A HANDBOOK ON THE TEETH OF GEARS, THEIR CURVES, PROPERTIES. AND PRACTICAL CONSTRUCTION. By GEORGE B. GRANT, M.E. ADVERTISEMENT OF THE LEXINGTON GEAR WORKS, GEO. B, GRANT, Proprietor. GEAR WHEELS AND GEAR CUTTING OF EVERY DESCRIPTION, \A/E wish to send our new 1890 GEAR BOOK to every manufacturing and nnechanical concern in the country. We do not distribute these valuable pamphlets broadcast, and cast most of them away, but when one is sent for it is evidence enough that it is wanted. We charge ten cents to parties not in business, and refund with first order. LEXINGTON GEAR WORKS, MASS. LIBRARY OF CONGRESS. : ©ujnjnjigt Ifn* Slielf:Afi::l-6 UNITED STATES OF AMERICA. A HANDBOOK TEETH OF GEAES, THEIR OUEYES, PEOPEETIES, AKD PRACTICAL CONSTRUCTIQ]^ George B. Geai^t, M. E. THIRD EDITION. Copyright, 1890, by Geo. B. Grant. PUBLISHED BY THE LEXINGTON GEAR WORKS, Lexington, Mass. 3 \'S '^ "k^ .?^a®fe5(^?_. PRESS OF C. A. PINKHAM A CO. »89 CONGRESS ST., BOSTON. ^"-^^^^^^^(^^'^ THE TEETH OF GEAR WHEELS. INTRODUCTION. Few mechanical subjects have attracted the attention of scientific men to such an extent, or are so intimately connected with mathematics, as the proper construction of the teeth of gear wheels, and, as a consequence, few can show such an advance as has here been made, from the rough cog wheel of not many years ago, to the neat cut gear of the present day. It is not apparent wherein much further improvement is needed in our knowledge of the theory of the subject, but it is evident that much remains to be done towards its practical application, and to induce the working mechanic to understand and use the improvements that have been developed by the mathematician and the inventor. The theory seems to be full and well nigh perfect, but the mill-wright and the machinist still clings to imperfect rules and clumsy devices that should have been forgotten years ago, and few workmen have a clear knowledge of even the rudiments of the science which it is their business to apply to practical purposes. It is the mathematical and scientific character of the subject that makes it so difficult to the practical man, who can understand but little of it as it is commonly presented in elaborate treatises or encyclopaedias, and who takes but little interest in the study of a matter that bristles with strange characters and technical terms. I have here undertaken to address the workman as well as the man of science, and have felt obliged to leave out nearly everything that cannot be treated in a plain, descriptive manner, to use language that any intelligent man can understand, and to refer to more pretentious works than this for demonstrations, or unessential details. A volume of a thousand pages would not properly present the whole subject, and this little pamphlet can deal only with the main principles and prominent points. It is not a treatise, it is a hand-book that does not pretend to cover the whole ground, and its principal object is to present the new odontographs, which I believe to be superior to those heretofore in use for the purpose of designing the teeth of gear wheels. FIRST PRINCIPLES. The original gear wheel had pins or projections for teeth, of any form that would serve the general purpose and communicate an unsteady motion from one wheel to another. The perfect gear wheel is the friction wheel, communicating a smooth, uniform, rolling motion, by means of the frictional icontact of its surface. It is, in fact, a gear wheel with a 'great many very small, weak, nnd irregular teeth. The whole aim and object of the science of the teeth of gear fic.2. FRICTION wHEELs.wheelsisto increase the size and strength of these teeth with- out destroying the uniformity of the motion they transmit, and this is accomplished by studying the shape of the teeth, and giving their bearing surfaces the curved outline that is found to produce the desired result. There are an infinite number of curves that will meet the requirement, but only two, the epicycloid and the involute, are of any practical impor- tance, or in actual use. THE EPICYCLOIDAL TOOTH. The epicycloidal or double curve tooth has its bearing surface formed of two curves, meeting at the pitch line P, which corresponds to the working ch- cle of the perfect gear wheel of fig. 2. If a small circle,a,be rolled around on the outside of the pitch circle, p, a fixed ^ tracing point, a, in its edge, will trace • out the dotted line called an epicycloid, { and a small part of this curve near the I pitch line, usually one sixth of its full .;j)»» height,f orms the face of the tooth. j^y Similarly, if a small circle, B, be rolled around on the inside of the pitch line, its tracing point, b, will describe the internal epicycloid, or hypocycloid, a small portion of which is used for the fio.». THE ep.cyc'loidal TOOTH. flank of thc tootli. .0^ ICYCLOIOAL TEETH. If a projection be formed on the friction wheel fig. 4, the curved outline of which is a whole epicycloid E, and a depression be formed in the wheel N having a whole hypocycloid H for its outline, then, if both curves have been formed by the same describing circle B, it can be mathemati- cally demonstrated that the two curves will just touch and slide on each other, without separating or intersecting, while the two friction wheels roll together. The reverse of this fact is also true, that, if one wheel drives another by means of an epicycloidal projection on it working against a hypocycloidal depression in the other, both curves being formed by the same describing circle, the t-wo wheels will roll together as uniformly as if driven by frictional contact, and it is this peculiar property of the epicycloid that gives it its value for the purpose in hand. The pressure acting between the two curves is in the direction of the line dg, is direct only at the start, and becomes more and more oblique, until, when the middle points, q q, come together, and beyond, there is no driving action at all. This defect forbids the use of the whole curve and we can use but a small portion of it near the pitch line. Another projection and depression must be formed so near the first that they will come into work- ing position before the first pair are out of contact, thus forming the theo- retically perfect but incomplete gears of fig. 5. Practical requirements still further modify the apparent shape of the tooth, for it is desirable that the wheels shall work in either direction, and that they shall be interchangea- ble, so that any one of a set of several shall work with any other of that set. This can be accomplished only by making the curves face both ways, and by putting both projections and depressions on each gear, thus form- ing the familiar tooth of fig. 3. no. B. INCOMPLETE EPICYCLOIDAL TEETH. THE INTERCHANGEABLE SET. If all the curves of a set of several gears, both the faces and the flanks of each gear, are described by the same rolling circle, the set will be interchangeable, and any one will work perfectly with any other. This is a property of the greatest practical importance, and interchangea- ble sets should come into as universal use on heavy mill work as with cut p-ear- ing. It is the only system that will allow the use of a set of ready made cutters, and is therefore essential to the economical manufacture of cut gear wiieels. The diameter of the rolling circle is usually made half the diameter of the smallest gear of the set, and that gear will have straight radial lines for flanks. The set in almost universal use and adopted for all the odontographs, has twelve teeth in its smallest gear, but there is a tendency to change this well established system, and create confusion for which the writer can see no adequate excuse, by the adoption of a pinion of fifteen teeth as the base or smallest gear. It may be admitted that as large a base as possible should be used, but the change from twelve to fifteen seems to be unwarranted in view of the confusion it creates by the abrupt change from an old and good rule to a new one that is a mere shade better, and the trouble it makes wdth small pinions of eight to twelve teeth. RADIAL FLANK TEETH. If the internal curves, or flanks, of a pair of gears that are to run together are on each radial straight lines described by a rolling circle of half its pitch diameter, and the rolling circle that describes the flanks of one gear is used to describe the faces of the other gear, then, the two gears will form a pair fitted to each other and not interchangeable with other gears. This style of gear is very often used under the erroneous impression that it is the best possible form, and will give the least possible friction and thrust on the bearings, but the saving in friction over the interchangeable form would be an exceedingly difficult thing to measure by any practicable method, although it can be mathematically demonstrated to be a fact, and the slender roots of such teeth make them weaker and much inferior to the others. The odontograph figures show both a pair of these gears, and the same pair on the interchangeable plan, also, by the dotted lines on the former figure, the shapes as they would be on the interchangeable plan. It is plainly seen that the interchangeable faces are but a shade more rounding, while their flanks are so curved that the teeth are much stronger at the roots. The larger the describing circle, the less the theoretical thrust and friction, and if the flanks Avere formed by a describing circle of more than half the diam- eter of the gear, the teeth would be undercurved, the friction less, and their strength less, than that of the radial flank tooth. In practical matters it is a good plan to give first place to practical points, and not to take too much notice of minute theoretical advantages, and there is no good reason, that will bear the test of experiment, for adopting the radial flank, non-interchangeable, and weak tooth, in preference to the strong tooth of the interchangeable system. THE PITCH. The pitch is a term used to designate the size of the tooth, and is either circular or diametral. THE CIRCULAR PITCH or more properly the circumferential pitch, is the actual distance from tooth to tooth measured along the curve of the pitch line, and is expressed in inches, as | inch pitch, 1^ inch pitch, etc. The table gives the proper pitch diameter of a gear of any given number of teeth, and one inch circular pitch. The tabular numbers must be multi- plied by any other pitch that is in use. Formerly, the circular pitch was the only one known, but it has deser- vedly gone out of use on cut gears, and it is hoped may soon be abandoned altogether. It is a clumsy, awkward, and troublesome device on either large or small Avork, having its origin in the ignorance of the past, and owing its existence not to any perceptible merit, but to habit, anci the natural per- sistence of an established custom. With the circular pitch the relation between the pitch diameter of the gear, and the number of teeth on it, is fractional. If the diameter is a convenient quantity, such as a whole number of inches, the pitch must be an inconvenient fraction, and if the pitch is a handy part of an inch, the diameter will contain an unhandy decimal. With the circular pitch there is no one length of tooth that is better than any other, and consequently there is no agreement upon that point. Each maker is at liberty to chose his own distance at random, and whatever he choses is as good as any other. Its worst feature is that it leads to endless errors, for the average mechanic appreciates convenience more than accuracy, and will stretch his figures to suit his facts, with a botch as the common result. A millwright figures out a diameter of 22.29 inches for a gear of one inch pitch and 70 teeth, and failing to make such a clumsy figure fit his work or his foot rule, and thinking a quarter of an inch or so to be of no importance, he lets it go at 22 whole inches. The same process on its mate of 15 teeth gives a 5 inch gear instead of one of 4.78 inches diameter, and the pair will never run or wear together properly. His only alternatives are to adopt the clumsy true diameters, or else use the clumsy figure .988 inch for his pitch. Again, he is apt to apply a carpenter's rule directly to the teeth of the gear he is to repair or match, and naturally takes the nearest convenient fraction of an inch as his measurement, when the real pitch may be just enough diiferent to spoil the job. There is no reason whatever for using the circular i^itch, unless the work to be done is to match work already in use. THE DIAMETRAL PITCH is an immense improvement on the old fashioned circular pitch. It is not a measurement, but a number, or ratio. It is the number of teeth on the gear, for each inch of its pitch diameter, and its merit is that it establishes a convenient and manageable relation between these two principal elements, so that the calculations are of the simplest description and the results convenient and accurate. The product of the pitch and the pitch diameter is equal to the number of teeth, and the number of teeth divided by the pitch is equal to the pitch diame- ter. A gear of 15 inches diameter and 2 pitch has 30 teeth, and a gear of 27 teeth of 4 pitch has a pitch diameter of 6f inches. The rule that the length of the tooth is two pitch parts of an inch, | or ^ an inch for 4 pitch, f or 1 inch for 2 pitch, etc. is so simple and so much bet- ter than any other that it is never disputed, and is in universal use. The circular and diametral pitches arc connected by the relation cXp=8.141G. or, the product of the circular and the diametral i)itch is the number 3. 1416. THE ADDENDUM. ^or reasons expressed above we can use but a small part of the epicy- cloidal curve near the pitch line, limiting it by a circle drawn at a distance inside or outside of the pitch line called the addendum. The outside limit need not be the same as the inside limit, but it is customary to make them equal. When the diametral pitch is used, the length of the addendum is always one pitch part of an inch, as Jthincli for 4 pitch, ird inch for 3 pitch, etc. If we use the same proportion for circular pitches the addendum will be 3 xir^ circular pitch, and the value ^rd of the circular pitch may be adopted as the most convenient for use. THE CLEARANCE. Theoretically, the depression formed inside the pitch line should be only as deep as the projection outside of it is high, but to allow for practical defects in the making or in the adjustment of the teeth, and to provide a place for dirt to lodge, the depression is always deeper than theory requires by an amount called the clearance. The amount of the clearance is arbitrary, but the sixteenth part of the depth of the tooth is a convenient and customary measure, or ^^:fth of the circular pitch, and 1 divided by 8 times the diametral pitch. The following tables will be convenient and save calculation : CLEARANCE FOR CIRCULAR PITCHES. Circular pitch. Clearance. CLE/ .02 f .03 f .03 i .04 1 .04 .05 H .05 If .06 .06 If .07 2 .08 .09 .10 .12 FRANCE FOR DIAMETRAL PITCHES. Diametral pitch. Clearance. 6 .02 5 .03 4 .03 H .04 .04 3 .04 2| .05 2i .05 2i .06 2 .06 If .08 H .09 .10 1 .12 1 THE BACKLASH. When wooden cogs or rough cast teeth are used, the inevitable irregular- ities require that the teeth should not pretend to fit closely, but that the spaces should be larger than the teeth by an amount called the backlash. The amount of the backlash is arbitrary, but it is customary to make it about equal to the clearance. Cut gears should have no allowance for backlash, and involute teeth need less backlash than epicycloidal teeth. PITCH DIAMETERS. waiti ONE I^^^C]EI circxjil.^1?, jpitch. For Any Other Pitch, Multiply by that Pitch. T. P.D. T. P.D. T. P.D. T. P.D. 10 3.18 1 33 10.50 56 17.83 79 25.15 11 3.50 34 10.82 57 18.15 i 80 25.47 12 3 82 35 11.14 58 18.47 81 25.79 13 4.14 36 11.46 59 18.78 82 26.10 U 4 46 37 11.78 60 19.10 83 26.43 15 4.78 88 12.10 61 19.42 84 26.74 27.06 16 5.09 39 12.42 62 19.74 85 17 5.40 40 12.74 63 20.06 86 27.38 18 5.73 41 13.05 64 20.38 87 27.70 19 6.05 42 13.37 65 20.63 88 28.02 20 6.37 43 13.69 66 21.02 89 28.34 21 6 69 44 14.00 61 21.33 90 28.65 22 7.00 45 14.33 68 21.65 91 28.97 23 7.32 46 11.65 69 21.97 92 29.29 24 7.64 47 14.96 70 22.29 93 29.60 25 7.96 48 15.28 71 22.60 94 29.93 26 8.28 49 15.60 72 22.92 95 30.25 27 8.60 50 15.92 73 23.24 96 30.56 28 8.90 51 16.24 74 23.56 97 30.88 29 9.23 52 16 56 75 23.88 98 31.20 30 9.55 63 16.87 76 24.20 99 31.52 31 9.87 54 17.19 77 24.52 100 31.84 32 i 10.19 55 17.52 L 78 24.83 ( The Epicycloid. THE EPICYCLOID. THEORETICAL FORMATION. Tlie true epicycloid, shown by fig. 6, is perpendicular to the pitch line at the origin a, and forms an endless series of lobes about it, as in fig. 3. The most convenient and simple process for drawing it, is to step it ofit' with the dividers. Several describing circles, M^ to M% are drawn at ran- dom ; steps are made, as shown by the figure, from the origin a^ to past each tangent point, a^ to a^, and then the same number back, around each circle, to locate the several points, b^ to b^, on the curve, which is then drawn by hand through the points, and is accurately in place if the steps are small. By the mechanical method for drawing the curve, the describing circle, B, is rolled around the pitch circle A, and a tracing point or pencil P, draws the curve. A steel ribbon s, is fastened to the templets at each end, and assists in keep- ing them in place. This process is the main principle of the epicy- cloidal engine, which carries a scribing tool, or a rotary cutter at p, to trace or cut out a tem- plet that is then used in forming gear teeth or gear cutters. It is, of course, the most accurate method known, but it is not available for ordinary pur- poses, for unless the templets are well made and skillfully handled, the resulting curve will be poorly drawn, and the method, although simple in principle, may be consid- ered difficult in its practical application. PRACTICAL FORMATION. Of course nothing but the perfect curve will answer its purpose with per- fect accuracy, but the epicycloid is a peculiar curve which cannot be accu- rately drawn by any simple process, or with common instruments, particu- larly when the teeth are small, and it is customary to use arcs of circles or other curves, which approximate as nearly as possible to the true curve. Such an arc can be made to agree with the curve so closely that it is a need- less refinement to be more particular for most practical purposes, such as drafting teeth, making wooden cogs or patterns for cast teeth, or even the templets for shaping gear cutters and planing bevel gear teeth. Some makers of rough cast or heavy planed gearing go to great expense to construct the (supposed to be) theoretically true epicycloid, by means of rolling circles. This practice looks very much indeed like accuracy, but if he had an absolutely true curve as a templet, supposing he could make such a thing, the maker of this class of work could not produce from it a work- ing tooth more nearly perfect than if the templet was properly constructed of circular arcs. It is labor lost to lay out teeth to the thousandth of an inch, that must be constructed with ordinary hand or machine tools, or shaped with a chisel and mallet. Furthermore, it is a question if the delicate processes and epicycloidal engines used for the finest cut gear work, can serve practical purposes and construct templets to work from, better than intelligent and skillful hand-work. It is a fact that the best work in this line is made from tem- plets that are laid out by theory, but dressed into shape and perfected by hand and eye processes. Fig. 7. Epicycloidal Engine. ODONTOGRAPHS. Many arbitrary or "rule of thumb" methods for shaping gear teeth have been propo ed, but they are generally worthless, and reliance should be placed only on such as are founded on the mathematical principles of the curve to be imitated. Of these only three are known to the writer. y ;>"' __.«i:t \ ^ c'^ J,--;--^' ^ -X ^ H" ctrei/lar pltth :'** / \\ c*/, \ THE WILLIS ODONTOGRAPH is a method for finding the center m of the circle which is tangent to the epicycloid a b c, at the point b, where it is cut by a line bm, which passes through the adjacent pitch point k, and makes the angle gkf=75^ with the radial line kf. The radius used, is not the line m b, but the more convenient line m a. The instrument is nothing whatever but a piece of card or sheet metal cut to the angle of 75", which is laid against the radial line kf, as a guide for drawing the line km. The center distance km, to be laid off along the line thus draAvn is given by a table that accompanies the instrument. '^o instrument is necessary, for the line k m may be placed by drawing the arc f g with a radius of one inch, and laying off the chord f g=1.22 inch. The tabular distance k m can be readily computed from k, m, =: ko m. in which c is the circular pitch in inches, and t is the number of teeth in the gear. The Willis odontograph, as found in use, is confined to the single case of an interchangeable series running from twelve teeth to a rack, but for any possible pair of gears the angle becomes g k f = 90" — 1^ s and k, m, =: ^ . 4_ . sin. 180! ' 6.28 t + s s 6.28 t — s s in which t is the number of teeth in the gear being drawn and s the number in the mate. The accuracy of the Willis circular arc will be examined further on. THE IMPROVED WILLIS ODONTOGRAPH. EPICYCLOIDAL TEETH. TWELVE TO RACK. INTERCHANGEABLE SERIES. FOR ONE For ONE INCH NUMBER OF DIAMETRAL PITCH. CIRCULAR PITCH. TEETH For any other pitch, divide For any other pitch, mul- IN THE QBAB. by that pitch. tiply by that pitch. Faces. Flanks. Faces. Flanks. 1 Exact. 12 Intervals. Rad. Dis. .15 Rad. Dis. OO Rad. .73 Dis. Rad. Dis. 12 2.30 oo .05 OO OO 13i 13-14 2.35 .16 15.42 10,25 .75 .05 4.92 3.26 15i 15-16 2.40 .17 8.38 3.86 .77 .05 2.66 1.24 m 17-18 2.45 . .18 6.43 2.35 .78 .06 2.05 .75 20 19-21 2.50 .19 5.38 1.62 .80 .06 1.72 .52 23 22-24 2.55 .21 4.75 1.23 .81 .07 1.52 .39 27 25-29 2.61 .23 4.31 .98 .83 .07 1.36 .31 33 30-36 2.68 .25 3.97 .79 .85 .08 1.26 .26 42 37-48 2.75 .27 3.69 .66 .88 .09 1.18 .21 58 49-72 2.83 .30 3.49 .57 .90 .10 1.10 .18 97 73-144 2.93 .33 3.30 .49 .93 .11 1.05 .15 290 145-rack. 3.04 .37 3.18 .42 .97 .12 1.01 .13 THE IMPROVED WILLIS ODONTOGRAPH. I have carefully calculated the distances nii Uj and nig Ug of the circles of centers from the pitch line, and also the radii ai m^ and ag ni2, and have arranged them in the table above, so that the data resulting from the usual process can be obtained without the usual labor. This improved Willis process will produce exactly the same circular arc as the usual method, with the same theoretical error, but its operation is simpler and less liable to errors of manipulation. By the usual process it is necessary to draw two radial lines, and to lay off a line at an angle with each. The tabular distances laid off on these lines, will locate the two centers. The two circles of centers are then drawn through them, and the dividers set to the radii to be used. By the new process the circles of centers are drawn at once without pre- liminary constructions, at the tabular distances from the pitch line, and the table also gives the radii to be taken on the dividers. No special instru- ment is required, no angles or special lines are drawn to locate the centers, and the chance of error is much less. This process, however, is not as correct, and is no simpler or more con- venient than the new odontographic process given further on. ROBINSON'S TEMPLET ODONTOGRAPH. This ingenious instrument, the invention of Prof. S. W. Eobinson of the Ohio State University at Columbus, is based on the fact that some part of a certain curve of uniformly increasing curvature, called the logarithmic spiral, can be made to agree with the true curve of a gear tooth with a degree of approximation that is very precise. It is a sheet metal templet having a graduated curved edge a c, shaped to a logarithmic spiral, and a hollow edge a b shaped to its evolute, an equal logarithmic spiral. To apply the instrument, draw a radial line from the pitch point d on the pitch line, and another from e, the center of the tooth, and then draw tangents d g and n e f , square with the radial lines. The instrument is then so placed that a certain graduation, given by accompanying tables, is at the point h on the tangent nef, while the. grad- uated edge ac, is at the pitch point d,and the hollow edge ab, just touches the tangent line nef at k, and then the face of the tooth is drawn with a pen along the graduated edge. The flank is similarly located by placing the instrument so that a certain other graduation is at the pitch point d, while its hollow edge touches the tangent line g d. The full theory of this instrument would be out of place here, but may be found in No. 24 of Van ISTostrand's Science Series, or in Van Nostrand's Mag- azine for July, 1876. A NEW ODONTOGRAPH. Having frequently to apply the "Willis Odontograpli, it occurred to me that the process would be much simplified and much time and labor saved if the location of the circles of centers and the lengths of the radii were computed and tabulated, thus forming the improved Willis method already described. It was then evident that the process would be precisely the same, and the result much improved, if the centers tabulated were the centers of the near- est possible approximating circles, rather than of the Willis circles, and 1 have embodied this idea in the following tables. I have carefully computed, by accurate trigonometrical methods, and have tabulated the location of the center of tlie circular arc that passes through the three most important points on the curve, at the pitch line a, fig. 9, at the addendum line k, and the point e, half way between. The tables locate this center directly, giving its distance from the pitch line, and from the pitch point. The circles of centers are drawn at the tabular distances " dis" inside and outside the pitch lines, and all the faces and flanks are drawn from centers on these circles, with the dividers set to the tabular radii "rad." The tables are arranged in an equidistant series of twelve intervals. For ordinary purposes the tabular value for any interval can be used for any tooth in that interval, but for greater precision it is exact only for the given "exact" number, and intermediate values must be taken for inter- mediate teeth. The tables are arranged for both the diametral and circular pitch sys- tems. The former is much the more manageable and should be used when the work is not to interchange with work already made on the latter system. The first table, giving an interchangeable set, from twelve teeth upwards, is the one for general use. The second, or radial flank table, is inserted because teeth are sometimes drawn that way, but, as before explained, they are weak, not interchange- able, and but a mere shade more direct in their action than the interchange- able style. ACCURACY OF THE ODONTOGRAPH. The assertion is often made that no circular arc can be made to do duty for the epicycloid, except for rough work, but it can be shown that the state- ment is not true if applied to the new method, for few mechanical processes can be made to work closer to a given example, than this arc is close to the true curve. Figure 9 shows the true curve, and both the new and the Willis aj^proximating arcs, the actual proportions being exagerated to show the errors more clearly. The Willis arc runs altogether within the true curve, while the new arc crosses it twice. We will take, for an example, the case of a twelve tooth pinion, Avhich will show the errors at their greatest, and calculate them with great care for a tooth of three inch circular pitch, which is twice the size of the figure on page 13, and may be considered a very large tooth. y ^^~' The distance from pitch line to addendum line is divided into eight equal spaces by parallel cir- cles, and the distance along each circle, in ten thousandths of an inch, from the true curve to each odontographic arc, is as follows : GKANT. WILLIS. At a .0000 .0000 inches " b +.0088 +.0175 " •' c +.0091 +.0244 " u ^ +.0050 + .0283 " " e .0000 +.0288 '• U f -.0086 +.0297 " u -.0061 +.0308 " " h -.004(5 +.0342 " " k .0000 +.0397 " rt = 12 a 20 a 40 <' 100 a 300 Average, .0042 .0260 " It is seen that the new arc is in no place one hundredth of an inch in error, and that for a tooth of four pitch, a large size for cut work, its average error is one thousandth of an inch. A greater accuracy than this would be of no practical value. The twelve tooth gear, for which the errors of both arcs were com- puted, shows them at their maximum value, for, as the number of teeth in the gear increases, the errors diminish, and for several locations their values for the new arc at c, which is the point of greatest error, are as follows : c — .009 inches. " .008 " '• .006 " " .004 " " .002 '• and the errors of tne Willis arc are subject to the same rule. The error of the Willis arc is plainly shown, at its greatest value, by the figure on page 13, where the dotted faces of the pinion teeth are correctly located by the Willis method. To further test the accuracy of the new method, construct the same tooth face several times by the same;iprocess, using either the method by points, or the usual Willis process. Unless the work is most carefully performed, it wall be found that the several results will not agree with each other by amounts that are noticeable, while by the new^ method they will be sub- stantially the same curve. The new arc is most nearly correct at the most important point, the upper part of the curve, just where the Willis arc is most out of place, or w^iere the true curve, unless drawn by some delicate and costly apparatus, in most likely to be out of place. CIRCULAR AND DIAMETRAL PITCHES COMPARED. CIR. P. DM. P. e .52 ^h .58 5 .68 4i .70 4 .78 H .90 3 1.05 21 1.15 2i 1.25 H 1.40 2 1.57 11 1.80 Ih 2.10 n 2.50 1 3.14 1 4.20 i 6.28 DM. P. CIR. P. k 6.28 1 4.20 1 3.14 H 2 50 U 2.10 11 180 2 1.57 2h 1.25 3 1.05 3i .90 4 .78 5 .63 6 .52 7 .45 8 .39 9 .35 10 .31 THE NEW ODONTOGRAPH. GENERAL DIRECTIONS. Draw the pitch line and divide it for the pitch points mag. Take from the tables, multiply or divide, as the case may require, by the pitch in use, and lay off, the addendum a b and a c, the clearance e f , the backlash g g', the face distance a d, and the flank distance a c. Draw the addendum line through b, the root line through e, the clearance line through f, the line of face centers through d, and the line of flank centers through c. Set the dividers to the face radius, and draw all the faces ab from centers A. Set to the flank radius, and draw all the flanks a k from centers B. Round the flanks into the clearance line. The flanks of a gear of twelve teeth are straight radial lines. ODONTOGRAPH TABLE. EPICYCLOIDAL TEETH. INTERCHANGEABLE SERIES. From a Pinion of Twelve Teeth to a Rack. 1 FOR ONE FOR ONE INCH \ NUMBER or TEETH DIAIV For ar [ETRAL P ITCH. vide by CllB Fori LCULAR PITCH. 1 ly other pitch, di my other pitch, multiply 1 IN THE GEAR. that pitch. by that pitch. | Faces. Flanks. Faces. Flanks. | Exact. Intervals. Rad. Dis. Rad. Dis. Rad. Dis. Rad. Dis. 12 12 2.01 .06 CO CO .64 .02 CO oo 134 13-14 2.04 .07 15.10 9.43 .65 .02 4.80 3.00 15h 15-16 2.10 .09 7.86 3.46 .67 .03 2.50 1.10 m 17-18 2.14 .11 6.13 2.20 .68 .04 1.95 .70 20 19-21 2.20 .13 5.12 1.57 .70 .04 1.63 .50 23 22-24 2.26 .15 4.50 1.13 .725 .05 1.43 .36 27 25-29 2.33 .16 4.10 .96 .74 .05 1.30 .29 33 30-36 2.40 .19 3.80 .72 .76 .06 1.20 .23 42 37-48 2 48 .22 3.52 .63 .79 .07 1.12 .20 58 49-72 2 60 .25 3 33 .54 .83 .08 1.06 .17 97 73-144 2.83 .28 3.14 .44 .90 .09 1.00 .14 290 145-rack. 2.92 .31 3.00 .38 .93 .10 .95 .12 A PRACTICAL EXAMPLE OF THE WORK OF THE NEW ODONTOGRAPH. Fig. 10. INTERCHANGEABLE SERIES. Example. — A gear of 24 teeth, and a gear of 12 teeth, of li circular pitch. Data. — Take from the table the numbers to be used, which are as follows when multiplied by li. For 24 teeth, face rad, = 1.08 face dis, := .07. " 24 " flank " — 2.15 flank '' — .54. " 12 " face " = .96 face '* = .03. " 12 " flank "=00 flank " = co Also take from the proper tables the pitch diameters 5.73 and 11.46 inches, the addendum, .5 inch, and clearance, .06 inch. Process. — Draw the two pitch lines, and divide for the pitch points. Draw the addendum, root, and clearance lines of both gears. Draw the circles of centers, .03 inside the pitch line of the 12 tooth gear, and .07 inside of that of the other. Draw the circles of flank centers, ..54 outside the pitch line of the 24 tooth gear, and draw straight radial flanks for the 12 tooth gear. Draw the faces of the 12 tooth gear with the radius. 96, and draw the faces of the 24 tooth gear with the radius, 1.08, and the flanks with the radius 2.15. Round the flanks into the root line, and allow backlash by thinning the teeth according to judgement. The dotted faces of the 12 tooth gear show them as they would be laid out by the Willis odontograph, and the figure also shows the two centers RADIAL FLANK SYSTEM. TEETH NOT INTERCHANGEABLE. Gears on this system must work together in pairs, each gear being fitted to its mate and to no other. See page 3. The process is the same that has been described on page 12 for the interchangeable set. Fig. 11. RADIAL FLANK SYSTEM. ExPLAXATiox OF THE TABLE. — The uppcr number in each square is the face radius, the lower is the center distance. The centers are mostly insid the pitch line, but some are on^the line, and those ha^dng the negative sign are outside of it. The tabular numbers are for one inch circular pitch, and must be multi- plied by any other circular pitch in use. For the value for any diametral pitch, multiply the tabular number by 3.14, and then divide by the diame- tral pitch in use. Example. — A gear of 12 teeth, paired with a gear of 24 teeth. Circular pitch li inches. Data. — Take from the table for 12 teeth into 24, face radius =.68 and cen- ter distance = 0, and for 24 teeth into 12. radius = 72, and distance = .05. These multiplied by 1^ give the values for use on the drawing, 12 rad. =1.02, 12 dis = 0, 24 rad. = 1.08, and 24 dis. = .07. The addendum is one third the pitch, = i inch, and the proper tables give the clearance =.06, and the pitch diameters = 5.73 and 11.46 inches. Process. — Draw the two pitch lines 5.73 and 11.46 inches in diameter and space them for the teeth. Lay ofE the addendum, .5 inch, and the clearance, .06 inch, and draw the addendum, root, and clearance lines. , . , i- Draw all the faces of the twelve tooth gear, from centers on its pitch hne, with the radius 1.02. Draw all the faces of the 24 tooth gear from centers on a line .07 inch inside its pitch line, with the radius 1.08 inches. Draw straight radial lines for the flanks of all the teeth. ODONTOGRAPH TABLE. EPICYCI.01DAI. TEETH. RADIAL FLANK TABLE. FOR ANY POSSIBLE PAIR OF GEARS, NOT INTERCHANGEABLE. Multiply by the Circular Pitch. Divide by the Diametral Pitch, and then multiply by 3.14. NUM TF.ETH BEING Exact. BER OF JN GEAK DRAWN. Intervals NUMBER OF TEETH IN THE MATE. ,o 13 15 . 17 19 22 25 30 37 49 73 145 ^2 14 16 18 21 24 29 36 48 72 144 rack 12 12 .64 .02 .64 .01 .65 .01 .66 .01 .67 .68 .69 -.01 .70 -.01 .71 -.02 .73 -.02 .74 -.03 .75 -.03 13^ 13-14 .65 .02 .66 .02 .67 .01 .68; .69 .01 .01 .70 .72 .74 -.01 .75 -.01 .76 -.02 .78 -.02 .79 -.03 15i 15-16 .67 .03 .68 .02 .69 .02 .70 ' .72 .01 .01 .74 .01 .75 .78 .79 -.01 .82 -.02 .84 -.02 .84 -.03 m 17-18 .68 .04 .70 .03 .71 .02 .73 i .75 .02 .01 .77 .01 .78 .01 .82 .84 -.01 .87 -.01 .89 -.02 .90 -.03 20 23 19-21 .70 .04 .72 .04 .74 .03 .76 .79 .02 j .02 .81 .01 .83 .01 .87 .90 .93 -.01 .96 -.02 -.03 22-24 .72 .05 .74 .04 .76 .04 .79 .03 .82 .02 .85 .03 .84 .02 .87 .01 .91 .01 .94 .98 -.01 1.01 -.02 1.03 -.03 27 25-29 .74 .05 .76 .05 .79 .04 .82 .04 .87 .02 .92 .02 .96 .01 .99 1.03 -.01 1.07 -.02 1.10 -.03 33 30-36 .76 .0(i .79 .05 .83 .05 .86 .04 .90 .03 .94 .03 .98 .02 1.02 .01 1.06 1.11 1.17 -.01 1.23 -.02 42 37-48 .79 .07 1 .83 .86 .06 .05 .90 .05 .96 .04 .98 .04 1.03 .03 1.08 ' 1.14 .03 1 .02 1.20 1.25 1.37 -.01 58 49-72 .83 .08 .87 .07 .91 .07 .96 .06 1.02 .06 1.05 .05 1.10 .04 1,17 1.24 .04 .03 1.30 .02 1.43 1.58 97 73-144 .90 .09 .93 .08 .97 .08 1.01 .07 1.07 .07 1.11 .06 i:i8 .06 1.28 1.34 .05 .04 1.47 .03 1.65 .02 2.03 290 145 rack .93 .10 .96 .09 1.00 .09 1.05 .09 1.10 .08 1.16 .08 1.24 .07 1.37 1 50 .07 .06 1.70 .04 2.12 .03 2.90 .02 THE INVOLUTE TOOTH. With the exception of tlie epicycloid^ the only curve in extensive use for the working face of a gear tooth, is the involute. THE INVOLUTE CURVE. As the rolling circle A of fig. 3 increases in size, it finally, when of infinite ^ ^- diameter, becomes the straight line d g of fig. 15, while the epicycloid traced by a fixed point in the circle becomes the involute. The involute is, therefore, not a new or sep- arate curve, but simply a particular case of the epicycloid. It is the infinite form of the epicy- cloid.* As the rolling circle of infinite diameter is the same thing as a straight line, the involute can be formed by a fixed tracing point in a cord which is unwound from a circle, called its " base circle," which has been wrapped or "involved" FIG. 15. THE INVOLUTE. 1^ ^ ^^^ from tlils propcrty i,t derives its name. ITS UNIFORM ACTION. If the two circles A and B, fig. 16, are separ- ated by the distance ab, and work together by means of two external epicycloids C and D, the motion communicated will be irregular, for the conditions of uniformity are that the two cir- cles shall touch, and that the external curve of one shall work with the internal curve of the other. See page 2 and figure 4. The amount of this irregularity will depend on the proportion between the separating dis- tance a b and the diameter of the rolling circle which describes the epicycloids. If the pro- portion is very small, the irregularity will be very small, and if the rolling circle has an in- finitely great diameter, the proportion and the irregularity will be infinitely small, that is, zero. Therefore, involutes will work together with perfect regularity and are suitable curves for gear teeth. ITS ADJUSTIBILITY. If the rolling circle is infinitely large, the proportion between the separat- ing distance and it will always be zero, and it will not be altered by any finite alteration of the former, and therefore the uniformity of the action of involute teeth is not in any way dependent upon, or affected by any change of the separating distance. The action will be perfect as long as the curves remain in contact, and this is a property of the greatest practical value, which gives the involute a great advantage over every other known or pos- sible curve. The curve of any gear tooth must of necessity be a " rolled curve " formed by a fixed object attached to the plane of or moving with some curve that rolls upon the base curve of the tooth, and, as the involute is the infinite form of any rolled curve, it is the only form that can possess this property of adjustibility. *The exact nature of the involute curve is more fully treated of in a paper in the appendix, on " The Normal Theory ot the Gear Tooth Curve." EPICYCLOIDS. ITS UNIFORM PRESSURE AND FRICTION. The point of contact of the two involutes C and D will always be upon the straight line of action mn, the common tangent of the two base circles, commencing at its point of tangeucy with one circle, and ending at the same point on the other. The direct pressure between the two teeth will always be in the direction of the line of action, and uniform both in direction and in amount, a prop- erty that is peculiar to the involute curve, and which contributes greatly to the smooth action and even wear of involute teeth. Friction is substan- tially in proportion to direct pressure, and when the pressure is uniform, the friction will be uniform, and no part of the curve will be more likely to wear away than any other part. The durability of a tooth, particularly when doing heavy work, depends on the uniformity of the friction as well as upon its absolute amount. THEORETICAL CONSTRUCTION. To draw the involute curve through the pitch point a of two pitch circles A and B, draw the line of action m n at any desired angle with the line of centers, usually 75°, and then draw the base circles C and D, touching the line of action at e and d, where the perpendicular radial lines e g and f d meet it. From a, step off any num- ber of . short steps along the line of action and around the base line to any point s, then draw any number of tangent lines b c, t v, then step olf the distances sbc, stv, sb, etc., each equal to s d a, and the points c, v, b, etc., will be points of the curve. Any line, as w c X drawn through c at right angles to he, will be tangent to the curve. The working part of the curve must not be extended beyond the circle k e p through the point of contact of the line of action m n and the base line C, for beyond that point it will interfere with the radial flank of the tooth it works with. The curve is generally limited by the addendum line z y, at an arbitrary distance from the pitch line B, and ends at b on the base line D, where it :» perpendicular to the base line. It is continued within the base line by a. radial line as far as the root line zy, and is then rounded into the clearance line. The matter under epicycloidal teeth, pages 3, 4, and .5, regarding the pitch, addendum, clearance, and backlash, will apply as well to involute teeth. ANGLE OF ACTION. The angle mag may be less, but not greater, than the value found from the formula .180° m a g = 90<^ — -— - in which s is the number of teeth in the smallest gear in the pair. If the angle is greater than this the motion will not be continuous, as each pinion tooth will pass out of action before the next one is in position to act. INTERCHANGEABLE SETS. Any number of involute gears from base circles of different diameters will work together correctly and interchangeably if all are of the same pitch, and have the same angle of action. If we put s = 12 teeth, we find ISO" m ag = 90° — -^ = 75" :ONSTRUCTICN. the value for the common twelve to rack interchangeable set, and if we use fifteen as the smallest number of teeth in the set, we have an angle of action of 78°. PRACTICAL CONSTRUCTION. When the involute is to be brought into use, we meet with the same diffi- culties as with the epicycloid, for its theoretically correct construction is not easily and accurately accomplished, and we must adopt some short cut of approximative accuracy. The principle of the epicycloidal engine of fig. 7 may be applied to the construction of the involute, the ribbon s being drawn tight and straight as it is unwound from the base circle, but the same difficulties prevent its use for ordinary purposes. THE OLD RULE. A defective rule in common use drawls the whole curve from base line to addendum line, as one circular arc. The angle mag is laid olf at 75", sometimes at Ib^^, the distance a c is made equal to one quarter of the pitch radius a g, and the tooth curve is drawn from c as a center. This rule is simple, to be sure, but it gives the faces shown by the dotted lines of the figure on page 23, and is abominably wrong and worth- less. If it would round off the points of the teeth of a large gear, it would be useful to correct interference, but it greatly rounds the teeth of a small gear that needs little or no correction, and gives the curf e on a large gear in nearly its theoretical position, without the allowance for interference that must be made. It is not to be wondered that the involute tooth is in small favor with practical mechanics who use this bungling method, and who do not under- stand that the trouble is not in the involute system, but in its defective application. A NEW METHOD. In devising a method for drafting the involute tooth, I have borne in mind that a minute degree of accu- racy is not the essential requirement, for although substantial accuracy must be secured, simplicity and con- venience are qualities that must also be considered. The method, in general terms, and given in full on pages 22 and 23, is to give, by a table, the distance of the base circle B, see fig. 19, inside the pitch circle P, and to give by the same table, the distances or radii ac and ad F.c. 13. THE NBw METHoo. f^om thc pitch polut a to ccuters c and d on the base Ime. The face arc a w is drawn from the center d and the flank arc a v from the center c. The table, page 22, is for one diametral pitch, and covers the common twelve to rack interchangeable set. THE OLD RULE. INTERFERENCE. As indicated above, the involute face will interfere with the radial flank of the mating tooth if the addendum is greater than a certain amount, and ris the addendum in common use for the interchangeable set generally i^xceeds this limit, we must gererally make corrections to avoid this trouble. INTERFERENCE Interference Table 7 inch cir- For one diametral pitch and 3 cular pitch. Angle of action, 75°. Number of Teeth in the Mate. 13 15 17 19 14 12 16 18 21 Figure 20 shows the interference, its effect, and i ts correction. The working' face of the involute should be limited at i by the circle k p through the tangent point e, but if the usual addendum continues it beyond that line, to s, the extension si will interfere with the radial flank c f , and the uniformity of the action will be destroyed. To correct it we must either weaken and spoil the shape of the mate tooth by undercutting the flank c f by an epitrochoidal line c g, or we may, and much better, round off the point of the tooth by an epi- cycloidal curve i h. The amount of this interference will depend on, and increases with, the angle of action, and also depends upon the number of teeth in each gear. It is greatest on a large gear or rack that runs in a small pinion, and least on a pinion running in a large gear. When the angle of action is 75° there is no interference when both gears of a pair have thirty or more teeth, or when an equal pair have twenty-one or more teeth. When onj gear has more, and the other has less than thirty teeth, the larger may need correction, but the smaller never will. The amount of the interference, the correction to be made by rounding off the point of the tooth, is very small and may generally be neglected on small pinions. It is given by the lower figures in the table, which shows that it is never more than a sixteenth of an inch on a large tooth of one diametral, or three inch circular pitch, and not over two or three hundreths of an inch on a gear of that pitch having few teeth. The table also shows by the upper figures the limit point or distance i x above the pitch line where the interference commences. The tabular numbers must be divided by the diametral pitch that maybe in use. and for any circular pitch it is sufficient to divide the tabular number by 3 and then multiply by the pitch. The table takes no notice of an inter- ference of less than a hundredth of an inch on a tooth of three inch circular pitch. When, as is usually and should always be the case, the gear being drawn belongs to the twelve to rack interchangeable set, the interference should be computed for a mate gear of twelve teeth , or by the first vertical column of the table. In this case the error will not be perceptible if the limit distance to point of first interfer- ence be always assumed to be half the addendum. When the work is upon a rough cog- wheel or mill gear, or upon a pattern for a cast gear, the only correction needed for interference, is a slight rounding olV of the points if it is a rack or very larger gear, and a mere touch on the point of a gear of fcAv teeth. 12 13-14 I 1.5-16 17-18 19-21 22-24 c 25-29 o 30-36 u 37-48 Si 3 49-72 2 73-144 145-00 .58 .01 .56 .02 .54 .02 .53 .02 .51 .02 .50 .02 .49 .03 .47 .03 .45 .03 .44 .04 .42 .05 .40 .06 .67 .01 .66 .01 .64 .01 .62 .02 .60 .02 .58 .02 .57 .02 .55 .02 .53 .02 .52 .03 .49 .04 .46 .05 .75 .01 .72 .01 .69 .01 .67 .01 .65 .02 .63 .02 .61 .02 .59 .02 .56 .03 .53 .04 .75 .01 .72 .01 .69 .01 .66 .02 .63 .02 .60 .02 .73 .01 .70 .01 .67 .01 EPICYCLOIDAL vs. INVOLUTE TEETH. A COMPARISON. The epicycloidal tooth is in much greater use and favor than the involute form, particularly for heavy work, both writers and mechanics generally preferring it, and seldom giving the preference to its rival. It is difficult to account for this favor except, as in the case of the circular pitch system, on the ground that the epicycloid was adopted in the infancy of mechanical science, and holds its place by virtue of prior possession, for the involute has certainly the advantage from every practical point of view. Space will not permit an extended discussion with the necessarily bulky demonstrations, but, if the two curves be closely and carefully examined under the same conditions within the limits of either the twelve tooth or the fifteen or higher tooth interchangeable series, with the customary adden- dum, which limitation will cover nine-tenths of the gears in actual use, it will be found that they compare as follows : I. Adjustibility. Involute teeth alone can possess the remarkable and practically invaluable property, that they are not confined to any fixed radial position with respect to each other, for, as long as one pair of teeth remains in action mitil the next pair is in position, the perfect uniformity of the action of the curve is not imj)aired. The shafts may be at the proper distance apart, or not, as happens, and they may change position by wearing, or variably as when used on rolls, or may be forced together to abolish backlash, and, in fact, the curve is won- derfully adapted to the variable demands, and will accommodate itself to errors and defects that cannot be avoided in practice. Epicycloidal teeth must be put exactly in place and kept there, and the least variation in position, from bad workmanship in mounting, or by wear or alteration of the bearings in use, will destroy the uniformity of the motion they transmit. When perfectly mounted and carefully kept in order, epicycloidal teeth are as good as any in this respect, but for most practical purposes they are decidedly inferior. This virtue of the involute is always recognized by writers, but is seldom given the position its importance demands, for it is only as a result of exj^e- rience in making and using gears, that its importance can be seen at its full value. II. Uniformity. The direct force exerted by involute teeth on each other, is exactly uniform, both in direction and in amount, and this property ensures a uniform wearing action of the teeth, a nearly uniform thrust on the shaft bearings, and a steadiness and smoothness of action that cannot be claimed for epicycloidal teeth under any circumstances. The direct pressure acting between epicycloidal teeth is variable in amount and very variable in direction, and consequently the friction and wearing action between the teeth, as well as the thrust on the bearings, is variable between wide limits. III. Friction. The measure, for purposes of comparison, of the loss of power by friction, is the product of the direct pressure between the teeth, multiplied by their rate of sliding motion on each other. This measure is always in favor of the involute by a decided advantage, although the advantage is usually claimed for the epicycloid, both as to maximum values and average values, and as this is an important point, it should have great weight in deciding between the two forms of teeth, for the element of friction is of chief importance in determining the life of a gear in continual and heavy service. The epicycloid is mostly in use for heavy gearing from a mistaken view of this point, it being generally taught that its friction is the least. IV. Thrust ON Bearings. Here the advantage is with the epicycloidal tooth, but not by a large amount, and not in a matter of first consequence. The thrust on the bearings due to the action of the teeth on each other is but a fraction of the whole thrust due to the power being carried, and as the average thrust of the teeth is but little in favor of the epicycloid, and as the maximum thrust is always from that form of tooth, the two forms may be said to be well balanced in this respect. Moreover, the thrust of the involute is but slightly variable, while that of the epicycloid varies from large values at the points of first and final action to nothing at all at the line of centers, and must give rise to a rattling and uneven action. V. Strength. The weakest part of a tooth is at its root, and as the involute tooth spreads more than the epicycloidal tooth, it is stronger at that point and has a considerable advantage. YI. Appearance. This is a small point and a matter of opinion, but is worth mention. The involute is a simple and graceful single curve, while the epicycloid is a double and not mechanically a neat curve, and, as gener- ally drawn, has a decided bulge or even a plain corner where the two halves join at the pitch line. In General. As the involute has the advantage of the epicycloid, in Ijine actual cases out of ten, with respect to adjustibility in position, in uniformity of wear and action, in loss of power and change of shape by friction, in strength, and in appeaiance, and is but a shade, if any, inferior with regard to the thrust on the bearings, it may be, and should be accorded first place for any and every practical purpose. The writer can imagine no possible case, unless it be in connection with a pinion of very few teeth, where the epicycloid would have either a theoretical or a practical advan- tage over the involute. ODONTOGRAPH TABLE, INVOLUTE TEETH. Corrected for Interference, Interchangeable Set. DIVIDE BY THE MULTIPLY BY THE DIAMETRAL CIRCULAR TEETH, PITCH. PITCH. Face Flank Face Flank Radius. Radius. Radius. Radius. 12 2.70 .83 .86 .27 13 2.87 .93 .91 .30 14 3.00 1.02 .95 .33 15 3.15 1.12 1.00 .36 16 3.29 1.22 1.05 .40 17 3.45 1.31 1.09 .43 18 3.59 1.41 1.14 .46 19 3.71 1.53 1.18 .50 20 3.86 1.62 1.22 .53 21 4.00 1.73 1.27 .57 22 4.14 1.83 1.32 .60 23 4.27 1.94 1.36 .63 25 4.56 2.15 1.45 .70 28 4.82 2.37 1.54 .77 31 5.23 2.69 1.67 .88 34 5 77 3.13 1.84 1.00 38 6.30 3.58 2.01 1.16 44 7.08 4.27 2.26 1.38 52 8.13 5.20 2.59 1.70 64 9.68 6.64 3.09 2.18 83 12.11 8.93 3.87 2.90 115 16.18 12.80 5.16 4.15 200 25.86 22.30 8.26 7.30 For intermediate teeth use proportionally intermediate values when great accuracy is desired, but for drafting purposes use the nearest value, thus : — 35 is at one-quarter of the distance from 34 to 38, and the proper values for accurate work are : face radius, 5.90 inches, and flank radius 3.24 inches. The table is not carried beyond 200 teeth, as the higher numbers are rarely used and the radii are then very great. For drafting purposes use values for 200 teeth for all higher numbers. The base distance, the distance from pitch line to base line, is always one- sixtieth of the pitch diameter. SPECIAL PROCESS FOR RACK TEETH. See the cut on the opposite page. The flank of the tooth and one-half of the face is a straight line at an angle of 75 degrees, five-sixths of a right angle, with the pitch line. Draw the outer half of the face of the tooth, one-quarter of its whole length, as a circular arc from a center on the pitch line and with a radius of 2.10 inches divided bv the diametral pitch. .67 inches multiplied by the circular pitch. The point must be rounded over in this way to avoid interference, if the- rack is to mesh with any pinion having less than 28 teeth. A PRACTICAL EXAMPLE. INVOLUTE TEETH. INTERCHANGEABLE SET. Example. — A rack, and a pinion of twelve teeth, of two diametral pitch. Pinion. — From the tables we have, after dividing by 2, the face radius 1.35 inches, flank radius .42 inches, and clearance .06 inches, The pitch diameter is 6 inches, and the addendum is .5 inches. The base distance, one-sixtieth of the pitch diameter, is .10 inches. Draw the pitch line and divide it for the pitch points, allowing for backlash if required. Lay off the addendum and the clearance, and draw the adden- dum line, root line, and clearance line. Draw the base line .10 inches inside the pitch line. With the face radius, 1.35 inches, and from centers d on the base line, draw all the face curves from addendum line to pitch line. With the flank radius, .42 inches, and from centers h on the base line, draw all the flanks from the pitch line to the base line. The flanks inside the base line are stra'ight radial lines. For fifty or more teeth draw the flank curve from pitch line to root line. Rack. — Draw by the special rule, the radius for the point being 1.05 inches. Note. — The dotted lines on the pinion teeth show the work of the common rule for involute teeth, as explained on page 18 and given by most of the " gear charts " and works on practical mechanism. The same rule draws the rack tooth with a point that is not rounded. The " old rule " is as worthless as it is simple. /y ^^ r o v- \P BEVEL GEARS. In layinpj out the teeth of a bevel gear but one new point needs to be con- sidered. The working pitch diameter a b c is not to be used, but the teeth are to be drawn on the conical pitch diameter ad c, developed or rolled out as in fig. 25. The conical diameter a d c may be found from a drawing, or if the gears are of some common proportion, from the following table by multiplying the true pitch diameters by the tabular numbers given for that proportion, TABLE OF CONICAL PITCH DIAMETERS OF BEVEL GEARS. Proportion. Larger Gear. Smaller Gear. 1 tol 1.41 1.41 2 " 1 2.24 1.12 3 " 2 1.80 1.20 3 " 1 3.16 1.05 4 " 3 1.67 1.25 4 " 1 4.12 1.03 5 " 4 1.60 1.28 5 " 3 1.94 1.17 5 " 2 2.69 1.08 5 « 1 5.10 1.02 6 " 5 1.56 1.30 6 " 1 6.08 1.01 7 " 1 7.07 1.01 8 " 1 8.06 1.01 9 " 1 9.06 1.01 10 '' 1 10.05 1.01 Examples.— A miter gear, proportion 1 to 1, of 4 pitch, 6'' diameter, and 24 teeth, has a conical diameter of 6" x 1.41 = 8.46'', and there are 24 x 1.41 = 33.8 teeth on the full circle of the developed cone. A pair of bevel gears of 3 to 1 proportion, 48" and 16'' diameters, 36 and 12 teeth, have conical diameters 48" x 3.16 = 151.68", and 16" x 1.05= 16.80", and there are 36 x 3.16 = 113.76, and 12 x 1.05 = 12.60 teeth on the full cir- cles of the developed cones. ^^^^ wm "'^^ij^ 4. ._'^''^V INTERNAL GEARS. The iuterD'i gear, sometimes called the "annular" gear, is drawn by the rules for spur gears, the teeth of a spur gear being the spaces between the teeth of an internal gear of the same pitch diameter, with the backlash and clearance reversed in position. Involute teeth should end at the base line, the radial part of the flank being omitted, or well rounded over if it is desirable to preserve the appear- ance of the full tooth. Internal teeth will interfere, even if properly drawn, unless the gear is considerably larger than the pinion running in it. If drawn for the common twelve to rack interchangeable set, there should be at least twelve more teeth in the gear than in the pinion, and if the difference is less, the teeth must be " doctored " or rounded over until they will pass, and there must be a difference of two teeth in any case. Involute teeth have a decided advantage over epicycloidal teeth for inter- nal gearing, their action being much more direct, with less sliding Jitid friction. STRENOTH ANDIHOESE-POWEE OF OAST GEAES. There are about as many different rules for this purpose, and contradictory re- sults, as there are writers upon the subject. I have preferred not to discuss the theory, but to adopt without question the method given by Thomas Box in his Prac- tical Treatise on Mill Gearing, because that engineer has most carefully considered the practical points in view, and because his formulae agree almost exactly with a great many cases in actual practice. STRENGTH OF A TOOTH. —For worm gears, crane gears, and slow-moving gears in general, we have to consider only the dead weight that the tooth can lift with safety. If we allow the iron to be subjected to but one tenth of its breaking strain, we can use the formula: — W = 350 c f , in which W is the dead weight to be lifted, c is the circular pitch, and f the face, both in inches. For the wooden cogs of mortise wheels, use 120 instead of 350 as a factor in the fonnula. When the pinion is large enough to insure that two teeth shall always be in fair contact, the load, as found by this rule, may be doubled. Example. — A cast-iron gear of 3" circular pitch and 6" face will lift W = 350X3X6 = 6300 lbs. HORSE-POWER OF A GEAR. — For very low speeds we can use the formula, HP for low speed = .0037 d n c f , in which d is the pitch diameter, c the circular pitch, and f the face, all in inches, and n is the number of revolutions per minute. Example. — The horse-power of a gear of three feet diameter, three inch pitch, and ten inch face, at eight revolutions per minute, is, HP = .0037 X 36 X 8 X 3 X 10= 32. For ordinary or high speeds, where impact has to be considered, it is found that the above formula gives too high results, and we must use the formula, HP at ordinary speeds = .012 c^ f -\/du. Example. - A gear of three feet diameter, three inch pitch and ten inch face, at one hundred revolutions per minute, will carry but HP = .012 X 9 X 10 X v'lOO X 36 = 65 horse-power, instead of the 400 horse-power found by the rule for low speeds. At ordinary or high speeds a wooden cog, on account of its elasticity, will carry as much as or more power than a cast-iron tooth, and we can use .014 instead of .012 in the formula. When in doubt as to whether a given speed is to be considered high or low, com- pute the horse-power by both formulae, and use the smallest result. For bevel gears the same rules will apply, if we use the pitch diameter and the pitch at the center of the face. Some rules in use take no account of the face of the gear, but assume that the tooth should be able to bear the whole strain upon one corner. A tooth that does not bear substantially along its whole face, at several points at least, is a very poor piece of work, and it would be better to straighten the tooth than to force the rule to follow it. HORSE POWER OF CUT GEARS. The rules giveu above for the horse power of gears apply only to gears with rough cast teeth ; and in applying them we must consider the speed of the gear as well as its real strength. One of the chief sources of weakness in a cast gear, is that the continual pounding of the teeth on each other crystalizes the metal so that its strength is gone long before it is worn out. There are no recorded tests on the horse power of cut gears, but it is gen- erally agreed among those not personallj^ interested in the sale of cast gear- ing, that a cut gear is much more durable, and that it will carry more power than a cast gear, with the same factor of safety. In the absence of experimental data, we can only proceed by judgment and inference. It is well settled that the continual pounding of cast gearing is a source of weakness that must be allowed for, and it may be assumed that that source is avoided in the use of cut gears having a smooth and even action. Until practical tests have been made we can consider that the rule that applies to cast gears for slow speeds where impact need not be considered, can safely be applied at higher speeds to cut gears where there is no impact to be allowed for ; and we have the formula : — Horse power of cut gears at ordinary speeds = .0037 dncf. Applying this formula to the case of a gear of 36 inches diameter and 3 inch circular pitch, at 100 revolutions per minute, it is found that the cut gear will safely carry six times the power that can be trusted to the cast gear. But it must be admitted that all that is known concerning the real horse power of a cut gear is a matter of inference, and it is to be hoped that the growing use of cut gearing for conveying heavy powers will furnish data of a more practical and trustworthy nature. Until such data is at hand it may safely be assumed that a cut gear has from two to three times the carrying power of a rough cast gear of the same size. CONFUSION OF RULES. The disagreement of standard authorities and the thorough confusion of rules on this subject, is well shown in an interesting paper by J. H. Cooper, in the Journal of the Franklin Institute for July, 1879, in which that engineer has industriously collected twenty-four formulas from Tredgold, Buchanan, Fairbairn, Box, Molesworth, Haswell, Nystrom and others, and applied them to the practical case of a gear of 60 inches diameter, 7k inches face, and 3 inches pitch, at 60 revolutions per minute. Cooper found twenty-two differ- ent results for this one example, as follows: — 46.31, 47.06,50.27, 53.18,56.09, 56.55, 63.62, 66.17, 66.27, 67.96, 68.56, 73.49, 80.78, 84.37, 86.75, 86.80, 86.96, 138.23, 147.27, 163.00, 294.53, and 295.59. Here is variety to suit all tastes, and if a gear is not strong enough for a given purpose according to Fair- bairn, it will certainly fill the bill according to Haswell. Diligent enquiry by myself among the cast gear makers of the United States gave the same result as to variety and confusion. I could get little but opinions that were not founded on experiment, and the opinions were of the most in- definite and unsatisfactory character. All cast gear makers are agreed that a cast gear is more durable than a cut gear, and all cut gear makers are equally certain that a cut gear is more du- rable than a cast gear. The stock argument of the makers of cast gearing is that the one-hundredth of an inch thickness of hard scale on a cast tooth makes it more durable than a cut gear from which the scale has been removed. But, from that point of view, they find it very hard to explain why a mortise gear, with soft hickory cogs, is quite as durable as a cast gear with hard teeth. CHART AND TABLES FOR BEVEL GEARS. A NEW, SIMPLE, AND ACCUEATE 3IETH0D EOR FINDING THE ANGLES AND DIAMETERS OF ANY PAIR OF BEVEL GEARS BY SIMPLE CALCULATION, AND WITHOUT DRAFTING INSTRU- MENTS OR SPECIAL TOOLS. J^ot one machinist in a dozen will admit that he does not knosv how to properly shape a bevel gear blank, but when put to the test, not one in a score can do it well without an amoimt of fussing with drafting instru- ments, and a deal of studying and figuring that looks ridiculous to one who has studied the subject and knows how simple it really is when it is once thoroughly understood. The average bevel gear blank can be relied upon to be wrong in i ts face angle, or its outside diameter, or both, even when it has been shaped by a competent and intelligent general workman, and the simple explanation is that the only reliance is generally a hurried and poorly made drawing from which the angles and diameters must be found by measurement, and used with many chances of error. This method proceeds by simple calculation, avoiding the use of drafting instruments, and it will be found to be not only much more accurate, but at tlie same time much easier and quicker than any other method. The work- man wlio can remember the numbers 1.41 and 81 and the angle 45° needs no further assistance on miter gears, and on other proportions needs the table only to supply equally simple data, while two to live minutes is suffi- cient time for any set of calculations after the method has been learned. This matter is of more importance than is generally supposed, for bevel gears unlike spur gears must be exactly correct in diameters and angles, or no amount of perfection in the cutter or care in the cutting will prevent a botch. To be learned this Chart must be studied. If it is not worth while to give it two or three hours of careful attention it is not worth while to keep it at all. It is simple and easy to learn but it cannot be taken in at a glance, or comprehended in ten minutes. EXPLANATION OF THE METHOD. The measures that must be right angles at with A, and the common involute system will bo formed. Therefore the involute tooth is a special form, the infinite form, of the segmental tooth. The segment has exactly the valuable prop- erties of the involute at the pitch line, and approximately away from it, the approxima- tion being closer as the radius A\b longer. The involute tooth is often, but not prop- erly, regarded as the special case of the cycloidal tooth for a rolling circle of infinite diameter. Kegarded simply as a curve, the involute is an infinite cycloid, but regarded as a gear tooth curve it is not, for, as shown by Fig. 37, infinite cycloids have a mathe- matical but not a practicable contact, and cannot bear properly, unless the conditions of the movement are so far strained that one is IHE NORMAL THEORY OF THE GEAR TOOTH CURVE. reversed on its pitch line, and then the pitch lines are separated. FoBM OF THE SEGMENT Aii TooTH. — The face of any segmental tooth, on any circle with center at C, will be a lobe, d' f , where e' d' =e d, and ^ f —0 ef. It is always at right angles at with a. The flank curve D' F' is at right angles at 6> with A^ has the same height D' E\ and the same base E F' at the rack face. If a, O and A are in the same line, the face and flank will join at 0, and be a single curve. no cusp will be apparent, but the slightest increase of the proportion will separate the points. In the most convenient system, where A0E=1^° 28' 40", and sin. AOE=\, the cusps will appear whenever the segment and pinion radii are in the proportion ^=:^/.i= 1.687. As the proportion OA 00 increases, the sec- ond branch (^ R will increase so that the curve will take the form OQ' B' D", and when OA Fig. 25 Cusps of Segmental Flanks Cusps of Segmental Flank. — When the ra- dius J., Fig. 25, is small, compared with the radius OC, the pinion flank takes the form shown by Fig. 24; but as the pro- OA portion j^n increases , a value will be reached , OA when j^ = Y sin. AOE, at which a double cusp, Q' i?, will form. At exactly that point the two points Q' and R will coincide, and is infinite the second branch, Q' Z, then an involute, is infinite. The Segmental Delineatoe, — The seg- mental curve can be formed by the * * conjuga- tor" previously described, and shown by Fig. 13, and it can be drawn by the special delineator, shown by Fig 26. A thin wooden wheel, C, turns on a pin at its center, and a rack, B, rolls on it, being held to it by a strip, aOc, of thin brass or THE NORMAL THEORY OF THE GEAR TOOTH CURVE. strong paper attached to both. It is kept in position by a guide, H. A fixed bar O, projecting over the wheel, carries a pin 0, placed exactly at the point of contact of wheel and rack. A rule E turns about a pin A in the rack B, and carries a pointed tracing pin or pencil point at P, The pins A and P are in line, and all three are always at the same distance from the straight edge of the rule E. The pin P will pass undei and come in line with the pin 0. As the rack is rolled on the wheel, the rule will turn about the pin A and slide on the pin 0. The point P will trace segment of The action is practicable until the point of contact arrives at the first cusp Q' of Fig. 27 ; but beyond that, when it is on the second branch Q i?, the flank curve is inside the rack face, and the action is impracticable. There will also be an actual interference with the first branch. When the point of contact is on the second branch, the rack face will cross the first branch at J, and therefore the addendum must terminate the rack tooth at the point Q that conjugates with the cusp Q'. The difference between theoretical and practical contact is illustrated by the two ma- a circle P 8 with respect to the rack, but on the wheel will trace out the segmental flank 0' Q R D'. If the pin A is carried by an arm on the other side of the rack pitch line, the face of the pinion tooth will be drawn, but, as the form of the face is very simple, the utility of the instrument is confined to the flank curve. Inteefeeence of Segmental Teeth. — The action of the segmental rack tooth on a flank that is conjugate to it, when the proportion is such that a cusp is formed, is always mathe- matically perfect, but not always practicable or capable of mechanical use. Fig, 26 Segmental Delitieator chines, the conjugator of Fig. 13 and the de- lineator of Fig. 26. A full rack tooth on the conjugator will form the first branch correct- ly, but when the cusp is reached will return on it and cut it away, while the delineator, having but one acting point, will follow the theory and trace out all three branches. Least Numbee of Teeth. — Therefore, if the addendum is fixed, and it usually is, interfer- ence will generally set a limit to the diameter of the smallest pinion with which a rack tooth having the given addendum will work, with- out bearing on the second branch of the pinion flank. The diametral pitch being unity, a the ad- THE NORMAL THEORY OF THE GEAR TOOTH CURVE. dendum EX, Fig. 25, 5, the segment radius OA, and F, the angle of obliquity AOE, the smallest possible number of teeth is 2a sin V (l+sinv)' (1) If 5= 00 for the common involute system , a=l, and sin. V=.25, this formula gives t= Interference of Segmental Teeth 32. Therefore, the common involute system cannot have an addendum of unity on an in- terchangeable set having gears with less than thirty -two teeth. When the set includes 12 teeth, as is usual, the addendum must be shortened, or the points must be rounded over, as at QQ^fiT, Fig. 27. If we have given the angle of obliquity, the addendum, and the number of teeth in the smallest pinion, the largest possible seg- ment radius that can be used is h=- a V 2a sin. Y —sin. V (2) This, for the common case, where «=1, i= 12, and sin. F— .25, gives Z> = 10,34: as the ra- dius for the usual twelve-tooth system. A ra- dius of 13.91 will allow 15 teeth, 16.95 will allow 16.95 teeth, 23.58 will allow 20 teeth, and a short radius of 8.44 will admit a 10- tooth pinion. The Natubal Set. — There is one particular proportion of segment radius to pinion ra- dius, that might be considered the natural limit to the interchangeable system, and that is the proportion at which the cusp first ap- pears. If that, or a smaller proportion is chosen, there is no limit set to the addendum, and no interference is possible, for the second branch of the curve never appears. For that point we have the relation b=^ t sin. V, (3) so that, by choosing some value of t as the lower limit, we can find h for the whole set. If sin. F=.25, we find 5=f|«, giving &=8xV for a ten-tooth set, 5=10^ for a twelve-tooth set, 5=12|| for a fifteen-tooth set, 6=27 for a thirty-two-tooth set, and so on. If we use the plan previously explained, and calculate by formula (2), we can get a greater value for it, but in that case the ad- dendum is limited to its chosen value. As the addendum is always limited in prac- tice, almost always being unity, formula (2) appears to be better adapted to practical purposes than formula (3). CoBRECTED INVOLUTE TooTH. — We havc secu that the true involute tooth, when sin. V— .25, cannot be used for an interchangeable set Stunted Involute that includes gears with less than thirty -two teeth, if the addendum is unity. It is, however, customary to use the full addendum on a set that includes twelve teeth with the result that it must be corrected (?) for interference by rounding over the corners as in Fig. 27. THE NORMAL THEORY OF THE GEAR TOOTH CURVE. Formnla (1) will apply to the involute if b=cc . In that case -r=o, and (4) If t=12, and sin. F=.25, we have a=f, so that the common involute, Fig. 28, is limited to the addendum «^=f , and the additional bc=^ of the full addendum must be cut off or got out of the way by rounding off as much of it as would interfere with a twelve-tooth pinion. This additional five- eighths is not inter- changeable, is not a tooth curve, and is kept on merely to give the appearance of a whole tooth. What usually appears to be a full ad- dendum, is really stunted to but little more than its third part. pitch line <* Action of Corrected Involute Tooth The only true correction, the only device that will allow of a full addendum of unity, retain the true involute for any part of it, and permit a rack to run in a pinion of less than thirty-two teeth, would be to correct the rack tooth by rounding over the point as in Fig. 29, to give the flank the same correction, so that the condition of interchangeability is satisfied, and then to form a conjugate set from the corrected rack tooth. The result would be a mixed action : true involute, near the pitch line, and epicycloidal or otherwise away from it. This plan would have the serious defect that the corrected part bd must be a very defective odontoid, with a jerky and very nearly im- practicable action ; for, to obtain the neces- sary correction between b and d the curve must turn so quickly that its normal intersec- tions vrith the pitch line must be crowded within a narrow limit mn. Therefore, it would not appear to be advisable to correct the involute at all, for low-numbered pinions, but to discard it altogether, or to keep up appearances, as at present, by a merely ornamental and deceptive extension to nearly three times its effective length. If it is discarded, its valuable peculiarities will be lost, and therefore its substitute should be the curve that is nearest like it, and most nearly has its properties. Evidently, the nearest possible approach to the involute, is the segment that has the same angle of obliquity, and the longest radius that will admit the required addendum on the required smallest pinion, as found by formula (2). Figs. 29 and 30 serve to compare the action of the corrected involute with the segmental tooth. The action of the segment. Fig. 30, is exactly the same as that of the involute at pitch line Action of Segmental Tooth a, and its rapidity gradually increases to the finish at n. The corrected involute action, Fig. 29, is uniform from a to m, and finishes with a sudden jerk from mio n. THE NORMAL THEORY OF THE GEAR TOOTH CURVE. If the correction should be curved enough to cause any of the normals to meet on or outside of the pitch line, the action would be wholly impracticable. The Ctcloldal System. — The cycloidal system is generally, but not properly, called the * ' epicycloidal " system. It is no more epicycloidal than it is hypocy- cloidal, for the faces are of the one form, and the flanks of the other. It is simpler and easier, as well as more correct, to apply the name cycloidal to both face and flank, and to the whole system, as is sometimes done. As before stated, the cycloidal system can be more easily developed and studied by the "rolled curve" theory than by the normal angles at and F, with a base, OF, equal in length to the circumference of the roller. As with all rolled curves, the line PE, from the tracing point tc the tangent point, is al- Fig. 31 The Cycloid ways a normal to the curve, and therefore, to draw a normal to any given point P, strike a circle through the point having a diameter Fig. 32 Cycloidal Teeth theory, because its roller, the circle, is the simplest of all curves. But, in this place, the former theory will not be used further than to define the nature of the cycloid, which is the generating odontoid that forms the system. The Cycloid. — If a circle A. Fig. 31. is? rolled on the straight line OF, a fixed point P, in it will trace out a transcendental curve OPDF, called the cycloid. It is a lobe, having a height, DE, equal to the diameter of the roller, and is at righ'. equal to the height DE, and tangent to the base line, and draw the normal PK to the point of tangency. As the arrangement of the normals is con- secutive, the curve is an odontoid, and all curves formed from it will be similar odon- toids that will work interchangeably with it. The simple process for drawing the normal makes it easy to form the conjugate face or flank belonging to any pitch circle. The flank cycloid Odf, Fig. 32, forms a face on the pinion, which is always a lobe Od' f bX THE NORMAL THEORY OF THE GEAR TOOTH CURVE. right angles at with the pitch line, and meeting it again at/', where Oe f' = Oef. The face cycloid ODF forms a flank, OD' F\ on the pinion, that is at right angles with the pitch line at 0, meets it again at F\ where OD' F=ODF, and which takes vari- ous forms, according to the size of the pin- ion compared with that of the cycloid. When the radius 0(7, of the pinion, is greater than the height ED of the cycloid, the flank will be a concave lobe, OD' F' . When the radius OCis equal to the height ED, as in Fig. 33, the flank will be a straight diameter OF' . When the radius is less than the height, as in Fig. 34, the flank will be a convex lobe. As an undercurved flank, as in Fig. 34, is weak, it is customary to so limit the radius of the pinion that it shall never be less than the height of the originating cycloid. As the proportion of EDio OC still further increases, the flank is still more undercurved, until when 0C= ^ ED, we have the base, OF, equal to the circumference of the pinion ; and the flank is concentrated to a single point at 0. The wearing action is also concentrated at the single point, and such a tooth, al- though practicable, is quite useless. If the height of the cycloid is greater than the diameter of the pinion, Fig. 35, the flank is a lobe, entirely external to the pitch line; and although the contact is still mathematically perfect, it is no longer practicable, for it is on the inside of the cycloid, as shown at P\ Cycloidal Involute If the proportion is carried to its extreme, the height being infinite as compared with the diameter of the pinion, as in Fig. 36, the THE NORMAL THEORY OF THE GEAR TOOTH CURVE. cycloid becomes the straight line OD, and the pinion flank is the involute OD' . From this it is plain that the involute, in the form of an infinite cycloidal odontoid, is not a practi- cable gear tooth curve, the action between two gears being, as in Fig. 37, always on the straight line AOB at the crossing of the two tooth curves. Infinite Cycloidal Teeth CTOLOiDAii Lines of Contact. — The pri- mary lines of contact, in the case of the cycloid, are circles. Fig. 38, of the same diameters as the heights of the originating cycloids. The secondary lines of contact are also circles. The diameter is equal to the pitch diameter plus or minus the height of the cycloid. Inteenal Intereebenge. — With cycloidal teeth we cannot have a partial interference, as with segmental teeth, which can be remedied by truncation of the teeth, for the exterior secondary of the pinion is either entirely inside the interior secondary of the gear or entirely outside of it, except in ence between the pitch diameters is greater than the sum of the heights of the originat- ing cycloids, there can be no interference, but if it is less there will be a continual inter- ference that can be remedied only by the en- tire removal of the face of one of the teeth. The condition of non-interference can be conveniently expressed by the rule that the difference between the numbers of teeth on the gears must not be less than the half sum of the numbers of teeth on the base gears. This, for the common interchangeable sys- tem, requires that there should be a differ- ence between the gears at least as large as the base gear. For example, in the fifteen tooth set there must be at least fifteen more teeth in the gear than in the pinion.* Double Internal Contact. — When the condition of non-interference is exactly satis- fied, there is a case of double contact, for then the two secondaries coincide. In a case of double contact of interchangeable teeth, the coinciding secondaries must exactly bisect the chord c d of Fig. 20, for the primaries are equal, and their chord 6 / is exactly bisected in that case. As the circle is the only curve that will bisect all the chords c d, it follows that the cycloidal system is the only one that can have double contact, and at the same time be interchangeable. Pbactical Construction. — The practical application of the conjugating process, in the case of the cycloid, presents the difficulty that the originating rack tooth must be an exact cycloid, which condition can be mei only by special mechanism. The originating segmental tooth has an outline that is formed of arcs of circles, and that of the involute is composed of straight lines, and both can be easily shaped. But the difficulty in the practical application of the cycloidal system is not by any means the greatest objection to it in comparison with its simpler and superior rival. the case of entire coincidence. If the differ * The discovery of the law of internal interference, as far as it relates to cycloidal teeth, is generally credited to Prof. C. W. MacCord ; but, in claiming that discovery, the professor could not have been aware of its previous publication, by A. K. Mansfield, in the Journal of the Franklin Institute for Januaiy, 1877. THE NORMAL THEORY OF THE GEAR TOOTH CURVE. Lines of Contact The comparative EFFICIENCY of the TEETH of GEARS. [Reprinted from the Journal of the Franklin Institute, May, 1887.] The effect of friction between the teeth of gears is not well understood, and the popular impression, even among educated engineers, concerning the comparative efficiency of the two forms of teeth in common use — the involute and the cycloidal — is that the latter is much the most economical, and, therefore, much better adapted for use for the transmission of heavy power. This impression is entirely wrong, the reverse of the provable facts, and it is based not entirely on fancy but partly on the teaching of authorities that are undoubtedly competent. It is with no small feeling of timidity, that I venture to contra- dict the declared and apparently proved opinions of such high authorities as Reuleaux, Herrmann and others, and I would not dare to assert a contrary view if I did not feel able to prove it, by evidence that will bear the closest examination. I will give the demonstration in great detail, so that it can be followed by any one who is familiar with the common processes of analysis. By the work done by a gear wheel, I mean the work done by the friction of sliding between the teeth. I shall leave out the small rolling friction between the teeth, and I ' shall not consider the friction of the shaft bearings. The work lost by the rubbing of two surfaces on each other is the product of the normal force acting between the two surfaces, by the distance through which the resistance is overcome, and by the coefficient of friction for the material in use. To determine the work done by a pair of gear teeth, we must determine these three factors or their product, and this may be done in two different ways : by a graphical process, and by an analytical method. The two processes are entirely independent of each other beyond the given premises, and their agreement upon a common result is a substantial proof of the accuracy of both. Graphical Process. — In Fig, J, the two tooth curves have rubbed upon each other, while the point of contact between them has moved from C to ^ on the line of action, AD, and they have done work that is the product of the coefficient of friction,/, by the difference, zy, of the lengths of the curves that have passed the point of contact, and, for graphical purposes, of the average force, /S", that has acted between the two teeth. If we make a drawing, showing the two teeth in several posi- tions, preferably at equal intervals of their action, we can deter- mine the work done within the limits of each interval by multi- plying together the factors as found by measurement. The total work done between any two points is the sum of these products for all the intervals between the points. In Fig. 2, this process is applied to an exaggerated example of a pair of cycloidal teeth. The gears, with radii h and hy have ten Fig. I. Analytical Process. and twenty teeth, the tangential force, P, between the two gears is assumed to be constant and unity, and the coefficient of friction is assumed to be one-tenth. The describing circle, with radius 0M= Ty has three teeth, so that a gear of six teeth would have radial flanks and be the base or smallest gear of the interchange- able set to which the two gears belong. The pitch and describing circles are divided into equal inter- vals, Oa, ah, he, etc., of one-twelfth of the whole tooth arc, or cir- cular pitch, Olj commencing at the line of centres, and the v/ork done over each of these small intervals is to be determined. Make a templet of an epicycloid on the gear h, and of a hypo- cycloid within the gear h, and draw curves from each of the divi- sions of the pitch lines. Each pair of curves should meet on the corresponding division of the describing circle. Measure the dif- ferences between the lengths of these curves (see column 2 of the table), and by subtracting each total difference from the next larger, find the partial length of curve passed over during each interval, as tabulated at column 3. Draw a line at an arbitrary distance, representing unity, from the line of centres and parallel with it, and draw lines, OSaj OSb, OSc, etc., through the centres of the intervals. The length of each line (column 4) can, with small error, be assumed to be the average normal force for its interval. These normal forces can be very Fig. 3. Graphical Process. easily computed, for each one is the reciprocal of the cosine of the angle POS. The angle for the first normal is 2^°, and there are 5° between each of the following normals: Multiplying together the normal for each interval, the partial curve for that interval, and the coefificient of friction, we obtain the loss for each interval as tabulated at column 5. By summa- tion we obtain the total loss to and including each interval, as tabulated at column 6. For the involute tooth, we have a constant normal force, ^=r 1*15, the total work done, column 10, up to any interval is the product of that force by the total curve, column 9, for that interval. The figure is so similar to Fig. 2^ that it need not be given here. The graphical process will determine the general result, and show that while the two curves are substantially equal in efficiency, the advantage is a very little in favor of the involute. If we wish a precise comparison between these two curves, no graphical process can be used, and we must resort to analysis. Analytical Process, — In Ftg. /, the two tooth curves are odon- toids of any possible form, and they will secure a uniform velocity ratio between the pitch lines. They slide on each other, the point of contact moving along the line of action, AD. At any time they are at a distance A = h from the pitch point, 0, and are pressed together with a normal force, St which is equal to the constant tangential force, P, divided by the cosine of the angle of obliquity, P A= V, and this normal force is always in the direction of the pitch point 0. While the normal, O A, turns through an elementary angle, the arc of which is d V, the two curves will rub on each other over an elementary distance, A B = A ' d V=b ' d V, and they will do the elementary work dW = fP'AB'S'==f-^'b'd V. cos V At the same time the wheel h will turn through the elementary angle, the arc of which is dx = -^dV in which the positive sign is for external, and the negative sign is for internal contact. Therefore, we have the total work done by friction, while the wheel h is turning through an angle, the arc of which is x. TTT f p h ± h rb dx k •^^cos V o and this cannot be carried further until we know the form of tooth curve to be used, and can determine b and cos Fin terms of x. First take the involute tooth. The distance ^ = 6 is equal to hx . cos V, and we have the total work done z I = fF.t^hfxdx, o which integrates to or, if we use the arc on the pitch line, w = hx,we have 1 = ^-_ . — — — w^ 2 kh for the value of the work done by the friction of involute teeth while moving from the pitch point over any arc, v), on the pitch circle. It is a singular fact that this loss of power is the same for all values of the angle of obliquity. All involute systems are equal in efficiency, without regard to the angle of obliquity. Then take the cycloidal tooth. We have b = 2 r . sin — x, and cos V = cos — x, giving 2r ' 2r ' the total work. X E=fP.^i^2rCian ^dx, •^ kh ^^ 2r ' o which integrates to js; = — the value of the total work of a pair of cycloidal teeth. E = — /P . ^r/ 4 r^ nat log cos ^, •^ kh ^ 2r To compare the cycloidal with the involute tooth for the same arc of action from the pitch point, divide E by /. Sr^ nat log cos — - E 2 V' T w" As this is unity for w = and greater than unity for any finite value of w, it follows that the efficiency of the involute is mathematically superior to that of the cycloidal curve, in all cases and under all circunistances, without regard either to the angle of obliquity of the involute, the size of the describing circle of the cycloidal curve, or the arc of action, and provided only that the comparison is made over the same arc of action. (See column 13 of the table.) In both of these formulae it is seen that h and h can exchange places without affecting the result for external contact, and there- fore the work done is the same, for the same arc of action, on both sides of the line of centres, the tangential force being constant. For a comparison between external and internal gears, we have Ji. ^ lor E Ext ^ h ^h B Im-EInt. k — h so that the internal gear is much the most economical, particularly when the two gears are nearly of the same size. When k = 2h WQ have -g- = 3. That is, if the internal gear is twice the size of its pinion, the work lost is but one-third of that lost when both gears are external. Small improvement can be made by putting a small pinion in- side, rather than outside of a large gear, as is often done at great expense on boring mills and large face plate lathes. A six-inch pinion and a six-foot gear will give -^ = 1-18 an advantage of no Jo great value. It is seen from the above that the work being done increases very rapidly with the arc of action ; with the square of that arc in the case of the involute, and in a still greater proportion for cycloidal teeth, and hence that arc should always be made as small as possible. Strength should be secured by a wide face rather than by a large tooth, for the face of the gear has no influence on its effi- ciency. The two formulae for E and Jean be very easily applied to any particular example, and the results obtained much more quickly, as well as more accurately than by the graphical method. For application to the given example, where h = 10, k = 6y y = 1, and P = 1, we have E = 6-21 70 [0 — log cos (5 n)^] I = -01028 n^ in which n is the number of any interval, C, is the characteristic with the sign changed, and hg cos contains only the mantissa of the common logarithmic cosine of 5 n^. It is seen from the tabulated value of E and J obtained by com- putation, columns 7 and 11, that the graphical and analytical pro- cesses agree very closely, the errors being shown by columns 8 and 12. As before stated, this agreement is a strong indication of the accuracy of both. Prof. Reuleaux* finds that the two curves are exactly equal when compared over the same arc of action, and Prof. Hermannf finds the same result by a different process. In both cases the result was arrived at by making an approximation, for reasons not given but probably to simplify the work. If the actual determination of the work done is the end in view, the approximations can be allowed, as the result is then close enough for all practical purposes. But, if the object is a close com- parison between the two curves, the slightest difference must be accounted for, and neither Reuleaux's nor Herrmann's formulae will answer the purpose. Herrmann remarks, *< It is evident, moreover, that the friction of involute teeth will be somewhat greater than that of cycloidal teeth, the angle y being smaller for the former than for the latter." This may be " evident," but it is not provable, and the state- ment that the angle y, which is the complement of the angle of obliquity, is smaller for the involute, is not correct. Up to the half tooth point it is so, but beyond that point the reverse is true. At the half tooth point the two forms always have the same angle of obliquity if they belong to interchangeable sets which have the same base gear. Further, it does not follow that the work of friction is the greater when the angle of obliquity is the greater, for the work of friction depends on two variable factors, the normal pressure, which indeed increases with that angle, and the length of the curve that is rubbed over. Within the half tooth point this curve is the shortest for the involute, so that the work done is the smallest although the other factor is the greatest. * Transactions of the American Society of Mechanical Engineers , vol. viii, 1886. The result, without the demonstration, is also given in Reuleav^'s Kon- sirukteur, § 213. t Klein's translation of Herrmann's revision of Weisbach's Mechanics of Engineering and Machinery ^ vol. iii, § 79. As Herrmann states, " This difference is insignificant for the tooth profiles ordinarily employed,'* but the general impression, which it is the object of this paper to contradict, is that the differ- ence is very significant and in favor of the cycloidal tooth. Reuleaux goes further, and, after finding that the two curves are exactly the same for the same arc of action, gives several practical examples, which show the involute to be decidedly inferior, the difference being from sixty to eighty per cent. This result is correct for the conditions of Reuleaux's examples, but it seems to me that those conditions are not correct if the object is to compare the two curves, for he does not take them on the same terms. He takes the involute with a long arc of action, and compares it with a cycloidal tooth having a short arc, and of course the involute is then inferior. Example for h - = lo- /& = = 5- r = 1-5 /= = -I AND P = I ^ Cycloidal Tbkth. Involute Teeth. Obliquity, 30°. S= 1-15. i Total Curve Par- tial Curve Normal Force. Par- tial Work Toial Work. Total Curve Total Work. g Graph Anal's Error Graph Anal's Error. E I I •oio •010 1-0009 -0010 -ooii •00103 •0001 •02 •0023 •00103 •0013 1-002 2 •035 •025 1*0087 -0025 •0036 -00413 •0005 •04 -0046 -0041 1 •0005 I 004 3 •085 •050 10243 -0510 •0086 •00937 -0008 -08 •0092 •00924 I -013 4 •155 •070 1-0485 •0735 -0160 -01679 -0008 •15 •0173 -01645 -0008 I -021 5 •245 •090 1-0824 •0975 -0257 •02656 •0009 •22 0254 •02570 •0003 1033 6 ■355 •no 1-1274 •1240 •0382 •03884 •0006 •32 •0370 •03701 1-049 7 •485 •130 1.1857 •1540 -0536 •05386 •0003 •44 •0508 •05038 •0004 i^o69 8 •630 •145 1-2604 •182s •0718 •07196 •0002 •57 •0658 -06580 1-094 9 •790 •160 1-3563 •2170 •0935 •09357 •0001 •72 •0831 •08328 •0002 I -124 lO •965 •175 1-4802 •2590 •I 194 •11932 -oooi •90 •1039 •10281 •oon 1-161 ZI X-150 •185 1-6426 -3040 •1498 •15008 •0003 x-09 •1259 •12440 •0015 1206 Z2 1-350 •195 1-8615 •3630 •1861 •18715 •001 1 i^30 -1501 •14805 -0020 1-264 ' * 3 4 5 6 7 8 9 10 zz Z2 X3 The work done increases rapidly with the distance of the point of contact from the line of centres, and the result of Reuleaux's method is to compare one curve that is at work a considerable dis- tance from the line with another that is nearer to it. This is clearly shown by the figures of Reuleaux's comparative examples, for in each case the losses are almost exactly proportional to the arcs of action. For the purpose of comparison, the two teeth should be taken under precisely the same circumstances, and they should commence work and stop work together. They should have the same arc of action rather than the same addendum, for the addendum has very little to do with the gear except by its effect on the maximum arc of action. When taken under similar circumstances ^ involute and cycloidal gear teeth are practically equal with regard to the work done by fric tion, the difference being always slightly in favor of the involute. THE LIMITING NUMBERS OF GEAR TEETH. The treatment of the subject of the limiting numbers of gear teeth is usually so difficult that the student is obliged to take the results as he finds them ; for it is a great work of time and patience to follow out the process, and prove the results to be either true or false. The following processes are easily derived from the trigonometrical condi- tions of the problem, but I will here give the results only.* Assume tbe arc of recess to be a times, and the thickness of the tooth to be b times the circular pitch, and the diametral pitch to be unity. Let d be the number of teeth in the driver, and/ the number in the follower. For the cycloidal system, assume the diameter of the describing circle to be q times the diameter of the follower, and the limiting numbers of teeth will be involved in the following equation: — fQ 1 d . r360° / b\ , 360° a n (1) . 360° sin ^0°/ b \ d^V—r) which is insoluble in general terms, but from which either/ or q can easily be separated, for any particular case, by a few numerical trials. For a common example, assume the driver to have six teeth, the arc of re- cess to be equal to the pitch, the tooth to be equal to the space, and the flanks of the follower to be radial. This gives a = l, b — ^, q — J, and d = 6; so Miat the formula becomes f= ' ^^ 20° ^ (2) ) ~? To solve this, put / equal to two numbers as near truth as can be estimated, say 140 and 160. This gives 140 = 140.171, and 160 = 159.193, the opposite errors showing that / is between the two chosen points. Intei-polating in proportion to the two errors, we get 143.5 as our first approximation. Trying 143 and 144 in the same way, we get 143.491 as a second approxima- tion, and 144 as the required nearest larger integer. If the chosen points had been 120 and 180, the first approximation would have been 144.5, and a second trial would have fixed 144 as the nearest integer. When the driver is a rack, we must use the formula Z TT (^-^) ^ . 360° a ^^^ q sm — r- ■ — and when the rack is driven we must use o d 360-/ h \ which are simpler than the unlimited formula. *This subject I have treated in full, with illustrations and examples, in a paper in the bcientihc American Supplement, Vol. XXIII., 1887. When the involute system is to be treated, the problem is a double one; for the action on one side of the line of centers will set one limit, while that on the other side will set another. If we know Q, the angle of obliquity, we have /=2«7rcot^ (6) so that the problem is reduced to finding the value of Q for the given condi- tions. The solution is exact, and not dependent, as with cycloidal teeth, on a pro- cess of trial and error. On the approach side we have the formula tan « = ^0-'^) (6) and on the recess side the formula cos Q = / j>cot TT-f ^ -f V^;^ cot W-p'-i-i ^^^ in which P=2^ (8) and F = 2^ (a - -|-) (9) The approach will set a minimum value for Q, and the recess will determine a maximum. The maximum must evidently be no less than the minimum. When the involute rack follows, we have the same case as for a cycloidal pinion and rack, see (4) ; but when the rack drives we can use cos ^ = ^ 1 _ ' (10) The direct solution is somewhat tedious in application, and may be simplified by the use, on the recess side, of the formula ^ p sin W ,,_ cos Q = — ,^, , — ^j7- (11) ^ cos {Q-{- W) ^ ' which can be easily worked by the above-described process of trial and error. This supposes the involute to be for the interchangeable system, but when it can be allowed to be non-interchangeable the angle of obliquity on the approach need not be the same as that on the recess. The interchangeable involute tooth will not permit as small pinions as the non-interchangeable cycloidal tooth, but when both forms are taken on the same terms, both non- interchangeable, the advantage of the cycloidal tooth is destroyed. CONIC PITCH LINES. The utility of the conic sections, used as the pitch lines of gear wheels, lies /n the fact that under certain conditions they will roll together in perfect rolling contact when mounted upon tlxed centers. We can put all the conic sections under one law as to each of several fea- tures when rolling together, as follows : Any two equal conic sections will roll together in perfect rolling contact when fixed on centers at their opposite foci. Their moving foci will move at a fixed distance apart. The two curves will make a continuous and complete revolution on each other. The point of contact of the the two curves will be at the intersection of the line of the fixed foci with the line of the moving foci. The common tangent to the two curves at their point of contact will pass through the point of intersection of the two axes. There are four conic sections, varying principally as to their focal distance. The circle, having an infinitely small focal distance : the ellipse, having a finite and positive focal distance; the parabola, having an infinitely great focal distance ; and the hyperbola, having a finite and negative focal distance. Any two curves that will roll together may be used as the pitch lines of gear wheels, and therefore we can have gears with either circular, elliptic, parabolic, or hj'perbolic pitch lines. In either case the moving foci may be connected by a link that will hold the two gears together when in motion, and this link will act in the most di- rect and advantageous manner when most needed, when the action of the teeth becomes so oblique as to be of little service. The four cases are illustrated by the four figures Fig. I CIRCULAR GEAP.S Case I. — When the focal distance is infinitely small the curves are circles, as in figure I. The lipk is here simply a fixed bar connecting the two centers, for the two foci are combined in one point at the center. ving focus Fig. IT ELLIPTIC GEARS Case IT. "When the focal distance is finite and positive, the curves are ellipses that will roll together if fixed on centers at their opposite foci, as in Fig. II. The link is a moving bar connecting the two moving foci. / Us ICig. Ill JPAHABOLIC GEARS Case III. When the focal distance is infinitely great the curves are parab- olas. One parabola turns about its focus while the other turns about its opposite focus, but, as the opposite focus is at an infinite distance the sec- ond parabola must move in a straight line at right angles to the line of centers, as in Fig III. The link becomes a bar of infinite length, and cannot be prac- tically applied. The revolution is complete but of infinite extent, so that it cannot be practically accomplished. Case IV. Wlien the focal distance is finite and negative, the curves are hyperbolas. The opposite focus about which one hyperbola turns is now on tlie other side of the curve, which becomes a negative or internal pitch line, as in Fig IV. The linli is of finite length and can be practically applied. Tlie revolution is complete, for as soon as one pair of curves separate, the other pair come together, and the motion is continued. The utility of circular gears is universal, and elliptic gears have many applications, but no use is apparent for parabolic or hyperbolic gears. A use for them will probably be found when their existence and properties become well known, and they are certainly cf interest to the student of mechanism' LIBRARY U^^„„V',i; 020 204 312 4