### 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.

- Section 3.1: 26
- Section 3.2:
- 4
- 6
- 9
- 10
- 13
- 27

- Section 3.4:
- 11
- 18
- 19
- 26

- Consider the torsional system illustrated in the following
figure.

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

where is the shear modulus, is the mass moment of inertia of
the disk, is the length of the shaft, is the diameter of the
shaft and the polar (area) moment of inertia of the shaft is given by

- Determine the equation of motion for this system.
- Determine the natural frequency.

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

- Determine the equation of motion for the system if is
measured from the
*unstretched* position of the spring.

- Determine the equation of motion for the system if is
measured from the
*equilibrium* position of the system, *i.e.,* is measured from the position

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

- In class we showed that the solution to

with the appropriate definitions of and was

Let and .
On the same plot, plot the
solution for and
for to .

*Last updated: September 3, 2003.*

B. Goodwine (*goodwine@controls.ame.nd.edu*)

J. Lucey (*jlucey@nd.edu*)