THE DISTURBED MOTION OF AN 



AEROPLANE 



By W. BEVERLEY, M.Sc. 



In the following pages I have attempted a mathematical account 

 of the forces at work in restoring equilibrium to an aeroplane 

 possessing dynamical stability and disturbed from steady motion 

 by periodic gusts of wind. Damping effects have also been 

 found. 



We take the centre of mass as origin and three mutually 

 perpendicular directions through it as axes fixed relative to the 

 aeroplane. 



Let W = mass of machine in lbs. (also weight in lbs.-wt.). 



A, B, C, D, E, F = moments and products of inertia. 



u, v, w ; and p, q, r = components of translational and angular 

 velocities respectively. 



We have F = — lbs.-wt. as a standard equation, where 



m = mass in lbs. and a = acceleration in ft./sec. 2 X, /x, v are the 

 components of angular momentum. 



X = Ap - Fq - Er = Ap - Fq, 

 fx = Bq - Dr - Fp = Bq - Fp 

 v = Cr - Ep - Dq = Cr, 



,' I if D = O 



In steady horizontal flight the axis of x is that of flight, the 

 axis of y being vertically downwards, and the axis of z to the 

 left for a right-handed system. For all cases of flight we take 

 the direction of flight as the " x " axis and the others fixed 

 relatively to it as above. In most aeroplanes z = o is a plane 

 of symmetry and D = O = E. 1 



When the aeroplane is tilted downwards through an angle 



1 N.B. — We assume that the aeroplane has two propellers rotating in opposite 

 directions, so that gyrostatic effects annul each other. 

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