Resonance
Resonance occurs when periodic driving couples efficiently to a system's natural motion. It can produce a large response, but only when frequency, damping and the way the force couples to the motion all line up.
Natural motion and periodic driving
A free oscillator has one or more natural frequencies. For an ideal mass–spring system, ω₀ = √(k/m) and f₀ = ω₀/(2π). A periodic external force introduces a driving frequency f.
When the driving frequency is near a natural frequency, energy transfer can become especially efficient and the steady-state amplitude can rise.
Damping controls the resonance peak
Real oscillators lose energy through friction, electrical resistance, sound radiation or other mechanisms. Damping prevents the amplitude from growing without limit.
Weak damping produces a narrow, high resonance peak. Strong damping produces a lower, broader response. The exact frequency of maximum amplitude can also shift slightly below the undamped natural frequency.
The familiar resonance curve describes the steady-state response after transients have faded. When driving begins, the system initially contains a mixture of its own free oscillation and the driven response. Damping removes the transient component over time.
Driven-oscillator response
Explore the dimensionless steady-state amplitude factor for a damped oscillator. Let r = f/f₀ and ζ be the damping ratio.
Resonance is repeated energy transfer
A push is effective when it arrives with the right phase relative to the motion. This is why a swing can be driven to large amplitude by modest pushes given at the right times.
Driving at the same nominal frequency is not enough if the force couples poorly to the relevant mode—for example, pushing at a node of an ideal standing-wave mode has little effect on that mode.
Mechanical, acoustic and molecular examples
Structures have mechanical modes; air columns have acoustic modes; electrical RLC circuits have resonant frequencies; molecules have rotational and vibrational modes that interact selectively with electromagnetic radiation.
The common pattern is not “everything vibrates wildly.” It is selective response: the system responds strongly only to frequencies and force patterns that couple to an allowed mode.
Systems with many degrees of freedom have several normal modes rather than one natural frequency. A guitar string, a molecule and a building can therefore show multiple resonances, each with its own spatial pattern and coupling strength.
Quality factor and bandwidth
The quality factor Q describes how lightly damped a resonance is. A high-Q resonance stores energy for many cycles and has a narrow frequency response; a low-Q resonance loses energy quickly and responds over a broader band.
For a weakly damped resonance, the useful approximation Q ≈ f₀/Δf relates Q to the full-width bandwidth Δf around the resonance peak.
Worked examples
1. Mass–spring natural frequency
A 0.50 kg mass is attached to a spring with k = 200 N m⁻¹. Find the undamped natural frequency.
Solution
ω₀ = √(k/m) = √(200/0.50) = 20 rad s⁻¹. Thus f₀ = ω₀/(2π) = 3.18 Hz.
2. Driving far from resonance
An oscillator has natural frequency 5 Hz but is driven at 0.5 Hz. Should a strong resonance response be expected?
Solution
No. The driving frequency is far below the natural frequency, so the repeated forcing is not timed to build a large resonant response.
3. Effect of damping
Two otherwise identical oscillators are driven near resonance, but one has much stronger damping. Which has the lower peak amplitude?
Solution
The more strongly damped oscillator. It dissipates energy more rapidly, so the steady-state resonance peak is lower and broader.