pully which receives the varied motion from the uniformly rotating cam pully, which is the driver of the combination, as B is the follower. Ap Bq This pully receives the varying motion from the constant uniform rotation of the cam pully by means of an endless band, pqtsr, and is therefore the follower pully. The disk C being supposed to revolve clockwise, the portion of band pq will pull the lower circumference of B, and the velocity ratio will be equal to where Ap and Bq are the respective perpendiculars upon the direction of the band, which is always a common tangent to the cam pully and the follower pully B. But as the former turns, the length of Ap varies, while that of Bq is constant. It is therefore impossible to employ an ordinary endless band. The bard is therefore carried over the upper parts of the two pullies, and brought down as shown by the dotted lines, and carried under a pully attached to the end of an arm, ht, which swings on a pin at h, and carries a weight W to stretch the band. It is apparent, therefore, that the position of the dotted part of the band has no influence on the velocity ratio, and also that the perpendiculars Ap, Bq, being always in the same direction, although varying in length, the directional relation is constant. In this figure the direction of the perpendiculars are both downwards on the band. But by carrying the band tangentially over A and B instead of under, the perpendiculars would be both upwards, and the part pq would become the loop over t. Fig. 189. S B Otherwise the band pq might pass under A and over B, or vice versâ. But whichever course has been given to the band, the directional relation remains constant. 274. In fig. 189 the cam pully is fixed to the disk in a position entirely beyond the center of rotation A. Hence in each revolution the entire cam pully is carried over and under the center, as shown in the positions 1 and 2. In the first the whole cam pully is travelling to the left, and the band pig, pulling the follower pully B clockwise with velocity ratio. But as the cam pully Ba is carried upwards, the perpendicular Ap, diminishes, and when it has risen so far that the common tangent of the circle B and the pully passes through A, Ap, vanishes for an instant, and the velocity ratio=0. But the motion of the cam pully to 2 now obtains a perpendicular Ap1, in the opposite direction, which gives out cord to the follower Bq. The cord, however, is kept tight by the stretching pully below, and thus the motion produced is, that each revolution of the great disk communicates one back and forward motion to the follower Bq. In this case, therefore, the velocity ratio and directional relation both vary. 275. In fig. 190, three positions of the cam are shown, numbered 1, 2, 3. The angle of the salient point is measured by that of its tangents qr, rv, and the cam is so fixed to the disk that the point coincides with the center of rotation of the disk. Beginning with position 1, the velocity ratio is Ap As the Bq motion of the disk goes on, the cam turns upon its salient point r, and the perpendicular rp, dimi nishes, and finally vanishes, when Fig. 190. the common tangent qs of the cam and follower is brought into coincidence with qr, and the cam into the position 2, in which the salient tangents are rv, rq The cam now turns on the center of the disk r, and therefore gives out no cord to B, until it reaches the position 3, where the tangent rq of the salient angle qru coincides with the direction of the cord. The cam proceeding from the position 3 towards 1 will now press with its lower edge upon S rq, and communicate motion to the follower, gradually increasing as the common tangent of the cam and follower is removed from the diametral direction Aq, and the angle App, increased. The motion in one revolution of the disk of this arrangement has an interval of perfect rest of the follower, succeeded by an oscillation, which begins gradually, reaches its maximum, and ends gradually. The angle of rest is measured by the passage of the tangent Aq, to the position Aq. Let 0=the angle of salience and angle of rest.. =π-0. By this adjustment, therefore, we have directional relation constant, with intermission of motion. Fig. 191. 276. In the above figures it is evident that by the rotation of the curvilinear pully A the stretching pully D receives a varied motion upwards and downwards. If, therefore, this pully be attached to a sliding piece or to an oscillating arm, a varied or intermittent motion will be communicated to this piece or arm by the rotation of the curvilinear pully. b. C. For example, if the pully be an excentric circle whose center is m, mb will be constant, and the motion the same as that produced by a crank with radius Am and link bm. If the pully have straight parallel sides and be terminated by semicircles whose centers are e and ƒ, and radii the same as that of the small pully d; and if C the center of motion of the large pully be midway between e and f, then Cd will be the radius of the ellipse whose foci are e and ƒ, and center the center of motion of the pully; so that the vertical sliding motion of d will be determined by the equation of this ellipse round its center. CHAPTER XI. ELEMENTARY COMBINATIONS. DIVISION D. COMMUNICATION OF MOTION BY LINK-WORK. CLASS A. { DIRECTIONAL RELATION CONSTANT. VELOCITY RATIO CONSTANT. 277. WE have seen that when two arms revolving in the same plane about fixed centers are connected by a link (Art. 30), their angular velocities are inversely as the segments into which the link divides the line of centers. This relation is constantly changing, as the arms revolve, unless the point of intersection T (fig. 6), can be thrown to an infinite distance, by making PQ parallel to AB, in all positions, which can only be effected by making the arms equal, and the link equal in length to the distance between the centers. In this case the angular velocities will become equal, and their ratio consequently constant. Fig. 192. a 278. This produces the arrangement of fig. 192. D, B are centers of motion, Bd= Df the arms, df (=BD) the link. If Bd be carried round the circle, BdfD will always be a parallelogram, and consequently the angular distances of Bd and Df from the line of centers the same, and their angular velocity the same. B But as in any given position of one of the arms Bd, there are two possible corresponding positions of the arm Df, found by describing with center d, and radius df, an arc which will necessarily cut the circular path of ƒ round D in two points ƒ and A A (Fig. 8, p. 19); therefore AD is also a position of the arm corresponding to Bd, in which the link dA intersects the line of centers in a point C'; and if Bd be moved, the point C will shift its place, and consequently the angular velocity of AD will not preserve a constant ratio to that of Bd. D pully which receives the varied motion from the uniformly rotating cam pully, which is the driver of the combination, as B is the follower. This pully receives the varying motion from the constant uniform rotation of the cam pully by means of an endless band, pqtsr, and is therefore the follower pully. The disk C being supposed to revolve clockwise, the portion of band pq will pull the lower circumference of B, and the velocity ratio will be equal to Ap, where Ap and Bq are the respective perpendiculars upon Bq the direction of the band, which is always a common tangent to the cam pully and the follower pully B. But as the former turns, the length of Ap varies, while that of Bq is constant. It is therefore impossible to employ an ordinary endless band. The bard is therefore carried over the upper parts of the two pullies, and brought down as shown by the dotted lines, and carried under a pully attached to the end of an arm, ht, which swings on a pin at h, and carries a weight W to stretch the band. It is apparent, therefore, that the position of the dotted part of the band has no influence on the velocity ratio, and also that the perpendiculars Ap, Bq, being always in the same direction, although varying in length, the directional relation is constant. In this figure the direction of the perpendiculars are both downwards on the band. But by carrying the band tangentially over A and B instead of under, the perpendiculars would be both upwards, and the part pq would become the loop over t. Otherwise the band pq might pass under A and over B, or vice versâ. But whichever course has been given to the band, the directional relation remains constant. 274. In fig. 189 the cam pully is fixed to the disk in a position entirely beyond the center of rotation A. Hence in each revolution the entire cam pully is carried over and under the center, as shown in the positions 1 and 2. In the first the whole cam pully is travelling to the left, and the band p19, pulling the follower pully B clockwise with velocity ratio= AP. But as the cam pully Bq |