THEORY OF VIBRATIONS 29 



The equation (9) is still satisfied if we add to (15) terms 

 representing a free oscillation; and these added terms are 

 necessary in order to constitute a complete solution capable of 

 adjustment to arbitrary initial conditions. The free vibration 

 dies out, however, asymptotically, so that after the lapse of a 

 sufficient time the forced vibration (15) is alone sensible. 



The circumstances which affect the amplitude and phase of 

 this forced vibration require careful attention. The amplitude 

 is a maximum when R? is least, i.e. when 



(-!-. a) 



and the maximum amplitude is accordingly 



In most cases of interest 6 2 /oc is a small quantity of the 

 second order; the maximum is then C/nb, and occurs when 

 p = n, very approximately. 



Again, it appears from (15) and (16) that the phase of q lags 

 behind that of the disturbing force by an angle a, which lies 

 between and JTT, or between TT and TT, according as p 2 is less 

 or greater than c/a, i.e. according as the imposed frequency 

 is less or greater than the natural frequency. If, keeping p 

 constant, we diminish the dissipation-coefficient b, a. tends to 

 the limit or TT, respectively, in accordance with 8, where we 

 found exact agreement or opposition of phase in the absence 

 of resistance. But if, keeping b constant, we make p approach 

 the value n ( = \/(c/a)) which determines the frequency in the 

 absence of dissipation, a tends to the limit ^TT, and the phases 

 of q and Q differ by an amount corresponding to a quarter-period. 

 This means that the maxima of the disturbing force are now 

 synchronous with the maxima of the velocity q. 



Some light is thrown on these relations if we examine 

 the case of a pendulum whose bob receives equal positive 

 and negative instantaneous impulses alternately at regular 

 intervals. It is seen at once from Fig. 13 that an impulse in 

 the direction of motion accelerates or retards the phase of an 

 otherwise free vibration, according as it precedes or follows 



