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Unformatted text preview: Math245 Final Project  Part I Due: Thursday, Apr 16 Objective: An understanding of the response of 1storder and 2ndorder 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 “personallygenerated” visual picture of these basic responses as a function of the key parameters. Problem 1
Consider the 1storder ODE dy + ky = u(t − 1) − u(t − c) dt with the initial condition y (0) = 0 and with parameters k and c. (a) Simulate the equation with k = 1 and three diﬀerent values of c = 2, 4, 8 (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 c = 4 and vary the parameter k using the three diﬀerent values k = 1/2, 1, 3 and plot the function y (t) for all of these simulations on another single graph of y vs. t. (c) Find the analytical solution by use of the Laplace transform for arbitrary values of the parameters k and c. Leave the answer in terms of k and c. Problem 2
Consider the 2ndorder ODE d2 y dy + 2λ + y = u(t − 1) 2 dt dt subject to initial conditions y (0) = y (0) = 0. (a) Simulate the equation for the values of the parameter λ = 0.1, 0.2, 0.4, 0.7, 1.0, 1.3, 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 (< 1). Leave the answer in terms of λ. General Requirements
• Attach Matlab programs along with the simulation results. • Analytical problems must be solved without use of the computer algebra system (Maple or Mathematica). • Students are expected to work on the problems independently. As for the ﬁnal project, therefore, collaborative working is not permitted. • Late turnin is not accepted unless otherwise excused to the instructor before the due date. ...
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This note was uploaded on 06/01/2009 for the course MATH 245 taught by Professor Alexander during the Spring '07 term at USC.
 Spring '07
 Alexander
 Math

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