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solmidterm05_mae143b

solmidterm05_mae143b - Name Student Midterm MAE143B Fall...

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Midterm - MAE143B, Fall 2005 Name: Student #: Solutions November 15, 2005 11:00am-11:50pm, HSS1330 open-book and open-notes midterm exam use the available space to derive your results, attach extra paper if necessary use of any electronic equipment (calculator, phone, PDA) not allowed during exam Consider a linear dynamic system G characterized by the transfer function model y ( s ) = G ( s ) u ( s ) , G ( s ) = 1 s s 2 + 2 s + 1 (1) 1. Show that the system is stable and has a right half plane zero. Derive the value of the undamped natural frequency and the damping ratio of this system. [15pt] The poles are found by solving s 2 + 2 s + 1 = 0 , and by using the quadratic equation the roots can be shown to be located at s = 1 2 ± 1 2 j . Therefore, the system is stable since the real part of the poles is negative. The dampening ratio β and the undamped natural frequency ω n can be found via the following comparison: ω 2 n = 1 2 βω n = 2 . This gives β = 1 2 . The zero is found by solving 1 s = 0 , and therefore a zeros is located are the point s = 1, which is in the RHP. 2. Compute the impulse response y ( t ) of this dynamcal system. [15pt] y ( s ) = 1 s s 2 + 2 s + 1 = 2 1 2 ( s + 1 2 ) 2 + 1 2 2 s 1 2 ( s + 1 2 ) 2 + 1 2
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