Exeentrically-loaded Overhung Shajt. 517 



motion of the C.Gr. o£ the flywheel is now easily obtained 

 on reference to fig. 2. Remembering that the plane of the 

 diagram in fig. 2 is to be supposed at the small angle to 

 the vertical plane of the paper, we easily see that the velocity 

 of the C.G. has components : — 



(i.*) b + r(<f> cos + co) sin cf> vertically downwards, 



(ii.) zco — r(<f) cos + eo) cos <£, right to left, 



(iii.) rcj) sin sin cf>, parallel to undeflected axis of shaft. 



The directions referred to are those in fig. 2, and r is the 

 (small) distance which the flywheel is displaced excentrically. 

 If M be the mass of the flywheel, the kinetic energy of 

 translation of the wheel is accordingly : — 



M • 



— \'z 2 + sV + r\(j> + ft)) 2 4- r 2 ^0 2 sin 2 <£ + 2Sr(# + ft)) sin 



— 2c?'o>(0 + a)) cos j-. . . . (i.) 



Here cos has been taken as unity, and sin as 0. 



We have now to discuss the kinetic energy of rotation of 

 the flywheel, considered as rotating about an axis through 

 its C.G. The resultant angular motion of the wheel is made 

 np of (i.) an angular velocity co about the axis of the shaft 

 at the constrained end ; (ii.) an angular velocity about an 

 axis perpendicular to the plane of bending ; (iii.) an angular 

 velocity (p about the axis of the shaft at the free end. The 

 axes in (iii.)? (ii-), and an axis at right angles to these may 

 be taken as the principal axes at the C.G. If we denote 

 the angular velocities about these axes by O^ fl 2 , and 3 

 respectively, we have 



Q 1 z= ( p-\- co cos 6 ; fl 2 = #; D. 3 = (o sin 0. . (ii.) 



Denoting the corresponding moments of inertia of the wheel 

 by I 1? I 2 , and I s , we get for the kinetic energy of rotation 

 about the C.G., by a well-known theorem, 



JI^ + ft) cos Of + il 2 2 + il 3 ft) 2 sin 2 0. . (iii. ) 



We have I 3 = I 2 , and if for simplicity we confine ourselves 

 to the case of a circular disk, so that I t = 21 2 , expression (iii.) 

 reduces to 



i 2 -S</) 2 +ft) 2 + 2^ft)-^ft)^ 2 +^ 2 -ft) 2 ^;-. . . (iv.) 



We have here replaced sin by 0, and cos by l — 2 ;2. 



