ASE 365 - Lecture 17

ASE 365 - Lecture 17 - Mass suspended in a moving frame for...

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Unformatted text preview: Mass suspended in a moving frame . for EOMs Derive m 3 4 5 k k k m ) ( t x ) ( t y ) ( t x ) ( t y ? if vibrate does How ) ( ) ( = = t y t x m 3 4 5 k k k m ) ( t x ) ( t y ) ( t x ) ( t y gravity) ignore m eq' to (relative : FBD- 1 S F 2 S F 3 S F ) ( 1 x x k F S- = ) ( 2 y y k F S- = )] ( 6 . ) ( 8 . [ ] ˆ ) ( ˆ ) [( ) ˆ 6 . ˆ 8 . ( 3 y y x x k y y x x k k F S- +- =- +- ⋅ + = ⋅ = j i j i elongation )] ( 6 . ) ( 8 . [ 8 . ) ( 8 . 3 1 y y x x k x x k F F F x m S S x- +--- =- = = ∑ )] ( 6 . ) ( 8 . [ 6 . ) ( 6 . 3 2 y y x x k y y k F F F y m S S y- +--- =- = = ∑ = + y x k y x k y x m m 36 . 1 48 . 48 . 64 . 1 36 . 1 48 . 48 . 64 . 1 : EOMs = + y x k y x k y x m m 36 . 1 48 . 48 . 64 . 1 36 . 1 48 . 48 . 64 . 1 : EOMs . assume ; : vibration Free ) cos( ) ( ) ( ) ( ) ( φ ϖ - = = = t y x t y t x t y t x =- + - ) cos( 36 . 1 48 . 48 . 64 . 1 1 1 2 φ ϖ ϖ t y x k m : become EOMs . if only solution Nontrivial ) det( 2 =- ≠ M K y x ϖ ) 2 )( ( 2 3 ) 36 . 1 ( 48 . 48 . ) 64 . 1 ( ) det( 2 2 2 2 4 2 2 2 2 =-- = +- =-- =- k m k m k mk m m k k k m k M K ϖ ϖ ϖ ϖ ϖ ϖ ϖ ? in about What . , ) cos( ) ( ) ( 2 2 2 2 1 φ ϖ ϖ ϖ- = = = t y x t y t x y x m k m k - = = = - 1 75 . 36 . 48 . 48 . 64 . ) ( 1 1 1 2 1 y x y x k k k k y x M K : From ϖ = = -- = - 75 . 1 64 . 48 . 48 . 36 . ) ( 2 2 2 2 2 y x y x k k k k y x M K : From ϖ . conditions initial by determined s ' and s ' with : form the in be must vibration Free i i B A t B t A t B t A t y t x ) sin cos ( 75 . 1 ) sin cos ( 1 75 . ) ( ) ( 2 2 2 2 1 1 1 1 ϖ ϖ ϖ ϖ + + + - = Modify the previous example… k m m ) k x2 x 1 k = +-- + + ) 1 ( ) 1 ( 2 1 2 1 x x k k k k x x m m α α α α ) 1 ( ) 1 ( ) det( , 2 2 2 =- +--- + =- ≡ γ α α α γ α ϖ ϖ γ k M K k m If [ ] 2 1 ) 1 ( 2 ) 1 ( 2 2 1 2 2 2 2 2 2 = + + +- = =- + +- + + =- + α γ α γ α γ α γ α α α γ α- ) (1 gives m k m k ) 2 1 ( , 2 1 , 1 1 ) 2 1 ( ) 1 ( 1 2 2 2 1 2 1 2 2 , 1 α ϖ ϖ α γ γ α α...
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ASE 365 - Lecture 17 - Mass suspended in a moving frame for...

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