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Sandbox Physics

M064 · Oscillations / frequency response

Driven Resonance & Phase Lag

A sinusoidal force actuator drives a spring and dashpot. Drag the frequency cursor, inspect startup transients and measure amplitude and phase over three complete drive cycles. Sweep nineteen independently integrated experiments against the exact response curve; track drive work, dissipation and stored energy.

Interactive modelDriven Resonance & Phase Lag
Elapsed time0 s0\,\mathrm{s}
Displacement from equilibrium0 m0\,\mathrm{m}
Velocity0 m s−10\,\mathrm{m\,s^{-1}}
Undamped reference period0 s0\,\mathrm{s}
Mechanical energy relative to equilibrium0 J0\,\mathrm{J}
Integrated dissipation0 J0\,\mathrm{J}
Integrated drive work0 J0\,\mathrm{J}
Energy ledger defect0 J0\,\mathrm{J}
Position error against exact solution0 m0\,\mathrm{m}
Endpoint change with half step0 m0\,\mathrm{m}
Actual integration step0 ms0\,\mathrm{ms}
Exact steady amplitude0 m0\,\mathrm{m}
Exact steady phase lag0∘0{}^{\circ}
Three-cycle measured amplitudeCollecting\text{Collecting}
Three-cycle measured phase lagCollecting\text{Collecting}
Harmonic projection residual RMSCollecting\text{Collecting}
Exact steady absorbed power0 W0\,\mathrm{W}
Absorbed-power half-maximum bandwidth0 Hz0\,\mathrm{Hz}
Quality factor00
Measured frequency points00
Displacement peak locationCollecting\text{Collecting}

Physics tutorial

Separate resonance from startup

BackgroundA periodic actuator continually supplies energy while damping removes it. The resulting motion combines a decaying startup transient with a steady response at the drive frequency. Amplitude alone misses the phase and power balance.

Why it mattersOpenStax University Physics Volume 1, section 15.6 supplies the forced-oscillation reference. Nineteen independent numerical trajectories and a three-cycle Fourier measurement put the response curve through an actual measurement pipeline.

Start with the essentials

Focus question
Where is the displacement peak, where is the power peak, and has the measured response settled?
One-sentence intuition
Frequency response depends on the measured quantity. Wait for transients, inspect projection residuals and compare phase as well as amplitude.

Core mathematical model

Complex frequency response

mx¨+cx˙+kx=F0sin⁡(ωt),A=F0(k−mω2)2+c2ω2m\ddot x+c\dot x+kx=F_0\sin(\omega t),\quad A=\frac{F_0}{\sqrt{(k-m\omega^2)^2+c^2\omega^2}}

The exact amplitude describes the infinite-time particular solution. The actual trajectory also contains a homogeneous transient set by the initial displacement and velocity.

Phase and displacement peak

ϕ=atan2⁡(cω,k−mω2),ωpeak=ω01−2ζ2(ζ<1/2)\phi=\operatorname{atan2}(c\omega,k-m\omega^2),\quad\omega_{\mathrm{peak}}=\omega_0\sqrt{1-2\zeta^2}\quad(\zeta<1/\sqrt2)

Displacement lags force. Above the stated damping threshold amplitude falls monotonically from zero frequency. The peak action is restricted to the displayed scan band, so it can select its lower boundary.

Power resonance and its bandwidth

P‾=F02cω22[(k−mω2)2+c2ω2],ΔωP=cm,Q=mω0c\overline P=\frac{F_0^2c\omega^2}{2[(k-m\omega^2)^2+c^2\omega^2]},\quad\Delta\omega_{P}=\frac cm,\quad Q=\frac{m\omega_0}{c}

Absorbed power peaks at natural frequency even when displacement peaks elsewhere. The exact power half-maximum width is not the displacement half-height width; the displayed bandwidth converts angular frequency to cycles per second.

Three-cycle projection and energy ledger

A^=as2+ac2,ϕ^=atan2⁡(−ac,as),E−E0−Wdrive+Q=0\widehat A=\sqrt{a_s^2+a_c^2},\quad\widehat\phi=\operatorname{atan2}(-a_c,a_s),\quad E-E_0-W_{\mathrm{drive}}+Q=0

Sine and cosine projections use only the last three complete drive cycles, with 128 uniform phase samples per cycle. Startup transients contaminate this projection and appear in the nonharmonic residual. Work and dissipation are integrated independently.

Common difficulties

Two different resonance peaks

Typical misconceptionDisplacement and absorbed power must peak at the same frequency.

Better mental modelIncrease damping and locate the displacement peak. Power still peaks at natural frequency; strong damping removes an interior displacement peak.

A complete window can still be transient

Typical misconceptionThree complete periods automatically measure steady state.

Better mental modelMeasure early and late. The Fourier projection is defined at both times, but the early residual and phase can reveal startup contamination.

Bandwidth needs a stated observable

Typical misconceptionThe power bandwidth is the displacement half-height width.

Better mental modelThe displayed width belongs to absorbed power and is exact for this viscously damped model. Do not apply it to another response curve.

Run the experiment

  1. 01

    Resolve startup

    Scrub through the first three drive cycles, then measure the current frequency at the endpoint.

    What to observe: A measurement is unavailable before a full window; its later amplitude and phase approach the steady reference.
  2. 02

    Build a response curve

    Measure nineteen frequencies and drag the amplitude cursor across the peak.

    What to observe: Each blue point comes from an independent solved experiment, not a copy of the analytical curve.
  3. 03

    Separate amplitude and power

    Increase damping, find the amplitude peak and compare absorbed power at natural frequency.

    What to observe: The displacement peak shifts down; power resonance remains at natural frequency.
  4. 04

    Check the energy budget

    Compare drive work with dissipated energy and stored energy late in the run.

    What to observe: Work can exceed the current stored energy because the dashpot continuously removes energy.