ME3514_Laplace_Transform_presentation

ME3514_Laplace_Transform_presentation - ME3514 Laplace...

This preview shows pages 1–10. Sign up to view the full content.

ME3514 Laplace Transform Topics Laplace transform Inverse Laplace Transform • Laplace Transform Table • Solving Differential Equations Using Laplace Transform • Final Value Theorem artial Fraction Expansion (by hand and using MATLAB) • Partial Fraction Expansion (by hand and using MATLAB) M. Remillieux 1

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

View Full Document
ME3514 Laplace Transform • Method to solve linear differential equations: What is it used for? EOM:      t f t cx dt t dx b dt t x d a 2 2 I.C.: 0 0 x x   0 0 0 x x dt t dx t How? • It converts the differential equation into an algebraic equation. M. Remillieux 2
ME3514 Laplace Transform Definition: • Assume f(t) is a function of time. Then, the Laplace transform is    0 st f tF s f te d t    where s is a complex variable • There exist an inverse Laplace transform, -1 -1 Another very complicated integral       t f s F __________ __________ • No need to solve the inverse Laplace transform integral. he pair is unique s ft M. Remillieux 3 The pair is unique     Fs

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

View Full Document
ME3514 Laplace Transform Does exist?     t f 3 conditions must be satisfied:   t f 1) for  0 t f 0 t t 2) Continuous on finite intervals of t>0 iecewise continuous)   t f (Piecewise continuous) 3) There exists a real positive constant σ such that,   0 lim t f e t t t M. Remillieux 4
ME3514 Laplace Transform How do you represent a function that is zero for negative time?   t f a) Graphically tan a  t b) Mathematically   1 t We will use this form where is the unit step function.  1( ) f ta t t  t 1 Another option is t 0 M. Remillieux 5 0 0 t ft at t

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

View Full Document
ME3514 Laplace Transform Example: Existence Check • Unit step Function t   ) ( 1 t t f   t f e 1 1 t t      0 lim lim t t t t e t f e Valid for any σ >0 M. Remillieux 6 exists  t f
ME3514 Laplace Transform xponential Function   1 () t f tA e t 0 with Example: Existence Check Exponential Function t   t f e A 1 t t        0 lim lim lim t t t t t t t Ae e Ae t f e Valid for any σ >0 M. Remillieux 7 exists  t f

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

View Full Document
ME3514 Laplace Transform xponential Function ) t t A e t ith Example: Existence Check Exponential Function   1( ) ft  0 with t   t f e A 1 t t        0 lim lim lim t t t t t t t Ae e Ae t f Ae Valid for σ > α M. Remillieux 8 exists  t f
ME3514 Laplace Transform xponential Function 2 ) t t e t Example: Existence Check Exponential Function   1( ) ft  t   t f e 1 1 t t      2 2 lim lim lim t t t t t t t t e e e t f e M. Remillieux 9

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

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

{[ snackBarMessage ]}

Page1 / 37

ME3514_Laplace_Transform_presentation - ME3514 Laplace...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online