Loaded Shaft supported in Three Bearings. 661 



Hence eliminating w 1? i* l5 we find, if — — 2~ y - =$>, 



! 46-</> 9 ™° 



18 23-20 

 0^=48-2 or 9-3. 

 o) 2 ={63-7 or 330-3} x^t. 



§ 18. When the number of separate loads is great and we 

 require to find approximate values of the first and second, 

 or perhaps only of the first whirling speed, a less exact 

 method may be used. 



In the general case of a shaft of varying section supported 

 in three bearings, the general form of the deflexion curve is 

 given in §§6, 7, 9. We note that the loads are due to 

 centrifugal action and are therefore dependent on the dis- 

 placements. The approximate method consists in assuming 

 a displacement curve of algebraic type, and determining its 

 coefficients by making it agree closely with the exact dis- 

 placement curve. Thus we may assume an algebraic curve 

 u = f(z), and compel this curve to lit the terminal conditions 

 at the bearings, and also coincide with the exact displace- 

 ment curve at one or more points, e. g. at the middle points* 

 of the spans. We make the assumed curve satisfy 



M = at c = and /; v = at z r =0 and V '; 



du_dv \ ~ jd 2 u _ __^j/d 2 v 



dz ~~ d'J " ~~ ' dz 2 ~ ~dz^ ' "~~ ' 



d 2 u d^v , , 



- 7 ^-=0at~ = /; -7-7^ = at z = I : 

 dz 2, dz 2 



u = iii at ~ = i/ ; v = i\ at :' — i/'. 



Thus assume 



u = az -f bz 2 + c: 3 + dz 4 + ez 5 + fz*, 



and v = az'--z' 2 + cz ,s -dz n + ez'*-fz 6 . 



We require 



al + U 2 + cl 3 + dl* + el* +// 6 = I ! - 



aV- tp + cl't-d!'** el /5 - #' 6 -=0, 

 2&-fG^ + 12^ 2 + 20^ 3 + 30/J 4 =0. 

 2&_6 c i' + 12<fc' a -20«l' 3 +3q/Z' 4 = 0. 



