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Damping and Resonance Investigations Using Laplace Transforms

Part 3: A Periodic External Force

  1. Now use your helper application to do a similar investigation of the case of a periodic external force,

    f(t) = 901 cos(3t).

    Plot the oscillations and the envelope curves.

  2. Your solution should be a combination of a steady periodic oscillation of the form

    A cos(ωt) + B sin(ωt)

    with an exponentially damped transient motion of the form

    C1 e-Lt cos(ωt) + C2 e-Lt sin(ωt).

    Plot each of these motions separately along with its envelope curve. (For more information about this sort of motion you may want to look at the "Background" section of the Gain and Phase Shift module.)

  3. Plot the sum of the amplitude for the steady periodic oscillation and the amplitude for the transient motion, along with the amplitude of the total solution. Do the two curves agree? Explain why or why not.

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