530_343lecture10

530_343lecture10 - ME 530.343 Design and Analysis of...

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Unformatted text preview: ME 530.343: Design and Analysis of Dynamic Systems Spring 2009 Lecture 10 - Introduction to the Laplace Transform Week of February 23, 2009 Today’s Objectives • Introduction to the Laplace Transform Reading: Chapter 19 The Laplace Transform Converts a function of a real variable (always t in this class) into a function of a complex variable, s . “Laplace transform of f ”: L [ f ( t )] = F ( s ) Time domain t → frequency domain s Defining equation: F ( s ) = ∞ f ( t ) e- st dt (Another convention is F ( s ) = ∞- f ( t ) e- st dt ) s is a complex quantity s = σ + iω (Note that some texts use j instead of i .) e- st = e- σt e- iωt Representing functions in the frequency domain Step Function f ( t ) = A for t > f ( t ) = 0 for t ≤ recall that F ( s ) = f ( t ) e- st dt L [ A ] = ∞ Ae- st dt 1 L [ A ] = Ae- st- s | t = ∞ t =0 L [ A ] = 0- A · 1- s L [ A ] = A s for a unit step function, U(t) L [ U ( t )] = 1 s Exponential function f ( t ) = e- at L [ e- at ] = ∞ e- at e- st dt L [ e- at ] = ∞ e- ( s + a ) t dt L [ e- at ] = e- ( s + a ) t- ( s + a ) | t = ∞ t =0 L [ e- at ] = 1 s + a Ramp function f ( t ) = t for t > L [ t ] = ∞ te- st dt Use formula for integration by parts b a udv = uv |...
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530_343lecture10 - ME 530.343 Design and Analysis of...

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