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so that
V(y) = Uo -ry
(4.601)
ql - ;:(Uo - r y~ (4.602)
N ~:(Uo2 - 2r UoY) (4.603)
The lift acting on the elemental strip RT is given by
dL = ~:p(Uo2 _ 2r Uoy)ao(y)ac(y) dy (4.604)
Then the total rolling moment due to the elemental strip RT is given roy
dRM = _-}:p(Uo2 _ 2r Uoy)ao(y)ac(y)y dy (4.605)
The rolling moment due to the right wing is given by
RMR=-~p[,'(Uo2-2rUoy)ao(y)ac(y)ydy (4.606)
A: :
'-1
i,l: .
N
.. .
;: I
k: .
i- .
!
i:,
414 PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Similarly; the rolling moment due to the left wing (change y to -y) is given by
RM,=~p[,'(Uo2+2rUoy)ao(y)ac(y)ydy (4.607)
The total or net rolling moment, which is the sum of the right and left wing rolling
moments, _
We have
so that
RM = 2p Uor [,' ao(y)crc(y)y2 dy (4.608)
C,= pRUMSb
C,r = a~a~
(4.609)
(4.610)
(Cu)w = S~, [,' ao(Y)ctc(y)y2 dy (4.611)
For an untwisted rectangular wing of constant chord and constant airfoil section
all along the span, Eq. (4.611) reduces to
CL
(qr)W= 3 (4.612)
Thus, for positive angles of attack, (C,r)W > O, which implies thatif an aircraft
experiences a positive yaw rate then the wing induces a positive rolling moment
that tends to raise the left wing and lower the right wing.
The strip theory estimate ignores the induced drag effects and mutual interfer-
ence between adjacent wing sections. Thus, the strip theory predictions will be in
error for wings with small aspect ratio. For more accurate prediction of (Clr)W at
subsonic speeds, the following Datcon/ formula can be used:
where
(CLr)W=C,(g),,=O.At+(~-)r7rad (4.613)
' (4.614)
(g ).,=o.)w = NDe (g ).,=O,M=O
A(l - B2) + 2 cos A,
Num -l+2B AB+2 osA~4 +(~~~4,os~ t)(_a 8At )
(4.615)
Den-.l+ ~:j::~/)( 8A ) (4.616)
EQUATIONS OF MOTION AND ESTIMATION OF STABILITY DERIVATIVES 415
SUBSONIC SPEEDS
Ac/4 (deg)
TAPER
RATIO
0 15 30 45 60 x
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