M063 · Oscillations / system identification

Damped Oscillations & Ringdown

When can the peaks of a filmed oscillation give you its damping and stiffness? When the motion is underdamped and each peak is clearly resolved. The steady shrink from one peak to the next, measured on a log scale, gives the damping rate; the spacing between peaks gives the damped frequency. With the mass known, these become the damping coefficient and the stiffness. Picking peaks from noisy frames can bias both.

Release a mass on a spring with a viscous dashpot, record noisy camera frames, and fit a straight line to the logarithms of the positive peaks you can resolve. The data give the decay rate and the damped frequency; with the calibrated mass you can then work out the damping and the stiffness. Compare an underdamped spiral, a critically damped return and an 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 balance defect $0\,\mathrm{J}$

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

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

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

Signed spring force along motion $0\,N$

Acceleration along motion $0\,\mathrm{m/s^2}$

Instantaneous actuator power $0\,W$

Instantaneous dissipated power $0\,W$

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 $1$

Film a ringdown and fit its damping and stiffness

Identify a fading system

Physical force vectorsSpring, drive, damping, gravity and track reaction forces are drawn to one common scale at each instant.Velocity vectorBlue arrow with its own scale, separate from force. Positive motion is to the right or downward.Recent acquired pathShows only the last two reference periods; future motion is never drawn.Exact trajectory referenceHide the exact curve to look at the camera data 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\,\%

Rotate the 3D view around the simulated spring and dashpot. Drag the mass in the coordinate view, or use the sliders from the keyboard. Switching to the energy view does not advance the record. A new capture changes only the random noise. Blue dots are captured frames, and the gold line is the fit to the log of the peaks. The fit never uses frames that come later in the record.

The damping is linear and viscous, and a massless Hooke spring acts on the mass along a horizontal track. The 3D apparatus shows the simulated motion, not the fitted or noisy camera positions. Its length is scaled to fit the whole computed swing; the hardware is schematic, and collisions and wire deformation are not modeled. The simulated camera takes frames at exactly even times and adds independent Gaussian noise to each position. A peak fit needs three positive peaks above eight noise standard deviations and at least eight frames per nominal damped period. The known mass and nominal stiffness set the peak spacing the detector expects and the sampling threshold. Peak selection, finite sampling and noise all bias the fit, and the residual of the log fit is not a confidence interval. Motion that does not oscillate, or peaks that cannot be resolved, give no peak fit.

Physics tutorial

Read the disappearing motion

BackgroundA viscous dashpot drains energy all the time. An underdamped oscillator still swings through equilibrium while its peaks shrink. Critically damped and overdamped systems have no oscillating peaks, so they have to be read a different way.

Why it mattersOpenStax University Physics Volume 1, section 15.5, covers damped oscillators. Here the camera and the peak detector are visible parts of the experiment, so changing the sampling or the noise can make the measurement fail even when the physical model is correct.

Start with the essentials

Focus question: Can peaks seen by a camera give back two physical coefficients, and when should the method give no answer?
One-sentence intuition: How fast the logarithm of the peaks falls gives the damping, and the spacing of the peaks gives the damped frequency. The known mass turns these into coefficients, but picking peaks out of noise can bias them.

Core mathematical model

Three kinds of damping

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 give the decay rate

\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 straight-line fit to the log of the peak heights and the spacing between successive positive peaks give two parameters from the camera data. Quadratic interpolation pins down each sampled peak, and the detector rejects peaks close to the noise floor.

Recover stiffness and damping

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

The mass is calibrated separately. The nominal mass and stiffness set the peak spacing the detector expects and the sampling threshold; they never stand in for the fitted decay rate or frequency. Picking peaks out of noise can bias the result.

Energy lost to damping, integrated separately

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

The energy lost is found by integrating the damping power, not by taking whatever mechanical energy is missing. Any leftover imbalance tests the numerical solution and its interpolation between steps.

Common difficulties

Critical does not mean oscillating quickly

Typical misconceptionCritical damping has the shortest oscillation period.

Better mental modelCritically damped motion does not oscillate at all. The peak-spacing method does not apply, although the simulation itself is still valid.

More noisy peaks can make the bias worse

Typical misconceptionEvery visible bump improves the decay fit.

Better mental modelLate peaks may just be noise. The detector requires peaks to clear a noise floor and to be a minimum distance apart, and choosing peaks this way can itself bias the log amplitudes.

Sampling and integration are different clocks

Typical misconceptionA fine solver step guarantees the camera resolves oscillation.

Better mental modelWith higher stiffness and a lower frame rate, the camera's sampling check can fail while the exact reference and the energy check still pass.

Run the experiment

01
Start without noise
Set Position noise standard deviation to zero, press Capture a new ringdown and compare the fitted stiffness with the value you set.
What to observe: The resolved peaks give back the damping and stiffness, up to the error from finite sampling.
02
Change the kind of damping
Compare the Resolved ringdown, Critical damping and Overdamped return presets. Rotate the view around the dashpot, pause and use Fine motion step to compare the damping force with the velocity.
What to observe: The spiral disappears, and motion that does not oscillate gets no peak fit.
03
Make the camera work harder
Choose Imperfect camera, capture several times, then raise the stiffness and lower Camera frame rate.
What to observe: Noise changes the estimate. When the sampling is too coarse, the Lab gives no fit at all instead of a false coefficient.
04
Track the lost energy
Switch to Energy & power, drag late into the ringdown and compare the stored energy, the integrated loss and the remaining imbalance. The 3D motion is the simulated state; the fitted coefficients still use only the camera peaks.
What to observe: The energy lost to damping matches the mechanical energy that disappears, and the numerical imbalance stays small.