20 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 62 



Let the aerodynamical forces along the axes x, y, c be denoted by 

 X, Y , Z and expressed in pounds force per unit mass/ The moments 

 about these axes are L, M, N in pounds-feet per unit mass. Angular 

 velocities about the axis are p, q. r in radians per second. Let angles 

 of pitch, roll, and yaw away from the normal attitude be 6, </>, i/' in 

 radians. Signs are positive in the directions xy, ys, and sx. 



The radii of gyration about the axes x, y, z are K^, Kb, Kc in feet. 

 The mass of the aeroplane is //; in slugs. The products of inertia are 

 D, E, F. Two are zero for reasons of symmetry, and one is small in 

 ordinary aeroplanes. 



In normal flight in still air, the apparent wind blows in the posi- 

 tive direction of the axis of x. Let this velocity be produced by 

 the forward velocity U of the aeroplane in normal flight, f/ is a 

 negative number of feet per second. 



Let small changes in velocity components along the axes x, y, s 

 be u, V, w when any departure is made from the normal flying attitude. 



In normal flight it is assumed that the power available maintains 

 the aeroplane at such a speed that the weight is sustained and also 

 that the normal attitude is that proper for the speed. 



§9. EQUILIBRIUM CONDITIONS AND DYNAMICAL 

 EQUATIONS OF MOTION 



Let the inclination ' of the flight path to the horizontal be 0^^. Since 

 normal flight takes place in a straight line, if/^,z=(fi^ — o. There is no 

 oscillation and pQ = qQ = rQ = o, and Lq — No = o. 



If the propeller thrust Tq be exerted in a line above or below the 

 center of gravity h feet, then 



M,= -ToK 

 To=-gsm6o-Xo, 

 Z^ = g cos e^. 

 In this aeroplane li = o, and hence Mq = o. 



If any accidental cause slightly disturbs the normal attitude of the 

 aeroplane, the relative wind is no longer symmetrical and the aero- 

 dynamical forces and moments are .Y^ Y, Z, L, M, N. 



In general, the aerodynamical forces and moments caused by the 

 deviation from " normal attitude " depend upon the relative motion 

 of the aeroplane through the air, which motion is defined by U, it, 

 V, XV, p, q, r. Thus X = f{U, u, v, w, p, q, r) where the form of the 

 function / is not known : and five similar expressions for Y, Z, 

 L, M, N. 



' Unit mass is the slug of 32.2 pounds weight. 



^ Consider ^0 positive for an upwardly inclined path as when climbing. 



