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Colorado School of Mines CHEN403 Laplace Transforms ...

Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 1 - Copyright 2017 April 23, 2017 Laplace Transforms Laplace Transforms .. 1 Definition of the transform .. 2 Properties of Laplace transform .. 2 Transforms of Simple Functions .. 3 Translation of Transforms .. 5 Transforms of Derivatives .. 6 Transforms of Integrals .. 6 Final-Value & Initial-Value Theorems .. 7 Inverse Laplace transform Conversion to Partial Fractions .. 8 Collection of Terms .. 9 Specific Values of s .. 11 Heaviside Expansion Method .. 12 More Complicated Partial Fractions Repeated Roots .. 13 Heaviside Expansion for Repeated Roots .. 14 Specific s Values for Repeated Roots .. 14 More Complicated Partial Fractions Quadratic & Higher Order Terms.

Colorado School of Mines CHEN403 Laplace Transforms . = = = . ¦ ¦ ¦

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Transcription of Colorado School of Mines CHEN403 Laplace Transforms ...

1 Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 1 - Copyright 2017 April 23, 2017 Laplace Transforms Laplace Transforms .. 1 Definition of the transform .. 2 Properties of Laplace transform .. 2 Transforms of Simple Functions .. 3 Translation of Transforms .. 5 Transforms of Derivatives .. 6 Transforms of Integrals .. 6 Final-Value & Initial-Value Theorems .. 7 Inverse Laplace transform Conversion to Partial Fractions .. 8 Collection of Terms .. 9 Specific Values of s .. 11 Heaviside Expansion Method .. 12 More Complicated Partial Fractions Repeated Roots .. 13 Heaviside Expansion for Repeated Roots .. 14 Specific s Values for Repeated Roots .. 14 More Complicated Partial Fractions Quadratic & Higher Order Terms.

2 15 Repeated Roots .. 15 Complex Conjugate Roots .. 16 Inverting Quadratic Term Completing the Square .. 19 Steps for Completing the Square .. 19 Example #1 .. 20 Numerical Root Finding & Partial 21 All Real Roots .. 21 Complex Conjugate Roots .. 23 Solution of Initial Value ODEs .. 27 Solution of Systems of Initial Value ODEs .. 28 Translation of transform .. 31 Further Examples Finding transform of Piece-Wise Driving Function .. 32 Further Examples Inverse transform With Time Delay .. 34 Further Examples Arbitrary Driving Function in ODE .. 36 Use of Mathematical Software to Work with Laplace Transforms .. 39 Mathematica .. 39 MATLAB .. 42 Microsoft Excel .. 46 POLYMATH .. 50 Laplace Transforms provide an efficient way to solve linear differential equations with constant coefficients.

3 These are traditional method to solve process control problems. Use algebra instead of calculus. Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 2 - Copyright 2017 April 23, 2017 Procedure: ftSolution in t domain fsSolution in s domain Definition of the transform Definition: 0stf sf tf t edtL where L is the Laplace operator. Converting back to the t domain requires the inverse Laplace operator, -1L: 1f tf sL. The inverse Laplace operator does have an integral definition: 112istif tf sf s e dsi L where i is 1 and is some real constant that exceeds the real part of all singularities of ()fs. We will not be dealing directly with this inversion formula in this class but will instead be using simplified techniques.

4 Properties of Laplace transform Properties of the Laplace Transforms : fs contains no information for 0t. fs does not necessarily exist for all ft. This should not be a problem for our work here. Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 3 - Copyright 2017 April 23, 2017 L is linear: 1212af tbf taf tbf tLLL L removes the t variable by integration but replaces it with s. Proof of linearity: 12120120012stststaf tbf taf tbf tedtaf t edtbf t edtaf tbf tLLL Example of function with transform let atf te where 0a. fs only exists when 0as, since: 00a s tatatstee edtedtL Transforms of Simple Functions Common Transforms shown in the text book.

5 There are 99 functions listed on pp. 296-299 of Handbook of Mathematical Formulas and Integrals. The most needed in this class are part of these notes. Let s derive some of them. First, we need to remember how to integrate by parts. Since: d uvudvvduudvd uvvdu then, the integration gives us: udvd uvvduudvuvvdu and: bbbaaaudvuvvdu Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 4 - Copyright 2017 April 23, 2017 Unit Step 00110tf tf sts 0001111011stststf sedtedstsesss (t) Ramp 20010tf tf stts 02022111011ststf st edtestsss (t) Quadratic 230020tf tf stts 202022000021111200212stststststf stedtt d est eed tsstedtsssss (t) Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 5 - Copyright 2017 April 23, 2017 Unit pulse 0011/00 Astef tAtAf sAstA 000111stAstAstAsf sf t edtedtAeAseAs (t) Unit impulse (requiring the use of L Hopital s rule) 00011limlimlim1 AsAAAsAef tf sAAsses (t)

6 Can proceed onward to show that, in general: 100!0nntnf tf stts Other formulas can be derived (some easily, some not). Translation of Transforms 1atatef tf saf saef tLL This most useful when doing the inverse translation. Examples: 22cosstsL so 22cosatsaetsaL Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 6 - Copyright 2017 April 23, 2017 21fssa since 21tsL, then 121attesaL Transforms of Derivatives 0df tsf sfdtL 22200td f tdfs f ssfdtdtL 323232000ttd f tdfd fs f ss fsdtdtdtL 2112210000nnnnnnnnntttdf tdfdfdfs f ssfssdtdtdtdtL Note that we can change an n-th order ODE in t into an n-th order polynomial in s.

7 This is why Laplace Transforms have been used. Also note that for an n-th order ODE in t we need n initial conditions: 0f, 0/tdf dt, .., 110/nntdf dt. Proof of transform of 1st derivative using integration by parts: 00000000stststststdf tdf tedtdtdtedf tef tf t d efsf t edtfsf sL Transforms of Integrals 01tfdf ssL Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 7 - Copyright 2017 April 23, 2017 Proof: 00000000000001111111001ttsttstttstststfd fdedtfdd esefdedfdssefdfdf t edtssfsssfssL Final-Value & Initial-Value Theorems 0limlimtsf ts f s for fs that exists over the entire range of 0s.

8 0limlimtsf ts f s With these two theorems, we can determine the limiting values of a function only knowing the Laplace transform . Example #1: 6123sfss sss 066limlim11236tssftsss 201616limlimlim0123123111tsssssftssssss Example #2: Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 8 - Copyright 2017 April 23, 2017 345123sssfss sss 034560limlim101236tssssftsss 0345111345limlimlim1123123111tssssssssft ssssss Example #3: 345123sssfss sss 0345111345limlimlim1123123111tssssssssft ssssss We would think that we could determine the final value as: 034560limlim101236tssssftsss.

9 However, the final value theorem cannot be applied since fs does not exist at 3s, so this equation cannot be applied. This is more readily apparent from the form of the function transformed to the t domain: 23259328lim2105ttttf teeef t. Inverse Laplace transform Conversion to Partial Fractions When doing the math in this class we will generally want to invert Laplace expressions that are ratios of polynomials: 22102210 QPmmmnnnsb sb sb sbfssa sa sa sa. Colorado School of Mines CHEN403 Laplace Transforms John Jechura - 9 - Copyright 2017 April 23, 2017 We will invert the Laplace Transforms by matching forms in tables. The functions we want to invert are generally more complicated than those listed in the reference tables.

10 We can do this by splitting up the Laplace function in terms of partial fractions, each of which can be matched up to the tables. There are three techniques to break the Laplace function apart into partial fractions: 1. Collection of Terms 2. Specific Values of s. 3. Heaviside Expansion. The procedure can be shown with a series of examples. Let s start with: 21212 ABCfss sssss where the A, B, and C constants to be determined so that they give an identity for any and all values of s. Once in the simpler form the Laplace function can easily be inverted as: 1111221212ttABCfts sssssf tA BeCeLLLL Note that we are going in the opposite direction of what we more typically do in algebra we want to end up with series of fractions with different denominators instead of combining in a single fraction with a common denominator.


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