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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