%282%29%20Modeling

(2 Modeling - Modeling Chapter 2 Laplace Transform Review Laplace Transform Review Why Many engineering systems are represented mathematically by

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Modeling Chapter 2
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
aplace Transform Review Laplace Transform Review hy? Why? Many engineering systems are represented mathematically by differential equations. Differential equations are difficult to model as a block diagram. ut e refer stem presentation y lock iagram But we prefer system representation by block diagram where there are distinct inputs-outputs and separate parts. utput System G(s) Input output R ( s ) C ( s ) Laplace Transform 2
Background image of page 2
Laplace Transform Review Example of Modeling ( acceleromter within rocket-propelled sled, 6345 mph, - US Air Force ) bk yyy M x M M   (c) 2010 Farrokh Sharifi 3
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
aplace Transform Review Laplace Transform Review hen? When? When coefficients of differential equations (model parameters) are LINEAR, TIME-INVARIANT (LTI). 11 10 () nn mm nm dct d ct drt d rt aa a c t b b b r t dt dt dt dt    Many systems are not LTI but can be simplified to LTI stems systems. In this course we will focus on LTI systems. utput Input output 4 (c) 2010 Farrokh Sharifi
Background image of page 4
aplace Transform Review Laplace Transform Review he aplace ransform efined The Laplace Transform is defined as: where , is called the Laplace transform f jw s   ) ( s F ) ( t f of . Question: what does the notation in the lower limit mean? e? The Inverse Laplace Transform which allows to find iven given is: where: ) ( t f ) ( s F 5 (c) 2010 Farrokh Sharifi
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Laplace Transform Review e n erive ble r We can derive a table for conventional functions: amp l e d l f f t Example : find the Laplace Transform of ) ( ) ( t u Ae t f at 6 (c) 2010 Farrokh Sharifi
Background image of page 6
Laplace Transform Review Laplace Transform Theorems: 1 0 1 (0 ) (0 ) 0 ) ff df  0 () t f dt   7
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
aplace Transform Review Laplace Transform Review xample ind e verse aplace ansform f Example : Find the inverse Laplace transform of: ased n ble e verse f e aplace ansfer 2 1 ) 3 ( 1 ) ( s s F Based on table 2.1, the inverse of the Laplace transfer function is . ) ( ) 1 ( ) ( 2 1 t tu s L t f 2 1 )) ( ( ) ( s t f L s F However, we have which can be written as: where a =3. 2 1 ) 3 ( 1 ) ( s s F 2 ) ( 1 ) ( a s a s F From table 2.2 we know the inverse transform of ence ) ( a s F at 1 at is , hence: ) ( t f e ) ( )) ( ( 1 t tu e s F L 8 (c) 2010 Farrokh Sharifi
Background image of page 8
aplace Transform Review Partial fraction expansion Convert complicated functions into a sum of simpler terms for which Laplace Transform Review we know the Laplace transform. Always make sure the order of nominator is less than denominator. Otherwise divide and make it less. ) () s Gs Gs For example: Use partial fraction 12 3 () Gs Gs Gs expansion Apply known aplace transform 1123 ) Fs Gs G s Gs g tg t G sG s  Laplace transform equires additional 123 1 2 3 ( () L  Requires additional expansion 9 (c) 2010 Farrokh Sharifi
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
aplace Transform Review ase oots f e enominator f re al Laplace Transform Review Case 1: Roots of the Denominator of F ( s ) are real and Distinct : () ) ( ) ( ) Ns Fs s spsp sp  Algorithm :
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/20/2011 for the course MEC 709 taught by Professor Sharifi during the Fall '11 term at Ryerson.

Page1 / 105

(2 Modeling - Modeling Chapter 2 Laplace Transform Review Laplace Transform Review Why Many engineering systems are represented mathematically by

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online