---
title: Damped Oscillators
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 3
order: 703
summary: >
  Every real oscillator eventually stops: friction, drag, and internal loss
  drain its energy, so free motion is a decay rather than a permanent swing.
  Adding a velocity-proportional resistance to the spring-mass equation produces
  one dimensionless number, $b/(2\sqrt{mk})$, that decides whether the mass rings
  down through many cycles, returns once without overshoot, or crawls back
  slowly. We solve the three regimes, tie the observed decay to the power balance
  $b\dot x^2$, and turn a measured ring-down into the decay rate and quality
  factor of the apparatus — reading damping off the data instead of assuming it.
topics: [Oscillations]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 14 — Oscillations; §14-4"
---

## Damping models and free solutions

A free oscillator evolves after its external drive has been removed. Its subsequent
motion exposes the mechanisms that remove mechanical energy: viscous drag in a
liquid, internal loss in a spring, friction in a pivot, electrical loss in a
transducer, or radiation into an attached medium. A linear model is appropriate
when the resisting force is proportional to velocity over the measured range:

$$
F_d=-b\dot x.
$$

The coefficient $b$ has SI units $\mathrm{kg\,s^{-1}}$, equivalently
$\mathrm{N\,s\,m^{-1}}$. Its sign follows from the velocity. A mass moving toward
positive $x$ experiences a negative damping force; a mass moving toward negative
$x$ experiences a positive damping force. The force therefore performs negative
work on every nonzero-velocity interval.

Newton's second law for a mass $m$ attached to a linear spring of stiffness $k$
gives

$$
m\ddot x+b\dot x+kx=0.
$$

The sign convention matters. The restoring term $kx$ and the damping term
$b\dot x$ appear on the same left-hand side because their forces are
$-kx$ and $-b\dot x$. A recorded displacement that grows in a nominally passive
ring-down warns of a reversed coordinate in one channel,
the source may still be connected, or the fitted model may omit an energy input.

$$
% caption: A mass sliding on a horizontal surface, tied to a fixed wall by a spring
% and a viscous damper in parallel. Displacement is measured from static equilibrium;
% the spring force grows with displacement while the damper force opposes velocity, so
% the damper drains mechanical energy on both halves of a cycle.
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\node at (1.78,1.46) {damper};
\node[above] at (5.15,2.62) {$x$};
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$$

Check the force signs at one instantaneous state. Take $x>0$ and $\dot x>0$;
then both $-kx$ and $-b\dot x$ point toward negative $x$. Reversing the velocity
flips the damping arrow while leaving the spring arrow unchanged.

Two combinations of $m$, $b$, and $k$ organize the analysis:

$$
\omega_0=\sqrt{\frac{k}{m}},
\qquad
\gamma=\frac{b}{2m},
\qquad
\zeta=\frac{\gamma}{\omega_0}
=\frac{b}{2\sqrt{mk}}.
$$

$\omega_0$ is the angular frequency of the corresponding undamped oscillator.
$\gamma$ has units of inverse time and sets the exponential amplitude-decay rate.
The damping ratio $\zeta$ is dimensionless. Rewriting the differential equation
with these parameters produces

$$
\ddot x+2\gamma\dot x+\omega_0^2x=0
\qquad\text{or}\qquad
\ddot x+2\zeta\omega_0\dot x+\omega_0^2x=0.
$$

The three parameters describe different features of the apparatus. The dimensionless
comparison $b/(2\sqrt{mk})$ includes mass and stiffness as well as the damper.
Increasing the mass changes the damping ratio of the same physical damper. A
laboratory report is clearer when it states either $(m,k,b)$ with units or
$(\omega_0,\gamma,\zeta)$, then converts between the two sets explicitly.

The mechanical energy of the spring--mass system is

$$
E=\frac12m\dot x^2+\frac12kx^2.
$$

Differentiate and use the equation of motion:

$$
\frac{\d E}{\d t}
=m\dot x\ddot x+kx\dot x
=\dot x(m\ddot x+kx)
=-b\dot x^2.
$$

The right side is nonpositive when $b>0$. It vanishes only at a turning point,
where velocity is momentarily zero, or when the system is at rest. The relation
ties the observed decay to a power balance: for linear damping, energy loss is
largest near the equilibrium crossing, where speed is largest, and smallest near
an extreme displacement.

The linear viscous model applies over a limited operating range. Dry sliding friction
can have approximately constant magnitude over part of a stroke. Aerodynamic drag
at high Reynolds number often scales closer to $\dot x|\dot x|$. A spring can
exhibit frequency-dependent internal loss, and a sensor can add electrical loading.
Such effects alter the envelope, frequency, or both. Residuals and repeated
measurements establish whether the linear model is adequate for the selected range.

### Roots, regimes, and free solutions

Try a solution $x=e^{rt}$. Substitution gives the characteristic equation

$$
r^2+2\gamma r+\omega_0^2=0,
\qquad
r=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}.
$$

The discriminant separates the free response into three regimes. Classify the
response by $\zeta$, using a numerical damping ratio rather than an informal
description.

- **Underdamped response, $\zeta<1$.** The roots are a complex-conjugate pair
  $-\gamma\pm i\omega_d$. The displacement oscillates while its envelope decays.
- **Critical damping, $\zeta=1$.** The characteristic equation has a repeated
  real root. Return to equilibrium is nonoscillatory and is fastest within this
  ideal linear family for a release with the same initial state.
- **Overdamped response, $\zeta>1$.** Two distinct negative real roots produce
  a sum of decaying exponentials. The slower root controls the late-time tail.

$$
% caption: Root locations for the three free-response regimes. Underdamped roots
% share a negative real part and nonzero imaginary parts; critical damping has one
% repeated negative root; overdamped roots lie separately on the negative real axis.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.70,1.60)--(5.95,1.60) node[right] {real part};
\draw[->,black] (3.40,.50)--(3.40,2.80) node[above] {imaginary part};
\draw[fill=acc!12,draw=acc,thick] (2.75,2.40) circle (2.6pt);
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\node[below] at (1.45,1.52) {over};
\node[below] at (3.05,1.52) {over};
\end{tikzpicture}
$$

For $\zeta<1$, define the damped angular frequency

$$
\omega_d=\sqrt{\omega_0^2-\gamma^2}
=\omega_0\sqrt{1-\zeta^2}.
$$

The displacement can then be written in either amplitude--phase form or in a form
that exposes the initial values:

$$
x(t)=A e^{-\gamma t}\cos(\omega_dt+\phi)
$$

and

$$
x(t)=e^{-\gamma t}
\left[
x_0\cos(\omega_dt)
+\frac{v_0+\gamma x_0}{\omega_d}\sin(\omega_dt)
\right].
$$

Here $x_0=x(0)$ and $v_0=\dot x(0)$. The symbol $v_0$ denotes initial
velocity in this equation, whereas $\omega_0$ denotes undamped angular frequency.
Keeping the symbols visually distinct avoids a common transcription error in
ring-down fitting notebooks.

The observed period is

$$
T_d=\frac{2\pi}{\omega_d}.
$$

Weak damping changes the frequency only at second order in $\zeta$:

$$
\omega_d
=\omega_0\sqrt{1-\zeta^2}
\approx\omega_0\left(1-\frac{\zeta^2}{2}\right).
$$

Consequently, an accurate frequency measurement alone has little sensitivity to a
weak damper. The envelope or peak-to-peak logarithmic decrement constrains
$\gamma$ much more strongly.

$$
% caption: An underdamped displacement record with exponential envelopes. Successive
% positive peaks lie on $+Ae^{-\gamma t}$ and negative peaks lie on
% $-Ae^{-\gamma t}$; their spacing measures $\omega_d$, while their height ratio
% measures the decay rate independently of the phase at release.
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\node[black,above] at (4.30,2.06) {upper envelope};
\node[black,below] at (4.30,1.14) {lower envelope};
\end{tikzpicture}
$$

At critical damping, the general solution has the form

$$
x(t)=(C_1+C_2t)e^{-\omega_0t}.
$$

At overdamping, use the two real rates

$$
r_{\pm}=-\gamma\pm\sqrt{\gamma^2-\omega_0^2},
\qquad
x(t)=C_+e^{r_+t}+C_-e^{r_-t}.
$$

Both exponents are negative for positive $m$, $b$, and $k$. The term with the
smaller magnitude exponent survives at late times. Increasing $b$ beyond the
critical value can lengthen this late tail, which explains why more damping can
produce a slower settling trace.

$$
% caption: Displacement records after the same initial release. The underdamped
% trace crosses equilibrium repeatedly, the critical trace reaches the baseline
% without crossing and avoids a long tail, and the overdamped trace returns more
% slowly because its slow exponential component remains visible.
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\node[black] at (4.74,1.62) {over};
\end{tikzpicture}
$$

## Energy, quality factor, and phase space

The underdamped envelope is $A(t)=Ae^{-\gamma t}$. Since the cycle-averaged
mechanical energy is proportional to amplitude squared,

$$
\langle E(t)\rangle=E_0e^{-2\gamma t}.
$$

Many texts define the energy time constant

$$
\tau_E=\frac{1}{2\gamma}=\frac{m}{b},
$$

so that $\langle E\rangle=E_0e^{-t/\tau_E}$. The amplitude time constant is
$\tau_A=1/\gamma=2m/b$. A statement such as “the decay time is $3\ \mathrm{s}$”
is incomplete unless it identifies whether the fitted quantity is amplitude, energy,
or a voltage whose relationship to displacement has been established.

$$
% caption: Amplitude and energy envelopes for the same underdamped oscillator.
% Energy falls twice as rapidly on a logarithmic scale because it is proportional to
% amplitude squared; the two marked time constants therefore differ by a factor of
% two even though both describe one physical ring-down.
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\definecolor{acc}{HTML}{4A6FA5}
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\end{tikzpicture}
$$

The quality factor $Q$ gives a dimensionless measure of weak loss:

$$
Q=\omega_0\tau_E=\frac{\omega_0}{2\gamma}
=\frac{\sqrt{mk}}{b}
=\frac{1}{2\zeta}.
$$

The relation applies to the linear free oscillator and connects a local parameter
$b$ with a directly measurable decay rate. For $Q\gg1$, the fractional energy
loss per cycle is approximately

$$
\frac{|\Delta E|}{E}\approx\frac{2\pi}{Q}.
$$

The approximation assumes that energy changes little during one cycle. It should
not be used near critical damping, where a cycle is no longer a recurring
unit of motion.

The logarithmic decrement estimates damping from separated peaks. If $X_n$ and
$X_{n+N}$ are same-sign extrema $N$ cycles apart,

$$
\delta_N=\frac{1}{N}\ln\left(\frac{|X_n|}{|X_{n+N}|}\right)
=\gamma T_d
=\frac{2\pi\gamma}{\omega_d}.
$$

For weak damping,

$$
Q\approx\frac{\pi}{\delta_N}.
$$

Using peaks several cycles apart suppresses relative reading error when individual
peak heights are noisy. It also increases sensitivity to drift in the equilibrium
position or damping coefficient. The separation $N$ should therefore be stated
along with the peak indices and the fitting interval.

$$
% caption: Logarithmic decrement from two same-sign peaks. The ratio uses peak
% heights measured relative to the local equilibrium baseline, not raw sensor
% voltages; choosing peaks several cycles apart reduces height-readout noise while
% retaining the exponential-envelope relation.
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\node[below] at (1.22,1.34) {$X_n$};
\node[below] at (4.82,1.34) {$X_{n+N}$};
\end{tikzpicture}
$$

### Phase-plane geometry and initial conditions

A displacement trace records only one of the two state variables. The pair
$(x,\dot x)$ gives a complete state for the linear free model: specifying both
quantities at one instant fixes the subsequent trajectory. In the undamped limit,
the state trajectory is a closed ellipse,

$$
\frac{x^2}{A^2}+\frac{\dot x^2}{\omega_0^2A^2}=1.
$$

Its horizontal intercepts are turning points, where kinetic energy is zero. Its
vertical intercepts occur at equilibrium crossings, where spring potential energy is
zero. The direction around the curve identifies the sign of velocity. Rescaling
vertical velocity as $u=\dot x/\omega_0$ changes the ellipse of an ideal harmonic
oscillator into a circle, so equal-energy contours have the same radial scale.

Damping turns the closed ellipse into an inward spiral. The state crosses the
displacement axis once per turning point and returns closer to the origin after every
cycle. The rate of radial contraction depends on energy loss, while the angular
advance depends on $\omega_d$. A phase portrait therefore separates frequency
from decay in a compact way. A noisy record may look oscillatory in time while its
phase portrait exposes a drifting baseline, saturation near a sensor limit, or
different damping on the two sides of equilibrium.

$$
% caption: State-space trajectory for an underdamped free response. Each revolution
% corresponds to one displacement cycle, and the shrinking radius records the
% decreasing mechanical energy; the marked turning point has zero velocity while the
% marked center crossing has the largest speed for that cycle.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.60,1.58)--(5.98,1.58) node[right] {displacement};
\draw[->,black] (3.30,.42)--(3.30,2.80) node[above] {scaled velocity};
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\draw[black,thin] (5.26,1.66)--(5.46,1.98);
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\draw[black,thin] (3.36,2.45)--(3.80,2.62);
\node[anchor=west] at (3.80,2.62) {crossing};
\end{tikzpicture}
$$

Initial-condition formulas should be applied at the instant assigned as $t=0$.
A release from rest at displacement $X$ uses $x_0=X$ and $v_0=0$.
A launch through equilibrium with measured speed $V$ uses $x_0=0$ and
$v_0=V$. A data-acquisition trigger that fires several milliseconds after release
requires a phase offset or a fitted time origin. Treating the trigger timestamp as
the physical release point shifts the fitted phase and can bias the amplitude decay
when the record begins near a steep slope.

A displacement sensor often reports voltage $V_s$, not length. An affine
calibration

$$
x=a(V_s-V_{\rm base})
$$

requires both a scale factor $a$ and a baseline $V_{\rm base}$. Differentiate
only after filtering or fitting the displacement record. Finite differences magnify
sample-to-sample noise by a factor proportional to the inverse time step. A fitted
sinusoid has an analytic derivative and retains the correlation between phase,
frequency, and velocity. When a separate velocity probe is available, compare it
with the derivative of the calibrated displacement trace over the same clock base.

$$
% caption: Displacement and velocity over a single underdamped cycle. Velocity is
% zero at the two displacement extrema and reaches opposite extrema at the center
% crossings; the quarter-cycle offset provides a direct sign and timing check for
% dual-channel sensor data.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.66,1.54)--(5.92,1.54) node[right] {time};
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\node[below] at (1.37,.60) {peak};
\draw[black,dashed] (2.72,.60)--(2.72,2.44);
\node[below] at (2.72,.60) {crossing};
\node[acc] at (5.42,1.92) {$x$};
\node[black] at (5.42,.80) {$v$};
\end{tikzpicture}
$$

A phase-plane plot also distinguishes linear damping from a constant-magnitude
sliding force. Linear damping produces a smooth inward spiral whose local radial
shrinkage is greatest near the velocity extrema. Coulomb friction produces nearly
piecewise ellipses with a more abrupt reduction in amplitude at reversals. Compare
axes with consistent units or normalize them by characteristic displacement and
velocity scales. Quantitative estimates of $b$ still require an envelope or
full-trace reduction.

## Ring-down measurement and model discrimination

A ring-down experiment begins with a controlled initial state and ends before the
signal approaches the sensor noise floor. A practical apparatus contains a
spring--mass or pendulum system, a release mechanism, a displacement sensor, a
clocked acquisition channel, and a calibration reference. The release should avoid
an impulsive sideways force. A latch or electromagnet can hold the mass at a stated
displacement; the triggering circuit records the release event and the sample-clock
relationship. A pendulum can use a stop at a defined angle for the same purpose if
its removal does not strike the bob.

Sampling frequency should exceed the damped oscillation frequency by enough margin
to preserve phase and peak shape. A minimum of ten to twenty samples per cycle is
often adequate for a clean sinusoid, whereas a differentiating or peak-timing
analysis benefits from more. A sampling rate close to the signal frequency can
alias the motion into a slower apparent oscillation. Anti-alias filtering must be
considered with its phase delay; a filter whose response is not modeled can modify
the early transient and imitate extra damping.

Record length is bounded at both ends. The initial segment can contain
release transients, sensor clipping, or a short interval of continuing drive. The
late segment can be dominated by baseline noise, quantization, and nonlinear
sensitivity. Plot the raw trace, its local maxima and minima, and the residual after
a preliminary fit. A single numerical fit statistic cannot diagnose clipping at
early peaks or a slowly moving equilibrium baseline.

The full displacement model for an underdamped trace can include a baseline and a
time origin:

$$
x_{\rm model}(t)
=x_c+A e^{-\gamma(t-t_c)}
\cos\!\bigl[\omega_d(t-t_c)+\phi\bigr].
$$

The fitted parameters are correlated. Increasing $\gamma$ lowers late peaks;
raising $A$ compensates at early times. A small error in $\omega_d$ accumulates
as phase mismatch across a long record. A nonlinear least-squares fit over all
retained samples uses this covariance more efficiently than a single pair of peaks,
provided the residual pattern is inspected. Peak-ratio estimates remain valuable as
an independent diagnostic because they rely on a different data reduction.

Weighted fitting is appropriate when measurement variance changes across the record.
For example, a position sensor with approximately constant voltage noise has
approximately constant displacement variance after calibration, whereas a
logarithm of peak height has increasing relative uncertainty as peaks approach the
baseline. Report either the noise model used for weighting or the reason for an
unweighted fit. Parameter uncertainties from a fitting routine should be enlarged
when repeat runs disagree beyond their nominal statistical intervals.


### Damping mechanisms and model discrimination

The velocity-proportional law $F_d=-b\dot x$ is supported when the force--velocity
relation is approximately a straight line through the origin over the motion range.
A dashpot moving through oil can approach this behavior at low speed. The same form
also represents many small-loss systems after linearization about a working point.
The fitted $b$ then characterizes the stated operating range.

Dry contact produces a different idealization,

$$
F_{\rm C}=-F_c\,\sgn(\dot x),
$$

where the opposing force has approximately constant magnitude away from reversals.
The velocity sign changes abruptly at a turning point. Energy lost per cycle is then
closer to a fixed amount proportional to amplitude than to the quadratic-velocity
loss of a viscous damper. The peak sequence tends to decrease by nearly equal
amounts over a moderate range, whereas an exponential envelope gives a nearly
constant ratio of successive same-sign peaks.

At sufficiently high speed, fluid drag can be closer to

$$
F_q=-c_q\dot x|\dot x|.
$$

The large-amplitude part of a ring-down then decays faster than a constant-$b$
model predicts. A fit performed only on early peaks can return a larger apparent
damping rate than a fit performed only on late peaks. A single quoted $Q$ is
therefore meaningful only after the amplitude range and the damping model are
identified.

$$
% caption: Three resisting-force laws as functions of velocity. Linear viscous loss
% is a straight line, dry sliding loss changes sign near zero speed, and quadratic
% drag grows more rapidly at large speed; the slopes and curvature determine the
% shape of a free-decay envelope.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
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\draw[black,very thick] (3.30,1.04)--(5.00,1.04);
\draw[black,very thick,densely dashed] plot[domain=1.44:5.00,samples=200] (\x,{1.54-.02*(\x-3.22)-.16*(\x-3.22)^3});
\draw[black,thin] (1.62,2.28)--(1.56,2.58);
\node[black,above] at (1.56,2.58) {quadratic};
\draw[black,thin] (2.70,2.04)--(2.70,2.44);
\node[black,above] at (2.70,2.44) {sliding};
\draw[black,thin] (4.60,1.06)--(4.60,.64);
\node[acc,below] at (4.60,.62) {linear};
\end{tikzpicture}
$$

A direct diagnostic compares the observed peak sequence with two simple reductions.
Let $A_n$ denote the magnitude of the $n$-th same-sign peak after baseline
subtraction. For viscous damping, plot $\ln A_n$ against peak time. A straight
trend supports an exponential envelope. For sliding loss, plot $A_n$ itself
against cycle number. A straight trend supports an approximately constant peak
decrement. Neither graph proves a microscopic mechanism, but the pair makes the
model assumption visible.

The two reductions should use the same extrema. Missing a peak, mixing positive and
negative extrema with unequal offsets, or taking logarithms before subtracting the
baseline produces a false curvature. When a sensor clips at the first one or two
peaks, omit those samples from the accepted window and record the instrumental
limit. A report should state the accepted and rejected intervals together with the
reason for each exclusion.

Internal material loss introduces further possibilities. A metal spring can lose
energy through microscopic strain cycles; a rubber element can have a loss tangent
that depends on frequency and temperature; a pendulum pivot can combine rolling
resistance with air drag. An attached coil can generate current through motional
emf and dissipate energy in a load resistor. These mechanisms can be approximated
locally by an effective $b$, but the approximation should be tested when the
amplitude spans a wide range or when the ambient conditions change.

Temperature sweeps can isolate environmental contributions to $\gamma$. An increase in oil
viscosity, for example, can change $\gamma$ without changing the spring stiffness
substantially. A shift in $\omega_d$ alongside $\gamma$ can indicate that the
spring or support stiffness also changed. Repeat each condition with the same
release amplitude, timing rule, and sensor range. Comparing arbitrary traces from
different initial amplitudes mixes a nonlinear damping effect with an initial-energy
effect.

An equilibrium offset can mimic asymmetric damping. Suppose a temperature change
moves the sensor baseline upward while the oscillator decays. Positive peaks then
appear larger relative to a fixed zero line, and negative peaks appear smaller. Fit
a baseline model only if the data show one; a constant baseline is normally the
first model. A linear baseline $x_c(t)=x_{c0}+rt$ can be justified by a slow support
drift, but it should improve residual structure and remain small relative to the
ring-down amplitude. Higher-order baselines can absorb real decay behavior and
should be avoided without an independent physical basis.

## Critical damping and settling design

Critical damping separates oscillatory and nonoscillatory free motion. The required
coefficient is

$$
b_c=2\sqrt{mk}=2m\omega_0.
$$

The result applies to a single coordinate with linear stiffness and linear damping.
A vehicle suspension, door closer, balance mechanism, or positioning stage can have
several masses, nonlinear springs, geometric constraints, and rate-dependent
dampers. In those systems, $b_c$ is a local target for one mode and requires a
separate multi-mode design analysis.

A release from rest above equilibrium gives different traces in the three regimes.
An underdamped system crosses equilibrium and may overshoot repeatedly. A critically
damped system approaches without a crossing. An overdamped system also avoids a
crossing, yet its slow exponential can extend the settling time. The comparison
depends on how settling is defined. A common engineering convention specifies that
the magnitude of the deviation remains below a stated fraction of the initial
displacement, such as $2\%$ or $5\%$, for all later times.

$$
% caption: Settling-band comparison after an identical release. The shaded band marks
% a stated tolerance around equilibrium; the critical trace enters and remains in the
% band sooner than the overdamped trace, while the underdamped trace may leave it
% after an earlier crossing.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.66,1.25)--(5.92,1.25) node[right] {time};
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\draw[black,very thick] plot[domain=.92:3.10,samples=150] (\x,{1.25+1.00*(1+2.50*(\x-.92))*exp(-2.50*(\x-.92))});
\draw[black,very thick,densely dashed] plot[domain=.92:4.30,samples=160] (\x,{1.25+1.00*(1.20*exp(-.95*(\x-.92))-.20*exp(-3.00*(\x-.92)))});
\node[acc] at (1.55,2.50) {under};
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\node[black] at (3.86,1.64) {over};
\node[black,below] at (5.10,1.12) {tolerance};
\end{tikzpicture}
$$

A settling calculation begins with the actual initial state. For the critical
solution,

$$
x(t)=(C_1+C_2t)e^{-\omega_0t},
$$

the coefficient $C_2$ depends on both $x_0$ and $v_0$. A mechanism released
from rest and a mechanism struck by an impulse have different critical trajectories.
Quoting a universal critical settling time such as “four time constants” can be
reasonable as an initial estimate, but the final specification must evaluate the stated
tolerance with the stated initial condition.

The dimensionless damping ratio is a design coordinate:

$$
\zeta=\frac{b}{b_c}.
$$

Values much smaller than one preserve oscillation for many cycles. Values near one
reduce oscillation sharply. Values much greater than one can create slow return.
The map applies to the dominant mode. A real multi-mode system may show a quick
initial return followed by a small low-frequency tail from a flexible support. A
single-coordinate fit to the early response can overlook that tail, especially if
the sensor is mounted on the moving support rather than on an inertial reference.

A damper can add unwanted stiffness or friction. Hydraulic devices can have
temperature-dependent viscosity. Friction seals can introduce a breakaway force
near zero speed. An electronic feedback system can imitate a viscous damper, but
sensor delay and sampling introduce phase lag. The effective damping may become
negative at some frequency if the feedback phase is incorrect. Validate the
closed-loop response through a small-amplitude ring-down, a controlled disturbance,
and a frequency response around the intended operating point.

## Ring-down reductions and validation

A spring--mass apparatus has $m=0.500\ \mathrm{kg}$ and
$k=80.0\ \mathrm{N\,m^{-1}}$. A displacement sensor is calibrated in millimetres.
After the release transient, the first accepted positive peak is
$X_0=40.0\ \mathrm{mm}$. Eight cycles later, the accepted positive peak is
$X_8=16.0\ \mathrm{mm}$. The measured time between those peaks is
$3.974\ \mathrm{s}$. Baseline drift over that interval is below
$0.10\ \mathrm{mm}$, so a constant baseline is adequate for this reduction.

The measured damped period is

$$
T_d=\frac{3.974\ \mathrm{s}}{8}=0.4968\ \mathrm{s},
\qquad
\omega_d=\frac{2\pi}{T_d}=12.65\ \mathrm{rad\,s^{-1}}.
$$

The peak ratio gives

$$
\delta_8
=\frac18\ln\left(\frac{40.0}{16.0}\right)
=0.1145.
$$

Thus

$$
\gamma=\frac{\delta_8}{T_d}
=0.2305\ \mathrm{s^{-1}},
\qquad
b=2m\gamma
=0.2305\ \mathrm{kg\,s^{-1}}.
$$

The undamped angular frequency is obtained from the measured damped frequency and
decay rate,

$$
\omega_0=\sqrt{\omega_d^2+\gamma^2}
=12.65\ \mathrm{rad\,s^{-1}},
$$

which agrees with the independently known value
$\sqrt{k/m}=12.65\ \mathrm{rad\,s^{-1}}$ at the displayed precision. The
damping ratio and quality factor are

$$
\zeta=\frac{\gamma}{\omega_0}=0.0182,
\qquad
Q=\frac{\omega_0}{2\gamma}=27.4.
$$

The small damping ratio supports the use of the underdamped model. The approximation
$Q\approx\pi/\delta_8$ gives $27.4$ as a cross-check. The exact fractional
cycle energy loss is $1-e^{-2\delta_8}=0.205$; the weak-loss approximation
$2\pi/Q=0.229$ is nearby but visibly less accurate at this $Q$.

$$
% caption: Data reduction for the worked ring-down. Eight same-sign peaks provide
% the amplitude ratio and elapsed time; using the full eight-cycle interval reduces
% sensitivity to a single peak-height reading while retaining the measured damped
% period needed to obtain the decay rate.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.66,1.38)--(5.92,1.38) node[right] {time};
\draw[->,black] (.66,.42)--(.66,2.86) node[above] {displacement};
\draw[acc,very thick] plot[domain=.92:5.10,samples=320] (\x,{1.38+1.08*exp(-.229*(\x-.92))*sin(720*(\x-.92))});
\draw[fill=white,draw=black,thick] (1.045,2.43) circle (2.4pt);
\draw[fill=white,draw=black,thick] (5.045,1.80) circle (2.4pt);
\draw[<->,black,thick] (1.045,2.62)--(5.045,2.62);
\node[above] at (3.045,2.62) {eight cycles};
\node[above left] at (1.045,2.43) {$X_0$};
\node[above right] at (5.045,1.80) {$X_8$};
\end{tikzpicture}
$$

The calculation has several internal checks. The units of $b$ are
$\mathrm{kg\,s^{-1}}$, so $b\dot x$ has units of force. The inferred $\gamma$
is much smaller than $\omega_0$, consistent with the visible long sequence of
oscillations. The independently measured $m$ and $k$ reproduce the inferred
natural frequency. A mismatch among these checks would prompt inspection of the
calibration, peak selection, time base, or assumption of linear stiffness before
reporting a damping coefficient.

A full trace fit can refine the estimates. Use the peak reduction to initialize
$(A,\gamma,\omega_d,\phi)$, then fit the calibrated samples in the accepted
window. The fitted $\gamma$ should agree with the peak-based value within the
combined uncertainty. A meaningful disagreement has diagnostic value: it may
indicate an amplitude-dependent loss law, an offset drift, a missed peak, or
correlated noise from the sensor.

The stored mechanical energy at the first accepted peak is

$$
E_0=\frac12kX_0^2
=0.0640\ \mathrm{J}.
$$

Eight cycles later the peak energy is

$$
E_8=\frac12kX_8^2
=0.0102\ \mathrm{J}.
$$

The difference is energy transferred to the damper and other loss channels during
the measured interval. Dividing that total difference by elapsed time gives an
interval-average dissipated power, but the instantaneous power remains
$b\dot x^2$ under the linear model and varies within every cycle.


### Uncertainty, model validation, and reporting

A damping result requires a measurement model as well as a fitted number. The
measurement model maps sensor output, timing, baseline, and peak-selection rules to
the reported $\gamma$, $b$, $\zeta$, or $Q$. Each element contributes a
different uncertainty pattern. Random height noise broadens repeated peak estimates.
A calibration-scale error changes every displacement amplitude together. A time-base
error changes the measured period and decay rate. Baseline drift changes late peaks
much more strongly than early peaks. A generic “instrument error” label gives no
basis for choosing the improvement that would reduce the final uncertainty.

Independent peak-height standard uncertainties $\sigma_{X_n}$ and
$\sigma_{X_{n+N}}$ give the following first-order propagation for a peak-ratio estimate:

$$
\sigma_{\delta_N}^2
\approx
\frac{1}{N^2}
\left[
\left(\frac{\sigma_{X_n}}{X_n}\right)^2
+
\left(\frac{\sigma_{X_{n+N}}}{X_{n+N}}\right)^2
\right].
$$

The relative uncertainty of the smaller late peak often dominates. Increasing $N$
reduces the factor $1/N$, but it also lowers the later amplitude and increases
susceptibility to baseline drift. An optimum separation balances those effects. The
choice should follow a trial analysis of several separations, for example
$N=3,5,8$, rather than a default value selected before viewing the signal.

A baseline uncertainty deserves explicit treatment. Let the raw positive peaks be
$P_n$ and $P_{n+N}$, and let the equilibrium reading be $B$. The logarithmic
decrement uses $X_n=P_n-B$ and $X_{n+N}=P_{n+N}-B$. An error $\Delta B$ has
a relative effect of approximately $\Delta B/X$, so the late peak receives the
larger fractional change. Record several samples with the oscillator at rest before
and after each run. A changing rest value establishes a baseline drift that must be
modeled or included as a systematic uncertainty.

Time uncertainty has two common sources: the sample clock and the determination of
peak time. A stable digitizer clock can be accurate far beyond the mechanical
precision of the apparatus. Peak timing can still be poor when there are only a few
samples around a broad maximum or when a filter distorts the local waveform.
Parabolic interpolation through a local three-point maximum can improve timing for
a clean sampled sinusoid, but it should be compared against a global fit rather than
accepted uncritically. A shared clock removes one relative timing error. An unknown
analog filter delay still requires measurement or modeling.

A full-trace fit returns a covariance matrix for fitted parameters. Read its
off-diagonal entries or correlation coefficients. A strong correlation between
amplitude and decay rate signals limited independent constraint on the two
parameters. Extending the record can help until the sensor noise floor dominates.
Improving calibration may have little effect on a dimensionless $Q$ estimated from
ratios, whereas a longer stable record can reduce its statistical uncertainty
substantially.

Use residual plots before quoting a covariance-derived interval. Residuals should be
plotted against time, displacement, velocity estimate, and cycle number. Each view
tests a different failure mode:

- **Residual versus time** exposes an omitted baseline drift or a second slowly
  decaying mode.
- **Residual versus displacement** exposes nonlinear stiffness, sensor saturation,
  and geometric misalignment.
- **Residual versus velocity** exposes drag laws that differ from the assumed
  linear relation.
- **Residual versus cycle number** exposes changes in the loss mechanism as
  amplitude falls or temperature changes during the run.

Residuals that alternate in sign near every peak can arise from a slightly wrong
frequency. Residuals with the same sign at both positive and negative large
displacements can arise from a cubic stiffness term. A late-time scatter increase
can reflect a fixed sensor noise floor. These patterns matter even when the
root-mean-square residual is small because a low average can conceal a structured
deviation.

Residual structure tests the response model; it does not include calibration scale,
clock error, or uncertainty in the independently measured mass and stiffness. Those
contributions enter the fitted decay rate through separate measurement paths and
therefore require a distinct uncertainty ledger.

Repeated releases provide a second uncertainty layer. Hold the apparatus at the
same initial displacement, use the same release protocol, and acquire independent
traces. The spread of independently fitted $\gamma$ values measures short-term
repeatability. Change one condition at a time for a reproducibility study:
temperature, initial amplitude, sensor range, mounting stiffness, or external load.
A systematic shift with amplitude is evidence about the damping model, whereas an
unstructured run-to-run spread may point to release variability or ambient
disturbance.

The uncertainty statement should distinguish a statistical interval from a
systematic allowance. If the mean of repeated $\gamma$ estimates has a standard
error of $0.003\ \mathrm{s^{-1}}$, and an independently assessed baseline effect
contributes $0.005\ \mathrm{s^{-1}}$, combine them according to the stated
laboratory convention and show both components. An unexplained factor-of-two spread
between early and late interval fits should be reported as model inadequacy. The
result requires a revised loss model or an explicitly limited amplitude range.

A complete result includes the following information in prose, a table, or a
captioned figure:

- apparatus geometry and the coordinate used for displacement;
- measured or independently calibrated $m$, $k$, sensor scale, and time base;
- release state, trigger convention, sample rate, and record duration;
- accepted fitting or peak window, baseline treatment, and excluded data;
- fitted $\omega_d$, $\gamma$, $b$, $\zeta$, $Q$, with units where
  applicable;
- uncertainty method, number of repeated runs, and dominant systematic terms;
- residual plots or a concise description of their observed structure;
- amplitude range over which the linear damping statement is supported.

The term “quality factor” names a ratio between stored energy and loss rate under a
specified model. High $Q$ describes slow fractional energy loss per cycle. A
high-$Q$ sensor supports narrow-band measurement and can lengthen settling time.
The appropriate value follows from the required bandwidth, settling time, signal
strength, and environmental stability.

## Laboratory protocol and acceptance checks

The free-damping equation isolates one degree of freedom with a constant linear
restoring force and a linear dissipative force. Several extensions alter that
assumption:

$$
m\ddot x+b\dot x+kx+\alpha x^3=0,
$$

$$
m\ddot x+b\dot x+kx+c_q\dot x|\dot x|=0,
$$

and, for two coupled coordinates with mass, damping, and stiffness matrices
$M$, $C$, $K$ acting on the coordinate vector $\vec q$,

$$
M\ddot{\vec q}
+C\dot{\vec q}
+K\vec q=\vec 0.
$$

A cubic stiffness changes frequency with amplitude. Quadratic drag changes decay
rate with speed. Coupled coordinates create several modal frequencies and decay
rates. The evidence for these effects comes from the record: a changing period,
curved log envelope, beat-like modulation, or residual frequencies that persist
after a single-mode fit.

A driven oscillator introduces an external force $F_0\cos(\omega t)$ and requires
separate treatment of steady-state amplitude, phase, bandwidth, and power supplied
by the source. Coupled oscillators require modal coordinates and energy exchange.
Those analyses use the free ring-down parameters developed here, yet their physical
questions differ. Keeping free decay separate prevents a resonance curve or beat
pattern from being treated as evidence for a single free-mode damping coefficient.

A ring-down can still support later driven-response work. The measured $\gamma$
sets the expected linewidth scale for weak damping, and the measured $\omega_0$
sets a starting frequency range. The driven system must then be checked independently
because a source can add nonlinearities, a support can move, and a transducer can
alter the effective stiffness or damping. Carry data from a freely decaying
oscillator forward as input parameters with their uncertainties. Measure the driven
response separately.

The same reasoning applies to electrical, acoustic, and optical resonators. An
$RLC$ circuit exchanges electric and magnetic energy while resistance removes
energy. A vibrating string loses energy through internal damping and radiation into
air. A cavity stores electromagnetic energy and loses it through wall absorption or
coupling ports. The coordinate and loss mechanism change, but the practice remains
the same: state the stored-energy variable, specify the loss law, measure a
time-resolved decay, inspect residuals, and report the range over which the model
holds.


### Laboratory protocol and independent checks

A defensible ring-down begins before the first release. Measure the moving mass that
belongs to the selected coordinate. A spring--mass rig may include the carriage,
sensor flag, added masses, and a fraction of the spring mass. A pendulum requires
its moment of inertia about the pivot. A flexible support can add an effective mass
or introduce a second coordinate. State the chosen model and how the corresponding
inertial parameter was obtained.

Measure stiffness independently whenever the apparatus allows it. Apply a sequence
of known static forces, allow the system to settle, and record the equilibrium
displacement. The slope of the force--displacement relation estimates $k$ for a
linear spring:

$$
F=kx+F_{\rm offset}.
$$

Use both positive and negative loads when the mount permits. A straight relation
with the same slope on loading and unloading supports a linear, low-hysteresis
spring over the tested range. A loop between loading and unloading points indicates
material loss, friction, or a drifting reference. The dynamic ring-down can still be
analyzed, but the static calibration documents a possible source of model error.

Sensor calibration requires more than matching an arbitrary screen scale. Move the
sensor target through several known positions that span the ring-down range. Record
the sensor output after it settles at each location. Fit the scale and offset,
inspect residuals, and retain the calibration uncertainty. An optical sensor can
change gain with target angle or surface reflectivity. A magnetic sensor can change
response with gap. A potentiometer can have end-region nonlinearity. The calibration
range should contain the largest accepted displacement but avoid the sensor's
mechanical stop and electrical saturation.

Clock verification can use a traceable frequency source, a calibrated pulse train,
or a comparison with a known oscillator over a long interval. The goal is an
uncertainty statement appropriate to the decay measurement. If the period is about
$0.5\ \mathrm{s}$, a one-part-per-million clock error is negligible beside a
millisecond-scale release or peak-location uncertainty. That conclusion follows from
a quantitative comparison, not from the brand name of the recorder.

The release protocol should leave the oscillator with the intended $x_0$ and
$v_0$. During a static release, hold at the target displacement until lateral motion
has stopped, trigger the recorder, and remove the restraint with a repeatable
mechanism. A hand release often gives a small initial velocity and can vary between
runs. That variation is acceptable when it is measured and incorporated in the
full-trace fit; it is unsuitable when the procedure assumes $v_0=0$ without
verification. A before-and-after video record or a high-rate sensor segment can
document the release transient.

Run a pilot trace before collecting the final set. The pilot establishes the sensor
range, expected period, baseline stability, and required record duration. Set the
acquisition range so the largest accepted peak uses a substantial portion of the
available resolution without clipping. Set the duration long enough to include many
cycles but short enough that late samples remain interpretable. Preserve the raw
pilot data; it identifies choices that would otherwise be invisible in a polished
final graph.

For every final run, retain raw samples, calibration records, analysis settings, and
a versioned reduction script or worksheet. A figure exported from an oscilloscope
screen cannot reproduce a peak-selection rule or a baseline correction. Data
provenance is part of the result because damping estimates can change when a later
review corrects a calibration or notices a missed release transient.

Independent parameter checks make the model more reliable. The frequency measured
from zero crossings can be compared with frequency measured from same-sign peaks
and with the full-trace fit. The damping rate from a logarithmic decrement can be
compared with the envelope from a global fit. The natural frequency inferred from
$m$ and $k$ can be compared with
$\sqrt{\omega_d^2+\gamma^2}$. Agreement among reductions that use different
features of the trace is stronger evidence than repeated agreement from one
spreadsheet formula.

The final report benefits from a compact graphical record. Include one calibrated
time trace with the accepted interval marked, one envelope or log-peak plot, and one
residual plot. Each axis needs physical units. State whether an amplitude means
peak displacement, RMS displacement over a cycle, or a fitted sinusoidal coefficient.
The word “amplitude” otherwise carries an avoidable factor-of-$\sqrt2$ ambiguity
when results are compared with electrical or acoustic resonators.

A final numerical check compares the observed energy decrease with the integrated
viscous-loss prediction. A fitted displacement trace gives

$$
E_{\rm loss}
=\int_{t_1}^{t_2}b\dot x^2\,\d t.
$$

The energy difference

$$
\Delta E
=
\left[
\frac12m\dot x^2+\frac12kx^2
\right]_{t_1}
-
\left[
\frac12m\dot x^2+\frac12kx^2
\right]_{t_2}
$$

should agree within the stated measurement and model uncertainty. Evaluate the
comparison over an interval that begins and ends at well-resolved states. If the
integral systematically underestimates observed energy loss, the damping law is too
weak, the stiffness or mass is miscalibrated, or an unmodeled loss channel is
present. If it overestimates the loss, check the velocity estimate and time-base
scaling before modifying the physical interpretation.

A ring-down result is complete when a reader can reconstruct the coordinate, the
damping model, the data window, the parameter reduction, and the principal checks.
That standard keeps the decay constant tied to a measured oscillator rather than
to a generic shrinking curve.


### Acceptance checks for a linear ring-down

A completed analysis should pass several numerical checks before the parameter values
are reused elsewhere. The checks are short, but each tests a different part of the
measurement chain.

- The fitted $\gamma$ is positive, and the fitted envelope decreases throughout
  the accepted interval.
- The measured $\omega_d$ is real and the inferred
  $\omega_0=\sqrt{\omega_d^2+\gamma^2}$ agrees with the independent
  mass--stiffness estimate within uncertainty.
- The fitted baseline remains small compared with the early accepted amplitude and
  is compatible with rest readings before and after the record.
- Peak-ratio and full-trace estimates of $\gamma$ agree to the stated precision.
- The residual scale is consistent with sensor noise, with no persistent second
  frequency, drift, clipping, or amplitude-dependent curvature.
- Repeated releases at the same nominal initial state produce a spread consistent
  with the stated repeatability and systematic allowances.

A failed check identifies the next measurement or model revision. A high-quality
record with a curved log envelope can support a nonlinear damping study; report its
range-dependent loss instead of a single misleading $b$. A clean exponential
record with a poor frequency check points toward mass, stiffness, clock, or unit
handling. This separation between evidence and interpretation keeps ring-down
measurements informative when the first model is an approximation.