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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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