The crank pin then is acted on by a force S. S sin TPO at right angles to OP, S cos TPO along PO. Then the first is the only one which produces any useful turning effect, it is therefore the crank effort, and Crank Effort S sin TPO, sin TPO This result verifies then, in this case, the Principle of Work, and also by this method of proceeding we have traced the effect of P, showing how it first acts to cause a stress between crosshead and gudgeon pin, and then this stress is transmitted along the rod, finally causing the crank effort. Mean Crank Effort.-Representing the mean crank effort by Rm we have It is usual to complete the curve of crank effort by drawing a circle to represent Rm. The radius will be 2.OP/π, and thus the dotted circle of mean velocity (Fig. 128) is also the circle of mean crank effort. Linear Curve of Crank Effort.-After obtaining the polar curve of velocity ratio, we (page 175) drew a linear curve on the piston stroke as base, and we could similarly draw a curve of crank effort. But such a curve would be of little use, and we will draw a linear curve in a different manner. The crank effort exerts energy on the crank pin; and, although it moves in a circular path, yet its direc A tion is always along the path. Referring then to chap. iii., page 71, if we straighten out the path of the pin, and set up at every point an ordinate representing the effort, when the pin is at that point; we shall obtain a curve which will give us, not only the crank effort for any position of the pin, but also the energy exerted by the effort between any given positions. This latter will be represented as shown in chap. iii. by the area between the two given ordinates. In Fig. 135 we have drawn a polar curve of crank effort, OA representing P (the piston pressure). B 8 Now, in Fig. 136, lay off the crank circle unrolled as a base. The points ABA'B' 12 being named identically in the two figures. 13 22 14 21 15 16 20 19 XIX 178 17 &XVIII B' Fig. 135. Divide the circle in 135 into a number of equal parts, say 24; and also mark off the same parts on the unrolled circle in 136. Now measure in 135 the efforts, and set them up as ordinates at the corre 2 3 4 5 6 7 8 9 10 11 A' 13 14 15 16 17 18 19 20 21 22 23 A B B' Fig. 136. sponding points of 136 as shown; II', 211", etc., being identical with OI, OII, etc. Finally, in 136, draw the curve AI'II’MA'NA through the tops of the ordinates. The curve AMA'NA is now a linear curve of crank effort. But it gives the efforts at points of the path of the crank pin, not of the piston. We complete by drawing a line, parallel to the base, at a distance 2P/π or 20A/π from it, showing the value of the mean crank effort. The full curves drawn are the true curves, taking account of obliquity, the actual case they refer to being n=4 (n being connecting rod-crank ratio). If we neglect obliquity the polar curves, as we have seen, are circles (Fig. 128); and the derived linear curve is obtained as above. Looking now at the shape of the curve, we see that, even with a constant steam pressure, the driving effort is very irregular. Thus, by reference to Fig. 128, it will be found to vary from o at A and A' to a maximum equal to P at B and B', i.e. from o to π/2 times the mean value-when obliquity is neglected; and from o to 1.57 times the mean, when obliquity is allowed for (Fig. 135). With a shorter rod the effect would be still worse, but the ratio 4 to I is about the smallest ever practically used. The result of altering the length of the rod is shown in the table on page 195. Motion of the Crank Shaft.-If the crank shaft revolved with exact uniformity, it would be necessary that, at every instant, the moment Ra of the crank effort should be balanced by an exactly equal resisting moment; and in the first method of finding the ratio of R to P (page 183) we assumed this to be the case. But the assumption there made was necessary, only that we might take the whole engine as the body acted on-or at least the whole of the moving parts, i.e. piston and rod, connecting rod, and crank shaft. But now applying the first method to the body consisting of piston and rod, and connecting rod only; we shall find the same value for R as before, quite irrespective of what the resisting moment applied to the shaft is. Or looking at the second method (page 184) of obtaining R, that is, as it stands, independent of the resisting moment. Now, in actual practice, the resisting moment does not vary at all according to the crank effort, but depends on entirely different considerations; and, as a rule, the moment is very nearly uniform. It follows then that, the driving effort being irregular and the resistance uniform, there must be produced an irregularity of motion in the shaft. What the magnitude of this effect will be we have so far not the means of calculating; but we can see that the effect will be an irregularity. For example—at A and A' there is no effort at all, and hence if means be not provided for moving the crank over these points, the engine will stop; hence the name dead points. The irregular application of effort also causes a greater tendency to break, in the parts, than a regular application would. Two-Crank Engine.-One way in which the irregularity of action can be decreased is by using two cylinders, working on cranks at right angles, on the same shaft. Then, for example, when one crank is on the dead point the other will have almost its maximum turning effect. We will now see how to represent the combined turning effect of the two cylinders. R1 In Fig. 137 we have the two crank efforts R1 Fig. 137. and R2. These produce turning moments R1a, Ra, both clockwise. Hence, no matter what the relative directions of the R's are, their combined effect is a clockwise turning moment, R1a+Ra. .. Combined moment = (R1+R2)a. Now this moment we can repre sent graphically by assuming it to consist of a crank effort R1+R2, acting either on OA CHAP. IX TЯLOWER DIVISION FORCES-CRANK EFFORTS 189 or OB, because it is actually equivalent to such an effort. We can now draw a curve by setting off the value of the combined crank effort along either OA, the leading, or OB, the following crank. It is quite immaterial which we choose, but we must keep to the same one all through the construction. The process then will be as follows :— In Fig. 138 we have drawn the ordinary single diagram of crank effort in dotted lines. And we will choose the leading crank as that along which to set off the combined effort. Commence with the leader at OA. The follower is at OB'. |