ASE 365 - Lecture 21 - Vibration absorbers K M F m k t i p...

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Unformatted text preview: Vibration absorbers K M F m k t i p t i e M K F x Fe Kx x M 2 ϖ ϖ ϖ- = → = + t i e F x x k k k k K x x m M 2 1 2 1 ϖ = -- + + Or... 1 2 2 2 1 2 kx kx x m kx kx x m = + = +- : Rearrange . : equation Lower ! , and If ∞ →- = ≠ = m k kx x x m k 2 1 2 1 2 ϖ ϖ . ) det( 2 2 2 2 ≠- =---- + =- k m k k k M k K ϖ ϖ ϖ But . or resonance, at is system the that indicates this But M K finite. is so frequency, resonant a not is that learned we Also, ". ! , and If " said, We 2 2 1 2 1 2 x m k kx x x m k ϖ ϖ ϖ ∞ →- = ≠ = zero! be to is that therefore, conclude, We required 1 x response.) zero a produce we , resonance" " a g introducin (By . 2 1 2 1 2 1 t i t i Fe kx x e F x x k k k k K x x m M ϖ ϖ- = = = -- + + get we , since : EOM 1st From O bj1 6 5 ? and with system the of s frequencie natural the are What m k s. frequencie natural the are roots two whose in polynomial quadratic a is ) : Consider 2 2 det( ϖ ϖ =- M K . ) ( ) det( . ) det( . ) det( 2 2 2 2 2 2 2 2 2 2 →- <- =- = = =- = ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ Mm k m k Kk M K M K M K , large For , When , When . above frequency another at and and zero between frequency one at zero equals Therefore 2 ) det( ϖ ϖ ϖ M K- . above one and quickly) through passed be must (which below resonance one is There ϖ ϖ Nonperiodic response (undamped) . ) ( ) ( ) ( v x x x F x x = = = + , ICs with , : EOMs t K M N η η F η η η x η x = ′ + ′ ⇒ = + = = K M U KU U MU U U t U t t U t T T T T by multiply and , let : approach Modal : ) ( ) ( ) ( ) ( 2 , 1 ) ( = = ′ + ′ i t N k m i i i i i , : EOMs Modal η η systems. SDOF undamped for EOMs to form in identical are EOMs modal These ). ( ) ( . 2 , 1 ) ( 2 , 1 ) ( t U t i t i t N k m i i i i i i η x = = = = ′ + ′ using back transform Then , for , : EOMs modal Solve η η η t t d t m N t i i i i i i t i i i i ϖ ϖ η ϖ η τ τ ϖ ϖ τ η sin ) ( cos ) ( ) ( sin 1 ) ( ) ( + +- ′ = ∫ : case general in response Modal ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( 1 1 x η η x x η η x -- = → = = → = U U U U : from available are ICs Modal etc....
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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 at Austin.

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ASE 365 - Lecture 21 - Vibration absorbers K M F m k t i p...

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