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Unformatted text preview: AE 352: Aerospace Dynamics II, Fall 2008 Homework 3 Due Friday, September 19 Problem 1. Find the free and forced responses of the following system: d 2 y dt 2 + 4 dy dt + 4 y = 3 du dt + 2 u where u ( t ) = 0 for t < 0 and u ( t ) = e 3 t for t ≥ 0. Solution. There are multiple ways to solve this problem. We could use the method of undetermined coe ffi cients, or the convolution integral (impulse re sponse) method. To use the impulse response method, we must first find the free response, that is, solve: d 2 y dt 2 + 4 dy dt + 4 y = To solve, assume a solution y ( t ) = Ce at . Substitute this solution (and its deriva tives) back into the homogeneous ODE: C ( a 2 4 a + 4) e at = Solving for a we get a = 2 , 2. When there is a repeated root, the solution becomes: y ( t ) = C 1 e 2 t + C 2 te 2 t In terms of the initial conditions: C 1 = y , and C 2 = v + 2 y . Thus the free response of the system is: y h ( t ) = y e 2 t + ( v + 2 y ) te 2 t Next, we must find the particular solution (forced response) of the system. WeNext, we must find the particular solution (forced response) of the system....
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This note was uploaded on 09/16/2009 for the course AE ae352 taught by Professor Sri during the Spring '09 term at University of Illinois at Urbana–Champaign.
 Spring '09
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