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HW11 - MAE 326 Spring 2008 HW 11 due Wed 16 April Problems...

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MAE 326 Spring 2008 HW 11, due Wed., 16 April Problems 1-4 are for review purposes and are optional extra credit, but recommended if you had trouble with problems 1 or 3 in Prelim 2. 1. A = ˆ 6 2 0 - 2 ! (1) Find the eigenvalues and eigenvectors of A . Do the calculation by hand. 2. B = 1 2 3 0 2 - 1 0 0 1 (2) One of the eigenvalues of this matrix is λ = 2. The associated eigenvector is v = 2 1 0 Find all solutions, x to the equation B x = 2 x . 3. For the matrix A from problem 1, if ˙ x = A x , sketch the trajectories of solutions in the neighborhood of the origin. 4. Consider two masses, m 1 = 1 kg and m 2 = 0 . 2 kg connected by a spring with stiffness k = 1 N/m as shown in figure 1. The masses are immersed in a viscous fluid that provides damping, b 1 = b 2 = 0 . 05 N/ ( m/s ). The masses and springs are constrained to move in one dimension. Input to this system is a force f ( t ) applied to mass 1 in the direction that would cause the spring to initially go into compression. Outputs of this model are the position and velocity of mass 1.
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