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Unformatted text preview: Problem 1. (7 points) Consider the following massdamper system: where u ( t ) is a forcing velocity, m is the mass of each of the three mass elements, b is the resistance of each of the dampers, and v 1 , v 2 , v 3 are respectively, the velocities of, the left, middle, and right mass, in the rightward reference direction. Derive the state space model with u ( t ) as the input, v = [ v 1 , v 2 , v 3 ] T as the state, and the force acting on the middle mass (in the rightward direction) as the output y . Solution: Applying Newtons law to the three masses, we can write three equations in terms of v 1 , v 2 , v 3 , and u m v 1 = b ( u v 1 ) + b ( v 2 v 1 ) m v 2 = b ( v 1 v 2 ) + b ( v 3 v 2 ) m v 3 = b ( v 2 v 3 ) bv 3 . We are concerned with the force applied to the middle mass therefore y = b ( v 1 v 2 ) + b ( v 3 v 2 ). Rearranging the four equations we get v 1 = 2 b m v 1 + b m v 2 + b m u v 2 = b m v 1 2 b m v 2 + b m v 3 v 3 = b m v 2 2 b m v 3 y = bv 1 2 bv 2 + bv 3 ....
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 Winter '09
 CALLAFON
 Damper

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