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Unformatted text preview: 1 AE 383 SYSTEM DYNAMICS Lecture Notes 2 (July 22 nd 2006) Laplace Transform Transformations are frequently used to simplify an analytical problem. Taking logarithms, for example, is a transformation that transforms the multiplication of two numbers into addition. Laplace transformation transforms a Linear Time Invariant Ordinary Differential Equation into an Algebraic equation. Let the time function f(t) = 0 for t < 0. Laplace transform of f(t) is defined as The reverse process of finding the time function f(t) from F(s) is called the inverse Laplace Transform The Laplace Transform of f(t) exists if the Laplace integral converges. The mathematical condition for the integral to exist is that f(t) is sectionally continuous in every finite interval in t > 0 and it is of exponential order. This means that e- σ t |f(t)| approaches zero as t b ∞ If the functions increases faster than the exponential function, such as 2 t e and 2 t te , Laplace Transform does not exist. If both f 1 (t) and f 2 (t) are Laplace Transformable then [ ] [ ] [ ] ) ( ) ( ) ( ) ( 2 1 2 1 t f L t f L t f t f L + = + Exponential Function: ≥ =- ) ( t for Ae t for t f t α p A and α are constants. α α + =- s A Ae L t ] [ (Exercise) Step Function: ≥ = ) ( t for A t for t f p s A A L = ] [ (Step is a special case of exponential with α = 0) 2 Unit Step Function: ≥ =- 1 ) ( 1 t for t for t t p Ramp Function: ≥ = ) ( t for At t for t f p 2 ] [ s A At L = Sinusoidal Function: ≥ = sin ) ( t for wt A t for t f p Note that sinjwt = (e jwt- e-jwt ) / 2 j 2 2 1 1 2 ) ( 2 ] sin [ w s Aw jw s jw s j A dt e e e j A wt A L st jwt jwt...
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