Damped Harmonic Response

Damped Harmonic Response - CE 573: Structural Dynamics...

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CE 573: Structural Dynamics Response to Harmonic Forcing: Damped Systems Complementary Solution + Particular Solution () 2 sin 2 sin o o F mu cu ku F t u u u t m ϖ φξω ω ++= +⇔+ + = + ±± ± ± φ Complementary solution: [ ] cos s in t cd ut e A t B t ξω d =+ Particular solution: Notice that the assumption ( ) s p ut U t = + will not work because of the presence of the damping term 2 x ± . Try a new assumption: ( ) ( ) () () 22 s s s p p p ut C t D t ut C t D t C t D t ϖφ ϖϖ + + ⇒= + + + + ± Put this into equation of motion: ( ) ( ) ( ) ( ) sin cos 2 cos 2 sin sin cos sin 2 20 o o Ct Dt F t m F CD C m DC D ξ ϖξω −+ − + + + ++ ⇒− + = + + = + Solving for C and D : ( ) ( ) ()() 2 2 2 1 21 (2 ) 2 = 2 oo FF r mk C rr r D ωϖ ξωϖ −− == = 2 Writing the particular solution in a form that looks like the form we used for the undamped system:
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Damped Harmonic Response - CE 573: Structural Dynamics...

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