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hw2_solution - Homework Set#2 MAE 4720/7720 – Modern...

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Unformatted text preview: Homework Set #2: MAE 4720/7720 – Modern Control, Fall 2008 Solution The governing equation for a simple mass-spring system is 4 = + z z a) the SSR is shown below: [ ] T z z = x - = 4 1 A Eigenvalues: ( 29 4 4 1 det det 2 = + = - =- λ λ λ λ A I Therefore, the eigenvalues are j 2 ± = λ (two purely imaginary numbers) The system (unforced) response is jt jt e K e K t z 2 2 2 1 ) (- + = = t a t a 2 cos 2 sin 2 1 + since t j t e jt 2 sin 2 cos 2 + = (Euler’s Theorem). So, we expect the unforced (natural) response to be a harmonic solution (sine/cosine functions) with no damping. b) STM using Laplace method: ( 29 [ ] + +- + + = - =- = Φ----- 4 4 4 4 1 4 4 1 ) ( 2 2 2 2 1 1 1 1 1 s s s s s s L s s L A sI L t Using Laplace tables: - = + +- + + = Φ- t t t t s s s s s s L t 2 cos 2 sin 2 2 sin 5 . 2 cos 4 4 4 4 1 4 ) ( 2 2 2 2 1 Note: if t = 0, the STM is a 2x2 identity matrix (as it should be) STM using diagonalization method: 1 ) (- = Φ P Pe t Dt The diagonalizing matrix is...
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hw2_solution - Homework Set#2 MAE 4720/7720 – Modern...

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