48090_q2spr05soln

# 48090_q2spr05soln - Massachusetts Institute of Technology...

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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
± ± ± ± ± ± ± 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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