M063 · Oscillations / system identification

Damped Oscillations & Ringdown

Release a spring and viscous dashpot, acquire noisy camera frames, then regress the logarithms of resolved positive peaks. Estimate decay rate and damped frequency from observations; use calibrated mass to reconstruct damping and stiffness. Compare underdamped spirals, critical return and overdamped relaxation.

Elapsed time $0\,\mathrm{s}$

Displacement from equilibrium $0\,\mathrm{m}$

Velocity $0\,\mathrm{m\,s^{-1}}$

Undamped reference period $0\,\mathrm{s}$

Mechanical energy relative to equilibrium $0\,\mathrm{J}$

Integrated dissipation $0\,\mathrm{J}$

Integrated drive work $0\,\mathrm{J}$

Energy ledger defect $0\,\mathrm{J}$

Position error against exact solution $0\,\mathrm{m}$

Endpoint change with half step $0\,\mathrm{m}$

Actual integration step $0\,\mathrm{ms}$

Damping regime $\text{Collecting}$

Envelope / slow-pole timescale $0\,\mathrm{s}$

Nominal damped frequency $0\,\mathrm{Hz}$

Decay rate from camera peaks $\text{Collecting}$

Damping from camera peaks $\text{Collecting}$

Stiffness from camera peaks $\text{Collecting}$

Logarithmic peak residual RMS $\text{Collecting}$

Peaks used by fit $0$

Acquired camera frames $1$

RELEASE / CAPTURE / IDENTIFY

Identify a fading system

Exact trajectory referenceHide truth to inspect the acquired observations alone.Moving mass

m

1.00\,\mathrm{kg}

Spring stiffness

k

20.00\,\mathrm{N\,m^{-1}}

Release displacement

x_0

0.40\,\mathrm{m}

Initial velocity

v_0

0.00\,\mathrm{m\,s^{-1}}

Damping ratio

\zeta

0.06\,

Camera frame rate

f_s

40.00\,\mathrm{Hz}

Position noise standard deviation

\sigma_x

0.00\,\mathrm{m}

Maximum integration step

\Delta t_{\max}

20.00\,\mathrm{ms}

Inspect experiment time

t/t_{\mathrm{end}}

0.00\,\%

Drag the mass or use keyboard sliders. A new capture changes only the noise realization. Blue dots are acquired frames; gold is the log-peak fit. The fit sees no future frames.

Linear viscous damping and a massless Hookean spring. The synthetic camera has exact uniform timestamps and independent Gaussian position noise. Peak inference requires three positive peaks above eight noise standard deviations and at least eight frames per nominal damped period. Known mass and nominal stiffness configure the detector separation and sampling gate. Selection, finite sampling and noise bias the fit; the logarithmic residual is not a confidence interval. Nonoscillatory or unresolved data produce no peak fit.

Physics tutorial

Read the disappearing motion

BackgroundA viscous dashpot drains energy continuously. An underdamped oscillator still crosses equilibrium while its peak envelope decays; critical and overdamped systems require a different interpretation because oscillatory peaks disappear.

Why it mattersOpenStax University Physics Volume 1, section 15.5 supplies the damped-oscillator reference. Here the camera observation model and the peak detector are visible parts of the experiment, so changing sampling or noise can invalidate an otherwise correct physical model.

Start with the essentials

Focus question: Can camera peaks recover two physical coefficients, and when must that method refuse to answer?
One-sentence intuition: Logarithmic peak decay identifies damping while peak spacing identifies damped frequency. The known mass converts them to coefficients; noisy peak selection can create bias.

Core mathematical model

Three damping regimes

m\ddot x+c\dot x+kx=0,\quad\gamma=\frac c{2m},\quad\zeta=\frac c{2\sqrt{mk}},\quad\omega_d=\sqrt{\omega_0^2-\gamma^2}

The damped frequency is real only below critical damping. At critical damping the two decay poles coincide; above it two real decay rates replace the oscillation.

Resolved peaks identify decay

\log x_{\mathrm{peak},i}=a-\gamma t_i+r_i,\qquad\widehat\omega_d=\frac{2\pi(N-1)}{t_N-t_1}

A log-amplitude line and successive positive peak spacing estimate two parameters from camera data. Quadratic interpolation localizes each sampled peak; the detector rejects peaks near the noise floor.

Recover stiffness and damping

\widehat c=2m\widehat\gamma,\qquad\widehat k=m(\widehat\omega_d^2+\widehat\gamma^2)

Mass is calibrated independently. Nominal mass and stiffness configure detector separation and the sampling gate; they do not replace the fitted decay rate or frequency. Noise selection can bias the result.

Dissipation is an independently integrated ledger

Q(t)=\int_0^t c\dot x^2\,\mathrm dt,\qquad E(t)-E(0)+Q(t)=0

Dissipated energy is integrated from damping power, rather than defined as the missing mechanical energy. The remaining imbalance tests the numerical solve and dense output.

Common difficulties

Critical does not mean oscillating quickly

Typical misconceptionCritical damping has the shortest oscillation period.

Better mental modelIt is nonoscillatory. The peak-spacing estimator is inapplicable even though the solver remains valid.

More noisy peaks can increase bias

Typical misconceptionEvery visible bump improves the decay fit.

Better mental modelLate peaks can be noise. The detector imposes a noise floor and a minimum separation; selection itself can bias log amplitudes.

Sampling and integration are different clocks

Typical misconceptionA fine solver step guarantees the camera resolves oscillation.

Better mental modelRaise stiffness and lower camera frame rate. The sampling gate can fail while exact-reference and energy checks still pass.

Run the experiment

01
Recover the noiseless system
Set noise to zero, capture a ringdown and compare fitted stiffness with the control.
What to observe: Resolved peaks recover damping and stiffness within finite sampling error.
02
Change the damping regime
Compare resolved, critical and overdamped presets.
What to observe: The spiral disappears and no oscillatory peak fit is reported for nonoscillatory motion.
03
Challenge the camera
Use the imperfect camera, repeat capture, then raise stiffness and lower frame rate.
What to observe: Noise changes the estimate; under-resolved sampling suppresses the fit instead of reporting a false coefficient.
04
Account for the lost energy
Scrub late into the ringdown and compare stored energy, integrated dissipation and the ledger defect.
What to observe: Dissipation replaces the lost mechanical energy; the numerical defect remains small.