Chapter III: THE TEETH OF GEAR-WHEELS (continued) (2)

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 spaces, 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.

In the best practice the true curve is marked by the rolling segments already described, and the compass 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 substitute. But in coarse 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 illustration, and it will be noted that the error is greatest in the flank, although the circle has three points of coincidence.

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 coarsely 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 compasses 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 compass 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 compasses 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 compasses 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 compasses 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.

It will be noted that all the points serving as centres for the face curves, in Fig. 130, fall within a space; 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 compass points.

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

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, passing 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 compasses to strike the face curve.

Another method of finding the face curve, with compasses, 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 compasses 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.

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.

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 compasses, 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.

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.

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 Space 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 Space 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.

From these tables may be found a tabular value which, multiplied by the pitch of the wheel to be marked (as stated at the head of the table), will give the setting number on the graduated edge of the instrument, the procedure being as follows:--

For the teeth of a pair of wheels intended to gear together only (and not with other wheels having a different number of teeth).

For the face of such teeth where the flanks are to be radial lines.

Rule.--Divide the pitch circle radius of the wheel to have its teeth marked by the pitch circle radius of the wheel with which it is to gear: or, what is the same thing, divide the number of teeth in the wheel to have its teeth marked by the number of teeth in the wheel with which it is to gear, and the quotient is the "ratio." In the ratio column find this number, and look along that line, and in the column at the head of which is the number of teeth contained in the wheel to be marked, is a number termed the tabular value, which, multiplied by the arc pitch of the teeth, will give the number on the graduated edge by which to set the instrument to the tangent line.

Example.--What is the setting number for the face curves of a wheel to contain 12 teeth, of 3-inch arc pitch, and to gear with a wheel having 24 teeth?

Here number of teeth in wheel to be marked = 12, divided by the number of teeth (24) with which it gears; 12 ÷ 24 = .5. Now in column of ratios may be found 1/2 = .500 (which is the same thing as .5), and along the same horizontal line in the table, and in the column headed 12 (the number of teeth in the wheel) is found .34. This is the tabular value, which, multiplied by 3 (the arc pitch of the teeth), gives 1.02, which is the setting number on the graduated edge. It will be noted, however, that the graduated edge is marked 1, 2, 3, &c., and that between each consecutive division are ten subdivisions; hence, for the decimal .02 an allowance may be made by setting the line 1 a proportionate amount below the tangent line marked on the wheel to set the instrument by.

Required now the setting number for the wheel to have the 24 teeth.

Here number of teeth on the wheel = 24, divided by the number of teeth (12) on the wheel with which it gears; 24 ÷ 12 = 2. Now, there is no column in the "number of teeth sought" for 24 teeth; but we may find the necessary tabular value from the columns given for 20 teeth and 30 teeth, thus:--opposite ratio 2, and under 20 teeth is given .30, and under 30 teeth is given .38--the difference between the two being .08. Now the difference between 20 teeth and 24 teeth is 4/10; hence, we take 4/10 of the .08 and add it to the tabular value given for 20 teeth, thus: .08 × 4 ÷ 10 = .032, and this added to .30 (the tabular value given for 20 teeth = .33, which is the tabular value for 24 teeth). The .33 multiplied by arc pitch (3) gives .99. This, therefore, is the setting number for the instrument, being sufficiently near to the 1 on the graduated edge to allow that 1 to be used instead of .99.

It is to be noted here that the pinion, having radial lines, the other wheel must have curved flanks; the rule for which is as follows:--

CURVED FLANKS FOR A PAIR OF WHEELS.

Note.--When the flanks are desired to be curved instead of radial, it is necessary to the use of the instrument to select and assume a value for the degree of curve, as is done in the table in the column marked "Degree for flank curving;" in which

1.5 slight--a slight curvature of flank.
2 good--an increased curvature of flank.
3 more--a degree of pronounced spread at root.
4 much--spread at root is a distinguishing feature of tooth form.
6--still increased spread in cases where the strength at root of
pinion is of much importance to give strength.
12--as above, under aggravated conditions.
24--undesirable (unless requirement of strength compels this degree),
because of excessive strain on pinion.

Rule.--For faces of teeth to have curved flanks.

Divide the number of teeth in the wheel to be marked by the number of teeth in the wheel with which it gears, and multiply by the degree of flank curve selected for the wheel with which that to be marked is to gear, and this will give the ratio. Find this number in ratio column, and the tabular number under the column of number of teeth of wheel to be marked; multiply tabular number so found by arc pitch of wheel to be marked, and the product will be the setting number for the instrument.

Example.--What is the setting number on the graduated edge of the odontograph for the faces of a wheel (of a pair) to contain 12 teeth of 2-inch arc pitch, and to gear with a wheel having 24 teeth and a flank curvature represented by 3 in "Degree of flank curving" column?

Here teeth in wheel to be marked (12) divided by number of teeth in the wheel it is to gear with (24), 12 ÷ 24 = .5, which multiplied by 3 (degree of curvature selected for flanks of 24-teeth wheel), .5 × 3 = 1.5. In column of ratio numbers find 1.5, and in 12-teeth column is .25, which multiplied by pitch (2) gives .5 as the setting number for the instrument; this being the fifth line on the instrument, and half way between the end and mark 1.

FOR CURVED FLANKS.

Rule.--Assume the degree of curve desired for the flanks to be marked, select the corresponding value in the column of "Degrees of flank curving," and find the tabular value under the number of teeth column.

Multiply tabular value so found by the arc pitch of the teeth, and the product is the setting number on the instrument.

Example.--What is the setting number on the odontograph for the flanks of a wheel to contain 12 teeth and gear with one having 24 teeth, the degree of curvature for the flanks being represented by 4 in the column of "Degree of flank curvature?"

Here in column of degrees of flank curvature on the 3 line and under 12 teeth is .20, which multiplied by pitch of teeth (2) is .20 × 2 = 40, or 4/10; hence, the fourth line of division on the curved corner is the setting line, it representing 4/10 of 1.

FOR INTERCHANGEABLE GEARING (THAT IS, A TRAIN OF GEARS ANY ONE OF WHICH WILL WORK CORRECTLY WITH ANY OTHER OF THE SAME SET).

Rule--both for the faces and for the flanks. For each respective wheel divide the number of teeth in that wheel by some one number not greater than the number of teeth in the smallest wheel in the set, which gives the ratio number for the wheel to be marked. On that line of ratio numbers, and in the column of numbers of teeth, find the tabular value number; multiply this by the arc pitch of the wheel to be marked, and the product is the setting number of the instrument.

Example.--A set of wheels is to contain 10 wheels; the smallest is to contain 12 teeth; the arc pitch of the wheels is four inches. What is the setting number for the smallest wheel?

Here number of teeth in smallest wheel of set is 10; divide this by any number smaller than itself (as say 5), 10 ÷ 5 = 2 = the ratio number on ratio line for 2; and under column for 12 is .17, which is the tabular value, which multiplied by pitch (4) is .17 × 4 = 68, or 6/10 and 8/100; hence, the instrument must be set with its seventh line of division just above the tangent line marked on the wheel. It will be noted that, if the seventh line were used as the setting, the adjustment would be only the 2/100 of a division out, an amount scarcely practically appreciable.

Both for the faces and flanks, the second number is obtained in _precisely_ the same manner for every wheel in the set, except that instead of 10 the number of teeth in each wheel must be substituted.

RACK AND PINION.--_For radial flanks_ use for faces the two lower lines of table. _For curved flanks_ find tabular value for pinion faces in lowest line. For flanks of pinion choose degree of curving, and find tabular value under "flanks," as for other wheels. For faces of rack divide number of teeth in pinion by degree of curving, which take for number of teeth in looking opposite "rack." Flanks of rack are still parallel, but may be arbitrarily curved beyond half way below pitch line.

INTERNAL GEARS.--For tooth curves within the pitch lines, divide radius of each wheel by any number not greater than radius of pinion, and look in the table under "flanks." For curves outside pitch line use lower line of table; or, divide radii by any number and look under "faces." In applying instrument draw tangents at middle and side of _space_, for internal teeth.

INVOLUTE TEETH.--For tabular values look opposite "Pinion," under proper number of teeth, for each wheel. Draw setting tangent from "base circle" of involute, at middle of tooth. For this the instrument gives the whole side of tooth at once.

In all cases multiply the tabular value by the pitch in inches.

BEVEL-WHEELS.--Apply above rules, using the developed normal cone bases as pitch lines. For right-angled axes this is done by using in place of the actual ratio of radii, or of teeth numbers, the square of that ratio; and for number of teeth, the actual number multiplied by the square root of one plus square of ratio or radii; the numerator of ratio, and number of teeth, belonging to wheel sought.

When the first column ratio and teeth numbers fall between those given in the table, the tabular values are found by interpolating as seen in the following examples:

EXAMPLES OF TABULAR VALUES AND SETTING NUMBERS.

_Take a pair of 16 and 56 teeth; radii 5.09 and 17.82 inches respectively; and 2 inches pitch._

+----------------+------+----------------+------+---------------+-------+
| |Number} | | First Column | Tab. |
|Kind of Gearing.| of } Kind of Flank. |Ratio | Ratio. | Val. |
| |Teeth.} |Radii.+--------+------+---+---+
| | | | | Flank. |Face. |[A]|[B]|
+----------------+------+----------------+------+--------+------+---+---+
|Epicycloidal, }|Small |Radial | .29 |Radial | .29 |.. |.44|
|Radial Flanks }|Large |Radial | 3.5 |Radial | 3.5 |.. |.44|
|Epicycloidal, } |Small |Curved 2 deg. | .29 | 2 | .87 |.63|.36|
|Curved Flanks.} |Large |Curved 3 deg. } | 3.5 | 3 | 7. |.82|.30|
|Epicycloidal, }|Small |"Sets," Divide} | 2. | 2 | 2. |.63|.26|
|Interchange'bl.}|Large |Radii by 2.55 } | 7. | 7 | 7. |.40|.30|
|Epicycloidal, } |Pinion|Curved 2 deg. | | 2 |Pinion|.63|.44|
|Internal. } |Wheel |Int. face 7 deg.| 3.5 |Pinion | 7[8] |.84|.39|
|Epicycloidal, }|Pinion|Curved 2 deg. | | 2 |Pinion|.63|.44|
|Rack & Pinion. }|Rack |Parallel | |Parallel|Rack |.. |.31|
|Involute } |Small |Face and Flank | | Pinion. | .44 |
|Gearing. } |Large |One Curve | | Pinion. | .84 |
+----------------+------+----------------+------+--------+------+-------+

Legend: A = Flank.
B = Face.

[8] The face being here internal, the tabular value is to be found
under "flanks." If bevels, use ratio radii .082 and 12.25; and teeth
numbers 16.6 and 203.8 respectively.

WALKER'S PATENT WHEEL SCALE.--This scale is used in many manufactories in the United States to mark off the teeth for patterns, wherefrom to mould cast gears, and consists of a diagram from which the compasses may be set to the required radius to strike the curves of the teeth.

The general form of this diagram is shown in Fig. 139. From the portion A the length of the teeth, according to the pitch, is obtained. From the portion B half the thickness of the tooth at the pitch line is obtained. From the part C half the thickness at the root is obtained, and from the part D half the thickness at the point is obtained.

Each of these parts is marked with the number of teeth the wheel is to contain, and with the pitch of the teeth as shown in Fig. 140, which represents part C full size. Now suppose it is required to find the thickness at the root, for a tooth of a wheel having 60 teeth of one inch pitch, the circles from the point A, pitch line B and root C being drawn, and a radial line representing the middle of the tooth being marked, as is shown in Fig. 142, the compass points are set to the distance F B, Fig. 140--F being at the junction of line 1 with line 60; the compasses are then rested at G, and the points H I are marked. Then, from the portion B, Fig. 139 of the diagram, which is shown full-size in Fig. 141, the compasses may be set to half the thickness at the pitch circle, as in this case (for ordinary teeth) from E to E, and the points J K, Fig. 142, are marked. By a reference to the portion D of the diagram, half the thickness of the tooth at the point is obtained, and marked as at L M in Fig. 142. It now remains to set compasses to the radius for the face and that for the flank curves, both of which may be obtained from the part A of the diagram. The locations of the centres, wherefrom to strike these curves, are obtained as in Fig. 142. The compasses set for the face curve are rested at H, and the arc N is struck; they are then rested at J and the arc O struck; and from the intersection of N O, as a centre, the face curve H J is marked. By a similar process, reference to the portion D of the diagram, half the thickness of the tooth at the point is obtained, and marked as at L M in Fig. 142. It now remains to set the compasses to the radius to strike the respective face and flank curves, and for this purpose the operator turns to the portion A, Fig. 139, of the diagram or scale, and sets the compasses from the marks on that portion to the required radii.

It now remains to find the proper location from which to strike the curves.

The face curve on the other side of the tooth is struck. The compasses set to the flank radius is then rested at M, and the arc P is marked and rested at K to mark the arc Q; and from the intersection of P Q, as a centre, the flank curve K M is marked: that on the other side of the tooth being marked in a similar manner.

Additional scales or diagrams, not shown in Fig. 139, give similar distances to set the compasses for the teeth of internal wheels and racks.

It now remains to explain the method whereby the author of the scale has obtained the various radii, which is as follows: A wheel of 200 teeth was given the form of tooth curve that would be obtained by rolling it upon another wheel, containing 200 teeth of the same pitch. It was next given the form of tooth that would be obtained by rolling upon it a wheel having 10 teeth of the same pitch, and a line intermediate between the two curves was taken as representing the proper curve for the large wheel. The wheel having 10 teeth was then given the form of tooth that would be obtained by rolling upon it another wheel of the same diameter of pitch circle and pitch of teeth. It was next given the form of tooth that would be given by rolling upon it a wheel having 200 teeth, and a curve intermediate between the two curves thus obtained was taken as representing the proper curve for the pinion of 10 teeth. By this means the inventor does not claim to produce wheels having an exactly equal velocity ratio, but he claims that he obtains a curve that is the nearest approximation to the proper epicycloidal curve. The radii for the curves for all other numbers of teeth (between 10 and 200) are obtained in precisely the same manner, the pinion for each pitch being supposed to contain 10 teeth. Thus the scale is intended for interchangeable cast gears.

The nature of the scale renders it necessary to assume a constant height of tooth for all wheels of the same pitch, and this Mr. Walker has assumed as .40 of the pitch, from the pitch line to the base, and .35 from the pitch line to the point.

The curves for the faces obtained by this method have rather more curvature than would be due to the true epicycloid, which causes the points to begin and leave contact more easily than would otherwise be the case.

For a pair of wheels Mr. Walker strikes the face curve by a point on the pitch rolling circle, and the flanks by a point on the addendum circle, fastening a piece of wood to the pitch circle to carry the tracing point. The flank of each wheel is struck with a tracing point, thus attached to the pitch circle of the other wheel.

The proportions of teeth and of the spaces between them are usually given in turns of the pitch, so that all teeth of a given pitch shall have an equal thickness, height, and breadth, with an equal addendum and flank, and the same amount of clearance.

The term "clearance" as applied to gear-wheel teeth means the amount of space left between the teeth of one wheel, and the spaces in the other, or, in other words, the difference between the width of the teeth and that of the spaces between the teeth.

This clearance exists at the sides of the teeth, as in Fig. 143, at A, and between the tops of the teeth and the bottoms or roots of the spaces as at B. When, however, the simple term clearance is employed it implies the side clearance as at A, the clearance at B being usually designated as _top and bottom clearance_. Clearance is necessary for two purposes; first, in teeth cut in a machine to accurate form and dimensions, to prevent the teeth of one wheel from binding in the spaces of the other, and second, in cast teeth, to allow for the imperfections in the teeth which are incidental to casting in a founder's mould. In machine-cut teeth the amount of clearance is a minimum.

In wheels which are cast with their teeth complete and on the pattern, the amount of clearance must be a maximum, because, in the first place, the teeth on the pattern must be made taper to enable the extraction of the pattern from the mould without damage to the teeth in the mould, and the amount of this taper must be greater than in machine-moulded teeth, because the pattern cannot be lifted so truly vertical by hand as to avoid, in all cases, damage to the mould; in which case the moulder repairs the mould either with his moulding tools and by the aid of the eye, or else with a tooth and a space made on a piece of wood for the purpose. But even in this case the concentricity of the teeth is scarcely likely to be preserved.

It is obvious that by reason of this taper each wheel is larger in diameter on one side than on the other, hence to preserve the true curves to the teeth the pitch circle is made correspondingly smaller. But if in keying the wheels to their shafts the two large diameters of a pair of wheels be placed to work together, the teeth of the pair would have contact on that side of the wheel only, and to avoid this and give the teeth contact across their full breadth the wheels are so placed on their shafts that the large diameter of one shall work with the small one of the other, the amount of taper being the same in each wheel irrespective of their relative diameters. This also serves to keep the clearance equal in amount both top, and bottom, and sideways.

A second imperfection is that in order to loosen the pattern in the sand or mould, and enable its extraction by hand from the mould, the pattern requires to be _rapped_ in the mould, the blows forcing back the sand of the mould and thus loosening the pattern. In ordinary practice the amount of this rapping is left entirely to the judgment of the moulder, who has nothing to guide him in securing an equal amount of pattern movement in each direction in the mould; hence, the finished mould may be of increased radius at the circumference in the direction in which the wheel moved most during the rapping. Again, the wood pattern is apt in time to shrink and become _out of round_, while even iron patterns are not entirely free from warping. Again, the cast metal is liable to contract in cooling more in one direction than in another. The amount of clearance usually allowed for pattern-moulded cast gearing is given by Professor Willis as follows:--Whole depth of tooth 7/10, of the pitch working depth 6/10; hence 1/10 of the pitch is allowed for top and bottom clearance, and this is the amount shown at B in Fig. 143. The amount of side clearance given by Willis as that ordinarily found in practice is as follows:--"Thickness of tooth 5/11 of the pitch; breadth of space 6/11; hence, the side clearance equals 1/11 of the pitch, which in a 3-inch pitch equals .27 of an inch in each wheel." Calling this in round figures, which is near enough for our purpose, 1/4 inch, we have thickness of tooth 1-1/4, width of space 1-3/4, or 1/2 inch of clearance in a 3-inch pitch, an amount which on wheels of coarse pitch is evidently more than that necessary in view of the accuracy of modern moulding, however suitable it may have been for the less perfect practice of Professor Willis's time. It is to be observed that the rapping of the pattern in the founder's mould reduces the thickness of the teeth and increases the width of the spaces somewhat, and to that extent augments the amount of side clearance allowed on the pattern, and the amount of clearance thus obtained would be nearly sufficient for a small wheel, as say of 2 inches diameter. It is further to be observed that the amount of rapping is not proportionate to the diameter of the wheel; thus, in a wheel of 2 inches diameter, the rapping would increase the size of the mould about 1/32 inch. But in the proportion of 1/32 inch to every 2 inches of diameter, the rapping on a 6-foot wheel would amount to 1-1/16 inches, whereas, in actual practice, a 6-foot wheel would not enlarge the mould more than at most 1/8 inch from the rapping.