202 



2jtnR* 



~da 

 dr 



= — 2jtiiR* 



R 



+ 





The differential equation for the free vibration is: 



d*a da 



K [- L ^ MazzzO 



dt* dt 



(23) 



givinfj- foi' the natural frequencies of the system the equation 



Kn^ + Ln -[-i/ = (24) 



The quantity L here contains V^n. 

 If we put L = P -{- Q ^n, where 



P = 4.T nR* and Q = 2ji (iRR' 



(24) assumes the form : 



Kn' + (P 4- Q i/«) n -L- M=0 (25) 



Equation (25) is bi-quadratic in z = y^ii. On further examination 

 it is found to have 2 complex roots 2 in the right hand portiori of 

 the complex plane and 2 in the left portion, only the former of 

 which we can use (comp. equation (22)); hence the system has but 

 one natural frequency. Further z" = /i is found to contain a negative 

 real term, as indeed could not be expected otherwise. 



