SYSTEMS WITH NON LINEAR REACTANCE 



433 



siitisficd. Thus the ratio of reactance to resistance must he the same 

 for the plate at co,„ and for the electric circuit at w,i- This condition is 

 satisfied if each is resonant at its ])articular fre(iuency, but resonance is 

 not a necessary condition. All that is necessary is that there be a pair 

 of frequencies, whose sum is equal to that of the electric generator, 

 for which the impedances have the same phase ant^le. If there are an 

 electric and a mechanical resonance such that the sum of the resonant 

 freciuencies is nearly equal to the generator frequency, and there is a 

 marked difference in the sharpness of the two resonances, then the 

 oscillations will fall closer to the sharper resonance. This is due to the 

 fact that the phase angle of the impedance changes more rapidly with 

 frequency in the neighborhood of a sharp resonance. 



From (22) we see that the amplitude of the current at the generator 

 frequency depends only on this frequency, the constants of the system, 

 and the new frequencies. It is independent of the amplitude of the 



5 6 7 8 9 10 II 12 13 

 IMPRESSED POTENTIAL (RMS) IN VOLTS 



Fig. 1 — Alternating displacement of plate as a function of generator voltage. 



generator voltage, of the amplitudes of the new frequencies, and of the 

 impedance of the electric circuit at the generator frequency. This 

 equation, while it tells us what happens when the oscillations are 

 present, tells us nothing about the conditions for their existence. 

 These are to be found by noting under what conditions the expression 

 (23) for the amplitude at the new frequency, co^, is real. We have two 

 cases to consider which are determined by the sign of cos {<pm + <P(i)- 



Assume first that this is positive, as would be the case if the plate is 

 resonant at co^ and there is any dissipation at Ug, such as would be 

 caused by resistance in the electric circuit. The first term in (23) is 

 negative and V^ can be real only if the second term exceeds it in abso- 

 lute value. This condition reduces to 



Eg > Zgig = 



^ [Z,„a;„,Z.co,]i/2. 



(29) 



This shows that there is a threshold value of the generator voltage, 



