289. If the two directions of motion be not in one plane, let ad, cb, fig. 200, be these lines; find their common perpendicular de; draw ce parallel to ad, and in the plane bce construct the required crank, as in Art. 286, of which let B be the center, Bb, Be the arms respectively perpendicular to be and ce.

Fig. 200.

B

d

с

Ba

Draw BA a common perpendicular to Bb and Be, and equal to de. Draw a parallel and necessarily equal to Be, then will AB be the axis, Aa and Bb the arms required to change the small motion in ad into the requisite motion in cb.

By a similar construction we can effect the e change of a small motion in a given direction, into another equal motion in the same direction parallel to the first; which has been shown to be impossible by the bellcrank in one plane, although the motions themselves are in one plane.

In the mechanism of organs, in which the transmission of such small motions is of frequent occurrence, the crank is termed a backfall when its arms are in one horizontal straight line, and a square when they are at right angles.

An armed axis like fig. 200 is a roller, and the links are stickers when they act by compression or pushing, and trackers when by tension or pulling.

CHAPTER XII.

ELEMENTARY COMBINATIONS.

DIVISION D. COMMUNICATION OF MOTION BY LINK-WORK.

CLASS C.

{

DIRECTIONAL RELATION AND VELOCITY RATIO
CONSTANT OR VARYING.

Fig. 201.

290. THE general definition of link-work, given above in the first chapter, Art. 29, has shown that it derives its name from the employment of an intermediate piece termed a link,'* which is a rigid bar connected to each of the pieces, between which it acts as a transmitter and modifier of motion at a constant point of itself and of the piece. In the majority of cases these pieces rotate on parallel axes, and thus the varieties of motion may be investigated by assuming that the pieces and the connecting link are simple radii turning on fixed points at one end and jointed to the respective extremities of the link at the other; the entire combination being thus reduced to four lines in a plane, forming a trapezium ABPQ with variable angles but constant sides, of which AB fixed in the plane is termed the 'line of centers,' AP, BQ the radii' capable of rotating in the plane about the fixed points A and B, and PQ the 'link,' which is compelled to move in the plane so that its extremities P and Q can only travel in the circles described about A B by the extremities P and Q of the radial arms to which they are jointed. In the formula by which the laws of motion of these movable parts are expressed, the length AB of the line of centers is designated by d, the link by 1, the greater and smaller radii by R and r respectively.

P

A

B

It has been already shown that the angular velocities of the

* Bielle, Fr.

radii are inversely as the perpendiculars from the fixed centers upon the link.

The most general motion for which link-work is used is to enable the rotation of one axis to communicate a reciprocating motion to the other. The path of the reciprocating piece is very commonly rectilinear, and this case is brought under the general principle by supposing the rectilinear path to be an arc of a circle of infinite radius. The motion of piston-rods for pumps, steamengines, &c., or the travelling platforms of printing presses, planing machines, the tool bars of slotting machines, and so on, may be quoted as examples of rectilinear reciprocation.

The axes may be required to revolve continuously with constant or varying velocity ratios, or finally, they may be connected so as to admit only of alternate reciprocations.

We may now proceed to examine these four cases in detail, taking them in the order of (1) Rectilinear reciprocation. (2) Rotative reciprocation. (3) Alternate reciprocation. (4) Continu

ous rotation.

(1) Rectilinear reciprocation.—In the four following diagrams, the bar, table, or other sliding piece is omitted, as its motion is a Fig. 202.

simple translation in which every point moves in a path parallel to that of the extremity of the link, and with a velocity equal to that

7

Fig. 203.

A

K

L

extremity, the direction of whose path usually passes through the axis or center of rotation of the driving radius, as in figs. 202, 203, 204.

In these figures, that radius is shown in two positions, AP, Ap, and the portion of the path KL to which the course of the extremity

Q is limited, is determined by setting off from A, in opposite directions distances AK, Ak=1-r and AL, Al=l+r.

In fig. 202 the distance of Q from A= Qd± Ad= √l2 — r2sin20 ±r cos ◊. (1); where the positive sign is used when dis between Q and A, and the negative when d is beyond QA.

In the small triangle Prs, Ps, rs are respectively perpendicular to AP, Pd, therefore we have velocity of P_PS_AP

=

=

velocity of d T'S Pd

Consequently, if P travel uniformly, the velocity of p vanishes at the extremities of its course gG; is at a maximum at the point where Pd=AP; and is the same at any two points taken at equal distances from A on opposite sides.

If the link be very long with respect to AP, its inclination may be practically neglected, and the distance Qd be supposed equal to PQ. Therefore the motion of Q will be the same as the motion of d, arriving at the middle point M of its course KL when Pis at the middle of its semi-rotation from g to G, and having its velocities symmetrically equal on opposite sides of the center point of KL.

=

But the effect of the inclination of the link is to draw the point Q nearer to A than it would be if I were infinitely long, by a space PQ-Qd=l-√l-r2sin20. (2) which when P is at the middle of its semi-rotation as at Ap (figs. 202, 203) becomes mq=l— √ l2 — r2. (3) The segments of the course lq, qk, described by the motion of the radius through the respective quadrants gp, pG, near and remote, are lm±qm=r±(l√l2 — r2) (4). In fig. 204 the link PQ is equal to the radial arm AP, and consequently AP and PQ constitute in all positions an isosceles triangle, of which the base AQ is the line of motion or groove of the pin which connects the radial arm with the sliding bar or piece.

Produce AP to R, making PR= AP, and with that radius describe a circle, LRwl, which is plainly twice the diameter of the inner circle.

Fig. 204.

If the rotating radius AP be pinned at P to the link PQ it will move the pin Q in the line or groove AQ until it arrives at A, the isosceles triangle APQ gradually becoming more and more acute at the apex P until Q is brought into coincidence with A, after which AP, PQ, being folded into a single radius will rotate about A. But if the link PQ be produced to W, we have PW=PQ and QW= AR. Also the figure AQRW is a rectangle, of which AR and QW are equal diagonals and P the

center.

Thus the link PQ is extended from P to W, and when Pits center, is rotating, the respective extremities travel in the crossed diameters IL, wx, like the pencil bar of a trammel.

Each revolution will cause the point or joint pin Q to travel from L to 1, and the point W from x to w. And thus the radial arm AP will move the bar through a course of twice the length due to its radius.

Now as AQ 2 Ad in all positions, it follows that the law of the motion of Q in the line AL is the same as the motion of d, with twice its velocity, and thus the point Q and the bar to which it is attached move with velocities symmetrically equal on opposite sides of the center point A. The left side of the figure shows that the radius Ap makes an acute angle Apq with the vertical diameter which compels the link pq to push the slide point q at an obtuse angle pq4, which would generate jamming friction of a magnitude that would prevent the motion of the bar from taking place (vide Chapter on Friction). To overcome this difficulty short grooves uS, xs, are fixed to the frame of the machine to receive a pin fixed to the extremity W, of the prolonged link. Thus, as W is carried upwards by the rotation of P and its lower end Q guided horizontally by the sliding piece, so, when the angle PQA has nearly reached a degree of obliquity that generates injurious friction, the upper end W of the link enters the guide groove. Its pin acts as a fulcrum against the side of the groove as at w, and the joint pin p of the radius acts transversely on the link so as to press the sliding piece in the direction of the longitudinal motion required.

PROB. To determine the motion of a slide when the path of the end of the link travels in a line that does not meet the

axis.

Fig. 205.

Let A be the center of motion of a revolving driving arm AP (r), PQ a link (1) jointed to AP at P. Its extremity Q is compelled to move in a right line LK, which for comparison with the previous formulæ may be considered as a circle of infinite radius,