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~
1.5586 x~0 4 S5+0 00~5 S4+0.~22S3+0 00~6 S2+6 6266 x~0 s s
(6.260)
586 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We note that
Arls) s A~(s)
A8.(s~ = A8 (s~
Ap(s) s A~(s)
A8a~s~ = A8 (s~
(6.261)
(6.262)
The transfer functions for the general aviation airplane for the rudder input are as
follows:
_Atj(s)
A8r~S~ =
1.1093 x 10-5 S3 + 8.2661 x 10-4 S2 + 0.0064 s - 2.3475 x 10-4
15586 x~O~ss+0~5 S4+0 0022S3+00. 0 6S2+6 6266 x~0 s s
(6.263)
Ap(s)
A8 (s~ =
-7.2441 x 10-4 S3 - 0.~61S2 + 0.0015 s - 0.0018
1.5586 x~0 4 S5 +0.~~5 S4 +0.0022 S3 +0 00 6S2+6 6266 x~0 S S
(6.264)
A~(s)
~8~s~ =
4.0151 x 10-4 S3 - 0.~~4 S2 - 0.0102 s
1.5586 x~0- S5+0 0 5S4+0.0022S3+0 076S2+6 6266 x~0 5S
(6.265)
Approximate /ateral-directiona/ transfer functions.
Rollsubsidence. Taking the Laplace transform ofEq. (6.206) for aileron input
with rudder held fixed (A8r = O) and rearranging the terms, we get
so that
A4(s) Cl8a
Aba~s~ = s(lx S~pb )
(6.266)
p(s) s A~(s)
A8r(s~ = A8a~s) (6.267)
Transfer functions for Dutch-roll approximation. Taking the Laplace trans-
form of Eqs. (6.212) and (6.213) and using Cramer's rule, we obtain the following
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 587
transfer functions for aileron defiections:
Ap(s)= Np
Aba~s~ = Alar.dr
A~(s)= Nv,
A8a (s) Alat.ttr
where
(6.268)
(6.269)
Np = Izl C,.8r S2 + (bi Cn8r C}.r - Cy8r Cnrbi - ITI.I Cn8r)S (6.270)
Alar.dr - IZI(mi - biC,.B)S3 - [C:.p/zi + Cn bi(mi - bi C,.B)
+ C"Bbi (mi - b] Cyr)]S2 + [Cyp Cnrbi + Cnp(mi - bi C:-r )JS
(6.271)
Nv, = [C,ibr(ITII - bi C:.p) + Cy8r CnBbi]s + (CYar Cnfl - Cr18r C)j3) (6.272)
Spira/ approximation. Taking the Laplace transform of Eqs. (6.224) and
(6.225) and using Cramer's rule, we obtain the following transfer functions for
aileron defiections:
Ap(s) = ApS2 + Bps
(6.273)
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