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           - Cyp(lzl - Irl)(,l;l - I;l)Czcx - I7221Cma(l;l - I;I)]
             + Cyp(Cza Cmq CICnrbi  - mi Cmtr Cnr bi) + lTl I Cnp Cza Cmq CI  - lTl21Cmct Cntt
             (7.42)
       Specialcase, Po = O.    Itisinteresting to note thatin Eqs. (7.38-7.42),wherever
the roll rate Po appears,it is having only the powers of 2 and 4. For very small
values of roll rates pg N pg N 0. Thus, assuming Po ~ O, Eq. (7.36) reduces to
    0
(Czcr - mis)
 Cma
    0
(Cyp - ITl IS)
     0
     0
 Cnp
     O
  mi
(Cmq CI  - Iy IS)
     0
 -mi
     O
     O
(C"rbi - cls)
-.0 (7.43)
Expanding tlus determinant, we get
                                    [-(Czor  - lTl ls)(Cmq CI  - I),is) + m.] Cmty]
                                       x [(Cyp - mis)(Cnr bi - Izis) + mi Cnpl = O              (7.44)
I:I = qjySb
I;1 = qjSb
634            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Expanding the first term, we get
S2-(C +CI )S+ C'rC'm,    )=0  (7.45)
        mllyl
Comparing this equation with Eq. (6.51), we recognize that the two equations are
identicalif we note that we haveignored the damping term Cnur. Thus, the ftrst term
in Eq. (7.44) represents the conventional short-period oscillation of the aircraft
Similarly, the second term reduces to
SZ_ miCnrbi+Cypl;i)s+( Cp+mi n  Cy nr ~pmi n   yCnrb)=0  (7.46)
    '  mil;i          mtlzi
This equation is identical with Eq. (6.219) for the Dutch-roll approximation if we
note that we have ignored the Cyr term.
     Thus, for Po = O, the characteristic Eq. (7.37) has a pair of complex roots, one
pair for the short-period oscillation and another pair for the Dutch- roll oscillation.
     Stabdity criterion for a stead17y ro//ing aircraft.    According to Routh's nec-
essary condition for stability, all the coefficients of the characteristic Eq. (7.37)
must be positive or must have the same sign. If any one of the coefficients is neg-
ative while at least one other coefficient is positive, the system is likely to become
unstable. Normally, the coefficients Ai, Bi, Ci, and Di  are likely to be positive.
The only coefficient that is most likely to become negative is Ei. However, the
 expression for Ei as given by Eq. (7.42)is stiU_ too complicated to derive a criterion
for the stability of a steadily rolling aircraft  Therefore, we need to introduce some
more simplifications. As done by Philips,l we assume that C",q  = Cnr  = Cyp  = 0.
With these assumptions, Eq. (7.42) reduces to
= lTI~ [(Izl - /ci)(~ - I;i)pg - p~(Cnp(lzl - IJrl)
- Cmct(l;,l  - I;  )) - Cmtr Cnp]
From Eq. (6.55), we have
coo :=
(7.47)
(7.48)
Here, coo denotes the natural frequency of the short-period mode. With Cmq - 0,
this equation reduces to
to9 -
-Cmct = tDglyl
Similarly, using Eq. (6.222) with Cnr  = Cyp = Cyr  - O, we get
Cnp = co:j,lzl
(7.49)
(7.50)
(7,51)
INERTIA COUPLING AND SPIN
635
Here, coy,  denotes the natural frequency of the Dutch-roll mode. Using Eqs.  (7.50)
and (7.51), Eq. (7.47) can be expressed as
=/yllzllTl:?j(  (I  I,,I  ) pg -  g-l [(
= /yllz,, ~[(/ I J) plj -  g]
I;l - cl
[(/ I,/)
pg - --~V]  (7.52)
pg-,p] (7.53)
     Thus, for Ei  < O (instability), one of the two terms in the square brackets must
be negative. Both terms should not become negative at the same time. Thus, we
have two conditions for instability. The first condition is given by
(I I,I ) pg >
　
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