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Damping and Resonance Investigations Using Laplace Transforms

Part 1: A Sample System

This lab uses Laplace transforms to investigate the behavior of a mass-spring-dashpot system

m y'' + c y' + k y = f(t), y(0)=0, y'(0)=0,

in response to a variety of possible external forces.
  1. Enter the differential equation with the starting parameter values m = 25, c = 10, and k = 226.

  2. To illustrate a sample system, we will use the external force

    f(t) = 900 t e-t/5 cos(3t).

    Enter this force in your helper application. Use the helper application to find Y(s), the Laplace transform of the solution to the differential equation.

  3. Factor the denominator of Y(s). The cubed quadratic factor would be difficult to handle manually, but you can use your helper application to find the inverse Laplace transform.

  4. You may know that a damped oscillating function of the form

    A(t) cos(ωt) + B(t) sin(ωt)

    can also be written in the form

    C(t) cos(ωt - δ)

    where

    C(t) = sqrt(A(t)2 + B(t)2).

    The function C(t) is called the time-varying amplitude of the function. Define this for y(t).

  5. Finally, plot y(t) between the "envelope curves" defined by the time-varying amplitude. Notice that the resonance resulting from the repeated quadratic factor consists of a temporary buildup before the oscillations are damped out.

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