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- Cxa Czu I).l + Cxd Czu Cmq C~ - Cxcr ITIl CmuCI
- Cxq Cl(Czu Cmd Cl + CmurTi\ - Cm i Cza CI) (6.23)
Da = -m] Czq CI Cmu - C.u (Cza Cmq CI - mi Cma - Czq Cmdr C~)
+ Cxa Czu Cmq CI - Cxtrmi Cmu - Cxdr Cf Cmu Czq
- CxO(Cz,,Cm ,cl + Cm,mi - Cm"CzaCI)
. + Cxq CI(Czu Cm" - Czcr Cmu) (6.24)
E8 = -Cxa Cmu Czq CI - CxO(Cz"Cm r - Cz"C,"") ' (6.25)
AIRPLANE RESPONSE AND CLOSED-LOOP CONTROL 541
Steady jy
┏━━━━━━━━━━━━━━━━━━┓
┃ dy Crmvergei ┃
┣━━━━━━━━━━━━━━━━━━┫
┃t ┃
┗━━━━━━━━━━━━━━━━━━┛
xo
┏━━━━┓
┃XA~ ┃
┃~J ~ ~ ┃
┣━━━━┫
┃ ┃
┗━━━━┛
a) Steady convergence and divergence b) Undamped (constant amplitude) oscillation
xo
┏━━━━━━━━━━━┓
┃ \ "'7-R -- - -7~ - ┃
┃~ - - -\z- - - t ┃
┣━━━━━━━━━━━┫
┃ ┃
┗━━━━━━━━━━━┛
c) Damped oscillation
xo
┏━━━━━━━━━━━━━━━━━━┓
┃ - ┃
┃ ┃
┃ -- ┃
┃ -- ┃
┃\--7~ / ┃
┣━━━━━━━━━━━━━━━━━━┫
┃/----~j t . ┃
┃ - ┃
┃ . ┃
┃ . ┃
┗━━━━━━━━━━━━━━━━━━┛
d) Diverging osallation
Fiig.6.1 Typicaldynamicmotions.
Equation (6.20) is also called the characteristic equation of the longitudinal
motion and gives four values of the root A,. The natu;e of the free response can
be any combination of the typical free responses shown in Fig. 6.1. A steady
convergence corresponds to a real negative root, a steady divergence if the real
root is positive, an oscillatory motion of constant ampli:de if the root is zero, a
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PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL3(94)
