Modern Machine-Shop Practice Part 12

[Ill.u.s.tration: Fig. 127.]

To overcome this objection the template may be made to equal half the thickness of a tooth and its edge filed to represent a radial line on the wheel. But there are other objections, as, for example, that the template can only be applied to the wheel when adjusted on the arm shown in Fig. 126, unless, indeed, a radial line be struck on every tooth of the wheel. Again, to produce the template a radial line representing the radius of the wheel must be produced, which is difficult where segments only are used to produce the curves. It is better, therefore, to form the template as shown in Fig. 127, the projections at A B having their edges filed to coincide with the pitch circle P, so that they may be applied to a length of one arc of pitch circle at least equal to the pitch of the teeth.

The templates for the tooth curves being obtained, the wheel must be divided off on the pitch circle for the thickness of the teeth and the width of the s.p.a.ces, and the templates applied to the marks or points of division to serve as guides to mark the tooth curves. Since, however, as already stated, the tooth curves are as often struck by arcs of circles as by templates, the application of such arcs and their suitability may be discussed.

MARKING THE CURVES BY HAND.

In the employment of arcs of circles several methods of finding the necessary radius are found in practice.

[Ill.u.s.tration: Fig. 128.]

In the best practice the true curve is marked by the rolling segments already described, and the compa.s.s points are set by trial to that radius which gives an arc nearest approaching to the true face and flank curves respectively. The degree of curve error thus induced is sufficient that the form of tooth produced cannot with propriety be termed epicycloidal teeth, except in the case of fine pitches in which the arc of a circle may be employed to so nearly approach the true curve as to be permissible as a subst.i.tute. But in coa.r.s.e pitches the error is of much importance. Thus in Fig. 128 is shown the curve of the _former_ or _template_ attachment used on the celebrated Corliss Bevel Gear Cutting Machine, to cut the teeth on the bevel-wheels employed upon the line shafting at the Centennial Exhibition. These gears, it may be remarked, were marvels of smooth and noiseless running, and attracted wide attention both at home and abroad. The engraving is made from a drawing marked direct from the _former_ itself, and kindly furnished me by Mr. George H. Corliss. A A is the face and B B the flank of the tooth, C C is the arc of a circle nearest approaching to the face curve, and D D the arc of a circle nearest approaching the flank curve. In the face curve, there are but two points where the circle coincides with the true curve, while in the flank there are three such points; a circle of smaller radius than C C would increase the error at _b_, but decrease it at _a_; one of a greater radius would decrease it at _b_, and increase it at _a_. Again, a circle larger in radius than D D would decrease the error at _e_ and increase it at _f_; while one smaller would increase it at _e_ and decrease it at _f_. Only the working part of the tooth is given in the ill.u.s.tration, and it will be noted that the error is greatest in the flank, although the circle has three points of coincidence.

[Ill.u.s.tration: Fig. 129.]

In this case the depth of the _former_ tooth is about three and three-quarter times greater than the depth of tooth cut on the bevel-wheels; hence, in the figure the actual error is magnified three and three-quarter times. It demonstrates, however, the impropriety of calling coa.r.s.ely pitched teeth that are found by arcs of circles "epicycloidal" teeth.

When, however, the pitches of the teeth are fine as, say an inch or less, the coincidence of an arc of a circle with the true curve is sufficiently near for nearly all practical purposes, and in the case of cast gear the amount of variation in a pitch of 2 inches would be practically inappreciable.

To obtain the necessary set of the compa.s.ses to mark the curves, the following methods may be employed.

First by rolling the true curves with segments as already described, and the setting the compa.s.s points (by trial) to that radius which gives an arc nearest approaching the true curves. In this operation it is not found that the location for the centre from which the curve must be struck always falls on the pitch circle, and since that location will for every tooth curve lie at the same radius from the wheel centre it is obvious that after the proper location for one of the curves, as for the first tooth face or tooth flank as the case may be, is found, a circle may be struck denoting the radius of the location for all the teeth. In Fig. 129, for example, P P represents the pitch circle, A B the radius that will produce an arc nearest approaching the true curve produced by rolling segments, and A the location of the centre from which the face arc B should be struck. The point A being found by trial with the compa.s.ses applied to the curve B, the circle A C may be struck, and the location for the centres from which the face arcs of each tooth must be struck will also fall on this circle, and all that is necessary is to rest one point of the compa.s.ses on the side of the tooth as, say at E, and mark on the second circle A C the point C, which is the location wherefrom to mark the face arc D.

If the teeth flanks are not radial, the locations of the centre wherefrom to strike the flank curves are found in like manner by trial of the compa.s.ses with the true curves, and a third circle, as I in Fig.

130, is struck to intersect the first point found, as at G in the figure. Thus there will be upon the wheel face three circles, P P the pitch circle, J J wherefrom to mark the face curves, and I wherefrom to mark the flank curves.

When this method is pursued a little time may be saved, when dividing off the wheel, by dividing it into as many divisions as there are teeth in the wheel, and then find the locations for the curves as in Fig. 131, in which 1, 2, 3 are points of divisions on the pitch circle P P, while A, B, struck from point 2, are centres wherefrom to strike the arcs E, F; C, D, struck also from point 2 are centres wherefrom to strike the flank curves G, H.

[Ill.u.s.tration: Fig. 130.]

It will be noted that all the points serving as centres for the face curves, in Fig. 130, fall within a s.p.a.ce; hence if the teeth were rudely cast in the wheel, and were to be subsequently cut or trimmed to the lines, some provision would have to be made to receive the compa.s.s points.

To obviate the necessity of finding the necessary radius from rolling segments various forms of construction are sometimes employed.

[Ill.u.s.tration: Fig. 131.]

Thus Rankine gives that shown in Fig. 132, which is obtained as follows.

Draw the generating circle D, and A D the line of centres. From the point of contact at C, mark on circle D, a point distance from C one-half the amount of the pitch, as at P, and draw the line P C of indefinite length beyond C. Draw a line from P, pa.s.sing through the line of centres at E, which is equidistant between C and A. Then multiply the length from P to C by the distance from A to D, and divide by the distance between D and E. Take the length and radius so found, and mark it upon P C, as at F, and the latter will be the location of centre for compa.s.ses to strike the face curve.

[Ill.u.s.tration: Fig. 132.]

Another method of finding the face curve, with compa.s.ses, is as follows: In Fig. 133, let P P represent the pitch circle of the wheel to be marked, and B C the path of the centre of the generating or describing circle as it rolls outside of P P. Let the point B represent the centre of the generating circle when that circle is in contact with the pitch circle at A. Then from B, mark off on B C any number of equidistant points, as D, E, F, G, H, and from A, mark on the pitch circle, points of division, as 1, 2, 3, 4, 5, at the intersection of radial lines from D, E, F, G, and H. With the radius of the generating circle, that is, A B, from B, as a centre, mark the arc I, from D the arc J, from E the arc K, &c., to M, marking as many arcs as there are points of division on B C. With the compa.s.ses set to the radius of divisions 1, 2, step off on arc M the five divisions, N, O, S, T, V, and V will be a point in the epicycloidal curves. From point of division 4, step off on L four points of division, as _a_, _b_, _c_, _d_, and _d_ will be another point in the epicycloidal curve. From point 3 set off three divisions on K, from point 2 two dimensions on L, and so on, and through the points so obtained, draw by hand or with a scroll the curve represented in the cut by curve A V.

[Ill.u.s.tration: Fig. 133.]

Hypocycloids for the flanks of the teeth may be traced in a similar manner. Thus in Fig. 134 P P is the pitch circle, and B C the line of motion of the centre of the generating circle to be rolled within P P, and R a radial line. From 1 to 6 are points of equal division on the pitch circle, and D to I are arc locations for the centre of the generating circle. Starting from A, which represents the supposed location for the centre of the generating circle, the point of contact between the generating and base circles will be at B. Then from 1 to 6 are points of equal division on the pitch circle, and from D to I are the corresponding locations for the centres of the generating circle.

From these centres the arcs J, K, L, M, N, O, are struck. From 6 mark the six points of division from _a_ to _f_, and _f_ is a point in the curve. Five divisions on N, four on M, and so on, give respectively points in the curve which is marked in the figure from A to _f_.

There is this, however, to be noted concerning the constructions of the last two figures. Since the circle described by the centre of the generating circle is of different arc or curve to that of the pitch circle, the chord of an arc having an equal length on each will be different. The amount is so small as to be practically correct. The direction of the error is to give to the curves a less curvature, as though they had been produced by a generating circle of larger diameter.

Suppose, for example, that the difference between the arc N 5 (Fig. 133) and its chord is .1, and that the difference between the arc 4 5, and its chord is .01, then the error in one step is .09, and, as the point V is formed in 5 steps, it will contain this error multiplied five times.

Point _d_ would contain it multiplied four times, because it has 4 steps, and so on.

The error will increase in proportion as the diameter of the generating is less than that of the pitch circle, and though in large wheels, working with large wheels (so that the difference between the radius of the generating circle and that of the smallest wheel is not excessive), it is so small as to be practically inappreciable, yet in small wheels, working with large ones, it may form a sensible error.

[Ill.u.s.tration: Fig. 134.]

An instrument much employed in the best practice to find the radius which will strike an arc of a circle approximating the true epicycloidal curve, _and for finding at the same time_ the location of the centre wherefrom that curve should be struck, is found in the Willis'

odontograph. This is, in reality, a scale of centres or radii for different and various diameters of wheels and generating circles. It consists of a scale, shown in Fig. 135, and is formed of a piece of sheet metal, one edge of which is marked or graduated in divisions of one-twentieth of an inch. The edge meeting the graduated edge at O is at angle of 75 to the graduated edge.

On one side of the odontograph is a table (as shown in the cut), for the flanks of the teeth, while on the other is the following table for the faces of the teeth:

TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE.

CENTRES FOR THE FACES OF THE TEETH.

Pitch in Inches and Parts.

+------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ |No. of|1/4|3/8|1/2|5/8|3/4| 1|1- |1- |1- | 2|2- |2- | 3|3- | |Teeth | | | | | | |1/4|1/2|3/4| |1/4|1/2| |1/2| |------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 12 | 1| 2| 2| 3| 4| 5| 6| 7| 9| 10| 11| 12| 15| 17| | 15 | ..| ..| 3| ..| ..| ..| 7| 8| 10| 11| 12| 14| 17| 19| | 20 | 2| ..| ..| 4| 5| 6| 8| 9| 11| 12| 14| 15| 18| 21| | 30 | ..| 3| 4| ..| ..| 7| 9| 10| 12| 14| 16| 18| 21| 25| | 40 | ..| ..| ..| ..| 6| 8| ..| 11| 13| 15| 17| 19| 23| 26| | | | | | | | | | | | | | | | | | 60 | ..| ..| ..| 5| ..| ..| 10| 12| 14| 16| 18| 20| 25| 29| | 80 | ..| ..| ..| ..| ..| 9| 11| 13| 15| 17| 19| 21| 26| 30| | 100 | ..| ..| ..| ..| 7| ..| ..| ..| ..| 18| 20| 22| ..| 31| | 150 | ..| ..| 5| 6| ..| ..| ..| 14| 16| 19| 21| 23| 27| 32| |Rack. | ..| 4| ..| ..| ..| 10| 12| 15| 17| 20| 22| 25| 30| 34| +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+

The method of using the instrument is as follows: In Fig. 136, let C represent the centre, and P the pitch circle of a wheel to contain 30 teeth of 3 inch arc pitch. Draw the radial line L, meeting the pitch circle at A. From A mark on the pitch circle, as at B, a radius equal to the pitch of the teeth, and the thickness of the tooth as A _k_. Draw from B to C the radial line E. Then for the flanks place the slant edge of the odontograph coincident and parallel with E, and let its corners coincide with the pitch circle as shown. In the table headed _centres for the flanks of the teeth_, look down the column of 3 inch pitch, and opposite to the 30 in the column of numbers of teeth, will be found the number 49, which indicates that the centre from which to draw an arc for the flank is at 49 on the graduated edge of the odontograph, as denoted in the cut by _r_. Thus from _r_ to the side _k_ of the tooth is the radius for the compa.s.ses, and at _r_, or 49, is the location for the centre to strike the flank curve _f_. For the face curve set the slant edge of the odontograph coincident with the radial line L, and in the table of centres for the faces of teeth, look down the column of 3-inch pitch, and opposite to 30 in the number of teeth column will be found the number 21, indicating that at 21 on the graduated edge of the odontograph, is the location of the centre wherefrom to strike the curve _d_ for the face of the tooth, this location being denoted in the cut at R.

[Ill.u.s.tration: Fig. 135.

TABLE SHOWING THE PLACE OF THE CENTRES UPON THE SCALE.

+--------------------------------------------------------------+ | CENTRES FOR THE FLANKS OF THE TEETH. | +--------------------------------------------------------------+ | PITCH IN INCHES AND PARTS. | +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ |Number| | | | | | | | | | | | | | | | of | | | | | | 1 | 1-| 1-| 1-| 2 | 2-| 2-| 3 | 3-| |teeth.|1/4|3/8|1/2|5/8|3/4| |1/4|1/2|3/4| |1/4|1/2| |1/2| +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ | 13| 32| 48| 64| 80| 96|129|160|193|225|257|289|321|386|450| | 14| 17| 26| 35| 43| 52| 69| 87|104|121|139|156|173|208|242| | 15| 12| 18| 25| 31| 37| 49| 62| 74| 86| 99|111|123|148|173| | 16| 10| 15| 20| 25| 30| 40| 50| 59| 69| 79| 89| 99|119|138| | 17| 8| 13| 17| 21| 25| 34| 43| 50| 59| 67| 75| 84|101|117| | 18| 7| 11| 15| 19| 22| 30| 37| 45| 52| 59| 67| 74| 89|104| | 19|...| 10| 13| 17| 20| 27| 35| 40| 47| 54| 60| 67| 80| 94| | 20| 6| 9| 12| 16| 19| 25| 31| 37| 43| 49| 56| 62| 74| 86| | 22| 5| 8| 11| 14| 16| 22| 27| 33| 39| 43| 49| 54| 65| 76| | 24|...| 7| 10| 12| 15| 20| 25| 30| 35| 40| 45| 49| 59| 69| | 26|...|...| 9| 11| 14| 18| 23| 27| 32| 37| 41| 46| 55| 64| | 28| 4| 6|...|...| 13|...| 22| 26| 30| 35| 40| 43| 52| 60| | 30|...|...| 8| 10| 12| 17| 21| 25| 29| 33| 37| 41| 49| 58| | 35|...|...|...| 9| 11| 16| 19| 23| 26| 30| 34| 38| 45| 53| | 40|...| 5| 7|...|...| 15| 18| 21| 25| 28| 32| 35| 42| 49| | 60| 3|...| 6| 8| 9| 13| 15| 19| 22| 25| 28| 31| 37| 43| | 80|...| 4|...| 7|...| 12|...| 17| 20| 23| 26| 29| 35| 41| | 100|...|...|...|...| 8| 11| 14|...|...| 22| 25| 28| 34| 39| | 150|...|...| 5|...|...|...| 13| 16| 19| 21| 24| 27| 32| 38| | Rack.| 2|...|...| 6| 7| 10| 12| 15| 17| 20| 22| 25| 30| 34| +------+---+---+---+---+---+---+---+---+---+---+---+---+---+---+]

The requisite number on the graduated edge for pitches beyond 3-1/2 (the greatest given in the tables), may be obtained by direct proportion from those given in the tables. Thus for 4 inch pitch, by doubling the numbers given for a 2 inch pitch, containing the same number of teeth, for 4-1/2 inch pitch by doubling the numbers given for a 2-1/4 inch pitch. If the pitch be a fraction that cannot be so obtained, no serious error will be induced if the nearest number marked be taken.

[Ill.u.s.tration: Fig. 136.]

An improved form of template odontograph, designed by Professor Robinson of the Illinois School of Industry, is shown in Fig. 137.

In this instrument the curved edge, having graduated lines, approaches more nearly to the curves produced by rolling circles than can be obtained from any system in which an arc of a circle is taken to represent the curve; hence, that edge is applied direct to the teeth and used as a template wherefrom to mark the curve. The curve is a logarithmic spiral, and the use of the instrument involves no other labor than that of setting it in position. The applicability of this curve, for the purpose, arises from two of its properties: first, that the involute of the logarithmic spiral is another like spiral with poles in common; and, second, that the obliquity or angle between a normal and radius sector is constant, the latter property being possessed by this curve only. By the first property it is known that a line, lying tangent to the curve C E H, will be normal or perpendicular to the curve C D B; so that when the line D E F is tangent to the pitch line, the curve A D B will coincide very closely with the true epicycloidal curve, or, rather, with that portion of it which is applied to the tooth curve of the wheel. By the second quality, all sectors of the spiral, with given angle at the poles, are similar figures which admit of the same degree of coincidence for all similar epicycloids, whether great or small, and nearly the same for epicycloids in general; thus enabling the application of the instrument to epicycloids in general.

To set the instrument in position for drawing a tooth face a table which accompanies the instrument is used. From this table a numerical value is taken, which value depends upon the diameters of the wheels, and the number of teeth in the wheel for which the curve is sought. This tabular value, when multiplied by the pitch of the teeth, is to be found on the graduated edge on the instrument A D B in Fig. 137. This done, draw the line D E F tangent to the pitch line at the middle of the tooth, and mark off the half thickness of the tooth, as E, D, either on the tangent line or the pitch line. Then place the graduated edge of the odontograph at D, and in such a position that the number and division found as already stated shall come precisely on the tangent line at D, and at the same time so set the curved edge H F C so that it shall be tangent to the tangent line, that is to say, the curved edge C H must just meet the tangent line at some one point, as at F in the figure. A line drawn coincident with the graduated edge will then mark the face curve required, and the odontograph may be turned over, and the face on the other side of the tooth marked from a similar setting and process.

For the flanks of the teeth setting numbers are obtained from a separate table, and the instrument is turned upside down, and the tangent line D F, Fig. 137, is drawn from the side of the tooth (instead of from the centre), as shown in Fig. 138.

It is obvious that this odontograph may be set upon a radial arm and used as a template, as shown in Fig. 126, in which case the instrument would require but four settings for the whole wheel, while rolling segments and the making of templates are entirely dispensed with, and the degree of accuracy is greater than is obtainable by means of the employment of arcs of circles.

The tables wherefrom to find the number or mark on the graduated edge, which is to be placed coincident with the tangent line in each case, are as follows:--

TABLE OF TABULAR VALUES WHICH, MULTIPLIED BY THE ARC PITCH OF THE TEETH, GIVES THE SETTING NUMBER ON THE GRADUATED EDGE OF THE INSTRUMENT.

+--------------------+-----------------------------------------------------+ | | Number of Teeth in Wheel Sought; or, Wheel for | | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | 8 | 12 | 16 | 20 | 30 | 40 | 50 | 60 | 70 | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Faces: Flanks Radial or Curved._ | | RATIOS.[7] | Draw Setting Tangent at Middle of Tooth.-- | | | Epicycloidal Spur or Bevel Gearing. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | 1/12 = .083 | .32 | .39 | .46 | .51 | | | | | | | 1/4 = .250 | .31 | .37 | .44 | .49 | .61 | .70 | .78 | .85 | .92 | | 1/2 = .500 | .28 | .34 | .41 | .46 | .57 | .66 | .73 | .80 | .87 | | 2/3 = .667 | .27 | .32 | .38 | .43 | .54 | .62 | .70 | .77 | .83 | | 1 | .23 | .28 | .34 | .39 | .49 | .58 | .65 | .72 | .78 | | 3/2 = 1.50 | .19 | .25 | .29 | .34 | .44 | .51 | .58 | .64 | .69 | | 2 | .17 | .22 | .26 | .30 | .38 | .46 | .53 | .59 | .63 | | 3 | | .16 | .19 | .23 | .31 | .38 | .44 | .49 | .53 | | 4 | | .14 | .17 | .20 | .26 | .33 | .38 | .42 | .46 | | 6 | | | | | .22 | .26 | .30 | .34 | .37 | | 12 | | | | | | .20 | .23 | .25 | .28 | | 24 | | | | | | | | | | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | Number of Teeth in Wheel Sought; or, Wheel for| | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | 80 | 90 | 100 | 120 | 150 | 200 | 300 | 500 | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Faces: Flanks Radial or Curved._ | | RATIOS.[7] | Draw Setting Tangent at Middle of Tooth.-- | | | Epicycloidal Spur or Bevel Gearing. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ | 1/12 = .083 | | | | | | | | | | 1/4 = .250 | .99 | 1.05| 1.11| 1.22| 1.36| 1.55| 1.94| 2.54| | 1/2 = .500 | .93 | 1.00| 1.06| 1.15| 1.29| 1.50| 1.86| 2.41| | 2/3 = .667 | .89 | .95| 1.01| 1.11| 1.24| 1.45| 1.79| 2.32| | 1 | .83 | .89| .94| 1.03| 1.15| 1.36| 1.65| 2.10| | 3/2 = 1.50 | .74 | .79| .84| .93| 1.05| 1.25| 1.53| 1.94| | 2 | .68 | .72| .76| .84| .95| 1.13| 1.40| 1.81| | 3 | .57 | .60| .63| .71| .82| .97| 1.23| 1.60| | 4 | .49 | .53| .56| .63| .73| .87| 1.08| 1.42| | 6 | .41 | .44| .47| .53| .61| .71| .90| 1.20| | 12 | .30 | .32| .34| .37| .42| .49| .60| .82| | 24 | | .19| .21| .23| .26| .31| .40| .57| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ +--------------------+-----------------------------------------------------+ | | Number of Teeth in Wheel Sought; or, Wheel for | | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | 8 | 12 | 16 | 20 | 30 | 40 | 50 | 60 | 70 | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Flanks, when Curved._ | | | Draw Setting Tangent at Side of Tooth.-- | | | Epicycloidal Spur and Bevel Gearing. | |D C | Faces of Internal, and Flanks of Pinion Teeth. | |e u +-----+-----+-----+-----+-----+-----+-----+-----+-----+ |g F r { 1.5 slight.| .77 | .98 | 1.18| 1.36| 1.75| 2.05| 2.31| 2.56| 2.75| |r l v { 2 good. | .44 | .54 | .63| .72 | .92| 1.09| 1.24| 1.38| 1.49| |e a a { 3 more. | .20 | .28 | .35| .40 | .54| .65| .76| .86| .95| |e n t { 4 much. | | .20 | .23| .25 | .34| .42| .51| .59| .66| | k u { 6 | | | .16| .17 | .26| .32| .38| .43| .48| |o r {12 | | | | | .19| .24| .28| .31| .34| |f e {24 | | | | | | | | | | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | Number of Teeth in Wheel Sought; or, Wheel for| | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | 80 | 90 | 100 | 120 | 150 | 200 | 300 | 500 | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | _For Flanks, when Curved._ | | | Draw Setting Tangent at Side of Tooth.-- | | | Epicycloidal Spur and Bevel Gearing. | |D C | Faces of Internal, and Flanks of Pinion Teeth.| |e u +-----+-----+-----+-----+-----+-----+-----+-----+ |g F r { 1.5 slight.| 2.92| 3.08| 3.24| 3.52| 3.87| 4.51| 5.50| 7.20| |r l v { 2 good. | 1.59| 1.79| 1.79| 1.98| 2.23| 2.67| 3.22| 4.50| |e a a { 3 more. | 1.02| 1.10| 1.18| 1.31| 1.46| 1.67| 2.08| 2.76| |e n t { 4 much. | .71| .77| .82| .92| 1.06| 1.25| 1.64| 2.15| | k u { 6 | .52| .56| .60| .66| .76| .93| 1.20| 1.54| |o r {12 | .36| .38| .40| .45| .52| .63| .80| .98| |f e {24 | | | .22| .25| .28| .33| .47| .60| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ +--------------------+-----------------------------------------------------+ | | Number of Teeth in Wheel Sought; or, Wheel for | | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | 8 | 12 | 16 | 20 | 30 | 40 | 50 | 60 | 70 | | +-----+-----+-----+-----+-----+-----+-----+-----+-----+ | _For Faces of Racks; and of Pinions for Racks and Internal Gears; for | | Flanks of Internal and Sides of Involute Teeth._ | | Draw Setting Tangent at Middle of Tooth, regarding s.p.a.ce as Tooth in | | Internal Teeth. For Rack use Number of Teeth in Pinion. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | Pinion. | .31 | .39 | .48 | .57 | .73 | .88 | 1.00| 1.10| 1.20| | Rack. | .32 | .38 | .44 | .50 | .62 | .72 | .80| .87| .93| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+-----+ | | Number of Teeth in Wheel Sought; or, Wheel for| | | Which Teeth are Sought. | | +-----+-----+-----+-----+-----+-----+-----+-----+ | | 80 | 90 | 100 | 120 | 150 | 200 | 300 | 500 | | +-----+-----+-----+-----+-----+-----+-----+-----+ | _For Faces of Racks; and of Pinions for Racks and Internal Gears; | | for Flanks of Internal and Sides of Involute Teeth._ | |Draw Setting Tangent at Middle of Tooth, regarding s.p.a.ce as Tooth in| | Internal Teeth. For Rack use Number of Teeth in Pinion. | +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+ | Pinion. | 1.30| 1.40| 1.48| 1.65| 1.85| 2.15| 2.65| 3.50| | Rack. | .99| 1.03| 1.08| 1.16| 1.27| 1.49| 1.86| 2.44| +--------------------+-----+-----+-----+-----+-----+-----+-----+-----+

[7] These ratios are obtained by dividing the radius of the wheel sought by the diameter of the generating circle.