242 SYMPOSIUM ON AERONAUTICS. 



The natural motion of the constrained machine is found from the 

 determinant 



A' = 833 = 36.7^' + 323-i^' + 77-88L> + 27.15 = 0. 



This is a cubic equation which has no positive root. 



The negative root is — 8.54. The quadratic factor remaining 

 after division by D + 8.54 is 



36.7/^^ + 8.746/5 + 3-18 = 0, 



of which the roots are 



D= — 0.1 19 ± 0.2692. 



The real part is negative and hence the motion is dynamically stable. 

 The introduction of the automatic device has removed the in- 

 stability in the lateral motion. As compared with the complex roots 

 in the free motion, these roots indicate a much slower period and a 

 considerably smaller damping. 



44. On the other hand suppose that the constraint had been such 

 as to keep the machine level, i. e., ^ = identically. The equations 

 would have been 



{D—Y,)v^ {U—Yr)r= Y,v, + Y„p, + F,r„ 



— LvV — Lrr=^LvV^ -\- Lpp^^LyV^ -\-F, (16) 



— N,.v ^{k^rD — Nr ) r = N,z.', + N,p, + Nrr,. 

 The natural motion would have been determined by 



A" = 802 = 70.6/)- +44.5!) + 109.9 = 0. 



The roots are 



/:> = — 0.31 5 ±0.237/. 



The machine is again stable. 



45. It follows that at high speed this Curtiss Tractor, which is 

 laterally unstable when free, becomes quite stable when constrained 

 either to remain on its course or to fly on even keel. 



If stabilizers against rolling and turning were provided, the 

 motion would reduce to 



(D — Y,)v=^ Y,v, + Y„p, + Yrr„ (17) 



and would be stable, D= Yv = — 0.248. 



