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ENME361 Quiz 6 Solution

ENME361 Quiz 6 Solution - B Balachandran ENME 361 Fall 2008...

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B. Balachandran ENME 361 Fall 2008 Solutions for Quiz #6 (October 23, 2008; Duration: 15 minutes) 1) Provide the definition of resonance. Solution : For a linear vibratory system with a single degree of freedom, the condition when the frequency of excitation ω equals the natural frequency ω n of the system; that is, ω = ω n , is called resonance. 2) Consider the following differential equation governing the vibrations of a single degree-of- freedom system subjected to a harmonic excitation: 2 2 j t o d x dx m c kx F e dt dt ω + + = Assume a solution of the form x ( t ) = X o e j ω t and show that the velocity is given by ( ) ( ) /2 ( ) ( ) j t o F x t H e k ω θ π ω Ω + = ± where ( ) ( ) 1 2 2 2 2 1 2 ( ) , ( ) tan , , and cos sin 1 1 2 jx n H e x j x ζ ω θ ω ζ Ω = Ω = Ω = = + − Ω − Ω + Solution : After substituting x ( t ) = X o e j ω t into the given equation and dividing throughout by the mass m , we get ( ) ( ) 2 2 0 0 0 2 2 2 2 0 2 2 0 2 2 2 j t j t j t j t o n n j t j t o n n o n n d X e dX e F X e e dt dt m F j j X e e m F j X m ω ω ω ω ω ω ζω
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