218 SYMPOSIUM ON AERONAUTICS. 



materially different from that of the system (supposed oscillatory) 

 than when the system and the force are nearly synchronous. 



12. The ordinary theory of simple resonance depends on the 

 equation 



(Z)2 + )^Z) + n)x = J sin pt. 



The particular solution 



is the imaginary part of the expression 



n — p' -\- kpi ' 



The amplitude of /j is the same as the modulus of the complex 

 value X. The modulus of e'P* is i ; that of x is 



T / 



amp. ix [(^^_^2)2_^^2^2]l/2- 



13. To make the denominator a minimum we have merely to 

 minimize 



(n — q)--]-k-q, q = p^-yo. 



We find q = n — ^k-, necessitating n^^ik-. If, then, « > fe", the 

 maximum amplitude of Ix is 



max. amp. ix = 



where the positive or negative sign must be taken according as k is 

 positive or negative. If n < ^k", the maximum amplitude for Ix 

 occurs when p = o and is J/n. 



The amplitude is large when k or {n — ■\k'^y^ is small; it is very 

 large when both conditions are satisfied. The largest possible value 

 occurs when n = ^k'^ and is V2//^". In this case the applied force 

 has an indefinitely small frequency where the natural oscillation has 

 the frequency k/\j2. The theory of the system here considered is 

 given by Webster (op. cit., p. 155). 



14. The case which corresponds to that in which we are in- 

 terested is where the system starts from rest at the position of 



