1. Engineering Analysis I
Laplace Transforms (LT)
Dr. Omar R. Daoud

2. Laplace Transforms (L.T.)
Introduction
Laplace Transforms (L.T.)
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3. Laplace Transforms (L.T.)
Introduction:
Laplace Transform in Engineering Analysis
Laplace Transforms (L.T.)
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4. Laplace Transforms (L.T.)
Introduction:
Laplace Transform in Engineering Analysis
Laplace Transforms (L.T.)
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5. Laplace Transforms (L.T.)
Table
Laplace Transforms (L.T.)
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6. Laplace Transforms (L.T.)
Table
Laplace Transforms (L.T.)
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7. Laplace Transforms (L.T.)
Table
Laplace Transforms (L.T.)
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8. Laplace Transforms (L.T.)
Table
Laplace Transforms (L.T.)
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9. Example. A force in Newton's (N) is given below
Example. A force in Newton's (N) is given below. Determine the Laplace transform.
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10. Example. A voltage in volts (V) starting at t = 0 is given below
Example. A voltage in volts (V) starting at t = 0 is given below. Determine the Laplace transform.
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11. Example. A pressure in pascals (p) starting at t = 0 is given below
Example. A pressure in pascals (p) starting at t = 0 is given below. Determine the Laplace transform.
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12. Inverse Laplace Transforms by Identification
When a differential equation is solved by Laplace transforms, the solution is obtained as a function of the variable s.
The inverse transform must be formed in order to determine the time response.
The simplest forms are those that can be recognized within the tables and a few of those will now be considered.
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13. Example. Determine the inverse transform of the function below.
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14. Example. Determine the inverse transform of the function below.
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15. Real poles of first order.
The roots of D(s) are called poles and they may be classified in four ways.
Real poles of first order.
Complex poles of first order (including purely imaginary poles)
Real poles of multiple order
Complex poles of multiple order (including purely imaginary poles)
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16. Partial Fraction Expansion Real Poles of First Order
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17. Example. Determine inverse transform of function below.
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18. Cont. Example.
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19. Example. Determine exponential portion of inverse transform of function below.
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20. Cont. Example.
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21. Partial Fraction Expansion for First-Order Complex Poles
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22. Example. Complete the inverse transform of the previous Example.
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23. Cont. Example.
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24. Second-Order Real Poles
Assume that F(s) contains a denominator factor of the form (s+)2. The expansion will take the form shown below.
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25. Example. Determine inverse transform of function below.
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26. Example 10-12. Continuation.
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27. Laplace Transforms (L.T.)
Steps involved in using the Laplace transform.
Laplace Transforms (L.T.)
Transform it to algebraic equation
Solve it
Determine the inverse transform
Solution
Differential Equation
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28. Significant Operations for Solving Differential Equations
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29. Procedure for Solving DEs
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30. Procedure for Solving DEs
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31. Example. Solve DE shown below.
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32. Cont. Example.
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33. Example. Solve DE shown below.
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34. Cont. Example.
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35. Cont. Example.
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36. Example. Solve DE shown below.
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37. Cont. Example.
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38. Example. Solve DE shown below.
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39. Cont. Example.
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40. Cont. Example.
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