{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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...

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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 ( )
This is the end of the preview. Sign up to access the rest of the document.

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) ) ( ) ( = = âˆ« dt t dy d y t Ï„ 9. For each of the following functions X(s), what can you say about x(t) (0 â‰¤ t â‰¤ âˆž ) without solving for x(t)? In other words, what are x(0) and x( âˆž )? 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...
View Full Document

{[ snackBarMessage ]}