TO THE DETERMINATION OF THE EFFICIENCY OF MACHINERY. 29 
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,. 
§ 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 compound 
machine of this class, fig. 33. In this frame the successive 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 m diminishing R on the driving side 
by a length depending 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 ie Let F, be the tension on 
the second rope. The couple required to unbend the rope has an effect which 
may be represented by shifting the line of action outwards to an amount s,, 
mv 
equal to pat each pulley a similar effect is produced, and as the value of the . 
: 3 
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 7 sin 9, 
where 7 is the radius of the shaft.- We then have the equation 
E (R —Prsingd —*) = F(R + 7rsing + zm) 
or E(R—7 sin 6) =F \(R+1r sin G) + 2m. 
From this equation we obtain F,, and by a similar equation we could from this 
obtain F,, &c. 2F=W, the weight which can be raised. The result is after a 
few turns to reduce F, to nil, after which no more plies can be of any service. 
The gradually diminishing efficiency of successive pulleys is very well shown 
by the diagram, fig. 33. 
§ 27. Loaded Dynamic Frame without Friction.—Let us now consider the 
effect of the mass and weight of the elements of a machine. We may feel sure 
that the effect of weight and inertia may be shown by means of a dynamic frame, 
for this frame consists of lines indicating bearing pressures at the joints, and 
the direction and magnitude of these bearing pressures are always determinate. 
In the case of a material element supported by two joints, the lines of bearing 
VOL. XXVIII. PART I. H 
