416 APPLIED MECHANICS: 
348. Balancing one Revolving Mass by two Others, all the Masses 
being in Separate Planes of Revolution.—Let A be the one mass, and B 
and C the other two masses (Figs. 682 and 683). Let the weights of — 
the masses be W,, W,, and W, 
respectively, and let the dis- A Cos 
tances of their centres of & 2 5 i 
gravity from the axis of = GY Ts; 
revolution be 7,, 7,, and 7, |ea | a —-C | ot 
respectively. Also let the —p—;4 S — ke Lol 
distances of the planes of = '-~|--€--|—>4 i i 
revolution of A and C from adem 3 ! é. 
the plane of revolution of B eit 4 Brat 
be a and c respectively. The &=s Cc 4 
centres of gravity of the three Fra. 682 Fic. 688 
masses must obviously lie in 
a plane containing the axis of revolution, and in Figs. 682 and 683 the 
plane of the paper has been taken as this plane. 
The centrifugal forces of the three masses A, B, and C are pro- — 
portional to W,7,, W.7,, and W,7, respectively, and the problem evidently 
reduces to the balancing of three parallel forces in the same plane. The 
conditions of equilibrium are, (1) for the case shown in Fig. 682, 
W,r, = Wer, + Wa73, and Wyr,a= Ware ; 
and (2) for the case shown in Fig. 683, 
W,r, + Wor3= Woo, and W,7r,a= W,r,¢. 
For each case there are therefore two equations from which two 
unknown quantities may be determined. 
If the planes of revolution of the masses B and C (Figs. 682 and 683) 
be the central transverse planes of the bearings of a revolving shaft 
carrying the given mass A, then when the given revolving mass is 
unbalanced by other revolving masses the centrifugal forces of the masses 
B and C determined as above will be the forces exerted by the bearings _ 
on the shaft due to the centrifugal force of the mass A. 
349. Balancing two or more Revolving Masses by two Others, 
the Masses being in Separate Planes of Revolution.—Let A and B 
(Fig. 685) be two given masses which have to be balanced by masses P 
1 
VR, 
‘ Aj rn 
if fig? 
Bi ch | ie AZ 
Fic. 684. BIg. 685. Fig. 686. 
and Q in the planes of revolution 1 and 2 respectively. Figs. 684 and — 
686 are face views of the planes 1 and 2 respectively, with the masses A — 
