248 Prof. Barton and Miss Browning on Coupled 



If the circuits when quite separate give free vibrations 

 proportional to sin mt and sin nt, we have 



m2 =m ■ ■ (*) 



and i 



n>= m ....... (5) 



On writing in (1) 



9=**, •• (6) 



we find ,j I \ ■ ' 



* = (m + mjt^K (7) 



Then, substituting (6) and (7) in (2) gives us the auxiliary 

 equation in #, viz., 



* 4 (l-7 2 ) + t f 2 (m 2 + w 2 )+mV = 0. ... (8) 



Let this be rewritten in the form 



x 4 + x 2 (p 2 + q 2 ) + P 2 g 2 = 0, .... (9) 



then ^ + 3 . = !!1±!L ( io) 



pv=^$> ai) 



and a= ±pi or j-^z, (12) 



where i= */( — 1). 



Thus the general solution of (1) and (2) may be written 



# = Esin(jrt + €)+Fsin(y/ + 0), . . . (13) 

 and 



s= A(i^^) sin( ^ +6 )-i(?-i) sin ^ + ^ 



• • ■ (14) 

 where E, e, F, <f> are arbitrary constants to be fixed by the 

 initial conditions and p and q are functions of m and n (for 

 the separate systems) and of 7 (their coefficient of coupling). 

 Let us now examine these functions. Dividing (10) by 

 (11) we obtain 



l + L = l+L. 



p 2 q 2 m 2 n 2 



On changing to the periods //, and v for the vibrations of 

 the separate systems and <r and t for those of the coupled 

 vibrations, this becomes 



- 2 + T 2 = /. 2 + Z, 2 (15) 



