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Unformatted text preview: AE 352: Aerospace Dynamics II, Fall 2008 Homework 4 Due Friday, September 26 Problem 1. Consider a mass resting on the floor, and attached to a wall with a spring and damper. Let m = 4, k = 4, c = 1. A force F acts on the particle. (Note this is a 1D problem) (a) Find the equations of motion for this system. Solution. Summing forces and applying Newtons law, we get m x + kx + c x = F Substituting in values for k , c , m , 4 x + x + 4 x = F (b) Find the undamped natural frequency , and the damping ratio . Solution. To find the damping ratio and undamped natural frequency, we rearrange the equation x + 2 n x + 2 n x = F and thus have n = 1, = 1 / 8. (c) Find the free (a.k.a. complementary) solution of this system. Is the system underdamped, critically damped, or overdamped? Plot the solution (i.e. plot x vs t ). Solution. Because < 1, we know the system is underdamped. To find the free solution, look at the EOM without forcing: x + x 4 + x = Now, using the method of undetermined coe ffi cients, assume a solution of the form x ( t ) = e at . The resulting solution is: x ( t ) = e- t / 8 " c 1 cos 3 7 8 t ! + c 2 sin 3 7 8 t !# Solving for c 1 , c 2 in terms of initial conditions, we have c 1 = x Page 1 of 5 AE 352: Aerospace Dynamics II, Fall 2008 and c 2 = 8 v + x 3 7 So the free solution becomes: x ( t ) = e- t / 8 " x cos 3 7 8 t !!...
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