ASE 365 - Lecture 19

ASE 365 - Lecture 19 - Harmonic response of 2DOF systems...

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Unformatted text preview: Harmonic response of 2DOF systems – Direct approach: – Or, modal approach using uncoupled modal EOMs – We’ll see resonances, modal cancellation ( 29 F X X x F x x = +- → = → = + ) ( ) ( ) ( 2 ϖ ϖ ϖ ϖ ϖ i K M e i t e K M t i p t i . So Define t i p e i Z t i Z i K M i Z ϖ ϖ ϖ ϖ ϖ ϖ F x F X 1 1 2 ) ( ) ( ) ( ) ( ) (-- = = → +- ≡ Harmonic response example 2k 2m m k x2 x 1 k t F x x k k k k x x m m ϖ cos 2 3 2 1 2 1 2 1 = -- + F1cos ωt t F k m k m k m k t F m k k k m k k m k m k t ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ cos 2 ) 2 5 )( ( 1 cos 2 3 2 ) 2 )( 2 3 ( 1 ) ( 1 2 2 2 1 2 2 2 2 2 --- = ----- = x ) 2 5 )( ( cos ) ( , ) 2 5 )( ( cos ) 2 ( ) ( 2 2 1 2 2 2 1 2 1 ϖ ϖ ϖ ϖ ϖ ϖ ϖ m k m k t kF t x m k m k t F m k t x-- =--- = 58 . 1 , 41 . 1 2 , 2 5 , 1 2 1 2 2 2 2 1 ≈ ≈ = = = ϖ ϖ ϖ ϖ ϖ ϖ ϖ numerator numerator m k m k m k so : s frequencie Important , , , , 2 1 2 2 1 2 2 1 1 2 1 1 < < < < < < < < < < X X X X X X X X numerator numerator : : : : ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ω/ω 1 X2 1.41 1.58 1.00 X1 1.00 1.58 1.41 ω/ω 1 N F η η F x x x η x ≡ = ′ + ′- = +- = +- = + = T T T T t i t i U K M KU U MU U U e K M K M e U t ) ( ) ( ) ( ) ( 2 2 2 ϖ ϖ ϖ ϖ ϖ : by multiply - pre and in Let : taken be can approach modal undamped, If i i i i m k N N N m k m k K M ′- ′ = ⇒ = ′- ′ ′- ′ ′ ′ 2 2 1 2 1 2 2 2 1 2 1 ϖ η η η ϖ ϖ : easy is solution the diagonal, are and Since [ ] t i T t i t i t i e U m k m k U e e e U t ϖ ϖ ϖ ϖ ϖ ϖ η η η η F u u u u η x 1 2 2 2 1 2 1 2 2 1 1 2 1 2 1 ) ( ) (- ′- ′ ′- ′ = + = = = Then (modal forces) [ ] m k m k U 2 5 , , 2 2 2 1 2 1 = = = ϖ ϖ u u : approach Modal N η η F η F x η x = ′ + ′- = +- ⇒ = +- = +- = ) ( ) ( ) ( : ) ( 2 2 2 2 K M KU U MU U U K M U K M U T T T ϖ ϖ ϖ ϖ in = ⇒ = ---- =-- 1 2 ) ( 2 1 2 22 12 2 1 2 2 2 u u u u k k k k M K ϖ = ⇒ =...
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This note was uploaded on 09/18/2011 for the course ASE 365 taught by Professor Staff during the Spring '10 term at University of Texas.

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ASE 365 - Lecture 19 - Harmonic response of 2DOF systems...

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