By Prof. C.W. MACCORD, Sc.D.
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 Aa. Then, since the rod is at that instant turning about the farther end A', we will have Dd 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 AOa. 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 aO, we have A'a' for the motion of the instantaneous axis. Therefore, by drawing a'd, cutting the normal at x, we determine Dx, 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 Ez at the other extremity of the axis of the curve. It will at once be seen that Ez is less than Dx, and that since the normal at P is vertical and infinite, the evolute of DPE will consist of two branches xN, zM, 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.
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 zM of the evolute in Fig. 23. The process of finding one upon the other branch xN 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.