Chapter 11 Project Solutions

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.

    At t=0 the drum is at its maximum negative vertical displacement. It looks like a bowl, and curves smoothly to the sides of the circular drum head, which are fixed at z=0. Since the drum reaches its maximum positive displacement 0.2 seconds later, at t=0.1 seconds the drum head is half way between its minimum and maximum. Thus the drum would be perfectly flat. At t=0.2 seconds the drum has reached its maximum positive vertical displacement so that the it would look like an upside down bowl, much like an inversion of the drum head at t=0 seconds.

  2. Describe the motion of the drum head.

    The drum, over time, will oscillate between the states described (and illustrated) in part 1.

  3. Describe the following cross sections:

    1. The drum head at t=0, with x=0.

      This cross-section is the profile of the drum head along the y axis at t=0 , and would look the graph below.

    2. How do these cross-sections change in time?

      In time, the cross-section would flatten, and then bulge upward. As t increases beyond t=0.2 seconds the bulge will flatten, depress and oscillate back and fourth.

    3. The center of the drum (x=0, and y=0).

      At x=0, y=0, we are looking at the height of the center of the drum head over time. From part 1 the drum head starts at it's maximum negative vertical displacement, at t=0.1 seconds the drum head is flat, and at t=0.2 seconds the drum head has reached it's maximum positive vertical displacement. This trend will continue, at t=0.3 seconds the drum head will be flat, and at t=0.4 the drum head will be back where it started at a maximum negative vertical displacement. Thus the center oscillates from -0.5 inches to 0.5 inches with a period of 0.4 seconds, so it would look like tex2html_wrap_inline107 , which is shown below.


    4. How does this cross section change as x is increased to x=1.

      A cross-section with tex2html_wrap_inline113 and tex2html_wrap_inline115 fixed looks at how the displacement changes in time at the point tex2html_wrap_inline117 . As we consider points closer to the edge of the drum (x=0) the amplitude should decrease. The point (1,0) is fixed at z=0, thus the amplitude should be zero at (1,0). The cross-sections for the points (0.25,0), (0.5,0), (0.75,0), and (1,0) are drawn below.



  4. What kind of symmetry do you expect the function describing the vertical displacement of the point (x,y) to have?

    1. From looking at the surface in 1, it is clear that the function z=f(x,y,t) should have circular symmetry in x and y (i.e. tex2html_wrap_inline143 is the only way x and y appear in the function.)
  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.

    We already know that z=f(x,y,t) should oscillate in time like tex2html_wrap_inline151 (see part 3c,d) the remaining part is to describe the amplitude as a function of x and y. This should have circular symmetry with a maximum amplitude at (0,0) and should decrease to zero amplitude at the drums edge ( tex2html_wrap_inline159 ). The maximum displacement is 0.5 inches and the function should start at the maximum negative displacement. Two such functions are:

    displaymath43

    and

    displaymath44

    Where x and y are measured in feet and the resulting amplitude is measured in inches. So the corresponding functions to describe the motion of the drum head would be:

    displaymath45

    and

    displaymath46

  6. Draw the following contour diagrams for the function you constructed in part 4, and label the contours corresponding to tex2html_wrap_inline167 tex2html_wrap_inline169
  7. sorry No contour diagrams yet

    1. Fix t=0
    2. Fix t=0.05
    3. Fix t=0.1
    4. Fix t=0.2
    5. Do your contour diagrams agree with your descriptions in part 1?
  8. How could you modify this model to account for friction (i.e. the drum will not vibrate forever)?

    One way is to include an exponential (in time) damping term in the amplitude: tex2html_wrap_inline179 . The first function would become

    displaymath47

    and the cross section for the middle of the drum (x=0 and y=0) would be