Forced Simple Harmonic Motion

Work through the following. Where you are expected to do some calculations do them, showing the working! When carrying out exercises using the programs on the pop-up windows give brief notes as to what you did and what your results were.

• Investigate how the system varies as you change the parameters:
  1. What happens as the damping term, b, is increased or decreased? (Answer here)
  2. How does the ratio (amplitude of oscillations)/(forcing amplitude) behave as as the forcing frequency varies? Hint: For larger frequencies you may find it easier to fix on some amplitude which correspond to one of the tick marks on the vertical axis and try and find the amplitude of the forcing which will give the oscillations this amplitude when they settle down to a steady state. (Answer here)
  3. What happens if the damping term b=0? Try finding the ratio (amplitude of oscillations)/(forcing amplitude) as the forcing frequency varies. Note:You will only be able to do this approximately as the response never settles down to a simple steady sinusoidal state. What happens when the frequency = 1? (Try a small forcing amplitude such as b=0.025.) (Answer here)
• Note: The possibility of changing c has been retained. This variable can be rescaled out from the governing equation. It has been retained here to allow the situation when omega >> 1 to be investigated. As the numerical algorithm has a fixed time step it may be more appropriate to fix omega and look at the small c case.
• The amplitude of the forcing, d has also been retained to allow for more direct comparisons with the next example, the forced simple pendulum. It also allows investigations of the large frequency limit when the amplitudes are typically small unless the forcing is large.
• The limits on the vertical axis are to enable comparisons to be made to the next example of a forced damped simple pendulum.

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