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Similarly,
=  p Ix - q Iyx - r Ixz + qr (,tz - Iy) - pq Izx
     + (r2 - q2)lyz + prI).x                                           (4.354)
M = q Iy -r Iyz - plyx + rp(lx - Iz) - qr Ixy
                                      + (P2 _ r2)lzx + pqlzy                                          (4.355)
                                N  = r Iz - p Izx - qIzy + pq(ly - Ix) - rplyz
                                       + (q2 _ p2)lxy +qrlxz                                          (4.356)
     For an aircraft with vertical plane of symmetry (OxbzZ> plane), Ixy = Iyz - 0.
Then Eqs. (4.354-4.356) assume the following form:
                                     L = plx - I,:z(Pq + r) +qr(lz - Iy)                      (4.357)
                                       M = qly +rp(lx - Iz) + (p2 _ r2  Ixz                       (4.358)
                                        N  = rlz - Ixz(P - qr) + pq(ly ~ Ix)                       (4.359)
The terms on the right-hand side having the form qr(lz - Iy) are called inertia
 coupling terms. We will study the problem ofinertial coupling in Chapter 7.
4.3.4   Equations of Motion with SmaIIDisturbances
   Equations (4.291-4.293) and (4.357-4.359) govern the motion of an aircraft
 with respect to an inertial frame of reference fixed to the surface of the Earth. In
deriving these equations, we have ignored the rotation of the Earth about its own
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 373
axis as well as its orbital motion around the sun. We have also assumed that the
aircraft has a vertical plane of symmetry. Furthermore, we have assumed that the
origin of the body axes system is located at the center of gravity of the aircraft.
Even after introducing these simplifying assumptions, it is still a formidable task
to obtain analytical solutions to these equations because they are a set of coupled,
nonlinear, differential equations.
    Another challenging taskis to evaluate the aerodynamic forces and moments as
functions ofmotion variables U,  V,  W (or a,  p),  p:uq"' r, and their time derivatives.
However, for the study ofstability, response, and automatic control ofconventional
airplanes, these equations can be farther simplified if we assume that the disturbed
motion is one of sufficiently small amplitudes in the disturbed variables. With
these assumptions, Eqs. (4.291-4.293) and (4.357-4.359) can be simplified and
separated into two sets of equations, one set of three equations for longitudinal
motion and another set of three equations for lateral-direlctional motion.
    For the disturbed mOtion, we assume
U -. Uo + AU      V : 'Vo + AV      W - Wo + AW
　
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