48090_q2spr05soln - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.011: Introduction to Communication, Control and Signal Processing QUIZ 2, April 14, 2005 Answer Booklet Your Full Name: Recitation Instructor & Time : at o’clock This quiz is closed book , but three sheets of notes are allowed. Calculators will not be necessary and are not allowed. This answer booklet has space for all answers, and for relevant reasoning. Check that the answer booklet has pages numbered up to 20. Neat work and clear explanations count; show all relevant work and reasoning! You may want to first work things through on scratch paper and then neatly transfer to this answer booklet the work you would like us to look at. Let us know if you need additional scratch paper. Only this answer booklet will be considered in the grading; no additional answer or solution written elsewhere will be considered. Absolutely no exceptions! There are three problems , weighted as indicated on the quiz . The quiz will be graded out of 50 points . DO NOT DISCUSS THIS QUIZ WITH 6.011 STUDENTS WHO HAVE NOT YET TAKEN IT TODAY! Problem Your Score 1 (16 points) 2 (16 points) 3 (18 points) Total (50 points) 1
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6.011 Quiz 2, April 14, 2005 Problem 1 (16 points) A particular object of unit mass, constrained to move in a straight line, is acted on by an external force x ( t ) and restrained by a cubic spring. The system can be described by the equation d 2 p ( t ) + kp ( t ) ±p 3 ( t )= x ( t ) , dt 2 where p ( t ) denotes the position of the mass and p 3 ( t ) is the cube of the position ( not its third derivative!); the quantities k and ± are known positive constants. 1(a) (4 points) Obtain a state-space model for the above system, using physically meaningful state variables; take x ( t ) to be the input and let the output y ( t ) be the position of the mass. q 1 ( t p ( t ) q 2 ( t )= ˙ p ( t ) q ˙ 2 ( t ( t )+ ±p 3 ( t x ( t ) State-space model: q ˙ 1 ( t q 2 ( t ) q ˙ 2 ( t {− k + ±q 2 1 ( t ) } q 1 ( t x ( t ) y ( t q 1 ( t ) 2
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± ± ± ± ± ± ± ² ³ ´ 6.011 Quiz 2, April 14, 2005 1(b) (5 points) Suppose x ( t ) 0 and the system is in equilibrium. You will find that there are three possible equilibrium conditions of the system. Determine the values of your state variables in each of these three equilibrium conditions, expressing your results in terms of the parameters k and ± .
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48090_q2spr05soln - Massachusetts Institute of Technology...

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