Damping, Driving & Resonance
Push at the right rhythm and small forces build large motion. The right rhythm is not quite where you would guess.
Introduction
Push an oscillator again and again near its own rhythm and the small pushes add up instead of cancelling: that is resonance, and it is how a radio tunes and a wine glass shatters.
An oscillator has natural frequency ω₀ and some real, non-zero damping. At which driving frequency is the steady-state amplitude largest?
Pushing at the right rhythm
Once the start-up transient dies away, the mass oscillates at the driving frequency, whatever its natural one, with an amplitude that peaks at
How damping shapes it
With no drive, the damping ratio decides how an oscillator settles. Below 1 it rings down inside a shrinking envelope. At exactly 1, critical damping, it returns fastest without overshooting, as a door closer should. Above 1 it creeps back slowly, wading through its own resistance.
A kg mass on a spring with N/m has damping N·s/m. What is its damping ratio?
Below and above resonance
When driven well above its natural frequency, how does the oscillator respond?
The end of the unit
You found a resonance peak sitting below its natural frequency, flattened it with damping, and located a natural frequency by its 90° lag. That is a first course in mechanics: forces when you need the details, energy and momentum when you only need the outcome, and rotational versions of all of it.
The same patterns come back under new names in every unit that follows. Thermodynamics is statistics applied to enormous numbers of particles. Electromagnetism replaces the inverse-square gravitational force with an inverse-square electric one, and adds something new. Waves are what you get when you couple a great many oscillators together.
Ideas this lesson took on
Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.
- “Resonance happens exactly at the natural frequency”
- “Damping only makes the response smaller”
- “The response is immediate”
Finished with this lesson?
Progress is stored on this device only. There is no account and nothing is sent anywhere.
Research behind this lesson
- Podolefsky et al. 2013: Guidance can be built into the affordances and constraints of an interface, so that learners explore productively without being told what to do, developed over 125 simulations and more than 600 student interviews.
- White & Gunstone 1992: The Predict–Observe–Explain cycle: committing to a prediction before observing makes a conflicting result impossible to ignore, which is what forces a belief to be revised rather than merely supplemented.