Mechanical Control System

Overview

Problem 1 Consider the ODE and IC: 𝑥̈ + 2𝑥̇ + 5𝑥 = 0,

𝑥(0) = 1, 𝑥̇(0) = 0. a) Use the trial solution method to calculate the

solution 𝑥(𝑡) analytically. b) Use Matlab to plot the analytical solution 𝑥(𝑡)

over the interval 𝑡 ∈ [0,6].

c) Calculate the solution numerically in Matlab and plot it in the same graph as the analytical solution. Submit the figure and your codes.

Problem 2.

Consider the ODE and IC: 𝑥̈ + 2𝑥̇ + 5𝑥 = 1 + 5𝑡 + 2𝑡2,

𝑥(0) = 1, 𝑥̇(0) = 0.

a) Use the trial solution method to calculate the homogeneous solution 𝑥H(𝑡) , use the method of undetermined coefficients to calculate the particular solution 𝑥P(𝑡) , and apply the ICs to obtain the solution 𝑥(𝑡) analytically.

b) Use Matlab to plot the analytical solution 𝑥(𝑡) over the interval 𝑡 ∈ [0,2].

c) Calculate the solution numerically in Matlab and plot it in the same graph as the analytical solution. Submit the figure and your codes. Problem 3

Consider the ODE and IC: 𝑥̇ + 3𝑥 = 𝑒−𝑡,

𝑥(0) = 2. a) Use the trial solution and variation of constants

methods to determine the solution 𝑥(𝑡) analytically. b) Use Matlab to plot the analytical solution 𝑥(𝑡)

over the interval 𝑡 ∈ [0,2].

c) Calculate the solution numerically in Matlab and plot it in the same graph as the analytical solution. Submit the figure and your codes.

Problem 4 Consider the ODE: 𝑥⃛ + 3 𝑥 ̈ + 9 𝑥 ̇ − 1 3 𝑥 = 0 .

a) Formulate the characteristic equation, determine the characteristic roots, and illustrate their locations in the complex plane.

b) Determine the stability of the system and determine whether solutions could oscillate.

c) Based on the characteristic roots, provide the solution of the ODE.