44] 



EFFECT OF TUNING. 



155 



and the displacement is in the same or opposite phase with the 

 force , according as h is greater than or less than p. In the latter 

 case the excursion is a maximum in one direction when the force is 

 exerting a maximum pull in the opposite direction. This need not 

 appear paradoxical, for consider the limiting case of a system with 

 very little stiffness in proportion to its inertia, that is li very small 

 and the natural period very great. Then the excursion is always 

 opposite in phase to the force on account of the inertia of the 

 system. In the opposite case of a system with very little inertia in 

 proportion to the stiffness, h is very large, and the excursion is in 

 phase with the force. In this case (that of complete agreement) we 

 have what is- called the equilibrium theory of 

 oscillation, the displacement being the same as 



(~F 1 \ 

 S = ph 



except that the force and displacement are varying 

 together. Such a theory was given by Newton 

 for the tides, which consist of a forced vibration 

 of the water covering the earth under the periodic 

 force due to the moon's attraction. The more 

 accurate theory taking account of inertia was 

 given by Lagrange. The relation of the dyna- 

 mical to the equilibrium theory is shown in Fig. 33. 

 The two points of distinction between free 

 and forced oscillations then are, first, that the 

 free vibration has its period determined solely 

 by the nature of the system, while the forced 



vibration 

 takes the 

 period of the 

 P. force, and 



secondly, 

 that if there 

 is damping, 



the free vibration dies away, while the forced vibration persists 

 unchanged. 



The theory of the forced vibration which we have given does 

 not take account of the gradual production of the motion from a 

 state of rest, but refers only to the motion after the steady state 

 has been reached. We may now complete the treatment and take 

 account of the motion at the start. Our previous solution is merely 

 a particular solution. According to the theory of linear differential 

 equations in order to obtain the general solution we must add to the 

 particular solution just obtained the solution of the equation 41) 



Fig. 33. 



