432 17 21 0.0872 0.0572 0.0826 0.0542 0.0785 0.0514 0.0654 0.0429 21 26 0.0872 0.0568 O.0S26 0.0538 0.0785 00511 0.0654 0.0426 26 35 0.0872 0.0565 0.0826 0.0535 0.0785 0.050S 0.0654 0.0424 35 55 0.0873 0.0562 0.0827 0.0532 0.0785 0.0505 0.0654 0.0421 55 135 0.0873 ■ 0.0558 0.0827 0.0528 0.0785 0.0502 0.0654 0.0419 135 Chordal Thicknesses and Addenda of Ge.\r Teeth of Diametral Pitch — Continued Number of 0/" /8 c. p. %" c. p. Vs" c. p. 1" c. p. Nuniber of Teeth. Thickness. Addendum. Thickness. Addendum. Thickness. 0.4347 Addendum. • 0.2997 Thickness. Addendum. ' Teeth. 8 0.3105 0.2142 0.3725 0.2570 0.4968 0.3426 8 9 0.3109 0.2125 0.3730 0.2550 0.4353 0.2976 0.4974 0.3400 Si 10 O.3112 0.21 1 2 0.3734 •02534 0.4357 0.2957 0.4978 0.3378 10 II 0.31 14 0.2100 0.3737 0.2520 0.4360 0.2941 0.4982 0.3360 I r 12 O.3116 0.2091 0.3739 0.2510 0.4363 0.2938 0.4986 0.3346 12 14 O.3118 0.2077 0.3741 0.2492 0.4366 0.2908 0.4988 0.?>?,2 2 14 17 0.3120 0.2061 0.3744 0.2473 0.4369 0.2886 0.4992 0.3298 17 21 0.3122 0.2048 0.3746 0.2457 0.4371 0.2868 0.4994 0.3276 21 26 0.3123 0.2036 0.3748 0.2443 0.4372 0.2851 0.4997 0.3258 26 35 0.3124 0.2024 0.3748 0.2429 0.4373 0.2833 0.4999 0.3238 35 55 0.3124 0.2011 0,3748 0.2414 0.4374 0.2816 0.4999 0.3218 55 135 0.3124 0.1999 0.3748 0.2398 0.4374 0.2798 0.4999 0.3198 135 Number of iM" c. p. I^" c. p. l^" c. p. 2" c. p. Number of Teeth. Thickness. Addendum. Thickness, Addendum. Thickness. Addendum. Thickness. Addendum. Teeth. 8 0.6210 0.4284 0.7450 0.5140 0.8694 0.5994 0.9936 0.685? 8 9 0.6218 0.4250 0.7460 0.5100 0.8706 0.5952 0.9948 0.6800 9 10 0.6224 0.4224 0.7468 0.5068 0.8714 0.5914 0.9956 0.6756 10 II 0.6228 0.4200 0.7474 0.5040 0.8720 0.5882 0.9964 0.6720 II 12 0.6232 0.4182 0.7478 0.5020 0.8726 0.5876 0.9972 0.6692 12 14 0.6236 0.4154 0.7482 0.4984 0.8732 0.5816 0.9976 0.6644 14 17 0.6240 0.4122 0.7488 0.4946 0.8738 0.5772 0.9984 0.6596 17 21 0.6244 0.4096 0.7492 0.4914 0.8742 0.5736 0.9988 0.6552 21 26 0.6246 0.4072 0,7496 0.4886 0.8744 0.5702 0.9994 0.6516 26 35 0.6248 0.4048 0.7498 0.4858 0.8746 0.5666 0.9998 0.6476 35 55 0.6250 0.4022 0.7499 0.4828 0.8748 0.5632 0.9999 0.6436 55 13s 0.6250 0.3998 0.7499 0.4796 0.8748 0.5596 0.9999 0.6396 135 Table 7 — Chordal Thicknesses and Addenda of Gear Teeth of Circular Pitch Boston Gear Works 44 AMERICAN MACHINIST GEAR BOOK INVOLUTE CUTTERS Until quite recently involute cutters were made in sets of eight, as follows: Number of Cutter 1 for 135 teeth to rack 2 for 55 to 134 teeth 3 for 35 to 54 teeth 4 for 26 to 34 teeth 5 for 21 to 26 teeth 6 for 17 to 20 teeth 7 for 14 to 16 teeth 8 for 12 to 13 teeth Modern conditions, however, require a more accurate tooth than can be produced by this number of cutters. A set of fifteen, utilizing the half numbers is now in common use. Number of Cutter 1 for 135 teeth to a rack i}4 for 80 to 134 teeth 2 for 55 to 79 teeth 23^ for 42 to 54 teeth 3 for 35 to 41 teeth 33^ for 30 to 34 teeth 4 for 26 to 29 teeth 4^ for 23 to 25 teeth 5 for 21 to 22 teeth 5^ for 19 to 20 teeth 6 for 17 to 18 teeth 63^ for 15 to 16 teeth 7 for 14 teeth 7^ for 13 teeth 8 for 12 teeth To produce accurate gears, templets for tooth thickness, made according to Tables 6 and 7, should be used instead of using one templet for each pitch and depending upon the workman's judgment as to how much shake to allow for different numbers of teeth. These templets, made up according to Tables 6 and 7, which are based on the use of eight cutters for each pitch, should be sufficiently accurate for all practical purposes. SECTION II Spur Gear Calculations To find the pitch diameters of two gears, the number of teeth in each and the distance between centers being given: Divide twice the distance between centers by the sum of the number of teeth: Find the pitch diameter of each gear separately by multiplying this quotient by its number of teeth. Example: Find the pitch diameters of a pair of spur gears, 21 and 60 teeth, for 2 5 -inch centers. 2 X 25 -—f- = 6.17284, 21 + 60 6.17284 X 21 = 12.96296 inches, or the pitch. diameter of the pinion 6.17284 X 60 = 37.03704 inches, or the pitch diameter of the gear The distance between the centers is one-half the sum of the pitch diameters. In the above example the center distance would prove to be: 12.96296 + 37.°3704 ^ ^5 i„^i^^^ . NOTATIONS FOR FORMULAS p = diametral pitch D^ = pitch diameter D = outside diameter V = velocity d' = pitch diameter d = outside diameter \- pinion V = velocity a = distance between the centers b = number of teeth in both 45 ^ gear These gears run together 46 AMERICAN MACHINIST GEAR BOOK TO FIND HAVING RULE N N D' d' a and p D' and d' b and p n V and V b V and V b V and V N V and T' p D' V and v N V and n pD' V and n n V and N a V and V a V and F The continued product of center dis- tance pitch and 2 One-half the sum of the pitch di- ameters Divide the total number of teeth by twice the pitch Divide the product of the number of teeth and velocity of pinion by the velocity of gear Divide the product of the total num- ber of teeth and velocity of pinion by the sum of the velocities Divide the product of the total num- ber of teeth and the velocity of gear by the sum of the velocities Divide the product of the number of teeth in gear and its velocity by the velocity of pinion Divide the continued product of the pitch, pitch diameter and velocity of the gear by the velocity of pinion. Divide the product of the number of teeth and velocity of gear by the number of teeth in pinion Divide the continued product of the pitch, pitch diameter and velocity of gear by the number of teeth in pinion Divide the product of the number of teeth in pinion and its velocity by the number of teeth in gear Divide the continued product of the center distance, velocity of pinion and 2, by the sum of the velocities. Divide the continued product of the center distance, velocity of gear and 2, by the sum of the velocities NV pD' V n V IT 2 av v+V 2a V V + V FORMULA EXAMPLE a p 2 15 X 3 X 2 = 90 D' + d' 20 -|- 10 2 2 - 15 b 2P 90 2X^ 15 n V 30 X 2 I = 60 b V v+ V 90 X 2 2 + 1 " = 60 bV 90 X I v^ V 2 + 1 ~ = 30 NV 60 X I V 2 = 30 pD'V 3 X 20 X I Ol-l 30 60 X I 30 3 X 20 X I 30 30 X 2 60 2 X 15 X 2 2 + 1 2 X 15X I = 2 = 2 2 + 1 = 20 = 10 Table 8 — Formulas for a Pair of Mating Spur Gears SPUR GJOAR CALCULATIONS 47 TO FIND The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. HAVING Pitch Diameter. Pitch Diameter. Pitch Diameter. Pitch Diameter Outside Diameter Outside Dianieter, Outside Diameter, Outside Diameter. Number of Teeth. Number of Teeth. Thickness of Tooth. Addendum. Dedendum. Working Depth. Whole Depth. Clearance. Clearance. RULE FORMULA The Circular Pitch.. The Pitch Diameter and the Number of Teeth The Outside Diame- ter and the Number of Teeth The Number of Teeth and the Diametral Pitch The Number of Teeth and Outside Diam- eter The Outside Diame- ter and the Diame- tral Pitch Addendum and the Number of Teeth . . The Number of Teeth and the Diametral Pitch The Pitch Diameter and the Diametral Pitch The Pitch Diameter and the Number of Teeth The Number of Teeth- and Addendum. . . . The Pitch Diameter and the Diametral Pitch The Outside Diame- ter and the Diame- tral Pitch Divide 3.1416 by the Circular Pitch .' Divide Number of Teeth by Pitch Diameter Divide Number of Teeth plus 2 by Outside Diameter Divide Number of Teeth by the Diametral Pitch Divide the product of Outside Diameter and Number of Teeth by Number of Teeth plus 2 . . . Subtract from the Outside Diam- ter the quotient of 2 divided by the Diametral Pitch Multiply Addendum by the Number of Teeth Divide Number of Teeth plus 2 by the Diametral Pitch Add to the Pitch Diameter the quotient of 2 divided by the Diametral Pitch Divide the product of the Pitch Diameter and Number of Teeth plus 2 by the Number of Teeth Multiply the Number of Teeth plus 2 by Addendum The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. The Diametral Pitch. Thickness of Tooth.. Multiply Pitch Diameter by the Diametral Pitch Multiply Outside Diameter by the Diametral Pitch and sub- tract 2 Divide 1.5708 by the Diametral Pitch Divide i by the Diametral Pitch, D' or s = N Divide r.157 by the Diametral Pitch Divide 2 by the Diametral Pitch. Divide 2.157 by the Diametral Pitch Divide 0.157 by the Diametral Pitch Divide Thickness of Tooth at pitch line by 10 P' D' N + 2 D = D' + P D = 0V+ 2) D' N D = {N+2)s N = D' p N = Dp- 2 1.5708 P s = ~ s+ f p W'+ f = I •157 2.157 0.157 p t 10 Table 9 — Spur Gear Calculations for Diametral Pitch 14 ^ Degree Standard 48 AMERICAN MACHINIST GEAR BOOK TO FIND The Circular Pitch. The Circular Pitch. The Circular Pitch. HAVING Pitch Diameter. Pitch Diameter. Pitch Diameter. Pitch Diameter. Outside Diameter. Outside Diameter, Outside Diameter. Number of Teeth. Thickness of Tooth. Addendum. Dedendum. Working Depth. Whole Depth. Clearance. Clearance. The Diametral Pitch. The Pitch Diameter and the Number of Teeth The Outside Diame- ter and the Number of Teeth The Number of Teeth and the Circular Pitch The Number of Teeth and the Outside Di- ameter The Outside Diame- ter and the Circular Pitch Addendum and the Number of Teeth. . The Number of Teeth and the Circular Pitch The Pitch Diameter and the Circular Pitch The Number of Teeth and the Addendum The Pitch Diameter and the Circular Pitch The Circular Pitch. . . The Circular Pitch. . . The Circular Pitch. . The Circular Pitch. . The Circular Pitch. . The Circular Pitch. . Thickness of Tooth. RULE Divide 3.1416 by the Diametral Pitch FORMULA Divide Pitch Diameter by the product of 0.3183 and Number of Teeth Divide Outside Diameter by the product of 0.3183 and Number of Teeth plus 2 The continued product of the Number of Teeth, the Circular Pitch and 0.3183 Divide the product of Number of Teeth and Outside Diameter by Number of Teeth plus 2 . . Subtract from the Outside Diam eter the product of the Circular Pitch and 0.6366 Multiply the Number of Teeth by the Addendum 3.I4I6 p D' 0.3183 N D p' = p' = ^ 0.3183 iV+2 D' = N p'o.siSs The continued product of the Number of Teeth plus 2 the Circular Pitch and 0.3183 .... Add to the Pitch Diameter the product of the Circular Pitch and 0.6366 Multiply Addendum by Number of Teeth plus 2 Divide the product of Pitch Di ameter and 3. 141 6 by the Cir cular Pitch One-half the Circular Pitch, Multiply the Circular Pitch by D' 0.3183, or s = -^ Multiply the Circular Pitch by 0.3683 Multiply the Circular Pitch by 0.6366 Multiply the Circular Pitch by 0.6866 Multiply the Circular Pitch by 0.05 One-tenth the Thickness of Tooth at Pitch Line D' = N D N + 2 D' = D - iP' 0.6366) D' = N s D = (N+2) / 0.3183 D = D -\- ip' 0.6366) D = s(N + 2) D's.1416 N = I = P' P' s = /?' 0.3183 s+ f = p' 0.3683 W = p' 0.6366 W = p' 0.6866 f = p 0.05 /■ = 10 Table 10 — Spur Gear Calculations for Circular Pitch 143^ Degree Standard SECTION III Speeds and Powers transmission of power by gearing with particular reference to spur and bevel gears SPEED RATIO The problem of finding the proper diameter or speed of a gear or pulley is simple enough when once thoroughly understood. The gear may be represented by its number of teeth, pitch diameter, pitch radius, or speed ratio, as the case may be. In the explanation to follow the number of teeth is used. The speed is in revolutions per minute. Rule: Divide the product of the speed and number of teeth of one gear by the speed or number of teeth of its mate to secure the lacking dimension. That is, if both the speed and number of teeth are known for one gear, mul- tiply the speed by the number of teeth, and divide this product by the known quantity of the mating gear to secure its number of teeth or speed, as the case may be. Or the same result may be obtained by proportion, the values being placed as follows: n : N : : R : r (i) n = number of teeth in pinion r = revolutions per minute of pinion N = number of teeth in gear R = revolutions per minute of gear Example: A gear having 60 teeth makes 300 revolutions per minute, what will be the speed of an engaging pinion having 1 5 teeth? n : N : R : r 15 : 60 : 300 : x Therefore, x = , or 1 200 revolutions per minute for pinion n To compute these values for a train of gears, use the continued product of the pinions and the continued product of the gears as a single gear and pinion and proceed as above. 49 50 AiMEKlCAx^ MACHINIST GEAR BOOK Example: In Fig. 41, the gear .Y has 100 teeth, N', 70 teeth, N'\ 60 teeth, n, 15 teeth; n\ 18 teeth; and n", 24 teeth. The gear iV makes 10 revolu- tions per minute. What will be the speed of the pinion n"? N, N^ and X" = 100 X 70 X 60 = 420,000 n, n' and n" = 15 X 18 X 24 = 6,480. n : N : : R : r 6,480 : 420,000 : : 10 : X Therefore, .v = — — h~~^ 5 <^r 648 revolutions per minute for i)inion }i^'. 6,480 riG. 41. GEAR TRAIN. FIG. 42. INTERMEDIATE GEAR DOES NOT AFFECT THE SPEED RATIO. The velocities of a train of gears may also be found as follows: .Y, N', N'' and n, n' , n" , etc., representing the number of teeth in the gears and pinions. R N N' N" n n n (2) _ r n n' n" ,. ~~ N N' N^' ^^^ The intermediate gear B, as shown in Fig. 42, while it changes the direction of the rotation of the gears, A and C does not alter their speed ratio, the cir- cumferential velocities of all three gears being equal. ARRANGEMENT OF GEAR TRAINS For compound reduction there must be four gears, as per Fig. 43, the gears B and C being keyed to an intermediate shaft, the power being transferred to the machine by the shaft-carrying gear D. When a great reduction is required, say 64 to i, there may be two inter- mediate shafts, as in Fig. 44. This reduction might be accompHshed by using a drive, as in Fig. 43, divid- ing the total reduction between two sets of gears, but a triple reduction is used by way of illustration. The best results are always obtained by dividing the reduction as evenly as possible among the different pairs of gears. For instance: for a double reduction, as in Fig. 43, the ratio of each pair should be made as SPEEDS AND POWERS 51 near the square root of the total reduction as possible. In case of the triple reduction, Fig. 44, the ratio of each pair should be the cube root of the total reduction, or v 64 = 4. That is, there are three sets of gears, each having a speed ratio of 4 to i. If double reduction had been used the reduction of each gear would have been V 64 = 8, or two sets of gears each having a speed ratio of 8 to I. Gear trains proportioned in this way give the highest possible efficiency. For instance: an unsuccessful single gear reduction of 16 to i might be 4 Motor \ \-P\P\ hww FIG. 43. DOUBLE GEAR REDUCTION. FIG. 44. TRIPLE GEAR REDUCTION. made efficient by substituting two pairs of gears, each having a ratio of 4 to i. Making the compound gears 8 to i and 2 to i would help, but would not be as efficient as the equal reduction. This will be especially noticeable in long leads in the lathe or miUing machines. POWER RATIO The relative powers of a train of gears are inversely proportional to their circumferential velocities. The circumferential velocity of each pair of gears in a train being equal, the driving pinion, as shown in Figs. 45 and 46, is ig- nored in the calculations for a single pair, the circumferential velocity and the load on the teeth being the same as for the mating gear. The problem is to determine the power ratio between the drum r and the gear R. 52 AMERICAN MACHINIST GEAR BOOK Ignoring friction, the values of this drive may be found by proportion, ar- ranged as follows: W : R : : F : r (4) Enough must be added to the load W or taken from the effective lifting force F to overcome the frictional resistance of the teeth and bearings. This loss must be estimated and the percentage of loss added to the load W, the ratio of R and r being determined according to this new ratio. FIG. 45, FIG. 46. POWER RATIO DIAGRAMS. Example: Referring to Fig. 45: if the radius of the gear i? is 18 inches, the radius of the drum r three inches, what power will be required at F to raise 300 pounds at W? W : R : : F : r 300 ; 18 .• ; jc .• 3 Therefore, x = o ? = 5° pounds required at F. lo Suppose the loss in efficiency to be 10 per cent and the radius of the gear R 18 inches. What must be the radius of the drum r to raise 300 pounds at W? 300 + 10 per cent = 330 pounds. W : R : : F : T 300 .' 18 .• .• 50 ; X Therefore, x = ^ — , = 2.7 inches for the radius of drum r. 330 For a train of gears, the continued products of the driving and driven gears may be considered as single gears. Or the powder ratio may be considered be- tween each pair inversely proportional to their velocity ratios. SPEEDS AND POWERS 53 Example: Referring to Fig. 47; what force is required at /*' to raise 2500 pounds at W, the loss in efficiency being 30 i)er cent? R R' R" = 20 X 18 X 10 = 3600 rr'r" = 6 X 8 X 5 = 240 W = 2500 + 30 per cent = 3250. ^ ■ '^ •■ •■ '' ■ ' 3250 X 240 , , 3250 : 3600 : : .V : 240, x = -^"^^^^^ , or 217 pounds at t. Also F = And W = W rr' r" R R! R" ' '' F RR' R" r r' r" 3600 3250 X 6 XJ X 5 20 X 18 X 10 217 X 20 X 18 X 10 6X8X5 , = 217 pounds. (5) , = 3250 pounds. (6) AN EXAMPLE IN HOIST GEARING Example: What gears will be required to lift a load of 2400 pounds at a uniform rate of speed, employing a 10 horse-power motor running 11 20 revolu- tions per minute, driving with a rawhide pinion 4 inches pitch diameter? See Fig. 48. F = 281 Pounds FIG. 47. POWER RATIO OF GEAR TRAINS. 2400 Pounds FIG. 48. EXAMPLE OF GEAR DRIVE FOR HOIST. Velocity of pinion in feet per minute, T^ = d' 0.2618 R. P. M. (7) HP X 33,000 The safe load, W + V (8) Therefore, F = 4 X 0.2618 X 1120 = 1173 feet per minute And the pinion T O ^^ "X T. 000 And W = ^-^ , or 281 pounds, w^hich is the load to be carried by 1173 54 AMERICAN MACHINIST GEAR BOOK Assuming thai 20 per cent is lost by the friction of the gear teeth, bearings,, etc., the real load to be raised by the force of 280 pounds at the pitch line of the driving pinion is: 2400 + 20 per cent. = 2880 pounds. The necessary velocity ratio of the gears to equal this ratio of power is, therefore: 2880 _ 10.25 ^281"'' ~ I * This reduction must be made between R' and r'\ and R and r\ the pinion r not being considered as its velocity is the same as that of the gear R, there- fore the load on the teeth will be the same. Since it is always best to make the reduction in even steps, and double re- duction is desirable for a ratio of 10.25 ^^ i take the square root of the total reduction, 10.25, which is approximately 3.2 to i for each reduction. Prac- tically, however, a reduction of -^ — and will answer. The ratio between R' and r" is made . Assuming the diameter of the drum r" to be 10 inches, the pitch diameter of the gear R' will be 3.4 X 10 = 34 inches. The ratio between R and r' is -^— , assuming the pitch diameter of the pinion r' to be 7 inches, the pitch diameter of the gear R will be 7 X 3 = 21 inches. The power or circumferential force of the gear R is, of course, that of the driving pinion r, 281 pounds. Therefore, the power of the pinion r', and con- sequently that of the gear R\ is 281 X 3 = 843 pounds. The problem is now reduced to two simple ones, that is — to design a pair of gears r and R to transmit a force of 281 pounds at a speed of 11 73 feet per minute, and a second pair r' and R' to transmit a force of 843 pounds at a speed of 390 feet per minute. It is necessary to assume a pitch judged to be suitable for the different drives and to try its value for carrying the required load by the Lewis formula, ob- taining the safe load per inch of face, and make the face sufficiently wide to transmit the power. For the first pair of gears, r and R, assume 4 diametral pitch — 0.7854-inch circular pitch — allowing 5000 pounds per square inch as a safe stress for raw- hide. Number of teeth in pinion r = 4X4= 16. Safe load per inch of face = spy —- , — zrz—. (See formula 24.) 600 -\- y SPEEDS AND POWERS 55 Or sooo X o.VcSi: X 0.077 . , = 100 pounds ])er inch 01 lace. ^ ' ^ 600 + 1173 Making the face of the gears r and R 3 inches it will safely carry 3 X 100 = 300 pounds, which is sufficient. For the second pair, r' and R' , try 3 diametral pitch — 1.0472-inch circular pitch — both gears of cast iron. Figure the strength of the pinion, as it is the weaker of the two. Allow 8000 pounds per square inch as a safe stress. For a pinion 7 inches pitch diameter, 3 pitch, the number of teeth equals 7X3 = 21 teeth. Factor y iox 21 teeth equals 0.092. IF = 8000 X 1.047 X O.OQ2 ^ , , or 472 pounds per inch of face. 600 + 390 Making the face 3 inches, the gears will carry a load of 3 X 472 = 141 6 pounds. These gears will therefore be heavier than necessary, but owing to the nature of the service this should be the case, especially as they are made of cast iron. From the ratio of this train of gears it will be found that the load will be raised at — ^- — = n S feet per minute, using the full speed of the motor. If 3 X 34 ^ ^ '6 ^ the load must be raised at a greater speed than 115 feet, a more powerful motor would be required, and if at a low^r speed there must be a greater gear re- duction. For instance, if the hoisting speed had been 80 feet per minute the 1 • 111 117^ 14-7 . 1 r IO-2 . ., 1 speed ratio would be — ^- - = ^, instead of as in the example. 80 I I The above problem is generally put before the designer in a different manner — that is the load and speed at which the load is to be raised are given, the size of motor and ratio of gearing, etc., to be determined. Example: A load of 2400 pounds is to be raised at the uniform rate of 115 feet per minute; what size motor and what gears wull be required? Assuming as before a loss of 20 per cent in efficiency in the driven gears, bearing, etc., this load will require: 2880 X 11=; , . , , 00 1 X — 10 horse power (2400 pounds + 20 per cent = 2880 pounds). 33,000 Using a rawhide pinion four-inch pitch diameter on the motor, we consider the problem in the same manner as in previous examples m'aking the ratio of , 2880 10.2c: the gears; ~^^ = -^^ The problem of determining the proper gears is the same. 56 AMERICAN MACHINIST GEAR BOOK RAILWAY GEARS Speed in feet per minute at rim of car wheel V = 88 X speed of car in miles per hour. (9) Speed in feet per minute at pitch line of gear V'= 88 X miles per hour X R. (10) _ pitch diameter of gear diameter of wheel Ratio of gear to wheel R (11) Force at pitch line of gear F = HP X 33^000 Kw X 44,102 , , , or — py . (12) V F V Fiber stress in tooth Traction effort at w^heel Horse power 5 = T = HP = P'fy r: TM 0-375 600 , (See formula 18.) (13) 600 + V Dia. of wheel X teeth in pinion X Speed of car in miles per hour M = revolution per minute of pinion. teeth in gear X 336 (14) (15) (16) Traction effort T = F or Pressure at Pitch Line =3553 Pounds. Tractive Effort = 5040 Pounds. 5040 X 26.8 From Formula 12, F= = 3550 Pounds FIG. 49. RAILWAY GEARS. teeth in gear X 24 X gear efficiency X torque of motor . (17) M X diameter of wheel Example: A car weighing 60 tons driven by four motors accelerating at the rate of i3^ miles per hour, per second, reaches the peak of its start- ing torque when at a speed of 20 miles per hour. The gears are 20 and 67 teeth 2}4 diametral pitch (1.26 inches circular pitch) 5M inch face. The diameter of the car wheels is 38 inches. It is required to know the maximum fiber stress in pinion tooth. The power exerted by motors at its peak is 400 kilowatts (800 amperes at 500 volts). See Fig. 49. SPEEDS AND POWERS 57 Kilowatts per motor Pitch diameter of gear Ratio of gear to wheel Kw = = 100 Kw D' = 67 = 26.8' 7? 26.8 R = — ~ = 0.705 38 Speed of gear in feet per minute Force at pitch line I' = 88 X 20 X 0.705 = 1 241 feet per minute. „ 100 X 44,102 , /* = =3553 pounds. 1241 3553 = 18,400 Fiber stress in pinion tooth S = , ^ , 600 poun s ^ 1.26X5.25X0.09X7 , per square 600+1241 inch. STRENGTH OF GEAR TEETH Lewis W = load transmitted in pounds (same as value F), p' = circular pitch, / = face, y = factor for different numbers and forms of teeth (Table 11), 6* = safe working stress of material, V = velocity in feet per minute, NUMBER 1 VALUE OF FACTOR V 1 1 NUMBER VALUE OF FACTOR y OF TEETH INVOLUTE 20° INVOLUTE 15° CYCLOIDAL RADIAL FLANKS ' OF TEETH INVOLUTE 20° INVOLUTE 15° CYCLOIDAL RADIAL FLANKS 12 0.078 0.067 0.052 27 O.III O.IOO 0.064 13 0.083 0.070 0-053 30 O.II4 0.102 0.065 14 0.088 0.072 0.054 34 O.I18 0.104 0.066 •15 0.092 0.075 0.055 38 0.122 0.107 0.067 16 0.094 0.077 0.056 43 0.126 O.IIO 0.068 17 0.096 0.080 0.057 50 0.130 O.II2 0.069 18 0.098 0.083 0.058 60 0.134 O.II4 0.070 19 O.IOO 0.087 0.059 75 0.138 O.I16 0.071 20 0.102 0.090 0.060 100 0.142 ' O.II8 0.072 21 0.104 0.092 0.061 150 0.146 0.120 0.073 23 0.106 0.094 0.062 300 0.150 0.122 0.074 25 0.108 0.097 0.063 Rack 1 0.154 0.124 0.075 Table ii — Values of Factor y for Lewis Formula 58 AMERICAN MACHINIST GEAR BOOK Safe working stress ^S" for 0.30 carbon steel = 15000 Safe working stress S for 0.50 carbon steel = 25000 Safe working stress S for cast iron = 8000 Safe working stress S for rawhide =^ 5000 AVERAGE VALUES FOR 6* Mr. Lewis' formula for the strength of gears originally read: W = S p' f y, a table being given in which the allowable stress of the material S was reduced as the speed of the gear was increased as follows: SPEED OF TEETH IN FEET PER MINUTE lOO OR LESS 200 300 600 900 1200 1800 2400 Cast Iron Steel 8,000 20,000 6,000 15,000 4,800 12,000 4,000 10,000 3,000 7,500 2.400 6,000 2,000 5,000 1,700 4,300 Safe Working Stresses 5 in Pounds Per Square Inch for Different Speeds Later Carl G. Barth introduced an equation, 600 which gives prac- 600 + F' tically the same result as the table w^hen added to the formula, the value 5 being the safe working stress per square inch of the material used, or 600 W = Sp'fy 600 + V Mr. Earth's equation is the one commonly accepted. It is evident, however, that this value will vary for different conditions, the design and workmanship being important factors in its proper determination. The load is reduced as the speed increases on account of impact. It is evident that an accurately spaced and generated gear should have a much higher value than one cut by ordinary methods. It is also evident that helical and herringbone gears, owing to the nature of their tooth contact, should have a much higher value, as they operate under entirely different conditions, therefore are capable of heavier loads at higher speeds for the same area of tooth contact. Rawhide gears should also have a higher factor, as rawhide w^ill absorb shocks that w^ould affect harder materials. In the absence of all vibration, and with an'indeflectable material, this equa- tion could be eliminated from the formula for strength and wear. These are conditions that can never be attained, but it is evident that this value will stand extended investigation. bo o .£ 2 S .2 ft •*-! 3 0) : £ :^£ .•J a 3 S? . > 0) 3 H 5 ^ rt ?* > j:: ft £ § c . 1 s s o ba ^ !» S 1 •£ ^ -g ^ o o is CO _W tn o y5 £ -^ ftb § E X ^ .9. ft c3 c5 ^ ^ ^ 2 1-1 Csl t-, ^- >w- hJ O *" o •^ 3 ft y 1-H < tn fl ■4-> , - CO i) t^ > rt S c £ o rt rt 2 u c <■ to ^ iJ bO> <^^ C/3 U U W sassaj^s -^^^JJ 6o AMERICAN MACHINIST GEAR BOOK Table 13 was prepared by Henry Hess and published in American Machin- ist to allow the pitch and the face to be found with very little arithmetic. It is based on Lewis' method of calculating the strength of gear teeth. As the bulk of gears in use are either 1 5 degrees involute or cycloid the table has been made'up of these forms. The arrangement of the table is such that, given the number of teeth in the gear, and the quotient obtained from dividing the working load per tooth by the greatest fiber stress, the pitch and face width can be directly picked out. The face width is given in inches and also as a ratio to the circular pitch; it is usual to make the face from two or three times the circular pitch — 2.5 is a fair average. For overhung gears 2 to 2.5 is proper; for gears supported on both sides 2.5 to 3 is good practice. For convenience sake these ratios are repeated at the tops of the vertical columns, while the face widths to the nearest sixteenth higher corresponding to each ratio for each pitch are given at the foot of the various columns. As diametral pitches are generally used for light work the table is arranged for these from 8 to 3.5, and for heavier work for circular pitches from i to 4 inches. The equivalent circular and diametral pitches are marked in lighter faced type. Directions and examples are given on page 61. The formula from which the tabular values were determined is which is but another way of writing the Lewis formula, which, with the notation changed to agree with formula above, is W = .//(o..,-^). the quantity in the parenthesis being the Lewis variable for cycloidal and 15 degrees involute teeth. The change in form is made by introducing for / its value — p' r — as per notation below: W = working load in pounds. S = greatest fiber stress in pounds per square inch. p' = circular pitch in inches. / = face width in inches. n = number of teeth in pinion. r = ratio of face width to circular pitch = -77-. p = diametral pitch. SPEEDS AND POWERS 6l To find the circular pitch p' or diametral pitch p and the face width f: Divide the known load W by the permissible greatest fiber stress S; find the resulting value k in the body of the table in line with the number of teeth in the pinion. Use the pitch given at the top and the face width given at the foot of the column. Example: A gear of eighteen teeth is loaded with 495 pounds per tooth; the permissible fiber stress is 3,000 pounds per square inch. Then = — ^^ = o 3,000 0.165. Opposite eighteen teeth find k = 0.167 under a diametral pitch 3.5 and over a face width of 2.25''. Or find k = 0.166 under a circular pitch of I inch and over a face width of 2 inches. Either solution will do. To find the greatt.t fiber stress S: In the column headed with the pitch used and marked with the face width used at the foot, find opposite the tooth number a constant. Divide this into the working load imposed on the tooth to get the greatest fiber stress. Example: The working load on a tooth of 4-inch circular pitch and lo-inch face in a loo-tooth gear is 28,000 pounds. Opposite 100 teeth, under p' = 4 inches and over/ = 10 inches, find 4.684. Dividing this into the load gives 28,000 4.684 5,970 pounds per square inch greatest fiber stress. Table 14 for the working strength of gear teeth has been furnished by the New Britain Machine Company. It is based on the Lewis formula, and, unlike other tables, gives the strength of the teeth when their size is indicated by diametral pitch. But one width of face is given for each pitch, this face being, as near as may be, that used by the makers of standard gears for the market. The figures in the body of the table give the working load in pounds for a speed of 100 feet per minute for cast-iron gears of the pitch and face found at the top of the corresponding column and of the number of teeth given at the left of the corresponding Hne. The left-hand figure at the top of each column gives the diametral pitch and the right-hand figure the face in inches. 600 For higher speeds the loads are to be reduced by the equation , — TTy used as a multiplier, and for steel gears the loads may be increased in proportion to the safe load for that material. 62 AMERICAN MACHINIST GEAR BOOK ro O ciOO OO O t^vO ^^"^ '0covO Ov Cl CO u^vo t^ O^ ci cO ■'^ u-^vO t^OO O cocO'^-'!J-'^-^'^'*u-)iou->iouoio lOvO vOvOvOvOvOvOvO t^ CO CS d lo Tt-00 vO O -^Cl u.)iocot--0 OOO Cl uo r/^ d M O OvOO r^ O H Cl CO -^vo t^ O O >-" c^ CO U-) \r)\C 00 OO m ci coco-^'O cocOcOCOcccococO'^Tt-'<^'^'sl--T^rt'^iiOU-)U-)iou->ir)io V:i H cl ro rJ-OO f^ Cl O u^J OvOO w Cl O ■^ QvOO 00 O u^rJ-vr^rO'^u-jroO M Cl ro lOvO r^OO O Cl oo •^ u-jvO 1^00 O m ci ro r^j- lovO t-~00 cococOcocococO'^'^'^'^'^'^'^'^upupLouou-jiou-jioio CO up Cl coci COM co'^Ovci Ot^O O r--.t^|-^OvvO tJ-ci OoOvO co O r^OO O O w vO vO X^OO \00 CO (N OvOO u-ju^M O OCI cioo -^coco CvvO cocoO^uomoO TfM t-~ O M Cl CO -^ lO '-0 t^OO OOOvQi-iM^co-^'^ uoo vO t^OO 00 Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl pococococococococococococo Cl CO MOvOU-)MOCvciOt^ciciOvOCii-HMvOcioOrt-OvOci O w M Cl co-'^-^-vO r^t^OO 0^0■0 M Cl cocO'i^'^u-jvOvO t^ Vi C) 00 u^jOoo coO i^r^u-jo uoroO u-jm t^uoO •rfCvcoOv'^O vO t^OO OOOOO'-iCicocO'^u-) u-)\0 vO t^OO 00 00 0^ O O w MI-IHMMCICIClClCICICICIClClCIClClCICIClCirOCO Cl TfO TfO '^Ovo -"tO T^oo ''^ O '^oo --i- O -^oo m vo O '^oo CO -^ '=*• u-> u-)0 vO 1^00 oooOOvOOOmcioicicoco.'J-.'I-'^ ^(^ 00 u- O 00 fO d fO Cl Qv'^ciO coO Ol^ COOO u-)CivO cioOvO civO O u->00 co Ov VO vO t^OO OOOvOi-ii-icicirO'<^T^io u->vO t-- I-^OO 00 00 0^ O- MMWMMMCICICICICICICICICICIOICICICICICICICI 1^ Cl M U-) OvvO Cl u^O\rot^M u-)Ov CO -^ -^ i^ u-;\0 vO t^OO OOOOOvOOOMClClCirocO'rf-^-^ Cl C^ OO covO w tJ-O •^OvOOO mvO m '^OO m t^ m ■^vO m co t^ O O M M Cl Cl cOco^-^^-^iO u-)\0 VO VO 1^ X^OO 00 00 Ov Ov Ov O MMMMMMIHMMMMI-IMMI-IIHMIHHI-IMMMCl H 00 -^ d PC ■^QvcoOvCioO coo u-jOx'^Ov u-joO co t^ covO m ■^OO ci vO 00 Ci Cl CO CO •^ •^ u-)0 vO vo r^ t^OO oOOOvOOmmmcicici \00 CON Cl up civO Ovci u^Ovm O O M >-> M Cl Cl cocO-^-^-^u-ju-jOvOvO r-~r^t^O0OO00 Ov Cl ^ Cl COVO 0\ Cl ir)00 Cl f-. 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Cl Cl Cl (V| (v-)COCococo<^'^coco-rl-Tt-^'^'^^^^3'^g" \00 II [X N M w Cl CO '^^ u^ uo i:^oo 000v0M-oo oo Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl cococococococococococpcpco ooooooooooooooooooooooqo 1^ ^ "^ p^ u u •S CTJ cr m Oh :3 o B :3 11 cij h. k\^ < Ml Pi o .Co H 0) > H-1 ?N ■^ C/3 O a g O u $^3 +-> c3 a (/3 >-i fH ai n CJ ;3 o II S-i •^ •^ SPEEDS AND POWJiKS 63 10 i oc ro t^O^ ''' 000 M OOOO c-i 00 "^OO Oi/r>lO 10 ^M J^TtOO M (N ro '^ ^ 100 :^00 O-OmcnNoo-^lo loo <^ t--00 O On 00 1 oO CN t-.-'^uoiOO lo^OCN l^roiotN 0\ro00 <0 f-~TtiO0 t^OO 0^ M CO CO ■^ 'O'^ t^OO O^O w M oO t^LOOO cs O'^O^'^On'^OO lO r^ r^oo oooOi-HCNCNroco'^io 100 O t^ r^oo 00 On O^ O v» HHMHHMC^(NC^ M On'^cn) r^o cNQ'-OfOOO (OM i/^OOO t^MO OOt^r^QCOOONOO'-ii-JOO COOO >0!N t^MOO 10 O^O t^ >-^ LO '^ On COOO LO 100 t^oo ooONOOMMOioto-^-^Lo LOO t^ t^ HH(HMHI-tMMO)l(NCNi CN 00 Tt-O coOncoiOO m OnOO -> >-* >-< 00 COOO LO CO 00 LO t^OO >0 CO O ■* >0 LO M LO •^ "JO CO)00 coo COOO CS LOt^M LOt^O ■^t^O cioo r^'^O M LoOO O coO lo O lo T}-00 tH LO r^ M LO coo 00 CIO OnC) loOnm Tj-O 00 >-i ■^O 0000 OnQnO^O I-I I-I I-I I-I CN CS 01 COCOCO'^'^'+'^LOLOIO 10 M O^O 000 t^O >-l CNO C^ LOt^LOCOLOM lOLO LOOO O l^ C) LiO 1^ M tJ-00 m CO LOOO M CO LOOO I-I CO LO t^ On M coo »- r^ t^ t^OO OOOOOnOnOOnOOOOi-ii-ii-ii-ii-ic^cicn M M CO ■^ ^ coo 'OLoOOOOO -^OO CI ^00 C4 '^OO ^0 CI On ■^OO CN CN -^00 TtO On hi -^O 00 000000 O^OnOnOnO I-I t-i M c< CI C< cocococOT^'TJ-T^'^ OOOc^woOOOc^MCiOOOooOOOr^t^i^oOO t^O C! LOt^O cor^O cj rj-r^o ci rj-O O ci coloi^Onm Tt r-. f>. t^ t^OO OOOOO-ONONONOOOOMMMMI-fMrOCO LO (N 000000 •'^O 00 cjo 0000 00 CI -rt-OO CN coo t-^ I-I "^O OnCn fOiOt^O I-I CO LOOO On M CO '^O t^ O LO LO LOO 0000 t^ t^ t>. t-»00 0000000000 O^OnOnOnOnm M V ^ 10 M M II II Hxaa i.0 '0 CO COCN t->.\00 CI O'aOO coo cnoO <0C4 O lolo>OLOi-i O I-I -^000 COO Oncn TfO O^O cOL^OO 1-1 rO'tOOO O ci rj- 0000 t--l^i--:^CO00O000 O^ONO^ONO M M M LO (N 10 CI LO COO r^O COCN .t^t^ c« CO ■* 100 t^OO On M CO LO 1^ -^00 coOOioOOOy MMMMHHMMCNC«C«COOJ5 ■^ CO d o O >^ u o H o o w > < 64 AMERICAN MACHINIST GEAR BOOK 1 tA Pi < o Q < < H o w a < Q z < Ph < « H W X r^oo O^O O HI ri ro'TtTt-u-5VD>o l^r^OO O^O^O O >-i m c) fs X o H ro Tf vnO 00 0^ M O NO M tJ-qO r^ r^ O fOOO fONO O- (nj looo f^ lo OMMMCNCiPOfO'^'^'^iO lONO NO O t^ t^ r^OO 00 00 On On X M 'to 'to ^OnO ^O Thoo Tl- O 'too Tl- O 'too M X 00 PO »0 C^ 't fN) 'tNO oO r^r^oO _<~0'^'t"^'^'t't't'^'^'t't M X HivOO O O LoO O lO't'tON^'t 'too rofOrO<~0<0.00 0000OnOnOnOn0000mmmmmcni(n