ENME361 Quiz 5 Solution

# ENME361 Quiz 5 Solution - ENME 361 Fall 2008 Solutions for...

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ENME 361 Fall 2008 Solutions for Quiz #5 (October 16, 2008; Duration: 15 minutes) 1) Given the differential equation 2 2 2 () 2( ) 0 nn dwt wt dt dt ζω ω + += where ζ is a constant such that 0 < < 1. If the initial conditions are w (0) = 0 and dw (0)/ dt = V o , find the response w ( t ) by using the Laplace transform technique. Some Laplace transform pairs are provided below. Some Laplace-transform pairs 1 1 (0 ) n k k k sGs s g −− = n n n dg gt dt = o st e s u ( t t o ) 22 1 2 ss ωω ++ 2 1 s in ( ) ; 1 n t dd n d e t == 2 2 n s 21 1 sin( ); 1 cos 1 n t n n d et ωϕωω ϕ −+ = = < 2 s 2 sin( 1 n t n n d ϕωω = Solution : Taking the Laplace transform of the given differential equation, we find that 2 2 ) ) 2 ( ) ) () 0 dw t sW s sw w W s dt = + = ------ (1) On substituting the initial conditions w (0) = 0 and dw (0)/ dt = V o into Eq. (1), we arrive at 0 (2 ) V Ws = From the given Laplace Transform pairs, we obtain the response

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2 0 () s in ( ) ; 1 n t dd n d V wt e t ζω ω ωω ζ == 2) Consider the displacement response of a linear, underdamped single degree-of-freedom system to an initial displacement shown in the figure below.
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ENME361 Quiz 5 Solution - ENME 361 Fall 2008 Solutions for...

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