G, draw the straight line F H, meeting the horizontal line AJ in H. Through the three points E, H, and F draw an arc. This arc will be the characteristic line of the gear, and includes the extremities of all the virtual eccentric arms for any position of the block in the link. Take any position of the gear, and divide the line EF in the same proportion that the block divides the whole length of the line. Referring to the diagrams, let X be the point of gear chosen. It is required to find the movement of the valve corresponding to this particular position. On the characteristic line EF mark the point W so that F W bears the same proportion

Valve Diagram for Stephenson's Link Motion.

to the line EF that XC bears to the whole length of the link O B. Join the points A and W. Then, AW is the virtual eccentric arm which produces approximately the same valve movement that results from the combined action of the two eccentrics A E and A F, acting through the medium of the link B C. On A W as diameter draw the valve circle. The maximum port opening, the periods of admission, cut-off, release, and compression are now clearly seen; and by choosing other positions of the link and repeating the above construction the action for any other point is

seen.

Consider the enlarged diagram for open rods, Fig. 54. The letters A, E, F, and H correspond with the letters of the previous diagrams;

and the characteristic line EF contains the extremities of all the virtual eccentric arms. The lap circle J K remains constant for all positions of the gear; the valve circles, however, are changed for every position of the gear, and cross the lap circle at different points, and, in the manner of the simple Zeuner diagram, indicate the percentages of admission and cut-off. The lead is also seen to

vary for all positions of the link, a fact which explains that common statement that in the shifting link gear, with open rods, the lead increases as the gear is brought into mid-position; but with crossed rods, the reverse is the case.

It can be shown that these statements are correct from an inspection of Fig. 52. The motion for both open and crossed rods is shown in mid-gear; the crank in each case is on the outer dead

Fig 54

The point

point, and the port is open for lead. Suppose the link BC moved so that the point B is brought to the centre line D G. B is constrained to move in an arc whose centre is E, and therefore the shifting of the link motion must pull the valve to the left, because E is fixed. This movement manifestly lessens the port opening and decreases the lead. Applying the same reasoning to the crossed rod gear, it is seen that the valve spindle moves to the right as the block in the link approaches full gear. Generally, the longer the eccentric rods and the shorter the valve travel, the less will be the variation of lead. The truth of these statements is at once verified by a glance at the figure.

Macfarlane Gray's Method of Describing the Characteristic Line. Macfarlane Gray has suggested a very

simple method of describing the line E F, which is as near the truth as practice demands. Describe with centre on AJ produced (Fig. 53) an arc passing through E and F with radius found as follows:

Radius = length of eccentric-rod from centre to centre × half the distance between the two eccentric sheaves the distance between the centres of the eccentric-rod pins on the link. ReDTX DD' ferred to Fig. 55, Radius

=

2T N

For open rods the arc is concave to the shaft; for crossed rods concave to the link.

It is instructive to apply the two methods given above, to one case of link motion, and observe the closeness of their agreement.

The above constructions emphasize the advantage of keeping the eccentric throws short, the eccentric-rods long, and the link short, in order that the lead disturbance shall be a minimum. The first and third conditions are to be met by employing double-ported valves; and to keep the eccentric-rods long, the link should be located as near the valve chest as circumstances permit.

At this point it may be well to explain that the diagrams just given are approximate only. When valves are operated directly from eccentrics, the movement can be predetermined with great exactitude, but with link motions it is not so. So many points step in to modify the valve movement, that to pre-arrange a link motion that shall satisfy all conditions is impossible. The best that can be done is to secure the best action for full forward and full backward gear, and then, with the link properly suspended, the eccentric-rods long, and the valve travel short in proportion, the action for intermediate gear will be almost exactly that shown by the characteristic line in the diagram, and will be found satisfactory.

The foregoing constructions are, as said, only approximate, but there is a method whereby the valve movement can be determined exactly when the leading dimensions have been settled upon.

Let the accompanying diagram (Fig. 55a) represent, in outline, a link motion in which it is desired to determine the exact movement of the link. Let AE and AF be the backward and forward eccentrics respectively, BC the link, and D G the suspension link, shown in position for mid-gear. Divide the eccentric circle EF into

any number of equal parts, say ten, starting from the point E for the backward eccentric, and from F for the forward eccentric. (See Fig. 55a). From E as centre, and with the length of the backward

eccentric as radius, describe an arc PQ; and from F as centre, and with the length of the forward eccentric as radius, describe an arc RS. Now, it will be apparent that when the eccentrics are at AE and A F, the centre of one pin hole of the link must be somewhere on PQ, assuming that the link is of the double bar form, shortly to be described; and the centre of the other pin hole must be somewhere on RS; whilst the third point is somewhere on the arc J K, the path of the end of the suspension link.

Now prepare a tracing paper templet of the link, drawn to scale, marking thereon the position of the pin holes and the point of suspension. In the figure, such a templet is shown, X being the backward pin hole, Z the forward pin hole, and Y the suspension point. Applying this templet to the arcs PQ, RS, and JK, so that X is on PQ, Y on J K, and Z on R S, a definite position of the link is given, which can be pricked through to the paper below, and numbered to correspond with the eccentric circle. By repeating this process for all the other positions of the eccentrics, the point-path of the pin holes is tracked out, discovering the exact movements of these points. Taking the case of the present gear, the extreme horizontal points of the point D are seen to be D and H; and since the valve spindle is in the direction A D, its travel will be exactly equal to DH; and, further, since the link is in mid-gear, D H is equal to twice the lap plus twice the lead, so that either of these being known, the other is found. The peculiar paths of the pin holes are known as "slip curves," and indicate the amount of slotting motion of the die in the link.

The above method of tracing the valve motion is open to one objection, namely, that if the point-paths are to be drawn full size, the drawing must be as large as the whole gear itself, a condition which is usually inconvenient; but, by the method about to be described, the point-paths may be found without drawing the eccentric-rods in position.

=

=

In the figure, suppose the eccentric circle EF transposed to the left a distance AT EB FC; and let a templet L M N be prepared, having the radius O equal to A T, and the side M N in the direction of the radius. If this templet be applied to the transposed circle, so that the point M coincides with the chosen. point in the circle, and the side M N is parallel to A D, the curved side L M contains the centre of the eccentric pin hole. The templet L M N being made of cardboard, or some other suitable material, the curve can be at once drawn on the paper. Similarly, placing the templet in position for the other eccentric, the curved side contains the centre of the other pin hole. Now, applying the link templet as before, so that each centre falls on the curves thus found, the link's position is discovered.* Referring to the figure, A valuable paper on Point-Paths in Mechanisms, found by the use of the templet," appeared in Engineering, vol. lvii., p. 439. The use of a templet in this way is applicable to many forms of link-work, and is especially valuable when the mechanism is too large to admit of all parts being drawn in position.

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