Resonance in Detail — Natural Frequency, Forced Vibration and Damping

The swing: resonance in everyday life

Everyone learns resonance on a playground swing. A single big shove gets you moving, but the real height comes from small, well-timed pumps: you lean back exactly as the swing moves forward, adding a little energy each cycle. Pump at the wrong moment and you kill the motion instead.

The swing is a driven oscillator: it has its own preferred tempo — its natural frequency — and it responds enormously when the pushes match that tempo. That selective, dramatic response is resonance, and it appears everywhere from musical instruments to radio tuners to skyscrapers.

Natural frequency: every system has a favorite tempo

Left alone after one disturbance, every oscillating system settles into its own rhythm. For a mass $m$ on a spring of stiffness $k$:

$$f_0 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$

A stiffer spring (larger $k$) raises the natural frequency; a heavier mass (larger $m$) lowers it. A simple pendulum of length $L$ has $f_0 = \tfrac{1}{2\pi}\sqrt{g/L}$ — longer pendulums swing slower, which is why grandfather clocks use long pendulums.

Crucially, the natural frequency belongs to the system, not to the push. Change the mass or the spring and $f_0$ moves; push however you like and $f_0$ stays put.

Forced vibration: pushing at the wrong tempo

Now drive the system with a periodic force of frequency $f$ — a motor shaking the spring's support, or your legs pumping the swing. The steady response depends on how $f$ compares to $f_0$:

• $f$ far below $f_0$: the mass simply follows the push, moving almost in step with it. Small amplitude.
• $f$ near $f_0$: the response explodes — this is resonance.
• $f$ far above $f_0$: the mass barely moves; it cannot keep up, and the motion lags far behind the push.

The lag matters too: below resonance the mass moves with the force, at resonance it lags by a quarter cycle (so the force always pushes along the velocity, doing maximum work), and far above resonance it moves opposite to the force.

The resonance curve

Plotting steady-state amplitude against driving frequency gives the resonance curve: flat and low on the left, a dramatic peak centered at $f_0$, then falling away on the right. The peak's position is set by the system's mass and stiffness; its height and width are set by damping.

Light damping (a swing with good bearings) gives a tall, needle-sharp peak: the system is extremely selective about frequency. Heavy damping (a car suspension) gives a low, broad hump: the system responds modestly over a wide range. Engineers choose the damping to suit the job — sharp for a radio tuner, broad for a shock absorber.

Damping and the Q factor

The sharpness of the peak is quantified by the quality factor $Q$:

$$Q = \frac{f_0}{\Delta f}$$

where $\Delta f$ is the bandwidth — the width of the peak between its half-power points. A tuning fork has $Q$ in the thousands: it rings at almost exactly one frequency and ignores everything else. A loudspeaker cone has low $Q$: it must respond evenly across a wide band.

$Q$ also measures energy bookkeeping: roughly, it is $2\pi$ times the energy stored per cycle divided by the energy lost per cycle. High $Q$ means the system hoards energy and loses little — which is exactly why the peak grows so tall.

Worked example: the 1 Hz spring

Take a 0.50 kg mass on a spring with $k = 20\ \mathrm{N/m}$. Its natural frequency is:

$$f_0 = \frac{1}{2\pi}\sqrt{\frac{20}{0.50}} = \frac{1}{2\pi}\sqrt{40} \approx \frac{6.32}{6.28} \approx 1.0\ \mathrm{Hz}$$

Drive it gently at 1.0 Hz and the amplitude climbs with every cycle until damping balances the energy input — a large steady oscillation from a small force. Now drive the same system at 2.0 Hz: the mass cannot follow, the pushes fight the motion half the time, and the response stays tiny.

Double the mass to 1.0 kg and $f_0$ drops to about 0.71 Hz — the 1.0 Hz drive that once caused resonance now produces only a modest response. Resonance follows the system, not the driver.

Resonance for good and for trouble

Resonance is a tool and a hazard. Instrument makers seek it: a violin's body is carved so its air and wood resonances enrich the strings' tone. Radio engineers exploit it: an LC circuit resonates electrically at one frequency, selecting a single station from the airwaves.

Structural engineers avoid it: every bridge and tall building has natural frequencies, and designers make sure that wind gusts, foot traffic and machinery never drive the structure at those frequencies. Damping systems — from tuned mass dampers in skyscrapers to shock absorbers in cars — exist to keep resonance peaks low and harmless.

Variables and units

Common mistakes

Real-world applications

• Radio tuners use an LC circuit whose electrical resonance selects one station's frequency and rejects all the rest.

• Musical instruments shape resonance deliberately — a guitar body's air cavity amplifies the string's vibrations at its resonant frequencies.

• Engineers map the natural frequencies of bridges and buildings so that wind and traffic never drive them at resonance.