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# Transform - ME375 Handouts Laplace Transform Motivation...

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ME375 Handouts 1 School of Mechanical Engineering Purdue University ME375 Laplace - 1 Laplace Transform Motivation Laplace Transform – Review of Complex Numbers – Definition – Time Domain vs s -Domain – Important Properties Inverse Laplace Transform Solving ODEs with Laplace Transform

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ME375 Handouts 2 School of Mechanical Engineering Purdue University ME375 Laplace - 2 Motivation Solving Differential Eq. Y ( s ): Solution in Laplace Domain y ( t ): Solution in Time Domain Differential Equations (ODEs) + Initial Conditions (ICs) ( Time Domain ) Algebraic Equations ( s-domain ) L [ ] L 1 [ ]
ME375 Handouts 3 School of Mechanical Engineering Purdue University ME375 Laplace - 3 The Many Faces of a Complex Number: – Coordinate Form : – Phasor (Euler) Form : Moving Between Representations – Phasor (Euler) Form Coordinate Form – Coordinate Form Phasor (Euler) Form Review of Complex Numbers z x jy z Ae A j j (cos sin ) 1 22 1 1 tan ( ) when is in the 1st or 4th quadrant atan2( , ) tan ( ) when is in the 2nd quadrant atan2( , ) tan ( ) when is in the 3rd quadrant y x y x y x z A x y y x z yx z sin cos cos j e A y A x j Real Img. z

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ME375 Handouts 4 School of Mechanical Engineering Purdue University ME375 Laplace - 4 Definition Laplace Transform – One Sided Laplace Transform where s is a complex variable that can be represented by s = j and f ( t ) is a continuous function of time that equals 0 when t < 0 . Laplace Transform converts a function in time t into a function of a complex variable s . Inverse Laplace Transform 0 ( ) ( ) ( ) st F s L f t f t e dt 1 1 ( ) ( ) ( ) 2 cj st f t L F s F s e ds j
ME375 Handouts 5 School of Mechanical Engineering Purdue University ME375 Laplace - 5 s-Domain vs Time Domain • Two Representations of a Signal (System Response) The response or input of a system can have two representations: – Time Domain f ( t ) Represents the value of the response at time t , which is a function of time.

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Transform - ME375 Handouts Laplace Transform Motivation...

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