Lecture07-Prof.Ju

# Lecture07-Prof.Ju - CEE M237A MAE M269A Lecture 7 MDOF...

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CEE M237A / MAE M269A Lecture 7: MDOF Analysis of Damped System: Modal Superposition Professor J. Woody Ju

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Damped Uncoupled E.O.M. y 2 () Must assume: Pre-multiply on both sides undamped eigenve 0 i. ctor If we then e., damping orthogonality assume 0 T mn T T n TTT nn n n T c mY c Y k Y p mv cv kv p t t cn m φφ φΦ φ Φ = ++= ++ =≠ = ± ±± ²² ² ±±± ± ² ±± ±± ± ± ± ± ± ± ± ± () ( ) 2 SDOF! unco or upled 2 n n n n n n n MY CY KY P t Pt YY Y M ξω ω = = ² ² orthog. orthog. ?
Damped Uncoupled Equation of Motion (Cont’d) y 3 ( ) () () 2 2 2 2 where 2 n nn n n n n n T T n n n n n T nnn T n n n T Pt YY Y M c Cc M M Mm Kk M p t ξω ω φ φφ ξ ++ = =⇒ = = ±± ± ² ³ ²² ² ³ ² ³ ² ³ ² ² nth mode “modal damping ratio”

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onditions for Damping rthogonality y 4 () 2 damped fre How can we e vibration u e ncou igen pled E.O value pr or ob thog lems .M. If not for , then we must solve i.e., no longer 0, but involving matrix! coupl onalit e satisfy ! d pr y 0? T mn km c c φ ω −= = ±± ± ± oblem very complex!
Damped Uncoupled EOM: Damping Orthogonality y y 5 ( ) () 1 01 (1) Rayleigh damping: special case of Caughey damping orthogonal w.r.t. , 0,1 O and ! K b b b mk ca m a k m a b ⎡⎤ =+ = = ⎣⎦ ±± ± ± ± ± 1 1 (2) Caughey damping: Damping is controlled throug where h !

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Lecture07-Prof.Ju - CEE M237A MAE M269A Lecture 7 MDOF...

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