Engineering Applications

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Engineering Applications

• Dr. Darrin Leleux
• Lecture 19 Laplace Transforms
• Chapter 9

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Introduction

• 9.1 Definition and Properties of the Laplace
  Transform
• 9.2 Computation of Inverse Laplace Transforms
• 9.3 MATLAB and Laplace Transforms
• 9.4 Applications to Differential Equations
• 9.5 Application of Laplace Transforms to Linear
  Systems
• 9.6 Relationship of Fourier and Laplace
  Transforms
• 9.7 Summary of Laplace Transform Properties

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Applications of Laplace Transforms
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Definition and Properties

• Fourier Transform used to analyze causal
  functions (t 0)

Laplace Transform
Inverse Laplace Transform
Example 9.1, page 419
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Properties of Laplace Transform
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Properties of Laplace Transform
Example 9.2, page 421
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Example Laplace Transforms
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Inverse Laplace Transform

• The primary mathematical use of the Laplace
  transform is to solve differential equations.

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Partial Fractions
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Inverse Transforms for Rational Functions
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Partial Fraction Expansions
Simple Factor
Example 9.3, page 427
Repeated Factor
Example 9.4, page 428
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Factors with Complex Roots
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Fix in book
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MATLAB Commands for Laplace analysis
Example 9.6, page 432
Example 9.7, page 433
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Solving Linear Differential Equations
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Laplace transform of the integral
Example 9.8 9.9, page 436
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Laplace Solution

• Solves initial-value problems uniquely
• Determines both the complementary and particular
  solutions

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Transfer Function Properties

• Is a mathematical model of the D.E. describing a
  system
• Depends only on the system it is independent of
  the input function or any initial conditions
• Possible to determine the transfer function of a
  system experimentally from the input and output
  functions

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Special Functions

• Input functions used to characterize systems
• Heavyside Unit Step Function (pg. 439)
• Impulse Function (pg. 439-440)
• Dirac Delta functionThe impulse is the
  derivative of the unit step response.

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Dirac Delta Function
Impulse function
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Laplace Transforms of Special Functions
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Impulse and Step Response
Example 9.10 Pg 442
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Linear Systems Analysis
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Laplace and Fourier pairs
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Properties of Laplace Transform