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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, well 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 + + = + + = : 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 splane, 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 timeshifted 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  + = = = = = = = = + = = : 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) =  =  = 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) =  =  = 1 1 ) ( ) ( ) ( ) ( 2 2 2 (29 ) ( 2 lim lim 2 2 2 2 = + = n s s s A s ds d s A : , 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 + = = n n n i s D i s C s B s A s s + + + + = + 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...
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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.
 Spring '10
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