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Damping and Resonance Investigations Using Laplace Transforms

Part 5: A Rather Complicated External Force

  1. The last external force we will consider is

    f(t) = 16200 t3 e-t/5 cos(3t).

    In this case you'll find that the transform Y(s) involves the fifth power of a quadratic factor, and its inverse transform by manual methods would be impossibly tedious. Plot the oscillations and the envelope curves.

  2. Where are the oscillations of the mass increasing in amplitude? Where are they decreasing?

  3. What is the maximum value of the amplitude function? Where does it occur?

  4. What happens to the oscillations as t goes to infinity?

  5. Compare the envelope curves from Parts 1, 4, and 5. Pay special attention to what happens as t approaches zero. Can you identify a factor in each of the external force functions f(t) that might be related to this?

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