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final08_mae143b - Final MAE143B Summer II 2008 September 5...

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Final - MAE143B, Summer II 2008 Name: Student #: September 5, 2008 3:00pm-6:00pm, Center 115 open-book and open-notes final 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 1 Consider the mechanical system depicted on the left with mass m = 1, spring stiffness k = 6, damping d = 5 and a position x ( t ). For this mechanical system we define the input u ( t ) as the force applied to the mass and the output y ( t ) is defined as the acceleration of the mass: u ( t ) = F ( t ) , y ( t ) = ¨ x ( t ) m F ( t ) x ( t ) k d single mass mechanical system 1a Show that the proper (but not strictly proper) transfer function G ( s ) = s 2 s 2 + 5 s + 6 (1) models the dynamics between input u ( s ) and output y ( s ) in the Laplace domain. [5pts] 1
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1b With G ( s ) given in (1), compute output y ( t ) due to unit step input signal u ( t ). [10pts] 1c Compute a state space model of the dynamic system based on the two-dimensional state vector [ x ( t ) ˙ x ( t )] T . [5pts] 2
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2 To improve the speed of the response of the mechanical system, a proportional feedback controller u ( t ) = K p [ r ( t ) y ( t )] , with K p > 0 (2)
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