ASE 365 - Lecture 16

ASE 365 - Lecture 16 - Recap x x = K M vibration free...

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Unformatted text preview: Recap x x = + K M : vibration free undamped for EOMs ). ( ) ( t f t u x = solution s" synchronou " for Looking other. each with consistent be must ODEs These function scalar the for ODEs of a to Leads ). ( t f one set . let if only energy in ve conservati be can solution whose , form the has one Each 2 ϖ = → ≥ = + C C Cf f . allow to need we which in , equations the give EOMs Then u u ≠ =- =- ) det( ) ( 2 2 M K M K ϖ ϖ Example k m m k x2 x 1 k = -- + --- =- +- = 2 2 ) ( ), ( 2 1 2 1 2 1 2 2 1 2 1 1 x x k k k k x x m m kx x x k x m x x k kx x m : EOMs = +- + =- + ⇒ = ) 2 ( ) 2 ( ) ( ) ( ) ( 2 1 2 2 1 1 2 1 2 1 f ku ku f mu f ku ku f mu t f u u t x t x , If . ) ( 2 2 . 2 2 ) 2 ( ) 2 ( 2 1 2 2 2 2 1 2 1 2 1 2 1 2 2 1 1 = -- + -- = +- = =- = +- + =- + f u u k k k k m m f f mu ku ku mu ku ku f ku ku f mu f ku ku f mu ϖ ϖ ϖ give EOMs and Then need we ODEs two the For ) is which , or , written be can t requiremen cy (Consisten . 2 2 2 2 2 2 1 2 2 1 2 2 2 1 1 2 2 1 u u M K u u m m u u k k k k mu ku ku mu ku ku ϖ ϖ ϖ ϖ = = -- = +- =- ) 3 )( ( 3 4 ) ( ) 2 ( 2 2 2 2 : ) det( 2 2 2 2 2 2 2 2 2 2 2 2 2 2 =-- = +- =-- =---- = - -- =- ≠ k m k m k km m k m k m k k k m k m m k k k k M K ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ need we , For u and roots two the has and in quadratic is equation The . 3 ) det( 2 2 2 1 2 2 m k m k M K = = =- ϖ ϖ ϖ ϖ . ) ( 2 u =- M K ϖ of solution nontrivial a is there these of each For . 1 1 2 2 1 2 1 2 1 2 1 2 1 = = -- = - -- = u u u u k k k k u u m m m k k k k k m k solution nontrivial the has equation the , For ϖ . 1 1 3 2 2 3 2 2 1 2 1 2 1 2 2 - = = ---- = - -- = u u u u k k k k u u m m m k k k k k m k solution nontrivial the has equation the , For ϖ . ing correspond the describing nts displaceme of vector a is there one, each For occur. can vibration free us) (synchrono which at 2 are there system, 2DOF this For : tion interpreta Physical vibration of mode natural s frequencie natural ....
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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 16 - Recap x x = K M vibration free...

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