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where Cy)S,V is given by Eq. (4.547).
For szupersonic speeds, no general method is available for the estimation of
(Cyr)V.7
Estimation of Ck. This derrvative is a measure of the rolling momentinduced
due to yaw rate experienced by the aircraft. Generally, the contributions ofthe fuse-
lage and the horizontal tail surfaces are small and can be ignored. The contribution
mainly comes from the wing and vertical tail surfaces so that
Clr = (Clr)W +(Clr)V
(4.600)
For approximate purposes, we can use the strip theory to estimate the wing con-
tribution atlow subsonic speeds as follows.
Because of yaw rate, the relative velocity experienced by the wing sections
varies along the span. Suppose the aircraft is experiencing a positive yaw rate.
Then the sections on the right wing experience a decrease in the relative velocitjr,
whereas those on the left wing experience an increase in the relative velocity. As a
result, the lift on right wingis smaller compared to that on the left wing. This gives
rise to a positive rolling moment. Thus, conceptually we see that (Clr)W > O.
The relative velocity experienced by the elemental strip RT (Fig. 4.27b) on the
right wing is given by
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: :
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i,l: .
N
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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-
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