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stability and control derivatives vary from one equilibrium condition to another.
   It is important to remember that the Taylor series expansion must include all
the motion variables on which the aerodynamic forces and moments are known
to depend. If this information is incomplete, estimated forces and moments in
the disturbed state will be incorrect. Any predictions based on such incomplete
aerodynamic data will also bein error. Therefore,itis the task ofthe aerodynamicist
to understand the physics ofthe problem,identify all the motion variables on wluch
the aerodynamic forces and moments show dependence, and correctly include all
of them in the Taylor series expansion.
   In the literature on airplane stability and control, it is customar)r to use the
short-hand notation to denote the stability and control derivatives. For example,
,   aq       acx       acm
.x"=a~ cxu=a  Cmcr=   (4.405)
    au       : ay      = a-
           acn
Cyp=88~ cl,,=aa2 Cp=ap  (4.406)
and so on. However, the partial derivatives with respect to variables such as
d, B, p, q, and r are deftned somewhat differently as follows:
C,d = aa(:c:.)
CyB = (, ~)
Cyp= a(:c..)
Cyr= a~c~
C q = aa(:c: )
CiB = (, A~,.,)
Clp = (, ::,)
Cmq = aa(c,:.)
Cnli = (, g)
(4.407)
(4,408)
    a cn                   (4.409)
a(~-)
       acn
C,r=~lC) C"r=a(l.) (4.410)

   .{
 : :'j
~
/;.&j
378               PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
This form of definition of stability derivatives invoMng time derivatives such as
av, p, and angular velocity components P, q, r is necessary to make these deriva-
tives nondimensional like all the other stability and control der:ivatives. Further-
more, the nondimensionalization helps us use the results obtained on scaled model
tests in wind tunnels for predicting the stability and response of the full-scale
airplane.ln view of this,itis important to note ffiat C,nq + acm/aq ~d so on.
    The stability and control derivarives with respect to variables such as ci, and p
are called acceleration derivatives, and those with respect to p, q, and r are called
rotary derivatives. Together, they are called dynamic stability-control derivatives.
   Using short-hand notation, we can rewrite the expressions for forces and mo-
ments in the nondimensional form as follows:
      ACx = CruU + Cxcr A2 + qr0AO + C., (~U ) + C^q (2qU )
                    + Cx8t A8e + Cxat A8r + - - .                                                                    (4.41 1)
ACy = Cyp Ap + Cy+A4 + C,B (~fjb) + Cyp (2pf;.) + Cyr (23 )
        + Cy&i A8a + Cy8r A8r + . - .                                                       (4.412)
       ACz = Cz,,u + Czcr Aa + Cza AO + Cz, Gu ) + Czq (2qU )
                     + Cz8e A8e + Cz8,A8r + * *.                                                                   (4.413)
  AC, = C,tr Ap + C,a c~ejb) + Ct*A4 + CrP (2pUb ) + C,, (2t; )
            + Cl8a A8a + CIBr A8r + . . .                                                           (4.414)
　
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