THE DISTURBED MOTION OF AN AEROPLANE 211 



It may be shown that X, Y, N are independent of p, q, w ; 

 and Z, L, M of u, v, and r. 



In a small change the angle of the tilt becomes = O + £ . 

 (S small) and cos# = cos# — £ sin# , sin0 = sin0 o + £ cos0 o . <f> is 

 small and is equal to sin</>. 



Then -77 = 3-. Substituting from the equations of 



equilibrium we have : 



Wdu 



~g d7 = Wsmd + H - X (since X, = o) 1. 



= Wcos0 o £ + 8H - uX u - vX v - rX r 

 Y\dT + r U ) = " Wsin^ £ - uY u - vY v - rY r 

 ~g{ dl - qU/ = - W<£cos0 o - wZ w - pZ p - qZ q (Z = o) 3 



A |rt ~ F Ju = - wL w - p l p - q L q ( L o = °) 4 



B £dt ~ F &t = " WM « - P M P - <l M q < M o = °) 5 



C dr 



-y t =-8Hh-uN u - vN v -rN r 6, 



Equations 1, 2, and 6 form a symmetrical group involving 

 (XYN) uvr , and 3, 4, 5 form an asymmetrical group of oscillations 

 — representing translations and rotations to the left or right of 

 the plane of symmetry, z = o — involving (ZLM) P 



/pqw 



Symmetrical Oscillations 



In disturbed steady horizontal flight we have 



d£ d<9 



^ = 0jfl = ^ 0+ e ) _ = — ==r (=x£) 



assuming u, v, r £ proportional to e Kt (\ to be found). For sim- 

 plicity we may take 8H = o — i.e. the thrust H is independent of 

 changes in the velocity. Substitute in the equations of motion 

 above. 



•'• (\V^ + X u )u + X v v + (X r - y cos0 o )r = 8H = O 

 Y u u + (w| + Y v )v + (^sin0 o + Ym. + y r )r = O 

 f u u+N v v + (c| + N r )r = 



N, 



