KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec8

KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec8

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1 Chapter 3 Solution of ODEs by Laplace Transforms Procedure: 1. Take the L of both sides of the ODE. 2. Rearrange the resulting algebraic equation in the s domain to solve for the L of the output variable, e.g., Y ( s ). 3. Perform a partial fraction expansion. 4. Use the L -1 to find y ( t ) from the expression for Y ( s ).
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2 Chapter 3
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3 Chapter 3 Example 3.1 Solve the ODE, ( 29 5 4 2 0 1 (3-26) dy y y dt + = = First, take L of both sides of (3-26), ( 29 ( 29 ( 29 2 5 1 4 sY s Y s s - + = Rearrange, ( 29 ( 29 5 2 (3-34) 5 4 s Y s s s + = + Take L -1 , ( 29 ( 29 1 5 2 5 4 s y t s s - + = + L From Table 3.1, ( 29 0.8 0.5 0.5 (3-37) t y t e - = +
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4 Chapter 3 Example 2: Example 2: system at rest (s.s.) Step 1 Take L.T. (note zero initial conditions) 0 0 0 0 0 0 2 4 6 11 6 2 2 3 3 at t= dt du )= ( y )= ( y )= y( u dt du y dt dy dt y d dt y d = + = + + + 3 2 6 11 6 ( ) 4 2 s Y(s)+ s Y(s)+ sY(s) Y s = sU(s) + U(s) + To find transient response for u(t) = unit step at t > 0 1. Take Laplace Transform (L.T.) 2. Factor, use partial fraction decomposition 3. Take inverse L.T.
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5 Chapter 3 Rearranging, Step 2a . Factor denominator of Y(s) Step 2b . Use partial fraction decomposition Multiply by s, set s = 0 input) ( 1 1
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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec8

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