Extra_homework_problems_Laplace_Solution

Extra_homework_problems_Laplace_Solution - 3.E – 1...

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Unformatted text preview: 3.E – 1 Laplace transform: Extra homework problems 1) Assume that the following system of two differential equations is given: u x x dt dx u x dt dx 1170 . 1 2381 . 2 8333 . 7 4048 . 2 2 1 2 1 1-- = +- = Furthermore y = x 2 . The initial conditions are x 1 (0) = x 2 (0) = 0. a) Demonstrate that the two equations can be rearranged to one second-order differential equation. Hint: First solve the second equation for x 1 , then take the derivative and substitute in the first equation. b) Apply the Laplace transform to the second-order differential equation obtained in a), and solve for Y(s). The initial conditions are: y(0) = dy(0)/dt = 0 and u(0) = 0. c) Considering that U(s) is a unit step input – i.e. U(s) = 1/s – find the correct expression for y(t). d) Calculate the value of y(t) for ∞ → t by applying the final value theorem. a) We first rearrange the second equation: u x dt dx x 1170 . 1 2381 . 2 8333 . 2 2 1 + + = + + = u x dt dx x 1170 . 1 2381 . 2 8333 . 1 2 2 1 u x dt dx x 3405 . 1 6858 . 2 20 . 1 2 2 1 + + = We then introduce the resulting expression for x 1 in the first equation: u u x dt dx dt u x dt dx d 7 3405 . 1 6858 . 2 20 . 1 4048 . 2 3405 . 1 6858 . 2 20 . 1 2 2 2 2 + + +- = + + u u x dt dx dt du dt dx dt x d 7 2235 . 3 4589 . 6 8858 . 2 3405 . 1 6858 . 2 20 . 1 2 2 2 2 2 2 +--- = + + ( ) ( ) u dt du x dt dx dt x d 2235 . 3 7 3405 . 1 4589 . 6 8858 . 2 6858 . 2 20 . 1 2 2 2 2 2- +- = + + + u dt du x dt dx dt x d 7765 . 3 3405 . 1 4589 . 6 5716 . 5 20 . 1 2 2 2 2 2 +- = + + u dt du x dt dx dt x d 1471 . 3 1170 . 1 3824 . 5 6430 . 4 2 2 2 2 2 +- = + + Or the 2 nd order differential equation can be rewritten as: u dt du y dt dy dt y d 1471 . 3 1170 . 1 3824 . 5 6430 . 4 2 2 +- = + + b) Taking the Laplace transform of each term (superposition principle), and considering that the initial conditions are: y(0) = dy(0)/dt = 0 and u(0) = 0, we obtain: ( ) ( ) ( ) ( ) ( ) s U s U s s Y s Y s s Y s ⋅...
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This note was uploaded on 11/20/2009 for the course CHME DTU-abroad taught by Professor Rafiqulgani during the Spring '09 term at Rensselaer Polytechnic Institute.

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Extra_homework_problems_Laplace_Solution - 3.E – 1...

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