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S6.6 Driving and damping

[A] An overview: the ins and outs of energy

[B] Feeding energy in: driving

  • Consider a simple harmonic oscillator of natural frequency ω subjected to a driving force which oscillates at frequency ωD
    (6.9)
    Fdriving=FDcos(ωDt)
  • The figure shows one example …but there are many others.
Graphic - No title - drivingfig

Examples

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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
 

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

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?

    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

  • Consider a simple harmonic oscillator of natural frequency ω subjected to a damping force proportional to the instantaneous velocity, and in the opposite direction to it:

    where b is a damping constant

  • The figure shows one example where the damping on the mass-spring system comes from the drag force exerted by the fluid …but there are many others.
Graphic - No title - dampingfig

Examples

Examples

  • Air resistance damping oscillations of a pendulum
  • Electrical resistance damping current oscillations in a circuit.
 

  • The resulting behaviour: x exhibits SHM close to the natural frequency ω but with an amplitude that decays exponentially with time

    (6.12)
    xm(t)=xm(0)e-γt/2

    where γb/m

Graphic - No title - ampdamped

Commentary

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 conditions

    A small untruth

    A 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 γω then the system never completes an oscillation.
    • If γω it completes many oscillations before coming to rest.
 

[D] Energy balance: damping and driving

Commentary

Commentary

  • The explicit solution is a little complicated

    Dare to take a peek?

    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)
    where
    \[ x_m = \frac{F_D}{m\sqrt{(\omega_D^2-\omega^2)^2 + \gamma^2\omega_D^2}} \]
    and
    \[ \tan\phi = \frac{\gamma\omega_D}{\omega^2-\omega_D} \]
    If 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:

     

    Graphic - No title - energyflow

    • 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 KU 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.

 
MDriving and Damping

Learning Resources

Textbook: HRW 15.8-15.9
Course Questions:
Self-Test Questions: