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Damping and Resonance Investigations Using Laplace Transforms

Part 6: Varying the Parameters

In this part, we will again investigate a force of the form

f(t) = f0 t3 e-a t / p cos(b t / p).

  1. To simplify our calculations, we will let

    m = p2

    2. c = 2 p a

    k = a2 + b2

    and

    f0 = 8 p b5 / 3 p4.

    Enter these formulas into your helper application.

  2. Enter the values p = 5, a = 1, and b = 15. Verify that this gives you the differential equation from Part 5.

  3. How do you think the solution will change if p is changed? Change p and repeat Part 5 using the new differential equation that you get. Plot the oscillations and the envelope curves. How has the solution changed? You may have to change your time scale in order to get a good picture.

  4. Return p to its original value. How do you think the solution will change if a is changed? Change a and repeat Part 5 using the new differential equation that you get. Plot the oscillations and the envelope curves. How has the solution changed? You may have to change your time scale in order to get a good picture.

  5. Return a to its original value. How do you think the solution will change if b is changed? Change b and repeat Part 5 using the new differential equation that you get. Plot the oscillations and the envelope curves. How has the solution changed? You may have to change your time scale in order to get a good picture.

  6. Can you explain these changes in terms of the physical characteristics of the system? Can you explain them in terms of the form of the solution?

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