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HW _8 MTR 500 Fall 2006

HW _8 MTR 500 Fall 2006 - (d 2 1 1 1 2 2 = s s s Y 8 Solve...

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MTR 500, Section 1 Fall 2006 Homework # 8 Dr. Ghaleb Husseini Name: ____________________________________ 1. Use Eq. 3-1 to show that Laplace transform of (a) 2 2 b) (s is sin ϖ ϖ ϖ + + - t e bt (b) 2 2 b) (s b s is t cos ϖ ϖ + + + - bt e 2. Find the Laplace transform x (t) = cosh (at) sin ) ( t ϖ by substituting for the hyperbolic function and then using Table 3.1. 3. An input function that is sometimes used to test the dynamics of processes is the so-called half-sine-wave pulse shown in the drawing. Here
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4. Calculate the Laplace transforms of the graphical input signals in the accompanying figure. 5. The start-up procedure for a batch reactor includes a heating step where the reactor temperature is gradually heated to the nominal operating temperature of 75 o C. The desired temperature profile T(t) is shown in the drawing. What is T (s)? 6. Using partial fraction expansion where required, find x (t) for (a) ) 4 )( 3 )( 2 ( ) 1 ( ) ( + + + + = s s s s s s X (b) ) 4 )( 3 )( 2 ( 1 ) ( 2 + + + + = s s s s s X (C ) 2 ) 1 ( 4 ) ( + + = s s s X (d) 1 1 ) ( 2 + + = s s s X
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7. Expand each of the following s-domain functions into partial fractions: (a) ) 1 ( ) 1 ( 6 ) ( 2 + + = s s s s Y (b) ) 9 ( ) 2 ( 12 ) ( 2 + + = s s s s Y © ) 6 )( 5 )( 4 ( ) 3 )( 2 ( )
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Unformatted text preview: (d) ) 2 ( ] 1 ) 1 [( 1 ) ( 2 2 + + + = s s s Y 8. Solve the following equation for y(t) using Laplace transform: 1 y(0) ) ( ) ( = = ∫ dt t dy d y t τ 9. For each of the following functions X(s), what can you say about x(t) (0 ≤ t ≤ ∞ ) without solving for x(t)? In other words, what are x(0) and x( ∞ )? Is x(t) converging or diverging? Is x(t) smooth or oscillatory? (a) ) 4 )( 20 9 ( ) 2 ( 6 ) ( 2 + + + + = s s s s s X (b) ) 2 )( 10 6 ( 3 10 ) ( 2 2 + +--= s s s s s X ( c ) 9 5 16 ) ( 2 + + = s s s X 3.13. Find the complete time-domain solutions for the following differential equations using Laplace transforms: (a) ) ( d , dt dx(0) 0, ) with x(0 4 2 2 3 3 = = = = + dt x e x dt x d t (b) x(0) 3 sin 12 = =-t x dt dx ( C) dt dx(0) 0, x(0) 25 6 2 2 = = = + +-t e x dt dx dt x d (d) ) ( x , ) ( sin 2 2 2 1 2 1 2 2 1 1 = = = +-=-+ x t x x dt dx t x x dt dx...
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