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386 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
We have
(dd ), = (ddV, )o - -I.I X Vl
-0-
zi li ki
001
50 25 0
= 2511 - 50/1m/s2
Example 4.9
An aircraft is initially in a dive and, at the bottom of the dive, the pilot effects a
steady pull-out with a constant pitch rate qo rad/s. Obtain the equations of motion
for small disturbances during the pull-out.
Solution. The equations of aircraft motion are
Fx -m(U +qW -r V)
Fy=m(V+rU -pW)
Fz =m(W +pV -qU)
L = plx - Lcz(Pq +r)+qr(lz - Iy)
M = ql,,+rp(L -Iz)+(p2 _ r2)l;cz
N -. r Iz - Ixz(P - qr)+ pq(ly - II)
We have
U-Uo+AU V -AV W -AW
p - Ap
Fx = AFx
L - AL
q = qo + Aq
Fy = AFy
M - AM
r : Ar
Fz -. AFz
N -. AN
Assuming that all the disturbance variables are small (note that qo is the steady-
state pitch rate and is not a disturbance variable) so that we can ignore higher order
terms invoMng products of smaIJ disturbance parameters, we get the following
equations:
AFx - m(AU + qo AW)
AFy = m(AV + UoAr)
AFz = m(AW - qo Uo - qo AU - Aq Uo)
AL = Aplx - Ixz(Apqo + Ar) + qoAr (Iz - Iy)
AM = Aqly
AN - IzAr - Ixz(Ap - qoAr) + Apqo(ly - Ix)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 387
We observe that the longitudinal and lateral-directional equations of motion are
now coupled. Here, qo = UolR where R is the radius of curvature. (See Chapter
2 for more information on steady pull-out from a dive in vertical plane.)
Substituting and simplifying, we get the following equations of motion for small
disturbances during the pull-out:
' d
(m,dd _ Cx,,),- (CxdC,dd + Cxa. -miqo)A - (CxqCld + CxO)AO
)
- Cx8e A8e + Cxit A8t
(-Czu +miqo)u + [(m,dd _ CzdCI dd ) - Cza] \ - (m,dd + CzqC,dd )AO
= miqo + Czae A8e + Cz8,A8t
(m,dd -bi C)JB dd _ Cyp) z:\p - (bi Cyp dd + Cy4) A4
+ (m, dd _ b,Cyr dd)A~ = Cy8r A8r + Cy8a A8a
(-C,p - b,C,B dd ) Ap + (~b,C,p dd _ Ixz,qo dd +/x, dd ),\4
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