THEORY OF VIBRATIONS 21 



When dealing with small motions in the neighbourhood of 

 a configuration of equilibrium we may neglect terms of the 

 second order as before. Hence, substituting the value of V 

 from 7 (5), we find 



aq+cq = Q ......................... (3) 



When Q is of simple-harmonic type, varying (say) as cos pt, 

 the forced oscillation is given by 



which is of course merely a generalized form of the last term in 



8 W- 



Two special cases may be noticed. When p is very small, 

 (4) reduces to q = Q/c. This may be described as the "equili- 

 brium" value* of the displacement, viz. it is the statical 

 displacement which would be maintained by a constant force 

 equal to the instantaneous value of Q. In other words, it is 

 the displacement which would be produced if the system were 

 devoid of inertia (a = 0). Denoting this equilibrium value by 

 q, we may write (4) in the form 



where, as in 7, n denotes the speed of a free vibration. 



When, on the other hand, p is very great compared with n, 

 (4) reduces to 



q = -Q/p*a, ..................... (6) 



approximately. This is almost the same as if the system were 

 devoid of potential energy, the inertia alone having any sensible 

 influence. 



When two or more disturbing forces of simple-harmonic 

 type act on a system, the forced vibrations due to them may be 

 superposed by mere addition. Thus a disturbing force 



Q =/ cos ( Pl t + d) +/ 2 cos (pj, + oj + ...... (7) 



will produce the forced oscillation 



a 2 )H-.... (8) 



* The name is taken from the theory of the tides, where the equilibrium 

 tide-height is denned as that which would be maintained by the disturbing 

 forces if these were to remain permanently at their instantaneous values. 



