FORCED OSCILLATIONS 353 



FORCED OSCILLATIONS 



285. The oscillations which have so far been considered are of 

 the type known as free vibrations, that is to say, the forces act- 

 ing arise entirely from the potential energy of the system itself. 



A second type of oscillation occurs when the system is acted on 

 by forces from outside, in addition to those arising from its own 

 potential energy. These oscillations are known as forced oscillations. 



Let us suppose that the potential and kinetic energies of the 

 system are given by equations (195) and (196), and that the sys- 

 tem of external forces acting at any instant is such that the work 

 done in a small displacement is 



Then Lagrange's equations for this system are 



1W_ - _^ + ^ 

 dt \9%J 0^ a^ 



which becomes 2 & ^ = - 2 a^ + (199) 



U/t 



in which W lt it must be remembered, is now a function of the 

 time. This equation can be solved according to the rules given 

 in any treatise on differential equations. If, as before, we take 



^ = -p the general solution is found to be 



<'= t 

 ^ = A, cos (\t - 6,) + ~= f W<= ,sin k, (t - t') dt r , 



the lower limit of integration being either t 1 = oo, or the instant 

 of which the external forces first came into operation. 



286. A case of extreme importance occurs when ^ is simply 

 periodic with respect to the time, say 



