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hw03sols - AE 352 Aerospace Dynamics II Fall 2008 Homework...

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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 = 0 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 = 0 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 0 , and C 2 = v 0 + 2 y 0 . Thus the free response of the system is: y h ( t ) = y 0 e - 2 t + ( v 0 + 2 y 0 ) te - 2 t Next, we must find the particular solution (forced response) of the system. We will calculate the forced response using the convolution integral. First, simplify
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