P2B' (ab)2 to A, and is equal to and is represented by a'b'. The tangential acceleration of B about A is not directly known, but if a line through b' be drawn perpendicular to AB, the point b' must lie on that line. But the acceleration of B can also be obtained in another way, namely, by considering the rotation P2B about P2. The acceleration of B over the base plate is clearly its centripetal acceleration about P2, combined with its tangential acceleration about the same centre. The former acts from B to P2, and is (pb)2 equal to and is represented by p'b'2. The tangential acceleration is not directly known, but the point b' must lie on a line through ba' perpendicular to P2B. The point b' must, therefore, be the point of intersection of this line and the line through l'1, and thus the acceleration diagram is p'a'b'b'b'2. In that diagram, p'a' represents the centripetal acceleration of A about P1, a'b', the centripetal, and b'b' the tangential accelerations of B about A; whilst p'b' represents the centripetal acceleration, and b'b' the tangential acceleration of B about P2. The angular acceleration of the shaft P2 is therefore the numerator being measured (b'ab') on the scale of accelerations, and the denominator on the scales of feet. The diagram shows that the acceleration is counter-clockwise, or that the shaft P2 is being retarded. If the shaft P1 rotate with varying angular velocity, the necessary modifications can be made as in § 165. § 169. Wigzell's Engine.*-As a quantitative example, take Wigzell's engine, a diagrammatic sketch of which is shown in Fig. 294. The engine has three cylinders, and the three pistons X, Y, Z, which move vertically along parallel lines in the same plane, operate on the same crank pin, A, through the triangular connecting-rod ACD.† The high-pressure piston rod (Y) is directly coupled to the rod at B; but the low and intermediate pressure piston rods (X and Z) are connected by short links, CE and DF. * For a description of this engine, see Engineering, September 7, 1900. The link AB is, kinematically, unnecessary, but is put in for greater clearness. The scale of feet is as shown. The stroke of the high-pressure piston rod is 2. PA; and the strokes of the intermediate and low pressure pistons may very approximately be obtained (§ 114) by neglecting the obliquity of the links CE and DF, and assuming C and D to move in vertical lines. The three strokes are B1B2, C1C2, D1D2, and, in the engine considered, are 15, 166, and 17·6 inches respectively. The diagram of velocities must first be drawn. The revolutions have been taken to 120 per minute, so that the circumferential velocity of the crank pin is (the crank radius being 7 inches) 7.84 feet per second. The velocity diagram is shown in Fig. 295, the scale of velocity being appended. To obtain it, pa is first drawn to represent the velocity of the crank pin, and the velocity of B is then obtained by drawing a perpendicularly to AB to meet the vertical through p in b. To find the velocity of C, it must be remembered that C relative to B moves perpendicularly to BC, and relative to A perpendicularly to AC; the point c is, therefore, at once obtained. The point d may be obtained in a similar manner, and the triangle abcd will be exactly similar to the triangle ABCD, as explained in § 146. Having obtained the points c and d, the velocities of E and F are obtained by drawing lines through c and d perpendicular to CE and DF respectively to meet the vertical line through p in the points e and f. Then pe, pb, pf represent the velocities of the pistons X, Y, Z; and are respectively 81, 465, and 0.64 feet per second. To draw the acceleration diagram, first take a length, p'a' (Fig. 296), to represent the centripetal acceleration of A. Since the velocity of A about P is 7.84 feet per second, and PA is 0.625 foot, the centripetal acceleration of A is 984 feet per second per second. The scale of acceleration is appended. The acceleration of B is obtained in the manner described in § 161. The velocity of B about A (namely, ab) is 6-8 feet per second, and AB is 2.835 feet; hence the centripetal acceleration of B about A, represented by a'b', is 16.3 feet per second per second. The acceleration of B, namely, p'b', is obtained by drawing b'b' perpendicularly to AB to meet the vertical through p' in b'. To find the accelerations of C and D, describe on a'b' a figure, a'c'b'd', exactly similar to the figure ACBD in the mechanism diagram (see § 160), care being taken that the sides of a'c'b'd' can, by rotation, be made parallel to the corresponding sides of ACBD. Having determined the point c', the acceleration of E (that is, of the piston X) is obtained by combining the acceleration of E about C with that of C over the base plate, namely p'c'. The centripetal acceleration of E about that is, 0.205 feet per (ce) 2 0.322 0·5' C is and in the engine taken is CE' second per second. This is so small that the length which represents it is inappreciable on the diagram, and has been omitted. The line drawn through perpendicular to CE to meet the vertical through p' in e', will therefore give the point e', and p'e' will be the acceleration of the piston X. Similarly, the centripetal acceleration of F about D is very small indeed, and d' is obtained by drawing d'f' perpendicular to DF to meet the vertical in f'; then p'f' represents the acceleration of Z. Thus, in the position of the mechanism sketched, the accelerations of the pistons X, Y, Ꮓ are 72-5, 97.5, and 127.5 feet per second per second respectively. 10 5 0 10 20 30 40 50 60 70 80 90 FT. PER SEC. PER SEC. FIG. 296. An approximate solution to the motion of the pistons X and Z may be obtained by the equivalent crank method described in § 154, the motion of the pistons being assumed to be the same as of the points C and D. This may be left as an example to the reader. |