Modern Machine-Shop Practice Part 16

[Ill.u.s.tration: Fig. 172.]

It now remains to draw in the top of the thread upon the curved surface of the half pattern; for this purpose take a piece of stiff card or other flexible material, wrap it around the pattern and fix it temporarily by tacks, we then trim off the edges true to the pattern, and mark upon the edges of the card the position of the tops of the thread upon each side; we remove the card and spread it out on a flat surface, join the points marked on the edges by lines as in Fig. 172, replace the card exactly as before upon the pattern, and with a fine scriber we p.r.i.c.k through the lines. The cutting out is commenced by sawing, keeping, of course, well within the lines; and it is facilitated by attaching a stop to the saw so as to insure cutting at all parts nearly to the exact depth. This stop is a simple strip of wood and may be clamped to the saw, though it is much more convenient to have a couple of holes in the saw blade for the pa.s.sage of screws. For finis.h.i.+ng, a pair of templates, P and Q, Fig. 173, right and left, will be found useful; and finally the work should be verified and slight imperfections corrected by the use of a form or template taking in three s.p.a.ces, as shown at R in Fig. 173. In drawing the lines on the card, we must consider whether it is a right or left-handed worm that we desire.

In the engraving the lines are those suitable for a right-handed thread.

Having completed one half of the pattern, place the two halves together, and trace off the half that is uncut, using again the card template for drawing the lines on the curved surface. The cutting out will be the same as before.

[Ill.u.s.tration: Fig. 173.]

As the teeth of cast wheels are, from their deviation from accuracy in the tooth curves and the concentricity of the teeth to the wheel centre, apt to create noise in running, it is not unusual to cast one or both wheels with mortises in the rim to receive wooden teeth. In this case the wheel is termed a mortise wheel, and the teeth are termed _cogs_. If only one of a pair of wheels is to be cogged, the largest of the pair is usually selected, because there are in that case more teeth to withstand the wear, it being obvious that the wear is greatest upon the wheel having the fewest teeth, and that the iron wheel or pinion can better withstand the wear than the mortise wheel. The woods most used for cogs are hickory, maple, hornbeam and locust. The blocks wherefrom the teeth are to be formed are usually cut out to nearly the required dimensions, and kept in stock, so as to be thoroughly well-seasoned when required for use, and, therefore less liable to come loose from shrinkage after being fitted to the mortise in the wheel. The length of the shanks is made sufficient to project through the wheel rim and receive a pin, as shown in Fig. 174, in which B is a blank tooth, and C a finished tooth inserted in the wheel, the pin referred to being at P. But, if a mortise should fall in an arm of the wheel, this pin-hole must pa.s.s through the rim, as shown in the mortise A. The wheel, however, should be designed so that the mortises will not terminate in the arms of the wheel.

[Ill.u.s.tration: Fig. 174.]

[Ill.u.s.tration: Fig. 175.]

Another method of securing the teeth in the mortises is to dovetail them at the small end and drive wedges between them, as shown in Fig. 175, in which C C are two contiguous teeth, R the wheel rim and W W two of the wedges. On account of the dovetailing the wedges exert A pressure pressing the teeth into the mortises. This plan is preferable to that shown in the Fig. 174 inasmuch as from the small bearing area of the pins they become loose quicker, and furthermore there is more elasticity to take up the wear in the case of the wedges.

[Ill.u.s.tration: Fig. 176.]

The mortises are first dressed out to a uniform size and taper, using two templates to test them with, one of which is for the breadth and the other for the width of the mortise. The height above the wheel requires to be considerably more than that due to the depth of the teeth, so that the surface bruised by driving the cogs or when fitting them into the mortises may be cut off. To avoid this damage as much as possible, a broad-face hammer should be employed--a copper, lead, lignum vitae, or a raw hide hammer being preferable, and the last the best. The teeth are got out in a box and two guides, such as shown in Figs. 176, 177, and 178, similar letters of reference denoting the same parts in all three ill.u.s.trations.

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.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: Fig. 178.]

The shanks, being correctly sawn, may then be fitted to the mortises, and let in within 1/8 of b.u.t.ting 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 pa.s.s farther through to the amount of the distance marked by the fork scriber. The holes for the pins which pa.s.s 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 s.p.a.ce 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.

[Ill.u.s.tration: Fig. 179.]

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 quant.i.ty, 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 circ.u.mference, 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 circ.u.mference 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 s.p.a.ce 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 circ.u.mstances, all that can be done when the circ.u.mstances require a definite circ.u.mference 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 f.l.a.n.g.es 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 s.p.a.ces so as to prevent the teeth from locking in the s.p.a.ces; and thirdly, because the teeth will not be subject to sudden impacts or shocks of tooth upon tooth by reason of back lash.

[Ill.u.s.tration: Fig. 180.]

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.

[Ill.u.s.tration: Fig. 181.]

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 cla.s.s 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 quant.i.ty 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 const.i.tute, 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.)