S6.6 Driving and damping
[A] An overview: the ins and outs of energy
- Thus far we have assumed that our oscillator is effectively isolated from anything else: its energy is thus constant.
- We must now allow for two possibilities:
- There is some mechanism that feeds energy into the oscillating system: we call this driving.
- There is some mechanism that draws energy out of the oscillating system: we call this damping.
- We shall initially consider these two mechanisms separately; then together.
- We shall explore, but not prove the main results.
[B] Feeding energy in: driving
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Examples
Reveal

Examples
- A glass being jiggled by a sound wave
- An electrical circuit subject to an alternating voltage
- The electrons in an atom responding to a light wave
The equation of motion:
- The resulting behaviour:
x exhibits SHM but - at the frequency ωD (not ω)
- with an amplitude that is determined by
- If ωD is close to ω the amplitude xm is large. This is called resonance.
Key Point 6.9
If a system of natural oscillation frequency ω is driven by some disturbance oscillating at frequency ωD the system will oscillate at ωD with an amplitude that is large if ωD is close to ω.
Commentary
- The solution to the equation of motion is, explicitly:
x=xmcos(ωDt)
- This describes the behaviour in the ’long term’
- Before that, the driving force feeds energy into the system building up the amplitude to the prescribed xm
Thereafter it feeds in no more energy.
But how can that be?
The energy fed into the system is the work done by the driving force. This work is positive if the driving force points in the same direction as the instantaneous velocity, and negative if it points in the opposite direction. Once things have ’settled down’, over any cycle the driving force does equal amounts of positive and negative work on the system and thus exchanges no net energy with it.- Resonance occurs widely across physical science …from a dangerous natural hazard to an invaluable experimental tool
- In a real system xm does not become infinite at resonance, because of damping
[C] Drawing energy out: damping
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- The equation of motion:
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Commentary
The solution to the equation of motion is, explicitly:
x(t)=xm(t)cos(ωt+φ)=xm(0)e-γt/2cos(ωt+φ)where xm(0) and φ are fixed by initial conditionsA small untruth
We have lied, here (to keep things looking reasonably simple). The frequency of the oscillation is actually shifted from the natural frequency ω by an amount that depends on the constant γ. This effect becomes important if γ is comparable with ω. In fact if γ=2ω one finds that the solution has no wiggly (oscillatory) bit at all; the system simply decays smoothly to equilibrium. This condition is known as critical damping: it is the condition which one attempts to fulfil in designing a shock absorber.- In the long term the system will be stationary, at the equilibrium position x=0.
- What happens before then depends on the
relative values of γ and ω
- If γ≫ω then the system never completes an oscillation.
- If γ≪ω it completes many oscillations before coming to rest.
[D] Energy balance: damping and driving
- Now we consider what happens in a system in which we have both driving (as in Equation 6.9 ) and damping (as in Equation 6.11.)
- The equation of motion:
- The resulting behaviour:
x exhbits SHM - at the frequency ωD (not ω)
- with a phase that is different from that of the driving force
- with an amplitude that is large, but not infinite at resonance.

Commentary
The explicit solution is a little complicated
Dare to take a peek?
Well, actually we are expecting you to do so in one of the xourse questions. You can start here:
The steady-state solution to the equation of motion (that is, the solution when ’things have settled down’) is
x(t)=xmcos(ωDt+φ)whereandIf you feel reasonably comfortable with this intensity of mathematics (or if you would like to try to make yourself more comfortable) you could ask yourself the following:- At what value of ωD is xm biggest?
- What is its value there?
- What is the value of φ when ωD is well below ω?
- What is the value of φ when ωD is well above ω?
But the underlying physical picture is simple:
- The damping force draws energy out at a steady rate
- The driving force feeds in energy at the same rate
- The total oscillator energy E is constant with the usual K↔U interchange
- The closer ωD is to ω the higher is E
- The oscillation frequency is that of the driving force, ωD
- The oscillation is not generally in phase with the driving force.
Learning Resources
![]() | HRW 15.8-15.9 |
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