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.

The requisite number on the graduated edge for pitches beyond 3¹⁄₂ (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 4¹⁄₂ inch pitch by doubling the numbers given for a 2¹⁄₄ 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.

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.

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.