16. Let a B, B D, D E, be a series of levers turning on the fixed centres C, Q, and R; then when the arcs through which the extremities A and E are moved are small the velocity ratio will be expressed by the following equality :

that is to say, the velocity ratio of P and P, is found by taking the product of the lengths of the arms lying towards P, and dividing by the product of those lying towards p ̧.

17. To find the velocity ratio of the rods A B and C D turning on the fixed centres, A and D; and connected by the link B C.

Through the centres A and D, draw the straight line D E A, cutting C B in E; and from A and D let fall the perpendiculars A G and DK upon CB, or it may be upon с в produced. Then

Fig. 7.

A

that is to say, the angular velocities of the rods D C and A B are to each other in the inverse ratio of the segments into which the link divides the line joining their axes.

These velocity ratios are obviously varying, depending upon the relative positions of the rods.

18. THE CRANK AND GREAT BEAM.-Let A B represent one half of the great beam of a steam engine, Do the crank, and BC the connecting rod. Putting ẞ for the angle D C B, and ß1 for the angle A B C; then

When the connecting rod B c is very long as compared with the length of the crank D C, then ẞ, is nearly constant, being nearly equal to 90°, in this case, eq. (1) becomes

The crank must be in the same straight line with the connecting rod, at the highest and lowest points of the stroke of the beam, and then ẞ= 0. In these positions the crank is said to be at its dead points.

The velocity ratio, expressed by eq. (2), will be a maximum when ẞ = 0, that is, the velocity of the crank will be a maximum when it is in its dead points. When ẞ = 90°, or when the crank is at right angles to the connecting rod, then the velocity of the crank is a minimum.

If R = A B, or one-half the length of the great beam; r = DC, the length of the crank; and a = the angular oscillation of the beam, or the whole angle described by the beam in one stroke; then

which expresses the length of the crank in terms of the radius of the beam and angle of its stroke.

A double oscillation of the beam produces one complete rotation of the crank, or conversely, taking the crank as the driver, each rotation of the crank produces a double oscillation in the beam.

=

From eq. (1) it follows, that the velocity of the crank is equal to the velocity of the beam, when B B, or angle D C B is equal to angle A B C ; that is, when the position of the crank is parallel to that of the beam.

By this form of the crank the reciprocating circular motion of the extremity of the beam is changed into a continuous circular motion; and conversely a continuous circular motion is changed into a reciprocating circular motion.

19. To determine the various relations of position and velocity of the CRANK and PISTON in a locomotive engine.

Here the connecting rod, D E, is attached to the extremity of the piston rod, P D, and the length of the stroke of the piston

Fig. 8.

P

내 D

E

is equal to double the length of the crank, F E. Moreover, the centre, F, of the crank is in the same straight line with the axis of the cylinder, or the direction of the piston rod.

Let =D E, the length of the connecting rod;

=P D, the length of the piston rod;

FE, the length of the crank;

k = F D, the varying distance of the extremity of the piston rod from the axis of the crank;

h = the corresponding height of the stroke of the piston; 0 = the varying angle, F E D, which the crank forms with the direction of the connecting rod.

(1.) The velocity ratio of the crank and piston is expressed by the following equality :

where ẞ in eq. (2) is put for angle E FD; that is, the angle which the crank makes with the direction of the piston rod. This latter form of the expression is the same as that given in eq. (2), Art. 18.

(2.) When the piston is at the bottom point of its stroke,

its distance from F = FE + E D + D P = r + 1 + ↳1; also FDFE + DE = r + l.

When the piston is at the middle point of its stroke, then FD = ED; that is to say, in this position of the piston D E F will be an isosceles triangle.

(3.) The position of the crank at any point of the stroke of the piston is determined by the two following general equations :

When the piston is at the middle point of its stroke, then h = r, and eq. (4) becomes

When the crank is at right angles to the connecting rod 0 = 90°, and then we find from eq. (4),

This expression is, obviously, less than r, or half the whole stroke of the piston. Hence it appears that the crank is at right angles with the connecting rod before the piston has attained the middle point of its upward stroke.

20. Fig. 9 shows how a rotation of the axis a is transmitted to another c, by means of the two equal cranks

Fig. 9.

B

D

A B and C D, connected by the connecting rod D B, whose length is equal to the distance a c, between the two axes. In all positions of the cranks, the figure A B C D will be a parallelogram, and the velocity of D will always be equal to the velocity of B, and the motion of the axis c will be exactly the same as that of the axis a.

21. Two sets of cranks may be placed upon the axes, having the cranks on each axis at right angles to each other, similar to the mode of connecting the wheels of a locomotive engine,

as shown in fig. 10, where the cranks are formed by bending, or loops made in the axes.

These

axes must be parallel to each other, and the connecting rods

must also be of equal lengths.

The advantage of this combination consists in maintaining a constant moving pressure, by which means an equable motion is sustained without the aid of the inertia of the machinery.

Fig. 10.

Fig. 11.

E

H

B

22. The double universal joint represented in fig. 11 furnishes another example of link-work for transmitting motion from one axis to another axis. This useful piece of mechanism should be constructed so that the extreme axes, A B and C D, would meet in a point, if produced, and the angles which they respectively make with the central line of the intermediate piece, E F H G, shall be equal to each other.

Fig. 12.

D

E

TO CONSTRUCT WATT'S PARALLEL MOTION.

23. This beautiful and useful piece of mechanism is formed by a combination of link-work