in a rigid body move in an assigned path by communicating certain motions to other points in that body. It is seldcm that one of these effects is produced without at the same time producing the other; but the classification of Willis depends upon which of those two effects, even supposing them to occur together, is the practical object of the mechanism. § 73. Differential Windlass.-The axis C (fig. 1 12) carries a larger barrel AE and a smaller barrel DB, rotatin as one E piece with the angular velocity ar in the c§ rection A AE. The pulley or sheave FG has a weight W A hung to its centre. A cord has one end made fast | to and wrapped round the barrel AE; it passes from A under the sheave FG, and has the other end wrapped round and made fast to the barrel BD. Required the relation between the velocity of translation 112 of W and the angular velocity al of the dijerential barrel.
In this case v2 is an aggregate velocity, produced F G by the joint action of the two drivers AE and BD, - transmitted by wrapping connectors to FG, and combined by that sheave so as to act on the fol- lower W, whose motion is the same with that of Fm H2 the centre of FG.
The velocity of the point F is
motion being considered positive.
of the point G is -a., . CB, downward motion Hence the instantaneous axis of the sheave FG is FG, at the distance
al . AC, upward
in the diameter
2 AC -l-BC
from the centre towards G; the angular velocity of the sheave is =, ,, ;e2Tj%J,1§ ;,
and, consequently, the velocity of its centre is FG AC -BC a, (AC -BC)
U2=Uf1 ' 2 ° = 2, 1
or the mean between the velocities of the two vertical parts of the cord. If the cord be fixed to the framework at the oint B, instead of being wound on a barrel, the velocity of W is half) that of AF. A case-containing several sheaves is called a block. A fall-block is attached to a fixed point; a running-block is movable to and from a fall-block, with which it is connected by two or more plies of a rope. The whole combination constitutes a tackle or purchase. (See PULLEYS for practical applications of these principles.) § 74. DL/ferential Screw.-On the same axis let there be two screws of the respective pitches pi and p2, made in one piece, and rotating with the angular velocit a. Let this piece be called B. Let the first screw turn in a fixedlnut C, and the second in a sliding nut A. The velocity of advance of B relatively to C is (according to § 32) alh, and of A relatively to B (according to § 57) -ap2; hence the velocity of A relatively to C is
Cl (Pl -P2) » (46)
being the same with the velocity of advance of a screw of the pitch pl-gg. This combination, called Hunter's or the dijerential screw, com ines the strength of a large thread with the slowness of motion due to a small one.
§ 75. Epicyclic Trains.-The term epicyclic train is used by Willis to denote a train of wheels carried by an arm, and having certain rotations relatively to that arm, which itself rotates. The arm may either be driven by the wheels or assist in driving them. The comparative motions of the wheels and of the arm, and the aggregate paths traced by points in the wheels, are determined by the principles of the composition of rotations, and of the description of rolling curves, explained in §§ 30, 31.
§ 76. Link Motion:-A slide valve operated by a link motion receives an aggregate motion from the mechanism driving it. (See STEAM-ENGINE for a description of this and other types of mechanism of this class.)
§ 77. Parallel Motions.—A parallel motion is a combination of turning pieces in mechanism designed to guide the motion of a reciprocating piece either exactly
A —~- - ~—~~——;-E or approximately in a straight line, " ' so as to avoid the friction which |
1 ~ D, -' E arises from the use of straight guides
- for that purpose.
} '. 1 Fig. II3 represents an exact
- parallel motion, first proposed, it is C;— ~'-"1 -~" -' B believed, by Scott Russell. The I If ' arm CD turns on the axis C, and
is jointed at D to the middle of the bar ADB, whose length is double
Q of that of CD, and one of whose
L .-' ends B is jointed to a slider, sliding ° in straight guides along the line FIG 113. CB. Draw BE perpendicular to CB, cutting CD produced in E, then
the bar ADB; and the direction of
motion of A is at every instant perpendicular to EA—that is, along E is the instantaneous axis of
the straight line ACa. While the stroke of A is ACa, extending to equal distances on either side of C, and equal to twice the chord of the arc Dd, the stroke of B is only equal to twice the sagitta; and thus A is guided through a comparatively long stroke by the sliding of B through a comparatively short stroke, and by rotatory motions at the joints C, D, B.
§ 78.* An example of an approximate straight-line motion composed of three bars fixed to a frame is shown in fig. 114. It is due D *E Q. //O
/' - “T
A, c' FIG. 114. FIG. II 5.
to P. L. Tchebichev of St Petersburg. The links AB and CD are equal in length and are centred respectively at A and C. The ends D and B are joined by a link DB. If the respective lengths are made in the proportions AC:CD:DB=1:1~3:o-4 the middle point P of DB wiii describe an approximately straight line parallel to AC within limits of length about equal to AC. C. N. Peaucellier, a French engineer officer, was the first, in 1864, to invent a linkwoik with which an exact straight line could be drawn. The linkwork is shown in fig. 115, from which it will be seen that it consists of a rhombus of four equal bars ABCD, jointed at opposite corners with two equal bars BE and DE. The seventh link AF is equal in length to hall the distance EA when the mechanism is in its central position. The points E and F are fixed. It can be proved that the point C always moves in a straight line at right angles to the line EF. The more general property of the mechanism corresponding to proportions between the lengths FA and EF other than that of equality is that the curve described by the point C is the inverse of the curve described by A. There are other arrangements of bars giving straight-line motions, and these arrangements together with the general properties of mechanisms of this kind are discussed in How to Draw a Straight Line by A. B. Kempe (London, 1877). § 79.* The Pantograph.—If a parallelogram of links (fig. 116), be fixed at any one point a in any one of the links produced in either direction, and if any straight
line be drawn from this point
to cut the links in the points
b and c, then the points a, b, c
will be in a straight line for
all ositions of the mechanism,
and) if the point b be guided
in any curve whatever, the
point c will trace a similar
curve to a scale enlarged
in the ratio ab zac. This
is utilized in the construction
Q b .. ° .
property of the parallelogram
of the pantograph, an instrument.
used for obtaining a copy of a map or diiawing on a different scale. Professor ]. ]. Sylvester discovered that this property of the parallelogram is not confined to points lying in one line with the hxed point. Thus if 11 (fig. 117) be any point on the link CD, and if a
point c be taken on the link DE such that the triangles CbD and DcE are
similar and similarly situated with regard to their respective links, then the ratio of the distances ab and
ac is constant, and the angle bac
is constant for all positions of the mechanism; so that, if b is guided in any curve, the point c will describe a similar curve turned through an angle bac, the scales of the curves being in the ratio ab to ac. Sylvester called an instrument based on this property a-plagiograph or a skew pantograph.
The combination of the parallelogram with a straight-line motion, for guiding one of the points in a straight line, is illustrated in Watts parallel motion for steam-engines. (See STEAM-ENGINE.) § 80.* The Reuleaux System of Analysis.-If two pieces, A and B. (fig. I 18) are jointed together by a pin, the pin being fixed, say, to A the only relative motion possible between the pieces is one of turning about the axis of the pin. Whatever motion the pair of pieces may have as a whole each separate piece shares in common, and this common motion in no way affects the relative motion of A and B The motion of one piece is said to be completely constrained relatively to the other piece. Again, the ieces A and B (fig. 119) are paired together as a slide, and the oni)y relative motion possible between them now is that of sliding, and therefore the motion of one relatively to the other is completely constrained. The pieces may be paired
will ' .1
Fic.. 1 1 7.