Abbildungen der Seite
PDF
EPUB

opposite on the other side of the great wheel; then these two levers acting by turns, will keep the motion tolerably uniform, and the power at N will have nearly a uniform action. The wheel CR is introduced for the sake of softening the friction: but it must be carefully adjusted to the magnitude of the waves, or else the motion will be hobbling and irregular. On this account the following method of obtaining a reciprocating motion is more usual.

Instead of making the axis AB (fig. 3. pl. II.) in one continued straight line, let it be bent at right angles in the points d, e, f, g, h, &c. so that the portions ef, gh, shall be parallel to AB, and in the course of a rotation of the lantern or trundle EF, they will describe cylindrical surfaces: if, then, pistons and their handles 1b, 1b, be hung upon the cranks 1, 1, as the rotatory motion of the trundle EF (when worked by another wheel) proceeds, the pistons are alternately forced up and down in the pumps; and thus make one complete stroke of each pump for every turn of the lantern. It will be advisable to place pulleys or rollers at a, b, a, b, for the handles or chains to work against, when the obliquity of the motion of the cranks 1, 1, carries them out of the vertical position.

Other methods of obtaining reciprocrating by means of circular motions may be seen under the articles Air-pump and Saw-mill. See also pl. XXXVIII. and XXXIX.

one.

7. To produce a rotatory motion by means of a reciprocating Suppose it is required to give to the wheel svTo (fig. 4. pl. II.) a rotatory motion about the centre c. In the plane of the wheel, attach to a fixed point F as a centre of motion a lever FO, which may move freely up and down: let a pin be fixed in the wheel as at R; and let an inflexible bar on hang upon the pin at R at one end, while the other end is attached to the lever Fo by a stirrup; the motion being quite easy at both ends. Then, while the point o is raised upwards the bar pulls upwards the pin R, and so continues to do until the points 9, K, and c, fall in a right line; at that time the effort of the bar to turn the wheel is nothing; but the wheel by its anterior rotation has acquired a quantity of motion which will carry it on in the same direction, till by the downward motion of the extremity of the lever, the bar begins to push forward the pin to which it is attached: thus the motion is continued till the points a, c, and R, are again in a right line, R being now the farthest from a: in this position the bar has no tendency to move the wheel along, but here the effort of momentum continues the motion, as before, till the bar begins to draw the point R upwards. And thus a reciprocating motion of the lever FR gives a complete rotation, to the wheel; and the velocity of the

circumference of the wheel may be made as rapid as we please, by making the distance CR so much the smaller in comparison of cv. If the lever ra be below the wheel, the general effect will be the same, but the particular circumstances of the motion will succeed each other in a contrary order. In practice it is common to substitute for the pin at R the handle of a bent winch, as represented by the dotted lines in the figure. It is not absolutely necessary that the lever and wheel should be in the same plane; but deviations from it are not often to be recommended, except in small machinery, such as a common spinning wheel worked by the feet, &c. When it is not required to have a complete rotation of the wheel, for every ascent and descent of the lever FQ, we may change the relation of the two motions in any proportion, by the intervention of tooth and pinion work.

8. To describe a rectilinear reciprocating motion, by means of an angular or circular reciprocating motion. Let it be proposed, for example, to move the end F of the beam FH to and fro in the line EC. Fix a beam AB (fig. 9. pl. I.) perpendicularly to the given line EC, and cut in that beam a groove CD equal in length to the beam FH: let the end н of the beam Fн be confined by a pin to run along the groove CD; and let two other pins be fixed, one at G the middle point of the beam FH, the other at c the lower point where the reciprocating motion of the point F terminates: take an iron bar CG equal in length to half FH, and let it move upon the pins c and G as joints. Then while the end G of the bar or guide CG moves through the quadrantal arc Lg GK; the point H of the beam will slide along the groove from D to c, and the point F along the line CE from c to E and when the guide returns from K by G to L, the end F of the beam will return along the line EC. For, when CG = GF GH, supposing a line drawn from C to F, the angle FGC = GCH + GHC = 2 GCH; and CGH = GCF + GFC=2GCF. Hence we have 2 GCF + 2 GCH = FGC + HGC 2 right angles, and consequently GCF + GCH = HCF = 1 right angle: that is, the point F falls in a right line drawn through c at right angles to CD. And when FH is in any other position, as fh, the same may be shewn.

9. To communicate motion in any direction by wheels, and to construct the wheels for that purpose. This may be done by placing the wheels so that their shafts or axles shall be inclined in given angles, as represented in figs. 1 and 7. pl. III. And in this case the wheels are seldom portions of cylinders, but most commonly portions of cones. When the wheels do not make an angle of 90°, the adjustment of the shape and magnitude of the conic frustums which constitute the wheels, is

known among millwrights by the name of bevel-geer work; a concise account of which is here added. If two cones A and B (fig. 2. pl. III.), whose surfaces always touch in a right line, as ae, revolve on their axes ab, ac, rolling the one upon the other; and if the bases and altitudes of these cones be equal, they will perform complete revolutions in one and the same time. For since the bases and altitudes are equal, circles on either cone parallel to the base, and at equal distances from the vertex, as the distances a2, a2, for instance, will be equal: and therefore while the surfaces of the cones roll one upon another, every point in the circumference of one of these circles will be brought successively into contact with a corresponding point on the circumference of the other, and they will both have revolved in an equal time. The same will hold of the corre sponding circles at any other equal distances from the vertex, al, a3, a4, &c. and consequently the two cones will perform their rotations in equal times.

Again, if the cone ade (fig. 3. pl. III.) have the diameter of its base double that of the cone adf while their slant heights are the same; and if these two cones turn on their axes ac, ab, their surfaces during the rotation always touching one another in a right line; then, since the circumference of the base de is double that of the base df, and the circumference of every circular section parallel to the former base, double that of every corresponding section parallel to the latter base, it follows that when the cone fd has performed one rotation, the cone ade will have made but half a rotation. The times of rotation being in the ratio of their bases.

In like mauner, if the cone aed, (fig. 4. pl. III.) have the diameter of its base, to the diameter of the base of adf, as m ton, the slant heights being the same; and if these cones turn upon their axes ac, ab, their surfaces being always in contact in some right line as ad; then will the time of a complete rotation of the cone aed, be to the time of rotation of adf, as m to n; and consequently the number of rotations of the former cone to the number of rotations of the latter in any given time, as n to m. And if these coues were fluted, the flutes diverging continually from the apex a to the base, they would become conical wheels, and constitute bevel-geer.

10. Thus, if Bb and вd (fig. 5. pl. III.) be the bases of two cones turning on their axes, having teeth cut in them diverging from the common vertex A to those bases, such teeth will work freely into one another from one end to the other: but, as such teeth would be very difficult of adjustment towards the point A, and because in practice the two axes could not both be properly

fixed to one and the same point; it is necessary to cut off a portion, as AFE, from the upper part of both cones, and apply the axles to the lower parts in the same manner as in common wheels. The great advantages of these conical wheels are, that their teeth may be made of any breadth, according to the stress they are to sustain; and that the friction will be small in comparison of that occasioned by most other methods of communicating motion in oblique directions.

11. Now, to determine the dimensions of two conical wheels to communicate motion in any oblique angle, the following graphic method may be used. Suppose ab (fig. 6. pl. III.) to represent the shaft or axle of one wheel, and de the axle of another wheel, the angler in which they intersect each other being equal to the angle in which the motion is proposed to be communicated: let it be required for the shaft de to revolve m times while the shaft ab revolves n times; and let the line i be drawn parallel to de at a distance equal to the radius of the base of the wheel whose axle is de. Then draw a line kk parallel to ab, and at a distance yg from it, which shall be to the distance yh as m to n: through the point of intersection of the lines first proposed, and y the intersection of the two lines i, kk, respectively parallel to the two former, draw the line xyw, which will be the pitch line of the two conical wheels, or the line in which the teeth of those wheels act upon one another; and gy, hy will represent the exterior radii of the wheels, which will work one against the other after the manner shewn in fig. 7, where the corresponding parts are marked by the same letters. A third shaft and wheel may easily be applied to communicate motion in a different direction from either of the former as the shaft and wheel rstv in fig. 7.

It is manifest from what is done above, that this is nothing more than to divide an angle bxh into two parts whose sines shall have a given ratio of m to n: a well-known problem which solved algebraically gives the theorem, 2 singxy = 2 sing.ch.

m

m + n

[ocr errors]

(Simpson's Select Exercises, pa. 138.). So that all which is required here may be easily calculated by the common rules of plane trigonometry; and thus the accuracy of the construction may be established.

12. Universal joints (invented by Dr. Hooke) are sometimes used to communicate motion obliquely, instead of conical wheels. ig. 8. pl. III. represents a single universal joint which may be employed where the angle does not exceed 40 degrees, and when the shafts are to move with equal velocity. The shafts A and B being both connected with a cross, will move on

A

the rounds at the points CE and DF, and thus if the shaft a is turned round, the shaft в will likewise turn with a similar motion in its respective position.

The double universal joint (fig. 9. pl. III.) conveys motion in different directions when the angle is between 50 and 90 degrees. It is at liberty to move on the rounds at the points G, H, 1, K, connected with the shaft B; also on the points L, M, N, T, connected with the shaft A: thus the two shafts are so connected that one cannot turn without causing the other to turn likewise. These joints may be constructed by a cross of iron, or with four pins fastened at right angles upon the circumference of a hoop or of a solid ball: they are of great use in cotton mills, where the tumbling shafts are continued to a great distance from the moving power: for by applying a universal joint, the shafts may be cut into convenient lengths, and so be enabled to overcome a greater resistance.

13. When the number of teeth in each of two wheels is given, and the diameter of one of them, the diameter of the other should be so found that one wheel may drive the other without shaking: and for this purpose there will be a different proportion of diameters or of radii, according to the number of teeth which are to be in contact. Let ADE, BDF (fig. 10. pl. III.) represent portions of the wheels, c the point where the teeth ought first to come into contact: draw CD perpendicular to AB the right line joining the centres of the wheels; and if this be reckoned the radius, CB will be the secant of the angle DCB, and AC the secant of the angle DCA. Consequently, CB : CA: secant DCB : secant DCA:: cosec. DBC: cosec. DAC. But, the number of teeth in each wheel being given, the angles DBC, DAC, vary as half the number of teeth in contact. Therefore, divide the arch of the semicircle, or 180 degrees, by half the number of teeth in each wheel, and proportion the radii of the wheel to the cosecants of the quotients, or of double, or of treble the quotients, according to the depth of the wheels running, viz. according as they are to have two, four, or six teeth, in contact; so shall the motion be regular and free from shaking.

In art. 147. of the first volume, we described the best forms for the teeth of wheels: in many cases, however, a small deviation from these perfect forms is not of great importance. But in cases where the utmost accuracy is required, as in the pallets of clocks and watches, the form of the teeth must be carefully attended to. See the article TEETH in this volume.

14. To regulate any motion and make it uniform, one of the most obvious methods is that by means of a pendulum and scapement. Thus, (fig. 5. pl. II.) as the pendulum AB vibra es, it causes EFG to vibrate also, about the axis FG: whilst the pendulum vibrates towards D a tooth of the wheel KL goes off the

« ZurückWeiter »