Exam Solutions 11-20

# Exam Solutions 11-20 - Part I theory(Closed Book Question 1...

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Part I: theory (Closed Book) Question 1 (weight:3) Given: Consider a linear, time invariant, strictly stable, system U ( s ) Y ( s ) G ( s ) ? with transfer function G ( s ) . Suppose to supply as an input a sinusoidal input equal to: u ( t ) = M sin ωt of magnitude M > 0 and frequency ω rad/sec. Asked: Prove that, waiting long enough, the output of the system for the given sinusoidal input will be y ( t ) M | G ( ) | sin( ωt + G ( ) ) I recall that: L{ sin( ωt ) } = ω s 2 + ω 2 , L b e αt B = 1 s α , G ( s ) = G ( s ) and the Euler formula sin( α ) = e e 2 j .

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Part II: exercises (Open Book) Question 2 (weight:3) Given: The input of the system illustrated below is the force F applied to the mass m 1 . F k x 1 x 2 m 1 m 2 b 1 b 2 The system is composed of two masses m 1 and m 2 , damper b 1 connects mass m 1 to the wall, while damper b 2 and spring k connect the two masses. The energy stored in the spring k is equal to E x ) = 1 2 · k 1 · Δ x 2 + 1 4 · k 2 · Δ x 4 where Δ x = x 1 x 2 . The damping effects can be modeled by the linear relation F b = b · v where for each damper the the accompanying damping coef£cient b and velocity v needs to be substituted. Asked:
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Exam Solutions 11-20 - Part I theory(Closed Book Question 1...

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