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the rack a cycloidal form, and making the face of the tooth of the pinion an epicycloid, as will be apparent if we conceive the diameter of the circle whose centre is C (see fig. p. 259.) to become infinite, the other two circles remaining unaltered. Any finite portion of the circumference of this infinite circle will then become a straight line. Let AE in the accompanying figure represent such a portion, and let PQ and PR represent, as before, curves generated by a point P in the circle whose centre is D, when all three circles revolve by their mutual contact at A. Then are PR and PQ the true forms of the teeth which would drive the circles as they are driven by their mutual contact at A (Art. 202). Moreover, the curve PQ is the same (Art. 199.) as would be generated by the point P in the circumference of APH; if that circle rolled upon the circumference AQF, it is therefore an epicycloid; and the curve PR is the same as would be generated by the point P, if the circle APH rolled upon the circumference or straight line AE, it is therefore a cycloid. Thus then it appears, that after the teeth have passed the line of centres, when the face of the tooth of the pinion is driving the flank of the tooth of the rack, the former must have an epicycloidal, and the latter a cycloidal form. In like manner, by transferring the circle APH to the opposite side of AE, it may be shown, that before the teeth have passed the line of centres when the flank of the tooth of the pinion is driving the face of the tooth of the wheel, the former must have a hypocycloidal, and the latter a cycloidal form, the cycloid having its curvature in opposite directions on the flank and the face of the tooth. The generating circle will be of the most convenient dimensions for the description of the teeth when its diameter equals the radius of the pitch circle of the pinion. The hypocycloidal flank of the tooth of the pinion will then pass into a straight flank. The radius of the pitch circle of the pinion is determined as in Art. 212., and the method of describing its teeth is explained in Art. 208.

215. The Teeth Of A Wheel Working With A Lantern Or Trundle.

In some descriptions of mill work the ordinary form of the toothed wheel is replaced by a contrivance called a lantern or

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trundle, formed by two circular discs, which are connected with one another by cylindrical columns called staves, engaging, like the teeth of a pinion, with the teeth of a wheel which the lantern is intended to drive. This combination is shown in the above figure.

It is evident that the teeth on the wheel which works with the lantern have their shape determined by the cylindrical shape of the staves. Their forms may readily be found by the method explained in Art. 200.

Having determined upon the dimensions of the staves in reference to the strain they are to be subjected to, and upon the diameters of the pitch circles of the lantern and wheel, and also upon the pitch of the teeth; strike arcs p AB and AC of these circles, and set off upon them the pitches Aa and Ab from the point of contact A of the pitch circles (if the teeth are first to come into contact in the line of centres, if not, set them off from the points behind the line of centres where the teeth arc first to come into contact). Describe a circle ae, having its centre in AB, passing through a, and having its diameter equal to that of the stave, and divide each of the pitches A« and Ab into the same number of equal parts (say three). From the points of division A, a, /3 in the pitch Aa, measure the shortest distances to the circle ae, and with these shortest distances, respectively, describe from the points of division y, 8 of the pitch Ab, circular arcs intersecting one another; a curve ab touching all these circular arcs will give the true face of the tooth (Art. 200.). The opposite face of the tooth must be struck from similar centres, and the base of the tooth must be cut so far within the pitch circle as to admit one half of the stave ae when that stave passes the line of centres.

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216. The Relation Between Two Pressures P, And Ps

APPLIED TO TWO TOOTHED WHEELS IN THE STATE BORDER-
ING UPON MOTION BY THE PREPONDERANCE OF P,.

Let the influence of the weights of the wheels be in the first place neglected. Let B and C represent the centres of the pitch circles of the wheels, A their point of contact, P the point of contact of the driving and driven teeth at any period of the motion, RP the direction of the whole resultant pressure upon the teeth at their point of contact, which resultant pressure is equal and opposite to the resistance Rof the follower to the driver, BM and CN perpendiculars from the centres of the axes of the wheels upon RP; and BD and CE upon the directions of P, and Ps.

BD=a„ CE = o2> Bm=m„ Cn = To2.

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p,( p^ radii of axes of wheels.

fv ip3=limiting angles of resistance between the axes of the wheels and their bearings.

Then, since P, and R applied to the wheel whose centre is B are in the state bordering upon motion by the preponderance of P„ and since a, and »(, are the perpendiculars on

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the directions of these pressures respectively, we have (equation 158)

where L, represents the length of the line DM joining the feet of the perpendiculars BM and BD.

Again, since R and P2, applied to the wheel whose centre is C, are in the state bordering upon motion by the yielding ofP2(Art. 163.),

where L2 represents the distance NE between the feet of the perpendiculars CE and CN. Eliminating R between these equations, we have

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Now let it be observed, that the line AP, drawn from the point of contact A of the pitch circles to the point of contact P of the teeth is perpendicular to their surfaces at that point P, whatever may be the forms of the teeth, provided that they act truly with one another (Art. 199.); moreover, that when the point of contact P has passed the line of centres, as shown in the figure, that point is in the act of moving on the driven surface Pp from the centre C, or from P towards p, so that the friction of that surface is exerted in the opposite direction, or from p towards P; whence it follows that the resultant of this friction, and the perpendicular resistance aP of the driven tooth upon the driver, has its direction rP within the angle aPp and that it is inclined (Art. 141.) to the perpendicular aP at an angle aPr equal to the limiting angle of resistance. Now this resistance is evidently equal and opposite to the resultant pressure upon the surfaces of the teeth in the state bordering upon motion; whence it follows that the angle RPA is equal to the limiting angle of resistance between the surfaces of contact of the teeth. Let this angle be represented by p, and let AP = *. Also let the inclination PAC of AP to the line of centres BC be represented by 0. Through A draw Are perpendicular to RP, and sAt parallel to it. Then,

m,=BM=~Bt + tM=Bt + An=BA sin. BAt + AP sin. APR. Also BA<=BOR=PAC + APR=fi + <p;

.-. OTl=r, sin. (9 + <p) + >. sin. p (239);

»»2=cn = Cs-*n = Cs-am=ca sin. CAs-AP sin. APR. But As is parallel to PR, therefore CAs=BOR=0 + p;

.•. mi=raam.(Q + f)— A.sin.<p (240.).

Substituting these values of mx and m3 in the preceding equation,

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