Loaded Shaft supported in 'Three Bearings. 



667 



general solution leads to a quadratic for co 2 . For two real 

 values of the whirling speed to be possible, both roots of the 

 quadratic must be positive. By assuming the load at an 

 appropriate position in the span the effects of X and M may 

 be separated. 



Thus in a shaft supported freely in two bearings,, if the 

 load is at the centre of the span, the normal whirling depends 

 on X simply and is due to a displacement of the centre of 

 the load from the line of bearings, without oscillation. On 

 the other hand, oscilialory whirling depends on M simply, 

 and is due to an oscillation of the load about a diameter, 

 without displacement of the centre of the load from the line 

 of bearings. 



§ 22. Vv 7 e will consider one simple case of oscillatory 

 vibration for a three bearing shaft. 



Let there he a single load of the nature of a wide and 

 heavy pulley on the shaft (fig. 5). 



Fie:. 5. 



Let it be given that the sp ins are equal and the shaft 

 uniform throughout its length. 



For simple oscillatory vibration the point at which the 

 load is fixed will remain on the line of bearings. We write 

 I 1 — I 2 for mk 2 in the formulae of § 15, and make z r = z 1 for 

 the load point, and wj = 0. The general equation for u 

 becomes 



(I 1 -I 2 )(J) i [^i{l-^-3(l-On 



+ jei(l-« l )(2-* 1 )"{3(l-«i) , -l)]=0, 



or 3* 1 I -12* 1 »-2* 1 + 4 = J 



which is satisfied by ^ = 0*531. 



Thus if the load be at ,c 1 = 0'531/, a simple separate 

 oscillatory vibration is possible. 



Also from § 15 we have 



6EI/d 



mi-c-o, 



[{l_3i 1 2 -3(l-0 2 WU-3(l-«i) 3 }{3(l-< 1 ) i, -l}] 



2 Y 2 



