List ofplatesVol. II.
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.
Fig. 1Fig. 1.
Fig. 1.
When the teeth are arranged to be parallel to the wheel-axis, as inFig. 1, the wheel is termed a spur-wheel. In the figure,arepresents 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.
Fig. 2Fig. 2.
Fig. 2.
Fig. 3Fig. 3.
Fig. 3.
Fig. 4Fig. 4.
Fig. 4.
Fig. 5Fig. 5.
Fig. 5.
When the wheel has its teeth arranged at an angle to the shaft, as inFig. 2, it is termed a bevel-wheel, or bevel gear; but when this angle is one of 45°, as inFig. 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 inFig. 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 inFig. 5, it is termed a lantern, trundle, or wallower; and each cylindrical piece serving as a tooth is termed astave,spindle, orround, and by some aleaf.
Fig. 6Fig. 6.
Fig. 6.
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 inFig. 6; hence the pinionpoperates 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.
Fig. 7Fig. 7.
Fig. 7.
When the teeth are arranged along a plane surface or straight line, as inFig. 7, the toothed plane is termed arack, and the wheel is termed a pinion.
Fig. 8Fig. 8.
Fig. 8.
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 inFig. 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 samepitch, as it is called.
Fig. 9Fig. 9.
Fig. 9.
The designations of the various parts or surfaces of a tooth of a gear-wheel are represented inFig. 9, in which the surfaceais the face of the tooth, while the dimensionfis the width of face of the wheel, when its size is referred to.bis the flank or distance from the pitch line to the root of the tooth, andcthepoint.his thespace, 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 circlepp.eis the depth of the tooth, andgits thickness, the latter also being measured on the pitch circlepp. When spoken of with reference to a tooth,p pis called the pitch line, but when the whole wheel is referred to it becomes the pitch circle.
The pointscand the surfacehare 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.
Fig. 10Fig. 10.
Fig. 10.
The pitch, however, may be measured in two ways, to wit, around the pitch circlea, inFig. 10, which is called the arc or circular pitch, and acrossb, 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.
The principle upon which diametral pitch is based is asfollows:—
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, as1⁄2,1⁄3,1⁄4, 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 of1⁄2,1⁄3, or1⁄4, 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 of1⁄2(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 fraction1⁄5) we have1⁄5× 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 11⁄2inches? 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 32⁄10outside 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 51⁄4inches, because the diametral pitch is1⁄8, and this multiplied by 2 gives1⁄4, which added to the pitch circle diameter of 5 inches makes 51⁄4inches, 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 be4⁄10inch; 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.
From this table we find that the nearest diametral pitch that will correspond to an arc pitch of4⁄10inch 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.
Fig. 11Fig. 11.
Fig. 11.
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 inthat 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 inFig. 11, in whicha,b,c,eare four wheels in gear,bandcbeing compounded, that is, so held together on the shaftdthat both make an equal number of revolutions in a given time. Hence the velocity ofcwill be less than that ofbin proportion as the diameter, circumference, radius, or number of teeth inc, varies from the diameter, radius, circumference, or number of teeth (all the wheels being supposed to have teeth of the same pitch) inb, although the rotations ofbandcare equal. It is most convenient, and therefore usual, to take the number of teeth, but if the teeth onc(and therefore those onealso) were of different pitch from those onb, 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 (ascin the figure) is of less radius than the followerb, 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 equals3⁄10of 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 (3⁄10of 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 asfollows:—
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 321⁄2inches; 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:—
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 183⁄8inches. Then the acting effective or operative radii of the pitch circles will bear the same proportion to the 183⁄8as 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 wheelis 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 by1⁄4or1⁄2inch at a step, and a pitch measuring about, say, 11⁄4inch chord pitch, would be known to be 11⁄4arc 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 11⁄8, or more often 11⁄4inch, and the difference between the arc and chord pitch of the smallest wheel would not amount to anything near1⁄8inch, hence there would be no liability to mistake a pitch of 11⁄8for 1 inch orvice 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 (as1⁄4,9⁄32,5⁄16, &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.
Fig. 12Fig. 12.
Fig. 12.
InFig. 12, leta,b, andcrepresent the pitch circles of three gears of whichaandbare in gear, whilecis compounded withb; letebe the shaft ofa, andgthat forbandc. Letabe 60 inches,b= 30 inches, andc= 40 inches in diameter. Now suppose that shaftesuspends 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 ofeand the 50 must be multiplied by 2, making 100 lbs. at the centre ofe. Then at the perimeter ofathis 100 will become one-thirtieth of one hundred, because from the centre to the perimeter ofais 30. One-thirtieth of 100 is 333⁄100lbs., which will be the force exerted byaon the perimeter ofb. Now from the perimeter ofbto its centre (or in other words its radius) is 15 inches, hence the 333⁄100lbs. at its perimeter will become fifteen times as much at the centregofb, and 333⁄100× 15 = 4995⁄100lbs. From the centregto the perimeter ofcbeing 20 inches, the 4995⁄100lbs. at the centre will be only one-twentieth of that amount at the perimeter ofc, hence 4995⁄100÷ 20 = 249⁄100lbs., which is the amount of force at the perimeter ofc.
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, thatahas 60 teeth,bhas 30 teeth, andchas 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 ona) becomes 333⁄100, which multiplied by 30 (the number of teeth onb) becomes 9990⁄100, which divided by 40 (the number of teeth onc) becomes 249⁄100lbs. 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.