512 PROFESSOR R. H. SMITH ON A NEW GRAPHIC 



if used to find b in the velocity diagram, and j3j $' i} &c., if used to find b' in the 

 acceleration diagram. 



In the displacement diagrams described above, the accented capitals 

 A'B'C &c, are suitable. 



The simplest mechanism is that with four bars and with two joints, P X P 2 

 in the base-plate, and two joints AB coupling the other three bars together. 

 An example is shown in fig. 4, the calculations being made for five different 

 phases of the periodic motion. 



The velocity of the crank-pin A is supposed known at each phase. From 

 any pole p, and to any convenient scale this velocity pa is plotted perpendicularly 

 to P t A. From p a line is drawn perpendicularly to P 2 B. Evidently the 

 extremity b of bp, the velocity of B, must lie in this line. But also pb pa 

 plus a velocity perpendicular to AB. Therefore from a a line is drawn 

 perpendicular to AB to meet the above line in b. Thus ^>& is determined. In 

 the example pa is taken of the same magnitude at all the five phases. 



To obtain the acceleration diagram we assume the acceleration of A. This 



is constant in magnitude =\P^L } on the supposition that pa is also constant in 

 magnitude and is wholly radial, since pa is taken as constant. 



From any pole p this acceleration ~\- —p'a is plotted parallel to AP 2 



P,A 



(not to PiA). 



The calculation of the magnitude is performed by the graphic construction 



(ph) 2 

 previously explained. By the same construction the magnitudes y-g- and 



(ad) 2 



-^ of the radial components of the accelerations of B round P 2 and round A 



are found and plotted off from p (as p $') and from a (a/B') parallel to BP 2 and 

 to BA. From these two points /3' thus obtained, lines are drawn perpendicular 

 to BP 2 and to BA. The point b' sought for must lie on both of these last lines, 

 and is, therefore, at their intersection. The acceleration pb' of the joint B 

 through the field of P is thus obtained for the five different phases of the 

 motion. The method of procedure is plain. Each joint of the mechanism is a 

 point in two different bars, and therefore the calculation for that joint may be 

 approached, as it were, from two different sides. In each of the two calcula- 

 tions there is one element missing, and the last stage of the calculation cannot 

 be completed directly ; for example, approaching the calculation of the accelera- 

 tion of B through A, we can calculate the radial component (parallel to BA) of 

 the acceleration that has to be added to that of A, but of the tangential com- 

 ponent the direction only is known, but this gives a line in which the desired point 

 must lie. ■ Another conditioning line being similarly found by approaching the cal- 

 culation in another way, the point is found at the intersection of these two lines, 



