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s03_ps07_spring04

s03_ps07_spring04 - Unied Engineering II Spring 2004...

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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 C L ( t ) = 2 πα ( t ) However, the airfoil does not respond instantaneously as the airfoil encounters the gust. If the airfoil encounters a “sharp-edged gust,” so that the apparent change in the angle of attack is a step function in time, α ( t ) = α 0 σ ( t ) then the
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