Chapter I: The Teeth of Gear-Wheels (1)

A wheel that is provided with teeth to mesh, engage, or gear with similar teeth upon another wheel, so that the motion of one may be imparted to the other, is called, in general terms, a gear-wheel.

When the teeth are arranged to be parallel to the wheel-axis, as in Fig. 1, the wheel is termed a spur-wheel. In the figure, A represents the axial line or axis of the wheel or of its shaft, to which the teeth are parallel while spaced equidistant around the rim, or face, as it is termed, of the wheel.

When the wheel has its teeth arranged at an angle to the shaft, as in Fig. 2, it is termed a bevel-wheel, or bevel gear; but when this angle is one of 45°, as in Fig. 3, as it must be if the pair of wheels are of the same diameter, so as to make the revolutions of their shafts equal, then the wheel is called a mitre-wheel. When the teeth are arranged upon the radial or side face of the wheel, as in Fig. 4, it is termed a crown-wheel. The smallest wheel of a pair, or of a train or set of gear-wheels, is termed the pinion; and when the teeth are composed of rungs, as in Fig. 5, it is termed a lantern, trundle, or wallower; and each cylindrical piece serving as a tooth is termed a _stave_, _spindle_, or _round_, and by some a _leaf_.

An annular or internal gear-wheel is one in which the faces of the teeth are within and the flanks without, or outside the pitch-circle, as in Fig. 6; hence the pinion P operates within the wheel.

When the teeth of a wheel are inserted in mortises or slots provided in the wheel-rim, it is termed a mortised-wheel, or a cogged-wheel, and the teeth are termed cogs.

When the teeth are arranged along a plane surface or straight line, as in Fig. 7, the toothed plane is termed a _rack_, and the wheel is termed a pinion.

A wheel that is driven by a revolving screw, or worm as it is termed, is called a worm-wheel, the arrangement of a worm and worm-wheel being shown in Fig. 8. The screw or worm is sometimes also called an endless screw, because its action upon the wheel does not come to an end as it does when it is revolved in one continuous direction and actuates a nut. So also, since the worm is tangent to the wheel, the arrangement is sometimes called a wheel and tangent screw.

The diameter of a gear-wheel is always taken at the pitch circle, unless otherwise specially stated as "diameter over all," "diameter of addendum," or "diameter at root of teeth," &c., &c.

When the teeth of wheels engage to the proper distance, which is when the pitch circles meet, they are said to be in gear, or geared together. It is obvious that if two wheels are to be geared together their teeth must be the same distance apart, or the same _pitch_, as it is called.

The designations of the various parts or surfaces of a tooth of a gear-wheel are represented in Fig. 9, in which the surface A is the face of the tooth, while the dimension F is the width of face of the wheel, when its size is referred to. B is the flank or distance from the pitch line to the root of the tooth, and C the point. H is the _space_, or the distance from the side of one tooth to the nearest side of the next tooth, the width of space being measured on the pitch circle P P. E is the depth of the tooth, and G its thickness, the latter also being measured on the pitch circle P P. When spoken of with reference to a tooth, P P is called the pitch line, but when the whole wheel is referred to it becomes the pitch circle.

The points C and the surface H are true to the wheel axis.

The teeth are designated for measurement by the pitch; the height or depth above and below pitch line; and the thickness.

The pitch, however, may be measured in two ways, to wit, around the pitch circle A, in Fig. 10, which is called the arc or circular pitch, and across B, which is termed the chord pitch.

In proportion as the diameter of a wheel (having a given pitch) is increased, or as the pitch of the teeth is made finer (on a wheel of a given diameter) the arc and chord pitches more nearly coincide in length. In the practical operations of marking out the teeth, however, the arc pitch is not necessarily referred to, for if the diameter of the pitch circle be made correct for the required number of teeth having the necessary arc pitch, and the wheel be accurately divided off into the requisite number of divisions with compasses set to the chord pitch, or by means of an index plate, then the arc pitch must necessarily be correct, although not referred to, save in determining the diameter of the wheel at the pitch circle.

The difference between the width of a space and the thickness of the tooth (both being measured on the pitch circle or pitch line) is termed the clearance or side clearance, which is necessary to prevent the teeth of one wheel from becoming locked in the spaces of the other. The amount of clearance is, when the teeth are cut to shape in a machine, made just sufficient to prevent contact on one side of the teeth when they are in proper gear (the pitch circles meeting in the line of centres). But when the teeth are cast upon the wheel the clearance is increased to allow for the slight inequalities of tooth shape that is incidental to casting them. The amount of clearance given is varied to suit the method employed to mould the wheels, as will be explained hereafter.

The line of centres is an imaginary line from the centre or axis of one wheel to the axis of the other when the two are in gear; hence each tooth is most deeply engaged, in the space of the other wheel, when it is on the line of centres.

There are three methods of designating the sizes of gear-wheels. First, by their diameters at the pitch circle or pitch diameter and the number of teeth they contain; second, by the number of teeth in the wheel and the pitch of the teeth; and third, by a system known as diametral pitch.

The first is objectionable because it involves a calculation to find the pitch of the teeth; furthermore, if this calculation be made by dividing the circumference of the pitch circle by the number of teeth in the wheel, the result gives the arc pitch, which cannot be measured correctly by a lineal measuring rule, especially if the wheel be a small one having but few teeth, or of coarse pitch, as, in that case, the arc pitch very sensibly differs from the chord pitch, and a second calculation may become necessary to find the chord pitch from the arc pitch.

The second method (the number and pitch of the teeth) possesses the disadvantage that it is necessary to state whether the pitch is the arc or the chord pitch.

If the arc pitch is given it is difficult to measure as before, while if the chord pitch is given it possesses the disadvantage that the diameters of the wheels will not be exactly proportional to the numbers of teeth in the respective wheels. For instance, a wheel with 20 teeth of 2 inch chord pitch is not exactly half the diameter of one of 40 teeth and 2 inch chord pitch.

To find the chord pitch of a wheel take 180 (= half the degrees in a circle) and divide it by the number of teeth in the wheel. In a table of natural sines find the sine for the number so found, which multiply by 2, and then by the radius of the wheel in inches.

Example.--What is the chord pitch of a wheel having 12 teeth and a diameter (at pitch circle) of 8 inches? Here 180 ÷ 12 = 15; (sine of 15 is .25881). Then .25881 × 2 = .51762 × 4 (= radius of wheel) = 2.07048 inches = chord pitch.

TABLE OF NATURAL SINES.

+--------+--------++--------+--------++--------+--------+
|Degrees.| Sine. ||Degrees.| Sine. ||Degrees.| Sine. |
+--------+--------++--------+--------++--------+--------+
| 1 | .01745 || 16 | .27563 || 31 | .51503 |
| 2 | .03489 || 17 | .29237 || 32 | .52991 |
| 3 | .05233 || 18 | .30901 || 33 | .54463 |
| 4 | .06975 || 19 | .32556 || 34 | .55919 |
| 5 | .08715 || 20 | .34202 || 35 | .57357 |
| 6 | .10452 || 21 | .35836 || 36 | .58778 |
| 7 | .12186 || 22 | .37460 || 37 | .60181 |
| 8 | .13917 || 23 | .39073 || 38 | .61566 |
| 9 | .15643 || 24 | .40673 || 39 | .62932 |
| 10 | .17364 || 25 | .42261 || 40 | .64278 |
| 11 | .19080 || 26 | .43837 || 41 | .65605 |
| 12 | .20791 || 27 | .45399 || 42 | .66913 |
| 13 | .22495 || 28 | .46947 || 43 | .68199 |
| 14 | .24192 || 29 | .48480 || 44 | .69465 |
| 15 | .25881 || 30 | .50000 || 45 | .70710 |
+--------+--------++--------+--------++--------+--------+

The principle upon which diametral pitch is based is as follows:--

The diameter of the wheel at the pitch circle is supposed to be divided into as many equal parts or divisions as there are teeth in the wheel, and the length of one of these parts is the diametral pitch. The relationship which the diametral bears to the arc pitch is the same as the diameter to the circumference, hence a diametral pitch which measures 1 inch will accord with an arc pitch of 3.1416; and it becomes evident that, for all arc pitches of less than 3.1416 inches, the corresponding diametral pitch must be expressed in fractions of an inch, as 1/2, 1/3, 1/4, and so on, increasing the denominator until the fraction becomes so small that an arc with which it accords is too fine to be of practical service. The numerators of these fractions being 1, in each case, they are in practice discarded, the denominators only being used, so that, instead of saying diametral pitches of 1/2, 1/3, or 1/4, we say diametral pitches of 2, 3, or 4, meaning that there are 2, 3, or 4 teeth on the wheel for every inch in the diameter of the pitch circle.

Suppose now we are given a diametral pitch of 2. To obtain the corresponding arc pitch we divide 3.1416 (the relation of the circumference to the diameter) by 2 (the diametral pitch), and 3.1416 ÷ 2 = 1.57 = the arc pitch in inches and decimal parts of an inch. The reason of this is plain, because, an arc pitch of 3.1416 inches being represented by a diametral pitch of 1, a diametral pitch of 1/2 (or 2 as it is called) will be one half of 3.1416. The advantage of discarding the numerator is, then, that we avoid the use of fractions and are readily enabled to find any arc pitch from a given diametral pitch.

Examples.--Given a 5 diametral pitch; what is the arc pitch? First (using the full fraction 1/5) we have 1/5 × 3.1416 = .628 = the arc pitch. Second (discarding the numerator), we have 3.1416 ÷ 5 = .628 = arc pitch. If we are given an arc pitch to find a corresponding diametral pitch we again simply divide 3.1416 by the given arc pitch.

Example.--What is the diametral pitch of a wheel whose arc pitch is 1-1/2 inches? Here 3.1416 ÷ 1.5 = 2.09 = diametral pitch. The reason of this is also plain, for since the arc pitch is to the diametral pitch as the circumference is to the diameter we have: as 3.1416 is to 1, so is 1.5 to the required diametral pitch; then 3.1416 × 1 ÷ 1.5 = 2.09 = the required diametral pitch.

To find the number of teeth contained in a wheel when the diameter and diametral pitch is given, multiply the diameter in inches by the diametral pitch. The product is the answer. Thus, how many teeth in a wheel 36 inches diameter and of 3 diametral pitch? Here 36 × 3 = 108 = the number of teeth sought. Or, per contra, a wheel of 36 inches diameter has 108 teeth. What is the diametral pitch? 108 ÷ 36 = 3 = the diametral pitch. Thus it will be seen that, for determining the relative sizes of wheels, this system is excellent from its simplicity. It also possesses the advantage that, by adding two parts of the diametral pitch to the pitch diameter, the outside diameter of the wheel or the diameter of the addendum is obtained. For instance, a wheel containing 30 teeth of 10 pitch would be 3 inches diameter on the pitch circle and 3-2/10 outside or total diameter.

Again, a wheel having 40 teeth of 8 diametral pitch would have a pitch circle diameter of 5 inches, because 40 ÷ 8 = 5, and its full diameter would be 5-1/4 inches, because the diametral pitch is 1/8, and this multiplied by 2 gives 1/4, which added to the pitch circle diameter of 5 inches makes 5-1/4 inches, which is therefore the diameter of the addendum, or, in other words, the full diameter of the wheel.

Suppose now that a pair of wheels require to have pitch circles of 5 and 8 inches diameter respectively, and that the arc pitch requires to be, say, as near as may be 4/10 inch; to find a suitable pitch and the number of teeth by the diametral pitch system we proceed as follows:

In the following table are given various arc pitches, and the corresponding diametral pitch.

+----------------+----------+----------+----------------+
|Diametral Pitch.|Arc Pitch.|Arc Pitch.|Diametral Pitch.|
+----------------+----------+----------+----------------+
| | | Inch. | |
| 2 | 1.57 | 1.75 | 1.79 |
| 2.25 | 1.39 | 1.5 | 2.09 |
| 2.5 | 1.25 | 1.4375 | 2.18 |
| 2.75 | 1.14 | 1.375 | 2.28 |
| 3 | 1.04 | 1.3125 | 2.39 |
| 3.5 | .890 | 1.25 | 2.51 |
| 4 | .785 | 1.1875 | 2.65 |
| 5 | .628 | 1.125 | 2.79 |
| 6 | .523 | 1.0625 | 2.96 |
| 7 | .448 | 1.0000 | 3.14 |
| 8 | .392 | 0.9375 | 3.35 |
| 9 | .350 | 0.875 | 3.59 |
| 10 | .314 | 0.8125 | 3.86 |
| 11 | .280 | 0.75 | 4.19 |
| 12 | .261 | 0.6875 | 4.57 |
| 14 | .224 | 0.625 | 5.03 |
| 16 | .196 | 0.5625 | 5.58 |
| 18 | .174 | 0.5 | 6.28 |
| 20 | .157 | 0.4375 | 7.18 |
| 22 | .143 | 0.375 | 8.38 |
| 24 | .130 | 0.3125 | 10.00 |
| 26 | .120 | 0.25 | 12.56 |
+----------------+----------+----------+----------------+

From this table we find that the nearest diametral pitch that will correspond to an arc pitch of 4/10 inch is a diametral pitch of 8, which equals an arc pitch of .392, hence we multiply the pitch circles (5 and 8,) by 8, and obtain 40 and 64 as the number of teeth, the arc pitch being .392 of an inch. To find the number of teeth and pitch by the arc pitch and circumference of the pitch circle, we should require to find the circumference of the pitch circle, and divide this by the nearest arc pitch that would divide the circumference without leaving a remainder, which would entail more calculating than by the diametral pitch system.

The designation of pitch by the diametral pitch system is, however, not applied in practice to coarse pitches, nor to gears in which the teeth are cast upon the wheels, pattern makers generally preferring to make the pitch to some measurement that accords with the divisions of the ordinary measuring rule.

Of two gear-wheels that which impels the other is termed the driver, and that which receives motion from the other is termed the driven wheel or follower; hence in a single pair of wheels in gear together, one is the driver and the other the driven wheel or follower. But if there are three wheels in gear together, the middle one will be the follower when spoken of with reference to the first or prime mover, and the driver, when mentioned with reference to the third wheel, which will be a follower. A series of more than two wheels in gear together is termed a train of wheels or of gearing. When the wheels in a train are in gear continuously, so that each wheel, save the first and last, both receives and imparts motion, it is a simple train, the first wheel being the driver, and the last the follower, the others being termed intermediate wheels. Each of these intermediates is a follower with reference to the wheel that drives it, and a driver to the one that it drives. But the velocity of all the wheels in the train is the same in fact per second (or in a given space of time), although the revolutions in that space of time may vary; hence a simple train of wheels transmits motion without influencing its velocity. To alter the velocity (which is always taken at a point on the pitch circle) the gearing must be compounded, as in Fig. 11, in which A, B, C, E are four wheels in gear, B and C being compounded, that is, so held together on the shaft D that both make an equal number of revolutions in a given time. Hence the velocity of C will be less than that of B in proportion as the diameter, circumference, radius, or number of teeth in C, varies from the diameter, radius, circumference, or number of teeth (all the wheels being supposed to have teeth of the same pitch) in B, although the rotations of B and C are equal. It is most convenient, and therefore usual, to take the number of teeth, but if the teeth on C (and therefore those on E also) were of different pitch from those on B, the radius or diameters of the wheels must be taken instead of the pitch, when the velocities of the various wheels are to be computed. It is obvious that the compounded pair of wheels will diminish the velocity when the driver of the compounded pair (as C in the figure) is of less radius than the follower B, and conversely that the velocity will be increased when the driver is of greater radius than the follower of the compound pair.

The diameter of the addendum or outer circle of a wheel has no influence upon the velocity of the wheel. Suppose, for example, that we have a pair of wheels of 3 inch arc or circular pitch, and containing 20 teeth, the driver of the two making one revolution per minute. Suppose the driven wheel to have fast upon its shaft a pulley whose diameter is one foot, and that a weight is suspended from a line or cord wound around this pulley, then (not taking the thickness of the line into account) each rotation of the driven wheel would raise the weight 3.1416 feet (that being the circumference of the pulley). Now suppose that the addendum circle of either of the wheels were cut off down to the pitch circle, and that they were again set in motion, then each rotation of the driven wheel would still raise the weight 3.1416 feet as before.

It is obvious, however, that the addendum circle must be sufficiently larger than the pitch circle to enable at least one pair of teeth to be in continuous contact; that is to say, it is obvious that contact between any two teeth must not cease before contact between the next two has taken place, for otherwise the motion would not be conveyed continuously. The diameter of the pitch circle cannot be obtained from that of the addendum circle unless the pitch of the teeth and the proportion of the pitch allowed for the addendum be known. But if these be known the diameter of the pitch circle may be obtained by subtracting from that of the addendum circle twice the amount allowed for the addendum of the tooth.

Example.--A wheel has 19 teeth of 3 inch arc pitch; the addendum of the tooth or teeth equals 3/10 of the pitch, and its addendum circle measures 19.943 inches; what is the diameter of the pitch circle? Here the addendum on each side of the wheel equals (3/10 of 3 inches) = .9 inches, hence the .9 must be multiplied by 2 for the two sides of the wheel, thus, .9 × 2 = 1.8. Then, diameter of addendum circle 19.943 inches less 1.8 inches = 18.143 inches, which is the diameter of the pitch circle.

Proof.--Number of teeth = 19, arc pitch 3, hence 19 × 3 = 57 inches, which, divided by 3.1416 (the proportion of the circumference to the diameter) = 18.143 inches.

If the distance between the centres of a pair of wheels that are in gear be divided into two parts whose lengths are in the same proportion one to the other as are the numbers of teeth in the wheels, then these two parts will represent the radius of the pitch circles of the respective wheels. Thus, suppose one wheel to contain 100 and the other 50 teeth, and that the distance between their centres is 18 inches, then the pitch radius or pitch diameter of one will be twice that of the other, because one contains twice as many teeth as the other. In this case the radius of pitch circle for the large wheel will be 12 inches, and that for the small one 6 inches, because 12 added to 6 makes 18, which is the distance between the wheel centres, and 12 is in the same proportion to 6 that 100 is to 50.

A simple rule whereby to find the radius of the pitch circles of a pair of wheels is as follows:--

Rule.--Divide number of teeth in the large wheel by the number in the small one, and to the sum so obtained add 1. Take this amount and divide it into the distance between the centres of the wheels, and the result will be the radius of the smallest wheel. To obtain the radius of the largest wheel subtract the radius of the smallest wheel from the distance between the wheel centres.

Example.--Of a pair of wheels, one has 100 and the other 50 teeth, the distance between their centres is 18 inches; what is the pitch radius of each wheel?

Here 100 ÷ 50 = 2, and 2 + 1 = 3. Then 18 ÷ 3 = 6, hence the pitch radius of the small wheel is 6 inches. Then 18 - 6 = 12 = pitch radius of large wheel.

Example 2.--Of a pair of wheels one has 40 and the other 90 teeth. The distance between the wheel centres is 32-1/2 inches; what are the radii of the respective pitch circles? 90 ÷ 40 = 2.25 and 2.25 + 1 = 3.25. Then 32.5 ÷ 3.25 = 10 = pitch radius of small wheel, and 32.5 - 10 = 22.5, which is the pitch radius of the large wheel.

To prove this we may show that the pitch radii of the two wheels are in the same proportion as their numbers of teeth, thus:--

Proof.--Radius of small wheel = 10 × 4 = 40
---- ----
radius of large wheel = 22.5 × 4 = 90.0

Suppose now that a pair of wheels are constructed, having respectively 50 and 100 teeth, and that the radii of their true pitch circles are 12 and 6 respectively, but that from wear in their journals or journal bearings this 18 inches (12 + 6 = 18) between centres (or line of centres, as it is termed) has become 18-3/8 inches. Then the acting effective or operative radii of the pitch circles will bear the same proportion to the 18-3/8 as the numbers of teeth in the respective wheels, and will be 12.25 for the large, and 6.125 for the small wheel, instead of 12 and 6, as would be the case were the wheels 18 inches apart. Working this out under the rule given we have 100 ÷ 50 = 2, and 2 + 1 = 3. Then 18.375 ÷ 3 = 6.125 = pitch radius of small wheel, and 18.375 - 6.125 = 12.25 = pitch radius of the large wheel.

The true pitch line of a tooth is the line or point where the face curve joins the flank curve, and it is essential to the transmission of uniform motion that the pitch circles of epicycloidal wheels exactly coincide on the line of centres, but if they do not coincide (as by not meeting or by overlapping each other), then a false pitch circle becomes operative instead of the true one, and the motion of the driven wheel will be unequal at different instants of time, although the revolutions of the wheels will of course be in proportion to the respective numbers of their teeth.

If the pitch circle is not marked on a single wheel and its arc pitch is not known, it is practically a difficult matter to obtain either the arc pitch or diameter of the pitch circle. If the wheel is a new one, and its teeth are of the proper curves, the pitch circle will be shown by the junction of the curves forming the faces with those forming the flanks of the teeth, because that is the location of the pitch circle; but in worn wheels, where from play or looseness between the journals and their bearings, this point of junction becomes rounded, it cannot be defined with certainty.

In wheels of large diameter the arc pitch so nearly coincides with the chord pitch, that if the pitch circle is not marked on the wheel and the arc pitch is not known, the chord pitch is in practice often assumed to represent the arc pitch, and the diameter of the wheel is obtained by multiplying the number of teeth by the chord pitch. This induces no error in wheels of coarse pitches, because those pitches advance by 1/4 or 1/2 inch at a step, and a pitch measuring about, say, 1-1/4 inch chord pitch, would be known to be 1-1/4 arc pitch, because the difference between the arc and chord pitch would be too minute to cause sensible error. Thus the next coarsest pitch to 1 inch would be 1-1/8, or more often 1-1/4 inch, and the difference between the arc and chord pitch of the smallest wheel would not amount to anything near 1/8 inch, hence there would be no liability to mistake a pitch of 1-1/8 for 1 inch or _vice versâ_. The diameter of wheel that will be large enough to transmit continuous motion is diminished in proportion as the pitch is decreased; in proportion, also, as the wheel diameter is reduced, the difference between the arc and chord pitch increases, and further the steps by which fine pitches advance are more minute (as 1/4, 9/32, 5/16, &c.). From these facts there is much more liability to err in estimating the arc from the measured chord pitch in fine pitches, hence the employment of diametral pitch for small wheels of fine pitches is on this account also very advantageous. In marking out a wheel the chord pitch will be correct if the pitch circle be of correct diameter and be divided off into as many points of equal division (with compasses) as there are to be teeth in the wheel. We may then mark from these points others giving the thickness of the teeth, which will make the spaces also correct. But when the wheel teeth are to be cut in a machine out of solid metal, the mechanism of the machine enables the marking out to be dispensed with, and all that is necessary is to turn the wheel to the required addendum diameter, and mark the pitch circle. The following are rules for the purposes they indicate.

The circumference of a circle is obtained by multiplying its diameter by 3.1416, and the diameter may be obtained by dividing the circumference by 3.1416.

The circumference of the pitch circle divided by the arc pitch gives the number of teeth in the wheel.

The arc pitch multiplied by the number of teeth in the wheel gives the circumference of the pitch circle.

Gear-wheels are simply rotating levers transmitting the power they receive, less the amount of friction necessary to rotate them under the given conditions. All that is accomplished by a simple train of gearing is, as has been said, to vary the number of revolutions, the speed or velocity measured in feet moved through per minute remaining the same for every wheel in the train. But in a compound train of gears the speed in feet per minute, as well as the revolutions, may be varied by means of the compounded pairs of wheels. In either a simple or a compound train of gearing the power remains the same in amount for every wheel in the train, because what is in a compound train lost in velocity is gained in force, or what is gained in velocity is lost in force, the word force being used to convey the idea of strain, pressure, or pull.

In Fig. 12, let A, B, and C represent the pitch circles of three gears of which A and B are in gear, while C is compounded with B; let E be the shaft of A, and G that for B and C. Let A be 60 inches, B = 30 inches, and C = 40 inches in diameter. Now suppose that shaft E suspends from its perimeter a weight of 50 lbs., the shaft being 4 inches in diameter. Then this weight will be at a leverage of 2 inches from the centre of E and the 50 must be multiplied by 2, making 100 lbs. at the centre of E. Then at the perimeter of A this 100 will become one-thirtieth of one hundred, because from the centre to the perimeter of A is 30. One-thirtieth of 100 is 3-33/100 lbs., which will be the force exerted by A on the perimeter of B. Now from the perimeter of B to its centre (or in other words its radius) is 15 inches, hence the 3-33/100 lbs. at its perimeter will become fifteen times as much at the centre G of B, and 3-33/100 × 15 = 49-95/100 lbs. From the centre G to the perimeter of C being 20 inches, the 49-95/100 lbs. at the centre will be only one-twentieth of that amount at the perimeter of C, hence 49-95/100 ÷ 20 = 2-49/100 lbs., which is the amount of force at the perimeter of C.

Here we have treated the wheels as simple levers, dividing the weight by the length of the levers in all cases where it is transmitted from the shaft to the perimeter, and multiplying it by the length of the lever when it is transmitted from the perimeter of the wheel to the centre of the shaft. The precise same result will be reached if we take the diameter of the wheels or the number of the teeth, providing the pitch of the teeth on all the wheels is alike.

Suppose, for example, that A has 60 teeth, B has 30 teeth, and C has 40 teeth, all being of the same pitch. Suppose the 50 lb. weight be suspended as before, and that the circumference of the shaft be equal to that of a pinion having 4 teeth of the same pitch as the wheels. Then the 50 multiplied by the 4 becomes 200, which divided by 60 (the number of teeth on A) becomes 3-33/100, which multiplied by 30 (the number of teeth on B) becomes 99-90/100, which divided by 40 (the number of teeth on C) becomes 2-49/100 lbs. as before.

It may now be explained why the shaft was taken as equal to a pinion having 4 teeth. Its diameter was taken as 4 inches and the wheel diameter was taken as being 60 inches, and it was supposed to contain 60 teeth, hence there was 1 tooth to each inch of diameter, and the 4 inches diameter of shaft was therefore equal to a pinion having 4 teeth. From this we may perceive the philosophy of the rule that to obtain the revolutions of wheels we multiply the given revolutions by the teeth in the driving wheels and divide by the teeth in the driven wheels.

Suppose that A (Fig. 13) makes 1 revolution per minute, how many will C make, A having 60 teeth, B 30 teeth, and C 40 teeth? In this case we have but one driving wheel A, and one driven wheel B, the driver having 60 teeth, the driven 30, hence 60 ÷ 30 = 2, equals revolutions of B and also of C, the two latter being on the same shaft.

It will be observed then that the revolutions are in the same proportion as the numbers of the teeth or the radii of the wheels, or what is the same thing, in the same proportion as their diameters. The number of teeth, however, is usually taken as being easier obtained than the diameter of the pitch circles, and easier to calculate, because the teeth will be represented by a whole number, whereas the diameter, radius, or circumference, will generally contain fractions.

Suppose that the 4 wheels in Fig. 14 have the respective numbers of teeth marked beside them, and that the upper one having 40 teeth makes 60 revolutions per minute, then we may obtain the revolutions of the others as follows:--

Revolu- Teeth Teeth Teeth Teeth
tions. in first in first in second in second
driver. driven. driver. driven.
60 × 40 ÷ 60 × 20 ÷ 120 = 6-66/100

and a remainder of the reciprocating decimals. We may now prove this by reversing the question, thus. Suppose the 120 wheel to make 6-66/100 revolutions per minute, how many will the 40 wheel make?

Revolu- Teeth Teeth Teeth Teeth
tions. in first in first in second in second
driver. driven. driver. driven.
6.66 × 120 ÷ 20 × 60 ÷ 40 = 59-99/100 =

revolutions of the 40 wheel, the discrepancy of 1/100 being due to the 6.66 leaving a remainder and not therefore being absolutely correct.

That the amount of power transmitted by gearing, whether compounded or not, is equal throughout every wheel in the train, may be shown as follows:--

Referring again to Fig. 10, it has been shown that with a 50 lb. weight suspended from a 4 inch shaft E, there would be 30-33/100 lbs. at the perimeter of A. Now suppose a rotation be made, then the 50 lb. weight would fall a distance equal to the circumference of the shaft, which is (3.1416 × 4 = 12-56/100) 12-56/100 inches. Now the circumference of the wheel is (60 dia. × 3.1416 = 188-49/100 cir.) 188-49/100 inches, which is the distance through which the 3-33/100 lbs. would move during one rotation of A. Now 3.33 lbs. moving through 188.49 inches represents the same amount of power as does 50 lbs. moving through a distance of 12.56 inches, as may be found by converting the two into inch lbs. (that is to say, into the number of inches moved by 1 lb.), bearing in mind that there will be a slight discrepancy due to the fact that the fractions .33 in the one case, and .56 in the other are not quite correct. Thus:

188.49 inches × 3.33 lbs. = 627.67 inch lbs., and
12.56 " × 50 " = 628 " "

Taking the next wheels in Fig. 12, it has been shown that the 3.33 lbs. delivered from A to the perimeter of B, becomes 2.49 lbs. at the perimeter of C, and it has also been shown that C makes two revolutions to one of A, and its diameter being 40 inches, the distance this 2.49 lbs. will move through in one revolution of A will therefore be equal to twice its circumference, which is (40 dia. × 3.1416 = 125.666 cir., and 125.666 × 2 = 251.332) 251.332 inches. Now 2.49 lbs. moving through 251.332 gives when brought to inch lbs. 627.67 inch lbs., thus 251.332 × 2.49 = 627.67. Hence the amount of power remains constant, but is altered in form, merely being converted from a heavy weight moving a short distance, into a lighter one moving a distance exactly as much greater as the weight or force is lessened or lighter.

Gear-wheels therefore form a convenient method of either simply transmitting motion or power, as when the wheels are all of equal diameter, or of transmitting it and simultaneously varying its velocity of motion, as when the wheels are compounded either to reduce or increase the speed or velocity in feet per second of the prime mover or first driver of the train or pair, as the case may be.

In considering the action of gear-teeth, however, it sometimes is more convenient to denote their motion by the number of degrees of angle they move through during a certain portion of a revolution, and to refer to their relative velocities in terms of the ratio or proportion existing between their velocities. The first of these is termed the angular velocity, or the number of degrees of angle the wheel moves through during a given period, while the second is termed the velocity ratio of the pair of wheels. Let it be supposed that two wheels of equal diameter have contact at their perimeters so that one drives the other by friction without any slip, then the velocity of a point on the perimeter of one will equal that of a point on the other. Thus in Fig. 15 let A and B represent the pitch circles of two wheels, and C an imaginary line joining the axes of the two wheels and termed the line of centres. Now the point of contact of the two wheels will be on the line of centres as at D, and if a point or dot be marked at D and motion be imparted from A to B, then when each wheel has made a quarter revolution the dot on A will have arrived at E while that on B will have arrived at F. As each wheel has moved through one quarter revolution, it has moved through 90° of angle, because in the whole circle there is 360°, one quarter of which is 90°, hence instead of saying that the wheels have each moved through one quarter of a revolution we may say they have moved through an angle of 90°, or, in other words, their angular velocity has, during this period, been 90°. And as both wheels have moved through an equal number of degrees of angle their velocity ratio or proportion of velocity has been equal.

Obviously then the angular velocity of a wheel represents a portion of a revolution irrespective of the diameter of the wheel, while the velocity ratio represents the diameter of one in proportion to that of the other irrespective of the actual diameter of either of them.

Now suppose that in Fig. 16 A is a wheel of twice the diameter of B; that the two are free to revolve about their fixed centres, but that there is frictional contact between their perimeters at the line of centres sufficient to cause the motion of one to be imparted to the other without slip or lost motion, and that a point be marked on both wheels at the point of contact D. Now let motion be communicated to A until the mark that was made at D has moved one-eighth of a revolution and it will have moved through an eighth of a circle, or 45°. But during this motion the mark on B will have moved a quarter of a revolution, or through an angle of 90° (which is one quarter of the 360° that there are in the whole circle). The angular velocities of the two are, therefore, in the same ratio as their diameters, or two to one, and the velocity ratio is also two to one. The angular velocity of each is therefore the number of degrees of angle that it moves through in a certain portion of a revolution, or during the period that the other wheel of the pair makes a certain portion of a revolution, while the velocity ratio is the proportion existing between the velocity of one wheel and that of the other; hence if the diameter of one only of the wheels be changed, its angular velocity will be changed and the velocity ratio of the pair will be changed. The velocity ratio may be obtained by dividing either the radius, pitch, diameter, or number of teeth of one wheel into that of the other.

Conversely, if a given velocity ratio is to be obtained, the radius, diameter, or number of teeth of the driver must bear the same relation to the radius, diameter, or number of teeth of the follower, as the velocity of the follower is desired to bear to that of the driver.

If a pair of wheels have an equal number of teeth, the same pairs of teeth will come into action at every revolution; but if of two wheels one is twice as large as the other, each tooth on the small wheel will come into action twice during each revolution of the large one, and will work during each successive revolution with the same two teeth on the large wheel; and an application of the principle of the hunting tooth is sometimes employed in clocks to prevent the overwinding of their springs, the device being shown in Fig. 17, which is from "Willis' Principles of Mechanism."

For this purpose the winding arbor C has a pinion A of 19 teeth fixed to it close to the front plate. A pinion B of 18 teeth is mounted on a stud so as to be in gear with the former. A radial plate C D is fixed to the face of the upper wheel A, and a similar plate F E to the lower wheel B. These plates terminate outward in semicircular noses D, E, so proportioned as to cause their extremities to abut against each other, as shown in the figure, when the motion given to the upper arbor by the winding has brought them into the position of contact. The clock being now wound up, the winding arbor and wheel A will begin to turn in the opposite direction. When its first complete rotation is effected the wheel B will have gained one tooth distance from the line of centres, so as to place the stop D in advance of E and thus avoid a contact with E, which would stop the motion. As each turn of the upper wheel increases the distance of the stops, it follows from the principle of the hunting cog, that after eighteen revolutions of A and nineteen of B the stops will come together again and the clock be prevented from running down too far. The winding key being applied, the upper wheel A will be rotated in the opposite direction, and the winding repeated as above.

Thus the teeth on one wheel will wear to imbed one upon the other. On the other hand the teeth of the two wheels may be of such numbers that those on one wheel will not fall into gear with the same teeth on the other except at intervals, and thus an inequality on any one tooth is subjected to correction by all the teeth in the other wheel. When a tooth is added to the number of teeth on a wheel to effect this purpose it is termed a hunting cog, or hunting tooth, because if one wheel have a tooth less, then any two teeth which meet in the first revolution are distant, one tooth in the second, two teeth in the third, three in the fourth, and so on. The odd tooth is on this account termed a hunting tooth.

It is obvious then that the shape or form to be given to the teeth must, to obtain correct results, be such that the motion of the driver will be communicated to the follower with the velocity due to the relative diameters of the wheels at the pitch circles, and since the teeth move in the arc of a circle it is also obvious that the sides of the teeth, which are the only parts that come into contact, must be of same curve. The nature of this curve must be such that the teeth shall possess the strength necessary to transmit the required amount of power, shall possess ample wearing surface, shall be as easily produced as possible for all the varying conditions, shall give as many teeth in constant contact as possible, and shall, as far as possible, exert a pressure in a direction to rotate the wheels without inducing undue wear upon the journals of the shafts upon which the wheels rotate. In cases, however, in which some of these requirements must be partly sacrificed to increase the value of the others, or of some of the others, to suit the special circumstances under which the wheels are to operate, the selection is left to the judgment of the designer, and the considerations which should influence his determinations will appear hereafter.

Modern practice has accepted the curve known in general terms as the cycloid, as that best filling all the requirements of wheel teeth, and this curve is employed to produce two distinct forms of teeth, epicycloidal and involute. In epicycloidal teeth the curve forming the face of the tooth is designated an epicycloid, and that forming the flank an hypocycloid. An epicycloid may be traced or generated, as it is termed, by a point in the circumference of a circle that rolls without slip upon the circumference of another circle. Thus, in Fig. 18, A and B represent two wooden wheels, A having a pencil at P, to serve as a tracing or marking point. Now, if the wheels are laid upon a sheet of paper and while holding B in a fixed position, roll A in contact with B and let the tracing point touch the paper, the point P will trace the curve C C. Suppose now the diameter of the base circle B to be infinitely large, a portion of its circumference may be represented by a straight line, and the curve traced by a point on the circumference of the generating circle as it rolls along the base line B is termed a cycloid. Thus, in Fig. 19, B is the base line, A the rolling wheel or generating circle, and C C the cycloidal curve traced or marked by the point D when A is rolled along B. If now we suppose the base line B to represent the pitch line of a rack, it will be obvious that part of the cycloid at one end is suitable for the face on one side of the tooth, and a part at the other end is suitable for the face of the other side of the tooth.

A hypocycloid is a curve traced or generated by a point on the circumference of a circle rolling within and in contact (without slip) with another circle. Thus, in Fig. 20, A represents a wheel in contact with the internal circumference of B, and a point on its circumference will trace the two curves, C C, both curves starting from the same point, the upper having been traced by rolling the generating circle or wheel A in one direction and the lower curve by rolling it in the opposite direction.

To demonstrate that by the epicycloidal and hypocycloidal curves, forming the faces and flanks of what are known as epicycloidal teeth, motion may be communicated from one wheel to another with as much uniformity as by frictional contact of their circumferential surfaces, let A, B, in Fig. 21, represent two plain wheel disks at liberty to revolve about their fixed centres, and let C C represent a margin of stiff white paper attached to the face of B so as to revolve with it. Now suppose that A and B are in close contact at their perimeters at the point G, and that there is no slip, and that rotary motion commenced when the point E (where as tracing point a pencil is attached), in conjunction with the point F, formed the point of contact of the two wheels, and continued until the points E and F had arrived at their respective positions as shown in the figure; the pencil at E will have traced upon the margin of white paper the portion of an epicycloid denoted by the curve E F; and as the movement of the two wheels A, B, took place by reason of the contact of their circumferences, it is evident that the length of the arc E G must be equal to that of the arc G F, and that the motion of A (supposing it to be the driver) would be communicated uniformly to B.

Now suppose that the wheels had been rotated in the opposite direction and the same form of curve would be produced, but it would run in the opposite direction, and these two curves may be utilized to form teeth, as in Fig. 22, the points on the wheel A working against the curved sides of the teeth on B.

To render such a pair of wheels useful in practice, all that is necessary is to diminish the teeth on B without altering the nature of the curves, and increase the diameter of the points on A, making them into rungs or pins, thus forming the wheels into what is termed a wheel and lantern, which are illustrated in Fig. 23.

A represents the pinion (or lantern), and B the wheel, and C, C, the primitive teeth reduced in thickness to receive the pins on A. This reduction we may make by setting a pair of compasses to the radius of the rung and describing half-circles at the bottom of the spaces in B. We may then set a pair of compasses to the curve of C, and mark off the faces of the teeth of B to meet the half-circles at the pitch line, and reduce the teeth heights so as to leave the points of the proper thickness; having in this operation maintained the same epicycloidal curves, but brought them closer together and made them shorter. It is obvious, however, that such a method of communicating rotary motion is unsuited to the transmission of much power; because of the weakness of, and small amount of wearing surface on, the points or rungs in A.

In place of points or rungs we may have radial lines, these lines, representing the surfaces of ribs, set equidistant on the radial face of the pinion, as in Fig. 24. To determine the epicycloidal curves for the faces of teeth to work with these radial lines, we may take a generating circle C, of half the diameter of A, and cause it to roll in contact with the internal circumference of A, and a tracing point fixed in the circumference of C will draw the radial lines shown upon A. The circumstances will not be altered if we suppose the three circles, A, B, C, to be movable about their fixed centres, and let their centres be in a straight line; and if, under these circumstances, we suppose rotation to be imparted to the three circles, through frictional contact of their perimeters, a tracing point on the circumference of C would trace the epicycloids shown upon B and the radial lines shown upon A, evidencing the capability of one to impart uniform rotary motion to the other.

To render the radial lines capable of use we must let them be the surfaces of lugs or projections on the face of the wheel, as shown in Fig. 25 at D, E, &c., or the faces of notches cut in the wheel as at F, G, H, &c., the metal between F and G forming a tooth J, having flanks only. The wheel B has the curves of each tooth brought closer together to give room for the reception of the teeth upon A. We have here a pair of gears that possess sufficient strength and are capable of working correctly in either direction.