laplace_transforms

Laplace_transforms - Laplace Transforms Rajesh Rajamani ME 4231 Department of Mechanical Engineering University Of Minnesota DEFINITION Consider

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Unformatted text preview: Laplace Transforms Rajesh Rajamani ME 4231 Department of Mechanical Engineering University Of Minnesota DEFINITION Consider any piecewise continuous function f (t ) Its Laplace transform is F ( s ) f (t )e st dt 0 For which functions does Laplace transform exist ? All functions for which the integral exists 1 CALCULATING LAPLACE TRANSFORMS Need to analytically evaluate the integral Example: Laplace of a Step function u (t ) f (t ) u (t ) t F ( s ) u (t )e st dt 0 e st dt 0 e st s 0 1 0 s 1 s CALCULATING LAPLACE TRANSFORMS Example: Laplace of f (t ) e at F (s) f (t )e 0 st dt e ( s a )t dt 0 e ( s a )t ( s a) 0 1 0 sa 1 sa 2 IMPULSE FUNCTION Definition of the impulse (t ) lim t 0 u (t ) u (t t ) t u (t ) - u (t ) u (t t ) t t 1 t = u (t t ) 0 t t Take limits as t 0 t t IMPULSE FUNCTION Laplace of the impulse function u (t ) u (t t ) t 0 t (t ) lim u (t ) u (t t ) F ( s ) L lim t t 0 u (t ) u (t t ) lim L t 0 t lim 1 t 0 s t 1 e t s (ts ) 2 ( ts )3 1 1 ts 2! 3! ( ts ) 2 ( ts ) 3 lim 1 s1 t t 0 2! 3! 1 lim 1 t 0 st 3 TABLE OF LAPLACE TRANSFORMS Function Laplace Transform f (t ) F (s ) Step function 1 s u (t ) Impulse function (t ) lim t 0 u (t ) u (t t ) t e at 1 1 sa TABLE OF LAPLACE TRANSFORMS sin(t ) cos(t ) e at sin(t ) e at cos(t ) s2 2 s s2 2 (s a) 2 2 sa (s a) 2 2 e at f (t ) F (s a) k1 f1 (t ) k 2 f 2 (t ) k1 F1 (s) k 2 F2 ( s) 4 TABLE OF LAPLACE TRANSFORMS df dt d2 f dt 2 sF ( s ) f (0) s 2 F ( s) sf (0) f ' (0) 0 t f (t )dt F (s) s f (t )dt s t G ( s ) H ( s ) where G (s ) and 0 H (s ) are Laplace of g (t ) g (t )h( )d and h(t ) APPLICATIONS Why should we learn Laplace transforms ? Can be used to find analytical solutions to ODEs Response to initial conditions Response to external inputs Most Important: We need to know Laplace transforms in order to learn Bode plots 5 ...
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This note was uploaded on 02/07/2012 for the course ME 4231 taught by Professor Staff during the Fall '08 term at Minnesota.

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