Engineering Analysis I Laplace Transforms (LT) Dr. Omar R. Daoud
Laplace Transforms (L.T.) Introduction Laplace Transforms (L.T.) 6/21/2012 Part III
Laplace Transforms (L.T.) Introduction: Laplace Transform in Engineering Analysis Laplace Transforms (L.T.) 6/21/2012 Part III
Laplace Transforms (L.T.) Introduction: Laplace Transform in Engineering Analysis Laplace Transforms (L.T.) 6/21/2012 Part III
Laplace Transforms (L.T.) Table Laplace Transforms (L.T.) 6/21/2012 Part III
Laplace Transforms (L.T.) Table Laplace Transforms (L.T.) 6/21/2012 Part III
Laplace Transforms (L.T.) Table Laplace Transforms (L.T.) 6/21/2012 Part III
Laplace Transforms (L.T.) Table Laplace Transforms (L.T.) 6/21/2012 Part III
Example. A force in Newton's (N) is given below Example. A force in Newton's (N) is given below. Determine the Laplace transform. 6/21/2012 Part III
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. 6/21/2012 Part III
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. 6/21/2012 Part III
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. 6/21/2012 Part III
Example. Determine the inverse transform of the function below. 6/21/2012 Part III
Example. Determine the inverse transform of the function below. 6/21/2012 Part III
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) 6/21/2012 Part III
Partial Fraction Expansion Real Poles of First Order 6/21/2012 Part III
Example. Determine inverse transform of function below. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
Example. Determine exponential portion of inverse transform of function below. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
Partial Fraction Expansion for First-Order Complex Poles 6/21/2012 Part III
Example. Complete the inverse transform of the previous Example. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
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. 6/21/2012 Part III
Example. Determine inverse transform of function below. 6/21/2012 Part III
Example 10-12. Continuation. 6/21/2012 Part III
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 6/21/2012 Part III
Significant Operations for Solving Differential Equations 6/21/2012 Part III
Procedure for Solving DEs 6/21/2012 Part III
Procedure for Solving DEs 6/21/2012 Part III
Example. Solve DE shown below. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
Example. Solve DE shown below. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
Example. Solve DE shown below. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
Example. Solve DE shown below. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III
Cont. Example. 6/21/2012 Part III