154 V. OSCILLATIONS AND CYCLIC MOTIONS. 



we see that energy is being alternately introduced into and withdrawn 

 from the system by the extraneous force. On the average however, 

 as we find by integrating the trigonometric terms with respect to 



the time, T 



/* 

 smptcos)tdt = 0, 







P 



/ 



the time average of the activity depends upon the last term containing 

 sin a, and this is always positive, consequently the extraneous force is 

 on the whole continually doing work on the system, which is being 



dissipated at the rate xl-^} This work is a maximum when a = -9 

 \dt/ 2 



when the system is in complete resonance. Thus the mechanical 

 effects producible by resonance are shown to be commensurate with 

 the causes acting, and the impossibility of the common story of the 

 fiddler fiddling down a bridge is demonstrated. 



The exactness of "tuning", or approach to exact coincidence of 

 period necessary for resonance is shown in Fig. 33, which is the 

 graph of the curve 



where y = -=?- is the ratio of the actual amplitude of equation 46) to 



T? 



the steady statical displacement p produced by a constant force E 

 (that is when p = 0), x = j- is the ratio of the frequencies of forced 



and free vibration, and 2 = ^- 1 ) The curves are drawn for values 



of the parameter a 2 equal to -01, -05, -10, -15, -20. Thus the magnitude 

 of the resonance for any particular case can be seen by a glance at 

 the figure. The resonance is sharper the smaller a. The maximum 



amplitude is not for perfect tuning, but for x = 1/1 - The value 

 of the maximum is nearly equal to 



If there is no friction , for p = h the vibration becomes infinite, 

 which means simply that in this case friction must be taken into 

 account. If there is no friction we have by 44), 



sinc = 0, cos a = 1 



1) This parameter a. is not the angle cc above. 



