ME3514_Laplace_Transform_presentation

ME3514_Laplace_Transform_presentation - ME3514 Laplace...

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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
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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
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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
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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
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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
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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
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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
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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
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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
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ME3514_Laplace_Transform_presentation - ME3514 Laplace...

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