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MIT2_017JF09_p22

# MIT2_017JF09_p22 - 22 AIRCRAFT IN WINDS 22 69 Aircraft in...

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22 AIRCRAFT IN WINDS 69 22 Aircraft in Winds This problem builds on the previous one: you will use the same wind gust spectrum, and study the response of an air vehicle that is being buffeted by it. Here is the spectrum of the wind again, along with a picture of the aircraft: S w + ( ω ) m 2 /s 0.00 0.50 0.65 1.00 2.80 3.10 2.80 2.00 1.60 1.20 0.80 0.60 0.50 0.40 0.20 0.00 wind fluctuations w I error e desired trackline U Frequency, cycles/hr 0 10 14 20 32 50 72 100 141 200 316 500 717 1000 1410 2000

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22 AIRCRAFT IN WINDS 70 The aircraft has a forward speed of U = 60 m/s , or about 120 knots. It employs GPS navigation and a simple feedback loop to stay on a given straight-line path. In particular, the feedback law is φ = 0 . 003 × e , where φ is the controlled heading of the vehicle relative to the desired line of travel, and e is the cross-track error, i.e., the position of the aircraft in the direction normal to the desired line of travel. Thus, a cross-track error of ten meters leads to a heading command of 0.03 radians, which points the vehicle back toward the track. We assume that wind gusts do not affect the heading of the craft directly, but that there is a very good heading controller, that will make the vehicle follow the desired φ very closely. 1. Making the linearizing assumption that φ is small, and including the feedback law, what is the differential equation relating wind gust velocity w ( t ) to cross-track error e ( t )? What is the transfer function E ( ) /W ( )? Note
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MIT2_017JF09_p22 - 22 AIRCRAFT IN WINDS 22 69 Aircraft in...

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