^Q VERSITY OF CALIFORNIA 356 s LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY 956 VERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY 9 p l VERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF TH ^se ! S j dM 9-i > ,WW> 3S.6 5ITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA LIBRARY OF TH 2!. i ^^^^3vt/^^S. i ~^ ^ ^-- A- JFig. 18. phical molding method 29 LINEAR PLANING PROCESS. A second planing process, quite distinct from the molding process of (27), is founded upon the fact that the tooth curves are in contact at a single point which has a progressive motion along the line of action. Therefore if a single cutting point p, Fig. 20, is caused to travel along the line of action with the proper speed relatively, to the speed of the piich line, it will trim the tooth out- line to the proper odontoidal shape. The figure shows the application to Fig. 20. fn^olu-^-e To Linear plan ing met ' OF THZ 12 Particular Forms. the involute tooth, the path of the cutting point being the straight line I a, and its speed being the speed of the base line b I. When the cutting point follows the circu- lar line of action with a speed equal to that of the pitch line, it will plane out the cycloid al tooth curve. This process is applicable to all possible forms of gear teeth, either spur or bevel, in either external or internal contact. When the curvature of the odontoid will permit, the milling cutter may take the place of the planing tool, and is the equivalent of it. ). THE BACK ORIGINATOR. The molding planing process of (28) sup- plies a means for easily and accurately pro ducing an interchangeable set of gears or cutters for gears, and it is best applied by means of the rack tooth as the originator. All four curves of the rack tooth being alike, the tooth is easily formed, particularly for the involute or the segmental systems, and it is a matter of less consequence that the curves shall be of some particular form, if care is taken that it is odontoidal. It has been taught, and it is therefore some- times considered, that any " four similar and equal lines in alternate reversion" will an- swer the purpose, but it is necessary that the four similar curves shall be odontoids. Four circular arcs, with centers on the pitch line, will answer the definition, but are not odontoids. 31. PARTICULAR FORMS OF THE ODONTOID. The odontoid, as so far examined, is un- defined except as to one feature of the ar- rangement of its normals, and to bring it into practical use it is necessary to give it some definite shape. This is most easily ac- complished by choosing some simple curve for the rack odontoid, and from that making an interchangeable set. A more correct but much more difficult method would be to choose some definite line of action, and from that derive the odontoids. If the rack odontoids are straight lines, Fig. 21, the common involute tooth system will be produced, and the line of action will be a straight line at right angles with the rack odontoid. For bevel teeth, as will be shown, the straight line odontoid produces the octoid tooth system, while to produce the involute system it is necessary to define the line of action as a straight line, and derive the system from that. If the rack odontoids are cycloids, as in Fig. 22, the resulting tooth system will be ihe cycloidal, commonly misnamed the " epicycloidal " system. The line of action will be a circle equal to the roller of the cycloid. If the rack odontoids are segments of cir- cles from centers not on the pitch line, but inside of it, as in Fig. 23, the tooth system Fig Segmental Fig. 23. Rolled Curve Theory. will be the segmental, and its line of action will be the loop of the " Conchoid of Nico- medes." If we choose a parabola for the rack tooth, as in Fig. 24, the parabolic system will be formed with its peculiar "hour glass" line of action. Only three of these tooth systems are in actual use, the involute and the cycloidal for spur gears, and the octoid for bevel gears only, and we will therefore confine the ap- plication of the theory to them. Only one of the sys'ems in common use for spur gears, the involute, should be in use at all, and we will pay principal attention to that. Parabolic Fig The segmental system would be superior to the cycloidal, and in many cases to the in- volute; but as there is already one system too many, we will not attempt to add another. 32. THE ROLLED CURVE THEORY. If any curve R, Fig. 25, is rolled on any pitch curve p I, a point p in the former will trace out on the plane of the latter a curve s p z, called a rolled curve. The line p q, from the tracing point p to the point of contact q, is a normal to the curve * p z, and, as all the normals are ar- ranged in "consecutive" oraer, that curve must be an odontoid. The converse of this statement is also true, that all odontoids are rolled curves ; but the fact is general^ . ery far fetched and of no practical importance. It is also a property of all such curves that are rolled on different pitch lines, that they are interchangeable. This accidental and occasionally useful feature of the rolled curve has generally been made to serve as a basis for the general theory of the gear tooth curve, and it is re- sponsible for the usually clumsy and limited treatment of that theory. The general law is simple enough to define, but it is so diffi- cult to apply, that but one tooth curve, the cycloidal, which happens to have the circle for a roller, can be intelligently handled by it, and the natural result is, that that curve has received the bulk of the atten- tion. For example, the simplest and best of Rolled curve Fig. 25. all the odontoids, the involute, is entirely beyond its reach, because its roller is the logarithmic spiral, a transcendental curve that can be reached only by the higher mathe- matics. No tooth curve, which, like the involute, crosses the pitch line at any angle but a right angle, can be traced by a point in a simple curve. The tracing point must be the pole of a spiral, and therefore the trac- ing of such a curve is a mechanical impossi- bility. A practicable rolled odontoid must cross the pitch line at right angles. To use the rolled curve theory as a base of operations will confine the discussion to the cycloidal tooth, for the involute can only be reached by abandoning its true logarithmic roller, and taking advantage of one of its peculiar properties, and the segmental, sinusoidal, parabolic, and pin tooth, as well as most other available odontoids, cannot be discussed at all. 33. MATHEMATICAL RELATION OP ODONTOID AND LINE OF ACTION. In Fig. 26 the odontoid on the pitch line by the relations P T = p t = y, and T 8 p I is connected with the line of action I a, j t = x, where P 8 is the normal to the 14 Mathematical Relation. odontoid at the point P, T 8 is a tangent to the pitch line at the intersection of the nor- mal, and P TMsa normal to the tangent. When any odontoid is given by its equa- tion, that of the line of action can be found by a process of differentiation, and when the line of action is given by its equation, that of the odontoid can be found by a process of integration. These processes, for the general case where the pitch line is curved, are quite intricate, Imt when the pitch line is a straight line, they are simple, and may be worked as follows. To get the equation of -the line of action from that of the given rack odontoid, ar- Tange the equation of the odontoid in the form x f(y), and put its differential co- efficient - equal to . Thus, the equation d y x of the straight rack odontoid of the involute system is y x tan. A, from which 3 12.566 depended upon as accurate. 36 40 .087 .079 ft 16.755 25.133 48 .065 T 1 5 50.266 39. ADDENDUM AND DEDENDUM. The tooth is limited in length by the circle j a I, Fig. 30, called the addendum line, and drawn outside the pitch line at a given distance, called the addendum. Its depth is also limited by aline r I, called the dedendum or root line, drawn at a given distance inside of the pitch line. The addendum and the dedendum are both arbitrary distances, but, for convenience in computation, they are nxed at simple fractions of the unit of pitch that is in use. When the circular pitch is used the ad> dendum is one-third of the circular pitch. When the diametral pitch unit is used the addendum is one divided by the pitch. It is customary to make the addendum and the dedeodum the same, except in certain cases where some special requirement is to be satisfied. Fig. 30. #1 Addenda Clearance Backlash Actual Sizes. 17 18 Actual Sizes. 2 Pitch. 2 I Pitch. 3 Pitch. Fig. 29. Items of Construction. 19 40. THE CLEARANCE. To allow for the inevitable inaccuracies of workmanship, especially on cast gearing, it is customary to carry the tooth space slightly below the root line to the clearance line c I, Fig. 30. The clearance, or distance of the clearance line inside of the root line, is arbitrary, but it is convenient and customary to make it one> eighth of the addendum. 41. THE BACK-LASH. When rough wooden cogs or cast teeth are used, the irregularities of the surface, and inaccuracies of the shape and spacing of the teeth, require that they should not pre- tend to fit closely, but that they should clear each other by an amount b, Fig. 30, called the back-lash. The amount of the back-lash is arbitrary, but it is a good plan to make it about equal to the clearance, one-eighth of the addendum. Skillfully made teeth will require less, back-lash than roughly shaped teeth, and". properly cut teeth should require no back- lash at all. Involute teeth require less back- lash than cycloidal teeth. 42. THE STANDARD TOOTH. The tooth must be composed of odontoids, preferably of odontoids of which the proper- ties are well known, and an advantage is gained if it is still further confined to a par- ticular value of that odontoid. If the teeth are to be drawn by an odontograph some standard must be fixed upon, since the method will cover but one proportion of tooth.. For example, the standard involute tooth is that having its line of action inclined at an angle of obliquity of fifteen degrees. For the cycloidal system the standard agreed upon is the tooth having radial flanks on a gear of twelve teeth. 43 . ODONTOGR APHS . The construction of the tooth is generally not simply accomplished by graphical means, as it is generally required to find points in the curve and then find centers for circular arcs that will approximate to the curve thus laid out. It is sometimes attempted to construct the curve by some handy method or empirical rule, but such methods are generally worth- less. An odontograph is a method or an instru- ment for simplifying the construction of the curve, generally by finding centers for ap- proximating circular arcs without first find- ing points on the curve, and those in use will be described. 44. THE When the teeth are laid out by theory there will be a portion of the tooth space at the bottom that is never occupied by the mating tooth. Fig. 31 shows a ten-toothed pinion tooth and space with a rack tooth in three of its positions in it, showing the un- used portion by the heavy dotted line. If this unused space is filled in by a " fillet "/ the tooth will be strengthened just where it needs it the most, at the root. The fillet is dependent on the mating tooth, and is therefore not a fixed feature of the tooth. If a gear is to work in an inter- changeable set, it may at some time work with a rack, and therefore its fillet should be fitted to the rack ; but if it is to work only FILLET. with some one gear it may be fitted to that. The light dotted line shows the fillet that would be adapted to a ten-toothed mate. The fillet to match an internal gear tooth would be even smaller than that made by the rack. 31. The fillet 20 Equidistant Series. When the tooth is formed by the molding process of (27), or by the equivalent planing process of (28), the fillet will be correctly formed by the shaping tool, but not so when the linear process of (29) is used. When the tooth is drawn by theory or by an odonto- graph the fillet must be drawn in, and can be most easily determined by making a mating tooth of paper, and trying it in several posi- tions in the tooth space, as in the figure. Except on gears of very few teeth the strength gained will not warrant the trouble of constructing the fillet. 45. THE EQUIDISTANT SERIES. When arranging an odontograph for drafting teeth, or a set of cutters for cutting them, we must make one sizing value do duty for an interval of several teeth, for it is impracticable to use different values for two or three hundred different numbers of teeth. The object of the equidistant series is to so place these intervals that the necessary errors are evenly distributed, each sizing value being made to do duty for several numbers each way from the number to which it is fitted, and being no more inaccurate than any other for the extreme numbers that it is forced to cover. This series is readily computed for any case that may arise, and with a degree of ac- curacy that is well within the requirements ^>f practice; by the formula . as , *-. + - in which a is the first and z is the last tooth of the interchangeable series to be covered; n is the number of intervals in the series, and * is the number in the series of any interval of which the last tooth t is required. For example, it is required to compute the series here used for the cycloidal odonto- graph, having twelve tabular numbers to cover from twelve teeth to a rack. Putting a = 12, z = infinity, and n = 12, the formula becomes 12 X 12 12 X 12 144 t = 12 s-f-0 ~ 12 and then, by putting successively equal to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12, we get the series of last teeth, 13 r ' T , 14|, 16, 18, 20|, 24, 28f, 36, 48, 72, 144, and infinity. These give the required equidistant series of inter- vals. 12 13 to 14, 15 to 16, 17 to 18, 19 to 21, 22 to 24, 25 to 29 30 to 36, 37 to 48, 49 to 72, 73 to 144, 145 to a rack ; and the method is as easily applied to any other practical example. This formula and method is independent of the form and of the length of the tooth, and therefore is applicable to all systems under all circumstances. This is proper and con- venient, for these elements can be eliminated without vitiating the results or destroying the "equidistant" characteristic of the series. The formula is an approximation based upon an assumption, but nothing more convenient or more accurate has so far been devised by laboriously considering all the petty elements involved. The sizing value, or number for which the tabular number is computed, or the cutter is accurately shaped, can best be placed, not at the center of the interval, but by considering the interval as a small series of two intervals, and adopting the intermediate value. The sizing value for the interval from c to d is given by the formula 2 cd Thus, the sizing value for the interval from 37 to 48 teeth should be 41.8, and that for the interval from 145 to a rack should be 290. It is sometimes the practice to size the cut- ter for the lowest number in its interval, on the ground that a tooth that is considerably too much curved is better than one that is even a little too flat. This makes the last tooth of the interval much more inaccurate than if the medium number was used. Friction of Approach. 46. THE HUNTING COG. It is customary to make a pair of cast gears with incommensurable numbers of teeth so that each tooth of each gear will work with all the teeth of the other gear. If a pair of equal gears have twenty teeth each, each tooth will work with the same mating tooth all the time; but if one gear has twenty and the other twenty-one teeth, or any two num- bers not having a common divisor, each tooth will work with all the mating teeth one after the other. The object is to secure an even wearing action; each tooth will have to work with many other teeth, and the supposition is that all the teeth will eventually and mysteriously be worn to some indefinite but true shape. It would seem to be the better practice to have each tooth work with as few teeth as possible, for if it is out of shape it will dam- age all teeth that it works with, and the damage should be confined within as narrow limits as possible. If a bad tooth works with a good one it will ruin it, and if it works with a dozen it will ruin all of them. It is the better plan to have all the teeth as near perfect as possible, and to correct all evident imperfections as soon as discovered. 47. THE MORTISE WHEEL. Another venerable relic of the last century is the "mortise" gear, Fig. 32, having wooden teeth set in a cored rim, in which they are driven and keyed. Where a gear is subjected to sudden strains and great shocks, the mortise wheel is better, and works with less noise than a poor cast gear, and will carry as much as or more power at a high speed with a greater dura- bility. But in no case is it the equal of a properly cut gear, while its cost is about as great. In times when large gears could not be cut, and when the cast tooth was not even ap- proximately of the proper shape, the mortise wheel had its place, but now that the large cut gear can be obtained the mortise gear should be dropped and forgotten. Mortise wheel Fig. 32. 48. FRICTION OF APPROACH. When the point of action between two teeth is approaching the pitch point, that is, when the action is approaching, the friction between the two tooth surfaces is greater than when the action is receding. This extra fric- tion is always present, but is most trouble- some when the surfaces are very rough, as on cast teeth, giving little trouble when the teeth are properly shaped and well cut. When the roller pin gear (93) is used, the friction between the teeth is rolling friction, and is no greater on the approach than on the recess. tion of roach rig. 33. 22 Efficiency. The difference in the friction is probably due to the difference in the direction of the pressure between the small inequalities to which all friction is due. When the gear D, Fig. 33, is the driver, the action between the teeth is receding, and the inequalities lift over each other easily, while if F is the driver, the action is approaching, and the inequalities tend to jam together. In the exaggerated case illustrated, it is plain that the teeth are so locked together that ap- proaching action is impossible, while it is equally plain that motion in the other direc- tion is easy. The same action takes place in a lesser degree with the small inequalities of ordinary rough surfaces. The action of the common friction pawl r which works freely in one direction and jam& hard in the other, is upon the same principle. A weight may be easily dragged over a rough surface that it could not be pushed over by a force that is not parallel to the surface. The extra friction of approaching action can be avoided by giving the driver the long- est face. When the driver has faces only, and the follower has only flanks, the action is particularly smooth. Teeth that are subject to excessive maxi- mum obliquity, such as cycloidal teeth, should not be selected for rough cast gearing, for it is the maximum rather than the average obli- quity that has the greatest influence. 49. EFFICIENCY OF GEAR TEETH. Much has been written, but very little has been done to determine the efficiency of the teeth of gearing in the transmission of power, and therefore but little of a definite nature can be said. The question is mostly a prac- tical one, and should be settled by experi- ment rather than by analysis. The only known experiments upon the fric- tion of spur gear teeth are the Sellers experi- ments, more fully detailed in (112), and but. one of these relates to the spur gear. From that one it is known that a gear of twelve teeth, two pitch, working in a gear of thirty- nine teeth, has an efficiency varying from ninety per centum at a slow speed to ninety- nine per centum at a high speed. That is, an average of five per centum of the power received is wasted by friction at the teeth and shaft bearings. This result is probably a close approximation to that for any ordinary practical case. Although theory can do nothing to de- cide such a question as this, it can do much to indicate probable results. If a pair of involute teeth, for example, move over a certain distance, w, either way from the pitch point, the distance being mea- sured on the pitch line, they will do work that is theoretically determined by the formula : / P k h work done = - . -= =- w 9 & K fl in which / is the coeflBcient of friction, P is the pressure, and k and h are the pitch radii of the gears. The positive sign is to be used for gears in external, and the negative sign, for those in internal contact. The loss by friction, as shown by the for- mula, decreases directly as the diameters in- crease, the proportion of the diameters being constant. The loss increases rapidly with the distance of the point of action from the pitch point When the contact is at the pitch point the teeth do not slide on each other, and there is no loss, but away from that point the loss is as the square of the distance in this case, and in a still greater proportion in the case of the cycloidal tooth. Therefore a short arc of action tends to improve the efficiency. It has been satisfactorily determined that the loss is greater during the approaching than during the receding action. This is not shown by the formula, but it may be laid to a variation in the coefficient /. The formula shows that the loss is inde- pendent of the width or face of the gear, and therefore strength can be increased by widening the face, without increasing the friction. If the work of internal gearing is com- pared with that of external gearing of the same sizes, the losses are in the proportion, k h k-\-h' Strength. 23 BO that the internal gear is much the more economical, particularly when the gear and pinion are nearly of the same size. If the gear is twice the size of the pinion the loss is but one-third of the loss when both gears are external. Small improvement can DC effected, by put- ting a small pinion inside rather than outside of a large gear. A six-inch pinion working with a six-foot gear has but 1.18 times the loss by the same gears, when the gear is in- ternal. Theoretical efficiency is discussed at great length in the Journal of the Franklin Insti- tute, for May, 1887: Also by Reuleaux, and again by Lanza, in the Transactions of the American Society of Mechanical Engineers for 1887, and the discussion has been carried far enough. A series of experiments with gear teeth oi various sizes and forms, of various metals, would add greatly to our knowledge of this important matter. A true determination of the efficiency of the rough cast gear, as compared with that of the cut gear, would tend to discourage the use of the former for the transmission of power, for experiment would undoubtedly show that the power wasted by the cast gear would soon pay the difference in cost of the better article. 50. STRENGTH OF A TOOTH. The strength of a tooth is the still load it will carry, suspended from its point, and is to be carefully distinguished from the horse-power, or the load the gear will carry in motion. The strength of a substance is not a fixed element, but will vary with different samples, and with the same sample under different circumstances ; allowance must be made for the amount of service the sample has seen, concealed defects must be provided against, and therefore nothing but an actual test will surely determine its character. Although no possible rule can be depended upon, the ultimate or breaking strength of a standard cast-iron tooth, having an addendum about equal to a third of the circular pitch, will average about three thousand five hun- dred pounds multiplied by the face of the gear and again by the circular pitch, both in inches. But a tooth should never be forced up to its ultimate strength, and the best practice is to give it only about one-tenth of the load it might possibly bear, so that the following rule should be used : Multiply three hundred and fifty pounds by the face of the gear, and again by the circular pitch, both in inches, and the product will be the safe working load of one tooth. Example : A cast-iron gear of one inch pitch, and two inches face, will safely lift 350 X 2 x 1 = 700 pounds, although it would probably lift 7,000 pounds. When there are two teeth always in work- ing contact, it is safe to allow double the load, but care must be taken that both teeth are always in full contact. A hard wood mortised cog has about one- third of the strength of a cast-iron tooth : steel has double the strength ; wrought-iron is not quite as strong. A small pinion generally has teeth that are weak at the roots, and then it will increase the strength to shroud the gear up to its pitch line, but shrouding will not strengthen a tooth that spreads towards its base, like an involute tooth, and when the face of the gear is wide compared with the length of the tooth the shroud is of /ittle assistance. It does not increase the strength of a tooth to double its pitch, for when the pitch is increased the length is also increased, and the strength is still in direct proportion to the circular pitch, wnile the increase has reduced the number of teeth fu contact at a time. Cut gears and cast gears are about equal as to actual strength, with the advantages in favor of the cut gear, that hidden d 3fects are likely to be discovered, and that it is not as liable to undue strains on account of defective shape. The rules for strength must not be used for gears running at any considerable speed, for they are intended only for slow service, as in cranes, heavy elevators, power punches, etc, J-f or se- Power. f 51. HORSE-POWER OP CAST GEARS. The horse-po\ver of a gear is the amount of power it may be depended upon to carry in continual service. It is very well settled that continual strains and impact will change the nature of the metal, rendering it more brittle, so that a tooth that is perfectly reliable when new may be worthless when it has seen some years of service. This cause of deterioration is particularly potent in the case of rough cast teeth, for they can only approximate to true shape required to transmit a uniform speed, and the continual impact from shocks and rapid variations in the power carried must and does destroy the strength of the metal. There are about as many rules for com- puting the power of a gear as there are manufacturers of gears, each foundryman having a rule, the only good one, which he has found in some book, and with which he will figure the power down to so many horses and hundredths of a horse as con- fidently as he will count the teeth or weigh the casting. Even among the standard writers on en- gineering subjects the agreement is no bet- ter, as shown by Cooper's collection of twenty-four rules from many different wri- ters, applied to the single case of a five-foot gear. See the "Journal of the Franklin Institute" for July, 1879. For the single case over twenty different results were ob- tained, ranging from forty-six to three- hundred horse-power, and proving conclu- sively that the exact object sought is not to be obtained by calculation. This variety is very convenient, for it is always possible to fit a desired power to a given gear, and if a badly designed gear should break, it is a simple matter to find a rule to prove that it was just right, and must have met with some accident. Although no rule can be called reliable, the one that appears to be the best is that given by Box, in his Treatise on Mill Gear- ing. Box's rule, which is based on many actual cases, and which gives among the lowest, and therefore the safest results, is by the formula: 12 c 2 / *J~dn Horse-power of a cast gear = THA^ J ,UUO in which c is the circular pitch, /is the face, the M is the diameter, all in inches, and n is the number of revolutions per minute. Example : A gear of two feet diameter, four inches face, two inches pitch, running at one hundred revolutions per minute, will transmit 12 X 2 X 2:x 4 X A/ 24 x 100 1,000 = 9.4 h. p. For bevel gears, take the diameter and pitch at the middle of the face. It is perfectly allowable, although it is not good practice, to depend upon the gear for from three to six times-the calculated power, if it is new, well made, and runs without being subjected* to sudden shocks and varia- tions of load. The influence of impact and continued service will be appreciated when it is con- sidered that the gear in the example, which will carry 9.4 horse-power, will carry seventy horse-power if impact is ignored, and the ultimate strength of the metal is the only dependence. A mortise gear, with wooden cogs, will carry as much as, or more than a rough cast- iron gear will carry, although its strength is much inferior. The elasticity of the wood allows it to spring and stand a shock that would break a more brittle tooth of much greater strength. And, for the same reason, a gear will last longer in a yielding wooden frame than it will in a rigid iron frame. 52. HORSE-POWER OP CUT GEARS. We know a little, and have to guess the rest, as to the power of a cast gear, but with respect to that of a cut gear we are not as well posted, for there are no experimental data upon which a reliable rule can be founded. Admitting, as we must, that impact is the chief cause of the deterioration of the The Involute Tooth. cast gear, we are at liberty to assume that a properly cut and smoothly running cut gear ris much more reliable. No definite rule is possible, but we can safely assume that a cut gear will carry at least three times as much power as can be trusted to a cast gear of the same size. i The great reliance of those who claim that a cast gear is superior to a cut gear is upon , the hard scale with which the cast tooth is covered. This scale is not over one-hun- dredth of an inch thick, is rapidly worn away, and is of no account whatever. From that point of view it is difficult to explain why a wooden tooth will outwear an iron one, I although it is softer than the softest cut iron. Assuming that a cut gear is about three ' times as reliable as a cast gear, we can com- pute its power by the formula : d n c'f I Horse-power of a cut gear = - in which c is the circular pitch, / is the face, and d is the pitch diameter, all in inches, and n is the number of revolutions per minute. 3. THE IKVOLUTE SYSTEM. 53. THE INVOLUTE TOOTH. The simplest and best tooth curve, theo- retic-ally, as well as the one in greatest prac- tical use for cut gearing, is the involute. The involute tooth system is based on the ' straight rack odontoid, (31) and Fig. 21, and \ it is illustrated by Fig. 34. If the four odon- ! toids of the rack outline are equally inclined i to the pitch line, the resulting tooth system will be completely interchangeable; but if, as in Fig. 35, the face and flank are inclined at different angles of obliquity, T S K and T S K', the system is not interchangeable, although otherwise perfect. The rack odontoid cannot have a corner or change of direction anywhere except at the pitch line, without causing a break in the line of action. As the normals p q are parallel, the line of action is a straight line W O W at right angles to the rack odontoid. The inter- changeable line of action is continued in a straight line on both sides of the pitch line, bui the non-interchangeable line changes di- rection at that line. In accordance with the universal custom we will consider that the involute tooth is 'always interchangeable, having a single angle ot obliquity. Z/ic involute tooth interchangeable Fiff. 34. 26 Involute Interference. 54. THE CUSP. As a circle t c, Fig. 34, can always be drawn tangent to the line of action at an interfer- ence point i, from the center b of any pitch line B, there will always be a cusp in the curve at the point c (16), and at that point the working part of the curve must stop. The working part of the rack tooth must end at the limit line i L through the interference point i. The working curves of any two teeth that work with each other must each end at the line drawn through the interference point of the other, Fig. 43, being limited by limit lines 1 1 and L L. r The second branch c m' of the curve is equal to the first branch c m, but is re- versed in direction. The second cusp is at infinity, and therefore has no practical ex- istence. The tangent circle i c, through the inter- ference point and the cusp, is called the "base line." It is customary to continue the flank of the tooth inside the base line by a straight radial line, as far as may be necessary to allow the mating gear to pass. 55. INTERFERENCE. When the point of the tooth is continued beyond the limit line it will interfere with and cut away a portion of the working curve of the mating tooth. Fig. 36 shows a rack tooth working with the tooth of a small pinion, and cutting out its working curve. This cut is not confined to the flank, but extends across the pitch line into the face, as shown by the line qmn. The rack tooth of the figure will not work with the pinion tooth unless it is cut off at the limit line 1 1 through the interference point i. The mathematical action still continues, and the figure shows the rack tooth in. action at k with the second branch of the curve. Effect of Interference . 36. 56. ADJUSTABILITY. An interesting and in many cases a valua- ble feature of the involute curve, and one that is confined to it, is the fact that its posi- tion as a whole with regard to the mating curve is adjustable. Two involutes, each with its base line, will work together in perfect tooth contact when they are moved with respect to each other, as long as they touch at all. The lines of action and the pitch lines will shift as the curves are moved, and will accommodate themselves to the varying position of the base lines. But this valuable feature of the involute curve is not always available, and involute gears are not, as commonly supposed, neces- sarily adjustable, for the conditions are often such that the teeth will fail to act when the centers are moved, except within very narrow limits. Care must be taken that the arc of action is not so reduced by separating the centers of the gears that it is less than the cir- cular pitch, for the former arc is variable and the latter is fixed. Care must also be taken that the working curve is not pushed over the limit line when the centers are drawn to- gether. In any limiting case, such as in Fig. 43, the centers are not adjustable. The gears of the standard set are either not adjustable at all or are so within very narrow limits, on ac- count of the correction for interference. Involute Construction. 27 57. CONSTRUCTING THE INYOLUTE BY POINTS. The simple involute curve can be con- structed by points by the general method of (24), but it is much better to take advantage of the property that it is an involute of its base circle, and construct it by the rectifica- tion of that circle. As in Fig. 37 any convenient small dis- tance A G is taken on the dividers, and the points on the curve located by stepping along the circle and its tangent from any given point to any desired point. This method is so accurate, if care is taken to step accurately on the line, that the curve seldom needs correction; but, when great ac- curacy is required, correction can be applied at the rate of one- thousandth of an inch to the step, if the length of the step is regulated by the diameter of the circle according to the following table: Diameter of Circle : 12345 678 9 10 11 12 Length of Step : .17 .26 .37 .46 .53 .60 .67 .73 .76 .79 .82 .84 For example: If the circle of Fig. 37 is Construction points Fig. 37. four inches in diameter, and the dividers are set to .46 inch, the true curve, A b' d', will be outside of the constructed curve A b d by .002 inch at b and .005 inch at d. From the table we can form the handy and sufficiently accurate rule that the length of the slep should be about one-tenth of the di- ameter of the circle, for a correction of about one-thousandth of an inch per step. Having thus found several points of the in- volute, we can draw it in by hand, or by con- structing a template, or by finding centers from which approximately accurate circular arcs can be drawn. 58. THE STANDARD INVOLUTE TOOTH. The tooth that is selected for general use, and the one that is the best for all except a few special cases and limiting cases, is the in- terchangeable tooth having an angle of ob- liquity of fifteen degrees, an addendum of one-third the circular pitch, or one divided by the diametral pitch, and a clearance of one- -eighth of the addendum. The standard to which involute cutters are made is slightly different, having an angle of 14 28' 40", the sine of which is one-quarter, .and a clearance of one-twentieth of the circu- lar pitch. If the obliquity is 15 the smallest possible pair of equal gears have 11.72 teeth, and therefore 12 is the smallest gear of the inter- changeable set. The base distance, the distance of the base line inside of the pitch line, is about one-tifty- ninth of the pitch diameter, and one-sixtieth is a convenient fraction for practical use. The limit points of the whole set must be determined by that of the twelve-toothed gear, for any gear of the set may be required to work with that one, and the working curve of each tooth must end at the point thus de- termined. As the limit point is always in- side of the addendum line there must always be a false extension on the tooth, the point being rounded over outside of the limit point. 59. THE INVOLUTE ODONTOGRAPH. As the base line must always be drawn, it is advisable, to save work, to locate the cen- ters of the approximate circular arcs upon that line. It is also necessary that the points -of the teeth shall be rounded over, to avoid interference. These requirements made it impracticable to compute the positions of the centers, and an empirical rule had to be adopt- ed instead. Teeth were carefully drawn by the stepping Ten arid Eleven Involute Teetli. method of (57) on a very large scale, one- quarter pitch, giving a tooth eight inches in length. These teeth were corrected for inter- ference by giving them epicycloidal points that would clear the radial flanks of the twelve-toothed pinion. Then the proper centers on the base line were determined by repeated trials, and tooth curves obtained that would agree with the true involute up to the limit point, and still clear the corrected point. The odontograph table is a record of these radii, which are be- lieved to be as nearly coyrect as the given : conditions will permit. It was found that separate curves were, required for face and flank up to thirty-six teeth, but that one curve would answer for teeth beyond. It was found necessary to devise a separate method for drafting the rack tooth. 60. TEN AND ELEVEN TEETH. Theoretically the twelve-toothed pinion is the smallest standard gear that will have an arc of action as great as the circular pitch, but ten and eleven teeth may be used with an error that is not practically noticeable. Fig. 38 shows a pair of ten-toothed gears in action. They can be in correct action only when the point of contact is between the two interference points i and J, but they will be in practical contact for a greater and suffi- cient distance Fig. 3S. OdontoyrapJiic pair 61. A BAD RULE. There is a simple and worthless rule for involute teeth that deserves notice only be- cause it is considerably in use. It constructs the whole tooth curve, face and flank, for all numbers of leeth, as a single arc from a center on the base line, and with a radius equal to one-quarter of the pitch radius, Fig. 39. This is wonderfully convenient, but the convenience is purchased at the expense of The Involute Odontograph. 29 'ordinary accuracy, for the rule is not even approximately correct. It is handy, and nothing else. Figs. 38 and 40 show the kind of teeth that are constructed by this rule on gears of ten and twelve teeth, where its error is the greatest, and it is reasonable that the invo- lute tooth should not be in great favor with those who have been taught to draw it thus. The error gradually decreases, until, for more than thirty teeth, it is tolerably correct, but it gives the rack with the straight, uncor- rected working face that would interfere, as shown at g, Fig. 40. As it is tolerable only for thirty or more teeth, and not good then, it may well be dropped altogether. bad rule Figl 39. 62. USING THE INVOLUTE ODONTOGRAPH. INVOLUTE ODONTOGRAPH. STANDARD INTERCHANGEABLE TOOTH, CENTERS ON (For Table of Pitch Diameters see 35.) b } j Divide by the Diametral Pitch. Multiply by the Circular Pitch. Teeth. Face Flank Face Flank Radius. Radius. Radius. Radius. 10 2.28 .69- .73 .22 11 2.40 .83 .76 .27 12 2 51 .96 .80 .31 13 2.62 .09 .83 .34 14 2.72 22 .87 .39 15 2,82 .34 .90 .43 16 2.92 .46 .93 .47 17 3.02 .58 .96 .50 18 3.12 .69 .99 .54 -19 3.22 .79 1.03 .57 20 3.32 .89 .06 .60 21 3.41 .98 1 09 .63 22 3.49 2.06 1 11 .66 23 3.57 2 15 1.13 .69 24 3.64 2.24 1.16 .71 25 3.71 2.33 1 18 .74 26 3.78 2.42 1 20 .77 27 3.85 2.50 .23 .80 28 3 92 2.59 25 .82 29 3 99 2.6T .27 .85 80 4.06 2.76 .29 .88 81 4.13 2.85 .31 .91 82 4.20 2.93 .34 .93 33 4 27. 3 01 36 .96 34 4.33 3 09 .38 .99 35 4 89 3 16 .39 1.01 36 4.45 3 23 .41 1.03 8740 4.20 1.34 41-45 4.63 1.48 46-51 5 06 1.61 62-60 5 74 1.83 61-70 6 52 2.07 71-90 7.72 246 91120 9.78 3.11 121-180 13.38 4.26 - r= ^JL^A l-o \ Fat, f?^ fJ ( .^)L_ 30 The Involute Odontograph. To draft the tooth lay off the pitch, ad- dendum, root, and clearance lines, and space the pitch line for the teeth, as in Fig. 40. Draw the base line one-sixtieth of the pitch diameter inside the pitch line. Take the tabular face radius on the divid- ers, after multiplying or dividing it as re- quired by the table, and draw in all the faces from the pitch line to the addendum line from centers on the base line. Set the dividers to the tabular flank radius, and draw in all the flanks from the pitch line to the base line. Draw straight radial flanks from the base line to the root line, and round them into the clearance line. Fig. 40. \. SPECIAL RULE FOR THE RACK. Draw the sides of the rack tooth, Fig. 40, as straight lines inclined to the line of centers c c at an angle of fifteen degrees, best found by quartering the angle of sixty de- grees, Draw the outer half a b of the face, one- quarter of the whole length of the tooth, from a center on the pitch line, and with a radius of 2.10 inches divided by the diametral pitch. .67 inches multiplied by the circular pitch. 64. DRAFTING INTERNAL GEARS. When the internal gear is to be drawn, the odontograph should be used as if the gear was an ordinary external gear. See Fig. 41. But care must be taken that the tooth of the gear is cut off at the limit line drawn through the interference point * of the pin- ion. The point of the tooth may be left off altogether or rounded over to get the appear- | ance of a long tooth. The pinion tooth need not be carried in to the usual root line, but, as in the figure, may just clear the truncated tooth of the gear. The curves of the internal tooth and of its pinion may best be drawn in by points (57), The Involute Odontograph. 31 for the odomographic corrected tooth is not as well adapted to the place as the true tooth, and no correction for interference is needed on the points of the pinion teeth or on the flanks of those of the gear. Care must be taken that the internal teeth do not interfere by the point a striking the point b, as they will if the pitch diameters are too nearly of the same size. Internal involutes 65. INVOLUTE GEARS FOR GIVEN OBLIQUITY AND ADDENDA. When the obliquity and addenda, as well as the pitch diameter and number of teeth in a gear are given, as is generally the case, we can proceed to draft the complete gear as follows: Draw the pitch line p I, Fig. 42, the ad- dendum line a I, the root line r I, and the clearance line c I, as given. Draw the line of action I a at the given obliquity W Z = K. Draw the base line b I tangent to the line of action. Find the interference point * by bi- secting the chord t. Draw the involutes i a m and t" a" in", and a a" will be the maximum arc of ac- tion. If the given arc of action a a' is not great- er than the maximum arc, the pitch line is to be spaced and the tooth curves drawn in from the base line to the addendum line. These tooth curves, when small, are best drawn as circular arcs from centers on or | near the b#se line, one center x for the flank from the base line to the pitch line, and another center y for the face from the pitch line to the addendum line. One involute i a m should be carefully constructed by ! points, and then the required centers can be 1 found by trial. One center and arc will ! often answer for the whole curve, and it is only when great accuracy is required that more than two centers will be necessary. Continue the flanks of the teeth toward center by straight radial lines, and round j these lines into the clearance line. If the interference point for the gear that the gear being drawn is to work with is at I, i within the addendum line, the limit line 1 1 i must be drawn through it, and the points of 32 Involute Special Cases. Fig. 42. Given obliquity and addendum the teeth outside of this limit must be slightly rounded over, to avoid interference (55). If a fillet / is desirable, to strengthen the tooth, it can be drawn in by the method of (44). 66. INVOLUTE GEARS FOR GIVEN NUM- ' BERS OF TEETH. When the numbers of teeth and the pitch lines are the only given details, the shape and action of the tooth depends upon the obli- quity, and the action will fail if the angle is too small. The principal object is to deter- mine the least possible angle that is permitted by the given pitch diameters and numbers of teeth. Draw the pitch lines P L and p I, Fig. 43, lay off the given pitch arc, as a straight line c d or C D, at right angles to the line of centers, and draw the line C d or c D. Then the required line pf action will be I a pass- ing through at right angles to c D or G d. The complete teeth can then be drawn in as previously directed. In this case, the obliquity WO Z being the least possible, the limit lines and the adden- dum lines must coincide, but the addenda may be reduced by increasing the angle. Fig. 43. la Given numbers of teeth Limiting Involute Teeth. 33 67. INVOLUTE GEARS FOR GIVEN OBLIQUITY. When the pitch diameters and the obliquity are the only given details, the lines C I and c i, Fig. 43, drawn from the centers at right angles to the line of action, will determine the limit lines. The maximum arc of action a a' may be found either by drawing the involutes i a and la', or by continuing the line G I to the line c d, and measuring the required distance c d. Any arc of action less than a a' may be used. The drawings should always be made to a scale of one tooth to the inch radius, so that the pitch arc will be 2*. If the scale is one tooth to the inch of diameter, the pitch arc will be ?r. 68. INVOLUTE GEARS WITH LESS THAN FIVE EQUAL TEETH. The "method of Fig. 43 and (66) will be found to apply to any given numbers of teeth not less than five, and to fail, if either gear has but three or but four teeth. Any external gear of five or more teeth will work with any external gear of five or more teeth, and with an internal gear of any number of teeth unless stopped by internal interfer- ence (64). For example, if a pair having four and five teeth, Fig. 44, is tried, the four-toothed pinion will fail, because its tooth will come to a point upon the line of action before it has passed over the required pitch arc. The difficulty cannot be remedied by increasing the obliquity, for an angle that would allow the four-toothed pinion to act would also cause the five-toothed pinion to fail. The practical limit is five teeth, but the mathematical limit is the pair having the fractional number 4. 62 teeth, Fig. 45. The four-toothed pinion will not work with any external gear, not even with a rack, but it will work with an internal gear that has ' about ten thousand teeth, and is practically a rack. It will work with any internal gear having less than ten thousand teeth, and Fig. 46 shows it working with an internal gear of six teeth. Internal interference will prevent its working with an internal gear of five teeth. The three-toothed pinion has no practical action. It has a mathematical action with in- ternal gears of 3.56 or less teeth, as shown by Fig. 47, but as its limit is less than four, it cannot work with any whole number. The figure shows the interference at a. The extreme mathematical limit may be said to be the gear of 2.70 teeth, which has a theoretical action with an internal gear of the same size, coinciding with it. 4.69 X 4.62 limit 'for equal teeth Fig. 45. Limiting Involute Teeth. Fig w 69. INVOLUTE GEARS WITH LESS THAN FIVE UNEQUAL TEETH. If we drop the condition that the pitch line must be equally divided into tooth and space arcs, we can make gears of three and of four teeth work with external gears by the method of (65). The failing case of Fig. 44 may be corrected by widening the failing tooth until it acts, and narrowing the other tooth to correspond, as shown in broken lines. In this way a four-toothed pinion will work with any number of teeth not less than 5.57, at which limit both gears have pointed teeth, as in Fig. 48. The three-toothed pinion will work with any gear having 10.17 or more teeth. Fig. 49 shows the 3x10.17 limiting pair, and Fig. 50 shows the three-toothed pinion working with an internal gear of five teeth. It will not work with an internal gear of four teeth, on account of internal interference, and there- fore the combination shown by Fig. 50 may be said to be the least possible symmetrical in- volute pair. A gear of 2.70 teeth will work with a rack, but there seems to be no way to make a pinion of two teeth work under any circum- stances. Fig. 47 5.57 teeth 4 teeth Fig. 48. Limiting Involute Teeth. 35 Fig. 49. 3 X 10. 17 Unequal teetU Fig. 50 70. THE MATHEMATICAL LIMITS. The above results for low numbered pinions can be obtained by graphical means, but that method is not accurate enough to determine the limits with great precision, and in any case is tedious and laborious. The mathematical process is not particu- larly difficult, and consists in repeated trials with given formulae. To determine the obliquity at which a limiting pinion will be pointed on the line of action, for tooth equal to space, we use the formulae : . . 27T M tan. h = M+n 90 ~ in which n is the given number of teeth in the pointed gear, Fig. 51, M is the number in the gear having the radius M, and h is the angle c I. Knowing n, we assume a value for M, and from that find a value for h by means of the first formula. This value of 7i, tried in the second formula, will give an error. A second assumption for M will give a second error, and if the two errors are not too great a comparison will nearly locate the true value of M. Knowing n and M, we find the obliquity from tan. K = M+n pointed pinion. Fig. 51. In this way the following values were de- termined : n M K 2.695 1.26 57 49! 3. 1.51 54 20' 4. 2.86 42 29' 4.62 4.62 34 11' 5. 6.75 28 8' 5.58 00 Having determined the obliquity for the pointed pinion, we can determine the least number of teeth it will work with by means of the following formula : 180 n 90 Angle B = -=? tan. K == -4- K IT JV -/V tan. B = -== tan. K-\- tan. K in which N is the required least number. 36 Limiting Involute Teeth. In this way it was found that a gear of four teeth will not work with a rack, but will work with an internal gear having a number of teeth not easily calculated with existing loga- rithmic tables, but which is approximately ten thousand. Also that a pinion of three j teeth will not work with an internal gear having more than 3.56 teeth. For unequal teeth we can use the formulae, 2 TT n tan. 7i = tan. H = in which N and n are the numbers of teeth in the pair of pointed gears. By these form- following results were determined, N K oo n 2.695 3. 4. 4.62 10.17 5.57 4*62 25 27' 33 17 34 11 71. MINIMUM NUMBERS FOR UNSYMMETRICAL TEETH. If we drop the condition that the fronts and backs of the teeth shall be alike we have an unimportant case that is similar to that already studied, but much more intricate. If we carry this case to its extreme, and adopt single acting teeth, we have no mini- mum numbers at all, for any two numbers of teeth will then work together. Fig. 52 shows one tooth working with three teeth, and any other combination can be obtained. The minimum obliquity for a given pair is obtained, as in (66), by laying off the known pitch arc, G D, at right angles to G c, and drawing the line of action at right angles to the line D c. The obliquity is also given by the formula : 2 7T tan. K = -=-. , N+n ' in which n and N are the numbers of teeth. When the obliquity is as great as is often Fig. 52. Unsymrnetrical teeth the case for very low numbers of teeth the action may be impracticable on account of the great friction of approach (48). The gears of Fig. 52 will not drive each other on the approach, unless the tooth surfaces are very smooth,and the power transmitted is almost nothing. 72. MINIMUM NUMBERS FOR GIVEN ARC OF RECESS. It has generally been assumed, although no good reason for the assumption has ever been given, that the minimum numbers of teeth occur when the tooth of one of the gears, Fig. 53, is pointed at the interference point /, and at the same time has passed over an arc of recess a that is a given part of the whole pitch arc a' a. The solution is simple enough, graphically by repeated trials, or by a formula that can be applied directly without the usual process by trial and error. But, as involute teeth have a uniform ob- liquity, there is no necessity for assuming Fig. definite arc of recess, and the condition on Involute Efficiency. 37 which the problem is based is unwarranted. I spur gears, in either external or internal con- No real limit is reached, and the matter is I tact, in the Journal of the Franklin Institute not worth examination at any length. The for Feb., 1888, and it has received more atten- problem is investigated, for both bevel and I tion than its slight importance entitles it to. . EFFICIENCY OF INVOLUTE TEETH. But little can be said in addition to the matter in (49), for both forms of teeth in common use are substantially equal with re- spect to the transmission of power. From the formula of (49), which is the formula for the involute tooth, it is seen that the loss from friction is entirely independent of the obliquity, and, therefore, all systems of involute teeth are independent of the ob- liquity in this respect. This is contrary to j the accepted idea that a great efficiency re- quires a small obliquity. It has been stated on high authority that the involute tooth is inferior to the cycloidal tooth in efficiency, but the statement is not true. The difference in efficiency is minute, a small fraction of one per centum, but what | little difference there is is always in favor of the involute tooth. 74. OBLIQUITY AND PRESSURE. The involute tooth action is in the direction of the line of action, and the obliquity is a constant angle. It is variable only when the shaft center distance is varied. As the pressure is always equal to the product of the tangential force at the pitch line multiplied by the secant of the obliquity, (26), it is constant for the involute tooth. Involute teeth, therefore, have a steady ac- tion that is not possessed by other forms; particularly by forms which, like the cy- cloidal, have a pressure and an obliquity that varies between great extremes. 75. THE ROLLER OF THE INVOLUTE. The involute odontoid, like all possible odontoids, can be formed by a tracing point in a curve that is rolled on the pitch line, and this roller is the logarithmic spiral with the tracing point at its pole, (32). This feature is, however, more curious than useful, and it is not of the slightest im- portance in the study of the curve. Neither is the operation of rolling the involute me- chanically possible, for the logarithmic roller has an infinite number of convolutions about its pole, and the tracing point would never reach the pitch line. The involute is often considered to be a rolled curve, because it can be formed by a tracing point in a straight line that rolls on its base line; but, although that is the fact, it is a special feature and has nothing to do with the rolled curve theory. The rolled curve theory requires that the odontoid shall be forme d by a roller that rolls on the pitch line only . . THE; CYCLOIDAIv SYSTEM. 76. THE CYCLOIDAL SYSTEM. If the curve known as the cycloid is chosen as the determining rack odontoid, (31), the resulting tooth system will be cycloidal. It is commonly called the " epicycloidal " system, because the faces of its teeth are epicycloids, but, as the flanks are hypocy- cloids, it seems as if the name "epihypo- cycloidal " would be still more clumsy and accurate. There is no more need of two different kinds of tooth curves for gears of the same pitch than there is need of two different kinds of threads for standard screws, or of two different kinds of coins of the same value, and the cycloidal tooth would never be missed if it was dropped altogether. But it was first in the field, is simple in theory, is easily drawn, has the recommendation of many well-meaning teachers, and holds its position by means of "human inertia," or the natural reluctance of the average human mind to adopt a change, particularly a change for the better. 77. THE CYCLOIBAL TOOTH. The cycloid is the curve A that is traced by the point p in the circle C that is rolled on the straight pitch line p I, Fig. 54. The normal at the point p is the line p q to the point of tangency of the rolling circle and the pitch line. The line of action is the circle I a, of the same size as the roller C. As no tangent arc can be drawn to the line of action from the pitch point as a center, no terminal point (18) exists. As there is no point upon the line of centers from which a circle can be drawn tangent to the line of action, there will be no cusps, (16) except on the pitch line. The cycloidal tooth can be drawn by the general method of (24), but there are several easier methods which will be described. There are numerous empirical rules and short cuts to save labor and spoil the tooth, which will not be de- scribed. When the pitch line is of twice the diame- ter of the line of action, the flank of the tooth is a straight line. If the pitch line is less than twice as large as the line of action, the flank of the tooth will be under-curved, as shown by Fig. 55, and it is customary to avoid the resulting weak tooth by limiting the line of action to a diameter not greatei than half that of the smallest gear to be used. Cycloid al Secondary Action. 78. SECONDARY ACTION. The secondary line of action (21) is a circle, Fig. 56, differing from the pitch circle by the diameter of the primary line of action, either inside or outside of it. When the internal secondary line of action of an internal pitch line coin- cides with the external secondary line of action of its pinion _ there will be secondary contact between the gears, the face of the gear working with the face of the pinion at a point of contact upon the combined secondaries. Fig. 57 shows this for the cycloidal tooth, the two faces working together at the point a. As both secondaries are cir- cles they must coincide, and the sec- ondary action will be continuous. When the teeth are also in contact at b on the primary line OL action, there will be double contact. Undcrcurved flanks Fig. 55. Secondary Fig 56. 40 Cycloidal Interference. 79. INTERNAL INTERFERENCE. If the secondary lines of action do not come together the teeth will not touch each other at all, but if that of the gear is smaller than that of the pinion the teeth will cross each other and interfere. The line c. Fig. 57, is the face of the gear tooth, and the line d is the face of the pinion tooth having a primary line of action equal to the difference between the pitch lines. The secondary line of each gear coincides with the pitch line of the other, and the faces interfere with each other the amount shown by the shaded space. The only remedy for internal interference is to reduce the diameter of the primary line of action to half the difference between the diameters of the pitch lines, or else to leave off one of the faces of the teeth. The discovery of the law of internal cycloid- al interference is due to A. K. Mansfield, who published it in the ' ' Journal of the Franklin Institute" for January, 1877. It was afterwards re-discovered by Professor MacCord, and most thoroughly applied and illustrated in his " Kinematics." When interference is avoided by omitting one of the faces of the teeth the primary line of action may be enlarged, but it must not then be larger than the difference between the pitch diameters. Fig. 58 shows on the right the action when the face of the gear is omitted, and on the left the action when the face of the pin- ion is left off. The teeth will just clear each other, each one touching the other at a single point a in its pitch line. As the contact at a is not a point of practi- cal action, care must be taken that the arc of action at the primary line of action is as great as the circular pitch, for otherwise, as in the figure, the gears will not be in continu- ous primary action. The rule for internal interference, simply stated, is that the diameters of the pitch lines must differ by the sum of the diameters of the lines of action if the teeth have both faces and flanks, and by the diameter of the acting line of action if the face of either gear is omitted. For the standard interchangeable system the gears must differ by twelve teeth Fig 58 if both teeth have faces, and by six teeth if one face is omitted. Fig. 62 shows the secondary contact in the case of a standard internal gear of twenty- four teeth working with a pinion of twelve teeth, and it is to be noticed that the teeth nearly coincide between the two points of contact. Where there is secondary contact the teeth practically bear on a considerable line instead of at a point. Cycloidal Odontograph. 41 80. THE STANDARD TOOTH. The standard tooth (42), selected for the cycloidal system, is by common consent the one having a line of action of half the diame- ter of a gear of twelve teeth, so that that gear has radial flanks. The standard adopted by manufacturers of cycloidal gear cutters is that having radial flanks on the gear of fifteen teeth, but it is not and should not be in use for other pur- poses. If any change is made, it should be made in the other direction, to make the set take in gears of ten teeth. It must be borne in mind that the standard adopted does not limit the set to the stated minimum number of teeth, but that it sim- ply requires that smaller gears shall have weak under-curved teeth. 81. THE ROLLED CURVE METHOD. It happens in this case, and in this case only, that the rolled curve method, which theoretically applies to all odontoids, can be actually put into practical use, for the generating roller is here the circle, the sim- plest possible curve. As in Fig. 59, roll a circle of the .diameter of the circle of action upon the outside of the pitch line for the faces, and upon the inside for the flanks, and a fixed point in it will trace the curve. The method can be used by actually con- structing pitch and rolling circles, but the same result can be reached more easily and quite as accurately by drawing several cir- cles, and then stepping from the pitch point along the pitch line, and back on the circles to the desired point. If the length of the Construction by rolling Fig. 59. step is not more than one-tenth of the diam- eter of the circle, the error will not be over one-thousandth of an inch for each step. This method is the best one to adopt, ex- cept for the standard tooth. 82. THE THREE POINT ODONTOGRAPH. It is a simple matter to draw the tooth curve by means of rolling circles, but such a method requires skill on the part of the draftsman. It is, moreover, nothing but a method for finding points in the curve for which approximate circular arcs are then determined. The "three point" odontograph is sim- ply a record of the positions of the centers of the circles which approximate the most closely to the whole curve of the standard tooth. The positions of two .points, a at the center of the face or of the flank, Fig. 60, and b at the addendum point or root point of the curve, were carefully computed, and then the position of the center C of the circle which passes through these two points and the pitch point 0, was calcu- lated. The circle that passes through these three points is assumed to be as accurately approximate to the true curve as any pos- sible circular arc can be. The odontograph gives the radius " rad." of the circular arc, and the distance "dis." of the circle of centers from the pitch line, for the tooth of a given pitch, and their values for other pitches are easily found by simple multiplication or division. The advantages of this method lie in the facts that the desired radius and distance are given directly, without the labor of find- ing them, and that as they are computed they are free from errors of manipulation. In point of time required, the advantage is \ 42 Cycloidal Odontograph. WeofJW* with the odontograph in the ratio of ten to one. The greatest error of the odonto graphic arc, shown greatly exaggerated by the dotted lines, is at the point c on the face, and it is greater on a twelve- toothed pinion than on any larger gear. For a twelve-toothed pinion of three- inch circular pitch, a large tooth, the actual amount of the maximum error is less than one one-hundredth of an inch, and its average for eight equidistant points on the face is about four-thousandths I that stated will be due to manipulation, and of an inch. Any error that is greater than ! not to the method. To apply the odontograph to any particu lar case, tirst draw the pitch, addendum 83. USING THE ODONTOGRAPH. distance "dis." inside of it. Take the face radius "rad."on the dividers, and draw in root, and clearance lines, and space the pitch line, Figs. 60 and 61. Then draw the line of flank centers at the tabular distance "dis." outside of the pitch line, and the line of face centers at the all the face curves from centers on the line of face centers; then take the flank radius "rad."and draw all the flank curves from centers on the line of flank centers. THREE POINT ODONTOGRAPH. STANDARD CYCLOIDAL TEETH. INTERCHANGEABLE SERIES. From a Pinion pf Ten Teeth to a Rack. For One /*** For One Inch DIAMETRAL PITCH. CIRCULAR PITCH. NUMBER OF TEETH For any other pitch divide by that pitch. For any other pitch multiply by that pitch. IN THE GEAR. Faces. Flanks. Faces. Flanks. Exact. Intervals. Rad. Dis. Rad. Dis. Rad. I is. Rad. Dis. 10 10 1.99 .02 8.00 ^ .62 .01 2.55 1.27 11 11 2.00 .04 11.05 6.50 .63 .01 -3.34 207 12 12 201 06 oo ' 00 .64 .02 00 oo : 1314 2.04 .07 15.10 9.43 .65 .02 4.80 3.00 15Vi 1516 2.10 .09 7.86 ' 3 46 .67 .03 2.50 1 10 ITji 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 .63 .50 23 22-24 2.26 .15 4.5Q V 1.13 .72 .05 .43 .36 27 25-29 2.33 .16 4.10 .96 .74 .05 30 .29 83 30-36 2.40 .19 3.80 .72 .76 .06 . .20 .23 42 37-48 2.48 .22 3.52 .63 .79 .07 .12 .20 58 4972 2.60 .25 3.33 .54 .83 .08 .06 , .17 97 73144 2.83 .28 3.14 .44 .90 .09 1.00 .14 290 145-300 2.92 .31 3.00 .38 .93 .10 .95 .12 00 Rack 2.96 .84 2.96 .34 .94 .11 .94 .11 Cyclo idal Odo n tograph . The table gives the distances and radii if the pitch is either exactly one diametral or one inch circular, and for any other pitch multiply or divide as directed in the table. Fig. 61 shows the process applied to a practical case, with the distances given in figures. Fig. 62 shows the c'ame process applied to an internal gear of twenty-four teeth work- ing with a pinion of twelve teeth. It illus- trates secondary action and double contact. It also shows the actual divergence of the Willis odontographic arp from the true curve. Odontographic example Fig. 61. Internal teeth Fig. 62. 44 Willis Odontograph. 84. THE WILLIS ODONTOGRAPH. This is the oldest and best known of all the odontographs, but it is inferior to several others since pro- posed, not only in ease of operation, but in accuracy of result. To apply it, find the pitch point* a and a' half a tooth from the pitch point 0, Fig. 63, draw the radii a c and a' c', lay off the angles cab and c' a' b', both 75, and Jay off the distances a b and a' b' that are given by table. The centers b and b' thus found are the centers of circular arcs that are tangent to the tooth curves at d and d'. The dividers are set to the radius b or b' to draw the curves. The Willis arc touches the true curve only at the pitch point 0, and its variation else- where is small, but noticeable. On the face of the tooth of a twelve-toothed pinion of three inch circular pitch, its error at the ad- dendum point is four-hundred ths of an inch, and it will average three times that of the three point method (82). The error is shown by Fig. 62. The greatest error of the method is due to manipulation. The angle is usually laid off by a card, and the center measured in by a scale on the card. The circle of centers is TJie Willis odontograph Fig. 63. then drawn through the center, and unless great care is used the chances of error are great. 180 The angle 90 c ab = JFs= , and the - sin. W, in which distance a b = 5 - 27T t is the number of teeth in the gear of the same set which has radial flanks, usually 12 ; c is the circular pitch, and t is the num- ber of teeth in the gear being drawn. The positive sign is used for the face radius, and the negative for the flank radius. 85. KLEIN'S CO-ORDINATE ODONTOGRAPH. This is a method of finding the positions of several points on the tooth curve by means of their co-ordinates referred to axes through the pitch point. Any point on the curve is found by laying off a certain dis- tance on the radius Y, Fig. 64, and then a certain distance at right angles to it, the distances being given by a table for a certain standard tooth. As many points as required are found by this method, and then the curve is drawn in by curved rulers, or by finding the approxi- mating circular arc. This odontograph is to be found in Klein's Elements of Machine Design. Coordinate odontograpli Fig. 64. Obliquity of Action. 86. THE TEMPLET ODONTOGRAPH. Prof. Robinson's templet odontograph is an instrument, not a method. It is a piece of sheet metal, Fig. 65, having two edges shaped to logarithmic spirals. It is laid upon the drawing, according to directions given in an accompanying pamphlet, and used as a ruler to guide the pen. It can be fastened to a radius bar, and swung on the center of the gear, to draw all the teeth. See Van Nostrand's Science Series, No. 24, for the theory of the instrument in detail. The templet odontograph Fig. 65. 87. OBLIQUITY OF THE ACTION. When the point of contact between two teeth is at the pitch point 0, Fig. 66, the pressure between the teeth is at right angles to the line of centers, but, as the point of con- tact recedes from the line, the direction of the pressure varies by an angle of obliquity which increases from zero until the point K, at the intersection of the addendum circle with the line of action, is reached. The angle K = K W, of the maximum obliquity, can be found by solving the trian- gle C c K, and for the standard set we have, cos. 2 K = ^ ^, 3 n-\- 18 in which n is the number of teeth in the gear. For the smallest gear of the set, the one having twelve teeth, K is 20 15', and for the rack it is 24 5', so that it will always be be- tween those two limits for external gears, and greater for internal gears. The friction between two gear teeth in- creases with the angle of obliquity, but not w Obliquity Fig. 66. in direct proportion. With the involute tooth the work done while going over a cer- tain arc from the line of centers is propor- tional to the square of the arc, and for cycloidal teeth the increase with the arc is still more rapid. Therefore it is the maxi- mum obliquity of the action that principally determines the injurious effects of friction. THE CUTTER LIMIT. When the number of teeth in the gear is less than that in the gear having teeth with radial flanks, the flanks will be under-curved, and when too much so they cannot be cut with a rotary cutter. The teeth of Fig. 55 could not be cut with a rotary cutter beyond the points where the tangents to the two sides are parallel. The limit is reached when the last point that is cut by the rotary cutter is also the last point that is touched by the tooth of the rack in action with it, not allowing for in- ternal gears. The diameter of the gear when this limit is reached is found by the formula, -" rt T C 46 Limiting Cycloidal Teeth. in which D is the diameter of the gear, d is the diameter of the circle of action, c is the circular pitch, and a is the addendum For the common addendum of unity divided by the diametral pitch this may be put in the shape, ^ n = s i in which * is the number of teeth in the radial flanked gear, and n is the number in the required cutter limit. For the common series, where 8 = 12, we have n = 8.26; and for the cutter standard of * = 15, we have n = 10.80, so that cutters could easily be made to cut gears with less than s teeth. When the rolling circle for the faces is of half the diam- eter of the pitch line of the mating gear, the flanks of both gears will be straight radial lines, as in Fig. 67. Such gears are fitted to each other in pairs, and are not interchangeable with other sizes. Their teeth are more easily made than those of standard gears. The maxi- mum obliquity is less, but the strength of the teeth is also less than usual. There is no reason for making such teeth in preference to the ). RADIAL FLANKED TEETH. standard, al- though, for that reason probably, they are used to a considerable extent. It would be Radial flanks Fig. 67. difficult to devise a form of tooth so whimsi- cal that it would find no one to adopt and use it. J 90. THE LIMITING NUMBERS OP TEETH. When the number of teeth in a driving gear is small, the point p, Fig. 68, of its pointed tooth may go out of action by leav- ing the line of action g before a certain definite arc of recess r has been passed over, and the problem is to find the smallest num- ber of teeth in the following gear that will just allow the given recess. This question, which is not a particularly important one, is discussed at length, and applied to both bevel and spur gears, in either external or internal contact, in an article in the "Journal of the Franklin Institute" for Feb., 1888, and we will here consider only the case of the common spur gear. The recess r is given as a times the cir- cular pitch, and the thickness a r of the tooth is given as b times the same. The diameter of the circle of action is q times Limiting ttftfth Fig. 68. that of the pitch line of the following gear. The number of teeth in the driving gear is d, and the number in the following gear is /. The Pin Tooth. 47 M is an auxiliary angle equal to -, and W is an angle / \. Then the required number / can be found by a process of trial and error with the formula, sin. (M -f W) sin. W -Tf- M For an example, let the recess be of the pitch, the tooth equal to the space, and the flanks of the follower to be radial. Let the problem be to find a follower for a driver of seven teeth. This gives a = , b = $,' q = i, d = 7, and the formula becomes gin. in. l^L + 25 43' \ \ J i sin. 25 43' If we put / at random, at 20, we shall get, +.134 = 0. Next, trying/ =10, we get, .132 = 0, and the opposite signs show that / is between 20 and 10. Trying 12 the result is positive, and for 11 it is negative, showing that 12 is the required value of /. That is, 7 teeth will not drive less than 12 teeth with radial flanks, unless it is allowed an arc of recess greater than f of the pitch. For another example, test MacCord's value of 382 as the least driver for a follower of 10 teeth, when recess equals the pitch and the follower has radial flanks. Trying d 382, the error is negative ; for 383 it is also negative, but for 384 it is positive, and there- fore the latter is the true number. Extensive and sufficiently accurate tables of limiting values are given by MacCord in his "Kinematics." 5. 1PITST TOOTH SYSTEM. 91. THE PIN GEAR TOOTH. The theory of the pin gear tooth is en- tirely beyond the reach of the " rolled curve" method of treatment, and, therefore, writers who have adopted that method have had to depend more on special methods adapted to it alone than on general principles. The re- sult is that its properties are often given in- correctly, or with an obscurity and complica- tion that is bewildering to the student. Although the tooth is one of the oldest in use, its theory is so difficult that its defect was not discovered until within a very few years, by MacCord, about 1880, and it was not until it was examined by means of its normals that a remedy for that defect was discovered. By treating the curve on the general prin- ciples here adopted, as a special form of the segmental tooth, it can be studied with ease, and its peculiarities developed in a complete and satisfactory manner. The method, in general terms, is to find the conjugate tooth curve of the gear, for the given circular tooth curve of the pinion, and it presents no new features or difficulties. 92. APPROXIMATE FORM OF PIN TOOTH CURVE. Considered roughly, but accurate enough for teeth of small size, the form of the gear tooth b, Fig. 69, is a simple parallel to the epicycloid E, formed by the center e of the pin, and is to be drawn tangent to any convenient number of circles having centers on the epicycloid. The action is practically all on one side of the line of centers, the face of the gear tooth working with the part of the pin that is The Pin Tooth. inside of its pitch line. It is, therefore, all approaching action when the pin drives and all receding action when the gear drives, and it is best to avoid the increased friction of the approaching action by always putting the pins on the follower. Zantern wheel Fig. 70. Pin gearing Fig. 69. 93. ROLLER TEETH. The pin gear is particularly valuable when the pins can be made in the form of rollers, Fig. 70, for then the minimum of friction is reached. The roller runs freely on a fixed stud, or on bearings at each end, and can be easily lubricated. The friction between the tooth and pin, otherwise a sliding friction at a line bearing, is, with the roller pin, a slight rolling fric- tion, and the sliding friction is confined to the surface between the roller and its bear- ings. When the roller pin is used there can be no increased friction of approach, and the pin wheel can drive as well as follow. For very light machinery, such as clock work, there is no form of tooth that is su- perior to the roller pin tooth, and, with the improvement to be explained, there is no better form for any purpose. 94. CUTTING THE PIN TOOTH. The pin gear tooth can be very easily and accurately shaped by mounting a revolving milling cutter M, Fig. 71, of the size of the pin, upon a wheel A, and causing it to roll with a wheel B, carrying the gear blank Q. The mill will shape the teeth to the correct form. Pin gear cutter Fig. 71. 95. PARTICULAR FORMS OF PIN GEARS. When the pins are supported between two plates, as in Fig. 70, the wheel is called a "lantern" wheel, and is the most common form of clock pinion. The pins are some- times called "staves," and are sometimes known as "leaves." Defect of Pin Tooth. 4 each other should be studied upon this normal surface. As the helix cannot be ^represented upon a plane figure it must be imagined, and as it is ob- scure it requires close attention. Any two spiral teeth will work together, provided their normal spiral sections are con- jugate (24), and, as the shape of the normal spiral section is independent of the angle of the spiral, two spiral gears will work to- gether, approximately, on shafts that are askew. This will be seen more clearly if the spiral section is imagined to be a flexible sheet-metal toothed helix, which can be coiled about the shaft of the gear, for it can evi- dently be coiled close or loose without affect- ing the shape of its teeth. If coiled close, with a short lead, it runs nearly at right angles to the shaft, and the gear approxi- mates to the spur gear, while if the lead is long the gear approximates to the screw. As the diameter of the spiral gear increases, the teeth straighten, and when the diameter is infinite and it is a rack, they are straight and in no way different from those of a com- mon rack. 104. THEOKY OF SPIRAL TOOTH ACTION. The Willis theory of the action of spiral teeth is the one generally accepted, but it is not correct. It assumes that the action be- tween the gears is upon a section by a plane through the axis of the gear and the common normal to the two axes, and that the section of the two gears made by the plane act to- gether like a rack and gear. When the axes are at right angles, and the spiral angle is great, this theory is apparently correct, the error being practically imper- ceptible, but, as the axes become more nearly parallel, the error is more apparent, until, when they are parallel, the error is plain enough. Willis applied his theory to worms and worm gears, on axes at right angles, and evidently did not consider the spiral gear in general. The action between spiral teeth is not upon the axial section, and it is not that of a rack and gear, but when there is any action at all it is upon the normal spiral section. See the AMERICAN MACHINIST for May 19th, 1888. When the axes are parallel the normal spiral sections, as well as the sections made by a plane normal to the axes, are conjugate, and therefore the action is correct and along a line of action. The action is also continu- ous when the axes intersect and the gears are bevel gears. When, however, the axes are askew, the normal spiral sections are not necessarily con- jugate, for they coincide only on one line, the common normal to the two axes. Therefore, there is no continuous tooth contact, except in one particular case, the teeth being in con- tact only for an instant as they pass the normal. The special case for which spiral teeth on askew axes have continuous tooth contact, is that case of the involute tooth when the base cylinders are tangent and the gears become spiraloidal skew bevel gears. See (175) and (176). In that particular case the teeth have a sliding conjugate action on each other. As the spiraloidal gear is fully described in its place, it will not be further considered here. This theory is corroborated by experiment- al gears made for the Brown & Sharpe Man- ufacturing Company, for whom Mr. O. J. Beale, to whom the theory of the spiral gear is much indebted, made a pair of theo- retically perfect spiral gears, exactly alike, with a spiral angle of 45, working on shafts at right angles, and of such a large size that the action of the teeth could be plainly ob- served. See Figs. 88 and 90. Beale's gears cannot be made to run to- gether properly at any shaft distance, but if their ends are brought to the common 54 Spiral Gearing. normal, and their base cylinders are in con- tact, they are skew bevel gears and show the action required by Olivier's theory. But, although the action of spiral gear teeth is intermittent, and their contact is the- oretically perfect at one instant only, when they are passing the common normal, they are very nearly in contact all the time and the action is practically perfect. Spiral teeth of ordinary sizes work together with a re- markably smooth action. 105. FORMATION OF THE SPIRAL TOOTH. As the spiral rack has an ordinary straight tooth, we can conveniently derive the spiral tooth in general from it by a method that is a form of the molding method of (27) for spur gears. If a plane is moved in any direction upon a cylinder it will move it, as if by friction, with a speed that depends upon the direction of the motion. If we imagine the same re- sulting motion between the plane and the pitch cylinder, and assume that the plane is provided with hard and straight teeth run- ning in any direction, it will mold the plastic substance of the cylinder and form spiral teeth upon it. All spiral teeth formed by the same rack will have normal spiral sections that are approximately conjugate to each other, and they will work together inter- changeably. This process may be put into practical shape by a modification of the process of (28) for spur gears, by substituting a planing tooth for the molding rack tooth. The tooth has the shape of the normal section of the rack, and, as it is reciprocated at an angle with the axis of the gear blank being shaped, both the tool and the gear blank receive the motion of the plane and pitch cylinder. The cutting face of the tool is normal to the direc- tion of its motion, which motion is tangent to the direction of the tooth spiral. The linear process of (29) may be used, the plane of Fig. 20 representing, approximately, the normal spiral section of the gear. Thus, if the planing tool or the equivalent milling cutter receives a motion as if in a plane roll- ing upon the base cylinder, the involute tooth will be produced. The spiral tooth may be formed by the linear planing process of (29), directly ap- plied on the principle that the spiral tooth is a twisted spur tooth. The planing tool re- ceives a planing motion in the direction of the axis of the gear blank, and both tool and blank receive the feeding rolling motion that would produce the spur tooth of the section that is normal to the axis. In addition, the blank receives a motion of rotation while the tool moves, that is repeated for every troke of the tool. The cutting edge of the tool is set normal to the axis of the gear. The spiral tooth may also be formed by a tool that is formed to the true shape of some section of the tooth, preferably its normal sec- tion, and which is guided in the tooth spiral. This is the process used to shape a worm, the tool being guided by a screw-cutting lathe. The process generally used to mill the teeth of the spiral gear is the equivalent of the operation last described. The milling cutter is shaped to the normal section of the tooth space, and is guided in the tooth spiral by a special feeding device that ro- tates the blank while the cutter is working in it. Of these processes the planing process of (28) is the best, as it produces the tooth with theoretical perfection, and because all gears formed with the same tool are conjugate and interchangeable. But the screw-cutting and milling processes are most in use, for the reason that they are more expeditious and better adapted to the common machine tools, and it is therefore necessary to study the shape of the normal section of the tooth with 1 some care. Spiral Gearing. oo 108. THE NORMAL PITCH. The real pitch of the spiral gear is meas- ured on a section that is normal to its axis, and, as in the case of the spur gear, it is found by dividing the number of teeth by the pitch diameter, but the shape of the tooth must be regulated by the normal pitch, or pitch of its normal section. The normal pitch is found by dividing the real pitch by the cosine of the angle made by the tooth spiral with the axis of the gear. Thus, if the pitch is 8, and the angle is 45, the normal pitch is 8, divided by .707, or 11.3. The normal circular pitch is found by mul- tiplying the real circular pitch by the cosine of the spiral angle. 107. THE ADDENDUM. The addendum of the spiral gear should not be determined by its real pitch, but by its normal pitch, for it is then usually possi- ble to mill the tooth with a milling cutter that is made for a standard spur gear. A gear of 8 pitch and 45 angle should have an 1 addendum of 11.3 = .089". If the addendum is determined by the true pitch when the angle is considerable, the tooth will be long and thin. Fig. 86 shows the normal pitch section of a rack to run with a pinion of 45 angle, while Fig. 87 shows the true pitch of the same rack. Fig. 88 also shows the true pitch of the pinion, and, although the tooth appears to be stunted, it is really of the standard shape. Spiral Rack, Fig. 87. Beetle's Experimental Gears. Fig. 88. 108. THE AXIAL PITCH. The section of the spiral gear by a plane through the axis is that of a rack, and the axial pitch, or pitch of the rack, is found by dividing the true pitch by the tangent of the spiral angle. Thus, if the angle is 45, the axial pitch is the same as the true pitch, but the axial pitch of a 70 spiral tooth is but .364 of the true pitch. 109. SHAPING THE TOOL. When the spiral gear is cut in a milling machine, or turned in a lathe, it is necessary to give the tool the shape of the normal sec- tion of the tooth to be cut, and this is most readily accomplished by shaping it for the spur gear that most nearly coincides with that normal section. The number of teeth in the gear that is osculatory to the normal spiral, and therefore most nearly coincides with it, is found by dividing the actual number of teeth in the gear by the third power of the cosine of the spiral anple. For example, if we are to cut a gear of 4" 56 Spiral Gearing. diameter, 6 pitch, and 24 teeth, at a spiral angle of 45, the cutter should be shaped to 24 24 cut a spur gear of = = 69 teeth of -=^ = 8.5 pitch. If the gear has 28 teeth of 4 pitch, and an angle of 10, the equiva- lent spur gear has 29 teeth of 4.08 pitch, as the gear varies but little from a spur gear. If the gear is of 5 pitch, and 15 teeth, with an angle of 80, the equivalent spur gear has 2,830 teeth of 28.7 pitch, and in general, when the gear has a great angle it is a worm, the section is practically that of a rack. Care must be taken, when the gear is a screw, and is turned in the lathe, that the tool should be set with its cutting edge nor- mal to the thread of the screw, if it is shaped by the above rule. If the tool is set in the axial section of the screw, and it generally is, it should be shaped to the axial section of the worm, and have the axial pitch and adden- dum. But when the lead of the thread of the screw is small compared with its diam- eter the difference between the normal and axial sections is not noticeable. 110. VELOCITY RATIO OP SPIRAL GEARS. The spiral gear does not follow the well- known rule of spur gears, that the velocities in revolutions in a given time are inversely proportional to the pitch diameters, but re- quires that ratio to be multiplied by the ratio of the cosines of the spiral angles. In the formula v I) cos. A ~V ~~ d cos. a D and d are the diameters of the gears, A and a are their spiral angles, and V and are their velocities in revolutions. If the angles are equal, the velocity ratio is the same as for spur gears of the same diameters. Fig. 88 shows a pair of gears B and G that are of the same size and have the same angle in opposite directions, requiring the shafts to be parallel. See also Fig. 89. The pair of gears A and B are exactly alike, with equal angles in the same direction, re- quiring the shafts to be at an angle equal to Fig. SO Spiral Spur Gears Equa Gears Fig. 00 twice the spiral angle. See also Fig. 90. The statement that like spiral gears will not run together is founded on the Willis theory of spiral gear contact, and is wrong. 111. SPIRAL WORM AND GEAR. When the shafts are at right angles, and the angle on one is so great that it is a screw, the combination is known as a worm gear and worm, Figs. 91 and 92, and is much used for obtaining slow and powerful motions. It is also too much used for wasting power and wearing itself out, for its friction is very great and consumes from one-quarter to two-thirds of thd power received . When the screw has a single thread, the velocity ratio is simply the number of teeth in the gear, and if th re are two or three threads it must be modified accordingly. The spiral worm is adjustable in its gear both laterally and longitudinally, so that it will change its position as required by wear in the shaft bearings. It is an excellent substitute for the hobbed worm and gear, and in most cases will serve practical purposes quite as well. Worm Gearing. 57 Spiral Worm Gear and Worm, Fig. 91. Fig. 92 Worm Gears 112. EFFICIENCY OF SPIRAL AND WORM GEARING. Unless the shafts are parallel the teeth of a pair of spiral gears are moving in different directions, and therefore they cannot pass each other without sliding on each other an amount that increases rapidly with the angle of divergence of the directions of motion, that is, the shaft angle. This sliding action creates friction and tends to wear the teeth, and to a very much greater extent than is generally supposed. The friction is so great, in fact, that such gears, particularly worm gears, should be used only for conveying light powers. They are extensively used, or rather misused, for driving elevators, and are even found in mill- ing machines, gear cutters, planers, and similar places, in evident ignorance that they waste from a quarter to two-thirds of the power received. The most extensive experiments on the efficiency of spiral and worm gears ever made were made by Wm. Sellers & Co., and they may be found described in great detail in a paper by Wilfred Lewis in the Transactions of the American Society of Mechanical En- gineers, vol. vii. Space will not permit ex- tensive quotations from this valuable paper, but the general result of the experiments is shown by the diagram, Fig. 93. The diagram shows that a common cast-iron spur gear and pinion on parallel shafts have an efficiency of from ninety to ninety-nine per cent., accord- ing to the speed at which they are working ; that a spiral pinion of 45, angle working in a spur gear, with shafts at 45, has an effi- ciency of from 81 to 97 per cent. ; that the efficiency decreases as the angle of the shafts increases, until, for a worm of a spiral angle of 5, at a shaft angle of 85, it goes as low as 34, and does not rise higher than 77 per cent. This includes the waste of power at the shaft bearings as well as that at the teeth of the gears. The efficiency is lowest for slow speeds, and rises with the speed. The diagram may be relied upon to give its true value, under ordinary conditions, within five per cent. The same experiments developed the fact that the velocity of the sliding motion of the cast-iron teeth on each other should not be over two hundred feet per minute in contin- uous service, to avoid cutting of the surfaces. It may be assumed that the efficiency will be higher when the worm is of steel, particu- larly when the gear is of bronze. Diagram, Fig. 94, shows the result of simi- 58 Worm Gearing. Velocity at Pitch Xine in feet per minute, o ,,^,,22 2 S g S -feq -S;., P. 20" 3 , #3, 6 4 , etc., negative multilobes a, a 3 , 4 , etc., will be formed about the original curve a,. All these multilobes, positive and negative, will roll together collectively about their fixed centers, in rolling contact at a common and shifting pitch point 0. Any two, of the same sign, will roll in inter- nal contact, and any two of opposite signs Train of tnultilobcs Fig. 110. will roll in external contact, so that they can be formed in train, Fig. 110. When it so happens, as it does with the ellipse revolving on its focus, or the logarith- mic spiral revolving on its pole, is taken, that the first derived pair of curves, or unilobes, are exactly alike, all the multilobes will be alike ; the positive trilobe like the negative trilobe, and so on, so that any two curves of such a set will work together in either inter- nal or external contact, Fig. 111. Conic Pitch Lines. 69 131. CONIC SECTION PITCH LINES. If two like conic sections are mounted upon their foci, they will roll together. Their free foci will revolve at a fixed dis- tance from each other, and may be connected by a link. The line of the free foci will in- tersect the line of the fixed foci at the point of contact of the pitch lines. Fig. 112 shows a pair of ellipses, Fig. 113 a pair of parabolas, and Fig. 114 a pair of hyperbolas. The elliptic pitch line is the only one known that will revolve with its equal, and make a practical and complete revolution. fixed Elliptic tnultilobea Fig. 111. Ellipticpitch lines Fig. 112. Parabolic pitch lines Fig. 113. Hyperbolic pitch Hues > Of IIVEE3ITY, 70 Logarithmic Pitch Lines. 132. THE LOGARITHMIC SPIRAL. If the radiants a, b, c, d, e, Fig. 115, make equal angles with each other, and each one is equal to the adjacent one multiplied by a constant number, their extremities will deter- mine a logarithmic spiral. If the first radiant a is given, with the con- stant multiplier ??, the second radiant will be na, the third will be n* a, the fourth will be n 9 a, and so OD. If the first and last radiants, a and e, are given, and there are p equal angles between them, the constant is so that it is a simple matter to construct a logarithmic spiral to connect any two given radiants at any given angle with each other. The curve possesses the singular property that all tangents, A or E, make the same angle with the radiants at their points of con- tact. The curves are always inclined to the line of centers at the constant angle. The curve continually approaches the center M, or "pole," making an infinite number of turns about it, but ntver reach- ing it. It also has the entirely useless property that the pole will trace an involute of the base circle if it is rolled upon the pitch circle (75). It possesses the property, not possessed by any other curve, that it will roll with an equal mate on fixed centers that can be varied in position. The curve H will roll with the curve C, whether its pole is at N, or at 8, or at F. Fig. 116 shows a pair of logarithmic spi- rals in internal contact. Logarithmic pitch lines Fig. 115. Internal logarithmic pitch lines Fig. 1W. 133. COMPOSITE PITCH LINES. Instead of drawing a curve at random, and finding the mate to run with it, Fig. 108, the complete pitch line may be built up of a number of curves, of which the properties are known. Thus, Fig. 117 shows composite gears, consisting of circular parts A and a, and an elliptic trilobe B, working with an elliptic bilobe b. Fig. 118 shows a combination of a pair of logarithmic spiral arcs A and a, a pair of elliptic bilobal arcs B and b, a pair of logarithmic spiral arcs D and d, and a pair of elliptic quadrilobal arcs E and e. An endless variety of combinations can be made in this way. It is not necessary that the component Composite Pitch Lines. 71 curves be tangent, if they succeed each other continuously. Fig. 119 shows a pair of equal logarithmic spirals with a break at ab, the action at b commencing just as it ends at a. Care should be taken to avoid salient points, breaks, and interruptions of the con- tinuity of the curve, for there must be defective tooth action at such points. The cuives should run smoothly into each other with gradual changes of curvature. Composite pitch lines Fig. Composite pitch lines Fig. Broken pitch lines Fig. 119. 134. TEETH OF NON-CIRCULAR PITCH LINES. The action of the teeth of non-circular pitch lines does not, at first sight, appear to follow the laws pertaining to circular lines, but there is really very little difference. If we consider the two pitch lines to be free, and to be so moved while they roll together that the pitch point 0, Fig. 107, is fixed, and so that the fixed line c C is always at right angles to both curves at their com- mon point 0, the laws of the tooth action will be almost precisely the same as laid down for the circular pitch line. Fig. 107 may be easily applied to (24) as illustrated by Fig. 15. When the centers are fixed, the same tooth action takes place, but the line of action and the pitch point continually change their positions. The teeth of non-circular pitch lines can therefore be formed either by conjugating a given odontoid, as in (24), or by the roiled curve theory of (32). By all means the most practicable method, when the circumstances will permit, is to make up the curve by joining approximating circular arcs, and to provide each circular arc with teeth in the ordinary way. See this process as applied to the elliptic pitch line at Figs. 129 -and 130. 135. TEETH AT SALIENT POINTS AND BREAKS. When there is a salient point, or other inter- ruption of the continuity of the action, as at q, Fig. 109, or at Mm, Fig. 118, there must be an interruption in the arrangement of the normals of any tooth curve, and a consequent failure of the tooth action. 1-2 Elliptic Pitch Lines. Fig. 120 shows a cycloidal tooth curve Jf, at a corner or salient point S, between two circular pitch arcs. There is a circular arc A on the odontoid made while the describing circle is turning about the point 8, and that arc can have no continuous tooth action. Therefore the tooth action will fail, unless the next tooth curve .2V springs from the salient point. If a tooth springs from the salient point, the tooth action will be correct, but mechani- cally imperfect, as the arc of action of two teeth cannot lap over each other to allow for practical defects. And then, as two tooth curves cannot spring from the same pitch point in opposite directions, such gears can run in but one direction, and are not reversi- ble. When there is a break, as at ab, Fig. 119, the teeth must be so cut off that they will The salient point Fig. 12O. separate at a just as they engage at b, for there is a sudden change in the velocity ratio. Such combinations are practicable, but in every way undesirable. 136. THE ELLIPTIC GEAR. The principal, and almost the only use of the irregular gear, is to produce a variation of speed between certain given limits, with- out conditions as to the variations of speed and details of the motion between the limits. When that is the only object, the elliptic pitch line is the only one that is required, and it is chosen because it is the only known con- tinuous closed curve that will work in roll- ing'contact with an equal mate, and because it is, next to the circle, the simplest known curve. Of the elliptic multilobes, the uni- lobe, or simple ellipse, revolving on one of its foci as a center, is the only one used to any appreciable extent, and therefore is the only one that requires examination in detail. The use of the elliptic gear is practically confined to producing a simple variation of speed between known limits, and to produc- ing a "quick return motion" for planers, shapers, slotters, and similar cutting tools, as well as for pumps, shears, punches, shingle machines, and others where the work is done mostly during one-half of the stroke of a reciprocating piece. The work of a planer tool or of the plunger of a single acting pump, is all done during the motion of the tool or of the plunger in one direction, and the only object on the return is to get the piece ready for the next useful operation in the quickest possible time. For an example, the bobbin of a spinning machine is to be wound in a conical form, the thread being fed to it through a moving guide, and the necessary variable motion of the guide, fast at the point of the cone, and slow at its base, is best given to it by a pair of elliptic gears. For another example, the motion of the platen of a printing press should be rapid when the press is open, and slow and powerful when the impression is being taken, and the object can be reached best by a pair of elliptic gears operating the platen. The practical uses of the elliptic gear are endless, and it would be in greater use and favor, if it were not for the fact that its pro- duction, by the means ordinarily in use for that purpose, is as difficult and costly as the resulting gear is unsatisfactory. Elliptic Pitch Lines. 73 137. THE To thoroughly understand the construc- tion and operation of the ellipse, it is neces- sary to learn but a few of its many proper- ties. The mechanical definition of the ellipse is that it is one of the " conic sections." If the cone, Fig. 121, is cut by a plane G at right angles with its axis, the outline of the section will be a circle; if the plane Ecuts the cone at an angle, the section will be an ellipse; if the plane P is parallel with the side of the cone, the section is a parabola, and if the plane H is at such an angle that it cuts both nappes of the cone, the section is a hyperbola. All these curves will roll together when fixed on centers at certain points called foci, but the ellipse, and its special case, the circle, are the only ones that are capable of continuous mo- tion. In the ellipse, Fig. 122, the point C is the center, the longest diameter, AA ', is the major axis, the shortest diameter, BB' , is the minor axis; A and A are the major apices, and B and B' are the minor apices. If an arc be drawn from the minor apex, with a radius equal to the major semi-axis, it will cut the major axis at points F and F, called the foci, and one focus must be chosen as the center, about which the curve is to re- volve if used as the pitch line of a gear. It is a property of the curve that the sum of the distances, PF and PF', from any point to the foci is equal to the major axis, AA', and this feature is used as a means of constructing the curve by points. Draw any arc at random from one focus with radius FP. Draw an arc from the other focus with a radius equal to A A FP, and it will cut the first arc at a point of the ellipse. When the point P is near either major apex, the arcs intersect at such a sharp angle that the method is nearly useless. Another, and much the best known method for constructing the ellipse by points, is to draw any radial line L, and also circular arcs W and V, from the center through the apices. From the intersections, w and v, of the radial line and the circles, draw lines parallel to the axes, and they will intersect, always at right angles, at a point u on the curve. This ELLIPSE. Conic sections Fiy. 121. The ellipse Fig. 122. method is very accurate, and has no failing position. Another valuable property of the ellipse is that if the line pab be so drawn that the dis- tance pa is equal to BC, and pb to AC, the point p will be upon the curve if the points a and b are upon the axes. The curvature of the ellipse is an important feature in connection with its use as a gear pitch line. It is sharpest at the major axis A, and flattest at the minor apex B, else- where varying between the two limits. The radius of curvature at either apex, that is, the radius of tne circle that most nearly coincides with the curve, is found by drawing the lines Bk and Ak at right angles 74 Elliptic Pitch Lines. with the chord AB. The distance Ch is the radius of curvature at the major apex A, and the distance Ck is the radius at the minor apex B. The normal PN to the curve at any point P bisects the angle FPF' between the focal lines, and the tangent PT is at right angles to the normal. 138. ELLEPTOGRAPHS. There are a multitude of elliptographs, or instruments for drawing the ellipse, but only two of them are of practical application in this connection. The simplest known elliptograph consists of a couple of pins, a thread, a pencil, and a stock of patience. The pins are inserted at the foci, as in Fig. 123, and the curve is drawn by moving the pencil with a uniform strain against the string. After a number of trials, depending in number on the skill of the draftsman, the curve may be induced to pass through the desired points. The best result will be obtained by the use of a well waxed thread running in a groove near the point of a hard pencil. The pencil should be long, .and held by the end, so that the strain on the string will be uniform, for the elasticity of the string is the greatest source of error. This " gardener's ellipse " will generally be accurate enough for a tulip patch, but should not be relied upon for mechanical pur- poses, unless one or more points between the apices are tested and found to be correct. If the two pins and the pencil are circular, and of the same diameter, ~the ac- curacy of the ellipse is independent ' of their diameter. The best elliptograph is the "trammel," Fig. 124, which takes a variety of shapes, but which in its simplest condition consists of a cross, with two grooves at right angles, and a bar D with two pins a and b, and a tracing point P placed in line. The distance Pb being set to the major semi-axis, and the distance Pa to the minor semi- axis, the point P will trace the ellipse if the pins are confined to move in the grooves. lf carefully made, the instrument works with great precision, is easily handled and set, and, if the curve drawn is not very flat, it may be inked. The cheap wooden Fig. 123. Gardener's ellipse. The trammel Fig. 124=. trammel should not be tolerated, for the string and two pins cost less and are more reliable. Elliptic Pitch Lines. 75 139. APPROXIMATE CIRCULAR ARCS. If a well-made trammel is not at hand, the best plan is to draw the ellipse with a string, through several constructed points, and then to ink it by finding centers for approximate arcs, as shown by Fig. 125. An arc from a center m on the major axis, will coincide very well with the curve near the major apex, a similar arc n from a center on the minor axis will serve near the minor apex, and a third center q can be found for an arc to join the first two. More than three cen- ters will seldom be required, and when the ellipse is not very flat the two centers on the axes will be sufficient. . 125. The elliptic involute 140. FOUR CENTER ELLIPSE. When the ratio of the axes is not less than eight to ten, as is generally the case, a prac- tically perfect ellipse may be drawn from four centers by the following method. Draw the line CL, Fig. 126, parallel to A'B, and construct the point u on the ellipse by the method of (137). Find a point a on the major axis, from which an arc from A will pass through u, and it will be the major center. It may be found by trial, or by drawing um at right angles to uA, and bisecting Am in a. Through a draw ac at right angles to AB, and its intersection with the minor axis will be the minor center b. Lay off Co,' and CV equal to Ca and Cb, and draw be', b'c", and b'c'". From the centers a draw the arcs cAc'", and e'A'c", and from the centers b draw the arcs cBc' and c"B'c'". Fig. 126. al JS' Four center method Lines that are parallel to the pitch line, such as the addendum, root, clearance, and base lines, are to. be drawn from the same centers. 141. ROLLING ELLIPSES. When two equal ellipses, Fig. 127, are arranged to revolve on their foci as centers, with a center distance equal to the major axis, they will roll together perfectly, and be fitted to act as the pitch lines of gear wheels. of the arrow d, it will drive the follower F by direct contact of the pitch ellipses, but when turning in the other direction with respect to the follower, as it must during half of its revolution, it has no direct driving action, and the follower must be When the driver D turns in the direction ! kept in contact by some other force. Spacing tJic Ellipse. As the two ellipses roll together, the free foci F% and F will always move at a con- stant distance apart, equal to the distance between the fixed foci, and therefore they may be connected by the link L. The center line of the link will always cross the fixed center line at the point of con- j tact of the ellipses, and the tangent T at that | point will pass through the intersection of the axes. 142. SPACING THE ELLIPSE. As the ellipses roll together it is essential | If the ellipse is drawn by means of the that the axes come in line, and therefore, if trammel, Fig. 124, it can be accurately spaced the teeth of one gear are fixed at random, by means of a graduated index circle /, hav- those of the other must be fixed to corre- j ing. a diameter equal to the sum of the diam- spond. If this requirement is satisfied, it j eters of the ellipse, for then the center line of makes no placed. difference where the teeth are It is, however, very desirable that the two the bar will pass over an arc on the ellipse that at the apices is exactly equal to half the arc passed over at the same time on the circle, gears shall be exactly alike, so that they can be cut at one operation while mounted together on an arbor through their focus holes, and to do this, it is necessary to start the teeth at different points, according to whether their number is odd or even. If the number of teeth is even, one tooth must spring from the major axis, as shown by Fig. 128. If the number of teeth is odd, the major axis must bisect a tooth and a space, as shown by Fig. 129. In this case, if one of the 1 and that is elsewhere very nearly in the same Fig. 128. gears can be turned over, or.if its other focus hole can be used as a center, it may have a tooth springing from the major axis. The simplest method of spacing the ellipse is to step about it with the dividers. If the curve is flat, the dividers should be set to less than a whole tooth, for equal chords will not measure equal arcs of the curve. But this stepping method, although it is sufficient and convenient for drafting pur- poses, is wholly unfit for mechanical pur- poses, and therefore we must have a method that is not dependent on personal skill. proportion. This method is not mathematically exact, but its accuracy is very far within the re- quirements of practice. The space on the quarter, at Q, will be greater than anywhere else, but the maximum error will in general be very minute. For an example, take an extreme practical case, a gear with axes eight and ten inches long, and with seventy-two teeth The max- imum error, the difference between the long- est and shortest tooth arcs, will be not over one five-hundredth of an inch. In the mor< Involute Elliptic Gears. 77 common practical case of a gear of nine and ten inches axes, and seventy-two teeth, the maximum error is about one two-thousandth of an inch. In both these cases, the differ- ence between the tooth arc at the major apex and that at the minor apex is too small to be readily calculated, but will be about one twenty-thousandth of an inch. In all cases likely to be met with in practice, the inevit- able mechanical errors are greater than the theoretical errors of the method, and it is serviceable on ellipses as flat as three to one. Involute elliptic teeth. Fig. 129. 143. INVOLUTE ELLIPTIC TEETH. As in the case of the circular gear, the best form of tooth for the elliptic gear is the involute, and for the same reasons. The base line of the involute tooth is any ellipse BE, Fig. 125, which is drawn from the same foci as the pitch ellipse ; the limit point i is the point of tangency of a tangent from the pitch point 0, and the addendum line a I of the mating gear must not pass beyond that point. The method of laying out the tooth and drafting it is so exactly like the process for the circular gear that the description need not be repeated. The centers of involute elliptic gears can be adjusted without affecting the perfection of the motion transmitted, but, as the focal distance remains fixed, the ratio of the axes will be altered. The work of drawing the teeth can be much abbreviated by the process illustrated by Fig. 129. Find the centers for approxi- mate circular arcs, preferably by the method of (140), and then consider the gear as made up of four circular toothed segments. It is then necessary to construct but two tooth curves, one for the major and one for the minor segment, and the flanks will be radii of the circular arcs. The line of action, la, Fig. 125, is not a straight line, and it is not the same for all the teeth. It is not fixed when the pitch point and the line of centers is fixed (134). Cycloidal Elliptic Gears. Cycloldal elliptic teeth. Fig. 130. 144. CYCLOIDAL ELLIPTIC TEETH. The cycloidal tooth is drawn, exactly as upon a circular pitch line, by a tracing point in a circle that is rolled on both sides of the pitch line. The line of action is not a circle, and it is not the same curve for all the teeth. That the flanks shall not be under-curved, the diameter of the rolling circle should not be greater than the radius of curvature at the tooth being drawn, and when, as usual, the same roller is used for all the teeth, its diameter should not be greater than the radius of curvature at the major apex, the distance Ch of Fig. 122. Fig. 130 shows a cycloidal gear drawn as four circular segments, by the methods of (140) and (83). 145. IRREGULAR TEETH. It is most convenient to draw all the teeth alike, with the same rolling circle, or from the same base line, and also to uniformly space the pitch line, but such uniformity is not essential. The only requirement is that each tooth curve shall be conjugate to the tooth curve that it works with, and if that condition is satisfied the teeth may be of all sorts and sizes. 146. FAILURE IN THE TOOTH ACTION. When the major axes are in line the action of the teeth on each other is nearly direct, but when the minor axes are in line the action is more oblique, as shown by Fig. 127. The teeth tend to jam together when the driver is pushing the follower, and to pull apart The Link. 79 when the follower is being pulled, and when the ellipse is very flat this tendency is so great that the teeth fail to act serviceably. At first glance it might appear that this difficulty in the tooth action of very eccentric gears might be overcome by making the teeth radial to the focus, as shown by Fig. 131, but examination will show that but little can be gained in that way. The teeth on the gear G were obtained by the method of (28) from the assumed tooth on the gear c, and the effect of the defective shape of one side of the assumed tooth was to cut away the conjugate curve of the derived tooth. Such teeth would not work as well as the ordinary form, and their construction would be very difficult. Uttrfial teeth Fig. 131. 147. THE LINK. When the teeth of the elliptic gear fail to properly engage, on account of the obliquity of the action, the difficulty can be entirely overcome by connecting the free foci by a link (141), as shown by Fig. 127. This link works to the best advantage when the teeth are working at the worst, and when it fails to act, as it passes the centers, the teeth are working at their best. There- fore gears that are connected by a link need teeth only at the major apices. When the tooth action is imperfect by rea- son of its obliquity, and the link is not avail- able or desirable, the difficulty can be over- come by using three or more gears in a train, as shown by Fig. 137, for then the same re- sult can be obtained by the use of gears that are much more nearly circular. 148. VARIABLE SPEED AND P.OWER. If the shaft c, Fig. 132, turns uniformly, the slowest speed of the shaft C will occur when the gears are in the position of the figure, and the proportion between the two speeds will be the proportion between the distances cO and CO. The greatest speed of the driven shaft will occur when the shafts have turned through a half revolution from the position of the figure, and the relative speed will be the same, reversed. Fig. 132. The ratio of speed, the ratio of the greatest variation of the axes to produce a decided speed to the slowest speed, is the square of the ratio between the speed of the driving shaft and the greatest or the least speed of the driven shaft, so that it requires but a slight variation of the speed. The following table will give the propor- tion of minor to major axes that will give any desired ratio of speeds. 80 Elliptic Quick Return Motion. Ratio of Speeds. 2 3 4 5 6 7... Ratio of Axes. 985 . .962 .907 .892 .878 .868 .854 .844 .8M .824 .817 .807 .800 The power is always inversely proportional to the speed. If the variable shaft is running twice as fast as the uniform shaft, it will ex- ert but one-half the force. When the gears are arranged in a train, as in Fig. 137, the speed ratio for the second, third, and following gears will be in the pro- portion of the first, second, third and follow- ing powers of the first ratio. Thus, the ratio for a pair of gears with axes in the proportion of .952 to 1 being 4 for the second gear, will be 16 for the third gear, 64 for the fourth gear, and so on. The use of gears of troublesome eccentric- ity can be avoided by this means. A train of three gears of .952 axes, Fig. 137, is equivalent to a single pair of very flat gears with .800 axes, Fig. 138, and, in general, three gears that are nearly circular are equiva- lent to a single very flat pair. 149. ELLIPTIC QUICK RETURN MOTION. If the gears are arranged with respect to the piece to be reciprocated, in the manner shown by Fig. 133, the time of the cutting stroke will be to the time of the return stroke, as the angle PEK is to the angle PEF, where J^and F are the foci of the ellipse. The following table will show the ratio of axes that must be adopted to produce a re- quired ratio of stroke to return. Quick Return. Ratio of Axes. J2tol 964 3 tol 910 4tol 861 5 tot 817 6 tol 778 To determine the ellipse that will give a required quick return, we lay off the angles PEK and PEF in the given proportion, and then find by trial a point P such that the length PE plus the length of the perpendicu- lar PF is equal to the known center distance Ee. F will be the other focus of the re- quired ellipse. When the driving gear has turned through the angle P'EF, from the position of the figure at the middle of the return, the varia- ble gear will have turned through the angle P"eO = P'FO, and we can study the action of the tool by drawing equi-distant radii about E, and finding the corresponding radii about F. Quick return Fig. 133. Fig. 134 shows the arrangement of the radii (P'F = F'e of Fig. 133) in the case of a four to one quick return, and it is seen, by the parallel lines, that the motion of the tool is very uniform, coming quickly to its maxi- mum speed, and holding a quite uniform speed until near the end of the stroke. Fig. 135 shows that the same motion derived from a simple crank is not as uniform. When the gears are arranged in a train, Fig. 137, the quick return ratios can be de- termined by the construction shown by Fig. 136. Draw Fc at right angles to A A', and draw cEd through the other focus. The quick return ratio of the second gear will be the ratio of the angles a a and b a . Draw dFe, and the ratio for the third gear will be Elliptic Trains. 81 Quick return crank . 134. Ordinary crunk Fig. 135. that of the angles 8 and &. Draw eEf, and 4 and Z> 4 will give the ratio for the fourth gear. And so on, in the same man- ner, as far as desired, the ratio being greatly increased by each gear that is added to the train. If carefully performed, the graphical pro- cess is quite accurate. The case of axes in the proportion of .98 to 1 gave a quick re- turn of 1.6 for the second gear, and 2.8 for the third gear, while their true computed values are 1.66 and 2.74. The chart will solve quick return train questions involving gears not flatter than .80, as accurately as need be. For example, the ratio of axes of .95 will give a quick return of 2.25 for the second gear, 4.85 for the third gear, 9.80 for the fourth ' gear, and 19.70 for the fifth gear. Again, the proportion of axes to give a quick return of 5 for the third gear is .948. Quick return train Fig. 130. Elliptic train Fig. 137 Fig. 138. Elliptic Gear Cutting Machine. Elliptic Quick Return Chart ,81 .82 .83 .84 .85 .87, *88 ..89 .90 .91 .92 .S Proportion of Axes .94 .95 .96 .97 .98 .99 1.00 150. THE ELLIPTIC GEAR CUTTING MACHINE. The conditions of the described -operation of drawing the ellipse by means of the trammel (138) may be reversed, the bar being held still while the paper and the cross are revolved, and it is evident that the xesult will be the same ellipse on the paper as if the bar is revolved as described. By thus reversing the process of describing the ellipse, and by adopt- ing the improved spacing device of (142), we can construct a machine for accurately cutting the teeth in an elliptic gear, the main features of which, omitting various unessen- tial details, are shown by Figs. 139 and 140. The blank to be cut is fastened upon a trammel stand, which cor- responds to the paper in the graphi- cal process, and revolves upon the fixed base. The adjustable trammel pins a and b are fixed in a slot in the bed, and they fit and slide in the slots M and N in the under surface of the stand. The cutter which corresponds to the tracing point is fixed with the pitch center of its Plan Fig. 139. Cutter Elevation Fig. 14O. Elliptic Bevel Gear. 83 tooth curve directly over the point P in the line of the pins. The index plate has a diameter equal to the sum of the axes of the ellipse, and it is held by an index pin p, which slides in the slot, and is always'in the line of the pins. Thus arranged, the machine will always cut its tooth in the true ellipse, and the teeth will be accurately spaced. The direction of the tooth will be sub- stantially at right angles to the pitch line, and a simple arrangement can be applied to make it exactly so. An index plate of a fixed diameter may be used for all sizes of gears, if the index pin is carried by an arm which swings about the center of the gear, and has an adjustable pin that slides in the slot. The tops of the teeth are trued by a cutter having a square .edge, and the line of the tops will be substantially parallel to the pitch line. The blank is held by an arbor through its focus hole, and the arbor is held by a slide, which slides in a chuck upon the stand, so that the focus can be accurately set in the major axis at the proper distance from the center. 151. CHOICE OF CUTTERS. Theoretically, the teeth are of different shapes, as they are in different positions upon the ellipse, and, therefore, each space should be cut with a cutter that is shaped for that particular space. But as this is impracticable, it is necessary to choose the cutter that will serve the best on the average. Strictly, the cutter should be the one that is fitted to cut a spur gear having a pitch radius equal to the radius of curvature of the ellipse at the major apex, but as that cutter will be much too rounding for the minor apex, it is better to choose the one that is fitted for the medium radius of cur- vature. The two radii of curvature are the dis- tances Oh and Ck, Fig. 122, and the cutter should be chosen for the radius half way between the two, approximately half the sum of the two. 152. THE ELLIPTIC BEVEL GEAR. An ellipse may be drawn on the surface of a sphere by means of a string and two pins, according to the method of (138), and a pair of such spherical ellipses will roll on each other while fixed on their foci, their free foci moving at a constant distance apart. Therefore we can have elliptic bevel gears that are very similar to elliptic spur gears, as shown by Fig. 141. The two gears revolve on radial shafts through their foci, and the link connects radial shafts through the free foci. The velocity ratio is the ratio of the perpendiculars a b and a c. The elliptic bevel gear is the invention of Pro- fessor MacCord. The spherical ellipse cannot be drawn by the trammel method of (138), and therefore the method of spacing of (142), as well as Elliptic bevel gears Fig. 141. the gear cutting machine of (150), does not apply. 84 Elliptic Calculations. 153. MATHEMATICAL TREATMENT. If the major semi-axis is a, and the minor semi- axis is b, the equation of the curve from the origin at G is a* y* -f&z=. a 8 b*, the major axis being the axis of X. The distance GF from the center to the focus will be c = _&2 = a ^ i _ in which TZ, is the ratio of axes = . a The radius of curvature at the major apex is , and that at the minor apex is . a b There is no practicable formula for the recti- fication of the curve, as the length is express- ible only by a series. The special spacing method of (142) is true only at the instant of passing either apex, for the tracing point describes half the arc described by the line of the bar on the index circle only when the bar is at right angles with the curve. The error will be at its maximum when the bar is at the maximum angle with the normal, which is at about an angle of forty-five degrees with the major axis. The difference between an ordinary tooth space at the major apex, and that at the minor apex, is very minute. A very careful calculation of the length of the chord of a gear of seventy-two teeth, and eight and ten inch axes, gave a chord of .41433" at the major apex, a chord of .41495" at 45 for the maximum, and a chord of .41441" at the minor apex. The difference between the chords at the apices is .00008", but as the cur- vature at the major apex is greater than at the minor apex, the difference between the arcs would be less, perhaps not over .00004". The ratio of speeds (148), is /JL+jv^-^lY \ i - vi K* i The ratio of quick return being given as qr, the value of n is n = y 2 V rf 8 -f d* 3d 8 , in which p. = X Shaft ang. 52.8 Teeth = 20) 66 (3.30 = .55 + 3.85 = face incr. i cut deer. Center angles ' = 35.80 3.30 17.00 3.30 Face angles = 39.10 20.30 Center angles deer .... = 35.80 ... 3.85 17.00 3.85 Cut angles = 31.95 13.15 Pitch = 5) 1.66 5) 1.91 Diam. incr -j- p. diams .33 4. .38 2. o. diams. . . . 4.33 2.38 92 Chords of Angles. 164. SHAFTS AT BIGHT ANGLES. 1st. Divide the pitch diameter by that of the other gear of the pair, or else the number of teeth by that of the other gear, to get the proportion. Enter the table by means of the proportion. All numbers for that pair will be found on the same horizontal line in the two columns. 2d. The center angles are given directly by the table at the proper proportion. 3d. Divide the tabular angle increment by the number of teeth in the gear, to get the angle increment. This need be done for but one gear of a pair, as the increment is the same for both. 4th. Add the angle increment to the cen- ter angle, to get the face angle. 5th. Increase the angle increment by one- sixth of itself, to get the cutting decrement, and subtract this decrement from the center angle, to get the cutting angle. 6th. Divide the tabular diameter incre- ment by the diametral pitch, to get the diameter increment, and add that to the pitch diameter, to get the outside diameter. Fig. 148 is a sample computation for shafts at right angles. 165. SHAFTS NOT The table cannot be entered by means of the proportion, and the numbers for the two gears of the pair will not be found on the same horizontal line, and it will be necessary to determine the center angles. As in Fig. 147, draw the axes, at the given shaft angle, and find the center angles, by the method described in (162). Then enter the table, for each gear by itself, by means of the center angles, and proceed as for shafts at right angles. The angle increment and decrement is the same for both gears of a pair. Fig. 149 is a sample computation applied AT BIGHT ANGLES. to the case of Fig. 147, the center angles be- ing found by means of the table of chords. If preferred, the center angles can be found by means of the formula, sin. 8 tan. C = in which C is the center angle of the gear, P is the proportion found by dividing the num- ber of the teeth in the gear by the number in the other gear, and 8 is the shaft angle. Having found one center angle, subtract it from the shaft angle to get the other center angle. 166. THE TABLE OF CHOBD8 AT SIX INCHES. When the lathesman is provided with a graduated compound rest which feeds the tool at any angle, nothing but the computa- tion is required; but when there is nothing but the common square feed, the faces must be scraped with a broad tool. A templet for guiding the work can easily be made by means of the table of chords at six inches. To lay out a given angle, draw an arc with a radius of six inches, draw a chord of the length given by the table for the angle, and then draw the sides oc and ob of the angle boc, Fig. 150. For tenths of a degree use the small tables. The chord of 37.5 is 3.81 + .05 = 3.86 inches. Fig. 151 shows the manner of using the angle templet at the lathe. This table of chords is very convenient for many purposes not connected with gearing, and it is more accurate than the common horn or paper protractor. 167. BILGBAM'S CHABT. A graphical method for determining the angle and diameter increments, the invention of Hugo Bilgram, is described in the AMEBI- CAN MACHINIST for November 10, 1883. It determines the required values by the inter- sections of lines and circles, and requires no computation. Chords of Angles. Chord of * angle Fig. 150. Using the templet TABLE OF CHORDS OF ANGLES, AT RADIUS OF SIX INCHES. Degrees. Chord. Tenths. Degrees. Chord. Tenths. Degrees. Chord. Tenths. 1 2 3 4 5 .10 .20 .31 .42 .52 31 32 33 34 35 3.20 3.31 3.41 3.51 3.61 61 62 63 64 65 6.10 6.19 6.28 6.36 6.45 6 7 8 9 10 .62 .73 .84 .94 1.04 36 37 38 39 40 3.71 3.81 3.91 4.01 4.10 66 67 68 69 70 6.54 6.62 6.71 6.80 6.89 11 12 13 14 15 .15 .26 .36 .46 .57 .1 .01 .2 .02 .3 .03 41 42 43 44 45 4 20 4.30 4.40 4.50 4.60 .1 .01 .2 .02 .3 .03 71 72 73 74 75 6.97 7.06 7.14 7.22 7.31 .1 .01 .2 .02 3 02 16 17 18 19 20 .67' .77 .87 98 2.08 .4 .04 .5 .05 .6 .06 .7 .07 .8 .08 46 47 48 49 50 4.69 4.79 4.88 4.98 5.08 .4 .04 .5 .05 .6 .05 .7 .06 .8 .07 76 77 78 79 80 7.39 7.47 7.55 7.63 7.71 .4 .03 .5 .04 .6 .05 .7 .06 .8 .06 21 22 23 24 25 2.18 2.29 2.39 2.49 2.59 .9 .09 51 52 53 54 55 5.17 5.26 5.35 5.45 5 54 .9 .08 81 82 83 84 85 7.79 7.87 7.95 8.03 8.11 .9 .07 26 27 28 29 30 2.70 2.80 2.90 3.00 3.10 56 57 58 59 60 5.63 5.72 5.82 5.91 600 86 87 88 89 90 8.18 8.26 8.34 8.41 8.48 Templet Planer. 168. ROTARY CUT BEVEL TEETH. The most common method of forming the ! ehape of the tooth changes, while that of the teeth of the bevel gear is by cutting them cutter is invariable. Therefore the result from the solid blank by the use of the com- mon rotary cutter. The cutter should be shaped to cut the tooth of the correct shape at the large end, and the small end must be shaped either by an- other cut with a different cutter, or with a file. It is impossible to cut the tooth correctly at both ends, for the simple reason that the must always be an approximation depending upon the personal skill and experience of the workman. It is a too common practice to make the teeth fit at the large ends, and to increase the depth of the tooth toward the point, so that the teeth will pass without filing, but such teeth can be in working con- tact only at the large ends. ). THE TEMPLET GEAR PLANER. The most common method of planing the teeth of bevel gears is by means of devices adapted to guide the tool by a templet that has previously been shaped, as nearly as may be, to the true curve. The arm that carries the tool is hung by a universal joint at the apex of the gear, so that all of its strokes are radial, and a finger placed in the line of the stroke of the cutting point of the tool is held against the templet. There are many different arrangements for the purpose, but they are all founded on the same princi- ples, and differ only as to details. The invention of the templet gear planer is commonly credited to George H. Corliss, who patented it in 1849, and was the first to use it in this country. But it was patented in France, by Glavet, in 1829, and may be even older. It is largely used for planing the teeth of heavy mill gearing, but has not been, and cannot be, profitably applied to common small gear work. Its product is, in any case, superior to the rough cast tooth, but its accu- racy is dependent on that of the templet, and is therefore dependent on personal skill. 9. THE SKEW BEVEL OEAR. 170. THE SKEW BEVEL GEAR. When a pair of shafts are not parallel, and do not intersect, they are said to be askew with each other, and they may be connected by a pair of skew bevel gears, having straight teeth, which bear on each other along a straight line. Such gears are to be carefully distinguished from spiral gears, used for the same purpose but having spiral teeth bearing on each other at a single point only. We will endeavor to describe the skew bevel gear so that its general nature can be understood, but it is impossible to do so in simple language. It is the most difficult ob- ject connected with the subject. The theory cannot even be considered as yet settled, for writers upon theoretical mechanism do not agree upon it, and there are points yet in controversy. In the theory of the bevel gear the surface of reference is the spherical surface upon which the tooth outlines are drawn, and upon which the laws of their action may be studied, for spheres of reference of two sep- arate gears may be made to coincide so that the lines upon one will come in contact with those upon the other. For the spur gear, the spheres become planes and the process is the same. But for the skew bevel gear there is no analogous process, for it is impossible to imagine a surface of such a nature that it can be made to coincide with a similar surface when both are attached to revolving askew shafts. There are spiral surfaces which will approximately coincide, and are analogous to the Tredgold tangent cones of bevel gears (161), but any tooth action developed upon such approximate surf aces must, of necessity, be not only approximate, but also very diffi- cult to define and formulate. Of all the skew tooth surfaces that have been proposed, there is but one, the Olivier involute spiraloid, that can be proved to be theoretically correct. 171. THE HYPOID. The pitch surface of the ?kew bevel gear is the surface known as the " hyperboloid of revolution," and it is so intimately connected with the subject that it must be thor- oughly understood before going further. The clumsy name may be abbreviated to "hypoid." If a line D, Figs. 152 and 153, called a generatrix, is attached to a revolving shaft A, so that it revolves with it, it will develop or ' ' sweep up " the hypoid H in the space surrounding the shaft. A section of the surface by any plane normal to the axis is a circle. The com- mon normal to the generatrix and the axis is the gorge radius G, and circular section through that line is the gorge circle. A section by a plane D, Fig. 152, parallel to the axis, at the gorge distance from the axis, will be the pair of straight lines d and d ', Fig. 153, either one of which is an element of the surface, and will form it if used as a generatrix. A section by any sections. Fig. 152. Hyperbolic sections. Fig. 153. other plane parallel to the axis will be a hyperbola, to which the elements d and d' Rolling Hypoids. are assymptotes, or lines which the curves continually approach, but reach only at in- finity. Fig. 153 shows at Q the hyperbolas cut by the plane Q of Fig. 152, and at R those cut by the plane R. The principal hyperbola H is the only one with which we are concerned . The hypoid is best studied as projected upon a plane parallel to the axis, as in Fig. 154, in which A is the projection of the axis, d is that of the generatrix, dGA is the skew angle, and //is the principal hyperbola. When the skew angle and the gorge radius are given, the hyperbola is easily constructed by points. Any line ab is drawn normal to the axis and the gorge distance be= Gg is laid off from b, the distance ab is made equal to ec, and a is then a point on the curve. The curve is to be drawn through several points thus constructed. The hyperbola. Fig. 154. To draw a tangent to the curve at any point a, draw a line am parallel to the assymptote d, lay off mn equal to Gm, and draw the tangent an. 172. THE PITCH HYPOIDS. The utility of the hypoid as the pitch sur- face of the skew gear depends upon the pe- culiar property that any number of such surfaces will roll together, and drive each other by frictional contact with velocity ratios in the proportions of the sines of their skew angles, if their gorge radii are in the propor- tions of the tangents of their skew angles. It is required to construct a pair of rolling hypoids that will transmit a given velocity ratio between two shafts that are set at a given angle with each other. In Fig. 155, A and .Bare the given axes, and AGB the given shaft angle. The directrix D is to be so drawn that the sines of the skew angles AGD and BGD are in the proportion of the given velocity ratio, and this is best done by drawing lines parallel to the axes, at distances from G that are in the given ratio, and drawing the directrix through their intersection D. In the figure the axes are situated one over the other at a distance GH called the gorge distance, and the directrix D is situated be- tween them so as to pass through the gorge line and divide the gorge distance into gorge radii, #TFand HW, which are in proportion to the tangents of the skew angles. This is Pitch hypoids. Fig. 155. best done by drawing cd normal to GD in any convenient position, laying off the gorge distance ce at any convenient angle with cd, and drawing de and gf parallel to it; cf will be the goige radius GW for the axis GA, Rolling Hypoids. 97 and fe will be the gorge radius HW for the axis OS. Then, if the directrix, thus situated, is at- tached first to one shaft and then to the oiher, and used as a generatrix, it will sweep up a pair of pitch hypoids that will be in contact at the directrix, and which will roll on each other. They will not only roll on each other in contact at the directrix, but they will also have a sliding motion on each other along that line, the two motions combining to form a resulting motion that must be seen to be understood. It is this sliding motion that makes all the difficulty in the construction of the teeth, for they must be so constructed as to allow it. It is also the cause of the great inefficiency of such teeth in action, for any possible form must have a lateral sliding motion, with the consequent friction and destruction. If we draw two diameters mn and m'n' through the same point C on the directrix, they will be the diameters of circles that will touch each other while revolving, and may be called pitch circles. If they are thin, and provided with teeth in the given velocity ratio, they will drive each other with a con- tact that is approximately correct, and if there are several pairs of such thin gears set so far apart that they do not interfere with each other, they will serve light practical purposes fairly well. If a face distance CE is laid off on the directrix and another pair of pitch circles constructed, the frustra of the hypoids in- cluded between the circles may be called pitch frustra, and they will roll together in, contact at the directrix. It is to be noticed that the pitch diameters thus determined are not, as in spur and bevel gearing, in the inverse proportion of the velocity ratio of the axes, and therefore if one diameter of a pair of skew gears to have a given velocity ratio is given, the other must be constructed. When the skew angles are equal, the pitch diameters are equal, but otherwise the proportion cannot be expressed in simple terms, and must be determined by making the drawing. 173. THE LOCUS OF AXES. The rolling hypoids may be examined from another and most interesting point of view. In Fig. 156 the gorge line G is nor- mal, and the directrix D is parallel to the plane of the figure. The plane P is normal to the directrix, and below is a front view of it. On the plane P draw any straight line L through the directrix. From any two points a and b on this line draw lines A and B normal to the gorge line G, and they will be axes of pitch hypoids that will roll on each other in contact at the directrix. Axes drawn from all points of the line L will form a continuous surface called a "hy- perbolic paraboloid," which will be the locus of all the axes of a set of hypoids that will mil together collectively in contact at the directrix. The locus of axes. Fif/. 156. 98 Olivier Skew Bevel Gears. 174. CYCLOID AL TEETH FOR SKEW GEARS. As any number of hypoids, on axes in the same locus of axes, will roll together in either external or internal contact at the directrix, it might be supposed that a tooth similar to the cycloidal tooth for bevel and spur gears might be formed by an element in an auxiliary hypoid X, Fig. 156, which rolls inside of one and outside of the other pitch bypoid. This is such a plausible supposition that it long passed for the truth, not only with its inventor, the celebrated Professor Willis, but with many other prominent writers, until shown by MacCord to be wrong. It serves to illustrate the confusion in which the whole subject has been and now is. The tooth surfaces which Willis supposed to be tangent at the generating element of the auxiliary hypoid really intersect at that line, and Fig. 157 shows a pair of such in- tersecting teeth. The curves of the figure were drawn by an instrument made for the Cycloidal tooth Curves Fig. 157. purpose, and are, therefore, a better proof of the intersection of the surfaces than solid teeth would be. The cycloidal tooth is examined at consid- erable length, and the instrumental proof of its failure is given in the AMERICAN MACHIN- IST for September 5th, 1889. 175. INVOLUTE TEETH FOR SKEW BEVEL GEARS. Herrmann's form of the Olivier spirrloidal tooth is constructed with the directrix of (172) as a generatrix, as follows : Suppose that cylinders are constructed upon the gorge circles of a pair of pitch hypoids, Fig. 158, and suppose a plane K to be placed between them. This plane will be tangent to both cylinders, and will contain the directrix, and if moved will move the ^cylinders as if by friction. Then imagine ithe plane to move in a direction normal to the directrix, and it will carry that directrix with it as a generatrix always parallel to its first position. It will sweep up the spiraloid tooth surfaces 8^ and S a imperfectly shown by the figure, or by Fig. 159, and they will be correct tooth surfaces always in tangent contact. Fig. 159 shows #, full involute tooth sur- face or " spiraloid," and Fig. 160 is a full Olivier skew bevel gear. The particular involute skew tooth above described is not the only possible form, but it has the least possible sliding action, and is, therefore, the best. Involute tooth action Fig. 158. If the plane K has a generatrix line at any angles with the axes of the gears, and is moved in a direction at right angles with that line, correct tooth surfaces will be swept up. In fact, any two spiraloids on any two cylin- ders will work correctly with each other, and therefore any two spiraloidal gears of the same normal pitch will work correctly to- gether. Beetle's Gears. Olivier Involute skew Bevel Gear Fig. 160. 176. HERRMANN'S LAW. Herrmann gives a law, and claims it to be universal, to the effect that the skew bevel tooth must be swept up by a straight line generatrix that is always parallel to the direc- trix. He mentions the Olivier tooth, and claims that it cannot be correct, evidently not understanding that Olivier's theory clearly includes the form he himself proposes. His form of tooth, claimed to be the only possi- ble form, is really only the best form of the Olivier tooth. We will not undertake to state wherein Herrmann's law is incorrect, but that it is wrong is clearly shown by the most con- vincing of all proofs, the reduction to prac- tice. Beale, for the Brown & Sharpe Mfg. Co., has made working Olivier gears on a large scale, which are directly contrary to Herrmann's law, but which work perfectly, and demonstrate the truth of Olivier's theory in a way that admits of no question. Indeed, the closest possible scrutiny of Olivier's theory, without the aid of Beale's experimental work, fails to detect a flaw in it. Herrmann's condemnation of it is not based on direct consideration, but simply on the fact that it does not agree with his own law. 177. BEALE'S SKEW BEVEL GEARS. . Beale's gears are the same as Olivier's gears in general theory, but the improvement in practical form and application is so great that they may be considered a distinct invention. Fig. 161 is a section through one axis, and at right angles to the other axis of a pair of Beale gears. Both surfaces of the teeth are true Olivier spiraloids of Fig. 159, and the gears will run in either direction. When corrected for interference they are reversible, like spur or bevel gears. The gorge cylin- ders are tangent to each other, and are so cut away inside as to allow the teeth of the ma- ting gear to pass. The Olivier theory requires the teeth to ! vanish at the gorge, as shown by the single j full tooth of Fig. 160, in order to pass, while i the Beale gear is cylindrical in form as a I whole, and passes the full tooth at the gorge, with action over its whole surface. The ; difference is practically very great. When in action a pair of uncorrected Beale ! gears must be placed as shown by Figs. 161 to 163, and Fig. 169, with one end of each at I the gorge, and they will not run together if placed at random. If either gear extends beyond the gorge line there is an interfer- ence between the involute spiraloids which is the same in kind as that between the in- volute curves of common spur gear teeth. 100 Beetle's Gears. In comparison, the Beale gear is taken so near the gorge that it is practical and service- able, having large teeth and small obliquity. Fig. 161. Section of Beale's Gears Each gear can drive in but one direction, depending upon the position of tbe gear and the direction of the spiral, and if turned backwards the action is intermit- tent and practically useless. The gears must be placed as in Fig. 162 for right-hand spirals, and as in Fig. 163 for left-hand spirals, and the direction of the rotation is shown by the arrow D, when the gear bearing the arrow is the driver. But, if the direction is to be re- versed, the gears can be arranged gorge c y u. as in Fig. 162a, or as in Fig. 163a. This resetting is the same in effect as turning the gears half around, except that opposite sides of the teeth are in contact in the two positions of the same gears. If, however, the interfering parts of the tooth surface are removed, the gears will run together per- fectly and in either direction when put together at random as in Fig. 168. In the cases shown by the figures, the spirals make the angles of forty- five degrees with the shafts, con- trary to Herrmann's law, but the action will be smoother, and the sliding of the teeth on each other will be less, if Herrmann's angles are adopted. These angles are the same as those made by the conical face of common bevel gears of the same proportion with the axes, and the best angles for the two-to-one proportion of figures are those of the line X of Fig. 162, making the angles 26 34' and 63 26' with the axes. The Olivier gear of Fig. 160 is perfect in theoretical action, but the teeth must be taken so far from the gorge that the obliquity of the ac- tion is excessive, and the arc of action is so limited that the teeth* must be small. The sliding and wedging The working length of each gear is as action is so great that the gears are practically determined by the line L of Fig. 161, and useless. ! the whole surface of the tooth within that 1 Fig. 162a. Beale Skew Bevel Gears. Right Hand Spiral at 45 i Fiy. 163a. Left Hand Spiral\at 45 Approximate Skew Teeth. 101 limit will be swept over "by the line of contact. It the length of each gear is equal to the radius of the other gear it will always be long enough. The action between two gears will be at the straight, equidistant, parallel lines a a, Figs. 161 and 162, in the plane of action tangent to both gorge cylinders. The shafts of a pair of skew bevel gears should be as near together as possible, just far enough apart to allow the shafts to T ass, so as to avoid the excessive sliding action. In that case both Beale and Olivier gears are practically useless, the former on ac count of the small size of the teeth, and the latter on account of the great obliquity of the action. The common bevel gear becomes the spur gear when the shaft angle becomes zero, but the analogous transformation of the skew bevel gear into a bevel gear by reducing the gorge distance to zero is not possible. The skew bevel gear becomes a spur gear if we imagine the axes to be brought parallel by removing the gorge to an infinite distance, for the spiraloids on the gorge cylinders then become involute surfaces on base cylinders. But, and it is a curious circumstance, when the shafts are brought parallel by imagining the shaft angles to become zero without changing the position of the gorge, the gorge cylinders become tangent and the gears do not become spur gears. Involute skew bevel gears do not appear to have any possible adjustment correspond- ing to the adjustment of the shaft distance of involute spur gears, or of the shaft angle of involute bevel gears, (56) and (157). Beale's gears are fully described in the AMERICAN MACHINIST of Aug. 28th, 1890. 178. TWISTED SKEW TEETH. As no two surfaces of reference attached to a pair of revolving askew shafts can be made to coincide with each other, like the planes of spur gears or the spheres of bevel gears, the twisted or spiral tooth is impossi- ble, for such a tooth would not permit the required sliding action. But, if a line is drawn upon one pitch hypoid of a pair, a corresponding line may be drawn upon the other, as if the given line could leave an impression. Therefore a tooth having edge contact (100) may be constructed, provided the twist is such that one pair of lines always crosses the directrix. These teeth are purely imaginary, but if the edges are thick they will have an action upon each other, at a single point of contact, that is closely approximate to the theoretical action, and they will serve the general pur- pose, if the power carried is inconsiderable. 179. APPROXIMATE SKEW TEETH. As the true involute skew tooth is diffi- cult to construct, and in many cases is of small practical utility, and all other proposed forms are incorrect, it follows that we must depend for practical purposes mostly upon some approximation, provided it is not pos- sible to avoid the skew gear altogether. The blanks can be constructed by a definite process Construct the frustra of the pitch hypoids by the method of (172) and Fig. 155. Consider the end sections mn and pq as ends of a frustrum of a pitch cone, and on this pitch cone construct the blank gear exactly as for a common bevel gear. Having constructed the blanks, the general direction of the tooth is to be marked upon them. Mount each blank as in Fig. 155, with its axis parallel with a plane surface Z. Set a surface gauge with its point at the line of the directrix W, and with it mark the po- sition of the directrix on the pitch line at each end of the blank. The tooth must then be cut so that its direction follows the directrix, and it is to be 102 Substitute Skew Trains, noticed that it is not only askew with the axis, but that the tooth outline twists. The appearance of the tooth on either rim, as well as upon any section between the two rims, is the same as upon a common bevel gear, symmetrical, and not canted to one side, as is sometimes taught. The approximate tooth is very similar to the twisted bevel tooth, see (155) and (99), with the twist following a straight line set askew with the axis, and as the line of the twist is not parallel with the conical face, that face should be as short as possible. The process of cutting is not capable of description, for it depends upon personal skill and judgment. The workman must imagine that he sees the twisted cut in the body of the blank, and then must persuade the cutter to follow it. Gear cutting ma- chines are seldom so made that the cutter can be turned while it feeds, and theretore i it must be set to a medium path, and reset | two or three times to get the desired form. The beginner will fail the first time, and there may be several failures. The best pos- sible result can be bettered with a file, after running the cut gears together to find where they interfere. In the hands of a skillful workman, a pass- able approximation can be reached, and if the axes are very near together compared with the diameters of the gears, the teeth are small, and the face is short, the result is satisfactory. In fact, when the conditions are favorable, this approximate tooth is more serviceable than the true tooth. Substitute train. Fig. 165. 180. SUBSTITUTES FOR THE SKEW BEVEL GEAR. When there is a chance to introduce an intermediate shaft, the skew bevel gear can be avoided, and it is not only better, but cheaper to avoid the objectionable gear at the cost of the extra mechanism. Fig. 164 shows how to place an inter- mediate shaft and gears, when the shafts are so far apart that the shortest or gorge distance can be used. Fig. 165 shows how the skew shafts can be connected by one pair of bevel gears and one pair of spur gears, and that is the best device for general purposes. Substitute train. Fig. 164. Skew Pin Gears. 105 181. SKEW PIN GEARING. Fig. 166 shows a pair of skew pin gears commonly called face gears. They will run together with a uniform velocity ratio if they are exactly alike and at right angles with shafts at a distance apart equal to the diameter of the pins. If the gears are not alike or not at right angles, the teeth on one may be straight pins, but those on the other must be shaped to correspond. Such gears are objectionable because they have but a single point of contact for each pair of teeth, at which they slide on each other with great friction. Face gearing in its various forms is thor- oughly examined in MacCord's Kinematics. At the present day they are not in use, and do not deserve much study. Skew pin Ttevel gears. Fig. 206. Beale gears corrected for interference. Fig. 168. Formation of Beale gear. Fig. 167. UncorrecUd Beale gears. Fig. 169. INDEX. SECTION. Addendum 39 Arc of Action 25 Axoids 7 Backlash 41 Base Circle 16, 54 Beale's Gears 177 Beale's Treatise 4 Begin at the Beginning 3 Bevel Gears 6,7,8, 152, 154 to 169 Box's Mill Gearing 4, 51 Brown & Sharpe Co.'s Treatise 4 Cast Gearing, Friction 49 Chart for Bevel Gears 163 Chordal Protractor 166 Clearance 40 Complete Tooth 22 Conic Pitch Lines 131 Conjugator 125 Consecutive Action 12 Construction by Points 23, 57 Cusp 16, 17, 54 Cutter Limit 88 Cutter Series 45 Cycloidal System 31, 76 to 90, 158, 174 Dedendum 39 Demonstrations Avoided 2 Disputed Points 9 Double Secondary Action 21 to 78 Double Terminal Action 18 Edge Teeth 100 Efficiency in Transmission 49, 73, 112 Elliptic Gears 136 to 153 Elliptic Gear Cutting Machine 150 Elliptic Pitch Lines 131 Elliptographs 138 Epicycloidal Teeth 76 Equidistant Series 45 Extent of the Subject 2 Fillet 28, 44 Friction 26 Friction of Approach , 48, 49 Gear Cutting Machines 27, 28, 29, 94, 102, 105, 122, 125, 150, 159, 168, 169 Gear Teeth, Theory 1 to 34 " Spur 35 to 52 " Involute 53 to 75 Cycloidal 76 to 90 Pin 91 to 97 Spiral 98 to 110 Worm Ill to 125 " Irregular 126 to 135 Elliptic 136 to 153 Bevel 154 to 169 SkewBevel. 170 to 181 SECTION. Gears, Beale's 177 Bilgram's 159 Composite 133 Elliptic Bevel 152 Herrmann's 175 Hindley 122 Hooke's 98 Hyperbolic 181 Lantern t>3 Mortise 47 Parabolic 181 Pin Bevel 1(50 " Skew Pin 1S1 Stepped JH Herrmann's Erroneous Law 17(1 Herrmann's Treatise 4 Hindley Worm Gear 122 Robbing Machines 124, 125 Hobbing Worm Gears 114, 115 Hooke's Gears 98 Horse Power of Gears 51, 52 Hunting Cog 46 Hyperbolic Pitch Lines 131 Hyperboloid of Revolution 7, 171 Hypoid 171 Integrater, Odontoidal 34 Interchangeable Odontoids 14, 22 Interchangeable Rack Tooth 22 Interference 1 6, 55 Interference, Internal 79 Interference, Worm 117 Internal Contact 15 Internal Double Action 21 Internal Gears 64 Internal Friction 49 Involute System 31, 53 to 75, 157, 175 Irregular Pitch Lines 126 to 128, 133 Kinematics Klein's Treatise Klein's Odontograph 85 Law of Tooth Contact 11, 12 Limiting Numbers of Teeth 66 to 72, 90 Limit Line 16 Line of Action 13 Literature 4 Logarithmic Pitch Lines 132 Logarithmic Spiral 32, 75, 132 MacCord's Treatise 4 Molding Construction 27 Mortise Gear 47 Multilobes 130 Natur Tooth Action 20 Normals 11 Normal Surfaces 8 SECTION. Obliquity of Action 26, 74r, 87 Octoid Teeth 31, 159 Odontics 6 Odontographs 43, 59, 62, 82, 83 Odontoid and Line of Action 33 Odontoids 12 to 34 Olivier's Spiraloidal Teeth 175 Parabolic Pitch Lines 131 Parabolic System 31 Pin Tooth System 91 to 97, 160, 181 Pitch, Circular 35, 119 Pitch Cylinders 10 Pitch Diameters, Table 35 Pitch, Diametral 36, 120 Pitch Lines 11 Pitch Point 11 Pitch Surfaces 7 Pitch Table 37 Planing Construction 28, 29 Quick Return Motion 149 Radial Flank Teeth 89 Rankine's Treatise 4 Rack Originator 30, 31 Retrograde Action 18 Reuleaux's Treatise 4 Robinson's Odontograph 86 Rolled Curve Theory 32, 75, 81, 91 Roller Teeth 93 SECTION. Secondary Action 21, 78 Segmental System 31 Sellers' Experiments 49, 112 Skew Bevel Gears 6, 7, 8, 170 to 181 Smallest Pitch Circle 17 Speed of Point of Action 19 Spiral Gears 98 to 103 Spur Gears 6, 7, 8, 35 to 153 Stahl & Wood's Treatise 4 Standard Teeth 42, 58, 80 Stepped Teeth 98 Strength of Teeth 49 Systems of Teeth 31 Templets 38 Terminal Point 18 Twisted Teeth .99, 101, 102, 15ft Unsymmetrical Teeth 22 Variable Speed 148 Willis' Odontograph 84 Willis' Treatise 4 Wooden Teeth 47 Worm Gears ..Ill, 113, 114, 116 Yale & Townes Experiments 112 MICHIGAN BRICK AND TILE MACHINE CO. MORENCI, MICH., Nov. 24th, 1891. GEORGE B. GRANT. Dear Sir : Two years ago you sent me one of your books on Teeth of Gears, and I have replaced all of the gears in our brick machinery with new ones from your in- volute odontograph table. I find that we now have the finest cast gears in the world. I do not understand why pattern makers don't catch on to your book. It is a sight to see the gear pat- terns that are made by some men who are called good pattern makers. O. S. STURTEVANT, Pattern Maker for M. B. & T. M. Co. ADVERTISEMENT IT'S NO USE TRYING TO GET ELLIPTIC GEARS OF ANY ONE BUT - \ CEO. B. GRANT, LEXINGTON, MASS. PHILADELPHIA, PA. -n] O ./*/ SHi/^/ UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. LD 21-100m-7,'52(A2528sl6)476 F THE UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA Q) - P ^ s i s 1 Ps YC 33177 E UNIVERSITY OF CALI fRNIA LIBRARY OF i UNIVERSITY o. E UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA I 1 E UNIVERSITY OF CALIFORNIA LIBRARY OF THE UNIVERSITY OF CALIFORNIA