s03_ps07_spring04 - Unified Engineering II Spring 2004...

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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.

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