ANALYSIS OF THE KINEMATICS OF MECHANISMS. 513 



In the ordinary steam-engine with guide bars, the radius bar BP 2 swinging 

 in the base-plate-bearing at P 2 is replaced by the cross-head sliding in straight 

 guides which form part of the base-plate. The effect is the same as if BP 2 were 

 infinitely long. On account of the cross-head joint being guided in a straight 

 line, passing through the crank journal centre, a symmetry is given to the 

 motion which materially lightens the labour of drawing complete velocity and 

 acceleration diagrams. Fig. 5 illustrates this. 



Here the four positions 1, 2, 3, and 4 of the crankpin A are taken equi- 

 distant from the dead-points O and O'. Therefore the two cross-head positions 

 B x and B 4 coincide, as do also B 2 and B 3 . Therefore also the four velocities pa u 

 pa 2 , pa 3 , and pa^ are equally inclined to the velocity line pb, and the four points 

 <h, «2> «3» « 4 are equidistant from the line pb. Also at 1 and 2 the connecting 

 rod has the same inclination to the centre line, which inclination is equal and 

 opposite to that at 3 and 4. Thus the lines aj) 4 and ajj x , and a 2 b 2 and a 3 b 3) 

 are equally inclined to pb ; and, therefore, the velocities pb 4 and pb x have equal 

 magnitudes, as also have pb 3 and pb 2 . Therefore also the radial accelerations 

 «i/3i, a' 2 fi 2 , a 3 fi 3 , a'^ have equal magnitudes, and are equally inclined to p'b'\ 

 while also the tangential accelerations, fi[b[, &c, are equally inclined to the 

 same line, and are of the same length, because a[, a 2 , a 3 , and a'± are equidistant 

 from pb' . Therefore, finally, b[ coincides with b' i} and b 2 with b 3 . The four 

 accelerations a'b' have equal magnitudes, but p'b 2 =p'b' 3 differs in magnitude as 

 well as direction fvom p>'bi=pb[. 



This symmetry is, of course, destroyed by want of uniformity in the rotation 

 of the crank. 



The joint lines of the bars of a mechanism, the velocity lines, and the 

 acceleration lines need be drawn in full for one position only. The results for 

 the other positions are indicated by numbered points on the three set of curves, 

 which are the loci of the corresponding points or extremities of lines. The first 

 set of curves are the paths of motion of the joints. The second series of curves 

 are the hodographs of the velocities of these same joints. The third series are 

 the loci of the extremities of the lines representing the velocity accelerations. 



Six-bar motion is nearly equally easy to deal with by this method. 



The first example given in fig. 6 is quite simple, because the velocity pa of 

 the joint A is assumed as known, the bar PjA being one of the quadrilateral 

 PiABP 2 . The determination of the velocity pb is, therefore, the same as that 

 given already. Thus, pb and ab are drawn perpendicular to P 2 B and AB, and 

 their intersection gives b. Then the triangle abc is made similar to ABC. pc? 

 is then drawn perpendicular to P 3 D, and col to CD, the intersection giving pd 

 the velocity of D. To find the velocity of E, there are drawn pe and de per- 

 pendicular to P 3 E and DE. The construction of the acceleration diagram here 

 offers no special difficulty. 



