WILSON— AEROPLANE ENCOUNTERING GUSTS. 219 



equilibrium. The motion is then defined by the equation 



J Vw - w~ 



X = r-. Ti^ (e"*'"' cos V« — IkH — cos V« — ^kH) 



k{n- Ik"-) 



4 



J { 4n- |F _,, . , ^ . , — \ 



+ ~7 TT^ I , . g ' sm V/z — \k-t — sm ^n — hk'H . 



2{n - ik-) \ Vm - ip ^ J 



Under normal conditions this quantity remains tolerably small 

 until the natural motion is nearly damped out or until that motion 

 has time to increase greatly (/e>o). Even if Ji^^k- -\- €~k'-, the 

 equation becomes 



2je 1 2j 



X = — (e~^^'* cos ^kt — cos ekt) + — (ee"*^' sin hkt — sin ekt), 



and the conclusion still holds. 



For the simple system ordinarily treated for resonance the state- 

 ment that the motion must be only slightly damped and the frequen- 

 cies of the natural and forced vibrations must be reasonably near 

 together, is therefore amply justified. The result holds even when 

 11 < ^k'-, in which case the maximum amplitude for Ij; (resonance) 

 occurs when p = o and is J/n. 



15. The next simplest case is like that which arises in treating 

 the constrained longitudinal motion (^ = 0) of the aeroplane (p. 69) : 



{D -\- a) u -\- bii' = — aii^ — bz^\, a = .i28, b^ — .162, 



cu^ (D-{- d)zv = — cu, — dii\, c=.S57, d = z-9S- 



The natural motion is given by 



A' = !)■- + {a^d)D-{- {ad — bc)=o, 



and in this case by /}- -|- 4.078Z) -}- .598 = 0. Here the roots are 

 both real, viz., — 3.93 and — 0.15. So far as the equation in D is 

 concerned we have the case where k is large and n is small. The 

 equations for the forced motion are 



A'u = — (aD -{- n)u^ — bDiv^, 



A't^.' = — ( dD -f- « ) z<.\ — cu-^. 



The question now arises : What is it that is to be a maximum ? 



