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ASE 365 - Lecture 13

ASE 365 - Lecture 13 - Response by Laplace transform...

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Unformatted text preview: Response by Laplace transform Laplace transform of F(t): [ ] tion. transforma the of kernel and variable, subsidiary where = = = =- ∞- ∫ st st e s s F dt t F e t F L ) ( ) ( ) ( To solve ODEs, we’ll need: [ ] [ ] [ ] ∫ ∫ ∫ ∫ ∫ ∞ ∞- ∞-- ∞ ∞- ∞-- ∞- +-- = + = = +- = + = = = = 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( s X s sx x dt t x e s t x e dt t x e t x L s X s x dt t x e s t x e dt t x e t x L s X dt t x e t x L st st st st st st st F kx x c x m = + + : EOM ( 29 ( 29 ) ( ) ( ) ( ) ( 2 s F s kX s sX x c s X s sx x m = + +- + +-- : transform Laplace Take 2 ) ( ) ( ) ( ) ( x m x c ms s F s X k cs ms + + + = + + : Or solution. s homogeneou with associated are terms IC k cs ms s X s F s Z + + = ≑ 2 ) ( ) ( ) ( : function Impedance ) 2 ( 1 1 ) ( ) ( ) ( 2 2 2 n n s s m k cs ms s F s X s G Ο– ΞΆΟ– + + = + + = ≑ : function Transfer Block diagrams: F(t) m,c,k x(t) F(s) G(s) X(s)=G(s)F(s) [ ] [ ] ) ( ) ( ) ( ) ( 1 1 s F s G L s X L t x-- = = : need we solution, For Requires line integral in complex s-plane, residue theorem. Or, in most practical cases, a table of Laplace transforms (e.g., Table B.8, p. 767), often with method of partial fractions. (time domain) ( s domain) Laplace transform of time-shifted function [ ] [ ] . then , if Also, ) ( ) ( ) ( ) ( ) ( ) ( 1 1 a t u a t f s F e L t u t f s F L as-- = =--- [ ] [ ] ? is what , If : Q ) ( ) ( ) ( ) ( ) ( a t u a t f L s F t u t f L-- = [ ] dt a t u a t f e dt a t u a t f e a t u a t f L a st st ) ( ) ( ) ( ) ( ) ( ) (-- =-- =-- ∫ ∫ ∞- ∞- [ ] [ ] ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( t u t f L e d u f e e d u f e a t u a t f L t a t d dt a t a t sa s sa a s- ∞-- ∞ +- = = =-- ∞ = β†’ ∞ = = β†’ = = + =- = ∫ ∫ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ Ο„ : so , and : Limits . and so Let Ramped step by Laplace transform [ ] ) ( ) ( ) ( , , , r r r r r r t t u t t t u t t F t t F t t t t F t F(t)--- = < < < = : excitation step" Ramped " F(t ) F0 tr t [ ] [ ] [ ] , and Since ) ( ) ( ) ( ) ( 1 ) ( 2 t u t f L e a t u a t f L s t tu L sa- =-- = [ ] ( 29 r r st r st r r r r e s t F s e s t F s F t t u t t t u t t F F(t)--- = - = β‡’--- = 1 1 ) ( ) ( ) ( ) ( 2 2 2 [ ] ( 29 r r st r st r r r r e s t F s e s t F s F t t u t t t u t t F F(t)--- = - = β‡’--- = 1 1 ) ( ) ( ) ( ) ( 2 2 2 (29 ) ( 2 lim lim 2 2 2 2 = + β†’ = β†’ β‹… n s s s A s ds d s A Ο– : , take , by multiply , For ( 29 (undamped) ) ( 1 ) ( ) ( ) ( 2 2 2 n r st s s mt e F s G s F s X r Ο– +- = =- n n n i s D i s C s B s A s s Ο– Ο– Ο– + +- + + = + 2 2 2 2 ) ( 1 : expansion fraction Partial 3 2 3 2 2 2 2 2 ) ( 1 lim 2 ) ( 1 lim 1 1 lim n n n n n n n n i i s s i s D i i s s i s C s s B Ο– Ο– Ο– Ο– Ο–...
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ASE 365 - Lecture 13 - Response by Laplace transform...

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