Eocoentrically-loaded Overhung Shaft. 521 



that the effect of torsional oscillations will be quite inde- 

 pendent of the motions just considered, provided they are 

 sufficiently small. At first sight, it 'might seem that cf> 

 would appear in equations taking into account torsional 

 vibrations. A little reflection, however, will show that the 

 torsional potential energy does not depend on 0, and for 

 •small displacements z and the torsional vibrations are 

 quite independent of the vibrations treated above. 



§ 10. It would here seem convenient to discuss the effect 

 of yielding at the bearings. This may have an appreciable 

 influence on affairs, if the framework is light. In the 

 problem considered above, we have only one hearing to 

 consider : this is taken for simplicity. 



Referring to fig. 3, we shall now let denote the angular 



Fie. 3. 



deviation of the flywheel end of the shaft from the direction 

 of the shaft at the bearing end. This latter will, in general, 

 not be quite constant in direction. Its variation (supposed 

 to be in the same plane as 6) will be denoted by l . 

 Similarly the deflexion at the bearing will be taken (in the 

 same plane as 0) as z\, whilst the deflexion at the flywheel 

 end will be taken as z + z^ Neglecting the mass of the 

 shaft itself, the kinetic energy will be the same as (v.), save 

 that 6 + 0i must be written for 0, and z 4- Z\ for z. As for 

 the effect on the potential energy, we may represent it by 



adding a term 



i(K 1 z 1 *+2K 2 z 1 1 + K z l 2 ),. • • (xxvii.) 

 to (vii.). 



The net effect is to give five equations of motion, instead 

 of the equations (viii.), (ix.), and (x.). A discussion of the 

 possible steady motions gives the same alternatives as these 

 equations led to. In particular, the stable motion is about 

 •</> = 7r. The actual equations in this case reduce to : — 



2 + *i-«'(f+^i>+3B(25-L^)/M =0, (xxvii..) 



'z + z l -(o 2 iz-{-z 1 )+K 1 z l + K 2 ] = 0, (xxix.) 



+ l + a> 2 (0i-0 i ) + B(-:hl J + 2L 2 0)/I 2 = O, (xxx.) 



d + 0*4 co\0 + l )K ? z l + K^ l =0, (xxxi.) 



r 2 <f> - 2r(i + i,)» + I 2 <£/M =0. (xxxii.) 



