Unformatted text preview: Math245 Final Project - Part I Due: Thursday, Apr 16 Objective: An understanding of the response of 1st-order and 2nd-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 1st-order 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 2nd-order 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 turn-in is not accepted unless otherwise excused to the instructor before the due date. ...
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