Physics · Engineering Physics · A2 Level

Resonance

Push a swing at exactly the right moment on every cycle and it climbs higher and higher with barely any effort. Drive an engineering structure or component at exactly its own natural frequency and something very similar — and often far more dangerous — happens. Resonance is one of the few phenomena in physics that is simultaneously prized by instrument designers and feared by structural engineers.

7 min read

What resonance is

The previous lesson introduced simple harmonic motion and the idea that every oscillating engineering component has its own natural frequency — the frequency it settles into if disturbed once and left alone. Resonance occurs when a system is driven by an external periodic force at, or very close to, that natural frequency. When this happens, the amplitude of the resulting oscillation grows to become dramatically larger than the driving force alone would seem to justify — often many times larger than the amplitude produced by the same-sized force at any other driving frequency.

This is a genuinely striking effect. A driving force with a small amplitude, applied at the wrong frequency, produces a correspondingly small, unremarkable response. The very same force, applied at the system's natural frequency, can build an oscillation large enough to be felt, heard, or — in engineering structures — large enough to cause damage. The frequency of the driving force matters just as much as its size.

Why energy builds up at the natural frequency

The mechanism behind resonance is about the timing of energy transfer, not the strength of the driving force itself. When a driving force acts on an oscillating system, it can do positive work on the system (feeding energy in) or negative work (taking energy out), depending on the phase relationship between the force and the system's own motion at that instant.

At the natural frequency, the driving force ends up acting in a fixed phase relationship with the system's velocity such that it is doing positive work on the system on essentially every cycle — pushing in the same direction the system is already moving, over and over. Energy is therefore fed into the oscillation continuously and cumulatively, cycle after cycle, rather than being fed in on some cycles and partially removed on others. In the idealised case of no damping at all, this accumulation has no natural limit: the amplitude would in principle grow without bound the longer the driving force continued to act.

Drive the same system at a frequency well away from its natural frequency, and the phase relationship between the driving force and the system's motion instead drifts continuously in and out of step. The force sometimes does positive work and sometimes negative work on the system, with the two roughly cancelling out over many cycles. The result is a small, steady oscillation at the driving frequency rather than a dramatically growing one — this is the general case of forced vibration away from resonance, covered fully in the next lesson.

A double-edged phenomenon

Resonance is neither simply good nor simply bad in engineering — it depends entirely on whether it is wanted. It is deliberately exploited in tuned systems such as certain sensors and measuring instruments, and in musical instruments, where a large, controllable, predictable response at a specific frequency is exactly the goal. But in structures, rotating machinery and vehicle components, an unplanned brush with resonance is potentially catastrophic: amplitudes can grow large enough to exceed a material's elastic limit, crack a component through fatigue, or destroy a structure outright. Engineers must actively identify the natural frequencies of anything they build and design either to avoid driving it near them, or to control the amplitude if that's unavoidable — a case study in exactly this kind of resonance-driven failure is covered later in this module.

The resonance curve

The relationship between driving frequency and response amplitude is usually summarised in a resonance curve — a graph of the oscillation's steady-state amplitude plotted against the frequency of the driving force. For a system with only light damping, this curve has a distinctive shape:

At driving frequencies well below the natural frequency, the amplitude is small and roughly constant, changing only gradually as frequency increases.
As the driving frequency approaches the natural frequency, the amplitude rises sharply, reaching a sharp, narrow peak very close to the natural frequency itself.
Beyond the natural frequency, the amplitude falls away again, dropping back down toward small values at high driving frequencies.

The lighter the damping present in the system, the taller and narrower this peak becomes — a very lightly damped system can show an enormous amplitude spike confined to a very narrow band of driving frequencies close to resonance. This matters directly for engineering design: it means a system with very little inherent damping is both capable of a very large resonant response, and vulnerable to it being triggered by only a small range of unlucky driving frequencies. The following lesson in this topic, on damping, returns to this curve directly: increasing the damping present in a system broadens and flattens this peak, trading away some of the sharpness of the resonant response in exchange for keeping the maximum amplitude under control.

A graph of amplitude against driving frequency showing three overlapping curves, all rising from near zero at low frequency to a peak near a marked frequency f-zero and falling back toward zero at high frequency. A solid curve labelled light damping forms a tall, narrow, sharply-pointed peak. A dashed curve labelled medium damping forms a shorter, wider, rounded peak. A dotted curve labelled heavy damping forms a low, broad, flat-topped peak. — The resonance curve for a lightly damped system is sharply peaked at the natural frequency f₀; increasing the damping broadens and flattens the peak, reducing the maximum amplitude reached at resonance.

Worked reasoningRecognising a resonance risk

A slender support bracket on a piece of factory machinery has a measured natural frequency of 48 Hz. The machine it supports contains a rotating component that will operate somewhere in the range 2400–3200 rev/min.

Rotating parts with any imbalance deliver a periodic driving force at their rotation frequency — this is developed fully in the next lesson.

Convert the operating range to hertz: $2400\text{ rev/min} = \dfrac{2400}{60} = 40\text{ Hz}$, and $3200\text{ rev/min} = \dfrac{3200}{60} \approx 53.3\text{ Hz}$.

Driving frequency in Hz is rev/min divided by 60 (seconds per minute).

The bracket's natural frequency, 48 Hz, falls inside this operating range.

Somewhere within its normal speed range, this machine will drive the bracket at or near its own natural frequency.

This is a resonance risk: the amplitude of vibration in the bracket could grow far larger than expected as the machine passes through roughly 48 Hz, unless the bracket is redesigned, damping is added, or the operating range is adjusted to avoid that frequency.

This is exactly the kind of check engineers carry out before a machine is put into service.

reasoning complete

Key takeaways

Resonance occurs when a system is driven at, or very close to, its own natural frequency of oscillation.
At resonance, the driving force stays in a phase relationship with the system's motion that lets it do positive work on essentially every cycle, so energy accumulates rather than partially cancelling.
In the idealised undamped case, resonant amplitude grows without limit the longer the driving force acts.
Resonance is a double-edged phenomenon: useful and deliberately exploited in tuned systems, sensors and musical instruments, but a serious risk to be designed against in structures, rotating machinery and vehicle components.
The resonance curve (amplitude of response against driving frequency) is sharply peaked at the natural frequency for lightly-damped systems.
Lighter damping produces a taller, narrower resonance peak; increasing damping broadens and flattens it, as the next lesson but one explores in detail.
Checking whether a component's natural frequency falls within a machine's normal operating range is a standard, essential engineering safety check.

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Mechanical Oscillations

Forced Vibrations