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ENME361 Quiz 2 Solution

# ENME361 Quiz 2 Solution - B Balachandran ENME361 Vibrations...

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B. Balachandran ENME361 Vibrations Fall 2008 Solution to Quiz #2 (September 11, 2008; Duration: 10 minutes) 1) Consider the differential equation 2 2 ( ) ( ) 2 ( ) ( ) d w t dw t w t u t dt dt ζ + + = where ζ is a constant such that 0 < ζ < 1 and u ( t ) is the unit step function. Given that the initial conditions are zero; that is, w (0) = dw (0)/ dt = 0, solve for the solution w ( t ) by using the Laplace transform technique. Some Laplace transform pairs are provided at the end. Taking Laplace transforms of the different terms on both sides, the differential equation is transformed into the algebraic equation s s w w s sW w sw s W s 1 ) ( )) 0 ( ) ( ( 2 ) 0 ( ) 0 ( ) ( 2 = + + ζ where L [ w ( t )] = W ( s ), the prime is used to indicate a time derivative, and we have made use of the table for the Laplace transform of the unit step function with t o = 0. Applying the initial conditions 0 ) 0 ( ) 0 ( = = w w and solving for W ( s ), we arrive at ) 1 2 ( 1 ) ( 2 + + = s s s s W ζ Making use of the table, the inverse Laplace transform L -1 [ W ( s )] is found to be w ( t ) = 2 1 1 sin( ); 1 ; cos ; 1 n t n d d n d e t ζω ω ω ϕ ω ω ζ

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ENME361 Quiz 2 Solution - B Balachandran ENME361 Vibrations...

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