Damped Oscillations & Ringdown

Can a position record measure damping even without visible peaks? Yes. Fit the whole position record to the viscous oscillator rather than requiring a train of peaks. The same model covers ringing and nonoscillatory returns. Check the residual and parameter uncertainty: near critical damping, noise may prevent you from distinguishing the two sides. Settling time also depends on the chosen displacement band and release.

Record a spring and viscous damper. Fit noisy positions across underdamped, critical and overdamped returns, then inspect uncertainty and settling time.

Fitted damping ratio ± standard error $0.12\pm1.7\times10^{-4}$

Second-order quality parameter $4.17$

Predicted final entry into 2% band $8.03\,\mathrm{s}$

Position-fit RMS $0.52\,\mathrm{mm}$

Model assumptions

A point mass slides horizontally with a massless Hooke spring and linear viscous damper. The apparatus shows the true simulated motion; red camera samples include independent Gaussian position noise at exact uniform times. Nonlinear least squares fits angular frequency, damping ratio, release displacement and equilibrium to those samples across all damping regimes. It receives the controlled initial velocity, but not the true stiffness or damping. The mass is known. Search bounds are 1–16 radians per second and damping ratios 0.005–3. Standard errors use a local linear covariance with four fitted parameters and omit time, mass, camera-calibration and model uncertainty. Near-critical results may not distinguish which side of critical damping applies. The quality readout is the second-order parameter; its energy-per-cycle interpretation applies only at weak damping. Settling means the final entry into a band of two percent of fitted initial displacement, predicted by the fitted model and possibly beyond the record. No dry friction, coil inertia, collisions or nonlinear drag is modeled.

Focus question: Can a noisy record distinguish damping regimes and predict when motion stays close to equilibrium?
One-sentence intuition: Fit the whole position record, then inspect its residual and parameter uncertainty. Near critical damping, noise may leave the regime unresolved even when a curve looks convincing.

Core mathematical model

One equation, three damping regimes

m\ddot q+c\dot q+kq=0,\quad\omega_0=\sqrt{k/m},\quad\zeta=\frac{c}{2\sqrt{mk}}

Damping ratios below, at and above one define underdamped, critical and overdamped motion. The model remains continuous through the transition.

Fit camera positions, not the ideal motion

x_i=x_e+q(t_i;\omega_0,\zeta,q_0,v_0)+\epsilon_i,\quad\min\sum_i\bigl[x_i-\widehat x(t_i)\bigr]^2

Release velocity is controlled and known; initial displacement, equilibrium, frequency and damping ratio are fitted. Search does not receive the true frequency or damping.

A named settling criterion

t_s=\inf\{t:\ |\widehat q(\tau)|\le0.02|\widehat q_0|\ \text{for every }\tau\ge t\}

This is final band entry, not the first crossing. Changing the tolerance or initial velocity changes the comparison between damping choices.

Dissipation and the quality parameter

E=\tfrac12m\dot q^2+\tfrac12kq^2,\quad\dot E=-c\dot q^2,\quad Q=\frac1{2\zeta}

The quality parameter is defined for the second-order system. Interpreting it as stored energy divided by loss per cycle is a weak-damping approximation, and an overdamped return has no oscillation period.

Common difficulties

A critical-looking curve is not exact critical damping

Typical misconceptionThe fitted damping ratio is near one, so critical damping is proved.

Better mental modelThe local confidence interval can straddle one. Collect more useful early-time data before naming a side of the transition.

More damping need not mean earlier settling

Typical misconceptionThe strongest damper always stops motion first.

Better mental modelA large damper can make the slow mode return very slowly. A finite band can also favour a slightly underdamped release over an exactly critical one.

Run the experiment

01
Read the fading swings
Use Many fading swings and compare the red observations with the fitted curve.
What to observe: Residuals vary around zero while the fitted damping ratio remains constrained.
02
Remove the peaks
Choose Return without ringing.
What to observe: The full-position fit still works; it does not need a logarithmic peak train.
03
Ask what the camera can resolve
Choose Critical neighbourhood and increase position noise in the details.
What to observe: Parameter uncertainty grows. If the record cannot constrain both nonlinear parameters, their readouts are hidden with an explanation.