The motion of the connecting rod of a reciprocating steam engine is very
clearly understood from the simple statement that one end travels in a
circle and the other in a right line. From this statement it is also
readily inferred that the path of any point between the centers of the
crank and crosshead pins will be neither circular nor straight, but an
elongated curve. This inference is so far correct, but the very common
impression that the middle point of the rod always describes an ellipse
is quite erroneous. The variation from that curve, while not conspicuous
in all cases, is nevertheless quite sufficient to prevent the use of this
movement for an elliptograph. To this there is, abstractly, one
exception. Referring to Fig. 22 in the preceding article, it will be seen
that if the crank OH and the connecting HE are of equal length, any point
on the latter or on its prolongation, except E, H, and F, will describe
an exact ellipse. But the proportions are here so different from anything
used in steam engines (the stroke being four times the length of the
crank), that this particular arrangement can hardly be considered as what
is ordinarily understood by a "crank and connecting rod movement," such
as is shown in Fig. 23.
The length DE of the curve traced by the point P will evidently be equal
to A'B', the stroke of the engine, and that again to AB, the throw of the
crank. The highest position of P will be that shown in the figure,
determined by placing the crank vertically, as OC. At that instant the
motions of C and C' are horizontal, and being inclined to CC' they must
be equal. In other words, the motion is one of translation, and the
radius of curvature at P is infinite.
To find the center of curvature at D, assume the crank pin A to have a
velocity A_a_. Then, since the rod is at that instant turning about the
farther end A', we will have D_d_ for the motion of D. The instantaneous
axis of the connecting rod is found by drawing perpendiculars to the
directions of the simultaneous motions of its two ends, and it therefore
falls at A', in the present position. But the perpendicular to the motion
of the crank pin is the line of the crank itself, and consequently is
revolving about O with an angular velocity represented by AO_a_. The
motion of A' is in the direction A'B', but its velocity at the instant is
zero. Hence, drawing a vertical line at A', limited by the prolongation
of _a_O, we have A'_a_' for the motion of the instantaneous axis.
Therefore, by drawing _a_'_d_, cutting the normal at _x_, we determine
D_x_, the radius of curvature.
Placing the crank in the opposite position OB, we find by a construction
precisely similar to the above, the radius of curvature E_z_ at the other
extremity of the axis of the curve. It will at once be seen that E_z_ is
less than D_x_, and that since the normal at P is vertical and infinite,
the evolute of DPE will consist of two branches _x_N, _z_M, to which the
vertical normal PL is a common asymptote. These two branches will not be
similar, nor is the curve itself symmetrical with respect to PL or to any
transverse line; all of which peculiarities characterize it as something
quite different from the ellipse.
[Illustration: FIG. 23.]
[Illustration: FIG. 24.]
[Illustration: FIG. 25.]
Moreover, in Fig. 22, the locus of the instantaneous axis of the trammel
bar (of which the part EH corresponds to the connecting rod, when a crank
OH is added to the elliptograph there discussed) was found to be a
circle. But in the present case this locus is very different. Beginning
at A', the instantaneous axis moves downward and to the right, as the
crank travels from A in the direction of the arrow, until it becomes
vertical, when the axis will be found upon C'R, at an infinite distance
below AB', the locus for this quarter of the revolution being a curve
A'G, to which C'R is an asymptote. After the crank pin passes C, the axis
will be found above AB' and to the right of C'R, moving in a curve HB',
which is the locus for the second quadrant. Since the path of P is
symmetrical with respect to DE, the completion of the revolution will
result in the formation of two other curves, continuous and symmetrical
with those above described, the whole appearing as in Fig. 24, the
vertical line through C' being a common asymptote.
In order to find the radius of curvature at any point on the generated
curve, it is necessary to find not only the location of the instantaneous
axis, but its motion. This is done as shown in Fig. 25. P being the given
point, CD is the corresponding position of the connecting rod, OC that of
the crank. Draw through D a perpendicular to OD, produce OC to cut it in
E, the instantaneous axis. Assume C A perpendicular to OC, as the motion
of the crank. Then the point E in OC produced will have the motion EF
perpendicular to OE, of a magnitude determined by producing OA to cut
this perpendicular in F. But since the _intersection_ E of the crank
produced is to be with a vertical line through the other end of the rod,
the instantaneous axis has a motion which, so far as it depends upon the
movement of C only, is in the direction DE. Therefore EF is a component,
whose resultant EG is found by drawing FG perpendicular to EF. Now D is
moving to the left with a velocity which may be determined either by
drawing through A a perpendicular to CD, and through C a horizontal line
to cut this perpendicular in H, or by making the angle DEI equal to the
angle CEA, giving on DO the distance DI, equal to CH. Make EK = DI or
CH, complete the rectangle KEGL, and its diagonal ES is, finally, the
motion of the instantaneous axis.
EP is the normal, and the actual motion of P is PM, perpendicular to EP,
the angle PEM being made equal to CEA. Find now the component EN of the
motion ES, which is perpendicular to EP. Draw NM and produce it to cut EP
produced in R the center of curvature at P.
This point evidently lies upon the branch _z_M of the evolute in Fig. 23.
The process of finding one upon the other branch _x_N is shown in the
lower part of the diagram, Fig. 25. The operations being exactly like
those above described, will be readily traced by the reader without
further explanation.
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