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An e--.pansion of the determinant results in a fourth-order algebraic equation of the
form  '
where
A8.iiuA5 + B8.larl4 + C8.ltuA3 + D8.larA2 + E8.Jar}. ~ 0               (6.190)
A8iac - (mi - biC),B) (Ixilzi - /2zi)                    (6.191)
B8iai = (-Cyp) (Ixi Izl - Ix2zl) _ (rTI,i - bi CyB)(lxl Cnrbi + Izl Ctpbi
         + Ixzl Ctrbi + Ixzl Cnpbi) - bi Cyp(CtBbl lzl + Cnpbi lxzl)
          + (mi  - biC).r)(Cirabllxzl + IxiCnBbi)                                          (6.192)
C81ru  = b:-(mi  ~ bi C}.B)(Ctp Cnr - Ctr Cnp) + bi Cyp(lxI Crrr + Izl Clp
           + Ixzl Ctr + Ixzl Cnp) - C),4b] (CiBIzl  + Cn B Ixzi) + bi Cyp
         x (CiBCr.rb:t - Ctplzl  - Ixzl Cnp - Clr Cn~b:) + (mi  - bi Cyr)
         x (Ctp Ixzl + Clp Cnpb~ - C,p Cn ib{- + I,i Cnp)                          (6.193)
      D8w = -Cypb~(Ctp Cnr - Ctr Cnr) + C).4 (CiBCnrb{- - Ctp Izl
               - Ixzl Cnp - CLr Cn~bf) + b~C),p(Ctp Cnr - Clr Cnp)
                  + bi (m i  - bi Cyr)(CtPC,,p - Clp C,,p)                                            (6.194)
                     E8iar = C).4bi(qpC,,r - C,,pCtr)                   (6.195)
Equation (6.190) is also called the lateral-directional characteristic equation and
gives five values of the root V One of them is the zero root. For conventional
airplanes, the other roots are two real roots and a pair of complex conjugate roots.
One real root is large and negative and corresponds to the heavily damped roll
subsidence mode. The other real root is small and may be positive or negative, and
the response associated with this root is called the spiral mode. If this real root is
positive, then the spiral mode is slowly divergent. The motion associated with the
pair of complex roots is called the Dutch-roll oscillation. The damping ratio and
frequency of the Dutch roll depend on the type of airplane.
      To illustrate the nature oflateral-directional response, let us consider the general
aviation airplane once again. In addition to the data given earlier, we have the
following data for the lateral-directional aerodynamic coefficients of this airplane:2
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL         569
    C).p = -0.564,  Cy8, = O,  Cy& = 0.157,  Cip = -0.074,  Cip = -0.410,  Clr =
0.107, CL8. = 0.1342, Ct& = 0.0118, Cnp = 0.0701, Cnp = -0.0575, Cnr - ~0.125,
Cn& = -0.00346, Cdr = -0.0717; CyB = O, qB = 0, qiB = 0, Cyp = 0, and Cyr  =
0. Note that all the derivatives age per radian. Furthermore, as before, we assume
M -.0.158 and p = 1.225 kg/m3.
    Substituting these values and other parameters for the general aviation airplane
in Eqs. (6.186) and (6.187) and assuming that the airplane is in equilibriun:level
flight (0o - O) before encountering a lateral-directional disturbance, we obtain
             -0.2557  0.1820    O    0 -1.(
   0 O 1.0000  '6.196)
            A -  -16.1572     0     -8.4481  0 . 2.2r
             O     O     0   0  1-0{
      .  4.5440   O  -0.3517 (

               0    0
　
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