laplace_transforms

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

## This note was uploaded on 02/07/2012 for the course ME 4231 taught by Professor Staff during the Fall '08 term at Minnesota.

Ask a homework question - tutors are online