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steady value at high supersonic or hypersonic Mach numbers. Therefore, if the
fiight Mach number exceeds 0.5, the derivative CD" should not be ignored. The
interested reader may refer to Datcom7 for information on the methods to evaluate
the derivative acDlaM.
Estimation of CLu-    This derivative, in a similar fashion to CDr,, can be ex-
pressed as
ac =Maaac "M
au
(4.481)
At low subsonic speeds (M  <. 0.5), the lift-curve slope CLa essentially remains
corisu~t so mat acLalaM  = 0. However, as the Mach number approaches critical
Mach number, the lift-curve slope starts increasing. At some Mach number in
the transonic/low supersonic range, it assumes a maximum value and then starts
decreasing with further increase in the Mach number. Therefore, this derivative
should be considered when M > 0.5.
    The data given in Chapter 3 can be used to obtain CLa as a function of Mach
number.
Estimation of Cmu.   This derivative can be expressed as
ac =MaaacmaM
au
(4.482)
       Like thelift-curve slope, the pitching- moment-curve slope also varies with Mach
number. Assuming that the airplane center of gravity remains fixed, the variation
of C,na with Mach number is caused by the rearward movement of the center of
pressure in the transonic and supersonic region. As a result, the aircraft becomes
more stable. With further increase in Mach number, especially as it approaches
the hypersonic range, the fuselage develops more and more lift, and the center of
pressure starts moving forward again. Thus, the derivative Cm" assumes signifi-
cance for M  > 0.5. The data given in Chapter 3 can be used to obtain Cm t as a
 function of Mach number.
Longitudinal rotary derivatives.
    Estimation of Cui.    This derivative is a measure of the effect of a steady pitch
rate on the lift coefficient. Because of the pitching motion, the effective angle of
392               PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
ic
Tail
Fig.4.19    InCRasein tailangle of attack during pitching motion.
attack of the fuselage, wing, and tail surfaces will be different from the steady-
state value. For a positive pitch rate, the sections of the fuselage and wing that
are ahead of the center of gravity experience a reduction in the angle of attack,
and those sections aft of center of gravity experience an increase in the local
angle of attack. The increase in angle of attack is particularly significant for the
aft-located horizontal tail because ofits large distance from the center of gravity.
For airplanes with short fuselages and a high-aspect ratio wing (small chord and
large span) located close to the center of gravity, the contribution of the fuselage
 and wing to CLq  can be ignored. For such configurations, the contributiori to CLq
mainly comes from the horizontal tail and can bc estimated as follows.
        The increase in the horizontal tail angle of attack (see Fig. 4.19) due to a positive
pitch rate q is given by
Acyr = qUt.
(4.483)
where l, is the distance between the center of gravity and the aerodynamic center
 of the horizontal tail and  Uo is the flight velocity. Usually,lt is called the taillength.
The resulting increase in the tail-lift coefficient is given by
                                        AC~r = at (SS )-7,(?jl )
Wittl CLq = acL]a(qc/2Uo), we obtain
(4.484)
(4.485)
(4.486)
where Vi - Stlt/Sc.
       For configurations withlong fuselages and short aspect ratio wings (alarge chord
and a small span), the contribution of the wing-body combination can become
significant so that the total value of CLq iS given by
CLq = (CLq)WB + (CLq)t
(4.487)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 393
The contribution of the wing~body combination can be estimated using the fol-
lowing expression/
                        (CLq)WB = [KWcB) + KBcW)](SS  )(C,q)e
                                 +(C q)B(S  S~f )                                   (4.488)
　
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