University of Notre Dame
Aerospace and Mechanical Engineering

AME 301: Differential Equations, Vibrations and Control
Homework 2

B. Goodwine J. Lucey Fall, 2003

Issued: September 3, 2003 Due: September 10, 2003

Unless otherwise indicated, all the course text, Boyce and DiPrima, Elementary Differential Equations (and Boundary Value Problems). Each problem is worth 10 points.

Unless otherwise indicated, if a problem calls for a plot of a solution, you can either plot the solution by hand, or use a computer package such as Matlab.

1. Section 3.1: 26
2. Section 3.2:
   1. 4
   2. 6
   3. 9
   4. 10
   5. 13
   6. 27

3. Section 3.4:
   1. 11
   2. 18
   3. 19
   4. 26
4. Consider the torsional system illustrated in the following figure.
   

   From solid mechanics, we know that twisting the shaft by an amount $\theta$ requires a torque given by
   

   $$M_t = \frac{G J \theta}{l},$$
   
   
   where $G$ is the shear modulus, $J_0$ is the mass moment of inertia of the disk, $l$ is the length of the shaft, $d$ is the diameter of the shaft and the polar (area) moment of inertia of the shaft is given by
   
   $$J = \frac{\pi d^4}{32}.$$
   
   
   1. Determine the equation of motion for this system.
   2. Determine the natural frequency.

5. Consider the mass-spring-damper system illustrated in the following figure.
   

1. Determine the equation of motion for the system if $x$ is measured from the unstretched position of the spring.

2. Determine the equation of motion for the system if $x$ is measured from the equilibrium position of the system, i.e., $x$ is measured from the position
   
   $$x_0 = \frac{m g}{k}.$$
   
   

   Hint: $g$ should not appear in the answer for the second case.

6. In class we showed that the solution to
   $$\begin{eqnarray*} m \ddot x + b \dot x + k x &=& 0\\ x(0) &=& x_0\\ \dot x(0) &=& \dot x_0 \end{eqnarray*}$$
   
   
   with the appropriate definitions of $\omega_n$ and $\zeta$ was
   
   $$x(t) = e^{-\zeta \omega_n t} \left( x_0 \cos\left( \sqrt{1 -... ...a_n} \sin \left(\sqrt{1 - \zeta^2} \omega_n t \right) \right).$$
   
   
   Let $x_0 = -1$ and $\dot x_0 = 0$.

   On the same plot, plot the solution for $\omega_n = 1$ and $\zeta = 0, 0.2, 0.4, 0.6, 0.8 \mbox{ and } 1$ for $t = 0$ to $t = 10$.

Last updated: September 3, 2003.
B. Goodwine (goodwine@controls.ame.nd.edu)
J. Lucey (jlucey@nd.edu)