lifornia Lonal lity Tecl^nical Drawing Series Essentials of Gear ANTHONY '1 Id TECHNICAL DRAWING SERIES THE ESSENTIALS OF GEARING A TEXT BOOK FOR TECHNICAL STUDENTS AND FOR SELF-INSTRUCTION, CONTAINING NUMEROUS PROBLEMS AND PRACTICAL FORMULAS GARDNER C. ANTHONY, A.M. Prokessor of Dravvint. in Tufts College; Dean of the Bromfield-Pearson School, Author of "Elements of Mechanical Drawing," and " Machine Drawing;" Member of American Society for the Promotion of Engineering Education; Member of American Society of Mechanical Engineers BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1897 Copyright, By Gardner C. Anthony, 1897. PEE FACE. The most feasible method for the acquirement of a working knowledge of the theory of gear-teeth curves is by a graphic solution of problems relating thereto. But it requires much time on the part of an instructor, and is very difficult for the student, to devise suitable exam- ples which, while fully illustrating the theory, shall involve the minimum amount of drawing. It is tlie aim of the author to overcome these difficulties by the presentation of a series of pro- gressive problems, designed to illustrate the principles set forth in the text, and also to encour- age a thorough investigation of the subject by suggesting lines of thought and study beyond the limits of this work. In this as in the other books of the series the author would emphasize the fact that the plates are not intended for copies, but as illustrations. A definite lay-out for each problem is given, and the conditions for the same are clearly stated. This is accompanied by numerous references to the text, so that a careful study of tlie subject is necessitated before performing the problems. Although specially addressed to students having no previous knowledge of the principles of kinematics, it is also designed to serve as supplementary to treatises on this subject. The methods and problems have already proved their usefulness in the instruction of stu- dents of many grades : and it is hoped that tlieir publication may promote a wider interest in, and more thorough studv of, the essentials of gearing. GARDNER C. ANTHONY. Tufts College, Sept. 24, 1897. 2065972 COISTTEI^TS. CHAPTER I. PAOK Introduction. — General Principles 1 I. Constant Velocity Ratio. 2. Positive Rotation. 3. (iearing. CHAPTER II. Odontoidal Curves • 4 4. Classes of Curves. 5. Cycloid. 6. Epicycloid. 7. Hypocycloid. 8. To Construct a Normal. 9. A Second Method for Describing the Cycloidal Cnrves. 10. Double Generation of the Epicycloid aud Hypocycloid. 11. Epitrochoid. 12. luvolute. CHAPTER III. Spur Gears and the Cycloidal System 8 13. Theory of Cycloidal Action. 14. Law of Tooth Coutact. 15. Application. 16. Spur Gears. 17. Circular Pitch. 18. Diameter Pitch. 19. Face or Addendum. 20. Flank or De- deudum. 21. Path of Contact. 22. Arc of Contact. 23. Arcs of Approach and Recess. 24. Angle of Obliquity or Pressure. 25. Rack. 26. Spur Gears Having Action on Both Sides of the Pitch Point. 27. Clearance. 28. Curve of Least Clearance. 29. Backlash. 30. Conditions Governing the Practical Case. 31. Proportions of Standard Tooth. 32. In- fluence of the Diameter of the Rolling Circle on the Shai>e and Efticiency of (iear Teeth. v VI rONTENTS. PAGE 33. Intercliaii,n'eal)le (xears. 34. Practical Case of Cycloidal Gearing. 35. Face of (iear. 36. Comparison of Gears, illustrated in Plates 4, 5, and 6. 37. Conventional Representa- tion of Spur Gears. CHAPTER TV. Involute System 26 38. Theory of Involute Action. 39. Character of the Curve. 40. Involute Limiting Case. 41. Epi- cycloidal J^xtension of Involute Teeth. 42. Involute Practical Case. 43. Interference. 44. Influence of the Angle of Pressure. 45. Method for Determining the Least Angle of Pressure for a Given Number of Teeth Having no Interference. 46. Defects of a System of Involute Gearing. 47. Unsymmetrical Teeth. CHAPTER V. Annular Gearing 38 48. Cycloidal System of Annular Gearing. 49. Limiting Case. 50. Secondary Action in Annular Gearing. 51. Limitations of the Intermediate Describing Curve. 52. Limitations of Exte- rior and Interior Describing Curves. 53. The Limiting Values of the Exterior, Interior, and Intermediate Describing Circles for Secondary Action. 54. Practical Case. 55. Summary of Limitations and Practical Considerations. 56. Involute System of Annular Gearing. CHAPTER VL Bevel Gearing 45 57. Theory of Bevel Crearing. 58. Cliaracter of Curves Employed in Bevel Gearing. 59. Tredgold Approximation. 60. Drafting the Bevel Gear. 61. Figuring the Bevel Gear with Axes at 90°. 62. Bevel (Jeai' Table for Shafts at 90°. 63- Bevel (iears with Axes at Any Angle. CONTENTS. VU CTIAPTKK VII. I'AGK Special Forms of Gears, Notation, Formulas, etc 57 64. Odoutograplis and Odoiitograph Tables. 65. Willis's Odoiitograph. 66. The Tliree-Poiiit Odoutograph. 67. The (iraiit Involute Oduntograph. 68. The llobinsuu Odoutograph. 69. The Klein C'ooidinate Odoutograph. 70. Special Forms of Odontoids. and Their Lines of Action. 71. Conjugate Curves. 72. \\'orni (Jeaiiug. "73. Literature. 74. Notation and Formulas. CHAPTER YIIL Problems ...,.,.,, ,68 75. ^Method to l>e Observed in Performing the Problems — PuoiiLK.M 1. Cycloidal Limiting Case. Face or Flanlc Only 69 '2. Cycloidal Limiting Case. Face and Flank 71 o. Cycloidal (iear. Practical Case 72 4. Involute Limiting Case , . . 74 5. Involute Practical Cases 75 6. Cycloidal Annular (iear 76 7. Involute Annular Gear 77 8. Cycloidal and Involute Bevel Gears. Shafts at !H)- 78 9. Cycloidal and Involute Bevel (iears. Shafts at Other than 90" 79 ERRATA FOR THE ESSENTIALS OF GEARING. 0.5 Page l(i, Hue 10, For o.oP read -— Page 18, line 10. For one-sixth read one-seventh. Page 18, line 11. For Fig. 6 read Fig. 9. Page 34, line 4. For — read ^^ Page 69, problem 1, column (i. For A read a. Plate 2. For A' on director circle read A. Plate 10. Erase "Pitch line" near H K. Plate 14. Fig. '2. The pinion is designed to have 5 teeth and the gear fi teeth. THE ESSENTIALS OF GEARING. CHAPTER I. INTRODUCTION. — GENERAL PRINCIPLES. 1. Constant Velocity Ratio. Motion may be transmitted between lines of shafting by means of friction surfaces ; and if there be no slipping of the contact surfaces, the circumference of the one will have the same velocity as the circumference of the other. The number of revo- lutions of the shafts will be inversely proportional to the diameter of the friction surfaces, and this ratio will l)e maintained constant under the condition of no slip. Such friction surfaces and shafts are said to have a co7istant velocity ratio. 2. Positive Rotation. In order to transmit force, as well as motion, and to insure its being positive, it will be necessary to place cogs, or elevations, on one of the friction sur- faces, and make suitable depressions in the other surface. 3. Gearing. The study of toothed gearing is a study of the shape of these cogs, teeth, or odontoids, which are designed to produce a positire rotation Mhile preserving the condition of constant velocity ratio. 1 GEARS CLASSIFIED. Fig. 1. Fig. 2. Fig. 3. Fig. 4. Gears may l)e classified as fol- lows : — 1. If the shafts are parallel, the friction surfaces would be cylinders (Fig. 1), and the gears designed to produce the same condition, as to the velocit}^ are called Spur G-ears (Fig. 2). 2. If the shafts intersect, the friction surfaces would he cones (Fig. 3), and the gears called Bevel Gears (Fig. 4). 3. If the shafts are neither in- tersecting nor })arallel, the friction surfaces will be hyperboloids of revolution (Fig. 5), and the gears called Ilt/perbolic, or Skeiv Grears (Fig. 6). In the preceding cases the ele- ments of the teeth are rectilinear, and the friction surfaces touch each other along right lines. 4. If the elements of the teeth GEARS CLASSIFIED. in either of the first three cases l)e made helical, an entirely dif- ferent class of gearing will result. The various forms are known as Tivisted, Sph-(d, Worm^ and Screw Crearing (Figs. 7 and 8). The action of the latter is analogous to that of a screw and nut. One of these forms is generally eni[)lo_yed as a siibstitnte for hy- perbolic, or skew gears, by reason of the ditficnlty experienced in cor- rectly forming the teeth of snch gears. 5. AnothtM-, allliough bnt little used, form, is that known as Fare (rf(irut(/. The teeth are iisnally pins secnred to the face of cir- cnlar disks having axes perpen- dicnlar. The action takes place at a point only. None of the latter forms can be represented by friction surfaces. Fig. 7. Fig. 8. 4 ODUNTOIDAL CUKVEtS. CHAPTER II. ODONTOIDAL CURVES. 4. The two classes of ciirves comnioiily employed in gear teeth are the cycloidal and the involute. A knowledge of their characteristics and methods of generating is essential to an understanding of their application in gearing. 5. Cycloid. Plate 1, Fig. 1. The cjcloid is a curve generated b}- a point in the circumference of a circle A\hich rolls upon its tangent. The circle is called the describing, or generating circle, and the point is known as the describing, or generating point. In Fig. 1, Plate 1, B is the describing point, and B D C E the describing circle, which rolls on its tangent E B'". Assume a point, C, on the describing circle, and conceive the motion of the circle to be from left to I'iglit. As it rolls upon its tangent, the arc E C will be measured off on E B'" until point C becomes a point of tangency at C. The center of the describing circle will now lie at A', in the perpendicular to E B"' at C\ From center A', with radius of describing circle, draw the ncAV position of describing circle. The generating point must lie in this circle at a distance from C equal to the chord B C. Therefore, with radius equal to this chord, from center C, describe an arc intersecting the new position of the describing circle. The line B' C is called the instantaneous radius, or normaU of the curve at B', it being a [)erpendicular to the tangent of the curve at this point. CYCLOIDAL (UKVES. 5 The normal at B" would be B" D'. The radius A' B' is known as the dt'scrilniif/, or f/enerating radius, and A' C is tlie contact radius, or the radius at the point of eontai-t. In like manner other positions of the describing pt)int may be found, and the curve connecting them will l)e the cycloid required. 6. Epicycloid. Plate 1, F'ig. 2. If the describing circle rolls upon the outside of an arc, or circle, called the director circle, the curve generated will be an epicycloid, Fig. 2, Plate 1. The method of descri1)ing this curve is similar to that for the cycloid, and the lettering is the same. It must be observed, however, that any contact radius, as A' C, is a radial line of the circle on which it rolls. 7. Hypocycloid. Plate 1, Fig. 3. If the describing circle rolls upon tlie inside of a cir- cle, the curve generated will be an hypocycloid. P'ig. 3 illustrates this curve, the same letter- ing Ijeing used as that of the preceding cases. If it be re(j[uired to draw a normal at any point of this, or the two preceding curves, the fol- lowing method may l)e employed : — 8. To Construct a Normal. From the given point on the curve, as a center, with radius of generating circle, describe an arc cutting the path described by the center of the generating circle. From this point draw the contact radius, thus obtaining the contact point. Connect this with the given point, and tlie line will be the required normal. 9. A Second Method for Describing the Cycloidal Curves. Plate 2, Fig. 1. A B C is a director circle, A D E, the generating circle for the epicycloid A A' A" H , and A K L the generating circle for the hypoc3'cloid A L C. b CYCLOIDAL ( UKVES. To describe the epicycloid, assume any point, D, on tlie generating circle, and lay off the arc A D' on the director circle, making it equal to arc A D. If A be the describing point, then A D will be the normal when D shall have become a contact point, as at D'. With L as a center, de- scribe the arc D A', llie describing point A must be in this arc when D shall be at D'. From D' as a center, with radius equal to the chord A D , describe an arc intersecting A' D , and thus deter- mine A', a point in the epicycloid. Similarly obtain other points, and draw the curve. The hypocycloid may be constructed in like manner, as shown by the same figure. This also illustrates a special case in which the hypocycloid is a radial line, A L C, and this is due to the diameter of tlie describing circle being equal to the radius of the director circle. The same method may also be employed in the construction of the cycloid. 10. Double Generation of the Epicycloid and Hypocycloid. Plate 2, Fig. 1. The epi- cycloid may always be generated by either of two describing circles, which differ in diameter by an amount equal to the diameter of the director circle. Thus in the case illustrated, the epicycloid A A' A" H may be generated by the circle A D E , with A as a describing point, or by the circle s T H , with H as a describing point. Similarly the hypocycloid is capable of being- generated by either of two rolling circles, the sum of which diameters must equal that of the director circle.* 11. Epitrochoid. Plate 2, Fig. 2, When the describing point does not lie on the cir- cumference of the generating circle, a curve, connnonly called an epitrochoid, is described. If the point lies without the circle, as at B , a looped curve, B B' B", called the curtate epitrochoid, * For tlie geoiiietriiMl (Iriuuij.sUatiuu of this ])r'>l)lein see tUe aiipeiulix of Professor jNIacCord's ■" Kinematics," page old. INVOLUTE CURVE. 7 is described: and if the point be within, as at D, the curve will be a prolate epitrot-hoid, as D D' D". To obtain a point in the former, assume any point, C, in the circumference of the describing circle, and determine its position, C, when it shall have become a contact point. Draw the contact radius A' C, and from C and A' as centei-s, with radii A B and C B, describe arcs intersect- ing at B', a point in the curve. B' C is the normal at this point. In like manner obtain the point D' in the })rolate epi trochoid. 12. Involute. Plate 1, Fig. 4. The involute is a curve generated by a point of a tan- gent right line rolling upon a circle, known as the base circle, or the describing point may be regarded as at the extremity of a fine wire which is unwound from a cylinder corresponding to the base circle. In Fig. 4, A B C D is the arc of a base circle, and A the point from which the involute is generated. Layoff arcs A B, B C, C D, preferabh' equal to each other, and from points B, C. and D, draw tangents equal in length to the arcs A B, A C, and A D. A line drawn through the extremity of these tangents will be an involute of the base cii-cle A B C D. b THE CYCLOIDAL SYSTEM. CHAPTER III. SPUR GEARS AND THE CYCLOIDAL SYSTEM. 13. Theory of Cycloidal Action. Plate 3, Fig. 1. ].iet H K and M L be the peripheries of two disks, having centers G and F, and S the center of a third disk, also revolving in contact with the arcs H K and M L. The largest disk will be known as disk 1, the second size as disk 2, and the smallest as disk 3, or the describing disk. Consider the ^peripheries of these disks in contact at A, so that motion imparted to one will produce an equal motion in the circumference of the other two, thus maintaining at all times an equal circumferential velocity, or constant velocity ratio. Imagine this to represent a model, disk 1 having a flange I extending below the other disks, and the describing disk as being provided with a marking point at A, each of the disks being free to revolve about their respective axes. Consider first the relation between the describing disk and disk 1, the marking point being at A. Suppose motion to be given disk 3 in the direction indicated by the arrow, so that the describing point shall move from A to A'. The point C of disk 1, which coincides with A when the describing disk is in the first position, will now have moved on the circumference H K, to C, an arc equal to A A'. During this time, the curve A' C will have been drawn upon the flange of disk 1 by the marking point. Next, suppose the marking point to move from A' to A", then, since the circumferences of these disks tra\erse equal spaces in equal times, C will have revolved to C", and the curve A' C will now (ic'fupy the position E" C". But, since the niarking point has continued to describe a, curve upon CYCLOIDAL ACTION. 9 the flange oi cli.sic 1. the curve E" C" will be extended to A". In like manner the marking- point moves to A"', continuing to describe a curve, as C" A" revolves to C" A'". If now the describing disk be freed from the axis on which we have supposed it to revolve, and Ije rolled on the cir- cumference H K, the marking point would describe the same curve, h'" E'" C", as that already drawn, which is an epicycloid. In the same manner, we may imagine the marking point to describe a curve upon disk 2, which curve, in its successive positions, would be shown by A' B', A" B", and A'" B'". For the same reason, too, the arc A A' A" A'" will equal the arc B B' B" B"' ; and if, in a manner similar to the preceding, we roll the describing disk on the inside of the arc M L, we shall describe the same curve A'" D'" B'", and find it to be an hypocycloid. Again, consider these curves, A'" C" and A'" B'", as l)eing traced at the same time l)y the describing point A. If we now observe any special position of the point, as A", it will be seen to be connnon to an epicycloidal, and a hypocycloidal curve, which have a common normal, A" A, intersecting the line drawn through the centers, F and G, at the point of tangency of the disks. This condition is true for all positions of the two curves. If these curves A'" C"\ and A'" B'", be now used as the outlines for gear teeth, as in Fig. 2, G and F being the centers and H K and M L the pitch lines, we shall have obtained a positive rota- tion with a uniform velocity ratio, for it was under this latter condition that the curves were generated, and the connnon normal to the curves at any point of contact will pass through the point A (the pitch point). Such curves are said to be conjugate. It is not necessary that the describing point be on the circumference of the circle, or that the describing curve be a circle, in order to obtain two curves which, acting together, shall pro- duce a constant velocity ratio. 10 LAW OF TOOTH CONTACT AM) .SPl'K (JEAHS. 14. Law of Tooth Contact. In order to preserve the eoiiditiou of coiistaiit velocity ratio, the tootli outlines which act in contact must he such that the common normals at the point of contact shall always cut the line of centers in the same point ; and in general, tiie curves must 1)6 such as may be simultaneously traced upon the planes of rotation of two disks, while re- volving, by a marking point which is carried by a describing curve, moving in rolling contact with both disks. 15. Application. Suppose action to take place between the odontoids, or gear teotli, shovvii in Fig. 2, Plate 3. Let 1 l)e the driver, and suppose motion to begin from the position shown in the figure, the contact being at A. As the motion takes place, points A', A", A'", will succes- sively come into contact, their common normals passing through the PITCH POINT, A, at the time of their contact, thus producing a constant velocity ratio, and the periphery, or pitch LINE, of 1, will have the same velocity as the periphery, or pitch line, of 2. But this uniform motion must cease when points k'" W" come into contact, and the velocity ratio will remain con- stant no longer, unless a second pair of curves begin contact at this moment. Plate 4 illustrates a pair of disks provided with a series of these curves arranged so as to continue the motion indefinitely in eitlier direction. 16. Spur Gears. Plate 4. F is the center of a pinion having twelve teeth, and G the center of a gear of eighteen teeth, only a segment of the latter being shown. A C K is the describing circle, carrying the marking point C, which descril)ed the epicycloid C D, and the hypocycloid C E. The depth of the pinion tooth must be made sufficient to admit the addendum of the gear tooth, but only that portion of the curve between C and E will engage CIRCULAR AND DIAMETER PITCH. 11 C D. The reinaiiider of the pinion Hank may l)e a continuation of tlio hypocN^chiicI, or any other curve which may not interfere with the action of the gear tooth. The oppoSi.te sides of the teeth are made alike in order that motion may take place in either direction. If the direc- tion be that indicated by the arrows, the pinion bemg the driver, the sluuled side of the teeth would have contact ; and if the direction be reversed the opposite faces would engage. In order to accurately reproduce the dedenda of the pinion, a scroll may be used in the fol- lowing manner : — Having selected one to match the tooth curve, C E, continue the curve of the scroll by the center F, from whicli a circle should l)e drawn tangent to the line of the scroll. jNlark that point of the scroll in contact with the pitch circle. Having laid off the pitch, and thickness of the teeth, place the marked point of the scroll to coincide with these points, and at the same time tangent to the circle already drawn. Draw such part of the curve as lies between the addendum and dedendum circles. Reverse the scroll for drawing the opposite side of the teeth. 17. Circular Pitch. The distance A D, or A E, measured on the pitch line between cor- responding points of consecutive teeth, is called the circular pitch, and is equal to the circumference ot pitch circle iiuiiiber ol teeth Let P' denote the circular pitch, D' the diameter of the pitch circle, and N the numl)er of teeth, then will P' = '^ (1), and, p- = ^ (2). 18. Diameter Pitch. In order to ex})ress in a more direct and simj)le manner the ratio between the diameter of the pitch circle and the number of teeth, and to easily determine the 12 TOOTH PARTS. proportions of the teeth, it hiis been found expedient to a})ply the term pitch, or more properly, diameter pitch, to designate the ratio between the number of teeth and tlie diameter of pitch circle. This is not an ahsolufe mrasvre^ hut a ratio ; and since it may usually be expressed by a whole number, the proportions of the parts of a tooth, which are commonly dependent on the pitch, may be more readily determined, and all the figuring of the gear simplified. Designating the diameter pitch l)y P, P = ^, (3). To obtain the relation between the diameter pitch and the circular pitch, compare formulas 2 and 3. -=— , — = ?; lience^, = P or P P' = 7r(4). 'J'liis last e(piation expresses the I'elation between tlie two pitches in a simple form which may ])e easily remembered. Illustration. — Tlie pinion represented in Plate 4 has 12 teeth, and is 3 inches in diameter, k; = P? T "" ^- '^'^^^ pitch, therefore, is 4. The circular pitch, P' = □ = —7 — = .7854. Having given any two of the terms, N, D', P, P', the other terms may l)e determined. 19. Face, or Addendum. Tliat portion of the tooth curve lying outside of the pitch circle is called the face or adtlendum, as C D, Plate 4. 20. Flank, or Dedendum. That portion of the tooth curve lying inside of tht^ pitch circle is called the flaidc or dedendum, as E H , Platp: 4. 21. Path of Contact. In Fig. 1, Plate 3, it will be observed that the contact between the two curves takes place in the arc A A' k" A'". This is called the path of contact, or line of action, and in the C3-cloidal system this line is an arc of the describing circle. ARCS OF CONTArT. 13 22. Arc of Contact. The tire (leseril)e(l l)y a point on the pitch lin(; during' the time of con- tact of two odontoids is called the arc of contact. It must not be less than the pitch. In this case the arc of contact would be measured by the arcs A D or A E, and these arcs being equal to the pitch, the case is called a limiting one. In ^)ractice it should be greater, wliicli w^ould be accomplished by lengthening the addendum. 23. Arcs of Approach and Recess. There are four cases of contact that may take place between the gear and pinion of Plate 4. 1. Gear as driver. Direction opposed to the arrows. Contact begins at A and ends at C. 2. Pinion as driver. Direction same as arrows. Contact begins at C and ends at A. In each of these cases the action will take place between the shaded portions of the teeth. 3. Gear as driver. Direction same as arrows. Contact begins at A and ends at L. 4. Pinion as driver. Direction opposed to the arrows. Contact begins at L and ends at A. In the last two cases there will be no contact between the shaded portions of the teeth. In the lii'st and third cases the contact takes place from the pitch point, and is called an arc of recess. In the second and fourth cases the contact takes place tow^ard the pitch point, ending at A, and is called an arc of approach. It should also be observed that in the case illustrated the arc of contact must be either one of approach or of recess; but had the teeth of each gear been provided with curves on both sides of the pitch-line, as in Plate 5, the arc of contact would have consisted of an arc of approach and of recess. (See Art. 30, page 10, for a further discussion of the relation between these arcs.) 14 ANGLE OF PRIiSSURE. RACK. 24. Angle of Obliquity, or Pressure. Tlie angle which the conimoii normal to a pair of conjugate teeth makes with the tangent at the pitch point, is called the angle of obliquity, or angle of pressure. The angle CAP, Plate 4, is the angle of greatest obliquity. The greater this angle, the greater the tendency to thrust the gears apart ; the friction will be increased and the component of force tending to produce rotation will be decreased. 25. Rack. If the diameter of the gear be indefinitely increased, the pitch circle will finally become a right line, and the gear will then be known as a rack. The rack shown in Plate 4 has teeth only on one side of the pitch line, like the pinion and gear, and the conditions of action are similar. The tooth-curve will be a cycloid, and the rolling circle, M N 0, must be the same as that used for the engaging pinion, in order to fulfil the general law for maintaining a constant velocity ratio (Art. 14, page 10). 26. Spur Gears having action on both sides of the Pitch Point, Plate 5. If we assume the diameters of pitch and rolling circles to be the same as before, and the arc of action, C A, un- changed, tlie addendum of gear and dedendum of pinion will be the same as those of Plate 4. This case, however, differs from the preceding in that the number of teeth is but half as great, and therefore the pitch will be doubled. This will require the arc of action to be doul)led, in order that it shall equal the pitch (Art. 22, page 13). Such increase in the arc of action may be made by continuing the path of contact to the other side of the pitch point, following the circumference of a rolling circle which may or may not be equal to the other rolling circle. Having laid off the arc A H equal to one-half the circular pitch, describe the curves H K and H L , with H as the generating point of the new rolling circle. The former of these curves will (UllVE OF LEAST CLEAIIAXCE. 15 beL-uiiiL' the addiMuliini of the pinion, ami the Litter the dedencliini of the gear tooth. The en- gaging gears ^vill then liave both faces and flanks, the action ^vill l)egin at C and end at H , the path of contact will be C A H, the arc C A being the path of approach, and A H the path of recess, their snm being equal to the circular pitch. In a similar manner the dedendum of the rack tooth may be described to engage the adden- dum of the pinion tooth, and the contact begun at N Avill end at 0, N M being the path of approach, and M the path of recess. That portion of the dedendum of rack tootli which engages the addendum of the pinion is indicated b}- sectioning, but it is necessary to continue the dedendum to a depth sufficient to allow the addendum of the engaging tooth to enter. 27. Clearance. The space between the addendum circle of one gear and the dedendum circle of an engaging gear is called clearance. Fig. 9, page 17. 28. Curve of Least Clearance. If the pitch circle of the gear ])e rolled on that of the pinion, and the epitrochoid of the highest point, C, of the gear tooth be determined, it will be the curve of least clearance. The successive positions of the tootli, when so revolved, are sliown by the dotted line in Plate 5, and the line connecting these points wotdd ])e the desired curve. This nui}^ be obtained as follows : Assume any point, R on the pitch circle of pinion, and la^• off ai'c A R' on the pitch circle of gear, equal to arc A R. From R, with radius R C, equal to R' C, describe an arc. Similarly describe other arcs, and draw a curve touching these arcs on the inside. This curve will be tlie curve of least clearance.* * See also the method of Akt. 71, page til. 16 CONDITIONS (;()A^Ei:\lN(; TIIK PRACTICAL CASE. 29. Backlash. In order to allow for unavoidable inaccuracies of workmanship and operat- ing, it is customary to make the sum of the thickness of two conjugate teeth something less than the circular pitch. This insures contact between the engaging faces only. 30. Conditions governing the practical case. From a consideration of the foregoing limiting cases, the following principles are deduced, to which are also added the limitations and modifi- cations established by practice. 1. The curves of gear teeth, which act to produce a constant velocity ratio, must be described by the same circle rolling in contact with their respective pitch circles. (Akt. 14, page 10.) Practical considerations limit the diameter of the describing circle to a maximum of about — , or equal to the radius of the pitch circle, and a minimum of about 1^ P', or 6.5 P. See also Art. 32, page 19. 2. The arc of contact must equal the circular pitch, and in practice exceed it as much as possible. 8. The addendum of a gear tooth engages the dedendum of the pinion, and the action between them either begins or ceases at the pitch point. Since the addendum and dedendum of any tooth are independent curves, they may be described by rolling circles differing in diameter. 4. In the limiting cases considered, the height of the tooth is dependent on the arc of con- tact, but in practice, the arc of contact is made dependent on the height of the tooth. While it is an almost universal custom to make the addenda of engaging teeth equal, there are special cases, in which very smooth-running gears are required, where it would be advan- tageous to make the addenda of the driver less than those of the driven gear, thus increasing the arc of recess, or decreasing the arc of approach. PROPORTIONS OF STANDARD TOOTH. 17 The approacliing uetion Ijeing- more (letrinientiil. by reason of the friction induced, it is common to de- sign clock gears so as to eliminate this by providing the driver with faces only, and the driven with flanks only. Or, if the gears are made with both faces and flanks, to so round the faces of the driven gear tluit no action may take place. 31. Proportions of Standard Tooth. The propor- tions most connnonly accepted for cut gears are those illustrated in Fig. !'. The dimensions are made depen- dent on the pitch, as follows : — Addendum, (S) = - ■. — diam. -pitch Dedendum, (s + f) = ~r. : — : + clearance = - + f . diam pitch r Thickness, (t) = ^ circular pitch = y = 7^ • Clearance, (f) = ^ addendum = |= ^ or, f = ^^ thick- t P' TT ness of tooth = —=—: = „„ „ • iO 20 20 P In assuming this value for the thickness of the tooth the backlash is taken as zero, but of course the Fig. 9. 18 INFLUENCE OF THE ROLLING CIRCLE. tooth must be slightly smaller than the space to permit of freedom in action. If there he any backlash the value of t will be e.rcuiar pitch -backlash ^ j,^ rough cast gears the backlash may be as great as ^Vth the circular pitch, but this amount is very excessive. It is, however, in- consistent to base the values for backlash, or clearance, on the pitch, since an increase in the size of the tooth, or pitch, does not necessarily mean a proportional increase in tlie allowance to be made for the inaccuracies of workmanship. Indeed, l)()th these clearances nuist be left to the judgment of the designer. Fillets. The circular arc tangent to the flank and dedendum circle is called the fillet. It is designed to strengthen the tooth by avoiding the sharp corner at the root of the tooth. A good rule is that of making the radius of fillet equal to one-sixth of the space between the teeth, measured on the addendum circle, as in Fig. 6. Tlie limit of size may be determined by obtaining the curve of last clearance. Art. 28, page 15. 32. Influence of the Diameter of the Rolling Circle on the Shape and Efficiency of Gear Teeth. If the height of the teeth be previously determined, any increase in the diameter of the describing circle will increase the path of contact and decrease the angle of pressure. Bat since an increase in the diameter of the describing circle produces a weaker tooth, by reason of the undercutting of the flank, as shown in Fig. 12, page 21, the maximum limit of the diameter is connnonly made equal to the radius of the pitch circle within which it rolls. As was shown iji Art. 9, page 5, this will generate a radial flank. In the case of gears designed to trans- mit a uniform force, and not subjected to sudden shocks, it is desirable that the teeth have radial flanks, and consequently the diameters of the rolling circles will be equal to the radii of the pitch circles within which they roll. If the force to be transmitted be irregulai-, and the INFLUENCE OF THE KULLING (IKCLF. 10 teeth required to sustain sudden strains, it is better that the flank be made wider at the dedendum circle, and a describing- circle chosen of a diameter sufficiently small to produce the desired result. In general, the diameters of the descril)ing circles D' 5 will lie between the values of -^ and - . The second i. r value was used for the describing circle of the gears in Plate 5, and would describe radial flanks for a gear having ten teeth. Fig. 10 illustrates the effect of a change in the di- ameter of the rolling circle on the path of contact and angle of pressure. Two gears of equal diameter are supposed to engage, and the teeth are described by roll- ing circles of equal diameter. K P is the addendum, and P L the dedendum of the tooth described b}- the rolling circles C P, and P D, which are of the same diameter, and equal to one- quarter of the pitch diameter. A C being the ad- dendum line of the engaging gear, C may be considered as the first, and D as the last, point of contact. The arcs C P and P D constitute the i)a-th of contact, and the angle C P H is the angle of pressure. Next consider the describing circle as increased, its Fig. 10. 20 INTERCHANGEABLE GEARS. Fig. 11. diuineter being equal to one-half of the diameter of the pitch circle. The form of the tooth will now be E P F, and the path of contact A P B. In the latter case the arc of contact will be greater, the maximum angle of pressure less, and the tooth weaker than in the former. The relation between the two cases may be more exactly stated as follows : — Diameter of describing curve, — — . Arc of contact, i.08 P' 1.35 P'. Maximum angle of pressure, 32.3° 20 3\ Again, the weakness of the tooth in the second case may be partially overcome by reducing the height of the tooth, and in general this would be advantageous, the so-called standard tooth being too high for the best results. 33. Interchangeable Gears. Since the same diameter of rolling circle must be used for the addendum of pinion tooth, and the dedendum of engaging gear tooth, it follows that for any system of interchangeable gears, the addenda and de- denda of all teeth nuist be described by the same descril)ing cur\^e. It is also necessary that the pitch, and proportion of the teeth, be constant. PRACTICAL CASE. 21 In pi-aetice, it is common to regard g'eai*s of twelve or t»f fifteen teeth as the hase of the system, and the diameter of the rolling circle is made equal to the radius of the corre- sponding pitch circle, thus describing teeth with radial flanks for the smallest gear of the set. If twelve be adopted as the smallest number of teeth in the system, the diameter of the pitch circle will be D' = - = — , and the diameter of the de- scribino- circle will be t-t; = -• & 2 P P Again, if a fifteen-toothed gear be used as the base of 7 5 the system, the diameter of the describing circle will be — . Figs. 11 and 12 illustrate a fifteen and a nine-toothed gear engaging a rack. The diameter of the rolling circle by which the teeth were described is — , which will equal 3.75 inches for a 2 pitch gear. The fifteen-toothed gear Mill have radial flanks, but the nine-toothed gear will have the flanks much undercut by reason of the diameter of the rolling circle exceeding the ra- dius of the pitt-h circle. 34. Practical Case of Cycloidal Gearing. Plate 6. Let F and G be the centers of pinion and gear having twelve and eighteen teeth respectively, and a diameter pitch of 4. The Fig. 12. 22 SPUR C4EARS. pitch diameters will equal 5=7"='" iiif'lit's, and p ^t =-l.V inches (Art. 18, page 12). If the tooth be of standard dimensions, the addendum and dedendum lines may be determined and drawn by Ai;t. 31, page IT. The diameter of the rolling circle is assumed to be 1^ inches for the addendum and dedendum of both gears. Since the teeth should usually l^e shown in con- tact at the })itch point, suppose the generating point of the describing curve to be at this point, and describe the curves by rolling the circles from this position, first on the inside of one pitcli circle, and then on the outside of the other pitch circle, thus obtaining the flank of one tooth, and the engaging face of a tooth of the other gear. An enlarged representation of these curves is shown in Plate (J. They may be di-awn by the methods of Arts. 6 and 7, l^age 5, or by Art. 9, page 5, but care should be used to draw them in their proper relation to each other, as sliown in the figure, so that it may not be neces- sary to reverse the curves in order to incor})orate them into tooth forms. The order for the drawing of the curves ma}* be A B, A T, A D, A S. Instead of reproducing the tooth curves by means of scrolls, it is sufficiently accurate, and much more rapid, to approximate them by circular arcs. Plate 2, Fig. 3, illustrates a simple method which closely approximates the curves of this system, and suffices for the ordinary drawing of a geai', but in no case should be used for descril)ing the curves for a templet. This method consists, Jirsf, in the construction of a normal for a point of the curve at a radial distance from the pitch line equal to two-thirds of the addendum or dedendum of the tooth; 8eco)id, in the finding of a center on this normal, such that an arc may Ije described through the pitch point, and the point of the tooth already determined. A P is the height of the adden- dum, and B a point radially distant from the pitch lint', equal to - A P, through which the arc PRACTICAL CASE. 23 B E is drawn. When the point E of the descril)ing cnrve shall have become a point of contact, as at E', the arc E' P being equal to E P, the })oint P will have moved to T, the chord T E' being equal to the chord E P. T will be a point in the addendum, and T E' the normal for this point. From a point, M , on this normal, and found l)y trial, describe the arc P T, limited by the ad- dendum line. Similarly the curve of the dedendum may be determined. Having determined such centers as may be required for describing the tooth curves, draw circles through these centers, as indicated in Plate 6, to facilitate the drawing of other teeth. The radius for the dedendum is often inconveniently great, and in such cases it is desirable to use scrolls, employing the method of Art. 1(3, page 11. Next divide the pitch circle into as many equal parts as there are teeth, beginning at the pitch point. From each of these divisions lay off the thickness of the teeth. If there be no backlash, this thickness will equal one-half the circular pitch ; but if an amount be determined for backlash, the thickness will equal P' — backlash 2 The circle of centers having been drawn, the tooth curves sliould be described. These will be limited l)y the addendum and dedendum circles already drawn. Finally draw the fillets. The maximum angle of pressure between the pinion and gear will be 24°, the arc of approach .52, the arc of recess .48, and tlieir sum, which is the arc of contact, 1 inch, or 1.27 times the circular pitch. The rack teeth would be similarly described. The pitch line being a right line, the circular pitch may be laid off directly by scale, or spaced from the pinion. The approximate method may be used for the tooth curves, and lines drawn parallel to the pitch line, for the centers of the arcs which ajjproximate the addenda and dedenda of the teeth. 24 GEAR FACE. Fig. 14. Fig. 15. 35. Face of Gear. In the previous consideration of gear teeth no attention has been paid to the width of the gear, or, as it is commonly termed, the face of the gear. Tliis dimension is one of the factors to be considered in determining the strength of the tooth, wliich is a subject apart from the kinematics of gearing. It should be ob- served, however, that the tooth having appreciable width, nuist be generated by an element of a rolling cylinder in place of the point of a rolling circle. 36. Comparison of Gears, illustrated in Plates 4, 5, and 6. In tlie three cases previously considered, the di- ameter of the pitch circles are equal, and only one diam- eter of rolling circle has been used. In Plates 4 and 5 the arc of contact is equal to the circular pitch ; 1)ut the pitch of the latter is twice as great as tlie former, hence there are but half as many teeth. In Plate 6 the arc of contact is made dependent on the height of the tooth, which is a standard so chosen as to permit of an arc of contact sufficiently long for a practical case. But in Plates 4 and 5 the height of the tooth is dependent on the arc of contact, which latter is made the least possible. COXYENTIOXAL KKPIIE^ENTATION OF SPUR (iEAPvS. ZO Till' number of teeili in the pinions of Plates 4 and 6 is the same; hut in tlie former the action is only on one side of the pitch point, there being no a(hlenda to the teeth, hence the limited arc of contact. In Plates 4 and 5 there is contact between only one pair of conjugate teeth, save at the instant of beginning and ending contact; while in the case of Plate 6, two pairs of conjugate teeth may be in contact during a part of the arc of contact. 37. Conventional Representation of Spur Gears. In making drawings of gears, it is usually best to represent them in section, as in Fig. 14. This enables one to give complete informa- tion concerning all details of the gear, save the character of the teeth. If the latter be special, an accurate drawing of at least two teeth and a space will be required. Should it be neces- sary to represent the geai-s on the plane of their pitch circles, as in Plate 6, they may be shown as in Fig. 13, thus avoiding the representation of the teeth. Again, if it be necessary to show a full face view of the gears, the method illustrated in Fig. 15 may be employed to advantage. This is simply a system of shading ; and no attempt is made to represent the proper number of teeth, or to obtain their projection from another view. 26 INVOLUTE 8Y8TEM. CHAPTER IV. INVOLUTE SYSTEM. 38. Theory of Involute Action. If the describing curves l)e other than circles we shall obtain odontoids differing in character from those already studied ; but so long as both pinion and gear are described by the same rolling curve, the velocity ratio will remain constant. The class of odontoids illustrated by Plate 7, Fig. 1, is known as the involute, or single-curve tooth. This curve cannot be described by rolling circles, but may be generated by a special curve rolling in contact with both pitch surfaces.* But as the curve may be described by a much more simple process, the above statement is of interest only as showing the conformity of the curve to the general law. (Art. 14, page 10.) F and G, Plate 7, Fig. 2, are the centers of two disks designed to revolve about their respective axes with a constant velocity ratio, which is maintained in the following manner: — Suppose tlie disks to be connected by a perfectly flexible and inextensible band, D C B A, which being wound on the surface of one, will be unwound from the other, after the manner of a belt, producing an equal circumferential velocity in the disks. Conceive a marking point as fixed to the band at A, so that during the motion from A to D, curves may be described on the extensions of disks 1 and 2, in a manner similar to that described for the generating of the cycloidal curves. When the point A, on tiie band, shall have moved to B, the curve Xj B will * For descTlption of this method, .see MaeCord"s Kinematics, page Ui5, Airr. 279. TIIK IX VOLUTE crRVE. 27 have been described on the exteii.siuii (if disk 2. and B Aj , on that of disk 1. When tlie motion of the marking point shall have continned to C, Xg Yj C will have l)een described on the extension of disk 2, and Ag B^ C. on that of disk 1. Finally, when the marking point shall have reached D, the curve Xg ¥3 Z^ D ^\ill have lieen described on tlie extension of disk 2. and Ag B^ Cj D on the extension of d'sk 1. If these curves be made the outlines of ffear teeth, and the former act aoainst the latter so as to produce motion opposed to that indicated by the arrows, a uniform velocity ratio will be maintained between the disks. On investigation, these curves will be found to be involutes, Ag D being an involute of the periphery of disk 1, and Xg D, an involute of disk 2. The curves may. thei'efore, be descrilied liy tlie method for drawing an involute (Art. 12, page 7), the path of contact, A D, being spaced off on the base circle from A to Ag, and the involute drawn from Ag; or the line A D may be conceived as wrapped about the base circle l)eginning the curve at D. 39. Character of the Curve. Plate 7, Fig. 1, represents the involute curve of Fig. 2 incorporated into gear teeth. It becomes necessary to continue the line of the tooth within the periphery of the disk, which will now l>e designated as the base circle, so as to admit the addenda of engaging teeth. This portion of the tooth is made a radial line. The pitch point being at B, (the intersection of the line of centers and the line of action), the pitch circles will be drawn through this point. The circles from wliich tlie involute curves are described, are called base Base Circle Defined. . , , ■ ^■ i i • iji i^it cirdi's. J heu' diameters liear the same ratio to each otiier as do the (ham- eters of the pitch circles. 28 CHARACTER OF THE INVOLUTE. The Path of Contact '^'^ ^^'^^ "^^ actioii, OF patli of coiitact, is ii right lliie tangent to the' Ija.se cir- a Right Line. (j[gy_ jj^ jg ^]jg jjj^g followetl b}' the marking point of the modeh Plate 7, Fig. 2. Since the path of contact is a right hne, and as tlie common normals at the point of con- constant Angle of ^'^•^^' i^^st alwajs pass through the pitch point (Art. 14, page 10), it fol- pressure. lows tluit the line of pressure, or angle of the normals, is constant. The action between the teeth of the gears in Fig. 1, begins at A, and ends at D, taking Limit of Action placc Only bctwccn the points of tangency of the line of action and l)ase circle. No involute action can take place within the, base circles. If the distance between the centers of the gear be increased or decreased, the angle of pres- sure, and length of the path of contact will be increased or decreased, but the involute curve, which is dependent on the diameter of the base circle only, will remain unchanged. Hence, any An Increase in the cen- c'liaugc iu the distauce bctwccn tliB ccuters of two involute gears will not '^Affec\'^^hrve°ioci"°'^ chaugc tlic vclocity ratio, provided the arc of action is (Mpial to the circular Ratio. pitch. The case illustrated by Fig. 1 is a limiting one ; and therefore an in- crease in the center distance would mean an increase in the height of the tooth, in order that the arc of action shouhl e(|ual tlie increased [)itch, an increase in the center distance necessitating an increase in tlie diameters of the pitch circles, and therefore in the circular pitch. But wliile the action between the teeth continued, the velocity ratio would remain constant. Since the angle of pressure is constant, and the paths of the elements of a rack tooth are right The Involute Rack b'lcs, it folh»ws tluit tlic tootli outliue of au iuvolute rack nnist be a right tooth, a Right Line. ]j,j^,^ perpcudicular to the angle of pressure. Plate 8 illustrates a rack for an inxohite gear, having an angle of pressure of about 30". (The section lined portions are not involute.) INVOLUTE LIMITIXO CASE. 29 40. Involute Limiting Case. Pi.ati-: S. Let the diameters of the pitch cireles, the angle of pressure, and the nnniljer of teeth, l)e given. Having drawn the })iteh cireles ahout their respective centers, F and G, ohtain the hase circles as follows : — Through the pitch point, B, draw A D. making an angle with the tangent at the pitch point equal to the angle of pressure, 'i'his will l)e the line of action ; and perpendiculars, F A and D G, drawn to it from centers F and G. will determine the radii of the base circles, and the limit of the action, or path of contact, at A and D. This is a limiting case, in that the path of contact is a maximum, and the arc of contact equal to the circular jjitch. Next determine the point, C, ])}• spacing the arc, D K C. e([ual to DA: A and C will he two points in the involute curve of the base circle, D K c, from which other points may be obtained. Similarly describe D P, the involute of the other l)ase circle, just beginning contact at D. The height of the teeth will be limited by the addendum circles drawn through D and A, from centers, F and G. The dedendum circles are made to admit the teeth without clearance. The pinion teeth are pointed, and the gear teeth till the space, having no backlash. The circular pitch may be found by divid- ing tlie circumference of the pitch circle into as many parts as there are teeth, or the teeth ma}' be spaced on the base circle.* The rack is made to engage the pinion in the following manner : — ■ being the pitch point of rack and pinion, the right line, R, drawn througli this point, and tangent to tlie base circle, will be the path of contact for motion in the direction indicated by the arrow. The contact will begin at R and end at S, the latter i)oint being that of the intersection of the [)ath of contact and addendum circle. The rack tooth will be perpendicular to the line of action, R S: and the thickness of tooth will equal that of the gear tooth, there * For further details coneeniiiig the construction of this pinion anJ gear, see Problem 4, Page 74. 30 EPICYCLOIDAL EXTENSION OF INVOLUTE TEETH. being no backlash in either case. The addendum of the rack tooth will be limited by the parallel to the pitch line drawn through the first point of contact, R ; and the dedendum made sufficiently great to admit the pinion tooth without clearance. 41. Epicycloidal Extension of Involute Teeth. The extent of the involute action between the gear and the pinion of Plate 8 is limited to the path D A ; for while an increase in the height of the gear tooth is possible, the limit of the engaging involute tooth is at A, since no part of an involute curve can lie within its own base circle. It is, however, entirely feasible to continue the contact by a cycloidal action, in the following manner : — The angle FAB being a right angle, the circle described on F B as a diameter must pass through A. This point may therefore be considered as a point in an epicycloid, described by the rolling circle FAB, and having A B for its normal, which is also the normal for the involute. I)ut this diameter of rolling circle being one-half the pitch circle within which it rolls, the hypocycloid will be a radial line, and the dedenda of the teeth wall be radial within the base circle. By rolling the same circle on the outside of the gear pitch circle, the addenda of the gear teeth may be extended, and the path of contact continued to N, which is a limit in this case, by reason of the gear tooth having become pointed. Similarly, the addendum of the rack tooth may be extended l)v the same describing circle. In the figure it is made sufficiently long to just clear the dedendum circle required for the pointed gear tooth. The action will now begin at Q, follow the I'oUing circle to R, and then, becoming involute, continue to S. 42. Involute Practical Case, Plates 9 and 10. Having given the number of teeth of engaging gears, and the diameters of their pitch circles, it is required to determine the curves for the involute teeth of a pinion, gear, and rack. INVOLl'TE PHACTICAL CASE. 31 The diameters of the (lescrihing circles would be of lii-st consideration in cycloidal gearing ; while in the involute system, the angle of pressure or line of action nnist first be established; and tangent to this the base circles may be drawn. By reference to Plate 7, Fig. 1, it will be seen that with a constant center distance, a decrease in the angle of pressure will necessi- tate an increase in the diameter of the base circles, and a corresponding decrease in the path of contact. That is to say, an increase in the possible length of the path of contact means an increase in the angle of pressure. In Plate 7, Fig. 1, this angle is too great for actual prac- tice, being about 30°; yet it cannot be lessened in this case, as the number of teeth is limited. Practice lias limited this angle to 14^° or 15°, which is, unfortunately, too small; but as one of these angles is generally adopted in the manufacture of gears, the latter will be used in the follo\\ing problem : — A pinion of 12 teeth is required to engage a gear of 30 teeth, and a rack, the diameter pitch being 1. The former is illustrated by Plate 9, and the latter by Plate 10. Pinion diameter = ci' = -=-=l2 inches. Gear diameter = D' = ^ = ?^ = 30 inches. (Airr. 18, page 11.) Since the teeth are to be of standaid dimensions (Art. 31, page 17), the addenda will Ije 1 inch, the dedenda li inches; and there being no backlash, the thickness of the teeth will be half the circular pitch, or ^. The circular pitch, P', =p = 3.1416. Draw tlie pitch circles. The line of action will pass through the pitch point, making tiie recpiired angle with the common tangent at this point. Next draw the base circles tangent to tliis line, and determine the points of tangeucy, D and A. Construct the involutes of these base circles in the manner OZ INTERFEREXCE. Fig. 16, indicated by Fig. l(i, and according to the nietliod for descril)- ing an involnte, Ai;t. 12, page 7. It will now l)e seen that the gear tooth will be limited by the arc drawn tlirougli D, the point of tangency of liase circle and line of action. If, however, the involnte cnrve be continued to the addendum circle, as shown by the dotted line, C E, it will interfere with the radial portion of the pinion flank, which lies within the base circle. The pinion tooth will have no such limitation, since the addendum circle intersects the line of action, D A, at L, a considerable distance from the limit of involute action, at the point A. Similarly, the rack tooth will be found to interfere with the pinion flank, if extended beyond the point C, Mhich comes into contact at the point D, the limit of involute ac- tion. But the pinion face may l)e extended indefinitely, so far as involute action is concerned. The remedy for this interference is treated of in the following article. 43. Interference. Since practical considerations demand the maintenance of a standard proportion of tooth, two schemes are adopted for avoiding or correcting this inter- ference, observed in Plates 9 and 10. The first is to hollow that j)art of the pinion flank lying INFLUENCE OF THE ANGLE OF PRESSTRE. 33 within the base circle so as to clear the interfering part of the gear, or rack tooth. In this case there will be no action beyond the point of tangency D. The second method consists in making the interfering portion of the addendum an epicycloid descril)ed by a circle of a diam- eter equal to the radius of the pinion pitch circle. Such a describing circle would generate a radial flank for that part of the curve lying within the base circle. By this means, the action will be continued and the velocity ratio maintained, although the action will cease to be involute. AuT. 41, i)age 30. 44. Influence of the Angle of Pressure. The interference may be entirely obviated by sufificiently increasing the angle of pressure ; but in the case cited (Plates 9 and 10) it would necessitate an angle of 24.1°, which is too great for general use. Had the number of teeth in the pinion been greater, the interference would have been less, and with 30 teeth in the pinion, there would have l)een no interference. See Art. 45. The angles of 14 i° and 15°, commonly adopted, are unfortunately small. There is, how- ever, a tendency to increase this angle, and gears for special machines have been made with a 20° angle of pressure. This latter angle will permit gears having 18 teeth to engage without interference, and the thrust due to this increase in the angle of pressure is an insignificant amount. A system based on this angle of pressure would unquestionably be an improvement over the present one. 45. Method for determining the Least Angle of Pressure for a Given Number of Teeth having no Interference. Fig. 17. Let A be the center of a gear having A B =^ R foi' the radius of pitch circle, and D B T the 34 LEAST ANGLE OF PRKSSURE WITHOUT INTERFERENCE. Fig. 17. angle of pressure to l)e (leterinined, the least number of teeth l)eing N. Suppose the gear to engage a rack having standard teeth, then will C = I D 2R - = — - . D will be the last point of con- P N N ^ tact, and A D = r, the radius of the base circle. A C = A C-R p-R N ~ N A C:A D::A D:A B, hence, A D2 = r2 = A BxAC = R'-2(N — 2) 1 ^ /N-2 The angle of pressure, D B T, is equal to an- gle D A B = p , and the cos. p = 1 = R V N = y/NZ2 V N Hence the COS. of the angle of pressure \/¥<''>- By substituting in the above formula, it will be seen that for a 12-toothed gear to engage a rack without interference, the angle of action must be 24.1°, and for 15 teeth the angle would be 21.4°. Again, if the angle be 15°, the least DEFE( T>s OF THE INVoLI'TE SY.STEM. OO iiuinbei- of teeth that will engage without interference will be 80, while with a 20" aiigie of pressure the least number would be 18. 46. Defects of a System of Involute Gearing. As in the case of the cycloidal system, it is desirable to make all involute gears having the same pitch to engage correctly. In cycloidal ge'dYs this was attained by the use of one diameter of rolling circle for all gears of the same pitch (AiiT. 33, page 20). In the involute system we assume an angle of obliquity, or pressure, which is constant for all geai-s ; but unless this angle be great, gears having so few as 12 teeth cannot be run together without interference. To obviate this difticulty we must adopt one of the two methods already described (Art. 43, page 32) ; namely, the undercutting of the interfering flanks, or the rounding of the interfering addenda. P^irst consider the hitter, which is illustrated l)y Plates 9 and 10. We have seen how that portion of the gear tooth adden- dum lying beyond the point C must be made epicycloidal in order to engage the radial part of the pinion flank which lies within the base circle ; also that the pinion addenda might be wholly involute since there would be nt) interference with the gear tooth flank, the action between the latter taking place without the base circle. But if a 12-toothed gear be taken as the base of the system, it will be necessary to round, or epicycloidally extend that portion of the pinion addendum lying beyond the point K, since this would be the last point of involute action between two 12-toothed geai-s. Tlierefore when the 12-toothed gear engages one having a greater number of teeth, that part of the addendum lying beyond this point will no longer engage the second gear, and the arc of contact will be greatly reduced. Again, suppose a pair of 30-toothed gears to engage (each being designed to engage a 12-toothed pinion), the only part of the tooth suitable for transmitting a uniform motion is that lying between the base 36 UNSYMMETRICAL TEETH. Pig. 18. circle and point C, Plate D, and the arc of contact would be but 1.05 of the circular pitch. Now, one of the claims made for the involute tooth is that the distance between the centers of the gears may be changed without changing the velocity ratio ; but in this latter case it cannot be done without making the arc of contact less than the circular pitch. If the system of midercutting the flanks be adopted, the addendum will be wholly involute; and in the case of Plate 9 all of the })inion addendum would have been available for action, Init the pinion flaidv, within the base circle would have been cut away so that there would have been no action of the gear addendum between C and E. If, however, the engagement had been between two 30- toothed gears, all of the tooth would have been available for action, and the arc of contact would have been equal to 1.91 of the circular pitch. Thus it will be seen that involute gears should be de- signed to engage the gears with which they are intended to run, if the best results would be attained. This would, of course, prevent the use of the ready-made gear or cut- ter, but would insure a longer arc of action between con- jugate teeth. UNSYMMETinCAL TKETH. 37 47. Unsymmetrical Teeth. Fiy-. IS. A very dcsiialiK', altliou^li little used, form of tooth is that known as the inisynimetrieal tooth, which usually combines the cycloidal and involute systems. Fig. 18 illustrates a pinion and gear haying the same number of teeth as those illus- trated l)y Plate 4, and the arc of contact is unchanged ; but the angle of pressure is much reduced, and the strength of the tooth increased. As the involute face of the tooth is designed to act only when it may be necessary to reverse the gears, and when less force would usually be transmitted, the angle of pressure may be made greater than ordinary. In this case the angle is 24.1°, which is sufficient to avoid interference in a standard 12-toothed gear (Art. 45, page 33). But this angle is no greater than the maximum angle of pressure in Plate 4. This reinforcement of the \ydck of the tooth makes it possible to use a much greater diameter of rolling circle ; and in the case illustrated, the diameter is one-third greater than the radius of the pitch circle. Tliis increase in the diameter of tlie rolling circle would have lengthened the arc of contact, had not the height of the tooth been reduced to maintain the same ai'c as that of Plate 4. The cycloidal action begins at C and cuds at H, making a maxinnuu angle of pressure of 17". The same i-oUing circle has been ust'd for the face and flank of each gear; Imt the one rolling within the pitch circle of the gear might have been nmch increased without materially weakening the gear tooth. The involute action would ))egin at D and end at B, making an arc of contact a little greater than the pitch. OO ANNULAR GP:AR1NG. CHAPTER V. ANNULAR GEARING. 48. Cycloidal System of Annular Gearing. If the center of the pinion lies within the pitch circle of the gear, the latter is called an internal, or annular gear. The solution of problems relating to this form of gearing differs in no wise from that of the ordinary external spur gear, save in the consideration of certain limitations which will be treated of. 49. Limiting Case. Plate 11 illustrates a pinion engaging an internal and an external spur geai'. The pinion has 6 teeth, and the gears have 13 teeth. The arc of contact is made equal to the circular pitch, and equally divided between recess and approach. The pinion has radial flanks, which therefore determines the diameter of the describing circle for the addenda of the gears. The second describing circle, 2, is governed by conditions which will appear later. It will be observed that the addenda of the annular gear teeth lie within, and the dedenda without, the pitch circle. The height of the teeth is governed by the arcs of approach and recess ; and the construction of the teeth does not differ from the limiting case considered in Art. 26, page 14, and Plate 5. The action between the pinion and annular gear begins at B, and ends at C, the pinion driving. 50. Secondary Action in Annular Gearing. We have already seen. Art. 10, page (>, that every epicycloid may be generated by either of two rolling circles, which differ in diameter by SECONDARY ACTION IN ANNULAR GEARING. 39 an amount equal to the diameter of the pitch circle. Also, that every hypocycloid may he generated by either of two rolling circles, the sum of the diameters of which shall equal that of the pitch circle within which they roll. Thus the addendum, C E, of the pinion, Plate 11, may he described by the circle 2, or the intermediate circle 3. But in this case the circles 1 and 2 are so chosen that the intermediate circle 3 is the second describing circle for the hypo- cycloid F G , as well as for the epicycloid C E ; consequently C E and F G will produce a uniform velocity ratio, the contact taking place from A to D. The addendum C E has contact also with the dedendum C F along the path A C ; hence, during a part of the arc of recess there must be two points of each tooth in contact at the same time. The plate illustrates the contact along the path A C as just completed ; but a second point of contact will be seen on circle 3, between F and E, and action along this path will be con- tinued to D. The case is therefore no longer a limiting one, inasmuch as the arc of contact is greater than the circular pitch. 'J'he additional contact takes place during the arc of recess, which is also advantageous. In order to ol)tain this secondary action, the sum of the radii of the inner and outer rolling circles must equal the distance hetween the centers of pinion and (/ear.* For, letting r,, r.^, and r:; be the radii of the inner, outer, and intermediate rolling circles, and Rp, Rg, the radii of pinion and gear, rg + r^ = Rg, (^6), and i-g — r.^ = Rp, (7), Art. 10, page (3. Subtracting the second equation from the fii'st, rj + r., = Rg — Rp ='C = center distance (8). Plate 12, Fig. 1, illustrates the same pinion and gear, the teeth having been described by the intermediate circle only. In this case the action takes place wholly during recess, the arc * The student is referred to Prof. MacCord's "Kinematics," images 104 to 10!) inclusive, for a very complete demonstration of this law, together with other limitations of aniudar gears. 40 LIMITATION OF INTERMEDIATE DESCRIBING CURVE. of recess being the same as before, about 1 j times the cireuhir pitch. Had the outer describing circle been used to describe the dedenda of the gear teeth, as in the })reeeding cases, a secondary action wouhl have taken place during the recess. Special notice should be taken of the reduced angle of pressure in the secondarv action of annular gearing, and of the possibility of obtaining a great arc of recess \vith little or no approacliing action. These advantages are very apparent in Plate 11, in which the pinion engages an external and an internal gear having an equal numl)er of teeth. 51. Limitations of the Intermediate Describing Circle. Plate 12, P'ig. 2, Suppose the inner describing circle, 1, Plate 11, to l)e increased luitil it equals the diameter of the pinion pitch circle, Of, the radius of the intermediate describing circle will then equal the center dis- tance, 5}^, and the outer describing circle, 2, would be but \'^ radius. For by sul)stituting Rp for rj in equations 6 and 8, Ai;t. 50, we shall obtain r, = Rg — Rp = c, and r, = C — Rp. Plate 12, Fig. 2, illustrates this case, the outer describing circle not being employed. Since the pinion pitch circle has now become a describing curve, there will l)e an appi'oach- ing action ; Ijut only one point of the pinion tooth will act, as the diameter of the describing circle and pitch circle being equal reduces the pinion flank to a point. But if any further increase be made in the diameter of the inner circle, whicli is equivalent to a decrease in the intermediate describing curve, an interference will take place during approaching action ; since the curves of gear and pinion teeth, generated by a circle greater than the pinion diameter, will cross one another, which would make action impossible. Hence, the radius of the intermediate describing circle ca)uiot be less than the line of centers. 52. Limitations of Exterior and Interior Describing Circles. Plate 12, Fig. 8. From LIMITATION OF EXTEIJIOR AND INTERIOR DESCRIBING CURVES. 41 Art. 50, page 39, it was seen that the sum of the radii of tlie exterior aiul interior descril)ing circles must equal the center distance if a secondary action be obtained. If either circle be decreased without decreasing the other, the secondary action ceases ; but if either circle be increased wdthout an equal decrease in the other, thus making the sum of their radii greater than the center distance, the addenda will interfere. Thus, iii Plate 11, a decrease in describ- ing circle 2 would produt-e a more rounding face, and C E would fail to engage F G; but had this describing circle been increased in diameter without a corresponding decrease in 1, C E would have interfered with F G . Hence, tlie limit of the sum of the radii of the exterior and interior describing circles is the center distance, Plate 12, Fig. •], illustrates a special case of the above condition, the interior describing circle being reduced to zero, and the radius of tlie exterior circle made equal to the center dis- tance, thus making the intermediate describing circle equal to the [)itch circle of the gear. There will be double contact during a portion of the arc of recess, the contact l^eginning at A, and following the outer describing circle to C, and the intermediate (or in this case the pitch circle of the gear) to D. Tliis design is ol)jectionable in that the secondary action takes place with only one point of the gear tooth. 53. The Limiting Values of the Exterior, Interior, and Intermediate Describing Circles for Secondary Action. !Since r.^H- ri = C, either radius will equal C, when the other becomes zero; l)ut if there l)e a secondary action, the minimum value of r., may not be zero, for r^ will be a maximum when r., is a mininnim. as rj + rs = Rg, r- is a minimum when equal to C (Art. 52), and substituting this value in the last equation, rj = Rg — C . Again suhstituting this value in the equation, r._, + rj = C , r. = C — (Rg — C) = 2 C — Rg. 42 LIMITING VALUES OF DESCIUBING CIRCLES FOR SECONDARY ACTION. Summary of the above limiting values and conditions governing secondary action: — rj maximum = Rg — C ; r^ minimum = ; rs + rj = Rg . (G) r.^ maximum = C ; r^ minimum = 2 C - Rg ; rs — r^ = Rp . (7) rg maxinmm = Rg ; h mininuun = C ; Rg - Rp = C . (8) 54. Practical Case. If annular gears be made interchangeable with spur gears, it will be necessary to have the number of teeth in the engaging gears differ b}- a certain number which will depend on the base of the system. This is due to the limitation in the sum of the radii of the describing circles, Art. 52, page 40. Thus, let 12 be the base of the system, and it is required to find the least number of teeth in the annular gear that will engage the pinion. If the pitch be 2, the diameter of the pinion will be 6, and that of the describing circles 8. But since the center distance cannot be greater than the sum of the radii of the describing circles (in this case 3), the diameter of the annular gear must be 12, and the least number of teeth in the annular gear will be 24. Using the notation of Plate 11, and Art. 17, page 11, n being the least number of teeth in the gear, and n the least number in the' pinion, or the base of the system: — n 1 „ „ r^ N n 1 n N n ^ ., C = 2 r^ = — , also C=Rg-Rp= — - — , hence ^ = ^ - ^^ or 2 n = N . The least number of teeth in the annular gear will be twice that of the base of the system. 55. Summary of Limitations and Practical Considerations, (a) The diameter of the inter- mediate describing circle is equal to the diameter of the pinion, plus the diameter of exterior describing circle, or diameter of gear minus interior describing circle. (Art. 10, page 6.) SUMMARY OF LIMITATIONS AND PRACTICAL CONSIDERATIONS. 43 (^) There will be .secondaiy action only when the sum of the radii of the exterior and interior deseril)ing cireles is e(|ual to tlie line of centers. (Art. 50, page 38.) ((?) The radius of the intermediate descril)ing circle cannot be less than the center distance. (Art. 51, page 40.) ((?) The sum of the radii of exterior an.d interior describing circles cannot be greater than the center distance. (Art. 52, page 40.) (^) The nundjer of teeth in any pair of gears of an interchangeable system must differ by an amount equal to the ])ase of the system. (Art. 54, page 42.) (/) If the pinion drives, the exterior describing circle should be the greater in order tliat the arc of contact may be chiefly one of recess. ( 9 .555 29.05 55.4 1.748 60.95 99.1 .971 — [il 2 .500 26.56 51.0 1.789 63.44 101.4 .894 --^ 4 9 .444 23.94 46.3 1.827 66.06 103.6 .812 BEVEL GEAR TABLE. 55 BEVEL GEAR TABLE FOR SHAFTS AT 90°. PROPORTION OF PINION TO GEAR. 1. 2. 3: 7 .428 2: 5 .400 3: 8 .375 1: 3 .333 3:10 .300 2: 7 .285 1: 4 .250 2: 9 .222 5 11 G 13 ; 7 ;15 .200 .181 .166 .153 .143 .133 1: 8 .125 2 17 .117 1 9 .111 1 10 .100 82.88 ! 112.2 83.30 1 112.3 83.67 112.4 84.30 1 112.6 .248 .233 .221 .200 -^ n ^ d' p-^. d' tan A = ^, n tan B 2 sin A 2.25 sin A tan C = 2 E = p cos A , 2 F = p sin A. 56 BEVEL GEARS WITH AXES AT ANY ANGLE. 63. Bevel Gears with Axes at any Angle. If the axes of the gears intersect at angles other than 90°, the drawing of the bhmks and devel- o[)nient of the teeth do not differ from the cases ah'eady descriljed. The figuring required is that inthcated in Fig. 27, those in the heavy face being used to determine the other vahies, and not a[)pearing on the finished drawing. tan A = + cos a tan A' = — + cos a N r, 2 sin A 2 sin A' tan B = or , r. 2.25 sin A 2.25 sin A' tan C = or E = - cos A ; E' = — cos A'. P P sin A F' = — sin A'. P Fig. 27. Or, the values for E, F, E', and F may be ob- tained from the table for shafts at 90°, pages 54 and 55 by determining the center angles A and A', and finding the values for 2 E and 2 F, corre- sponding to each gear separately. WILLIS S ODOXTOGllAril. 5/ CHAPTER VII. SPECIAL FORMS OF ODONTOIDS, NOTATION, FORMULAS, ETC. 64. Odontographs and Odontograph Tables. If tooth curves are to be drawn according to some estal)lislied system, in whieli the angle of pressure is constant, or but one diameter of rolling circle be used, it may be desirable to employ some of the approximate methods for shortening the operation. While it is unnecessary for the student to familiarize himself with the theor3% or even the details, of operating the various systems of approximating these curves, it is essential that a knowledge be had of the more useful tables and methods to which refer- ence may be made when required. Three methods are employed for approximating the odontoidal curves. First, by circular arcs, the centers and radii of which are given in tables, or established by instruments, designed for this purpose. Second, by curved templets from \\liich the curves may be traced directly. Third, by ordinates. 65. Willis's Odontograph. Among those of the first type, the oldest, best known, and least accurate, are tlie odontographs designed by Professor Willis. When used for gears having a large number of teeth, the error is very slight ; but in the case of involute teeth of small number it is very noticeable. Fig. 28 illustrates the application of this instrument to the 58 THE GRANT ODONTOCJRAPIIS. drawing of curves of the cycloidal system. The' centers for the circuhir arcs designed to approximate the curves are found on tlie straight edge, A B, and at a distance from the zero point of the scale to be found in tlie pul)lished table accompanying the instrument. 'Jlie theory and application of these odontographs is clearly treated of in the instructions accompanying these instruments, also in Stalil and Wood's " Elements of Mechanism," pages 113 to 122, and more briefly in MacConUs " Kinematics," pages 172 to 174. 66. The *' Three Point Odontograph," designed by Mr. Geo. B. Grant, is a table for face and flank radii and centers, figured for circular arcs passing through j \ \ \ ^^^ three most important points of the tooth curves ; / \ \ ! viz., at addendum or dedenduni circles, pitch circle, and a point midway between. This gives a very close approximation to the true curve for the systeni which has radial flanks for gears of twelve teeth. The tables and instructions are pul)lished in Grant's " Odontics," pages 41 and 42, and in Stahl and Wood's '•* Elements of Mechanism," pages 124 and 125. Fig. 28. 67. The Grant Involute Odontograph, designed by Geo. B. Grant, and puljlished in his "Odontics," pages 29 and 80, gives a very close approximation to the involute for 15° angle of pressure and epicycloidal extension, all gears being designed to engage a 12-tootlied gear without interference. THE ROBIXSOX AXD KLP:iN ODONTOCillAPHS 68. The Robinson Odontograph differs from the preceding in that it is an instrument hav- ing a curved edge which is used as a templet to trace the tooth curve, tables being used to determine the position of the instrument \^ith relation to the pitch circle. Fig. 29 illustrates the instrument in posi- tion. The curve B C A is a logarithmic spiral, and the curve B F H the evolute of the first, and therefore a similar and equal spiral. By means of this instrument, in connection with the pul>- lished tables accompanying it, involute teeth may be drawn as well as C3^cloidal, and a much larger range of the latter is possiljle than is afforded by the Willis odontograi)h. Tlu^ theory of this instrument is best treated by Professor Rob- inson in Van Nostrand's Eclectic Magazine for July, 18T(>, and Van Nostrand's " Science Series," No. 24. Also see Stahl and Wood's "Elements of Mechanism," pages 12(3 to 1-30. 69. The Klein Coordinate Odontograph. Fig. 30 is de- signed to eliminate the labor of drawing pitch circles of large radii by constructing the cuinc l)y ordinates from a radial line. 'I'he tables and ex2)laiiatioii of tlie method may l)e Fig. 30. 60 SPECIAL FORMS OF ODONTOIDS AND THEIR LINES OF ACTION. Fig. 33. found in Professor Klein's " Elements of ^Machine Design," page 50. 70. Special Forms of Odontoids and their Lines of Action. Gears maybe classified from the forms of rack teetli, as follows : System. Tooth Curve. Lixe of Action-. Involute, Fig. 31, A right line, .V riglit line. Cycloidal, Fig. o2, A cycloid, A circular arc. Segmental, Fig. 00. A circular arc. Conchoid of Xicomedes. In like manner other systems might be derived from, and classified by, the forms of their rack teeth. It is of interest to note in connection with the first two that any tooth of either system may be derived from a right line. In the cycloidal system the addendum of any gear tooth will properly engage the radial flank of some gear. If. there fore, the addenda of any gear tooth be made to fit the dedenda of teeth consisting of radial flanks, the resulting teeth must be cycdoidal. \ skilled mechanic with file and straight-edge could in this COXJUCIATE CURVES. manner })roduce the templet for any de- sired cyeloidal tooth without the aid of other mechanism. Of course such a method would require considerable skill in producing a perfect tooth, and it is not the best means to the end ; but it is of much interest to the student as illustrat- ing the relation between the mechanical and graphic methods of attaining the same end. In like manner we may produce templets for invo- lute teeth from the right line rack tooth of the system. 71. Conjugate Curves. — The curves of any pair of teeth being so related as to produce a uniform velocity ratio are called conjugate, or odontoids, and if any tooth curve of reasonal)le form be assumed, a second curve may be obtained which shall be conjugate to the first. By a reasonable form is meant the conformity to the following principle : — The normals to the curve must come into action consecutively, as in Fig. 34, and not as in Fig. 35, in which it will be seen that the normal E F will })ass through the pitch point M, and the point E come into 62 WORM GEARING. Elg. 37. action before the point C, which is impossible. Let C, Fig. 36, be any tooth form conforming to the above condition, and the periphery of disk A its pitch line. Suppose it is required to derive its conjugate having for its pitch circle the periphery of disk B. This may be obtained by a graphic process, as in Art. 28, page 15, or by the mechanical method known as the molding process of Fig. 36. C is a templet of the given tooth form, which is fastened to disk A , and revolving in contact with disk B, the disks maintaining a constant velocity ratio. The successive positions of C are then traced on the plane of disk B, and the tangent curve will be that of the required conjugate tooth. The method is applicable to all forms of spur gear teeth, but to only one form of bevel gear, the octoid. 72. Worm Gearing. A woi'm is a screw designed to oper- ate a gear, called a worm wlieel or gear, the axis of the latter being perpendicular to that of the worm. Art. 3, page 3. The section of a worm and gear made by a plane perpendicular to the axis of the gear, and including the axis of the worm, is identical with that of a rack and gear of the same system and pitch. The worm, or screw, may be single, double, etc. If single, the circular pitch corresponds with the pitfli of the LITERATURE. G3 thread : if double, the cireidai' iiitcli will he half the pitcli of the thread, etc. To avoid niis- uiiderstandiug, it is customary to speak t)f the pitch of the tluead as the lead. A drawing of the tooth foi-ni is required oiiiy in special cases of large cast gears, and the usual representation is that shown by Fig. 37. The diameter of the worm is connnonly made equal to foui' or live times the circular pitch, and the angle A varies from (iU ' to \H) . FoiiMULAS Full WoilISI AND (iEAll. L = Lead of worm ; m = Threads per iucli in worm; d = Outside diameter of worm; d' " Pitcli diameter of worm; W = Wliole diameter of gear; D =-- Tliroat diameter of gear; D' = Pitcli diameter of gear; L = — = P', for single tlireads, 2 L = — = 2 P'. for doiihie tlireads. etc.; m Tj and r., are dimensions reciiiircd for the liol), or cutter, emploved in cutting the worm gear; >.; r^ , ^ / ^ ' ' •' " • W = D + 2 rj — rj cos - C = Center distance; \ ^ 73. Literature. The following list of books and articles is published to assist the student who may wish to puisue the subject beyond its elementary stage. Only those treatises have been enumerated which are likely to be accessible and uscfid. The great works of Willis, p, TT U N + 2' 0..11-.^; D=^ + ^ d 2 •■i - 2 P' ^2 = ^1 + ^ P; r D + d ^ " 2 1 P' W = D + 2 ( rj — rj cos (;4 LITEHATUHK.. Kaiikiiie, and Reuleux are omitted, as the student will derive more beuetit from the interpreta- tion of these works by hiter authors tlian by a stndy of the original treatises. '^The Mechanics of the Machinery of Transmission," revised by Professor Herrmann, is Voh Iir., Part I., Sect. 1, of Weisbach's "Mechanics of Engineering." This work includes one of the most valualde treatises on the subject of gearing, bnt it is somewhat dithcnlt. Wiley, $')M. "■ Kinematics," by Pi'ofessor MacCord, is chiefly devoted to the snbject of gearing. It con- tains nuich original matter of importance. No student of the subject can afford to do witliout this treatise. Wiley, -f!5.0(). " Elements of Machine Design," l)y Professor Klein, was published for the students of Lehigh University. Several chapters are devoted to gearing, and include some excellent tables and problems. The Klein coordinate odontograph is fully illustrated and explained. J. E. Klein, Bethlehem, Pa., -16.00. " Odontics," by Mr. Geo. B. Grant, is one of the most valuable modern treatises on gearing. It is both theoretical and practical. It is concise, contains many useful tables, and is well illustrated. The subject cannot be })ursued to advantage without its use. Lexington Gear Works, Lexington, Mass., minimum = 2 C — Rg, Ai;t. 52, page 40; r3 maximum = Rg ; r^ minimum = C, Art. 51, page 40. REV?:r. Gears, Shafts at 90°, Airr. 6i, Page 5G. A = Tenter angle of piiiiuu; B = Angle increment; C = Angle decrement; E = ()ne-half the (iianictci' inciemenl foi- pinion: F ( )n('-lialf llic di;nii('tcr iiirrcinciit fur geai'. tan A d' _ n . ~ D' N' tan B 2 sin A. n tan C 2.25 sin A n- E 1 , = p cos A sin A. notation and formulas. 67 Bevel (iEaks, Shafts at Othei: than *J0^, Ai;t. 63, Page 5(3. a = Aniile of .sliafl>; ^ , sm a ° tan A = A = Center angle of pinion; A' = Cenlei angle of gear; ^^p ^' — B = Angle inerennMil : „ . , , _, 2 sin A 2 sin A C = Angle deeienienl : tan B = N — + cos a 11 sin a n N + COS a n N E = One-half the diameter inerenient for pinion; ^ 22.5 sin A 2 25 sm A' tan C = =- , n N E' = OiH'-half the diameter increment lor gear; ■, , E = _ cos A E' = p cos A'; F = Dimension i-e<|nireil for hacking of jiinion; c 74- • • 1 .■ 1 1 ■ i- r = „ sm A F = — sm A ; I- = Dimension re(iuued tor hacking ot gear. P P \Voi:m (iEAEs, Ajrr. 72, I'AciE (32. L = Lead of worm ; , 1 r^ . , , , L = — =^ P' tor .sinule threads; m nn = Threads per inch in worm; 2 L = — = 2 P' tor doiihle thread, etc., m d = Outside diameter of worm; k, m i 9 D'= - • D = "*" : P P d = Pitch diameter of worm: TT D ..,„,/ A' P' -- • W = D + 2 1^ - ri COS D = Thread diameter of gear; N + 2 y C = ^-±- - -• D' = Pitch diameter of gear; 2 P' W = AVhole diameter of gear. ^^ ~ 7 P ' "^i ~ ''i """ 3 '^^ 68 METHOD TO BE OBSERVED IX PEKEOKMJXC; THE PKOBLEMS. / CHAPTER VIII. PROBLEMS. 75. Method to be Observed in Performing the Problems. No attempt slionld be made to oraphically solve the following problems until the general principles involved are well under- stood. The first requisite to tliis is tlie mastery of C-hapter II., on Odontoidal Curves; and this can be best acquired l)y the drawing of the various curves, together A\'ith a study of their characteristics. No problems have been given on this topic, but the following course of study would be desirable : — Having prescribed diameters for rolling circles and director, or pitch circles, draw a cycloid, epicycloid, and hypocycloid, as described in Arts. 5, G, and 7, page 5. Obtain a sufficient number of points in each case to enable the curves to l)e drawn free-hand with considerable accuracy, after wliich they may be corrected by the use of scrolls. Next prescribe a point on each (not one already found), and draw^ normals to each by Akt. 8, page 5. The second method, Art. 9, page 5, is the more practical, and shonld also l)e studied by drawing a small part of each curve, beginning at a point on the director circle. It is also desirable that one of the epitrochoidal forms be drawn, and a normal determined. Art. 11, page 6. The problems are designed to be solved on a sheet which shall measure 10" by 14" within I PKOBLKM 1. CYf'LOIDAL LIMITING CASE. 69 the margin line, and the hiy-out of these sheets is given on Plates 14 and 15, there being four problems on each plate. ]\Ieasurements are from the margin line. It is unnecessary to represent all the teeth in a gear, but such as are shown should be drawn with the greatest accuracy attainable by the student. Without this care the study will avail one little, and the time consumed in discovering errors will be great. The inking of the curves may be omitted if time will not admit of its being w^ell done ; but in either case it is desirable to emphasize the curves, and distinguish clearly between the gears by making a very light wash of color on the inside of the curve, the width to be about one-quarter of an inch. One color may be used for the pinion, and a second for the rack and Cycloidal Limiting Case. Face or Flank only. d' gear. Problem i, Plate 14, Fig. Example. D' 1 10 2 10 3 121 4 5 10 6 lOL 7 8 12' N n A B 15 12 H 41 *2 12 H 4' 21 3 4 15 10 H 4i 15 H 4i 14 10 3] H 24 12 2'» •^4 4 21 12 31 4i Statement of Problem. Having given the diameters of pitch circles, number of teeth, and diameter of describing circle, it is re(]^uired to draw the teeth for pinion, gear, and rack, liaving arcs of contact equal to the pitch, and contact on one side of pitch point only. 70 PROBLEM 1. CYCLOIDAL LIMITIXC; CASE. Study Arts. 1 to 26 before performing this prol)lein. Operations. 1. By Art. 18, page 11, determine the value of N, n, D' or d, one of whicli is omitted from the table. Observe that - = -• d n 2. Draw center and pitch lines and describing circle. Lay off the circular })itch on each gear by spacing the circumferences into as many parts as there are teeth. 3. Obtain the first point of contact by laying off from the pitch point on the describing circle an arc equal to the circular pitch, the directif)n being determined b}^ the rotation required. Art. 16, page 10. Ar.t. 21, page 12. Arts. 22 and 23, page 13. 1. With the above describing point, generate the face and flank required. Arts. 14 and 15, page 10. 5. Draw the working faces of gear teeth, and assuming the gear teeth to l)e pointed, draw opposite side of each. Art. 16, page 10. 6. Draw the working flanks of the pinion teeth, observing that the depth must be sufificient to admit the gear teeth, but without clearance. Obtain the thickness, and draw the opposite sides. Art. 16, page 10. 7. Draw the describing circle for rack. Obtain the first point of contact between pinion and rack, and describe the cycloid for rack teeth. Construct rack teeth. Ar/r. 2o, i)age 14. Note that thickness of rack tooth must equal space between pinion teeth, or thickness of gear teeth, measured on the })itcli line. 8. To determine points of contact of conjugate teeth, assume any ])oint on face of gear tooth, and determine, first, its position when in contact with the pinion ; second, the jjoint of the pinion tooth engaging it. Since the contact must take place on the path of contact. Art. 21, page 12, the assumed })oint will lie at the intersection of this arc and one described PROBLEM 2. CYCLOIDAL LIMITING CASE. 71 through the given point from center of gear. To solve the second, describe an arc from the center of the pinion through the point previously determined, and its intersection with the pinion flank will be the engaging point required. Next construct the normals for each of these points. Art. 8, page 5. They should he equal to each other, and also to the distance from the pitch point to the point on the path of contact in which they engage. Art. 14, page 10. 9. Obtain the maximum angle of obliquity, or pressure, between gear and pinion, pinion and rack. Art. 24, page 14. Problem 2, Plate 14, Fig. 2. Cycloidal Limiting Case. Face and Flank. Study Arts. 2(3 to 30. Statement of Problem. The diameters of gears, number of teeth, and describing circles being given, it is required to draw the teeth for pinion, gear, and rack, when the arc of approach = the arc of recess = half the circular pitch, the flank of gear being radial. Operations. 1. Draw center lines, pitch lines, and rolling circles, the second circle being determined by Art. 9, page 0. Divide the pitch circle into the required parts to obtain the circular pitch. 2. Lay off arcs equal to — on each of the rolling circles to obtain the first and last points of contact, observing the direction of rotation prescriljed in Fig. 2. 3. With the point thus determined on small rolling circle, describe the addendum of gear tooth and dedendum of pinion tooth. With the point on the second describing circle generate the addendum of pinion tootli. The dedendum of gear tooth being radial may then be drawn. Make the dedenda of pinion and gear deep enough to admit the engaging addenda, but allow no clearance. 72 PROBLEM .?. CYCLOIDAL GEAR. 4. Draw the working faces of the pinion teeth and then the opposite faces to make the teeth pointed. Simihirly draw the gear teeth, making them pointed also. The sum of the thickness of tlie teeth cannot he greater that the circular pitch. Art. 29, page 16. In this case it will he found to be about one-hundredth of an inch less, which will be the backlash. An increase in the diameter of either rolling circle would make the solution impossible. 5. Draw the dedenda of pinion and gear teeth. 6. The describing circles for the rack teeth Avill be determined by Art. 14, page 10. Draw the circles with their centers on the line of centers, and obtain the first and last points of contact. These points should fall on the addendum and dedendum of pinion teeth already drawn, as in Plate 5 at M and . From these points describe the addenda and dendenda of the rack teeth. The thickness of these teeth must equal those of tlie gear. 7. Obtain the maximum angle of pressure for approach and recess between pinion and gear and pinion and rack. It would also be desirable to obtain the curve of least clearance in one case. Art. 28, page 15. Problem 3, Plate 14, Fig. 3. Cycloidal Gear. Practical Case. ('om})lete Chapter III. be- fore performing this problem. N n A a C 15 12 4 5 31 21 12 4 4 3| 20 16 3i 4 3 22 12 3i 5 4 16 12 4" 5 3i 20 12 3 4 4^ ^^X AMPLE d' 1 9 2 8 3 10 4 8 5 9 6 7 PROBLEM 3. CYC'LOIDAL GEAR. 73 Statement of Problem. Tlu' diameters of pitch circles and rolling circles being given, and the number of teeth known, it is required to draw the teeth for gear, pinion, and rack, to obtain the maximum angle of obliquity, and the arcs of approach and recess in each case. The teeth will be standard with j^^" backlash. Art. 31, page 17. Art. 71, page 61. Operations. 1. Figure the diameter of gear, circular, and diametral pitch. Arts. 17 and 18, page 11, and determine proportions of teeth. Art. 31, page 17. 2. Draw center lines, pitch lines, addendum, and dedendum circles, and rolling circles. Divide the pitch circle into as many parts as there are teeth, beginning to space at the pitch point. 3. Beginning at the pitch point, describe pinion flank, gear face, gear flank, and pinion face, by Art. 9, page 5. See also Art. 34, page 21. 4. Lay off thickness of teeth. Art. 31, page 17, and describe addenda of pinion and gear teeth by approximate method. Art. 34, page 22. Describe dedenda by Art. 10, page 11. Draw fillets. Art. 31, page 18. 5. Describe rack teeth. 6. Determine the following for gear, pinion, and I'ack in tei-ms of P'. Arts. 21 to 24 inclusive, pages 12, 13, and 14, Art. 32, page 18. Pinion and Gear. Pinion and Rack. Arc of approach Arc of recess Arc of contact Maxinunn angle of pressure ^ /4 PROBLEM 4. INVOLUTE LIMITING CASE. Problem 4, Plate 14, Fig. 4. Involute Limiting Case. Study Arts. 38 to 42. Statement of Problem. Number of teeth five and six. Pinion teetli pointed. No Ijacklasli or clearance. Arc of contact equal to the circular pitch. This problem being similar to that of Plate 8, reference will be made to that figure. The case being a limiting one, the distance between the points of tangency of base circles and line of pressure must equal one-sixth of the circumference of the gear base circle, or one- tifth of the circumference of the pinion base circle. The tangent of the angle of pressure will equal ^ = ^ = / ^ , but A D = D K C by construction, and D K G = tt. Also A F + D C = 5i, hence, —= — — = — = tan. of the angle of pressure. The angle corresponding to this tangent is 29° 44' 6". The distance between the centers will be ^ho'^ + A F + 'D~G^ = The angle of pressure and distance between centers could have been determined graphi- cally by laying off F A , in any direction, equal to the radius of pinion base circle, A D perpen- dicular to FA, and equal to one-fifth of pinion base circle. Finally, D G perpendicular to A D, and equal to the radius of gear l)ase circle. Operations. 1. Draw the line of centers, base circles, and line of pressure. Deter- mine the points of tangency, which limit the action in either direction, and through the pitch point, determined by the intersection of the line of centers and line of pressure, draw the pitch circles. It is desirable now to test A D by proving it equal to one-fifth of the pinion base circle, or one-sixth of the gear l)ase circle. 2. Draw the involute A C, Plate 8, of the gear, and D p of the pinion. Airr. 12, page 7. Art. 38, page 26. Determine the circular pitch, and lay off as many divisions as there are teeth to be drawn. Copy the curves already drawn. PROBLEM .-,. INVOLUTE PRACTICAL CASE. 75 3. Draw the opposite face of pinion tectli. making them pointed. To draw the opposite faces of gear teeth proceed as follows : Since contact between the opposite faces must take place along the line of action C E , Plate 8, the contact between the engaging teeth will be at E. At E draw arc E 1 from center G . Bisect this arc, and lay off M and H from this radial bisector equidistant with A and C. 'I'hrough these points describe the curve of opposite face, and draw the remaining teeth. That j)ortion of the teeth lying within tlie base circle will ])e radial, and extend sufficiently to admit the engaging teeth, but without clearance. 4. Construct two rack teeth. Art. 40, page 29. 5. Epicycloidally extend the gear teeth so as to make them pointed. Similarly extend the rack teeth, l)ut only as mucli as the clearance for tlie pointed gear tooth will permit. Art. 41, page 30. Problem 5, Plate 15, Fig. i. Involute Practical Cases. Complete the study of Chapter IV. State.mext of Pnop.LEMS. Several gears and racks are given to describe involute teeth of standard dimensions. 'J'o determine the interference, if there be any, and to correct the curves for the same. QPERATroxs. 1. Draw three or four teeth of gear A, and two teeth of engaging pinion B, the angle of pressure being 16°. Art. 42, page 32, Fig. 16. Make contact at pitch point in all cases. Correct for interference by epicycloidal extension. Art. 31, page 17. Art. 42, page 30. Art. 41, page 32. 2. Draw three or four teeth of o-ear A enofaofino- rack F . 3. Draw three teeth of pinion B engaging rack E, and correct rack teeth for interference. 76 PROBLEM 6. CYCLOIDAL ANNULAR GEAR. 4. Draw a portion of gear C and rack K , the angle of pressure being 20°. Test this for interference by Art. 45, page 33, as well as by graphic method. 5. Draw a few teeth of gear D, the angle of pressure being 15°. Determine the least number of teeth that will engage it without interference. Problem 6, Plate 15, Fig. 2. Cycloidal Annular Gear. Study Arts. 48 to 56. EXAMPLK. D' d' N n A a B 1 m 9 13 6 7 H H 2 191 9 13 6 6h 4 51 3 191 9 13 6 6 41 51 4 171 7 15 6 7 31 51 5 17i 7 15 6 7i 3 5h Statement of Problem. The number of teeth and diameters of pitch and describing circles being given, it is required to draw the tooth outlines, and determine the increased arc of contact due to secondary action. The arc of contact, not including that due to the secondary action, is equal to the circular pitch, and the arc of approach equals the arc of recess. Operations. 1. Draw the center and pitch lines and describing circles. 2. Determine the circular pitch, and lay off half this amount from the pitch point on each of the describing circles to determine the first and last points of contact. 3. Describe the curves of the teeth. PROBLEM 7. INVOLUTE ANNULAR GEAR. i I 4. Determine the intermediate describing cnrve, and draw tlie same to obtain the limit of secondary action. 5. Determine the maximnni angle of pressure for approach and recess. Also the angle of pressure for the last point of secondary action, and the increase in the arc of contact. Problem 7, Plate 15, Fig. 2. Involute Annular Gear. Complete Chapter V. Example. D' d' N n Angle of Pressure. B 1 15 -i 20 10 20° 61 2 15 G 30 12 15° 7 3 16 8 16 8 20° 6 4 20 8 30 12 15° 7 5 24 18 24 18 20° oh Stateiment of Problem. The pitch diameters, number of teeth, and angle of pressure being given, it is required to draw the tooth cui've, to determine if there Avill l)e any inter- ference when the addenda of pinion teeth are made standard, and finally the length of the arc of contact in terms of P'. Operations. 1. Draw center and pitch lines, line of pressure, and base circles. 2. Make addenda of pinion standard if a second engagement does not take place. Art. 56, page 43, and limit addenda of gear by Art. 56, page 43. 3. Determine the arc of contact in terms of P'. 78 PROBLEM 8. CYCLOIDAL AND INVOLUTE BEVEL GEARS. Problem 8, Plate 15, Fig. 3. Cycloidal and Involute Bevel Gears. Shafts at 90^. Study Arts. 57 to 63. IMPLE. , P N n Q K 1 3 18 15 31 u 2 4 24 20 31 n 3 2 16 12 H n 4 4 28 20 3 n 5 3 21 15 3 u 6 2 14 12 n ^ 7 3 21 18 •^8 u 8 2 18 14 4 n 9 4 20 16 H 1 10 3 21 18 3| u w 3 4 u 3 3 4 n 21 1 2 4' 1 91 "2 i li 2^ "^8 3 4 11 31 1 1^ 3i 3 4 2 4 f H 91 ~4 1 li 3i 1 li 3 4 n n 1^ 1 H 1 n 1 u 7 8 n- 1 la 3 4 1 I 1* I If involute, make angle of pressure 15°. If cycloidal, make diameter of rolling circles equal to the elements of normal cone of pinion. Statement of Problem. The proportions of the gear being given by the table, it is required to draw the gear blanks, describe the development of the teetli on the normal cones, and figure the gears. Operations, 1. Having determined the pitch diameters, draw the gear blanks. Art. 60, page 48. PROBLEM 9. CYCLOIDAL AM) INVOLUTE BEVEL (iEARS. 79 2. Describe two or three teeth of each gear on the developed surfaces of the outer and inner normal cones. Akt. 60, page 48. 3. Figure the gears, Art. 61, i)age 51. Problem 9, Plate 15, Fig. 4. Cycloidal and Involute Bevel Gears. Shafts at other than 90°. Study Art. 63. EXAMT 'Lt; a P N n Q J K L M H w 1 40° b 24 15 9 8^ 2h i li i u 2 45° 3 24 15 9 "s 2i i u 5 n 3 50° 4 34 24 m S 2 i n i n 4 55° 3 27 21 9 8 21 ^ li i IS 5 60° 2 20 12 8t. li 2n 5 s If 1 2 V Y 2.1 u 2 u 1^1 1* 1] Is 1§ 3^. i If involute, make angle of pressure 15°. If cycloidal, make diameter of rolling circles equal to the elements of normal cone of pinion. Statement of Problem. Tlie pro|)ortions of the gear being given by the table, it is required to draw the gear ])lanks, describe the teeth on the development of the normal cones, and figure the gear. Operations. 1. Determine the ])it{h diameters from al)ove tal)le, and draw tiie gear blanks. 2. Describe two or three teeth of each gear on tlie developed surfaces of the outer and inner normal cones. o. Figure the geai-s. I ]^ D E X„ Addendum defined, 12: proportion for, 17. Angle decrement, 52. Angle increment, 52. Angle of edge, 51 ; of face, 51. Angle of obliquity, or pressure, 14 ; affected by rolling circle, 18; constant, 28; for involute, 31; influence of, 33; method for determining, 33; reduced in annular gear- ing, 40. Annular gear, notation, and formulas, ()(>: epicycloidal prol)- lem, 76; involute problem, 77. Annular gearing, 38; secondary action in. ."W ; interchan- geable with spur gearing, 42; involute system of. 43. Approaching action detrimental, 17. Approximate cycloidal curves. 22. Approximation, Tredgold, 47 ; by circular arcs, 22 Arc of approach defined, 13. Arc of contact defined, 13; relation to circular pitch, ](>. Arc of recess defined, 13. Backing, .52. Back cone, 48. Backlash defined, Hi; dimen.sions for, 18. Base circle defined, 7, 27. Base of system, 21 ; in annular gearing, 42. Beale's "Practical Treatise on Gearing," 04. Bevel gear defined, 2: Theory of, 45; character of curves employed, 4(5; drafting the, 48; blank, 49; length of face, 49; figuring the, 51 ; table for, 53, 54, 55; cliart for plotting curves, 65 ; notation and formulas, 66 : prol> lems, 78, 79. Bevel gears with axes at any angle, 5(5. Bilgram, Hiigo, inventor of octoid tooth, 47; machine for cutting bevel gear teeth, 47, 65; exhibit, 65 Brown & Sharpe publications, (54. Circular pitch defined, 11. Character of curves in bevel gearing, 46. Clearance defined, 15; proportion for, 17. Clock gears, 17. Conchoid of Nicomedes, (50. Conditions governing the practical case, 16. Conjugate curves defined, 9, 61. Constant angle of pressure, 28. Constant velocity ratio defined, 1. Conventional representation of spur gears, 25. Contact, point of, 5; radius, 5; path of, 12; arc of, 13. Coordinate odontograph, .59. Crown gear, 47. Curtate epitrochoid, (i. Curve of least clearance, 15. 81 82 INDEX. Curves, odontoidal, 4. Cutting bevel gear teetli, 65. Cutting angle, 51. Cycloid, defined, 4; problem relating to, 68. Cycloidal action, Theory of, 8. Cycloidal curves, second method for describing, 5 ; approxi- mated, 22. Cycloidal system of annular gearing, 38. Cycloidal annular gear problem, 76. Cycloidal bevel gear i^roblem, 78, 7fl. Cycloidal limiting case problems, (59. 71. Cycloidal practical case problem, 72. Dedendum defined, 12; proportions for, 17. Defects of involute system, 35. Describing circle, defined, 4; a path of contact, 12; maxi- mum and minimum, 16; influence on sliape and effi- ciency of teeth, 18 ; relation to interchangeable gears, 20. Describing disk, 8. Describing point, 4. Describing cone, 45. Describing cylinder, 45. Describing radius, 5. Description of bevel gear table, 53. Developed pitch circle, 50. Development of normal cone, 49. Diameter pitch, 11. Director circle, 5. Double contact in annular gearing, 3fl. Double generation of epicycloid and hypocycloid, 6. Drafting bevel gears, 48. " Elements of Machine Design," (i4. " Elementary Mechanism," 65. Epicycloid, defined, 5; second method for describing, 5; double generation, 6; spherical, 45; problem relating to, 68. Epicycloidal extension, 30. Epitrochoid, defined, 6; curtate, 6; prolate, 7: problem re- lating to, 68. Exterior (outer) describing circle, 39; limitations of, 40, 41. Face gearing, 2. Face of gear, 24. Face of tooth, 12. Flank of tooth, 12; radial, 18. Figuring bevel gears, 51. Fillet, 18 ; size of, 18. " Formulas in Gearing," 64. Formulas for worm and gear, 63. Formulas, Notation and, 65. Gearing, 1. Gear arm proportions, 65. Gears, interchangeable, 20: face of, 24; comparison of, 24. Generating point, 4. Generating radius, 5. Grant, Geo. B., bevel gear chart, 53; three point odonto- graph, 58; involute odontograph, 58; " Odontics," 58, 64; epicycloidal and bevel gear generator, 65. Hyperboloid of revolution, 2. Hyperbolic gears, 2. Hypocycloid, defined, 5; second method for describing, 5; a radial line, 6: double generation, (!; si)lierical, 45; prob- lem relating to, 68. INDEX, 83 Influence of the angle of pressure, 33. Influence of the diameter of rolliiij; circle on shai>e and efficiency of teeth, 18. Inner desci'ibino; circle, 3i); limitations of, 40, 41. Inner normal cone, 48, 50. Instantaneous radius, 4. Intermediate describing circle, 39; limitations of, 40. Internal gear, see annular gear. Interference, 32; in annular gearing, 43. Interchangeable gears, 20. Involute, 4 ; defined, 7 ; system, 26 ; curves, character of, 27 ; rack, 28 ; system of annular gearing, 43 ; annular gear l)roblem, 77; bevel gear tooth, 46; bevel gear problems, 78, 79 ; limiting case, 29 ; limiting case jiroblem, 74 ; prac- tical case, 30; practical case problem, 75. Involute action, Theory of, 26; limit of, 28. Involute gearing, defects of system, 35. Involute teeth, epicycloidal extension of, 30. " Kinematics," MacCord's, (i4. Klein's coordinate odontograph, 59; Design," 64. Elements of Machine Law of tooth contact, 10. Lead of screw, 04. Least angle of i)ressure, method for determining, 33. Least number of teeth in annular gears, 42. Limit of involute action, 28. Limiting ca.se, cycloidal, 10. 14; involute, 29; ainiular gear- ing, 38. Limitations of intermediate, exterior, and interior describing circle, 40, 41. Line of action a great circle, 47. Literature, 63. Logarithmic spiral, 59. ISIacCord's "Kinematics," 64. Method for determining least angle of pressure, 33. Method to be observed in performing problems, 68. "Mechanics of Engineering," 64. " Mechanics of the Machinery of Transmission," 64. Normal, defined, 4 ; to construct, 5 : law Normal cone, 48 ; development of, 49. Notation and formulas, 65. governing, Gl. Obliquity, angle of, 14. Octoid bevel tooth, 47, 62. " Odontics," Grant's, 64. Odontoid defined, 1 ; special forms of, 60. Odontoidal curves, 4 ; problems relating to, (i8. Odontographs and odontograph tables, 57. Odontograph, "Willis, 57: Grant involute, 58; Grant Three point, 58; Robinson, 59; Klein, 59; coordinate, 59. Outer describing circle, 39; limitations of, 40. Outer normal cone, 48. Path of contact defined, 12; affected by rolling circle, 18; a right line, 28. Path of approach defined, 15. Path of recess defined, 15. Pitch cone, 48. Pitch line, 10. Pitch point, 9, 10, 27. Pitch, circular, 11 : diameter, 11. Planed bevel gear teeth, ()5. 84 INDEX. Positive rotation defined, 11. Practical case, conditions governing the, 16; cycloidal, 21 involute, 30; annular, 42. "Practical Treatise on Gearing," 04. Pressure, angle of, 14. Prolate epitrochoid, 7. Proportions for standard tooth, 17. Problems, method to be observed in performing, 68. Rack, 14 ; involute, 28 ; gears classified by, 60. Radial flank, 18 ; as base of system, 21, 60. Radius, describing, 5 ; contact, 5. Rankine, 64. Reuleux, 64. Robinson odontograph, 59. Rolling circle, see describing circle. Rotation, jiositive, 1. Screw gearing defined, 3. Scroll, use of, 11. Second method for describing cycloidal curves, 5. Secondary action in annular gearing, 38, 41. Segmental system, 60. Skew gear defined, 3. Spiral gear defined, 3. Special forms of odontoids, 45. Spherical epicycloid, 45. Spherical hypocycloid, 45. Spur gear defined, 2; illustrated, 10; having action on one side of pitch point, 10; having action on both sides of pitch point, 14; conventional representation, 25 ; inter- changeable with annular gears, 42 ; notation and formu- las, 05. Theory of cycloidal action, 8. Theory of involute action. 20. Thickness of tooth, 17. Three point odontograpli, 58. Tooth contact, law of, 10. To construct a normal, 5. Tredgold approximation, 47. Unsymmetrical teeth, 37. Use of bevel gear table, 53. Velocity ratio constant, 1 ; not aeffcted by increase of center distance in involute, 28. Weisbacli's "Mechanics," 64. Willis, odontograph of, 57 ; writings of, 64. Worm gearing defined, 3, 62; notation and formulas for, 67. Worm wheel, 62. I Plate I. Cycloid, Epicycloid, Hypocycloid and Involute curves. ItEFEIlENCES TO TEXT, Art. 4, Page 4. Art. 8, Pao-e 5. 5, ^ 4. 9, 5. G, 5. 12, 7. 7, 5. Plate 1 , B 1 i__^ B / "^ 5v kl?" v\ "7^ ^C^^l^^x j / V^ \\L^ /\ ^0. \ 1 / \ / 2V^ \ k^^^' N. \ \^ 5 ' \ i >^ i- 'xL A g~ — \- - Kr 'f- / ---At / ^^-\ — ~~~- 1 |l\v^^ r / " \ / 1 \ /^^ \ ^^ \ ' \ yc Y J "/ J \ \7c I 1/ A^,^ vV \ \ \, Ax ^ 1 ^ V!>< J^V^/ 3/v K 1^ ^ 'e c D' B'" Je /V ^'0 FIG 1 j / ^~~^^7<^:i -^^^Z""i ^y^-y^ /O^ ^\ /B,ic^ 0/ 1 / ./ \ E C 1 /' /— ""^/V^^^^ / /Fig 2 /^rVr -^/A^\ 1 >^ A" ^ \ / 1 /^ \ / A / \ 1 ^^^"--^ \ [ f^~~Jl n \ a""^^'^^'"' / / A' ^^ -^A. ri i\ I^nM-^ / / A {B / / . N v^ ^^ ^^,^#^A^/V]\^-4?^ p/ / / k / """^ Jf ~\//^'\ \ \ '\ \ 1 / i / j / i / \ i /' / /' / / / " F,G. / \ / 4 ■/// K ! i / X 1 1 Iky'- §' ^XXA"" P K Fig. 3 \0 Y p P P Plate 2. Epitrochoidal curves. Double generation of Epicycloid and Hypocycloid. Approximate method. REFERENCES TO TEXT. Art. 10, Page 6. 11, 6. 34, 22. Plate 2 Plate 3. Mechanical method for describing Odontoidal curves. REFERENCES TO TEXT. Art. 13, Page 8. 15, 10. 21, 12. Plate 3, Plate 4. Cycloidal Gear, Pinion and Rack having action on one side of pitch point. Limiting case. Airr, REFERENCES TO TEXT. IT). Page 10. Art. 21, Page 11. 10, ' 10. 25," 11. 18, 12. 26, 14. 19, 12. 36, 24. 20, 12. 47, 37. 23, 13. LATE 4. Plate 5. Cycloidal Gear, Pinion and Rack having action on both sides of the pitch point. Limiting case. ItEFEEENCES TO TEXT. Art. 23, Pao-e 1-3. Art. 3(3, Page 24. 20, ' 14. 41), 38. 28, 15. PK015. 2, 72. 32, 10. Plate 5. Plate 6. Cycloidal Gear, Pinion and Rack. Practical case. REFERENCES TO TEXT. Art. 34, Page 21. 36, 24. 37, 25. Plate 6. Plate 7. Involute Gear and Pinion. Limiting case. Mechanical method for describing the Involute. REFEREXCES TO TEXT. Art. 38, Page 26. 39, 27. 42, 31. Plate 7. Plate 8. Involute Gear, Pinion and Rack. Limiting case. REFERENCES TO TEXT. Art. 39, Page 28. Art. 41, Pao-e 30. 40, 29. Pror. 4, 74. PLATE 8. Plate 9. One Pitch Involute Gear and Pinion, showing Interference. REFERENCES TO TEXT. Art. 42, Page 30. Art. 44, Page 33. 43, 32. 46, 35. Plate 9. 1 PITCH INVOLUTE GEAR & PINION SHOWING INTERFERENCE Plate 10, One Pitch Involute Pinion and Rack, showing Interference. REFERENCES TO TEXT. Art. 42, Page 30. Art. 44, Page 33, 43, 32. 46, 35. Plate to. RACK 1 PITCH INVOLUTE PINION & RACK SHOWING INTERFERENCE Plate II. Annular Gearing. REFERENCES TO TEXT. Art. 49, Page 38. Ai;t. 52, Page 41. 50, 39. 54, 42. Plate 11 Plate 12. Annular Gearing. Special cases. REFERENCES TO TEXT. Art. 50, Page 39. 51, 40. 52, 40. Plate 12 12 3 116 1 I . I ililinlilililililililililihlililililililililihlililihiilil ' Plate 13. Bevel Gearing. REFERENCES TO TEXT. Art. 50, Page 47. Art. 60, Page 48. Plate 13 Plate 14. Problems i to 4 inclusive. EEFEREXCES TO TEXT. Art. 65, Page 69. Peob. 3, Page 72. Peob. 1, 69. 4, 74. 2, 71. Plate 14 I \ /<^ DRIVER T A t / / \ /o- \ / __B ^ ^/ \ -9/ \ v/ 9 .Fig. 1 ^i/ RADIAL FLANKS ON GEAR. \ \. ■Fig. 2' // / /.--~\.. //•' > > I \ / A / / I / >-a nJ; / ^/ \^ -■ -- -6.334 / '^^■^ch'linS. qt/- PITCH LINE OF RACK ._. Y^ \- / r\ / / //y \ / NX/ft -H-^-^: — # -^ ^V / "-^ /7V\ /I X< ^v/ / PITCH UNtlOF^RACK Fig. 3 / Fig. 4 Plate 15. Problems 5 to 9 inclusive. REFERENCES TO TEXT. Art. 65, Page 69. Prop.. 7, Page 77. Prob. 5, 75. 8, 78. 6, 76. 9, 79. Plate 15 o^^^^?-^e!^.o,. PlTCH'LiNE OF University of California SOUTHERN REGIONAL LIBRARY FACILITY Return this material to the library from which it was borrowed. B 000 003 239 • ^'A'f^^ ^ »> tQii^iii ^nn'7i^B'9 Univers Sout] Lib