Forced Spring Systems

Part 2: Beats

The following figures show the solution function and phase plane trajectory for the initial value problem

-- the case with which you ended Part 1. You may have seen the beginnings of a pattern showing up here that you did not see when w and k were farther apart. We explore that pattern in this part of the module.

Solution function y(t) (above) and phase plane plot (right).

1. Set w successively equal to 4.25, 4.5, 4.75, and plot the solution and trajectory for each case. Describe what you see in your own words. (You may want to plot over longer time intervals to confirm your observations.)
2. Show that, for any numbers F₀, k, and w (with |w| not equal to |k|), the unique solution of the initial value problem
   y'' + k² y = F₀ cos wt, y(0) = 0, y'(0) = 0
   is
   y = F₀ (cos wt - cos kt) / (k² - w²).
   You may use your helper application.
3. Explain why the amplitude of the oscillation increases as w gets closer to k.
4. Use the trigonometric identity
   cos A - cos B = 2 sin [(B - A)/2] sin [(A + B)/2]

   to write the solution in step 2 in another form.

5. There are two apparent "frequencies" in the solution function. Explain how those frequencies are related to k and w -- the frequencies of the system and on the driving force, respectively. In particular, explain why one of the periods gets longer as w gets closer to k.

The phenomenon you are observing here is called beats. In the Summary you will be asked to explain what you think is meant by this word.