ASE 365 - Lecture 4

ASE 365 - Lecture 4 - Prototype SDOF system k c m x F FBD:...

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Unformatted text preview: Prototype SDOF system k c m x F FBD: fS f D F m : EOM Separating equilibrium from dynamics c k/ 2 k/ 2 m y(t ) F(t ) FBD: F(t ) m g y c 2 ky 2 ky y c ky mg F y m --- = mg t F ky y c y m- = + + ) ( : EOM k mg y mg ky t F y y eq- = →- = → = = = ) ( : m Equilibriu ) ( t F kx x c x m = + + : Dynamics x(t ) m g 2 ky eq 2 ky eq F(t ) x c 2 kx 2 kx = + response dynamic m equilibriu + = + = ) ( ) ( t x y t y eq Damped free response ) ) ( ( 2 2 2 st n n Ce t x ω s ζω s = = + + : equation stic Characteri ( 29 1 2 2 2 1- ±- =- ±- = ζ ζ ζ n n n n n , ω ω ω ζω ω s 2 : Roots ed) (underdamp roots complex 2 : damped) y (criticall root real repeated : d) (overdampe roots real distinct 2 : : and nt discrimina on depends roots of Type 1 1 1 < → = → → ζ ζ ζ ζ Overdamped case (ζ>1) easier... be Can . : ICs For 2 1 ) ( C C x + = ) 1 sinh 1 cosh ( ) ( 2 2 2 1 t A t A e t x n n t n- +- =- ζ ϖ ζ ϖ ζϖ and , so , Use x x e x x e e e x e e x x x x x x x sinh cosh sinh cosh 2 sinh 2 cosh- = + =- = + =--- Overdamped case (ζ>1) (cont’d) ) 1 sinh 1 cosh ( ) ( 2 2 2 1 t A t A e t x n n t n- +- =- ζ ϖ ζ ϖ ζϖ . : ICs For 1 ) ( x A x = = [ ( 29 ] ) 1 cosh 1 sinh ( 1 ) 1 sinh 1 cosh ( ) ( ) ( 2 2 2 1 2 2 2 2 1 t A t A t A t A e t x n n n n n n t n- +-- +- +-- =- ζ ϖ ζ ϖ ζ ϖ ζ ϖ ζ ϖ ζϖ ζϖ [ ] 2 2 1 ) 1 ( ) ( ) ( v A A x n n =- +- = ζ ϖ ζϖ Critically damped case (ζ=1)...
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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 4 - Prototype SDOF system k c m x F FBD:...

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