590 Messrs. Cowley and Levy: Method of Analysis suitable 



in the first place that the law of formation is clearly bound up 

 with the relation 



Vn =\'<Lx\ ~ \ dx \ dxfll/n-1' 



Jo Jo K Jo Jo 



The four arbitrary constants A, B, C, D are as yet un- 

 determined, but will shortly be found from the boundary 

 conditions. 



Inserting the boundary conditions (7) in {12), the four 

 following equations for A, B, C, D are easily derived : 



o = 4/1(0) + b/ 2 (0) + c/ s (0) + D/;(0) ) 

 o = A/;'(i)+B /t (i)+ o/,(i)+D/ 4 (i) . 



M = A / /'(0) + B/ 2 "(0) + q/ 3 "(0)+D/'/'(0) j 



= A/ 1 "(l)+B/ 1 "(l) + C/ i "(l) + p/ 4 "(l) J 



(17) 



Referring now to the expressions for these functions, it is to 

 be noted that 



/i(0)-/i(0)*MO) = 0, /i(0) = l, 



/i"(0) = 0, / 2 "(0) = l/Bo, /,"(<>) = 0, //'(0) = 0. 



Hence it follows that B = M R and D-D, where A and C 

 are likewise easily determined. The interest of this problem 

 does not, however, lie, from the engineer's point of view, in 

 determining the deflected position of the shaft under the 

 applied bending moment EI M at the end, for any given 

 period of rotation 2ir/q, but rather in finding those particular 

 values of q for which the shaft whirls. Since the analysis is 

 based on the assumption that the deflexions are small, 

 whirling would involve a violation of the basic assumptions, 

 and the deflexion would appear from the equations to be 

 infinite. The condition that this should be so, and therefore 

 that whirling should occur, is easily derived from (17), viz. 



/i(0) 



M0) 



MO) 



M0) 







/id) 



A"(0) 



/»(i) 



/."(0) 



/•"(0) 



A"(0) 



= o, 



(18) 



fi"(i) 



/i"(D 



/."(i) 



//'(i) 







when th( 



) particular values for the 



functions 



are 



inserted, becomes 



/i(l)/i"(l) =/i"(l)/s(l) (19) 



