University of Notre Dame
Aerospace and Mechanical Engineering

AME 302: Differential Equations, Vibrations and Control
Homework 3

B. Goodwine Spring, 2004

Issued: February 9, 2004 Due: February 13, 2004

Unless otherwise indicated, all problems are from Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, seventh edition.

1. Consider
   
   $$A = \left[ \begin{array}{cc} -2 & 1 \ 1 & -2 \end{array}\right].$$
   
   
   1. Find the general solution to
      
      $$\dot \xi = A \xi.$$
      
      

   2. Find the solution to the previous equation if
      
      $$\xi(0) = \left[ \begin{array}{c} 1 1 \end{array}\right].$$
      
      

2. Consider
   
   $$A = \left[ \begin{array}{cc} 4 & -3 \ 8 & -6 \end{array}\right].$$
   
   
   1. Find the general solution to
      
      $$\dot \xi = A \xi.$$
      
      
      Note: this has a zero eigenvalue; however, the solution technique is as you would expect.
   2. Find the solution to the previous equation if
      
      $$\xi(0) = \left[ \begin{array}{c} 1 1 \end{array}\right].$$
      
      

3. Consider
   
   $$A = \left[ \begin{array}{cc} 2 & -5 \ 1 & -2 \end{array}\right].$$
   
   
   1. Find the general solution to
      
      $$\dot \xi = A \xi.$$
      
      

   2. Find the solution to the previous equation if
      
      $$\xi(0) = \left[ \begin{array}{c} 1 1 \end{array}\right].$$
      
      

4. Section 7.8, numbers 7, 8 and 10.

5. Consider the spring-mass system illustrated in the following figure. Let $k=m=1$.
   1. Derive the equations of motion for the system.
   2. Assuming two solutions of the form
      $$\begin{eqnarray*} x_1(t) &=& a_1 \cos \omega t\\ x_2(t) &=& a_2 \cos \omega t, \end{eqnarray*}$$
      
      
      determine two solutions for $\omega$ and the ratio $\frac{a_1}{a_2}$.
   3. Describe the two modes of the response of the system.
   4. What is the response of the system if $x_1(0) = x_2(0) = 1$?
   5. Write the equations of motion in the form of
      
      $$\dot \xi = A \xi.$$
      
      

   6. Compute the eigenvalues and eigenvectors of A (using a computer is allowed).
   7. Determine the general solution of the system in terms of real-valued functions.
   8. Verify your answer from part 5d using your solution from this problem.

Last updated: February 9, 2004.