Take a radius r and with centre O describe a circle. This circle cuts the curve in four points, either of which joined to O gives a crank position fulfilling the required condition. To determine which is the particu lar one we must know on which side of the centre the piston is, and in which direction it is travelling. Thus in Fig. 119 if the piston be In OA travelling to A', Or is the crank from A', 04 Fig. 119. Rod. We have seen that the piston position corresponding to a given crank position depends on the ratio of connecting rod to crank arm. Now to see in what way an alteration of connecting NM O NM O Fig. 120. rod length does affect the position, let us examine Fig. 120, where we have two examples drawn, differing only in length of connecting rod. In each figure O is the crank centre, OP a given crank, CP the connecting rod, and M the piston position, PM then being an arc with centre C. In each figure drop PN perpendicular to the line of stroke. In each figure the distance CN is less than CM or CP, the connecting rod length. The amount of the difference in each case depends on the angle PCO, i.e. on the obliquity of the connecting rod; and it varies from zero at the beginning and end of the stroke to a maximum somewhere between. Contrasting the two figures we see that MN is much less in (b) than in (a), due to the fact that the connecting rod is longer in (b). If we made the connecting rod still longer, the difference MN would be less still; and when the ratio of connecting rod to crank reaches say about 12:1, the points M, N would be practically identical. The process of dropping a perpendicular from P is a simpler one than that of finding the arc PM; and, moreover, the result obtained can be expressed in a simpler form. Hence, for many purposes, we treat the point N as the piston position instead of M; our error in doing so is MN; and we say MN is the error due to obliquity, and N is the piston position neglecting obliquity. We can very easily calculate the amount of the error due to obliquity. For with the usual lettering (Fig. 120) MN=CM-CN, =na-na cos PCN. PCN is usually denoted by and PON by 0, .. MN=na (1 - cos ), The greatest value of is when the crank is upright, i.e. 0=90°; and then It is quite near enough, ø being a small angle, to take M If n=4, the greatest error is only s/16, while if n=12 it is only s/48. There are two cases in which there would be no error due to obliquity. Ist. With an infinite connecting rod, for then M and IN would be identical. This is of course an impossi bility. 2d. With connecting rod of length zero. This latter can be in a way effected, and there is a certain practical type of direct acting engine, which we may say has a connecting road of zero length, and the movement of which is unaffected by obliquity. The figure shows a portion of the mechanism of such an engine, used as a pumping donkey. A is an outside view of the cylinder, with guides fastened to the top and bottom. B is the piston rod, the end being either forged solid or screwed into a crosspiece, thus forming a T head, in which is cut a slot at right angles to the line of stroke. C is a block sliding in the slot, which (C) in a way takes the place of the connecting rod and contains the crank pin brasses. O is the centre of the crank shaft, the crank circle being dotted. The shaft is only shown by its centre, and the remainder of the mechanism is, for clearness, omitted. Now, plainly this mechanism works as an ordinary direct actor neglecting obliquity. For taking the centre point of the slot as defining the motion of the piston (page 154), this point always is at N, the foot of the perpendicular from P, the centre of the crank pin, on to the line of stroke. This point is also the centre of the end of the piston rod, so that it is C in our former figures, thus C and N are identical; and this is why we say that in a way the connecting rod is of zero length. Having thus found that results obtained neglecting obliquity are not only approximate for the ordinary engine, but are also the actual results for another kind of engine, let us see what these results come to. To find the piston position, we simply drop PN perpendicular to AA'. But now let us proceed, as on page 160, to draw a curve. Then (Fig. 122), we make OQ= ON, and so on. Doing this, and drawing a curve through the Q's obtained, the student will find he obtains two curves, as in Fig. 122, which look very like circles, with OA, OA' as diameters. as can be readily proved. For Fig. 122. And in fact they are so, Therefore Q lies on a circle having OA for diameter, and similarly for the other side OA'. We see then that the determination, neglecting obliquity, of a piston position is by far simpler than the actual accurate determination; and thus for proportions of connecting rod to crank which do not make the error very large, we can save much labour by the use of this simpler diagram. There is one important case in which we utilise the diagram just obtained, and which we will now briefly consider. Motion of the Slide Valve.-Fig. 123 shows the ordinary eccentric motion by which the valve is driven. B Fig. 123. This we can easily see is identical with that of a direct actor, for A is the shaft, B is the crank pin, centre P. B is of course made very large, so that it actually embraces the whole of the crank arm in itself and projects on the other side of the shaft. But, as we have explained, the size of a bearing does not affect the motion, but only the position of its centre point. Since B however is so large it gets a new name, viz. Eccentric. Then we have the connecting rod, now called Eccentric Rod and Strap, but being nevertheless only to us one piece, as the connecting rod was. The centre line of this is PC. |