508 Mr. NJC. Krishnaiyar on Amplitude of Vibrations 



Expanding the even function cos 2 (pt-\- tan -1 B^/A^ by 

 Fourier's Series into a series of cosines of multiples of 



(pt + tan~ 1 B 1 /A 1 ) between the limits — '- and -, the 



term containing cos (pt + tan -1 B^Aj) is obtained as 



^/rW + B^) cos (^ + tan- 1 B 1 /A 1 ), 



and it can be written as 



^ r fA 1 y/ Af +•&!* coa pt- ^i> 2 Bi \/ A^ + B X * sin pt. 



Equating the collected coefficients of sinpt and cospt 

 separately to zero, we obtain 



Aii^-^ + fW+B^} 



= B 1 { a + [pk' ± ~p^k v/A7+B?] 



and 



= Aj j *—\pk'±.-; 

 Therefore 



8 



^ i2 _ p2 + |^(A 1 2 + B 1 2 )} 2 = a 2 -^f^ ± ^VA 1 2 + B 1 2 ] . 

 If A stands for the amplitude of the maintained vibration, 



vAV+B^A. 



.\ -Sf/3A 2 -(^ 2 -?i 2 )} 2 =:a 2 -p 2 r//±^^A] 2 . 



(1) The amplitude A is not symmetrical with respect to 

 p 2 — n 2 . So there will be no "peak" or maximum resonance. 

 The last term on the right involving the squares and product 

 of the small quantities k and k' will be of small importance 

 in the change of the value of A. So the relation between A 

 and p 2 — n 2 will be nearly parabolic. 



f 8 ~l 2 



(2) Since the right-hand side ot 2 —p 2 \k'± : — pkA is 



equal to a square and therefore cannot be negative, the 



