Physics › Periodic motion › Forced vibrations and resonance
Forced vibrations and resonance
Every oscillator has a frequency it wants. Drive it at exactly that frequency and the amplitude grows out of all proportion: resonance. Damping is the volume knob on the effect, and engineering is largely the art of turning it down.
Builds on SHM systems: pendulums and springs and Stationary waves.
IN THIS TOPIC
- Distinguish free vibrations at the natural frequency from forced vibrations at the driving frequency.
- Explain resonance as the response when driving frequency equals natural frequency.
- Describe how damping changes the sharpness of resonance, with mechanical and stationary-wave examples.
WHAT YOU PROBABLY THINK
Pushing harder is all that matters.
Free and forced
Displace a system and let go and it performs free vibrations at its own natural frequency, the one its mass and stiffness choose. Push it periodically instead and it performs forced vibrations at the driving frequency, whatever that is. The interesting physics lives in the relationship between the two frequencies.
Resonance
When the driving frequency equals the natural frequency, every push arrives exactly in step with the motion, feeding energy in at the most efficient possible rate, and the amplitude climbs to a maximum: resonance. This is the child on the swing: not harder pushes, but correctly timed ones. A mistimed shove partly cancels the motion; a small, punctual one builds it without limit until losses intervene.
Damping sets the sharpness
The amplitude-frequency curve is the topic's exam centrepiece, and damping controls its shape. Light damping: a tall, narrow spike at the natural frequency, a system that responds enormously but only to almost exactly the right note. Heavier damping: a lower, broader hump, a duller but safer response. Sketching the family of curves on one set of axes is a standard mark.
Engineering runs both directions. Where resonance is a menace, vehicle suspensions, buildings in wind or earthquakes, bridges under marching feet, designers add damping to flatten the peak. Where it is the whole point, a musical instrument's body, radio tuning circuits, damping is kept light so the response stays sharp.
Resonance with a shape
AQA's examples explicitly include situations involving stationary waves, and the stretched string is the cleanest. Drive it at one of its natural frequencies and the reflections reinforce, cycle after cycle, until a large stationary wave stands on the string: the resonance curve's tall peak, rendered as a shape you can see. The stationary-waves lesson's harmonics are, from this vantage, simply the string's menu of resonant frequencies.
THE EXAM BIT
- Definitions first: free vibrations happen at the natural frequency; forced vibrations happen at the driving frequency. Confusing whose frequency the system adopts is the standard slip.
- The resonance condition in one line: driving frequency = natural frequency, giving maximum amplitude because energy transfer from driver to oscillator is most efficient.
- Damping and the curve: more damping means a lower and broader peak. Sketch all requested curves on one set of axes with the natural frequency marked.
- Applications answer in the same shape both ways: name the driver, name the natural frequency, then say whether damping is added (suspension, buildings) or minimised (instruments, tuning).
- For a driven string, resonance appears as a large-amplitude stationary wave; the harmonics are its resonant frequencies. The cross-topic link is examinable.
CHECK YOURSELF
A washing machine vibrates violently at one particular spin speed and is calmer both above and below it. Explain the violence, and one design change that would reduce it.
Show a hint
One special frequency, and one knob that reshapes the response curve.
Show the answer
At that spin speed the drum's driving frequency matches the machine's natural frequency: resonance. Each cycle of the imbalance feeds energy in exactly in step, so the amplitude builds to a maximum.
Above and below, the pushes fall out of step with the motion and partly cancel, so the response is far smaller: the machine is off the peak of its resonance curve.
Adding damping, stiffer mounts or shock absorbers, lowers and broadens the peak, so even at the unlucky speed the amplitude stays modest.
Every system has a frequency it wants.
Damping decides how badly it wants it.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.