APPLICATION OF GRAPHIC METHODS 313 



frame, namely, the quadrilateral with t\vo diagonals. As an 

 example, we may take the hanging pulley and fixed pulley 

 combined, as in Fig. 32, on which is shown the dynamic frame 

 of the combination. When the ropes are nearly parallel, as in 

 Fig. 32a, the bars shown by thick black lines may be considered 

 as jointed to the six links, which would otherwise meet at the 

 joints B, A. and A P 



26. Reduplication of Cords. When a series of fixed and 

 hanging pulleys are employed as in the ordinary blocks and 

 tackle, it is found that, with the usual stiff ropes, no advantage 

 can be obtained by using more than 5 or 6 plies of rope. The 

 reason of this is shown clearly by the dynamic frame for a com- 

 pound machine of this class, Fig. 33, p. 315. In this frame the suc- 

 cessive pulleys and plies are arranged in one plane, so that the 

 diagram may be better followed than could be the case if the 

 pulleys were placed so as to be co-axial. Let the driving link 

 act between the rope a and the fixed support d, and let the force 

 applied by the driving link be called E, and the effective radius 

 of each sheave R. The effect of the rope a is to produce a 

 couple 7?i diminishing R on the driving side by a length depend- 

 ing on the stiffness of the rope, and inversely proportional to 

 the diameter of the pulley and to the tension on the rope. The 

 effect of this is shown in the diagram by a shifting of the line of 

 action of the force towards the centre of the pulley by a distance 



s equal to . Let F t be the tension on the second rope. The 

 L 



couple required to unbend the rope has an effect which may be 

 represented by shifting the line of action outwards to an amount 



SP equal to ~ ; at each pulley a similar effect is produced, and 



*i 



as the value of the tension in the rope diminishes at each pulley, 

 so the value of s increases at each pulley. The effect of friction 

 at the axle is shown by shifting the joint in the bar representing 

 each pulley towards the driving rope by a distance r sin <f>, 

 where r is the radius of the shaft. We then have the equation 



E (R-r M < />-|)=F 1 (Rfr sin <#>+ 



or E(R r sin <) = F,(R + r sin <f>) + 2m. 

 From this equation we obtain F 1; and by a similar equation we 



