Resonance Graph

Your good friend finds a swing in a playground and wants you to push her. You are tired but grudgingly agreed. She sits happily on the swing and you give her a gentle push on her back. "Harder!" she said. You could not find the energy. When she swings back to you, you give the same gentle push on her way out. To your surprise, you find that after a few times, she swings higher and higher, even though you have not pushed any harder.

This is an example of resonance. Left to itself, the swing would oscillate a few times at its own natural frequency, and stop because of friction. When you push your friend, you are applying a force to the oscillation. We call this a forced oscillation. A forced oscillation would not stop even if there is friction, because there is the force to keep it going. As you friend swings higher, the maximum displacement, or the amplitude, increases. It will reach some limit. This is because of the friction, and not because you could not push any harder. If there is no friction at all, you friend may keep going higher, even if you continue with you very gentle push! This is because the energy you put in accumulates. With no friction to convert it to heat, the energy can only increase the kinetic and potential energy. This means that the amplitude will just keep growing and growing.

By waiting for the swing to come back to you before you push again, you are actually adjusting your force to the same, natural frequency of the swing. What if you ignore how the swing moves and just push it with a different timing, say twice as often? You may push once when the swing moves away from you, and again when the swing comes back at you. If you push when the swing comes back at you, you would actually slow it down. This would obviously reduce the amplitude. So it is not such a good idea.

What if you push half as often? This means that you only push once every two swings of you friend. Then you friend only receives half your energy, and would also not swing as high. Lets summarise this behaviour:

  1. If the frequency of the driving force (your pushing) is the around the natural frequency, the amplitude of oscillation is largest.
  2. If the frequency of the force is higher than the natural frequency, the amplitude could be smaller.
  3. If the frequency of the force is lower than the natural frequency, the amplitude could also be smaller.

Just to be clear, natural frequency is the frequency of the swing on its on - like when you only push it once and don't push anymore.

If you find another friend to write down the frequency of your push, measure the amplitude of the swing, and plot a graph, it would look like this:

In the graph, as frequency of the driving force increases, the amplitude goes up to a maximum around the natural frequency fnatural, and comes down again. At very high frequency, like if you tap gently but rapidly on your friend's back, your friend would hardly move. The amplitude is very close to zero. At the other end of the spectrum, when the frequency is zero, you would keep you hand on our friend's back all the time. Zero frequency means a constant force, not repeated pushes. Then your friend would not be too happy - she would just be displaced by a fixed amount, and stay there. On the graph, this shows as a non-zero amplitude ('fixed displacement') when frequency = 0 Hz ('constant force').

If it is a very old swing, there is more friction, and the amplitude when you push would be smaller. If your friend again measure you pushing frequency and the swinging amplitude and plot a graph, it would look lower, and broader, as shown in the picture below. On the other hand, if you find a brand new swing with much less friction, then the energy from your gentle push would not get lost as heat. It would accumulate to give a larger amplitude. The resulting graph would look higher, and sharper.




Copyright 2010 by Kai Hock. All rights reserved.
Last updated: 15 May 2011.