570 Prof. A. L. .Narayan on Coupled Vibrations 



Substituting these values in (1) and (2) above, we have 

 Aj{ (m{K-i + m 2 a 2 )n 2 — (m-Ji 1 + m 2 a)g} + Bim 2 a/i 2 n 2 = 0, 



and • ' ■ ( 3 > 



A l aJ h n 2 +B l {(K 2 2 + h 2 2 )n 2 -gh 2 } = 0. . . (4) 



A 

 Eliminating the ratio ^, we obtain the determinants! 



equation : l 



= 0. 



[m^Yii + m 2 a 2 )n 2 — (mjix + m 2 a)g ; m 2 ali 2 n 2 ; 

 aA 2 n 2 ; (K 2 2 + /i 3 2 ) ?i 2 — a/i 2 



Put 



(miK^ + m^CK^ + V) =&, 

 A 2 (??i 1 K 1 2 + ??i 2 a 2 ) + (K 2 2 + /^ 2 2 ) (wjAx + m 2 a) — b ; 



we can write down the above determinant as 



™ 4 £(l- 7 2 ) -n 2 #& + % 2 (m 1 A 1 + ??? 2 a) = 0. . . (5) 



This is a quadratic equation in n 2 , thus giving two admissible 

 values of n 1 . 



The roots are given by 



2 _ i>g + \/b 2 g 2 — 4:h 2 g' 2 (^m 1 /i l + m 2 a)(l—<y 2 )k 

 U ~ J 2^(l- 7 2 ) ~' 



Calling the two roots p 2 and ^ 2 , we have 



}*. . (6) 



i 9 _ J ^~*~ V b 2 ~ ^k (»»A + m 2 a) ( 1 - 7 2 ) 

 2 1 6 — \/6 2 — 47i 2 A( wi/ii + wi 2 a) (1 — 7^" 

 Therefore the complete solution is 



= Axcos (jo£ + a) + A 2 cos(y + /3), . . .' (7) 



<£ = B 1 cos(j?£ + a)+B 2 cos(gtf + /3), . . . (8) 



where p and ^ depend upon the constitution of the system, 

 and Ai, A 2 , B x , B 2 , and a, and /3 are arbitrary and enable 

 us to satisfy any prescribed initial conditions. 



Thus the two pendulums act and react on each other, 

 creating in each a motion which is made up of two super- 

 posed S.H. vibrations of different periods. The difference 

 between the two frequencies depends upon the coefficient of 

 coupling (7) of the circuits. As in the case of the electrical 

 problem, if 7 is a small fraction the circuits may be said to 

 be loosely coupled, and when 7 is large, they may be said to 

 be closely coupled. When the coupling is loose and p and q 

 are nearly equal, we have a vibration of nearly constant 

 period but whose amplitude fluctuates between A! + A 2 and 

 Bi + B 2 , thereby presenting the well-known phenomenon of 

 beats. 



