Driven oscillators

Adding an external force

A driven oscillator is an oscillator acted on by an external time-dependent force. A child on a swing being pushed periodically, a speaker cone driven by an electrical signal, and an AC circuit driven by alternating voltage are all examples.

For a damped mass-spring oscillator driven by a sinusoidal force,

$F_{drive}(t)=F_0cos(omega t)$

the equation of motion is

mrac{d^2x}{dt^2}+brac{dx}{dt}+kx=F_0cos(omega t)

Here $omega$ is the driving angular frequency, which may differ from the oscillator's natural frequency $omega_0$.

Natural and driven frequencies

The natural frequency is the frequency at which the system would oscillate freely with little damping:

omega_0=sqrt{rac{k}{m}}

The driving frequency is set by the external force. The long-term motion of a damped driven oscillator occurs at the driving frequency, not necessarily the natural frequency.

This is important: after transients die away, the system follows the driver.

Transient and steady-state motion

The full motion has two parts. The transient part depends on initial conditions and resembles the damped free motion. It decays over time because of damping.

The steady-state part persists as long as the driving force continues. It has the same frequency as the driver:

$x_{ss}(t)=A(omega)cos(omega t-delta)$

Here $A(omega)$ is the response amplitude and $delta$ is the phase lag between the drive and the response.

Amplitude response

The steady-state amplitude is

A(omega)=rac{F_0}{sqrt{(k-momega^2)^2+(bomega)^2}}

This expression shows that response depends strongly on driving frequency. When damping is small, the amplitude becomes largest near the natural frequency.

At very low driving frequency, the system has time to follow the force almost statically. At very high frequency, inertia prevents large motion. Near resonance, the driver transfers energy efficiently.

Phase lag

The response does not generally stay in phase with the driving force. The phase lag satisfies

andelta=rac{bomega}{k-momega^2}

At low frequencies, displacement is nearly in phase with the force. Near resonance, the displacement lags by about $pi/2$. At high frequencies, displacement is nearly opposite in phase to the force.

Phase is crucial in understanding energy transfer. The driver does the most net work when the force and velocity are appropriately aligned.

Energy balance

In steady state, the oscillator's average energy no longer grows without bound because damping removes energy. Over each cycle, the energy supplied by the driving force equals the energy dissipated by damping.

Without damping, driving exactly at natural frequency would cause amplitude to grow without limit in the ideal linear model. Real systems always have some damping or nonlinear limit.

Examples

A radio receiver selects a frequency by responding strongly to a specific driving signal. A musical instrument body responds to string vibrations and amplifies certain frequencies. A building can sway strongly if driven by wind or ground motion near a resonant frequency.

Driven oscillators appear in mechanics, electronics, acoustics, optics, and quantum systems.

The big idea

A driven oscillator is controlled by both its own natural dynamics and the external driving frequency. Damping removes transient motion, leaving a steady-state response at the drive frequency. The amplitude and phase of that response depend on frequency, leading directly to resonance and frequency selection.