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SCREW GEARING-HOOKE'S GEARING.

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i, the obliquity of the threads to the pitch circles, and of the normal helix to the axis;

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473. Screw Gearing.-A pair of convex screws, each rotating about its axis, are used as an elementary combination, to transmit motion by the sliding contact of their threads. Such screws are commonly called endless screws. At the point of contact of the screws, their threads must be parallel; and their line of connection is the common perpendicular to the acting surfaces of the threads at their point of contact. Hence the following principles :

I. If the screws are both right-handed or both left-handed, the angle between the directions of their axes is the sum of their obliquities-if one is right-handed and the other left-handed, that angle is the difference of their obliquities.

II. The normal pitch, for a screw of one thread, and the normal divided pitch, for a screw of more than one thread, must be the same in each screw.

III. The angular velocities of the screws are inversely as their number of threads.

474. Hooke's Gearing is a case of screw gearing, in which the axes of the screws are parallel, one screw being right-handed and the other left-handed, and in which, from the shortness and great diameter of the screws, and their large number of threads, they are in fact wheels, with teeth whose crests, instead of being parallel to the line of contact of the pitch cylinders, cross it obliquely, so as to be of a screw-like or helical form. In wheelwork of this kind,

Fig. 206.

the contact of each pair of teeth commences at the foremost end of

the helical front and terminates at the aftermost end; and the helix is of such a pitch that the contact of one pair of teeth does not terminate until that of the next pair has commenced. The object of this is to increase the smoothness of motion. With the same object, Dr. Hooke invented the making of the fronts of teeth in a series of steps. A wheel thus formed resembles in shape a series of equal and similar toothed discs placed side by side, with the teeth of each a little behind those of the preceding disc. In such a wheel, let p be the circular pitch, and n the number of steps. Then the arc of contact, the addendum, and the extent of sliding, are those due to the smaller pitch, while the strength of the teeth is that due to the thickness corresponding to the entire pitch p; so that the smooth action of small teeth and the strength of large teeth are combined. Stepped teeth being more expensive and difficult to execute than common teeth, are used for special purposes only.

Fig. 207.

n

475. The Wheel and Screw is an elementary combination of two screws, whose axes are at right angles to each other, both being right-handed or both left-handed. As the usual object of this combination is to produce a change of angular velocity in a ratio greater than can be obtained by any single pair of ordinary wheels, one of the screws is commonly wheel-like, being of large diameter and many-threaded, while the other is short and of few threads; and the angular velocities are inversely as the number of threads.

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Fig. 208 represents a side view of this combination, and fig. 209 a cross section at right angles to the axis of the smaller screw. It has been shown by Mr. Willis, that if each section of both screws be made by a plane perpendicular to the axis of the large screw or wheel, the outlines of the threads of the larger and smaller screw should be those of the teeth of a wheel and rack respectively: BB

SLIDING OF SCREWS-OLDHAM'S COUPLING

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in fig. 208, for example, being the pitch circle of the wheel, and B., B, the pitch line of the rack.

The periphery and teeth of the wheel are usually hollowed to fit the screw, as shown at T, fig. 209.

To make the teeth or threads of a pair of screws fit correctly and work smoothly, a hardened steel screw is made of the figure of the smaller screw, with its thread or threads notched so as to form a cutting tool; the larger screw, or wheel, is cast approximately of the required figure; the larger screw and the steel screw are fitted up in their proper relative position, and made to rotate in contact with each other by turning the steel screw, which cuts the threads of the larger screw to their true figure.

476. The Belative Sliding of a Pair of Screws at their point of contact is found thus:-Let r1, 72, be the radii of their pitch cylinders, and i, i, the obliquities of their threads to their pitch circles, one of which is to be considered as negative if the screws are contrary-handed. Let u be the common component of the velocities of a pair of points of contact along a line touching the pitch surfaces and perpendicular to the threads, at the pitch point, and v the velocity of sliding of the threads over each other. Then

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and

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v = α11⋅ cos 12+ a2o1⁄2· cos i2 = u (cotan &+cotan i)........... (2.)

When the screws are contrary-handed, the difference instead of the sum of the terms in equation 2 is to be taken.

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477. Oldham's Coupling.-A coupling is a mode of connecting a pair of shafts so that they shall rotate in the same direction, with the same mean angular velocity. If the axes of the shafts are in the same straight line, the coupling consists in so connecting their contiguous ends that they shall rotate as one piece; but if the axes are not in the same straight line, combinations of mechanism are required. A coupling for parallel shafts which acts by sliding contact was invented by Oldham, and is represented in fig. 210. C1, C2, are the axes of the two parallel shafts; D1, D2, two crossheads, facing each other, fixed on the ends of the two shafts respectively; E, E, a bar, sliding in a diametral groove in the face of

D

E

Fig. 210.

D; E, E, a bar, sliding in a diametral groove in the face of D.; those bars are fixed together at A, so as to form a rigid cross. The angular velocities of the two shafts and of the cross are all equal at every instant. The middle point of the cross, at A, revolves in the dotted circle described upon the line of centres C, C, as a diameter, twice for each turn of the shafts and cross; the instantaneous axis of rotation of the cross, at any instant, is at I, the point in the circle C1 C2, diametrically opposite to A.

Oldham's coupling may be used with advantage where the axes of the shafts are intended to be as nearly in the same straight line as is possible, but where there is some doubt as to the practicability or permanency of their exact continuity.

SECTION 3.-Connection by Bands.

478. Bands Classed.—Bands, or wrapping connectors, for communicating motion between pulleys or drums rotating about fixed axes, or between rotating pulleys and drums and shifting pieces, may be thus classed :

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I. Belts, which are made of leather or of gutta percha, are flat and thin, and require nearly cylindrical pulleys. A belt tends to move towards that part of a pulley whose radius is greatest; pulleys for belts, therefore, are slightly swelled in the middle, in order that the belt may remain on the pulley unless forcibly shifted. A belt when in motion is shifted off a pulley, or from one pulley on to another of equal size alongside of it, by pressing against that part of the belt which is moving towards the pulley.

II. Cords, made of catgut, hempen or other fibres, or wire, are nearly cylindrical in section, and require either drums with ledges, or grooved pulleys.

III. Chains, which are composed of links or bars jointed together, require pulleys or drums, grooved, notched, and toothed, so as to fit the links of the chains.

Bands for communicating continuous motion are endless.

Bands for communicating reciprocating motion have usually their ends made fast to the pulleys or drums which they connect, and which in this case may be sectors.

479. Principle of Connection by Bands.-The line of connection of a pair of pulleys or drums connected by means of a band, is the central line or axis of that part of the band whose tension transmits the motion. The principle of Article 433 being applied to this case, leads to the following consequences :

I. For a pair of rotating pieces, let r, r,, be the perpendiculars let fall from their axes on the centre line of the band, a, a,, their angular velocities, and i, i, the angles which the centre line of the

BANDS-PULLEYS-DRUMS.

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band makes with the two axes respectively. Then the longitudinal velocity of the band, that is, its component velocity in the direction of its own centre line, is

u = r1 a1 sin 11 = r2 Ag sin is ;

whence the angular velocity-ratio is

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When the axes are parallel (which is almost always the case), i=ig,

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The same equation holds when both axes, whether parallel or not, are perpendicular in direction to that part of the band which transmits the motion; for then sin 1 = sin i̟ = 1.

II. For a rotating piece and a sliding piece, let r be the perpendicular from the axis of the rotating piece on the centre line of the band, a the angular velocity, i the angle between the directions of the band and axis, u the longitudinal velocity of the band, j the angle between the direction of the centre line of the band and that of the motion of the sliding piece, and v the velocity of the sliding piece; then

u = r a sin i = v cos j; and..........

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..(4.)

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When the centre line of the band is parallel to the direction of motion of the sliding piece, and perpendicular to the direction of the axis of the rotating piece, sin i = cos j = 1, and

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480. The Pitch Surface of a Pulley or Drum is a surface to which the line of connection is always a tangent; that is to say, it is a surface parallel to the acting surface of the pulley or drum, and distant from it by half the thickness of the band.

481. Circular Pulleys and Drums are used to communicate a

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constant velocity-ratio. In each of them, the length denoted by

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