Math 223 Homework 1
Spring 1997
Due: Friday, January 31, 1997
Instructions: Your homework must be typed. All figures can be hand drawn. If you use a calculator (or computer) you must state the calculator (or computer) you used and explain how you used it.
Consider a circular drum which is two feet in diameter, and has a maximum vertical displacement of 0.50 inches. The drum is struck dead center forcing the maximum vertical displacement. Time is measured from the instant the drum stick breaks contact with the drum head, and the drum reaches a maximum positive displacement 0.2 seconds after the drum is struck. Fix the origin at the center of the drum, with the undisturbed drum head in the xy-plane.
1. What is the shape of the drum at t = 0,0.1,0.2 Describe the shape and sketch it.
2. Describe the motion of the drum head.
3. Describe the following cross sections
a. The drum head at t = 0, with x = 0.
b. How do these cross sections change in time?
c. The center of the drum (x = 0, and y = 0).
d. How does this cross section change as x is increased to x = 1.
4. What kind of symmetry do you expect the function describing the vertical displacement of the point (x,y) to have?
5. Construct a function that might represent the motion of the drum head, make sure you account for the dimensions and the speed of the drum head.
6. Draw the following contour diagrams for the function you constructed in part 4, and label the contours corresponding to z = + / - {0,0.125,0.25,0.375,0.5\}
a. Fix t = 0
b. Fix t = 0.05
c. Fix t = 0.1
d. Fix t = 0.2
e. Do your contour diagrams agree with your descriptions in part 1?
7. How could you modify this model to account for friction (i.e. the drum will not vibrate forever)?