THEORY OF V1BKATIONS 31 



is greater than the natural period ; the maxima and minima 

 of the force now follow those of the velocity. The reader is 

 recommended to follow out in detail the argument here sketched, 

 and to examine the effect of substituting a continuous simple- 

 harmonic force for the series of disconnected impulses. An 

 explanation may also be found, on the same principles, of the 

 fact that a small frictional force varying as the velocity has no 

 sensible effect on the free period. 



We return to the analytical discussion. A difference of 

 phase between the force and the displacement is essential in 

 order that the disturbing force may supply energy to compensate 

 that which is continually being lost by dissipation. When, as 

 in 9, there is complete agreement (or opposition) of phase 

 between q and Q, the force is, in astronomical phrase, "in 

 quadrature " with the velocity q, that is, the phases differ by \ir, 

 and the total work done in a complete period is zero. Under 

 the present circumstances the disturbing force is at any instant 

 doing work at the rate 



Qq = -^D~ s i n (pt ~ cos^tf 



= g {sin a -sin (2^- a) j, ............ (19) 



the mean value of which is 



(20) 



The same expression is of course obtained as the mean value 

 of bq 2 , since the energy supplied by the disturbing force must 

 exactly compensate, on the average, that which is continually 

 being lost by dissipation, the mean energy stored in the system 

 being constant. 



It follows from (16) and (20) that the dissipation is greatest 

 when OL^TT, or p = n, i.e. when the imposed frequency coincides 

 with that of the free vibration in the absence of resistance. 

 The maximum value is %C*/b, being greater, of course, the 

 smaller the value of b. 



