Chapter III: THE TEETH OF GEAR-WHEELS (continued) (4)

In Fig. 176, X is a frame or box containing and holding the operative part of the tooth, and resting on two guides C D. The height of D from the saw table is sufficiently greater than that of C to give the shank G the correct taper, E F representing the circular saw. T is a plain piece of the full size of the box or frame, and serving simply to close up on that side the mortise in the frame. The grain of T should run at a right angle to the other piece of the frame so as to strengthen it. S is a binding screw to hold the cog on the frame, and H is a guide for the edge of the frame to slide against. It is obvious, now, that if the piece D be adjusted at a proper distance from the circular saw E F, and the edge of the frame be moved in contact with the guide H, one side of the tooth shank will be sawn. Then, by reversing the frame end for end, the other side of the shank may be sawn. Turning the frame to a right angle the edges of the cog shank can be sawn from the same box or frame, and pieces C, D, as shown in Fig. 177.

The frame is now stood on edge, as in Fig. 178, and the underneath surfaces sawed off to the depth the saw entered when the shank taper was sawn. This operation requires to be performed on all four sides of the tooth.

After this operation is performed on one cog, it should be tried in the wheel mortises, to test its correctness before cutting out the shanks on all the teeth.

The shanks, being correctly sawn, may then be fitted to the mortises, and let in within 1/8 of butting down on the face of the wheel, this amount being left for the final driving. The cogs should be numbered to their places, and two of the mortises must be numbered to show the direction in which the numbers proceed. To mark the shoulders (which are now square) to the curvature of the rim, a fork scriber should be used, and the shanks of the cogs should have marked on them a line coincident with the inner edge of the wheel rim. This line serves as a guide in marking the pin-holes and for cutting the shanks to length; but it is to be remembered that the shanks will pass farther through to the amount of the distance marked by the fork scriber. The holes for the pins which pass through the shanks should be made slightly less in their distances (measured from the nearest edge of the pin-hole) from the shoulders of the cogs than is the thickness of the rim of the wheel, so that when the cogs are driven fully home the pin-holes will appear not quite full circles on the inside of the wheel rim; hence, the pins will bind tightly against the inside of the wheel rim, and act somewhat as keys, locking and drawing the shanks to their seats in the mortises.

In cases where quietness of running is of more consequence than the durability of the teeth, or where the wear is not great, both wheels may be cogged, but as a rule the larger wheel is cogged, the smaller being of metal. This is done because the teeth of the smaller wheel are the most subject to wear. The teeth of the cogged wheel are usually made the thickest, so as to somewhat equalise the strength of the teeth on the two wheels.

Since the power transmitted by a wheel in a given time is composed of the pressure or weight upon the wheel, and the space a point on the pitch circle moves through in the given time, it is obvious that in a train of wheels single geared, the velocities of all the wheels in the train being equal at the pitch circle, the teeth require to be of equal pitch and thickness throughout the train. But when the gearing is compounded the variation of velocity at the pitch circle, which is due to the compounding, has an important bearing upon the necessary strength of the teeth.

Suppose, for example, that a wheel receives a tooth pressure of 100 lbs. at the pitch circle, which travels at the velocity of 100 feet per minute, and is keyed to the same shaft with another wheel whose velocity is 50 feet per minute. Now, in the power transmitted by the two wheels the element of time is 50 for one wheel and 100 for the other, hence the latter (supposing both wheels to have an equal number of teeth in contact with their driver or follower as the case may be) will be twice as strong in proportion to the duty, and it appears that in compounded gearing the strength in proportion to the duty may be varied in proportion as the velocity is modified by compounding of the wheels. Thus, when the velocity at the pitch circle is increased its strength is increased, and per contra when its velocity is decreased its strength is decreased, when considered in proportion to the duty. When, however, the wheels are upon long shafts, or when they overhang the bearing of the shaft, the corner contact will from tension of the shaft, continue much longer than when the shaft is maintained rigid.

It is obvious that if a wheel transmits a certain amount of power, the pressure of tooth upon tooth will depend upon the number of teeth in contact, but since, in the case of very small wheels, that is to say, pinions of the smallest diameter of the given pitch that will transmit continuous motion, it occurs that only one tooth is in continuous contact, it is obvious that each single tooth must have sufficient strength to withstand the whole of the pressure when worn to the limits to which the teeth are supposed to wear. But when the pinion is so small that it has but one tooth in continuous contact, that contact takes place nearer the line of centres and to the root of the tooth, and therefore at a less leverage to the line of fracture, hence the ultimate strength of the tooth is proportionately increased. On the other hand, however, the whole stress of the wheel being concentrated on the arc of contact of one tooth only (instead of upon two or more teeth as in larger wheels), the wear is proportionately greater; hence, in a short time the teeth of the pinion are found to be thinner than those on the other wheel or wheels. The multiplicity of conditions under which small wheels may work with relation to the number of teeth in contact, the average leverage of the point of contact from the root of the tooth, the shape of the tooth, &c., renders it desirable in a general rule to suppose that the whole strain falls upon one tooth, so that the calculation shall give results to meet the requirements when a single tooth only is in continuous contact.

It follows, then, that the thickness of tooth arrived at by calculation should be that which will give to a tooth, when worn to the extreme thinness allowed, sufficient strength (with a proper margin of safety) to transmit the whole of the power transmitted by the wheel.

The margin (or factor) of safety, or in other words, the number of times the strength of the tooth should exceed the amount of power transmitted, varies (according to the conditions under which the wheels work) between 5 and 10.

The lesser factor may be used for slow speeds when the power is continuously and uniformly transmitted. The greater factor is necessary when the wheels are subjected to violent shocks and the direction of revolution requires to be reversed.

In pattern-cast teeth, contact between the teeth of one wheel and those of the other frequently occurs at one corner only, as shown in Fig. 179, and the line of fracture is in the direction denoted by the diagonal dotted lines. The causes of this corner contact have been already explained, but it may be added that as the wheels wear, the contact extends across the full breadths of the teeth, and the strength in proportion to the duty, therefore, steadily increases from the time the new wheels have action until the wear has caused contact fully across the breadth. Tredgold's rule for finding the proper thickness of tooth for a given stress upon cast-iron teeth loaded at the corner as in Fig. 179 and supposed to have a velocity of three feet per second of time, is as follows:--

Rule.--Divide the stress in pounds at the pitch circle by 1500, and the square root of the quotient is the required thickness of tooth in inches or parts of an inch.

In the results obtained by the employment of this rule, an allowance of one-third the thickness for wear, and the margin for safety is included, so that the thickness of tooth arrived at is that to be given to the actual tooth. Further, the rule supposes the breadth of the tooth to be not less than twice the height of the same, any extra breadth not affecting the result (as already explained), when the pressure falls on a corner of the tooth.

In practical application, however, the diameter of the wheel at the pitch circle is generally, or at least often a fixed quantity, as well as the amount of stress, and it will happen as a rule that taking the stress as a fixed element and arriving at the thickness of the tooth by calculation, the required diameter of wheel, or what is the same thing, its circumference, will not be such as to contain the exact number of teeth of the thickness found by the calculation, and still give the desired amount of side clearance. It is desirable, therefore, to deal with the stress upon the tooth at the pitch circle, and the diameter, radius, or circumference of the pitch circle, and its velocity, and deduce therefrom the required thickness for the teeth, and conform the pitch to the requirements as to clearance from the tooth thickness thus obtained.

To deduce the thickness of the teeth from these elements we have Robertson Buchanan's rule, which is as follows:--

Find the amount of horse-power employed to move the wheel, and divide such horse-power by the velocity in feet per second of the pitch line of the wheel. Extract the square root of the quotient, and three-fourths of this root will be the least thickness of the tooth. To the result thus obtained, there must be added the allowance for wear of the teeth and the width of the space including the clearance which will determine the number of teeth in the wheel.

In conforming strictly to this rule the difficulty is met with that it would give fractional pitches not usually employed and difficult to measure on an existing wheel. Cast wheels kept on hand or in stock by machinists have usually the following standard:--

Beginning with an inch pitch, the pitches increase by 1/8 inch up to 3-inch pitch, from 3 to 4-inch pitches the increase is by 1/4 inch, and from 4-inch pitch and upwards the increase is by 1/2 inch. Now, under the rule the pitches would, with the clearance made to bear a certain proportion to the pitch, be in odd fractions of an inch.

It appears then, that, if in a calculation to obtain the necessary thickness of tooth, the diameter of the pitch circle is not an element, the rule cannot be strictly adhered to unless the diameter of the pitch circle be varied to suit the calculated thickness of tooth; or unless either the clearance, factor of safety, or amount of tooth thickness allowed for wear be varied to admit of the thickness of tooth arrived at by the calculation. But if the diameter of the pitch circle is one of the elements considered in arriving at the thickness of tooth requisite under given conditions, the pitch must, as a rule, either be in odd fractions, or else the allowance for wear, factor of safety, or amount of side clearance cannot bear a definite proportion to the pitch. But the allowance for clearance is in practice always a constant proportion of the pitch, and under these circumstances, all that can be done when the circumstances require a definite circumference of pitch circle, is to select such a pitch as will nearest meet the requirements of tooth thickness as found by calculation, while following the rule of making the clearance a constant proportion of the pitch. When following this plan gives a thinner tooth than the calculation calls for, the factor of safety and the allowance for wear are reduced. But this is of little consequence whenever more than one tooth on each wheel is in contact, because the rules provide for all the stress falling on one tooth. When, however, the number of teeth in the pinion is so small that one tooth only is in contact, it is better to select a pitch that will give a thicker rather than a thinner tooth than called for by the calculation, providing, of course, that the pitch be less than the arc of contact, so that the motion shall be continuous.

But when the pinions are shrouded, that is, have flanges at each end, the teeth are strengthened; and since the wear will continue greater than in wheels having more teeth in contact, the shrouding may be regarded as a provision against breakage in consequence of the reduction of tooth thickness resulting from wear.

In the following table is given the thickness of the tooth for a given stress at the pitch circle, calculated from Tredgold's rule for teeth supposed to have contact when new at one corner only.

+-------------------+--------------------+---------------------+
| Stress in lbs. at | Thickness of tooth | Actual pitches to |
| pitch circle. | in inches. | which wheels may be |
| | | made. |
+-------------------+--------------------+---------------------+
| 400 | .52 | 1-1/8 to 1-1/4 |
| 800 | .75 | 1-1/2 " 1-5/8 |
| 1,200 | .90 | 1-7/8 " 2 |
| 1,600 | 1.03 | 2 " 2-1/8 |
| 2,000 | 1.15 | 2-1/4 " 2-3/8 |
| 2,400 | 1.26 | 2-1/2 " 2-5/8 |
| 2,800 | 1.36 | 2-5/8 " 2-3/4 |
| 3,200 | 1.43 | 2-7/8 " 3 |
| 3,600 | 1.56 | 3-1/8 " 3-1/4 |
| 4,000 | 1.63 | 3-1/4 " 3-3/8 |
| 4,400 | 1.70 | 3-3/8 " 3-1/2 |
| 4,800 | 1.78 | 3-1/2 " 3-5/8 |
| 5,200 | 1.86 | 3-5/8 " 3-3/4 |
| 5,600 | 1.93 | 3-3/4 " 4 |
| 6,000 | 2.00 | 4 " 4-1/4 |
+-------------------+--------------------+---------------------+

In wheels that have their teeth cut to form in a gear-cutting machine the thickness of tooth at any point in the depth is equal at any point across the breadth; hence, supposing the wheels to be properly keyed to their shafts so that the pitch line across the breadth of the wheel stands parallel to the axis of the shaft, the contact of tooth upon tooth occurs across the full breadth of the tooth.

As the practical result of these conditions we have three important advantages: first, that the stress being exerted along the full breadth of the tooth instead of on one corner only, the tooth is stronger (with a given breadth and thickness) in proportion to the duty; second, that with a given pitch, the thickness and therefore the margin for safety and allowance for wear are increased, because the tooth may be increased in thickness at the expense of the clearance, which need be merely sufficient to prevent contact on both sides of the spaces so as to prevent the teeth from locking in the spaces; and thirdly, because the teeth will not be subject to sudden impacts or shocks of tooth upon tooth by reason of back lash.

In determining the strength of cut gear-teeth we may suppose the weight to be disposed along the face at the extreme height of the tooth, in which case the theoretical shape of the tooth to possess equal strength at every point from the addendum circle to the root would be a parabola, as shown by the dotted lines in Fig. 180, which represents a tooth having radial flanks. In this case it is evident that the ultimate strength of the tooth is that due to the thickness at the root, because it is less than that at the pitch circle, and the strength, as a whole, is not greater than that at the weakest part. But since teeth with radial flanks are produced, as has been shown, with a generating circle equal in diameter to the radius of the pinion, and since with a generating circle bearing that ratio of diameter to diameter of pitch circle the acting part of the flank is limited, it is usual to fill in the corners with fillets or rounded corners, as shown in Fig. 129; hence, the weakest part of the tooth will be where the radial line of the flank joins the fillet and, therefore, nearer the pitch circle than is the root. But as only the smallest wheel of the set has radial flanks and the flanks thicken as the diameter of the wheels increase, it is usual to take the thickness of the tooth at the pitch circle as representing the weakest part of the tooth, and, therefore, that from which the strength of the tooth is to be computed. This, however, is not actually the case even in teeth which have considerable spread at the roots, as is shown in Fig. 181, in which the shape of the tooth to possess equal strength throughout its depth is denoted by the parabolic dotted lines.

Considering a tooth as simply a beam supporting the strain as a weight we may calculate its strength as follows:--

Multiply the breadth of the tooth by the square of its thickness, and the product by the strength of the material, per square inch of section, of which the teeth are composed, and divide this last product by the distance of the pitch line from the root, and the quotient will give a tooth thickness having a strength equal to the weight of the load, but having no margin for safety, and no allowance for wear; hence, the result thus obtained must be multiplied by the factor of safety (which for this class of tooth may be taken as 6), and must have an additional thickness added to allow for wear, so that the factor of safety will be constant notwithstanding the wear.

Another, and in some respects more convenient method, for obtaining the strength of a tooth, is to take the strength of a tooth having 1-inch pitch, and 1 inch of breadth, and multiply this quantity of strength by the pitch and the face of the tooth it is required to find the strength of, both teeth being of the same material.

Example.--The safe working pressure for a cast-iron tooth of an inch pitch, and an inch broad will transmit, being taken as 400 lbs., what pressure will a tooth of 3/4-inch pitch and 3 inches broad transmit with safety?

Here 400 lbs. × 3/4 pitch × 3 breadth = 900 = safe working pressure of tooth 3/4-inch pitch and 3 inches broad.

Again, the safe working pressure of a cast-iron tooth, 1 inch in breadth and of 1-inch pitch, being considered as 400 lbs., what is the safe working pressure of a tooth of 1-inch pitch and 4-inch breadth?

Here 400 × 1 × 4 = 1600.

The philosophy of this is apparent when we consider that four wheels of 1-inch pitch and an inch face, placed together side by side, would constitute, if welded together, one wheel of an inch pitch and 4 inches face. (The term _face_ is applied to the wheel, and the term breadth to the tooth, because such is the custom of the workshop, both terms, however, mean, in the case of spur-wheels, the dimension of the tooth in a direction parallel to the axis of the wheel shaft or wheel bore.)

The following table gives the safe working pressures for wheels having an inch pitch and an inch face when working at the given velocities, S.W.P. standing for "safe working pressure:"--

+------------+------------+------------+------------+--------------+
| Velocity of| | | | |
|pitch circle| S.W.P. for | S.W.P. for |S.W.P. for | S.W.P. |
| in feet | cast-iron |spur mortise| cast-iron | for bevel |
|per second. |spur gears. | gears. |bevel gears.|mortise gears.|
+------------+------------+------------+------------+--------------+
| 2 | 368 | 178 | 258 | 178 |
| 3 | 322 | 178 | 225 | 157 |
| 6 | 255 | 178 | 178 | 125 |
| 12 | 203 | 142 | 142 | 99 |
| 18 | 177 | 124 | 124 | 87 |
| 24 | 161 | 113 | 113 | 79 |
| 30 | 150 | 105 | 105 | 74 |
| 36 | 140 | 98 | 98 | 69 |
| 42 | 133 | 93 | 93 | 65 |
| 48 | 127 | 88 | 88 | 62 |
+------------+------------+------------+------------+--------------+

For velocities less than 2 feet per second, use the same value as for 2 feet per second.

The proportions, in terms of the pitch, upon which this table is based, are as follows:--

Thickness of iron teeth .395 of the pitch.
" wooden " .595 "
Height of addendum .28 "
Depth below pitch line .32 "

The table is based upon 400 lbs. per inch of face for an inch pitch, as the safe working pressure of mortise wheel teeth or cogs; it may be noted that there is considerable difference of opinion. They are claimed by some to be in many cases practically stronger than teeth of cast iron. This may be, and probably is, the case when the conditions are such that the teeth being rigid and rigidly held (as in the case of cast-iron teeth), there is but one tooth on each wheel in contact. But when there is so nearly contact between two teeth on each wheel that but little elasticity in the teeth would cause a second pair of teeth to have contact, then the elasticity of the wood would cause this second contact. Added to this, however, we have the fact that under conditions where violent shock occurs the cog would have sufficient elasticity to give, or spring, and thus break the shock which cast iron would resist to the point of rupture. It is under these conditions, which mainly occur in high velocities with one of the wheels having cast teeth, that mortise wheels, or cogging, is employed, possessing the advantage that a broken or worn-out tooth, or teeth, may be readily replaced. It is usual, however, to assign to wooden teeth a value of strength more nearly equal to that of its strength in proportion to that of cast iron; hence, Thomas Box allows a wood tooth a value of about 3/10ths the strength of cast iron; a value as high as 7/10ths is, however, assigned by other authorities. But the strength of the tooth cannot exceed that at the top of the shank, where it fits into the mortise of the wheel, and on account of the leverage of the pressure the width of the mortise should exceed the thickness of the tooth.

In some practice, the mortise teeth, or cogs, are made thicker in proportion to the pitch than the teeth on the iron wheel; thus Professor Unwin, in his "Elements of Machine Design," gives the following as "good proportions":--

Thickness of iron teeth 0.395 of the pitch.
" wood cogs 0.595 " "

which makes the cogs 2/10ths inch thicker than the teeth.

The mortises in the wheel rim are made taper in both the breadth and the width, which enables the tooth shank to be more accurately fitted, and also of being driven more tightly home, than if parallel. The amount of this taper is a matter of judgment, but it may be observed that the greater the taper the more labor there is involved in fitting, and the more strain there is thrown upon the pins when locking the teeth with a given amount of strain. While the less the taper, the more care required to obtain an accurate fit. Taking these two elements into consideration, 1/8th inch of taper in a length of 4 inches may be given as a desirable proportion.

As an evidence of the durability of wooden teeth, there appeared in _Engineering_ of January 7th, 1879, the illustration shown in Fig. 182, which represents a cog from a wheel of 14 ft. 1/2 in. diameter, and having a 10-inch face, its pinion being 4 ft. in diameter. This cog had been running for 26-1/2 years, 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 perimeter of 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.

TABLE SHOWING THE HORSE-POWER WHICH DIFFERENT KINDS OF GEAR-WHEELS OF ONE INCH PITCH AND ONE INCH FACE WILL SAFELY TRANSMIT AT VARIOUS VELOCITIES OF PITCH CIRCLE.

+------------+------------+------------+-------------+-------------+
|Velocity of | | | | |
|Pitch Circle|Spur-Wheels.|Spur Mortise|Bevel-Wheels.|Bevel Mortise|
|in Feet per | H.P. | Wheels. | H.P. | Wheels. |
|Second. | | H.P. | | H.P. |
+------------+------------+------------+-------------+-------------+
| 2 | 1.338 | .647 | .938 | .647 |
| 3 | 1.756 | .971 | 1.227 | .856 |
| 6 | 2.782 | 1.76 | 1.76 | 1.363 |
| 12 | 4.43 | 3.1 | 3.1 | 2.16 |
| 18 | 5.793 | 4.058 | 4.058 | 2.847 |
| 24 | 7.025 | 4.931 | 4.931 | 3.447 |
| 30 | 8.182 | 5.727 | 5.727 | 4.036 |
| 36 | 9.163 | 6.414 | 6.414 | 4.516 |
| 42 | 10.156 | 7.102 | 7.102 | 4.963 |
| 48 | 11.083 | 7.680 | 7.680 | 5.411 |
+------------+------------+------------+-------------+-------------+

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, substituting 2/3rds 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 the _Cincinnati Artisan_ and 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 by _tens_; by multiplying by 100 we likewise have the pitch diameters from 100 to 900, increasing by _hundreds_.

TABLE FOR DETERMINING THE RELATION BETWEEN PITCH DIAMETER, PITCH, AND NUMBER OF TEETH IN GEAR-WHEELS.

+-----+------------------------------------------------------------------+
| | NUMBER OF TEETH. |
|Pitch.------+------+------+------+------+-------+-------+-------+-------+
| | 1. | 2. | 3. | 4. | 5. | 6. | 7. | 8. | 9. |
+-----+------+------+------+------+------+-------+-------+-------+-------+
|1 | .3183| .6366| .9549|1.2732|1.5915| 1.9099| 2.2282| 2.5465| 2.8648|
|1-1/8| .3581| .7162|1.0743|1.4324|1.7905| 2.1486| 2.5067| 2.8648| 3.2229|
|1-1/4| .3979| .7958|1.1937|1.5915|1.9894| 2.3873| 2.7852| 3.1831| 3.5810|
|1-3/8| .4377| .8753|1.3130|1.7507|2.1884| 2.6260| 3.0637| 3.5014| 3.9391|
| | | | | | | | | | |
|1-1/2| .4775| .9549|1.4324|1.9099|2.3873| 2.8648| 3.3422| 3.8197| 4.2971|
|1-5/8| .5173|1.0345|1.5517|2.0690|2.5862| 3.1035| 3.6207| 4.1380| 4.6552|
|1-3/4| .5570|1.1141|1.6711|2.2282|2.7852| 3.3422| 3.8993| 4.4563| 5.0134|
|1-7/8| .5968|1.1937|1.7905|2.3873|2.9841| 3.5810| 4.1778| 4.7746| 5.3714|
| | | | | | | | | | |
|2 | .6366|1.2732|1.9099|2.5465|3.1831| 3.8197| 4.4563| 5.0929| 5.7296|
|2-1/8| .6764|1.3528|2.0292|2.7056|3.3820| 4.0584| 4.7348| 5.4112| 6.0877|
|2-1/4| .7162|1.4324|2.1486|2.8648|3.5810| 4.2972| 5.0134| 5.7296| 6.4457|
|2-3/8| .7560|1.5120|2.2679|3.0239|3.7799| 4.5359| 5.2919| 6.0479| 6.8038|
| | | | | | | | | | |
|2-1/2| .7958|1.5915|2.3873|3.1831|3.9789| 4.7746| 5.5704| 6.3662| 7.1619|
|2-5/8| .8355|1.6711|2.5067|3.3422|4.1778| 5.0133| 5.8499| 6.6845| 7.5200|
|2-3/4| .8753|1.7507|2.6260|3.5014|4.3767| 5.2521| 6.1274| 7.0028| 7.8781|
|2-7/8| .9151|1.8303|2.7454|3.6605|4.5757| 5.4908| 6.4059| 7.3211| 8.2362|
| | | | | | | | | | |
|3 | .9549|1.9099|2.8648|3.8197|4.7746| 5.7296| 6.6845| 7.6394| 8.5943|
|3-1/4|1.0345|2.0690|3.1035|4.1380|5.1725| 6.2070| 7.2415| 8.2760| 9.3105|
|3-1/2|1.1141|2.2282|3.3422|4.4563|5.5704| 6.6845| 7.7986| 8.9126|10.0268|
|3-3/4|1.1937|2.3873|3.5810|4.7746|5.9683| 7.1619| 8.3556| 9.5493|10.7429|
| | | | | | | | | | |
|4 |1.2732|2.5465|3.8197|5.0929|6.3662| 7.6394| 8.9127|10.1839|11.4591|
|4-1/2|1.4324|2.8648|4.2972|5.7296|7.1619| 8.5943|10.0267|11.4591|12.8915|
|5 |1.5915|3.1831|4.7746|6.3662|7.9577| 9.5493|11.1408|12.7324|14.3240|
|5-1/2|1.7507|3.5014|5.2521|7.0028|8.7535|10.5042|12.2549|14.0056|15.7563|
|6 |1.9099|3.8196|5.7295|7.6394|9.5493|11.4591|13.3690|15.2788|17.1887|
+-----+------+------+------+------+------+-------+-------+-------+-------+

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 given pitch--

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, 1-1/2 inches pitch?

We find in line corresponding to 1-1/2 inch pitch--

Pitch diameter for 8 teeth 3.8197
" " 20 " 9.549
" " 100 " 47.75
--- -------
" " 128 " 61.1187

Or about 61-1/8". Answer.

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 given _number of teeth_.

Second, divide _given pitch diameter_ by the _pitch diameter_ for 1-_inch pitch_.

The quotient is the pitch desired.

Example.--What is the pitch of a wheel with 148 teeth, the pitch diameter being 72"?

First, pitch diameter for 148 teeth, 1-inch pitch, is--

8 teeth 2.5465
40 " 12.732
100 " 31.83
--- -------
148 " 47.1085

Second, 72/47.1 = 1.53 inch equal to the pitch.

This is nearly 1-1/2-inch pitch, and if possible the diameter would be reduced or the number of teeth increased so as to make the wheel exactly 1-1/2-inch pitch.

Rule 3.--Given ---- pitch and pitch diameter; to find ---- number of teeth.

First, ascertain from table the _pitch diameter_ for 1 _tooth_ of the given _pitch_.

Second, divide the _given pitch diameter_ by the _value_ found 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 2-1/2 inches?

First, the pitch diameter, 1 tooth, 2-1/2-inch pitch, is 0.7958 inches.

42
Second. ------ = 52.8. Answer.
0.7958

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 2-1/2 times 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 2-1/2 times those of the wheel on the fast-running shaft; so that 84 inches must be divided into two parts, one of which is 2-1/2 times the other, and these quantities will be the pitch radii of the wheels; that is, 84 inches are to be divided into 3-1/2 equal parts, 1 of which is the radius of one wheel, and 2-1/2 of which the radius of the other, thus 84"/3-1/2 = 24 inches. So that 24 inches is the pitch radius of pinion, pitch diameter = 48 inches; and 2-1/2 × 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 2-5/8 inches 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 2-1/2 times 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.

The forms of spur-gearing having their teeth at an angle to the axis, or formed in advancing steps shown in Figs. 183 and 184, 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 have _no manner of rubbing_, nor be more difficult to make than common wheel work."

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 in Fig. 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 in Fig. 183, is that it induces an end pressure tending to force the wheels apart _laterally_, and this causes _end_ wear on the journals and bearings.

To obviate this difficulty the form of gear shown in Fig. 186 is 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 in Fig. 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 in Fig. 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.

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 in Figs. 187 and 188; 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 wheel A upon 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 wheel B upon a vertical axis. Let it be upon the driving shaft and the motion will be changed in direction as if A and B were 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 in Fig. 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 in Figs. 187 and 188 may 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 about 1/4 inch, 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 the piece 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 and 2/6 of another inch, or 5-2/6 inches, which is the full diameter of the wheel, or the diameter of the addendum, as it is termed.

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. In Fig. 189 let the line A represent the circumference of the wheel, and B a line of equal length but at a right angle to it, then the line C, joining A, 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.

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) in Fig. 190, let A be the shaft of gear _h_, and B _b_ that of wheel _e_. Then a double gear-wheel having teeth on _f_, _g_ may be placed as shown, and the face _f_ will gear with _e_, while face _g_ will gear with _h_, the cone surfaces meeting in a point as at C and D respectively, hence the velocity will be equal.

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 in Fig. 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. 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, A being a spherical bearing, and B, C being cupped to fit to A.

Rotary motion variable at different parts of a rotation may be obtained by means of gear-wheels varied in form from the true circle.

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, in Fig. 193, A and B are the two fixed points, called the _foci_; L, E, F, G, P are points in the curve; and A F + F B = A E + E B. Also, A L + L B = A P + P B = A G + G B. From this it follows that A G = L O, O being the centre of the curve, and G the 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, as B, we describe an arc with a radius greater than B P and less than B L, for instance B E, and about A another arc with radius A 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.