NO. 5 .STABILITY OF AEROPLANES MUNSAKER AND OTHERS 3I 



§12. DAMPING 



The derivative Mq is the rate of change of pitching moment due to 



angular velocity, or rapidity of pitching q. For a pitch of velocity 



do 



-T. —q, there is a moment of qMq tending to resist such pitching. 



This is the damping due to the horizontal stabilizer, elevator flaps, 

 body, and all parts forward and aft of the center of gravity. The 

 pitching takes place about the center of gravity. The damping is 

 increased by a large tail and a long body. 



The damping of a surface should depend on the area of the sur- 

 face, the moment arm of that surface, the linear velocity with which 

 it swings through the air (which varies also as the moment arm), and 

 with the velocity of advance. Thus : qMq'-^ql^U, where / is a linear 

 dimension. 



If we can measure Mq for the model at any wind speed, we may 

 convert it to Mq for the full-scale aeroplane at its proper speed by 

 multiplying by the fourth power of the scale and the ratio of aero- 

 -plane speed to wind speed. Naturally this is an approximate method, 

 but it is the besf available since full-scale tests for Mq are not 

 practicalile. 



Similarly A',- and Lp may be obtained from model tests. These 

 refer to the damping of a yaw and a roll respectively. 



In order to measure Mq, Nr, and Lp a special oscillator was de- 

 signed, shown in the photograph in figure 12. By setting the appa- 

 ratus to oscillate in pitch, roll, or yaw the corresponding damping 

 coefficients can l)e computed from the observed decrement. The pho- 

 tograph (pi. i) shows the apparatus with model as used for pitching 

 oscillations. 

 Let: 



/ = moment of inertia of all oscillating parts in slug foot units, 

 m' = mass of all oscillating parts in slugs. 

 Mo = moment of air forces on model at rest, 

 Ms = moment of springs at rest, 

 i^^ = additional moment of springs when deflected, 

 c = center of gravity of entire apparatus above pivot, feet, 

 ^ = angle of pitch from normal attitude in radians, 



fjif^-. = damping" moment due to friction, 



/i^^ = damping moment due to wind on apparatus, 



do 

 fj-m^ = damping moment due to wind on model, 



cw'^ = static moment due to gravity. 



