ENME361 Quiz 6 Solution

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

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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 jt o dx d x mc k x F e dt dt ω ++ = Assume a solution of the form x ( t ) = X o e j t and show that the velocity is given by () () / 2 ( ) o F xt H e k ωθ π −Ω + =Ω ± where 1 2 2 2 2 12 ( ) , ( ) tan , and cos sin 1 jx n He x j x ζ θ Ω= = + + Solution : After substituting x ( t ) = X o e j t into the given equation and dividing throughout by the mass m , we get 2 2 00 0 2 22 2 0 0 2 2 2 o nn o o dXe F X ee dt dt m F jj X e e m F jX m ωω ζω ωζ = = −+ + = which we solve for X o and get 0 2 2 2 2 n FF X mj == ⎛⎞ ⎜⎟ ⎝⎠
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= () 0 2 1 12 F k j ζ −Ω+ = ( ) 2 0 2 2 2 j F
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ENME361 Quiz 6 Solution - B. Balachandran ENME 361 Fall...

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