Modern Machine-Shop Practice Part 6

It may here be shown that a worm-wheel may be made to work correctly with a square thread. Suppose, for example, that the diameter of the generating circle be supposed to be infinite, and the sides of the thread may be accepted as rolled by the circle. On the wheel we roll a straight line, which gives a cycloidal curve suitable to work with the square thread. But the action will be confined to the points of the teeth, as is shown in Fig. 81, and also to the arc of approach. This is the same thing as taking the faces off the worm and filling in the flanks of the wheel. Obviously, then, we may reverse the process and give the worm faces only, and the wheel, flanks only, using such size of generating circle as will make the s.p.a.ces of the wheel parallel in their depths and rolling the same generating circle upon the pitch line of the worm to obtain its face curve. This would enable the teeth on the wheel to be cut by a square-threaded tap, and would confine the contact of tooth upon tooth to the recess.

The diameter of generating circle used to roll the curves for a worm and worm-wheel should in all cases be larger than the radius of the worm-wheel, so that the flanks of the wheel teeth may be at least as thick at the root as they are at the pitch circle.

To find the diameter of a wheel, driven by a tangent-screw, which is required to make one revolution for a given number of turns of the screw, it is obvious, in the first place, that when the screw is single-threaded, the number of teeth in the wheel must be equal to the number of turns of the screw. Consequently, the pitch being also given, the radius of the wheel will be found by multiplying the pitch by the number of turns of the screw during one turn of the wheel, and dividing the product by 6.28.

[Ill.u.s.tration: Fig. 82.]

When a wheel pattern is to be made, the first consideration is the determination of the diameter to suit the required speed; the next is the pitch which the teeth ought to have, so that the wheel may be in accordance with the power which it is intended to transmit; the next, the number of the teeth in relation to the pitch and diameter; and, lastly, the proportions of the teeth, the clearance, length, and breadth.

[Ill.u.s.tration: Fig. 83.]

When the amount of power to be transmitted is sufficient to cause excessive wear, or when the velocity is so great as to cause rapid wear, the worm instead of being made parallel in diameter from end to end, is sometimes given a curvature equal to that of the worm-wheel, as is shown in Fig. 82.

[Ill.u.s.tration: Fig. 84.]

[Ill.u.s.tration: Fig. 85.]

The object of this design is to increase the bearing area, and thus, by causing the power transmitted to be spread over a larger area of contact, to diminish the wear. A mechanical means of cutting a worm to the required form for this arrangement is shown in Fig. 83, which is extracted from "Willis' Principles of Mechanism." "A is a wheel driven by an endless screw or worm-wheel, B, C is a toothed wheel fixed to the axis of the endless screw B and in gear with another and equal toothed gear D, upon whose axis is mounted the smooth surfaced solid E, which it is desired to cut into Hindley's[2] endless screw. For this purpose a cutting tooth F is clamped to the face of the wheel A. When the handle attached to the axis of B C is turned round, the wheel A and solid wheel E will revolve with the same relative velocity as A and B, and the tool F will trace upon the surface of the solid E a thread which will correspond to the conditions. For from the very mode of its formation the section of every thread through the axis will point to the centre of the wheel A. The axis of E lies considerably higher than that of B to enable the solid E to clear the wheel A.

[2] The inventor of this form of endless screw.

"The edges of the section of the solid E along its horizontal centre line exactly fit the segment of the toothed wheel, but if a section be made by a plane parallel to this the teeth will no longer be equally divided as they are in the common screw, and therefore this kind of screw can only be in contact with each tooth along a line corresponding to its middle section. So that the advantage of this form over the common one is not so great as appears at first sight.

"If the inclination of the thread of a screw be very great, one or more intermediate threads may be added, as in Fig. 84, in which case the screw is said to be double or triple according to the number of separate spiral threads that are so placed upon its surface. As every one of these will pa.s.s its own wheel-tooth across the line of centres in each revolution of the screw, it follows that as many teeth of the wheel will pa.s.s that line during one revolution of the screw as there are threads to the screw. If we suppose the number of these threads to be considerable, for example, equal to those of the wheel teeth, then the screw and wheel may be made exactly alike, as in Fig. 85; which may serve as an example of the disguised forms which some common arrangements may a.s.sume."

[Ill.u.s.tration: Fig. 86.]

In Fig. 86 is shown Hawkins's worm gearing. The object of this ingenious mechanical device is to transmit motion by means of screw or worm gearing, either by a screw in which the threads are of equal diameter throughout its length, or by a spiral worm, in which the threads are not of equal diameter throughout, but increase in diameter each way from the centre of its length, or about the centre of its length outwardly.

Parallel screws are most applicable to this device when rectilinear motions are produced from circular motions of the driver, and spiral worms are applied when a circular motion is given by the driver, and imparted to the driven wheel. The threads of a spiral worm instead of gearing into teeth like those of an ordinary worm-wheel, actuate a series of rollers turning upon studs, which studs are attached to a wheel whose axis is not parallel to that of the worm, but placed at a suitable inclination thereto. When motion is given to the worm then rotation is produced in the roller wheel at a rate proportionable to the pitch of worm and diameter of wheel respectively.

In the arrangement for transmitting rectilinear motion from a screw, rollers may be employed whose axes are inclined to the axis of the driving screw, or else at right angles to or parallel to the same. When separate rollers are employed with inclined axes, or axes at right angles with that of the main driving screw, each thread in gear touches a roller at one part only; but when the rollers are employed with axes parallel to that of the driving screw a succession of grooves are turned in these rollers, into which the threads of the driving screw will be in gear throughout the entire length of the roller. These grooves may be separate and apart from each other, or else form a screw whose pitch is equal to that of the driving screw or some multiple thereof.

In Fig. 86 the spiral worm is made of such a length that the edge of one roller does not cease contact until the edge of the next comes into contact; a wheel carries four rollers which turn on studs, the latter being secured by cottars; the axis of the worm is at right angles with that of the wheel. The edges of the rollers come near together, leaving sufficient s.p.a.ce for the thread of the worm to fit between any two contiguous rollers. The pitch line of the screw thread forms an arc of a circle, whose centre coincides with that of the wheel, therefore the thread will always bear fairly against the rollers and maintain rolling contact therewith during the whole of the time each roller is in gear, and by turning the screw in either direction the wheel will rotate.

[Ill.u.s.tration: Fig. 87.]

To prevent end thrust on a worm shaft it may have a right-hand worm A, and a left-hand one C (Fig. 87), driving two wheels B and D which are in gear, and either of which may transmit the power. The thrust of the two worms A and C, being in opposite directions, one neutralizes the other, and it is obvious that as each revolution of the worm shaft moves both wheels to an amount equal to the pitch of the worms, the two wheels B D may, if desirable, be of different diameters.

[Ill.u.s.tration: Fig. 88.]

[Ill.u.s.tration: Fig. 89.]

Involute teeth.--These are teeth having their whole operative surfaces formed of one continuous involute curve. The diameter of the generating circle being supposed as infinite, then a portion of its circ.u.mference may be represented by a straight line, such as A in Fig. 88, and if this straight line be made to roll upon the circ.u.mference of a circle, as shown, then the curve traced will be involute P. In practice, a piece of flat spring steel, such as a piece of clock spring, is used for tracing involutes. It may be of any length, but at one end it should be filed so as to leave a scribing point that will come close to the base circle or line, and have a short handle, as shown in Fig. 89, in which S represents the piece of spring, having the point P', and the handle H.

The operation is, to make a template for the base circle, rest this template on drawing paper and mark a circle round its edge to represent on the paper the pitch circle, and to then bend the spring around the circle B, holding the point P' in contact with the drawing paper, securing the other end of the piece of steel, so that it cannot slip upon B, and allowing the steel to unwind from the cylinder or circle B.

The point P' will mark the involute curve P. Another way to mark an involute is to use a piece of twine in place of the spring and a pencil instead of the tracing point; but this is not so accurate, unless, indeed, a piece of wood be laid on the drawing-board and the pencil held firmly against it, so as to steady the pencil point and prevent the variation in the curve that would arise from variation in the vertical position of the pencil.

The flanks being composed of the same curve as the faces of the teeth, it is obvious that the circle from which the tracing point starts, or around which the straight line rolls, must be of less diameter than the pitch circle, or the teeth would have no flanks.

A circle of less diameter than the pitch circle of the wheel is, therefore, introduced, wherefrom to produce the involute curves forming the full side of the tooth.

[Ill.u.s.tration: Fig. 90.]

The depth below pitch line or the length of flank is, therefore, the distance between the pitch circle and the base circle. Now even supposing a straight line to be a portion of the circ.u.mference of a circle of infinite diameter or radius, the conditions would here appear to be imperfect, because the generating circle is not rolled upon the pitch circle but upon a circle of lesser diameter. But it can be shown that the requirements of a proper velocity ratio will be met, notwithstanding the employment of the base instead of the pitch circle.

Thus, in Fig. 90, let A and B represent the respective centres of the two pitch circles, marked in dotted lines. Draw the base circle for B as E Q, which may be of any radius less than that of the pitch circle of B. Draw the straight line Q D R touching this base circle at its perimeter and pa.s.sing through the point of contact on the pitch circles as at D. Draw the circle whose radius is A R forming the base circle for wheel A. Thus the line R P Q will meet the perimeters of the two circles while pa.s.sing through the point of contact D at the line of centres (a condition which the relative diameters of the base circles must always be so proportioned as to attain).

If now we take any point on R Q, as P in the figure, as a tracing point, and suppose the radius or distance P Q to represent the steel spring shown in Fig. 89, and move the tracing point back to the base circle of B, it will trace the involute E P. Again we may take the tracing point P (supposing the line P R to represent the steel spring), and trace the involute P F, and these two involutes represent each one side of the teeth on the respective wheels.

[Ill.u.s.tration: Fig. 91.]

The line R P Q is at a right angle to the curves P E and P F, at their point of contact, and, therefore, fills the conditions referred to in Fig. 41. Now the line R P Q denotes the path of contact of tooth upon tooth as the wheels revolve; or, in other words, the point of contact between the side of a tooth on one wheel, and the side of a tooth on the other wheel, will always move along the line Q R, or upon a similar line pa.s.sing through D, but meeting the base circles upon the opposite sides of the line of centres, and since line Q R always cuts the line of centres at the point of contact of the pitch circles, the conditions necessary to obtain a correct angular velocity are completely fulfilled.

The velocity ratio is, therefore, as the length of B Q is to that of A R, or, what is the same thing, as the radius of the base circle of one wheel is to that of the other. It is to be observed that the line Q R will vary in its angle to the line of centres A B, according to the diameter of the base circle from which it is struck, and it becomes a consideration as to what is its most desirable angle to produce the least possible amount of thrust tending to separate the wheels, because this thrust (described in Fig. 39) tends to wear the journals and bearings carrying the wheel shafts, and thus to permit the pitch circles to separate. To avoid, as far as possible, this thrust the proportions between the diameters of the base circles D and E, Fig. 91, must be such that the line D E pa.s.ses through the point of contact on the line of centres, as at C, while the angles of the straight line D E should be as nearly 90 to a radial line, meeting it from the centres of the wheels (as shown in the figure, by the lines B E and D E), as is consistent with the length of D E, which in order to impart continuous motion must at least equal the pitch of the teeth. It is obvious, also, that, to give continuous motion, the length of D E must be more than the pitch in proportion, as the points of the teeth come short of pa.s.sing through the base circles at D and E, as denoted by the dotted arcs, which should therefore represent the addendum circles. The least possible obliquity, or angle of D E, will be when the construction under any given conditions be made such by trial, that the base circles D and E coincide with the addendum circles on the line of centres, and thus, with a given depth of both beyond, the pitch circle, or addenda as it is termed, will cause the tooth contacts to extend over the greatest attainable length of line between the limits of the addendum circles, thus giving a maximum number of teeth in contact at any instant of time. These conditions are fulfilled in Fig. 92,[3] the addendum on the small wheel being longer than the depth below pitch line, while the faces of the teeth are the narrowest.

[3] From an article by Prof. Robinson.

[Ill.u.s.tration: Fig. 92.]

In seeking the minimum obliquity or angle of D E in the figure, it is to be observed that the less it is, the nearer the base circle approaches the pitch circle; hence, the shorter the operative length of tooth flank and the greater its wear.

In comparing the merits of involute with those of epicycloidal teeth, the direction of the line of pressure at each point of contact must always be the common perpendicular to the surfaces at the point of contact, and these perpendiculars or normals must pa.s.s through the pitch circles on the line of centres, as was shown in Fig. 41, and it follows that a line drawn from C (Fig. 91) to any point of contact, is in the direction of the pressure on the surfaces at that point of contact. In involute teeth, the contact will always be on the line D E (Fig. 92), but in epicycloidal, on the line of the generating circle, when that circle is tangent at the line of centres; hence, the direction of pressure will be a chord of the circle drawn from the pitch circle at the line of centres to the position of contact considered. Comparing involute with radial flanked epicycloidal teeth, let C D A (Fig. 91) represent the rolling circle for the latter, and D C will be the direction of pressure for the contact at D; but for point of contact nearer C, the direction will be much nearer 90, reaching that angle as the point of contact approaches C. Now, D is the most remote legitimate contact for involute teeth (and considering it so far as epicycloidal struck with a generating circle of infinite diameter), we find that the aggregate directions of the pressures of the teeth upon each other is much nearer perpendicular in epicycloidal, than in involute gearing; hence, the latter exert a greater pressure, tending to force the wheels apart. Hence, the former are, in this respect, preferable.

It is to be observed, however, that in some experiments made by Mr.

Hawkins, he states that he found "no tendency to press the wheels apart, which tendency would exist if the angle of the line D E (Fig. 92) deviated more than 20 from the line of centres A B of the two wheels."

A method commonly employed in practice to strike the curves of involute teeth, is as follows:--

In Fig. 93 let C represent the centre of a wheel, D D the full diameter, P P the pitch circle, and E the circle of the roots of the teeth, while R is a radial line. Divide on R, the distance between the pitch circle and the wheel centre, into four equal parts, by 1, 2, 3, &c. From point or division 2, as a centre, describe the semicircle S, cutting the wheel centre and the pitch circle at its junction with R (as at A). From A, with compa.s.ses set to the length of one of the parts, as A 3, describe the arc B, cutting S at F, and F will be the centre from which one side of the tooth may be struck; hence from F as a centre, with the compa.s.ses set to the radius A B, mark the curve G. From the centre C strike, through F, a circle T T, and the centres wherefrom to strike all the teeth curves will fall on T T. Thus, to strike the other curve of the tooth, mark off from A the thickness of the tooth on the pitch circle P P, producing the point H. From H as a centre (with the same radius as before,) mark on T T the point I, and from I, as a centre, mark the curve J, forming the other side of the tooth.

[Ill.u.s.tration: Fig. 93.]

[Ill.u.s.tration: Fig. 94.]

In Fig. 94 the process is shown carried out for several teeth. On the pitch circle P P, divisions 1, 2, 3, 4, &c., for the thickness of teeth and the width of the s.p.a.ces are marked. The compa.s.ses are set to the radius by the construction shown in Fig. 93, then from _a_, the point _b_ on T is marked, and from _b_ the curve _c_ is struck.

In like manner, from _d_, _g_, _j_, the centres _e_, _h_, _k_, wherefrom to strike the respective curves, _f_, _i_, _l_, are obtained.

Then from _m_ the point _n_, on T T, is marked, giving the centre wherefrom to strike the curve at _h_ _m_, and from _o_ is obtained the point _p_, on T T, serving as a centre for the curve _e_ _o_.

A more simple method of finding point F is to make a sheet metal template, C, as in Fig. 95, its edges being at an angle one to the other of 75 and 30'. One of its edges is marked off in quarters of an inch, as 1, 2, 3, 4, &c. Place one of its edges coincident with the line R, its point touching the pitch circle at the side of a tooth, as at A, and the centre for marking the curve on that side of the tooth will be found on the graduated edge at a distance from A equal to one-fourth the length of R.

[Ill.u.s.tration: Fig. 95.]

The result obtained in this process is precisely the same as that by the construction in Fig. 93, as will be plainly seen, because there are marked on Fig. 93 all the circles by which point F was arrived at in Fig. 95; and line 3, which in Fig. 95 gives the centre wherefrom to strike curve _o_, is coincident with point F, as is shown in Fig. 95. By marking the graduated edge of C in quarter-inch divisions, as 1, 2, 3, &c., then every division will represent the distance from A for the centre for every inch of wheel radius. Suppose, for example, that a wheel has 3 inches radius, then with the scale C set to the radial line R, the centre therefrom to strike the curve _o_ will be at 3; were the radius of the wheel 4 inches, then the scale being set the same as before (one edge coincident with R), the centre for the curve _o_ would be at 4, and arc T would require to meet the edge of C at 4. Having found the radius from the centre of the wheel of point F for one tooth, we may mark circle T, cutting point F, and mark off all the teeth by setting one point of the compa.s.ses (set to radius A F) on one side of the tooth and marking on circle T the centre wherefrom to mark the curve (as _o_), continuing the process all around the wheel and on both sides of the tooth.

This operation of finding the location for the centre wherefrom to strike the tooth curves, must be performed separately for each wheel, because the distance or radius of the tooth curves varies with the radius of each wheel.

In Fig. 96 this template is shown with all the lines necessary to set it, those shown in Fig. 95 to show the ident.i.ty of its results with those given in Fig. 93 being omitted.

The principles involved in the construction of a rack to work correctly with a wheel or pinion, having involute teeth, are as in Fig. 97, in which the pitch circle is shown by a dotted circle and the base circle by a full line circle. Now the diameter of the base circle has been shown to be arbitrary, but being a.s.sumed the radius B Q will be determined (since it extends from the centre B to the point of contact of D Q, with the base circle); B D is a straight line from the centre B of the pinion to the pitch line of the rack, and (whatever the angle of Q D to B D) the sides of the rack teeth must be straight lines inclined to the pitch line of the rack at an angle equal to that of B D Q.