Proportions of Gear-wheels.

Thickness of rim below root = depth of tooth.

Width of Teeth.-The width of the faces of teeth is generally made from 2 to 3 times the circular pitch - from 6.28 to 9.42 divided by the diametral pitch. There is no standard rule for width.

The following sizes are given in a stock list of cut gears in "Grant's Gears:

Diameter pitch..

Face, inches..

3 and 4 2% 134 and 2 14 and 1% 34 and 1 1⁄2 and 5%

Rules for Calculating the Speed of Gears and Pulleys.The relations of the size and speed of driving and driven gear wheels are the same as those of belt pulleys. In calculating for gears, multiply or divide by the diameter of the pitch-circle or by the number of teeth, as may be required. In calculating for pulleys, multiply or divide by their diameter in inches.

If D= diam. of driving wheel, d= diam. of driven, R minute of driver, revs. per min. of driven.

If N

=

revolutions per

Rrd D; r =RD÷d; D= dr+ R; d= DR+ r.
number of teeth of driver and n = number of teeth of driven,
NnrR; n= NR + "'; Rrn N; r RN+n.

To find the number of revolutions of the last wheel at the end of a train of spur-wheels, all of which are in a line and mesh into one another, when the revolutions of the first wheel and the number of teeth or the diameter of the first and last are given: Multiply the revolutions of the first wheel by its number of teeth or its diameter, and divide the product by the number of teeth or the diameter of the last wheel.

To find the number of teeth in each wheel for a train of spur-wheels, each to have a given velocity: Multiply the number of revolutions of the driving-wheel by its number of teeth, and divide the product by the number of revolutions each wheel is to make.

To find the number of revolutions of the last wheel in a train of wheels and pinions, when the revolutions of the first or driver, and the diameter, the teeth, or the circumference of all the drivers and pinions are given: Multiply the diameter, the circumference, or the number of teeth of all the driving-wheels together, and this continued product by the number of revolutions of the first wheel, and divide this product by the continued product of the diameter, the circumference, or the number of teeth of all the driven wheels, and the quotient will be the number of revolutions of the last wheel. EXAMPLE.-1. A train of wheels consists of four wheels each 12 in. diameter of pitch-circle, and three pinions 4, 4, and 3 in. diameter. The large wheels are the drivers, and the first makes 36 revs. per min. Required the speed of the last wheel.

2. What is the speed of the first large wheel if the pinions are the drivers, the 3-in. pinion being the first driver and making 36 revs. per min.?

Milling Cutters for Interchangeable Gears.-The Pratt & Whitney Co. make a series of cutters for cutting epicycloidal teeth. The number of cutters to cut from a pinion of 12 teeth to a rack is 24 for each pitch coarser than 10. The Brown & Sharpe Mfg. Co. make a similar series, and also a series for involute teeth, in which eight cutters are made for each pitch, as follows:

In order that the teeth of wheels and pinions may run together smoothly and with a constant relative velocity, it is necessary that their working faces shall be formed of certain curves called odontoids. The essential property of these curves is that when two teeth are in contact the common normal to the tooth curves at their point of contact must pass through the pitch-point, or point of contact of the two pitch-circles. Two such curves are in common use-the cyloid and the involute.

The Cycloidal Tooth.-In Fig. 154 let PL and pl be the pitch-circles of two gear-wheels; GC and gc are two equal generating-circles, whose radii should be taken as not greater than one half of the radius of the smaller pitch-circle. If the circle gc be rolled to the left on the larger pitch-circle PL, the point O will describe an epicycloid, oefgh. If the other generatingcircle GC be rolled to the right on PL, the point O will describe a hypocycloid oabcd. These two curves, which are tangent at O, form the two parts of a tooth curve for a gear whose pitch-circle is PL. The upper part oh is called the face and the lower part od is called the flank, If the same circles be rolled on the other pitch-circle pl, they will describe the curve for a tooth of the gear pl, which will work properly with the tooth on PL.

The cycloídal curves may be drawn without actually rolling the generating-circle, as follows: On the line PL, from O, step off and mark equal distances, as 1, 2, 3, 4, etc. From 1, 2, 3, etc., draw radial lines toward the centre of PL, and from 6, 7, 8, etc., draw radial lines from the same centre, but beyond PL. With the radius of the generating-circle, and with centres successively placed on these radial lines, draw arcs of circles tangent to PL at 123,678, etc, With the dividers set to one of the equal divisions, as Q11

step off 1a and 6e; step off two such divisions on the circle from 2 to b, and from 7 to f; three such divisions from 3 to c, and from 8 to g; and so on, thus locating the several points abcdH and efgk, and through these points draw the tooth curves.

The curves for the mating tooth on the other wheel may be found in like manner by drawing arcs of the generating-circle tangent at equidistant points on the pitch-circle pl.

The tooth curve of the face oh is limited by the addendum-liner or r1.

and that of the flank oH by the root curve R or R1. R and r represent the root and addendum curves for a large number of small teeth, and R1r the like curves for a small number of large teeth. The form or appearance of the tooth therefore varies according to the number of teeth, while the pitch circle and the generating-circle may remain the same.

In the cycloidal system, in order that a set of wheels of different diameters but equal pitches shall all correctly work together, it is necessary that the generating-circle used for the teeth of all the wheels shall be the same, and it should have a diameter not greater than half the diameter of the pitchline of the smallest wheel of the set. The customary standard size of the generating-circle of the cycloidal system is one having a diameter equal to the radius of the pitch-circle of a wheel having 12 teeth. (Some gearmakers adopt 15 teeth.) This circle gives a radial flank to the teeth of a wheel having 12 teeth. A pinion of 10 or even a smaller number of teeth can be made, but in that case the flanks will be undercut, and the tooth will not be as strong as a tooth with radial flanks. If in any case the describing circle be half the size of the pitch-circle, the flanks will be radial; if it be less, they will spread out toward the root of the tooth, giving a stronger form; but if greater, the flanks will curve in toward each other, whereby the teeth become weaker and difficult to make.

In some cases cycloidal teeth for a pair of gears are made with the generating-circle of each gear,having a radius equal to half the radius of its pitchcircle. In this case each of the gears will have radial flanks. This method makes a smooth working gear, but a disadvantage is that the wheels are not interchangeable with other wheels of the same pitch but different numbers of teeth.

The rack in the cycloidal system is equivalent to a wheel with an infinite number of teeth. The pitch is equal to the circular pitch of the mating gear. Both faces and flanks are cycloids formed by rolling the generating circle of the mating gear-wheel on each side of the straight pitch-line of the rack.

Another method of drawing the cycloidal curves is shown in Fig. 155. It is known as the method of tangent arcs. The generating-circles, as before, are drawn with equal radii, the length of the radius being less than half the radius of pl, the smaller pitch-circle. Equal divisions 1, 2, 3, 4, etc., are marked off on the pitch circles and divisions of the same length stepped off on one of the generating-circles, as oabc, etc. From the points 1, 2, 3, 4, 5 on the line po, with radii successively equal to the chord distances oa, ob, oc, od, oe, draw the five small arcs F. A line drawn through the outer edges of these small arcs, tangent to them all, will be the hypocycloidal curve for the flank of a tooth below the pitch-line pl. From the points 1, 2, 3, etc., on the line ol, with radii as before, draw the small arcs G. A line tangent to these arcs will be the epicycloid for the face of the same tooth for which the flank curve has already been drawn. In the same way, from centres on the line Po, and oL, with the same radii, the tangent arcs H and K may be drawn, which will give the tooth for the gear whose pitch-circle is PL.

If the generating-circle had a radius just one half of the radius of pl, the bypocycloid Fwould be a straight line, and the flank of the tooth would have been radial.

The Involute Tooth.-In drawing the involute tooth curve, the angle of obliquity, or the angle which a common tangent to the teeth, when they are in contact at the pitch-point, makes with a line joining the centres of the wheels, is first arbitrarily determined. It is customary to take it at 15°. The pitch-lines pl and PL being drawn in contact at O, the line of obliquity AB is drawn through Onormal to a common tangent to the tooth curves, or at the given angle of obliquity to a common tangent to the pitch-circles. In

the cut the angle is 20°. From the centres of the pitch-circles draw circles c and d tangent to the line AB. These circles are called base-lines or basecircles, from which the involutes F and K are drawn. By laying off convenient distances, 0, 1, 2, 3, which should each be less than 1/10 of the diameter of the base-circle, small ares can be drawn with successively increasing radii, which will form the involute. The involute extends from the points F

and K down to their respective base-circles, where a tangent to the involute becomes a radius of the circle, and the remainders of the tooth curves, as G and H, are radial straight lines.

In the involute system the customary standard form of tooth is one having an angle of obliquity of 15° (Brown and Sharpe use 141⁄2°), an addendum of about one third the circular pitch, and a clearance of about one eighth of the addendum. In this system the smallest gear of a set has 12 teeth, this being the smallest number of teeth that will gear together when made with this angle of obliquity. In gears with less than 30 teeth the points of the teeth must be slightly rounded over to avoid interference (see Grant's Teeth of Gears). All involute teeth of the same pitch and with the same angle of obliquity work smoothly together. The rack to gear with an involute-toothed wheel has straight faces on its teeth, which make an angle with the middle line of the tooth equal to the angle of obliquity, or in the standard form the faces are inclined at an angle of 30° with each other.

To draw the teeth of a rack which is to gear with an involute wheel (Fig. 157).-Let AB be the pitch-line of the rack and AI-II' the pitch. Through

E

H

K

FIG. 157.

the pitch-point I draw EF at the given angle of obliquity, Draw AE and IF perpendicular to EF. Through E and F draw lines EGG' and FH parallel to the pitch-line. EGG' will be the addendum-line and HF the flankline. From I draw IK perpendicular to AB equal to the greatest addendum. in the set of wheels of the given pitch and obliquity plus an allowance for clearance equal to % of the addendum. Through K, parallel to AB, draw the clearance-line. The fronts of the teeth are planes perpendicular to EF, and the backs are planes inclined at the same angle to AB in the contrary direction. The outer half of the working face AE may be slightly curved. Mr. Grant makes it a circular arc drawn from a centre on the pitch-line