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Unformatted text preview: Uniﬁed Engineering II Spring 2004 Problem S3 (Signals and Systems) Note: Please do not use oﬃcial or unoﬃcial bibles for this problem. An airfoil with chord c is moving at velocity U with zero angle of incidence through
the air, as shown in the ﬁgure below: U The air is not motionless, but rather has variations in the vertical velocity, w. As the
airfoil ﬂies through this gust ﬁeld, the leading edge of the airfoil “sees” a variation
in the angle of attack. If w is small compared to U , then the angle of attack change
seen by the airfoil is α = w/U . Since the velocity proﬁle varies in space, the angle of
attack seen by the airfoil is a function of time, α(t).
One might expect that the lift coeﬃcient of the airfoil is just
CL (t) = 2πα(t)
However, the airfoil does not respond instantaneously as the airfoil encounters the
gust. If the airfoil encounters a “sharpedged gust,” so that the apparent change in
the angle of attack is a step function in time,
α(t) = α0 σ (t)
then the change in lift is given by
¯
CL (t) = 2πα0 ψ (t)
¯
where t = 2U t/c is
the step response of
considered to be the
output is considered
be approximated as ¯
the dimensionless time. ψ (t) is the Kussner function, and is
¨
the airfoil (neglecting multiplicative constants), if the input is
vertical gust at the leading edge as a function of time, and the
to be the lift as a function of time. The K¨ssner function can
u
� ¯
ψ (t) = ¯
0,
t<0
¯
¯
1 −t
1 −0.13t
¯≥ 0
− 2e , t
1 − 2e Assuming that the airfoil acts as an LTI system, determine and plot the lift
coeﬃcient, CL (t), and the gust velocity, w(t), for the following conditions:
c = 1m
U = 1 m/s
�
0 m/s,
t<0s
w(t) =
0.1 · (1 − e−2t ) m/s, t ≥ 0 s ...
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.
 Fall '05
 MarkDrela

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