different numbers of lobes, and a, a, b, b, be the respective apsidal distances, we should find m = αα bb 271. To employ rolling curves in practice. In fig. 124, let the upper curve be the driver, and let it revolve in the direction from T to t. Then since the radius of contact SP increases by this motion, and the corresponding radius PH decreases, the edge of the driver will press against that of the follower, and so communicate a motion to it of which the angular velocity ratio will be PH SP But when the point m has reached M, the radii of contact in the driver will begin to diminish, and its edge to retire from that of the follower, so that the communication of motion will cease. To maintain it, it is necessary to provide the retreating edge with teeth, as in fig. 129, which will engage with similar teeth upon the corresponding edge of the follower, and thus maintain the communication of motion until the point a has reached A, when the advancing side of the driver will come into operation, and the teeth be no longer necessary. 129 M H These teeth, however, necessarily destroy the advantage of no friction, and another practical difficulty is introduced. If the curves be not very accurately executed, it may happen that the first pair of teeth and spaces that ought to come together at M, m in each revolution, may not accurately meet, and that either the tooth may get into the wrong space, or become jammed against another tooth, by which the machinery may be broken. 272. To prevent this accident, a curved guide-plate n (fig. 130) may be fixed to one of the wheels, and a pin p to 130 the other. The edge of this plate must be made of such a form that the pin p may be certain of engaging with it, even if the wheels are not exactly in their proper relative position. When the pin has fairly entered the fork of the plate, it will press either on the right or left side, and so correct the position, and guide the first pair of teeth into contact. It is easy to see that the edge of this plate should be the epicycloid that would be described by p, if the lower plate were taken as a fixed base, and the upper made to roll upon it; but the outer edge of the plate must be sloped away from the true form, to ensure the entrance of the pin into the fork. 131 273. Another method is to carry the teeth all round the two plates, which effectually prevents them from getting entangled in the above manner, but at the same time entirely destroys the rolling action. This method, however, is the one always adopted in practice, as for example, in the Cometarium, and in the silkmills, and is an excellent Fig. 131 method of obtaining a varying velocity ratio. represents a pair of such wheels that were employed by Messrs. Bacon and Donkin in a printing machine. 274. The forms of the teeth to be applied to these rolling curves may be obtained by a slight extension of the general solution in Art. 82. For calling the rolling curves pitch curves, it can be shewn for them, precisely in the same manner as it is there shewn for pitch circles, that if any given circle or curve be assumed as a describing curve, and if it be made to roll on the inside of one of these pitch curves, and on the outside of the corresponding portion of the other pitch curve, that the motion communicated by the pressure and sliding contact of one of the curved teeth so traced upon the other, will be exactly the same as that effected by the rolling contact of the original pitch curves. 275. The Cometarium is a machine which has two parallel axes of motion carrying indices or clock-hands; one of which axes is the center of a circle, and the other the focus of an ellipse, which represents the orbit of a comet. The two axes must be connected by mechanism, so that when the first revolves uniformly, the second shall revolve with an angular velocity that will make it describe equal areas of its ellipse in equal times, and thus represent the motion of a comet round the sun*, for which purpose the machine is constructed. Now, according to what is termed Seth Ward's hypothesis, if one radius vector HP of an ellipse (fig. 124), revolve uniformly round the focus H, the other SP will describe equal areas round the focus S. This, although a very coarse approximation, is considered sufficient for the mechanical representation of planetary or cometary motions in this instrument, and is accordingly obtained by connecting the two axes with a pair of rolling ellipses, as in fig. 124. For by Art. 259, it appears that HP h P, and the angle SHP = sh P. The motion therefore of HP and h P with respect to the axis major of their respective ellipses is the same, and the ratio of the angular velocities of 8P and h P round their foci s and h is the same as those of SP and HP round S and H Also, since the corresponding radii &P, PH have been shewn to = • In any ellipse APM (fig. 124), we have Angular velocity of SP round S HP SP.HP CD2 Angular velocity of HP round H = = SP SP2 SP where CD is the conjugate diameter of the ellipse. If the ellipse be nearly a circle, CD may be supposed constant, in which case if the angular velocity of HP be uniform, that of SP will vary as which is the law of motion 1 SP2 of the radius vector of a planet. This is termed Seth Ward's hypothesis, but is a very coarse approximation. coincide with the fixed line of centers, it follows that the angular velocities of SH and sa round the centers H and s are respectively the same as those of HP and s P, that is, of HP and SP with respect to the major axes of the ellipses. 276. This machine was first introduced by Dr Desaguliers, and may be considered as the first attempt to employ rolling curves in machinery. He did not however furnish his ellipses with teeth, but connected them by means of an endless band of catgut, which embraced the circumference of each ellipse, lying in a groove in the circumference. The addition of teeth was a subsequent improvement. 277. When the required periodic variation in the ratio of angular velocity is not very great, a pair of equal common spur-wheels, with their centers of motion a little excentric, may be substituted for the equal ellipses revolving round their foci; but in this method the action of the teeth will become very irregular, unless the excentricity be very small. 278. The difficulty of forming a pair of rolling curves is sometimes evaded in the manner represented by fig. 132. A is a curved plate revolving round the center B, and having its edge cut into teeth. Ca pinion with teeth of the same pitch. The center of this pinion is not fixed, but is carried by an arm or frame, which revolves on a center D. So that as A revolves, the frame rises and falls to enable the 132 pinion to remain in geer with the curved plate, notwithstanding the variation of its radius of contact. To maintain the teeth at a proper distance for their action, the wheel A has a plate attached to it which extends beyond it, and is furnished with a groove de, the central line of which is at a * Vide Rees' Cyclopædia, art. Cometarium; or Ferguson's Astronomy. constant normal distance from the pitch line of the teeth equal to the pitch radius of the pinion. A pin or small roller attached to the swinging frame D and concentric with the pinion Crests in this groove. So that as the wheel A revolves, the groove and pin act together, and maintain the pitch lines of the wheel and pinion in contact, and at the same time prevent the teeth from getting entangled, or from escaping altogether. Let R be the radius of C, r the radius of contact of A, the angle between R and r; then it can be easily shewn But as the center of motion of C continually oscillates, and it is generally necessary to communicate the rotation of A to a wheel revolving on a fixed center of motion, a wheel E must be fixed to the pinion C, and this wheel must geer with a second wheel D concentric to the center of the swing-frame. When A revolves, the rotation of C will be communicated through E to F, but will also be compounded with the oscillation of the swing-frame, in a manner that will be explained under the head of Aggregate Motions, in the second Part of this work. 279. If for the curved wheel A an ordinary spur-wheel A, (fig. 133) moving on an excentric center of motion B, be substituted, a of course fixed to the extremity of its axis, to prevent the link from striking it in the course of its revolutions*. This From a machine by Mr Holtzapfel. |