week6 Solution1.pdf - University of Wollongong MECH321...

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1 n c ( t ) University of Wollongong MECH321 Autumn 2016 - Tutorial 6 - Solution 1. (a) Determine which of the following second order systems are critically damped, underdamped, overdamped, or undamped? (2 Marks) s + 3 7 2 s + 9 4 s + 1 G 1 ( s ) = 4 s 2 + 2 s + 8 , G 2 ( s ) = s 2 + 9 , G 3 ( s ) = s 2 + 9 s + 4 , G 4 ( s ) = s 2 + 6 s + 9 Answer: Damping is determined by the damping ratio( ζ ). Arrange the denominator into the format of the characteristic equation for a second order system ( ce = s 2 + 2 ζ ω n s + ω 2 ) and determine ζ . G 1 ( s ) , 0 < ζ < 1, underdamped, G 1 ( s ) , ζ = 0 , undamped, G 3 ( s ) , ζ > 1, over- damped, G 4 ( s ) , ζ = 1, critically damped. (b) Determine the location of the poles of the systems given below, sketch the shape of their unit-step responses. (2 Marks) 3 s + 1 s + 2 6 G 1 ( s ) = s 2 + 8 s + 1 , G 2 ( s ) = s 2 + 4 s + 4 , G 3 ( s ) = s 2 + 9 s + 49 HINT: Determine where the poles are in the s plane and from this sketch the generic shape of the step response, quan titativ e analysis is not required. Answer: Note: the poles are the roots of the denominator.
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