3.6.1.4 - Forced Vibrations and Resonance

Free and forced vibrations

A free vibration follows an initial disturbance with no continuing periodic driver. In the ideal undamped case the system oscillates at its natural frequency $f_0$, set by its inertia and restoring force. For a spring, $f_0=(1/2\pi)\sqrt{k/m}$: a stiffer spring raises it, and a larger mass lowers it. Light damping gradually removes energy and gives a frequency close to $f_0$.

A forced vibration is maintained by a periodic external force. After initial transients decay, a linear oscillator responds at the driving frequency $f_d$, even when that differs from $f_0$. The driver can replace energy lost through damping, producing a steady amplitude. “Free” does not mean no forces act: restoring and damping forces still act.

Imagine a hanging spring mass. Pulling it down once and releasing it produces a free response. Moving its support periodically drives a forced response. Turning the driver off leaves a decaying free response; changing the driver frequency changes the steady forced frequency, not the system's underlying stiffness or mass.

Resonance and energy transfer

Resonance is a large response when a periodic driving frequency is close to a natural frequency. For light damping, the displacement-amplitude maximum occurs very close to $f_0$, often treated as equal to it in A-level questions. Energy supplied in a favourable phase accumulates over successive cycles, increasing amplitude.

For a sinusoidal force driving a linear oscillator at $f_d=f_0$, force and velocity are in phase; displacement lags the force by a quarter cycle. The force therefore does positive work through the cycle. As amplitude grows, damping dissipates more energy. A steady state is reached when average input power equals average dissipated power. This is an energy-rate balance, not zero resultant force: the mass still accelerates throughout most of its motion.

For a lightly damped $0.12\,\mathrm{kg}$ spring oscillator resonating at approximately $2.4\,\mathrm{Hz}$, $k=m(2\pi f_0)^2=27.3\,\mathrm{N\,m^{-1}}$. This estimate explicitly assumes the resonance frequency is sufficiently close to the undamped natural frequency.

An ideal undamped model driven exactly at its natural frequency has no finite steady resonant amplitude. Real systems have losses and limits to the linear approximation, so “infinite amplitude” is not a physical prediction for an actual structure.

Mechanical systems and stationary-wave modes

A useful resonance explanation identifies the driver, the oscillating system and the frequency match. In Barton's pendulums, a driver pendulum transfers a periodic disturbance through a shared support. The pendulum of matching length has approximately the same natural frequency and responds most strongly. Others still undergo forced motion, but with smaller amplitude.

A stretched string driven by a vibration generator has several natural modes. Reflections at fixed ends require displacement nodes there, so only particular wavelengths fit the length. Driving close to one of the corresponding natural frequencies produces a large stationary-wave pattern. An air column also has allowed modes, but its open/closed end conditions differ; do not assume every air column has every integer harmonic.