588 Messrs. Cowley and Levy : Method of Analysis suitable 



various powers of the class variable to zero. A series of 

 simpler differential equations will then be obtained for each 

 of the coefficients, with simplified boundary conditions, 

 Consider the following example. The equation whose 

 solution is required at whirling is, expressed in non- 

 dimensional form, 



£( R 8)-^=o, .... (5) 



where R = I/I = the ratio of the moments of inertia at any 

 point x to that at any standard section ; 

 y = deflexion of the shaft axis at any point x. 



This quantity is assumed so small that it is legitimate to 



1 /V2 



write j j-4 for the curvature where I is the length of -the 



I d 



shaft. 



**=^\ ^^ (6) 



g EI ' ^ *>o 



where E = elastic constant, — =mass per unit length, and 



2irjq = period of rotation of shaft. The boundary con- 

 ditions that we may suppose imposed on the shaft are 

 given as deflexions and bending moments at the extremities 

 of the shaft. 



For the sake of deflniteness, let these conditions be 



x = y = 0, Bending moment = M EI , j 



* = 1 .y = 0, „ „ =o. j- ■ (7) 



The class variable in this case is clearly # 4 , while R and fi 

 are given non-dimensional functions of x. It will for the 

 moment be assumed that y can be expanded in ascending 

 powers of # 4 in the form 



y = y*+!hP+yf i +9ifi a - (8) 



Inserting this in equation (5) and equating coefficients of 

 powers of 6 to zero, 



^(*S0 



dx 



4 /*#o 



+*U*%) I -"* 



^U^S) \ 



o. 



