WILSON— AEROPLANE ENCOUNTERING GUSTS. 217 



As the natural motion is no longer periodic, there can hardly be 

 any such thing as resonance, in the usual acceptation of that term. 

 We can, however, ask what value oi p will make zu/J a maximum 

 and hence induce the maximum oscillation in the vertical motion. 

 To maximize 



P' I 



■ Q|- ■ 



(.598 - p'-y + 4-07S'P' 4-07^ +{P- ■59^:py 



take /)- = 0.598 or p^ 0.774. The value of iv/J is then 



and the amplitude of w is 0.136/. The amplitude of the oscillation 

 corresponding to the particular solution /„■ is 0.175/. 



Thus again it is seen that the steering device makes the motion 

 far easier than when the machine is free (p. 70). There seems to 

 be no need of carrying out the details of the integration. 



Note on Resonance. 



II. In defining, by implication, a state of resonance in the calcula- 

 tions above, I have assumed that it was the angle 9 which was to be 

 maximized by the proper choice of the frequency p of the applied 

 periodic force. It may be well to take up the theory of resonance in 

 a little greater detail, for there are complications in the kind of 

 system we have here to consider. 



A. G. Webster, in his "Dynamics of Particles, etc.," Teubner, 

 1904, p. 175, gives general formulas for resonance and shows that 

 if the damping coefficients are small and if the frequency of the im- 

 pressed force nearly coincides with that of the natural oscillation, 

 the amplitude of the forced vibration will be relatively large. 



This is not enough. For in the first place, the damping coeffi- 

 cients in the case of the aeroplane can hardly be regarded as small 

 (they sometimes exceed the frequencies) ; in the second place, we are 

 not even certain that the motion of the system is wholly oscillatory 

 (some of the roots may be real, and even positive if the machine has 

 a certain amount of dynamical instabihty) ; and in the third place, 

 under such conditions, the amplitude of the forced oscillation may be 

 considerably greater when the frequency of the applied force is 



