Fig. 182Fig. 182.
Fig. 182.
As an evidence of the durability of wooden teeth, there appeared inEngineeringof January 7th, 1879, the illustration shown inFig. 182, which represents a cog from a wheel of 14 ft.1⁄2in. diameter, and having a 10-inch face, its pinion being 4 ft. in diameter. This cog had been running for 261⁄2years, day and night; not a cog in the wheel having been touched during that time. Its average revolutions were 38 per minute, the power developed by the engine being from 90 to 100 indicated horse-power. The teeth were composed of beech, and had been greased twice a week, with tallow and plumbago ore.
Since the width of the face of a wheel influences its wear (by providing a larger area of contact over which the pressure may be distributed, as well as increasing the strength), two methods of proportioning the breadth may be adopted. First, it may be made a certain proportion of the pitch; and secondly, it may be proportioned to the pressure transmitted and the number of revolutions. The desirability of the second is manifest when we consider that each tooth will pass through the arcs of contact (and thus be subjected to wear) once during each revolution; hence, by making the number of revolutions an element in the calculation to find the breadth, the latter is more in proportion to the wear than it would be if proportioned to the pitch.
It is obvious that the breadth should be sufficient to afford the required degree of strength with a suitable factor of safety, and allowance for wear of the smallest wheel in the pair or set, as the case may be.
According to Reuleaux, the face of a wheel should never be less than that obtained by multiplying the gross pressure, transmitted in lbs., by the revolutions per minute, and dividing the product by 28,000.
In the case of bevel-wheels the pitch increases, as the perimeterof the wheel is approached, and the maximum pitch is usually taken as the designated pitch of the wheel. But the mean pitch is that which should be taken for the purposes of calculating the strength, it being in the middle of the tooth breadth. The mean pitch is also the diameter of the pitch circle, used for ascertaining the velocity of the wheel as an element in calculating the safe pressure, or the amount of power the wheel is capable of transmitting, and it is upon this basis that the values for bevel-wheels in the above table are computed.
In many cases it is required to find the amount of horse-power a wheel will transmit, or the proportions requisite for a wheel to transmit a given horse-power; and as an aid to the necessary calculations, the following table is given of the amount of horse-power that may be transmitted with safety, by the various wheels at the given velocities, with a wheel of an inch pitch and an inch face, from which that for other pitches and faces may be obtained by proportion.
In this table, as in the preceding one, the safe working pressure for 1-inch pitch and 1-inch breadth of face is supposed to be 400 lbs.
In cast gearing, the mould for which is made by a gear moulding machine, the element of draft to permit the extraction of the pattern is reduced: hence, the pressure of tooth upon tooth may be supposed to be along the full breadth of the tooth instead of at one corner only, as in the case of pattern-moulded teeth. But from the inaccuracies which may occur from unequal contraction in the cooling of the casting, and from possible warping of the casting while cooling, which is sure to occur to some extent, however small the amount may be, it is not to be presumed that the contact of the teeth of one wheel will be in all the teeth as perfect across the full breadth as in the case of machine-cut teeth. Furthermore, the clearance allowed for machine-moulded teeth, while considerably less than that allowed for pattern-moulded teeth, is greater than that allowed for machine-cut teeth; hence, the strength of machine-moulded teeth in proportion to the pitch lies somewhere between that of pattern-moulded and machine-cut teeth—but exactly where, it would be difficult to determine in the absence of experiments made for the purpose of ascertaining.
It is not improbable, however, that the contact of tooth upon tooth extends in cast gears across at least two-thirds of the breadth of the tooth, in which case the rules for ascertaining the strength of cut teeth of equal thickness may be employed, substituting2⁄3rds of the actual tooth breadth as the breadth for the purposes of the calculation.
If instead of supposing all the strain to fall upon one tooth and calculating the necessary strength of the teeth upon that basis (as is necessary in interchangeable gearing, because these conditions may exist in the case of the smallest pinion that can be used in pitch), the actual working condition of each separate application of gears be considered, it will appear that with a given diameter of pitch circle, all other things being equal, the arc of contact will remain constant whatever the pitch of the teeth, or in other words is independent of the pitch, and it follows that when the thickness of iron necessary to withstand (with the allowances for wear and factor of safety) the given stress under the given velocity has been determined, it may be disposed in a coarse pitch that will give one tooth always in contact, or a finer pitch that will give two or more teeth always in contact, the strength in proportion to the duty remaining the same in both cases.
In this case the expense of producing the wheel patterns or in trimming the teeth is to be considered, because if there are a train of wheels the finer pitch would obviously involve the construction and dressing to shape of a much greater number of teeth on each wheel in the train, thus increasing the labor. When, however, it is required to reduce the pinion to a minimum diameter, it is obvious that this may be accomplished by selecting the finer pitch, because the finer the pitch, the less the diameter of the wheel may be. Thus with a given diameter of pitch circle it is possible to select a pitch so fine that motion from one wheel may be communicated to another, whatever the diameter of the pitch circle may be, the limit being bounded by the practicability of casting or producing teeth of the necessary fineness of pitch. The durability of a wheel having a fine pitch is greater for two reasons: first, because the metal nearest the cast surface of cast iron is stronger than the internal metal, and the finer pitch would have more of this surface to withstand the wear; and second, because in a wheel of a given width there would be two points, or twice the area of metal, to withstand the abrasion, it being remembered that the point of contact is a line which partly rolls and partly slides along the depth of the tooth as the wheel rotates, and that with two teeth in contact on each wheel there are two of such lines. There is also less sliding or rubbing action of the teeth, but this is offset by the fact that there are more teeth in contact, and that there are therefore a greater number of teeth simultaneously rubbing or sliding one upon the other.
But when we deal with the number of teeth the circumstances are altered; thus with teeth of epicycloidal form it is manifestly impossible to communicate constant motion with a driving wheel having but one tooth, or to receive motion on a follower having but one tooth. The number of teeth must always be such that there is at all times a tooth of each wheel within the arc of action, or in contact, so that one pair of teeth may come into contact before the contact of the preceding teeth has ceased.
In the construction of wheels designed to transmit power as well as simple motion, as is the case with the wheels employed in machine work, however, it is not considered desirable to employ wheels containing a less number of teeth than 12. The diameter of the wheel bearing such a relation to the pitch that both wheels containing the same number of teeth (12), the motion will be communicated from one to the other continuously.
It is obvious that as the number of teeth in one of the wheels (of a pair in gear) is increased the number of teeth in the other may be (within certain limits) diminished, and still be capable of transmitting continuous motion. Thus a pinion containing, say 8 teeth, may be capable of receiving continuous motion from a rack in continuous motion, while it would not be capable of receiving continuous motion from a pinion having 4 teeth; and as the requirements of machine construction often call for the transmission of motion from one pinion to another of equal diameters, and as small as possible, 12 teeth are the smallest number it is considered desirable for a pinion to contain, except it be in the case of an internal wheel, in which the arc of contact is greater in proportion to the diameters than in spur-wheels, and continuous motion can therefore be transmitted either with coarser pitches or smaller diameters of pinion.
For convenience in calculating the pitch diameter at pitch circle, or pitch diameter as it is termed, and the number of teeth of wheels, the following rules and table extracted from theCincinnati Artisanand arranged from a table by D. A. Clarke, are given. The first column gives the pitch, the following nine columns give the pitch diameters of wheels for each pitch from 1 tooth to 9. By multiplying these numbers by 10 we have the pitch diameters from 10 to 90 teeth, increasing bytens; by multiplying by 100 we likewise have the pitch diameters from 100 to 900, increasing byhundreds.
The following rules and examples show how the table is used:
Rule 1.—Given —— number of teeth and pitch; to find —— pitch diameter.
Select from table in columns opposite the givenpitch—
First, the value corresponding to the number of units in the number of teeth.
Second, the value corresponding to the number of tens, and multiply this by 10.
Third, the value corresponding to the number of hundreds, and multiply this by 100. Add these together, and their sum is the pitch diameter required.
Example.—What is the pitch diameter of a wheel with 128 teeth, 11⁄2inches pitch?
We find in line corresponding to 11⁄2inchpitch—
Rule 2.—Given —— pitch diameter and number of teeth; to find —— pitch.
First, ascertain by Rule 1 the pitch diameter for a wheel of 1-inch pitch, and the givennumber of teeth.
Second, dividegiven pitch diameterby thepitch diameterfor 1-inch pitch.
The quotient is the pitch desired.
Example.—What is the pitch of a wheel with 148 teeth, the pitch diameter being72′′?
First, pitch diameter for 148 teeth, 1-inch pitch,is—
This is nearly 11⁄2-inch pitch, and if possible the diameter would be reduced or the number of teeth increased so as to make the wheel exactly 11⁄2-inch pitch.
Rule 3.—Given —— pitch and pitch diameter; to find —— number of teeth.
First, ascertain from table thepitch diameterfor 1toothof the givenpitch.
Second, divide thegiven pitch diameterby thevaluefound in table.
The quotient is the number required.
Example.—What is the number of teeth in a wheel whose pitch diameter is 42 inches, and pitch is 21⁄2inches?
First, the pitch diameter, 1 tooth, 21⁄2-inch pitch, is 0.7958 inches.
This gives a fractional number of teeth, which is impossible; so the pitch diameter will have to be increased to correspond to 53 teeth, or the pitch changed so as to have the number of teeth come an even number.
Whenever two parallel shafts are connected together by gearing, the distance between centres being a fixed quantity, and the speeds of the shafts being of a fixed ratio, then the pitch is generally the best proportion to be changed, and necessarily may not be of standard size. Suppose there are two shafts situated in this manner, so that the distance between their centres is 84 inches, and the speed of one is 21⁄2times that of the other, what size wheels shall be used? In this case the pitch diameter and number of teeth of the wheel on the slow-running shaft have to be 21⁄2times those of the wheel on the fast-running shaft; so that 84 inches must be divided into two parts, one of which is 21⁄2times the other, and these quantities will be the pitch radii of the wheels; that is, 84 inches are to be divided into 31⁄2equal parts, 1 of which is the radius of one wheel, and 21⁄2of which the radius of the other, thus84′′/31⁄2= 24 inches. So that 24 inches is the pitch radius of pinion, pitch diameter = 48 inches; and 21⁄2× 24 inches = 60 inches is the pitch radius of the wheel, pitch diameter = 120 inches. The pitch used depends upon the power to be transmitted; suppose that 25⁄8inches had been decided as about the pitch to be used, it is found by Rule 3 that the number of teeth are respectively 143.6, and 57.4 for wheel and pinion. As this is impossible, some whole number of teeth, nearest these in value,have to be taken, one of which is 21⁄2times the other; thus 145 and 58 are the nearest, and the pitch for these values is found by Rule 2 to be 2.6 inches, being the best that can be done under the circumstances.
Fig. 183Fig. 183.
Fig. 183.
Fig. 184Fig. 184.
Fig. 184.
The forms of spur-gearing having their teeth at an angle to the axis, or formed in advancing steps shown inFigs. 183and184, were designed by Dr. Hooke, and “were intended,” says the inventor, “first to make a piece of wheel work so that both the wheel and pinion, though of never so small a size, shall have as great a number of teeth as shall be desired, and yet neither weaken the wheels nor make the teeth so small as not to be practicable by any ordinary workman. Next that the motion shall be so equally communicated from the wheel to the pinion that the work being well made there can be no inequality of force or motion communicated.
“Thirdly, that the point of touching and bearing shall be always in the line that joins the two centres together.
“Fourthly, that it shall haveno manner of rubbing, nor be more difficult to make than common wheel work.”
Fig. 185Fig. 185.
Fig. 185.
The objections to this form of wheel lies in the difficulty of making the pattern and of moulding it in the foundry, and as a result it is rarely employed at the present day. For racks, however, two or more separate racks are cast and bolted together to form the full width of rack as shown inFig. 185. This arrangement permits of the adjustment of the width of step so as to take up the lost motion due to the wear of the tooth curves.
Another objection to the sloping of the teeth, as inFig. 183, is that it induces an end pressure tending to force the wheels apartlaterally, and this causesendwear on the journals and bearings.
Fig. 186Fig. 186.
Fig. 186.
To obviate this difficulty the form of gear shown inFig. 186is employed, the angles of the teeth from each side of the wheel to its centre being made equal so as to equalize the lateral pressure. It is obvious that the stepped gear,Fig. 184, is simply equivalent to a number of thin wheels bolted together to form a thick one, but possessing the advantage that with a sufficient number of steps, as in the figure, there is always contact on the line of centres, and that the condition of constant contact at the line of centres will be approached in proportion to the number of steps in the wheel, providing that the steps progress in one continuous direction across the wheel as inFig. 184. The action of the wheels will, in this event, be smoother, because there will be less pressure tending to force the wheels apart.
But in the form of gearing shown inFig. 183, the contact of the teeth will bear every instant at a single point, which, as the wheels revolve, will pass from one end to the other of the tooth, a fresh contact always beginning on the first side immediately before the preceding contact has ceased on the opposite side. The contact, moreover, being always in the plane of the centres of the pair, the action is reduced to that of rolling, and as there is no sliding motion there is consequently no rubbing friction between the teeth.
Fig. 187Fig. 187.
Fig. 187.
Fig. 188Fig. 188.
Fig. 188.
A further modification of Dr. Hooke’s gearing has been somewhat extensively adopted, especially in cotton-spinning machines. This consists, when the direction of the motion is simply to be changed to an angle of 90°, in forming the teeth upon the periphery of the pair at an angle of 45° to the respective axes of the wheels, as inFigs. 187and188; it will then be perceived that if the sloped teeth be presented to each other in such a way as to have exactly the same horizontal angle, the wheels will gear together, and motion being communicated to one axis the same will be transmitted to the other at a right angle to it, as in a common bevel pair. Thus if the wheelaupon a horizontal shaft have the teeth formed upon its circumference at an angle of 45° to the plane of its axis it can gear with a similar wheelbupon a vertical axis. Let it be upon the driving shaft and the motion will be changed in direction as ifaandbwere a pair of bevel-wheels of the ordinary kind, and, as with bevels generally, the direction of motion will be changed through an equal angle to the sum of the angles which the teeth of the wheels of the pair form with their respective axes. The objection in respect of lateral or end pressure, however, applies to this form equally with that shown inFig. 183, but in the case of a vertical shaft the end pressure may be (by sloping the teeth in the necessary direction) made to tend to lift the shaft and not force it down into the step bearing. This would act to keep the wheels in close contact by reason of the weight of the vertical shaft and at the same time reduce the friction between the end of that shaft and its step bearing. This renders this form of gearing preferable to skew bevels when employed upon vertical shafts.
It is obvious that gears, such as shown inFigs. 187and188may be turned up in the lathe, because the teeth are simply portions of spirals wound about the circumference of the wheel. For a pair of wheels of equal diameter a cylindrical piece equal in length to the required breadth of the two wheels is turned up in the lathe, and the teeth may be cut in the same manner as cutting a thread in the lathe, that is to say, by traversing the tool the requisite distance per lathe revolution. In pitches above about1⁄4inch, it will be necessary to shape one side of the tooth at a time on account of the broadness of the cutting edges. After the spiral (for the teeth are really spirals) is finished thepiece may be cut in two in the lathe and each half will form a wheel.
To find the full diameter to which to turn a cylinder for a pair of these wheels we proceed as in the following example: Required to cut a spiral wheel 5 inches in diameter and to have 30 teeth. First find the diametral pitch, thus 30 (number of teeth) ÷ 5 (diameter of wheel at pitch circle) = 6; thus there are 6 teeth or 6 parts to every inch of the wheel’s diameter at the pitch circle; adding 2 of these parts to the diameter of the wheel, at the pitch circle we have 5 and2⁄6of another inch, or 52⁄6inches, which is the full diameter of the wheel, or the diameter of the addendum, as it is termed.
Fig. 189Fig. 189.
Fig. 189.
It is now necessary to find what change wheels to put on the lathe to cut the teeth out the proper angle. Suppose then the axes of the shafts are at a right angle one to the other, and that the teeth therefore require to be at an angle of 45° to the axes of the respective wheels, then we have the following considerations. InFig. 189let the linearepresent the circumference of the wheel, andba line of equal length but at a right angle to it, then the linec, joininga,b, is at an angle of 45°. It is obvious then that if the traverse of the lathe tool be equal at each lathe revolution to the circumference of the wheel at the pitch circle, the angle of the teeth will be 45° to the axis of the wheel.
Hence, the change wheels on the lathe must be such as will traverse the tool a distance equal to the circumference at pitch circle of the wheel, and the wheels may be found as for ordinary screw cutting.
If, however, the axes of the shafts are at any other angle we may find the distance the lathe tool must travel per lathe revolution to give teeth of the required angle (or in other words the pitch of the spiral) by direct proportion, thus: Let it be required to find the angle or pitch for wheels to connect shafts at an angle of 25°, the wheels to have 20 teeth, and to be of 10 diametral pitch.
Here, 20 ÷ 10 = 2 = diameter of wheel at the pitch circle. The circumference of 2 inches being 6.28 inches we have, as the degrees of angle of the axes of the shafts are to 45°, so is 6.28 inches (the circumference of the wheels, to the pitch sought).
Here, 6.28 inches × 45° ÷ 25° = 11.3 inches, which is the required pitch for the spiral.
Fig. 190Fig. 190.
Fig. 190.
When the axes of the shafts are neither parallel nor meeting, motion from one shaft to another may be transmitted by means of a double gear. Thus (taking rolling cones of the diameters of the respective pitch circles as representing the wheels) inFig. 190, letabe the shaft of gearh, andbbthat of wheele. Then a double gear-wheel having teeth onf,gmay be placed as shown, and the facefwill gear withe, while facegwill gear withh, the cone surfaces meeting in a point as atcanddrespectively, hence the velocity will be equal.
Fig. 191Fig. 191.
Fig. 191.
When the axial line of the shafts for two gear-wheels are nearly in line one with the other, motion may be transmitted by gearing the wheels as inFig. 191. This is a very strong method of gearing, because there are a large number of teeth in contact, hence the strain is distributed by a larger number of teeth and the wear is diminished.
Fig. 192Fig. 192.
Fig. 192.
Fig. 192(from Willis’s “Principles of Mechanism”) is another method of constructing the same combination, which admits of a steady support for the shafts at their point of intersection,abeing a spherical bearing, andb,cbeing cupped to fit toa.
Rotary motion variable at different parts of a rotation may be obtained by means of gear-wheels varied in form from the true circle.
Fig. 193Fig. 193.
Fig. 193.
The commonest form of gearing for this purpose is elliptical gearing, the principles governing the construction of which are thus given by Professor McCord. “It is as well to begin at the foundation by defining the ellipse as a closed plane-curve, generated by the motion of a point subject to the condition that the sum of its distances from two fixed points within shall be constant: Thus, inFig. 193,aandbare the two fixed points, called thefoci;l,e,f,g,pare points in the curve; anda f+f b=a e+e b. Also,a l+l b=a p+p b=a g+g b. From this it follows thata g=l o,obeing the centre of the curve, andgthe extremity of the minor axis, whence the foci may be found if the axes be assumed, or, if the foci and one axis be given, the other axis may be determined. It is also apparent that if about either focus, asb, we describe an arc with a radius greater thanb pand less thanb l, for instanceb e, and aboutaanother arc with radiusa e=l p-b e, the intersection,e, of these arcs will be on the ellipse; and in this manner any desired number of points may be found, and the curve drawn by the aid of sweeps.
“Having completed this ellipse, prolong its major axis, and drawa similar and equal one, with its foci,c,d, upon that prolongation, and tangent to the first one atp; thenb d=l p. Aboutbdescribe an arc with any radius, cutting the first ellipse atyand the linelatz; aboutddescribe an arc with radiusd z, cutting the second ellipse inx; drawa y,b y,c x, andd x. Thena y=d x, andb y=c x, and because the ellipses are alike, the arcsp yandp xare equal. If thenbanddare taken as fixed centres, and the ellipses turn about them as shown by the arrows,xandywill come together atzon the line of centres; and the same is true of any points equally distant frompon the two curves. But this is the condition of rolling contact. We see, then, that in order that two ellipses may roll together, and serve as the pitch-lines of wheels, they must be equal and similar, the fixed centres must be at corresponding foci, and the distance between these centres must be equal to the major axis. Were they to be toothless wheels, if would evidently be essential that the outlines should be truly elliptical; but the changes of curvature in the ellipse are gradual, and circular arcs may be drawn so nearly coinciding with it, that when teeth are employed, the errors resulting from the substitution are quite inappreciable. Nevertheless, the rapidity of these changes varies so much in ellipses of different proportions, that we believe it to be practically better to draw the curve accurately first, and to find the radii of the approximating arcs by trial and error, than to trust to any definite rule for determining them; and for this reason we give a second and more convenient method of finding points, in connection with the ellipse whose centre isr,Fig. 193. About the centre describe two circles, as shown, whose diameters are the major and minor axes; draw any radius, asr t, cutting the first circle int, and the second ins; throughtdraw a parallel to one axis, throughsa parallel to the other, and the intersection,v, will lie on the curve. In the left hand ellipse, the line bisecting the anglea f bis normal to the curve atf, and the perpendicular to it is tangent at the same point, and bisects the angles adjacent toa f b, formed by prolonginga f,b f.
“To mark the pitch line we proceed asfollows:—