This document discusses driven harmonic oscillators. It explains that a damped oscillator will slow down over time due to friction, but an external driving force can counteract this and sustain the oscillations. Resonance occurs when the driving frequency matches the natural frequency of the system. Under resonance conditions, the amplitude of the oscillation will not decrease as the system is absorbing energy at the same rate it is losing it. The driving force can be represented by a harmonic function.

1. Driven Harmonic
Oscillators
Kate MacDonald Physics 101 Section 201

2. Definite motion
• In a world with no frictional force, a simple harmonic
oscillator set in motion will never lose energy and never slow
down.
• We do not live in that world, and for every oscillation that is
completed, frictional forces transform a fraction of the
oscillator’s kinetic energy into thermal energy.
• This transformation decreases the amplitude of the
oscillation.
• An oscillator losing kinetic energy in this way is called a
damped oscillator, and will eventually slow to a stop.

3. Sustaining motion
• We can counteract this inevitable stop by transferring energy
to the oscillator by means of an external mechanism.
• One example is a swing set.
• This externally applied force is called the driving frequency.
A child set in motion on a swing will eventually come to rest if
he or she simply sits. However, if a parent applies a force to
the child after every full oscillation, the motion can continue
indefinitely.

4. Resonance
• A mechanical system is in resonance with an externally
applied force when the driving force matches the natural
frequency of the system.
• The external force by the parent from the previous example is
in resonance with the swing system when the parent pushes
the child every time the swing comes back to the parents
hands
• The swing is absorbing energy at the same frequency it is
losing energy.
fdriving =
fo

5. Resonance
• Under resonance conditions, the amplitude of the oscillation
will not decrease.

6. Other driving frequencies
• The system can also absorb energy at frequencies that are not
in the resonance condition.
• If the swing were to be pushed after every other oscillation:
• If it were to be pushed after every third oscillation:
• and so on.
• It’s important to note that the conditions described on this
slide are not in resonance.
fdriving = (1/2)fo
fdriving = (1/3)fo

7. Harmonic driving force
• An oscillator subjected to a driving force is called a driven
oscillator.
• The oscillations that come as a result are called forced
oscillations or driven oscillations.
• The driving force can be represented and graphed in the
following form:
Fdriven = Focos(wdrivent)

8. Conclusion
• A simple harmonic oscillator subjected to a driving force
oscillates with the frequency of that force.
• Resonance occurs when the frequency of the driving force is
equal to the natural frequency of the system.