Math245_SimulProject_Part2_Fall08

# Math245_SimulProject_Part2_Fall08 - a = ¼ 1 4 and as in...

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Fall 2008 Math 245 Simulation Pro ject: Part 2. Date Due : Tuesday, Dec. 2 Note: An understanding of the response of 1 st -order and 2 nd -order systems to unitary inputs is fundamental to natural and controlled processes in many systems of physical relevance. The purpose of this project is force all students to encounter a “personally-generated” visual picture of these basic responses as a function of the key parameters. Problem #1. Consider the 1 st -order ODE ( 1) ( ) dy ay u t u t c dt + = - - - with the initial condition (0) 0 y = and with parameters a and c . a) Simulate the equation with 1 a = and three different values of 2,4,6 c = ( i.e. , perform three simulations with different values of the parameter c). Plot the function ( ) y t for all three simulations on the same graph y vs. t . b) Repeat the simulation with 4 c = and vary the parameter a using the three different values

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Unformatted text preview: a = ¼, 1, 4 and, as in part a), plot the function ( ) y t for all of these simulations on another graph of y vs. t . c) Find the analytical solution by use of the Laplace transform for arbitrary values of the parameters a and c . Problem #2. Simulate the 2 nd-order ODE 2 2 2 ( 1) d y dy y u t dt dt λ + + =-subject to the initial conditions (0) '(0) y y = = . You should note that the time has been scaled such that the natural frequency is ω = 1 ( i.e. , a dimensionless time is used equivalent to τ = ωt ). a) Simulate the equation for the values of the parameter = 0.1, 0.2, 0.4, 0.7, 1.0, 1.5 , and plot the results on a single graph of y vs. t . b) Find the analytical solution by use of the Laplace transform for arbitrary values of the parameter λ ....
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