Driven Resonance & Phase Lag

Do a driven oscillator's largest swing and largest power intake happen at the same frequency? The power peak stays at the natural frequency. Damping shifts the displacement peak lower and can remove it.

Sweep a force-driven carriage through resonance. Compare its motion with the drive, measure amplitude and phase, and see where energy is absorbed.

Acquired amplitude $\text{Loading}$

Acquired phase lag $\text{Loading}$

Acquired mean power $\text{Loading}$

Model quality factor $\text{Loading}$

A horizontal mass has a massless Hookean spring and linear viscous damping. A sinusoidal force acts on the mass; its spring anchor stays fixed. Hardware dimensions are schematic.

Focus question: Do displacement and absorbed power peak at the same frequency?
One-sentence intuition: At the natural frequency, velocity follows the force and absorbed power peaks. Damping shifts the displacement peak lower or removes it.

Core mathematical model

A force drives the mass

m\ddot x+c\dot x+kx=F_0\sin\omega t

The mass moves horizontally. The force actuator pushes the mass directly; the spring anchor remains fixed.

Steady amplitude and lag

\begin{gathered}A=\frac{F_0}{\sqrt{(k-m\omega^2)^2+(c\omega)^2}}\\\phi=\operatorname{atan2}(c\omega,k-m\omega^2)\end{gathered}

After the startup fades, motion follows the drive frequency. Lag is measured behind the applied sine force.

Average power and its bandwidth

\begin{gathered}\overline P=\tfrac12 c\omega^2 A^2\\\Delta f=\frac{c}{2\pi m},\qquad Q=\frac{m\omega_0}{c}\end{gathered}

These are exact steady results for linear viscous damping. Bandwidth is the half-maximum width of absorbed power.

The displacement peak shifts

\begin{gathered}\omega_{\rm peak}=\omega_0\sqrt{1-2\zeta^2}\\\zeta<1/\sqrt2\end{gathered}

At stronger damping there is no positive-frequency displacement peak. The power peak and a quarter-cycle displacement lag remain at the natural frequency.

Where drive work goes

\begin{gathered}E=\tfrac12 m\dot x^2+\tfrac12kx^2\\E-E_0=W-Q_{\rm heat}\end{gathered}

Work supplied and dashpot heat are integrated independently along the numerical trajectory.

Common difficulties

Two different peaks

Typical misconceptionThe mass must absorb the most power wherever displacement is largest.

Better mental modelPower depends on velocity and on when the force acts. Compare amplitude and mean power as damping grows.

Run the experiment

01
Sweep through resonance
Move Drive frequency ratio below and above one. Compare Build a resonance with Move against the push.
What to observe: The motion grows near the peak and changes from following the force to moving nearly against it.
02
Broaden the response
Increase Damping ratio, then choose Record sweep.
What to observe: The measured power peak broadens. Bandwidth and an estimated quality factor appear in the sweep diagnostics.
03
Watch a startup
Drag the gold carriage to prepare a release, or choose Startup.
What to observe: The initial free motion fades into a response at the drive frequency.
04
Keep the samples
Choose Raw CSV and open Simulated observations and sweep results.
What to observe: Each sweep point has its own sampled trajectory. Measurements and the steady reference can differ while a transient remains.