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= (Lo + AL)sin Aar _ (Do + AD)cos Aa + T - W sin Oo
(4.444)
Similarly,
so that
AF\- - Lo Aa - AD
aAFx 8Fx=Lo_aAD
aA~ = a~ = - d~ = ~0 -- 8aD
Clu, - CL - CDct
Fzo = -Lo + W cos Oo
(4.445)
(4.446)
(4.447)
(4.448)
Fz - Fzo + AFz
= -(Lo + AL)cos Aaf - (Do + AD)sin Aa- + W cos 00 (4.449)
- -Lo - AL - DoAcy + W cos 00 (4.450)
AFz ~ -AL - DoAa
aAFz aFz aAL
aA -= aa =-a~L _Do
= - ?dL _ Do
(4.451)
(4.452)
(4.453)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 383
or
It can be shown that
CZCL = -CLa - CD
Cxcr - CLa Acy - CDrr ~ -CDa
Cxq = CLq Act - CDq ~ -CDq
Cza = -CLd, - CDd Aa -CLa,
Czq = -CLq - CDq AcY ~ -CLq
Usually, CDcir and CDq are small and ignored.
(4.454)
(4.455)
(4.456)
(4.457)
(4~458)
Lateral-directiona/ equations of motion for sma// disturbance. Substitu-
ting for ACy, ACi, and ACn from Eqs. (4.412), (4.414), and (4.416) in Eqs.
(4.394), (4.396), and (4.398), we obtain
(m, dd _ b,CyB dd _ Cyp) Ap - (b,Cyp dd + Cy*) A4 + (m, dd _ b,Cyr dd) A k
= Cy8r A8r + Cyasa A8a (4.459)
(-C,p - bi CiB dd )AB + (-bi Cipdd + /x, dd )A4
+(_b,C,rdd _/z,dd )Ayr=Cl8rA8r+Cl8aA8a (4.460)
(-: C,,p - b,Cnp dd )Ap + (-bi Cnp dd _ /xz,dd )A4
+(_biCnrdd+Iz,dd )A =Cn8rA8r+Cn&1A8a (4.461)
where
d4
dt =P - (4.462)
d~
dt =r (4.463)
bi
Ixl =
(4.464)
(4.465)
b
= 2U
Ix
2~p U2Sb
384 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Iz
Izl = ;-p U2Sb
Ixz
lxzl = 2pU2Sb
(4.466)
(4.467)
Now let us evaluate various derivatives like Cyp, C),4, etc., appearing in Eqs.
(4.459-4.461).
Suppose the airplane is disturbed in bank angle A4 as shown in Fig. 4.17c. We
have
Fy = Fyo + AFy = W cos OoA4 + Yaero + AYacro (4.468)
where Yaem denotes the aerodynamic side force. We have Fyo = Yauo = 0 so that
AFy = W cos 0o A4 + AYauo
aAFy a Fy
a A~y ~ ajy = W cos0o
Similarly, it can be shown that
Cy4 = CL COS Oo
Cyp = (Cyp)aero
Cyp = (CyB)aero
Cyp = (Cyp)aero
Cyr = (Cyr)aero
(4.469)
(4.470)
(4.471)
(4.472)
(4.473)
(4.474)
(4.475)
and so on. Here, it is assumed that the side force Yauo has no dependence on the
bank angle ~.
It is important to bear in mind that Eqs. (4.417-4.419) and (4.459-4.461) are
applicable for small disturbance motion when the airplane is disturbed from a
steady, unaccelerated flight with zero sideslip. If this is not the case, then the
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