From A, rule the line AX in such a position that the arc AB makes the least possible departure from it. It will be found that this necessitates AX cutting the arc AB at a point a little past the middle of the arc. With the length of the crank and crank-arm, viz. 15 inches, in the compasses, from A as the centre cut the line AX at E, and this gives the position for the crank-shaft which will give the least possible eccentricity to the slay. This will be obvious, as the nearer the connecting pin moves on the straight line AX, the less will be the eccentricity of the slay.
That the movement of the slay in the back half of its stroke is slower than in the front half can easily be proved by taking the length of the crank-arm in the compasses, and, after bisecting the arc AB at C, from C marking off the points D and H on the crank circle. It will be seen that both these points are somewhat inside the top and bottom centres of the crank indicated by the dotted line, and therefore the slay moves from C to A and back, the front half of its stroke, in less time than it moves from C to B and back to C. The reason for this eccentricity or unevenness in the movement of the slay is that when the crank is moving from the back centre to the top centre the crank-arm is oscillating and opening an angle with AX while the slay is moving forward, and therefore while the crank is making this quarter of a revolution, the connecting pin of the slay will move something less than from B to C; and while the crank is moving from the top centre to the front centre, the crank-arm is straightening or closing the angle while the slay is moving forward, and thus the connecting pin will move a greater distance than from C to A while the crank is making this quarter of a revolution. When the crank moves from front to bottom centre the angle is opening while the slay is moving backwards, and therefore the pin will move a little more than from A to C; and when the crank moves from bottom to back centre the angle is closing while the slay moves backwards, thus retarding the velocity of the slay.
FIG. 59.
This will be better understood from [Fig. 59], where CD is the crank-arm, and ED the crank at the top centre. A is the position of the connecting pin when at the front of its stroke, and B its position when at the back of its stroke. The dimensions are as in [Fig. 58]—viz. 12-inch arm, 3-inch crank—and for simplicity we will assume the connecting pin moves on the straight line AE. From A to B is 6 inches, and therefore it is obvious that AC is something over 3 inches, and that the connecting pin moves this distance whilst the crank is making the quarter of a revolution from top to front centre. The distance AC can be obtained as follows. It is obvious that CDE is a right-handed triangle, and therefore CD2 is equal to CE2 + ED2. Therefore CD2-ED2 = CE2, and having obtained the length of CE, we can subtract this from AE, which leaves the length of AC. The formula will stand thus—
CD2
−
ED2
=