c. 



+ X,.M, + Kl( XuZ„ - X,oZu) 



NO. 5 STABILITY OF AEROPLANES HUNSAKER AND OTHERS 4I 



second term is but ^ of its high-speed vakie. The third term is 

 unimportant. 

 From 



Mu„ Ma 



we see by inspection that the principal reduction in C\ at low speed 

 is due to smaller values U, Mw, Z^, and Mo, which greatly reduce the 

 terms ZwMci and UMic- These two terms are the principal numerical 

 ones in the expression for C\. 



In general, £1= —gZuMiv will increase in value due to increase in 

 Zu and Mio, but the efifect on the motion is not great. On the other 

 hand, B-^= ~Mq — Kl(Xu + Zu->) will drop rapidly for large angles 

 of incidence due to drop in Mq and in Z,„. This is favorable to 

 stability. 



It is seen that the quantities U, Zuu and Mq preponderate in the 

 numerical values of the coefficients D^C-^ and E^B-^. For ordinary 

 speeds, or s])eeds above the speed of minimum power, we have, 

 approximately, 

 D,^-Xn{ZuMq-UM^) +ZuX^Mq= -Xu(ZrMa~UM^), 



C, = {Z„Mq- UM,,) +XuMq + Kl(XuZ,,-X,,Zu)= {Z^Mq-UM,,). 



B,^-Mq-Kl(Xu + Z,o) = -M,~K\Z,o, 



Z:,= —gZuMw 



The condition for damped motion then becomes : 



D,C\>E,B, or {Z,,Mq-UM,y> - ^^^^ M,,(Mq + K%Zrc), 



7 Z 



where "- = -^ and J/„; are nearly constant. Damping of the long 



oscillation is then favored by large values of Z^^, Mq, and U . That 

 is, by light wing loading, large damping surfaces, and high velocity. 

 As speed is reduced these quantities become smaller and the oscilla- 

 tion is less strongly damped. 



For very low speeds, including those below the speed for minimum 

 power, the value of Zn- nearly vanishes and 'Mq becomes small. Here 

 the approximate expressions would be written, 



D ^ — Xu UM^o + ZuX^uMq, 



C,= -UMu^, 



B,= -Mq-K%{Xu + Z,o), 



Ei=—gZuMw, 

 and 



— l^'^ M,,V + M^X,^>jj {Mq + Kl{Xu + Z^)). 



