WILSON— AEROPLANE ENCOUNTERING GUSTS. 



231 



The gust r^ corresponds to a column of air rotating about a vertical 

 line. 



This last is a common type of aerial disturbance, easily observed 

 on a warm day, often of very small diameter compared with the 

 spread of the wings of an aeroplane, and accompanied by a strong 

 rising current of air. Such a vertical vortex, if small, might strike 

 one wing of the machine alone, and, due to the rising current, heel 

 it over suddenly. It is, however, not this small local disturbance 

 which we can consider by our methods here, but the larger and more 

 gentle rotation in the air which might immerse the whole machine 

 many times over and which produces a yawing motion in the 

 machine rather than (primarily) a roll or bank. 



31. Place D = d/dt. Then the equations are 



(D-Y,)v-{-(g-Y,D)cl>-{-{U—Yr)r=Y,v,+Y,p,^Yrr„ 



—L,v-\-{k/D—Lp)D<i>—Lrr=L,z\-\-LpP,+Lrr„ (13) 



where k^'=A/m and k^^^=C/m. The determinant whose vanish- 

 ing determines the natural motion is 



D-Y, g-Y,D 

 A = -Lv kjcD' - LpD 

 - N. - Nr>D 



Let the cofactors of A be 



U-Yr 



-Lr 



kc'D - Nr 



5ii = 



5l2 = 



5l3 = 



Soi = 



522 = 



5o, = 



W- - LpD 

 -N,D 



-Lr 



-Lr 

 kr?D - Nr 



= 25922)3 + 23140D2 + 8478P, 



kc-D - X, - Nv 



- L„ k/D'- - LpD 



- N, - NpD 



- NpD kc-D - Nr 

 g-YpD U-Yr 

 D-Y, U-Yr \ 



- N, kc-D - Nr : 



- iV, - NpD 

 D-Y. 2-Yr,D 



= - 59-55D - 26.55, 

 = - 32.841^2 _ 280.7Z), 

 = - 2270Z) - 868.8, 

 = 70.6D- + 44.5Z) + 109.9, 

 = 28.76, 



