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.

[Ill.u.s.tration: Fig. 137. NEW ODONTOGRAPH Full Size]

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.

[Ill.u.s.tration: Fig. 138.]

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 a.s.sume 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 p.r.o.nounced 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.--a.s.sume 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 subst.i.tuted.

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 _s.p.a.ce_, 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 compa.s.ses may be set to the required radius to strike the curves of the teeth.

[Ill.u.s.tration: Fig. 139.]

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.

[Ill.u.s.tration: Fig. 140.]