US2031888A - Tooth curve for rotors and gears - Google Patents

Description

Feb. 25, 1936. M. F. HILL I TOOTH CURVE FOR ROTORS AND GEARS 2 Sheets-Sheet 1 Original Filed Aug. 24, 1928 ifiw TnR mwmm Feb. 25, 1936. M. F. HILL v 2,031,888

' TOOTH CURVE FORROTORS AND 8848s Original Filed Aug. 24, 192 3 2 Sheets-Sheet 2 INVENT UR Jvlu mfl 112w,

Patented Feb. 25, 1936 UNITED STATES PATENT OFFICE TOOTH CURVE FOR ROTORS AND GEARS Myron F. Hill, Lansdowne, Pa., assignor to Hill Engineering Company, Inc., a corporation of New York 36 Claims.

This case is a division of my application for patent Ser. No. 310,880, now Patent No. 1,833,993 and contains in addition to the matter divided out .of that patent, additional matter, the subject of further invention. I

My invention relates primarily to internal rotors, that is, two rotors one inside the. other; and having tooth divisions, one preferably having one less tooth division than the other, the teeth of each making continuous contact with the contours of the other during rotation at steady angular velocity.

My invention comprises broadly a specific form of rotor or gear contour in which the teeth have curves based upon a specific type of trochoid, that is, a trochoid generated by a point carried by one pitch circle as it rolls on the other pitch circle and outside of it. The use of the terms cycloid and trochoid in this case are intended to have difierent meanings. Some authors and dictionaries confuse them. The gear art at one time employed curves traced by points of circles rolling upon straight lines or on other circles, and these curves were generally known as cycloids. They were not known as trochoids. Therefore the term cycloid in this case refers to the curves thus well known. The term trochoid is limited to curves traced by points not on circles but carried inside or outside of them as they roll on other circles.

As the geometry upon which my invention is based is novel, new names help to define its factors.

Theterm circroid refers to that class of trochoids outlined by points carried by pitch or ratio circles rolling within or outside of other pitch orratio circles. There are hypocircroids and epicircroids.

An epicroid is a curve alongside of a circroid such as an envelope, developed by a master form travelling along the circroid. It is not necessarily equidistant from the circroid.

A "phypocroid is a curve which may be outlined by a circular form whose center follows a hypocircroid, the first p indicating parallel to or equidistant from.

A. prepicroid is an envelope parallel to or equidistant from a circroid, such as that developed by a circle whose center'follows a circroid, the letters pr suggesting parallel.

In Fig. I, the curve I2 is an epicircroid,- the curve 3 is an epicroid, and being equidistant from l2, also a prepicroid. In Fig. IX the curve 36 is-an epicircroid, and the curve 33 is a prepicroid.

RElSSUED JAN 9 1940 If the master form So, Fig. IX, is irregular, the curve 33 generated by it is merely an epicroid.

It is thus seen that this invention relates to certain species of trochoidal curves outlined by points moving with ratio circles as they roll with relation to other ratio circles, and that it involves pairs of curves, mainly of the epi system, in novel mated combinations.

In the drawings:

Fig. I illustrates one form of my rotors.

Fig. II shows a section of Figure I on line IIII.

Fig. III shows a milling cutter used as a master form in generating rotor contours.

Fig. IV shows another form of tool which may be used as a master form with which to generate a complete rotor contour.

Figs. V and VI show other forms of tools, with curves such as are generated by the master form on the inner rotor, which are capable of gen- In Fig. I are shown two rotors, the inner rotor I on the axis 4 having tooth crowns 2 and tooth spaces 3; and the outer rotor 5 having tooth crowns 6 and spaces 1, centered upon the axis 8. The teeth of these rotors make continuous travelling contact with the contours opposed to them during relative rotation at uniform angular speeds determined by their ratio of teeth. These speeds are proportional to their pitch circles A and B, as they roll in contact with each other with one outside of the other and without slip at the point of contact.

It will be noted that the pitch circles of these particular rotors lie well inside of their actual teeth. The crowns of the teeth of the outer rotor correspond to the radius of curvature of the master circle.

The master circle used for generation of the rotors in Fig. I may be in the form of a milling cutter 9 (Fig. III) shown in generating positions for generating the inner rotor I at l0 and II. As this cutter generates the curve 2, 3, of rotor l its center follows the trochoidal curve l2 called a circroid.

The master tool may also generate a tool form having a contour of the inner rotor on a blank of which may be used as a mating to'ol swinging around the center i of the inner pitch circle and carried with it during rotation, cutting the tooth crowns and spaces 6, l of the outer rotor directly in the outer rotor blank. Such tooth cro'wns are the same in curvature as the master circle.

If the outer rotor drives theinner rotor clockwise the teeth of the rotors engage each other at i5, i6, ll, I8 and near ill (the last contact just opening) at first with pressure relations everywhere noted; but after being burnished and ready for service with pressure only in the driving range represented by the double headed arrow. Elsewhere the contacts are without pressure; and friction and heat cease, yet sufiicient contact to one rotor hitting the bottoms of tooth spaces of the'other rotor at full mesh. A thousandth of an inch more or less in the bottom of the tooth spaces is enough for this purpose.

When the master form 9 is circular the contour 2, 3 is equidistant from the circroid B2.

The rotor curves in Fig. VII present advantages over those in Fig. I. It will be noted that the contours are intersected by the pitch circles of the two rotors Al and BI, centered at al and bl respectively, as they are based upon circroids external to them.

The master milling cutter is illustrated at 201 in the position of a tooth 2| of the outer rotor 22, the latter being centered upon the axis bl. The tooth spaces of this rotor are narrow as shown at 23. The inner rotor is generated by this milling cutter as it is carried by the pitch circle Bl, while a blank carried by the pitch circle AI has contoursgenerated 'on it providing teeth 24 and tooth spaces 25. In this case the teeth of the outer rotor have. greater angular length, andgeneration of tooth spaces is some- 1 times unimportant.

If desired, the spaces of the two rotors may also be undercut as illustrated, at 26 and 27, enough to prevent a contact with the top of a tooth crown of the other rotor at full mesh.

It will be noted-that the radial slide of a tooth 2 3 into a tooth space 23 in the'range of the double headed arrow is not much different from that between the teeth in Fig. I, yet the driving range in Fig. VII extends further across full mesh. Larger capacity results from a smaller number of teeth and greater eccentricity for the same diameter of rotor curves.

After rotors or gears have been machined they may be run in, that is, operated under service conditions until the rough surfaces have been polished oif, and the pressure outside of the driving range has .been mostly eliminated. In the form shown in Fig. I the surfaces on the two! range.

ti'ons are possible.

like that shown in Figs. IV, V, VI, the contours the metal to the point of freeing rotation until cool enough for use. In the form shown in Fig. VII the outer rotor teeth are preferably broached so as to acquire a smooth polished surface, and only the crowns and corners of the teeth of the inner rotor have to be burnished, that is, abraded and polished, to provide the pressureless slide peculiar to these rotors outside of the driving With the narrow teeth in Fig. VII the burnishing action is acomplished in a few minutes and the master form curve of the teeth of the outer rotor not affected. The teeth. of the rotor in Fig. VIII areenlarged. The corners 28 and 29 for broaching purposes may be a curve of small minimum radius, perhaps a thirty-second of an inch or even less. This curve is determined by the curtate or circroidal addition, the distance from Z to Y in Fig. IX. The greater the circroidal additional, other factors remaining constant the larger is the curve at the corners 28, 29. The top of the tooth at 30 is comparatively flattened, having a large maximum radius of curvature admirably cooperating with the enlarged curve of the teeth of the outer rotor at open mesh, as illustrated at 3!, Fig. IX, in broken lines, to maintain tightness between high and low pressure ports in a pump or motor.

These tooth relations-are interesting from another angle. In Fig. I, if the contours are reduced in diameter to the shapes shown in Fig.

VII (using the same master tool 9) the contours may be intersected by the pitch circles A and B. In other words the rotors have smaller contour diameters and hence smaller capacity. The capacity however may be restored by increasing the numbers of teeth of the rotors until the diameters in Fig. I are restored. This will increase the numbers of teeth and the numbers of rotor chambers between the teeth over those shown in Fig. I. In air compressors rotor chambers are separated from each other by tooth contacts on the closing side; and with a larger number of rotor chambers between atmospheric pressure and the discharge port, the difference in pressure of the by using a master form having the shape and size of a tooth of. the inner rotor, and generating the outer rotor with it, thereby producing the contours described in the outer rotor, and then using a tooth form of the outer rotor with which to generate the inner rotor contour. This is merely reversing the process after finding the inner tooth contour. The result is the same as heretofore described.

When starting to generate rotor curves on a blank, the mill 9 may be located on the center line thru the axes, though other starting posi- If it is started as at the bottom of Fig. I it is fed vertically toward the axes until it touches the point 3| of the top of a prospective tooth. If it is located at the top of a blank as at Z, Fig. I, it is fed vertically downward into the blank until it reaches the bottom of a prospective R. P. M. or the like.

preliminarygeneration on larger diameters is possible and often advisable,

I find cast iron of the gray type to be admirable for my rotors since in service is acquires a glaze that is hard and durable. ever may be used as well as other materials.

As to numbers of teeth, under otherwise equal conditions the fewer the teeth the greater the abrasion or wearing effect in the driving range, and the more quickly rotors deteriorate in ordinary service. I have found rotors having a five to six ratio about two inches in diameter, very serviceable in handling fuel oil at low pressures; anda six to seven ratio, four inches in diameter, wonderfully durable with gasoline at low pressures, both ratios running at motor speeds of 1725 A ten to eleven ratio in small size rotors two and a half inches in diameter for example, has been found practically indestructi-.

ble at these speeds and at hundreds of pounds pressure in fuel oil. V

A seven to eight ratio in an air compressor having a displacement of 50 cubic feet per minute at the same speed provides a durable construction for fluid pressures up to 150 lbs. The greater the number of teeth, other conditions being equal, the less the capacity, and the greater the durability. The area of the rotor chambers increases with either the eccentricity or the diameter.

In the generating machine in the patent of Hugo Bilgram and myself No. 1,798,059 the method of operation established by the gearing produces phypocroid contours as described above. The contours in this present invention are termed prepicroids and may be produced by the machine in Patent No. 1,798,059 by changing the gearing between the worm shafts so that they turn in opposite directions to each other. This is accomplished by removing the gears 38, 44, and 40 in Fig. 2 of that patent and substituting two gears meshed directlytogether. These substitute gears have the same relative pitches, as 38 and 40, and being mounted on and keyed to the shafts 31 and 39 of the worms, the shafts are rotated at the same relative speeds but in opposite directions to each other. Instead of a blank for the outer rotor being 'carried by shaft 23, a

blank for the inner rotor is substituted, the milling cutter l2 (corresponding to cutter 9 in this case) is fed radially inward until the right diameter for the desired tooth form is reached. Without limiting my invention to any specific numbers of teeth, master tool and size, the following dimensions have been used with good results. They apply to the rotors in Fig. VII. .The distance between the centers of the two rotor pitch circles is 3 when the outside diameter of the inner rotor is 44, and the master circle diameter is 26. The unit of measurement may be centimeters or inches or any other unit. In Fig. VII the unit is intended to be a sixteenth of an inch. In such rotors the radial depth of a rotor chamber when full open is.12.

The capacity of a pair of rotors is approximately the product of the thickness times the area of an annulus between two circles, one touching the crests of the tooth of that rotor which is attached to the driving shaft, and the other touching the' generated bottoms of the tooth spaces of the same rotor.

The geometry of my invention is illustrated in Fig. IX.

My rotor contours may have geometrical outlines produced by three elements, namely, two

rolling pitch or ratio circles which may be A2 Any other metal howand B2 and a master curve located upon a center Z which is upon the radius (extended) :0; this radius in this figure being also the center line thru the axes a2, D2 of the rolling circles. These elements are sometimes assisted by a fourth derivative element, a mating curve, such as the convex tooth top of a rotor contour generated by the said master curve, as D.

Let these two rolling ratio circles be located upon a plane, one outside of the other and tangent to it. Let their diameters be in proportion to the numbers of tooth divisions selected for the two rotors which preferably should differ by one.

' A master curve 9a to represent the desired tooth form of the outer rotor is selected to start the curve generation with, and preferably located upon the radius of the pitch circle A2 of the rotor which it represents the tooth of.

One pitch or ratio circle carrying the master form is rolled with the other ratio circle without slip at the point of tangency producing uniform angular motion of one with relation to the other, i

that is, of A2 to B2. This steady angular motion is determined by the relative diameters of the two rolling ratio circles which are proportioned to the numbers of teeth of the two rotors. The master form carried upon the radius of the outer circle A2 may be traced with relation to the inner circle B2 in each successive relative position that it assumes,'thus producing a series of overlapping curves, as illustrated at E. The contour 32,

v 33, is a curve drawn along the crests of these overlapping outlines,a curve of envelopment, which is the tooth and space contour of the inner rotor. In all such curves the instant centers of curvature must lie sufficiently outside of the ratio circle to prevent the generation of undercut curves on the driving surfaces of the gears and the fluid pressure holding surfaces of the rotors. The actual distance may be determined as heretofore described. A portion of this contour, as illustrated at F, preferably the convex portion of the tooth of the inner rotor, may then be carried upon the extended radius F! of the rolling circle 32 cen-' tered at b2, and the rolling action continued in the same way. Its form is outlined in each successive position which it assumes with relation to circle A2 as illustrated at G and a curve of envelopment drawn along the crests of the out lined curves. This curve of envelopment is the tooth and space contour 34, 35 of the outer rotor. Instead of a complete master circle 9a, a portion of it or any other master form may be used.

The rolling circles A2 and B2 are, of course, the pitch circles of the two rotors, having diameters proportional to the numbers of teeth. The distance of the center of the master curve Z to the pitch circle that carries it, A2, is termed the circroidal addition, since the circroidal path 36 followed by the center Z during the aforesaid generation is a species of 'trochoid; or a series of them, one for each tooth division. By locating the center further out on the radius 1' as at I In this figure, the inner circle rolls seven times toothcurves of the rotors or gears may occur.-

If the teeth of one are a thousandth of an inch wider than the tooth spaces of the othenthey would have to be driven together by force. Suchminor variations may be compensated by making the curves loose angularly, that is, by providing back lash, betweenv them.

One way by which this may be accomplished is byforming both rotor or gear curves as described above and then side stepping, or angularly turning, the generated rotor a few thousandths more or less and regenerating its contours. When the inner rotor is' so treated the mill describes a curve illustrated at ii, 62. The portion dl is in the air, no metal being removed; but the portion i2 is the result of removing metal,-so that the original toothcurve is shifted angularly around the center I)! to'this broken line position. The pinion that drives the gear on the worm shaft 39 of the Bilgram Hill generator after loosening the gear at that it carries, has been shifted with relation to the gear 3 turns and the gear refastened to the shaft to accomplish this result.

Either side of the teeth may then be employed in the driving action according to direction of r0- 7 tation and drive. The very middle of a tooth the broach may be provided with a plurality of might momentarily break contact slightly with the opposingrotor tooth in the position at 3| Fig. Di, If this parted contact in fluid'mechanism permits leakage in fluid operations, ports may, if desired, be correspondingly modified to prevent it. Rotors so made prevent foreign matter from wedging between the teeth. Leakage at open mesh for a few degrees at motor speeds is'not discoverable.

I find the broaching method a most-convenient one for manufacture of smaller rotors. The contour of the outer rotor is formed on a tool steel blank which is then operated as a broach, with which to broach the outer rotor.; The blank should be prevented from stretching during the breaching operation. The tool steel blank ior cutting edges formed in the same" way, some smaller than others for entering the hole in the blank in "accordance with common broaching practice. v 1

The inner rotor too may be broached, by cutting or generating the contour of the inner rotor on tool steel broaching blanks, trimming them to a burnished rotor curve, and employing them as masterbroaches with which to broach hollow dies assembled as a hollow broach with which to broach the inner rotor.

In order .to manufacture rotors or gears that may be too large for practicable broaching oper-' ations, a tool may have a curve representing in reverse, the contour of a single tooth division of the outer rotor. Preferably the curves wouldbe arranged like that in Fig. VI and correspond in form to the correct rotor curve. Such a tool may be mounted in a vertical indexing gear shaper and indexed around for the operation of cutting the successive tooth divisions in the outer step" the rotor curve as indicated at 63 and id amnes a "inwhich the convex tooth curve of the inner rotor conformed to a circle and other contours generated therefrom; another in which the convex tooth curve of either rotor was a simple cycloid or other form of varying radius of curvature, and other contours generated therefrom; and the third, the system, as illustrated in this case, in which the convex tooth form of the outer rotor has the circular characteristic. The first was claimed specifically in'that patent; the cycloid in my application 551,079; and the others including the prepicroid in this patent.

One great advantage of the 'prepicroid system ,over the others is that the tooth driving range,

when the middle portions of a tooth curve of the outer rotor and the space curve of the inner rotor coincide, extends across the centerline at full mesh as indicated by the double headed arrows in "Figs. I, and VII, reducing the radial slide and friction (as illustrated particularly in Fig. VII) between the teeth under driving pressure. Also the sharp convex curves of cycloids engaged in the driving range are avoided. This makes possible fewer teeth, greater eccentricity and larger capacity, in rotors having, the same outside diameter as noted. In other words the action in the driving range is nearer to a pure roll of the convex teeth of one rotor on the concave teethof the other.

The convex side of the teeth of the master rotor or gear engage the generated sides or the teeth of the mated rotor or gear across the driving range, the length from one tooth contact to 'the next one, in the full mesh region, presenting a driving and rolling action between them, entraining a lubricant which may distribute pressures between teeth over a wider area than is possible with convex gear teeth. The master form is generally convex, and the mated form generally concave. While the concave sides'of the mated gear or rotor, under certain relations and sizes of parts, may begin with circular portions near the middle of the tooth space, they nevertheless ratio circle of the other rotor is outside. After the tooth form has been generated on the mating rotor within it, it may be employed to generate the original master rotor curves whose tooth forms were adopted in the first place to start generation with. In my Patent 1,682,563, granted August 28, 1928, the system of hypof generation was specifically claimed, in which the ratio circle of the inner rotor carried a circular master form which was employed to generate a mated curve upon the inside of the outer rotor, the ratio circle of the inner rotor rolling around in the ratio circle of the outer rotor. The relations were such that about half the height of a tooth of the inner rotor, the outside half, was circular. while the tooth space, including the inner half of the height of the tooth of theinner rotor had a curvature of varying instant radii. That crown and space curve is hereby disclaimed in this case. Some of the results of. this invention however may be attained by reducing the size of the master circle, to perhaps aquarter or fifth of the diameter of that shown, varying the circroidal addition to suit, and then generating the curve of the outer rotor. With enough teeth a relation is arrived at in which all the driving relation becomes that of a convex against a concave tooth, which is a primary feature of this invention.

Nevertheless, the driving range stops short of the center line and is inferior to the driving range of the epi system, with a circular master form. For this reason the preferred curves are generated by the epi system. The hypo system of curves is automatically limited to internal gears or rotors, while the epi system is not so limited.

I have sometimesreferred to my preferable rotor contours as circular. Contours may be generated by master forms of many shapes, and the circular contours are those generated by a circular master form. In such case the curves 34, teeth of the outer rotor, are arcs of circles, and the curves 32 and 33 may be generated thereby; The curves 33 are not exactly circles, since the center Z of the circular master form 90. which generates them follows the epicircroid 36 and in so doing travels around on the cusps between the loops. It has been explained that the point Z must be outside of the pitch circle A2 a sufficient distance to prevent undercutting the curve 33. (This means that lines between the curve 32, 33, to the circroid 36, normal to both must not cross each other between the curves, and that with circular master forms the curves 32, 33 and 36 are equidistant.) The further away from A2 that the point Z lies, the flatter the cusps, and

the more angular motion the point Z makes while rounding the cusp, and the further away froma circle the curve 33 becomes as it is being generated by the circle 90.

In'my teeth the bases preferably are wider and,

the tops narrower so that the angles of pressure, which tend to break involute gears, are in my gears more strongly resisted.

The driving angle between the teeth in the driving range becomes. less and less easy, and if carried far enough would jam. Another disadvantage of larger cusps is the increased diam-' eters. As'thetooth curves lie ever farther from the pitch circles, they have to rub and slide to constantly increasing amounts. Hence the smaller cusps and corresponding inner rotor space contours are superior for service conditions. One important factor of the contours specified in this case is that the driving range, which in some of my earlier inventions stopped short of the center line thru the'rotor axes (at full mesh of course) now lies on both sides of the center line. Since the driving range which is always the finally effective factor, is but the length of a tooth di-. vision, this range, divided on both sides of the center line, involves less radial slide between the teeth since in thisrange the pitch circles are closer together. In a pump the rotor chambers contain far less liquid to be discharged thru the.

slowly opening tooth contacts beyond the center line, and require less driving powerthan the rotors in my earlier forms. In gearing this drive across the center line at full mesh more nearly ap proaches a rolling contact, capable of heavy driving power. This result is due not only to the specific circular formof the teeth of the outer rotor but also to their increased relative angular length.

Inmy Patent 1,683,563; and those patents hav-. ing the two next higher numbers, the generic type of rotors herein was claimed. Various specific forms also were claimed, particularly one in which driving range was entirely on one side of the center line at full mesh. In compressors in which the outer gear drives the inner, the driving range being the length of a tooth division, trapped more liquid and required more power than the contours in this case under otherwise even conditions.

The specific contours shown in the patents were phypocroids and based upon hypocircroids; and the preferred contours in this case are prepicroids andare based upon epicircroidsfI While I have described my invention as applied to gears for fluid pumps, I do not limit it to pumps since the same tooth curves are applicable in whole or in part to other gear and engine movements such as compressors, meters, engines, and gears of all classes.

While I have described tooth forms for rotors for pumps, with specific ratios for such mechanisms, I do not limit it to such ratios.

It has beennoted how the tooth form provides curves of generous radius, closely fitting each other at uniform angular speed over a driving distance at full mesh, a trifle less than a tooth division including a tooth and. tooth space. This may be true regardless of ratio or type of gears.

In cycloidal gear forms, the cycloids have instant radii diminishing to' zero toward the pitch or ratio circle. Such curves, because they are sharp just where the point of maximum drive occurs, wear rapidly. Involute curves engage between convex surfaces at points away from the center line where there is greater radial, slip, and with contacts tending to cut thru lubricants and concentrate loads on small areas. Variations in assembly aggravate this factor. the shape of the cutting tool due to sharpening do the same. My invention greatly improves these conditions.'-

First, it provides a plurality of driving convex curves engaging concave curves at uniform angular speed, with a better engagement away from the center line to ofiset greater radial slip.

Second, it makes possible the location of these contacts nearer to, or crossing the pitch or ratio circles,

Third, it so increases the angular length of the tooth base and the arc of tooth pressure as to allow a corresponding reduction in size of tooth and diameter.

Fourth, the lines or vectors representing the angles of pressure between teeth may, if desired, pass thru the base of the teeth in each gear. The teeth of both gears are then sublasted more to compression stresses than to breaking or shearing stresses, making further reduction in diameter of gears possible.

Fifth, variations in assembly and variations due to tool grinding have less efiect with such curves.

These results are attained to such a degree that my teeth may run silently and smoothly at all speeds with a durability heretofore unknown.

These various results are accomplished, among other factors, by the employment of the circroidal form of trochold as a basis for gear curve formacircles rolling on or within the ratio circles tend either toward loops or flattened teeth and hence are not suitable for gear or rotor curves.

- The ratio circles represent the gear ratios desired of course. The numbers of teeth may be the same as the ratio.

In internal gearing I find the best curves for gear teeth to be those outlined by a master form circle whose center is outside of the outer ratio circle, a distance just sufficient to describe a trochoidal curve whose smallest inside instant radius is at least equal to one half the radius of themaster form circle.

While I have described trochoids or circroids, and curves parallel thereto as the best means of arriving at the novel characteristics of my invention, I do not limit it to that geometrical system, since if variations of curves and describing forms may attain some or all of the novel results and even. the same exact curves, they lie within its scope.

What I claim is:

1. A rotary mechanical movement comprising two rotor members having internal andexternal teeth respectively, one member within, eccentric to, and characterized by one less tooth division than theother, the contours of the tooth divisions of each rotor characterized by curves described cular convex arcs, the teeth of the outer rotor being substantially wider than those of the inner rotor.

2. The combination claimed in claim 1 having the corners between the convex and concave contacting rotor'curves of slight radius.

3. A rotary mechanical movement comprising two rotor members having internal and external teeth respectively, one member within, eccentric to, and characterized by one less tooth division than the other, the contours of the tooth divisions of each rotor characterized by curves described or outlined by the form of the teeth of the other at relatively steady or uniform angular speed, having continuous travelling contacts between the teeth between the opening and closing chambers across both the open mesh and thefull mesh regions, said sliding contact curves on the inwardly convex teeth of the'outer rotor includ ing curves generated inwardly by approximately circular convex arcs.

4. The combination claimed in claim ,3 having the contact curves of the tooth crowns of the outer rotor substantially wider in angular direction than the sliding contact curves of the tooth crowns of the inner rotor.

5. In a rotary mechanical movement comprising two rotor members, one member within, ec-

centric to, and characterized by one less tooth division than the other, the contours of the tooth divisions ofthe. two rotor members having curves, each generated by the form of the other at relatively steady or uniform angular speed, providing continuous travelling contacts between them where needed for the performance of driving and pressure functions, the convex curves of one rotor being substantially wider than those of the other.

6. In a rotary mechanical movement comprising two rotor members, one within, eccentric to, and characterized by one less tooth division than the other, the contours of each rotor including curves outlined or described by those of the other at relatively steady or uniform angular speed providing continuous'contacts for operations on or by pressure, said curves of one of said rotor members being described'inwardly by a form substantially no- Wider and no deeper than an are described from a point on the radius of the pitch circle of the other rotor, said point being outside of said are and of said pitch circle.

7. In a rotary mechanical movement comprising two rotor members, one within, eccentric to, and characterized byone l-ess tooth division than the other, the contours of each rotor including .curves outlined or described by those of the other,

during relative rotation at steady or uniform angular speed, having continuous service travelling contacts on both sides of the center line thru the axes at full mesh' and having a driving relation across said center line at full mesh.

8. In a rotary mechanical movement cor'npris ingtwo rotor or gear members, a master gear and a mated gear, having dissimilar mated teeth, said members being eccentric to each other, the master teeth having selected curves engaging generative reversed curves of the mated gear at full mesh for driving for substantially a tooth division, the teeth of one having a generative relation to those of the other at uniform angular speed, the contours of the teeth of one having a minimum driving curvature at full mesh with a radius substantially greater than zero, and the driving portions of the other based upon or directed by an epitrochoidal curve described by a point carried byone ratio circle representing a master gear, as it rolls in relationto the ratio circle of the matinggear whose tooth form is being described, said point being outside of the ratio circle of said mating gear at the point of tangency of the ratio circles, and the master gear having tooth driving portions mating therewith at uniform angular speed, the tooth forms of the mated gear conforming to curves whosecrowns are characterized by varying instant radii. 9. The combination claimed in claim 8, said tooth contours paralleling said trochoidal curve. 10. In a pair of rotors or gears having a tooth ratio and a driving range, teeth on one gear having curves based upon trochoids described by a point carried by one ratio circle as it rolls relatively with the other ratio circle in contact with and outside of it, without slip at the point of tangency, said point being outside of said other ratio circle, the driving areas of the teeth of the other rotor or gear generated at uniform angular speed,

by said teeth travelling along trochoids, the driving range of said teeth characterized by curves including an engagement substantially at the full mesh point.

'11. The combination claimed in claim 10, having as said other ratio circle that of the inner gear or rotor of internal gears.

,12, The combination claimed in claim 10, having as the actual gear curves, lines along said trochoids such as are described by a body carried upon or directed by said point as it travels on said trochoids.

13. The combination claimed in claim 10, having as the actual gear curves lines equidistant to said trochoids described by the'arc of .a circle centered on said point while that point travels on said trochoids.

14. The combination claimed in claim 10, having as said other ratio circle that of the inner gear or rotor of internal gears and having said point carried by the outer ratio circle.

15. In combination, mating gears or rotors having a driving range, one having a circular master generating form of tooth, the other a mated generated form of tooth curvature, the ratio circle of the master gear lying outside of and tangent to the ratio circle of the mated gear, having as the actual gear or rotor curves lines having regular and repeated variations of curvature for each tooth division outlined at relatively uniform angular speed by said master form, the focal center of curvature of said master form being located outside of the ratio circle of the mated'gear when at the point of tangency of the ratio circles.

16. A pair of rotors, one within the other, and having a less number of teeth, said inner rotor having contours corresponding to those developed by a describing formhaving a substantially circular characteristic representing the tooth form of the outer rotor carried by a ratio circle of the outer rotor surrounding the ratio circle of the inner rotor and rolling upon it, said describing form carried by the outer ratio circle and so located upon it as to generate complete curves for driving upon the inner rotor.

' 17. The combination set forth in claim 16 having said location of the said describing form such as to generate complete convex and concave curves on the inner rotor without undercutting.

18. The combination claimed in claim 16 having said describing form of such relative size as to generate comparatively small tooth tops on the inner rotor to facilitate running them in to cooperate with the circular generating form with continuous contact. at uniform angular speed.

19. Rotors having teeth maintaining continuous contact over a range equal at least to a tooth division in length at fullmesh, one rotor having teeth whose driving areas correspond to a master form and have radii of curvature substantially greater than zero, the other rotor having teeth whose. driven areas are generative of the master form at uniform angular speed, said driving and driven areas of said teeth appreaching closer registration as the distance of the point of tooth engagement from the pointof tangency of their ratio circles increases towards the end of the driving range.

20. The combination claimed in claim 19, having the master form substantially a circular arc;

21. A pair of rotors, one within the other and.

having a lesser number-of teeth, said rotors have ing contours maintaining continuous pressure engagements at uniform angular speed, the outer-rotor having teeth of greater angular length than the teeth of the inner rotor, the inner rotor having its contours generative of the teeth of the outer rotor and adapted to wear to the curve of the outer teeth.

22. The combination claimed in claim 21 having the outer teeth provided with-driving areas substantially circular in form, and the inner rotor having one less tooth than the outer rotor.

23. In a pair of gears, a tooth curve on a first gear having a driving area such as is generated or outlined by the outside of a master form representing the form, size and relative location of the driving area of a tooth of a second gear, with a point fixed with relation to said master form following an epitrochoid developed on the ratio circle of the first gear, which epitrochoid is traced by said point while it is carried around said ratio circle of said first gear by the ratio circle of said second gear, as one rolls upon. the other without slip, said point located sufficiently outside of the ratio circle of the first gear to prevent undercutting, during said generation, of said driving areas on said first gear, the convex areas of the teeth of the first gear having varying radii of curvature as distinguished from circular arcs, and a minimum curvature substantially greater than zero as distinguished from .cycloids; the driving areas of the second gear corresponding to the master form convex curve, engaging concave driving areas of the first gear at uniform angular speed in said driving range, said engagements occurring inside and outside of said ratio circle.

24. The combination claimed in claim 23 having curves characterized by having the said point inside of said master form, and a center of its curvature, and travelling outside of 'the ratio circle of the first gear during complete generation.

25. The combination claimed in claim 23 having curves characterized by having the said point inside of said master form, and a center generation, and having the curve of the master form a circular arc and said point the center of said arc.

26. In a pair of gears, a tooth curve of a first gear generated or outlined by a curved master form representing the tooth form of the second gear, with a point on the inner side of the curvature of said master form following an epitrochoid developed on the ratio circle of the first gear, which epitrochoid is traced by said point while it is carried around said ratio circle of said first gear by the ratio circle of the second gear rolling upon the ratio circle of the first gear, said point located sufficiently outside of the ratio circle of the first gear to prevent undercutting the driving portions of the teethof said first gear, the convex crowns of the curves to which the teeth of said first gear conform having varying radii of curvature, the teeth of said two gears maintaining in their driving range continuous contact at uniform angular motion; the curve of the driving portion of the tooth curve of the first gear having a corresponding point on its concave side coinciding with the said point on the concave side of the master form on the center line thru the axes, at full mesh.

27. In gear pair, rotary toothed members, one adapted to drive the other, outwardly converging driving curves on said members, those on one convex and those on the other concave, providing a driving range over a distance not less than that between one tooth and the next one at full mesh, both inside and outside of their ratio circles, one member being a, master member, the other. a mated member, the driving curve on said mated member having a curvature conforming to that generated by the driving curve on said master member while the ratio circle of said master member. rolls upon the. outside of the ratio circle of the mated member without slip at the point of tangency, the driving curves on said master memher having curvatures whose instant radii are centered outside of its ratio circle and so located with relation to' it that said generated mated driving curves maintain continuous contact with said master driving curves at relatively uniform angular motion, said mated driving curves having curvatures such that when extended over the tops of the mated teeth said tops have curvatures of varying instant radii.

28. The combination set forth in claim 23 having a circular curve for said master form.

29. The combination set forth in claim 27 having for said master driving curve a circular are.

30. The combination set forth in claim 23 having the master form tooth curve conforming to a relatively long radius of curvature whereby said mated tooth convex curve is relatively narrow angularly.

31. The combination claimed in claim 27 having the driving range crossing the center line thru the two axes, the engagement between the driving curves increasing in intimacy of engagement from the center line to the end of the driving engagement.

v 32. In a gear pair, master convex driving tooth faces having continuous unreversed curves on one gear lying across the ratio circle of that gear, partly within and partly outside said circle, engaging continuous unreversed driving tooth surfaces of the other conforming to concave curves outlined by said convex master driving faces at constant relative speed determined by the tooth ratio of said gears, the centers of curvature of said master curvesdisposed at such distance outside of the ratio circle of the gear having concave driving curves as to prevent outlining curves having undercuts or driving areas out of contact with said master faces, in the full mesh region.

33. Two meshing gears having engaging faces and. flanks of forms determined by the system of generation at uniform angular speeds determined by the ratio of the teeth, wherein the ratio circle of the first gear is located outside the ratio circle of the second gear, wherein a master form is selected for a face of said first gear and located with relation to the second gear so that its instant center of curvature lies outside the ratio circle of that gear, at a sufiicient distance to outline, during generation, a continuous mating curve on said second gear.

34. The combination claimed in claim 33 having a circular master form.

35. A pair of meshing gears having teeth, one side of said teeth having a generated curve changing from concave to convex substantially outside of its ratio circle and nearer the outermost portions of the tooth curves than the middle portions thereof, whereby tooth widths on one rotor are so increased as to permit cutting additional tooth spaces in them, and the tooth space widths on the other rotor are so increased as to permit the insertion of additional teeth in them to coop erate with said additional tooth spaces. 36. In a gear tooth form, a tooth surface conforming to an epicroid curve, and comprising an envelope of a master form traveling along an epicir croid around the ratio circle of said gear, saidv circroid being disposed .outside of the ratio circle of said gear far enough to prevent undercutting.

MYRON F. I-HIL;

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