---
title: Driven Oscillators
module: Oscillations and Waves
moduleNumber: 7
lessonNumber: 14
order: 714
summary: >
  Drive a damped oscillator at a frequency you control and it eventually forgets its
  own: $m\ddot x+b\dot x+kx=F_0\cos\omega t$ settles into a steady response whose
  amplitude and phase depend sharply on how close the drive sits to resonance. We
  solve for that response, show how damping alone fixes the resonance width, the peak
  power, and the settling time, and treat base excitation as the same problem with a
  different input. The steady-state formulas hold only for constant $m$, $b$, and $k$;
  level-dependent peaks or hysteresis between up- and down-sweeps are how nonlinearity
  or an extra mode announces itself.
topics: [Oscillations]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 14 — Oscillations; §14-5"
---

## Forced motion and phase measurement

A shaker, rotating imbalance, loudspeaker coil, or moving support can supply a
periodic force to an oscillator. The force source supplies energy during part of a
cycle and can receive energy during another part. Dissipation determines the
long-term balance. A linear translating model with a sinusoidal force source is

$$
m\ddot x+b\dot x+kx=F_0\cos(\omega t).
$$

Here $x$ is displacement from the static equilibrium position, $m$ is the
moving mass, $k$ is stiffness, and $b$ is the viscous damping coefficient. The
source amplitude $F_0$ is a peak force, not an rms value. Every term has units of
force: $m\ddot x$ has units of $\mathrm{kg\,m\,s^{-2}}$, $b\dot x$ has
units of $\mathrm{kg\,s^{-1}}\,\mathrm{m\,s^{-1}}$, and $kx$ has units of
$\mathrm{N\,m^{-1}}\,\mathrm{m}$. That unit check catches a common error in
which a force amplitude is treated as a displacement amplitude.

$$
% caption: A force-driven oscillator. A spring and a viscous damper join the mass to a
% fixed wall; the actuator applies the sinusoidal drive $F_0\cos\omega t$ along the
% $x$ direction. Displacement is measured relative to the laboratory frame.
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\node[above] at (5.12,1.72) {drive};
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$$

The natural angular frequency, decay constant, and damping ratio are

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

The compact driven equation is

$$
\ddot x+2\gamma\dot x+\omega_0^2x=\frac{F_0}{m}\cos(\omega t).
$$

The source frequency $\omega$ is varied during a scan; the apparatus parameters
$m$, $b$, and $k$ are intended to remain constant. Warm actuator coils,
amplitude-dependent stiffness, and loose mounting hardware can shift an apparent
response curve between repeated runs.

**Phase channels and a defensible phase measurement.**

Amplitude alone discards a sensitive part of the response. The phase curve changes
rapidly through the resonant region even when a broad amplitude maximum makes the
peak location uncertain. A two-channel measurement keeps that information. Use the
source reference $r(t)=\cos\omega t$ and write the measured displacement as

$$
x(t)=X\cos\omega t+Y\sin\omega t.
$$

Comparison with $A\cos(\omega t-\delta)$ gives

$$
X=A\cos\delta,\qquad Y=A\sin\delta,
\qquad
A=\sqrt{X^2+Y^2},
\qquad
\delta=\atanTwo(Y,X).
$$

The $X$ channel is called in-phase because its waveform is aligned with the
reference. The $Y$ channel is in quadrature because it is shifted by one quarter
of a cycle. Sign conventions matter. Reversing the sensor leads changes the
measured displacement by a factor of minus one and adds $\pi$ to the reported
phase. Changing a digital reference from cosine to sine changes every reported
phase by a quarter-cycle. A data file needs a statement of the reference waveform,
the positive sensor direction, and whether lag is represented by positive or
negative angle.

The driven-oscillator model predicts a dimensionless phase relation in terms of
$r=\omega/\omega_0$:

$$
\delta(r)=\atanTwo\!\left(2\zeta r,\,1-r^2\right).
$$

At low $r$, the in-phase component is positive and the quadrature component is
small. At $r=1$, the in-phase component crosses zero while the quadrature
component is positive. At high $r$, the in-phase component is negative and the
phase approaches $\pi$. Those sign changes diagnose the response when an
amplitude calibration is uncertain. A phase trace that falls from zero to minus
$\pi$ can describe the same physical response if the instrument defines lag with
the opposite sign. A discontinuous jump at the phase-wrap boundary should be
unwrapped before a smooth model is fitted.

$$
% caption: Steady-state phase lag $\delta(r)=\operatorname{atan2}(2\zeta r,\,1-r^2)$ for
% three damping ratios. Each curve climbs from $0$ at low frequency toward $\pi$ well
% above resonance and passes through the quarter-cycle value $\pi/2$ exactly at $r=1$,
% independent of damping. Lighter damping makes the crossing steeper.
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$$

A sinusoidal least-squares fit estimates $X$ and $Y$ directly. Include a
constant term when the sensor has offset and include a slow linear term only when
the record has visible drift that cannot be removed by waiting for equilibrium.
With sample times $t_j$, measured values $x_j$, and known $\omega$, fit

$$
x_j=C+Dt_j+X\cos\omega t_j+Y\sin\omega t_j+\varepsilon_j.
$$

Uniformly spaced samples over an integer number of cycles make the sine and cosine
columns nearly orthogonal. A record that ends halfway through a cycle still admits a
fit, but the estimates become more correlated and more sensitive to offset or
drift. Triggering the acquisition from the same clock that synthesizes the drive
eliminates an otherwise invisible timing uncertainty between channels.

Timing error has a frequency-dependent phase effect. A delay error
$\Delta t$ produces

$$
\Delta\delta=\omega\Delta t.
$$

An uncorrected $10\,\mu\mathrm{s}$ channel delay is only
$0.00063\,\mathrm{rad}$ at $10\,\mathrm{Hz}$, but it becomes
$0.063\,\mathrm{rad}$ at $1\,\mathrm{kHz}$. The correction belongs in the
instrument calibration record, ideally measured with both channels connected to
the same electrical signal. A phase reference generated by software also needs an
account of digital filtering delay. Filters can shift phase even when their
amplitude response looks flat across the narrow frequency range of interest.

Random displacement noise also limits phase precision. With independent
zero-mean sample noise of standard deviation $\sigma_x$, a well-distributed
record of $N$ samples has a rough phase standard uncertainty

$$
u(\delta)\approx\frac{\sqrt{2}\,\sigma_x}{A\sqrt{N}}.
$$

The expression assumes a known drive frequency, stationary noise, and a response
well represented by one sinusoid. It becomes optimistic when samples are strongly
correlated or when the measurement window contains an unresolved transient. A
coherence calculation, repeat records, and residual inspection test those
assumptions more directly than a formula alone.

## Resonance, bandwidth, and force balance

Name the measured observable when reporting a resonance frequency. Displacement
amplitude, velocity
amplitude, absorbed power, spring force, and acceleration do not all reach their
maxima at the same drive frequency once damping is appreciable. A laboratory graph
labelled only “resonance frequency” leaves the measured quantity ambiguous. The
response formula distinguishes the measured quantities.

Square the displacement amplitude and omit the constant factor
$F_0^2/m^2$. The frequency-dependent denominator is

$$
D(\omega)=(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2.
$$

An amplitude maximum occurs where $D$ is minimum. Differentiation gives

$$
\frac{\d D}{\d\omega}
=4\omega\left(\omega^2-\omega_0^2+2\gamma^2\right).
$$

Apart from the endpoint at zero frequency, the displacement-peak frequency is

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

The expression is real only when $\zeta<1/\sqrt2$. A strongly damped linear
oscillator can therefore have a smooth decreasing displacement response with no
interior displacement peak. It still converts drive energy into heat and still has
a meaningful natural frequency. The absence of a pronounced displacement peak does
not establish the absence of a mode.

$$
% caption: Normalized displacement amplitude $A/A_{\mathrm{static}}=1/\sqrt{(1-r^2)^2+(2\zeta r)^2}$
% for three damping ratios. Light damping gives a tall interior peak just below $r=1$;
% heavier damping lowers and broadens it, and past $\zeta=1/\sqrt2$ the interior peak
% disappears though the same mode is present. All curves start at the static value $1$.
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$$

Velocity amplitude is $V=\omega A$. Its square is proportional to

$$
V^2(\omega)=\frac{(F_0/m)^2\omega^2}
{(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2}.
$$

Differentiating with respect to $\omega^2$ places the maximum at
$\omega=\omega_0$ for any positive $b$. That result has a mechanical
interpretation. At $\omega_0$, the elastic force and inertial force cancel in the
phasor balance, leaving the applied force to balance the damping force. The
velocity therefore aligns with the applied force and the energy transfer per cycle
is largest.

$$
% caption: Response peaks depend on the measured observable. Displacement peaks
% slightly below $\omega_0$, velocity peaks exactly at $\omega_0$, and acceleration
% peaks above it. The dashed guide marks $\omega_0$; the three maxima separate once
% damping is appreciable.
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$$

Instantaneous mechanical power from the source is

$$
p_{\mathrm{in}}(t)=F_0\cos(\omega t)\,\dot x(t).
$$

Average this product over a complete cycle in the steady state. Using
$\dot x=-\omega A\sin(\omega t-\delta)$ gives

$$
\overline P
=\frac12F_0\omega A\sin\delta
=\frac12b\omega^2A^2
=\frac12bV^2.
$$

The middle equality is the steady-state energy balance. The source supplies, on
average, the same power that viscous loss removes. Stored mechanical energy may
rise and fall within each cycle, while the average stored energy remains constant.
At $\omega_0$, the force is in phase with velocity and

$$
\overline P_{\max}=\frac{F_0^2}{2b}.
$$

The factor one-half arises because $F_0$ and $V$ are peak amplitudes. Values
entered as rms quantities require the corresponding rms identity
$\overline P=F_{\mathrm{rms}}V_{\mathrm{rms}}$ only when force and velocity
are in phase. Units remain a strong check. A force times a velocity has units of
$\mathrm{N\,m\,s^{-1}}=\mathrm{W}$.

Away from resonance, the force and velocity have a nonzero phase difference. The
power product then has alternating positive and negative portions. The positive
area exceeds the negative area only by the energy dissipated in the damper. A
source may therefore absorb energy briefly during a cycle even while it delivers
net energy over the cycle. A drive amplifier that cannot sink this returned energy
can clip, saturate, or alter the intended waveform near a reactive load.

An energy ledger makes the same balance visible without phasors. The spring holds
$U=\tfrac12kx^2$, the mass holds $K=\tfrac12m\dot x^2$, and the damper
converts energy at rate $b\dot x^2$. Over a complete steady cycle,

$$
\int_{t}^{t+T}F\dot x\d t
=\int_{t}^{t+T}b\dot x^2\d t,
\qquad T=\frac{2\pi}{\omega}.
$$

The equality is independent of the phase origin. It checks a
numerical integration or a measured force--velocity loop. The signed area enclosed
by a plot of force versus displacement is the work per cycle. Under a harmonic
force drive, that area equals the energy dissipated per cycle in steady operation.

**Bandwidth, half-power points, and quality factor.**

Power response is often more stable than displacement response near resonance,
because a force sensor and a velocity estimate give direct access to the energy
transfer rate. Divide the average-power expression by its value at
$\omega_0$:

$$
\frac{\overline P(\omega)}{\overline P_{\max}}
=\frac{4\gamma^2\omega^2}
{(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2}.
$$

Half-power frequencies $\omega_1$ and $\omega_2$ satisfy

$$
\overline P(\omega_1)=\overline P(\omega_2)
=\frac12\overline P_{\max},
\qquad \omega_1<\omega_0<\omega_2.
$$

Substitution reduces the half-power condition to

$$
|\omega_0^2-\omega^2|=2\gamma\omega.
$$

Solving the two signs gives the exact roots

$$
\omega_1=\sqrt{\omega_0^2+\gamma^2}-\gamma,
\qquad
\omega_2=\sqrt{\omega_0^2+\gamma^2}+\gamma.
$$

Their separation is exact for this linear viscous model:

$$
\Delta\omega=\omega_2-\omega_1=2\gamma=\frac{b}{m}.
$$

Apply that equality to the half-power power curve. It does not describe
arbitrary “half-amplitude” widths, whose locations depend on the plotted response
and on damping strength.

$$
% caption: Half-power bandwidth of the average-power response. The curve crosses
% $\tfrac12 P_{\max}$ at $\omega_1$ and $\omega_2$; their separation is the bandwidth
% $\Delta\omega=2\gamma=b/m$. The peak sits at $\omega_0$ in the linear model.
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$$

Quality factor compares the resonant energy stored with the energy lost during one
cycle. In a lightly damped oscillator,

$$
Q=2\pi\frac{E_{\mathrm{stored}}}{|\Delta E|_{\mathrm{cycle}}}
\approx\frac{\omega_0}{2\gamma}
=\frac{m\omega_0}{b}
=\frac{1}{2\zeta}.
$$

Combining the weak-damping approximation with the exact bandwidth identity yields

$$
Q\approx\frac{\omega_0}{\Delta\omega}.
$$

The approximation replaces the resonant frequency scale in the numerator with
$\omega_0$ and assumes the resonance is narrow enough that nearby frequency
factors can be treated as equal. State the criterion in a report. A device with
$Q=2$ has a broad response, so language meant for a sharply tuned resonator can
mislead; a device with $Q=200$ has a narrow peak and a much longer settling time.

Free ring-down gives an independent $Q$ estimate. In the underdamped viscous
model, energy falls as $E(t)=E_0e^{-2\gamma t}$. Over one near-resonant period,
the fractional loss is approximately $2\gamma(2\pi/\omega_0)$, giving the same
$Q\approx\omega_0/(2\gamma)$. Agreement between bandwidth and ring-down checks
the model. A bandwidth much wider than predicted by ring-down can signal frequency-
dependent damping, actuator loading, an uncalibrated force sensor, or a scan that
did not settle at each setting.

**Force balance and base excitation.**

At positive displacement and positive velocity, spring and damping forces point
toward decreasing coordinate. The applied force changes sign during a cycle, so it
cannot be absorbed into a constant restoring term.

Many laboratory systems are driven by moving the support rather than by applying a
known force directly. Let support motion be $y(t)=Y\cos\omega t$, absolute mass
motion be $x(t)$, and spring extension be $z=x-y$. The force balance becomes

$$
m\ddot x+b(\dot x-\dot y)+k(x-y)=0,
$$

or, in the relative coordinate,

$$
m\ddot z+b\dot z+kz=-m\ddot y.
$$

The right side has amplitude $m\omega^2Y$. Base excitation thus has the same
mathematical form as force drive after the drive amplitude is replaced by
$m\omega^2Y$, but its physical interpretation differs. A displacement transducer
across the spring reports $z$, whereas a camera aimed at the laboratory frame
reports $x$. Mixing those coordinates can produce a response curve with the wrong
high-frequency behavior.

$$
% caption: Base excitation defines two displacements. The support moves by $y$, the
% mass by $x$ in the laboratory frame, and the spring--damper link carries the relative
% extension $z=x-y$. A transducer across the link measures $z$, not $x$.
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\node[below] at (1.67,.56) {support};
\node[right] at (1.98,1.54) {link};
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$$

## Transients, phasors, and frequency response

The full solution is the sum of a homogeneous transient and a driven particular
solution. An underdamped apparatus has a transient term proportional to
$e^{-\gamma t}\cos(\omega_d t+\phi_t)$, where

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

Its amplitude depends on the release state and on the instant at which the drive
was switched on. The periodic steady state has the source frequency, even when that
frequency differs greatly from $\omega_0$. A record immediately after a frequency
change contains both parts. Fitting that record as though it were steady state can
inflate an amplitude estimate or assign an arbitrary phase lag.

$$
% caption: Starting the drive from rest, the response builds up to steady amplitude:
% the homogeneous transient decays as $e^{-\gamma t}$ while the particular solution
% grows in. The dashed envelope approaches the steady level; samples in the early
% shaded interval still carry transient motion, later ones give the frequency response.
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$$

The amplitude of a transient decays by $e^{-\gamma t_s}$ after a settling time
$t_s$. A practical acceptance threshold follows from the required fractional
contamination. For example, residual transient amplitude below $1\%$ of its
initial value requires $\gamma t_s>\ln 100\approx4.61$. If the sweep dwells at
one frequency for $t_s$, the total scan time can become large for a high-
$Q$ oscillator. A rapid continuous sweep has a separate risk: the frequency can
move appreciably before the stored energy adjusts to its local steady value.

The repeatable measurement sequence is therefore explicit: set the drive amplitude,
set a frequency, wait for the designated settling interval, record a time window
containing many cycles, estimate the in-phase and quadrature components, and then
change frequency. The number of cycles needed depends on noise and on the desired
phase precision. A phase estimate from one noisy peak-to-peak interval is generally
less stable than a sinusoidal fit over a long window.

The free-decay lesson calibrates the parameters $\omega_0$ and $\gamma$ before a
forced test begins. That calibration makes the driven scan a test of the complete
model instead of a curve-fitting exercise with every parameter unconstrained. A
free ring-down and a driven bandwidth should yield compatible $Q$ values when
linear viscous loss is a valid description.

**Phasors, dynamic stiffness, amplitude, and phase.**

Represent the force as the real part of $F_0e^{i\omega t}$ and seek a steady
response $x=\Re\{\tilde x e^{i\omega t}\}$. Substitution gives

$$
\left(k-m\omega^2+i b\omega\right)\tilde x=F_0.
$$

The complex dynamic stiffness is

$$
\tilde K(\omega)=k-m\omega^2+i b\omega,
\qquad
\tilde x=\frac{F_0}{\tilde K(\omega)}.
$$

The real part $k-m\omega^2$ combines elastic and inertial contributions. The
imaginary part $b\omega$ is associated with velocity-proportional loss. Their
units are both $\mathrm{N\,m^{-1}}$, so the magnitude of their vector sum can
be used as an effective stiffness for a sinusoidal test.

$$
% caption: Dynamic stiffness $\tilde K=k-m\omega^2+ib\omega$ in the complex plane. The
% horizontal leg is the elastic-minus-inertial part $k-m\omega^2$, the vertical leg is
% the loss part $b\omega$, the hypotenuse is $|\tilde K|=F_0/A$, and its angle at the
% origin is the displacement lag $\delta$.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.62,1.00)--(5.94,1.00) node[right] {elastic and inertial};
\draw[->,black] (1.18,.52)--(1.18,3.02) node[above] {loss};
\draw[acc,very thick,->] (1.18,1.00)--(4.76,1.00);
\draw[black,very thick,->] (4.76,1.00)--(4.76,2.52);
\draw[black,very thick,->] (1.18,1.00)--(4.76,2.52);
\draw[black] (1.98,1.00) arc (0:23:.80);
\node[below] at (3.10,.80) {elastic minus inertia};
\node[right] at (4.84,1.80) {loss};
\node[above] at (2.34,2.06) {resultant};
\node[right] at (2.02,1.20) {lag};
\end{tikzpicture}
$$

Taking the magnitude gives the displacement amplitude:

$$
A(\omega)=\frac{F_0}{\sqrt{(k-m\omega^2)^2+b^2\omega^2}}
=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(2\gamma\omega)^2}}.
$$

With $F(t)=F_0\cos\omega t$, write the response as

$$
x(t)=A\cos(\omega t-\delta),
\qquad
\delta=\atanTwo\!\left(b\omega,\,k-m\omega^2\right).
$$

The two-argument angle is essential. A one-argument inverse tangent cannot
distinguish $k-m\omega^2>0$ from $k-m\omega^2<0$ when the ratio has the same
value. The physical lag increases continuously from $0$ toward $\pi$ as the
source frequency crosses the resonance region; $\atanTwo$ returns the correct
quadrant.

At low frequency, inertia and damping are small relative to stiffness, so

$$
A\approx\frac{F_0}{k},\qquad \delta\approx0.
$$

The apparatus follows a slowly varying force as a spring balance would. At high
frequency, inertia dominates and

$$
A\approx\frac{F_0}{m\omega^2},\qquad \delta\approx\pi.
$$

The mass then moves almost opposite to the applied force, with an amplitude that
falls as $\omega^{-2}$. A response curve whose high-frequency tail falls only as
$\omega^{-1}$ often indicates that velocity, rather than displacement, has been
plotted or that the sensor calibration has been mixed between channels.

**Frequency-response measurement and parameter fitting.**

An actuator command is seldom a calibrated force. A shaker driven at constant
voltage can deliver a different force when the mechanical impedance of the specimen
changes, when the coil resistance warms, or when the amplifier approaches a current
limit. The response calculation requires the force actually applied to the moving
system. A force transducer mounted in series with the actuator, or an independently
calibrated actuator model, determines that quantity. Report the peak-versus-rms
convention for both the force channel and the displacement channel. Mixing a peak
force with an rms displacement shifts a fitted compliance by a factor of
$\sqrt2$.

The complex compliance, or displacement transfer function, is

$$
H_{xF}(\omega)=\frac{\tilde x(\omega)}{\tilde F(\omega)}
=\frac{1}{k-m\omega^2+i b\omega}.
$$

Its magnitude has units of $\mathrm{m\,N^{-1}}$; its angle is the response
phase relative to force. Plotting $H_{xF}$ instead of raw displacement removes
deliberate changes in drive force. A related velocity transfer function is

$$
H_{vF}(\omega)=i\omega H_{xF}(\omega),
$$

which has units of $\mathrm{m\,s^{-1}\,N^{-1}}$. A graph must identify which
transfer function appears on the vertical axis. The high-frequency slope and the
resonant peak change with that choice.

The frequency grid should resolve the narrowest feature. A constant increment that
looks dense across a wide scan may place only two samples within a high-$Q$
half-power bandwidth. Use a coarse grid away from the mode and a finer grid across
the phase turn and half-power crossings. Frequency points should be randomized or
scanned in both directions when heating, drift, or amplitude-dependent behavior is
plausible. Repeating a few anchor frequencies at the beginning, middle, and end of
the run tests for time-dependent changes that a one-way sweep can average away.

A weighted fit to both amplitude and phase uses more of the measured information
than a fit to peak height alone. The complex residual at setting $j$ is

$$
\rho_j=\tilde x_j-H_{xF}(\omega_j;m,b,k)\tilde F_j.
$$

When the channel uncertainties are known, minimize a weighted sum of
$|\rho_j|^2$. Weighting prevents a high-amplitude point near resonance from
dominating the rest of the curve through its larger absolute displacement
is larger. Alternatively, fit the in-phase and quadrature channels with their
covariance matrix. A fit restricted to amplitude can trade an error in damping
against a small frequency shift; the phase data constrains that trade.

Residuals carry more diagnostic content than a single goodness-of-fit number.
Alternating residual signs across the peak can indicate a small resonance-frequency
error. Residuals that grow with amplitude suggest sensor nonlinearity or a spring
whose stiffness changes with excursion. A narrow secondary feature often indicates
another mode in the mounting fixture, transducer, or nominally rigid support.
Retain the unaveraged time records so that a suspicious frequency point can be
checked for waveform clipping, harmonic distortion, or a transient.

Uncertainty needs both an instrumental and a procedural component. A calibration
certificate may give sensor scale uncertainty, but the force sensor alignment,
mount compliance, settling criterion, and phase-reference delay contribute their
own terms. Repeated complete scans give a direct estimate of reproducibility. When
fitting a response model, state whether uncertainty bars include only sample noise
or also variation between mounts and drive levels. The distinction matters when
results are used to predict a resonant displacement outside the measured run.

## Base excitation, transmissibility, and isolation choices

Support motion requires a separate response description because the applied
quantity is a displacement or acceleration of the mounting point rather than a
known force. Let $y=Y\cos\omega t$ describe the support, $x$ describe the
absolute mass coordinate, and $z=x-y$ describe spring extension. The relative
equation is

$$
m\ddot z+b\dot z+kz=-m\ddot y
=m\omega^2Y\cos\omega t.
$$

The relative displacement is therefore driven by an effective force with amplitude
$m\omega^2Y$. Introducing $r=\omega/\omega_0$ gives the relative-motion
ratio

$$
\left|\frac{Z}{Y}\right|
=\frac{r^2}{\sqrt{(1-r^2)^2+(2\zeta r)^2}}.
$$

At low frequency, $Z/Y$ scales as $r^2$. The mass follows the support, so the
spring extension is small. Near the natural frequency, relative displacement can
become large. At high frequency, $Z/Y$ approaches one; the mass remains nearly
inertial while the support moves beneath it, placing nearly the whole support
displacement across the spring and damper. Clearance and allowable stroke can
therefore set the practical upper frequency limit even when the absolute mass
motion is well isolated.

Absolute motion is usually the response of interest for vibration isolation. The
complex transfer ratio follows directly from the base-excitation balance:

$$
\frac{\tilde X}{\tilde Y}
=\frac{1+i2\zeta r}{1-r^2+i2\zeta r}.
$$

Its magnitude is

$$
T_x(r)=\left|\frac{X}{Y}\right|
=\frac{\sqrt{1+(2\zeta r)^2}}
{\sqrt{(1-r^2)^2+(2\zeta r)^2}}.
$$

The condition $T_x<1$ means the mounted mass moves less than the support. After
squaring and cancelling the damping term, the condition becomes $r>\sqrt2$.
The threshold separates an amplification region from an isolation region. Support
spectrum, travel limit, damper heating, payload mass, and desired attenuation
determine whether a design is acceptable.

$$
% caption: Absolute-motion transmissibility $T_x(r)=\sqrt{1+(2\zeta r)^2}/\sqrt{(1-r^2)^2+(2\zeta r)^2}$
% for two damping ratios. Both curves cross unity exactly at $r=\sqrt2$; below it the
% mass moves more than the support, above it less. More damping lowers the resonant
% peak but raises the high-frequency transmitted motion.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.60,.56)--(6.02,.56) node[right] {frequency ratio};
\draw[->,black] (.60,.56)--(.60,2.94) node[above] {motion ratio};
\draw[black] (.60,1.16)--(5.98,1.16);
\draw[acc,very thick] plot[domain=.92:5.70,samples=240]
  (\x,{.56+.60*sqrt(1+(0.3*((\x-.90)/1.6))^2)/sqrt((1-((\x-.90)/1.6)^2)^2+(0.3*((\x-.90)/1.6))^2)});
\draw[black,thick] plot[domain=.92:5.70,samples=240]
  (\x,{.56+.60*sqrt(1+(0.8*((\x-.90)/1.6))^2)/sqrt((1-((\x-.90)/1.6)^2)^2+(0.8*((\x-.90)/1.6))^2)});
\draw[black,dashed] (3.16,.56)--(3.16,2.76);
\node[black] at (5.35,1.30) {unity};
\node[below] at (3.16,.34) {isolation edge};
\node[acc] at (1.42,2.54) {light loss};
\node[black] at (4.25,1.88) {large damping};
\node[below] at (4.90,.34) {isolation};
\end{tikzpicture}
$$

The high-frequency asymptote makes the damping compromise explicit:

$$
T_x(r)\approx\frac{2\zeta}{r},
\qquad
\left|\frac{Z}{Y}\right|\approx1
\qquad (r\gg1).
$$

Increasing damping lowers the sharp resonance peak and reduces settling time after
a disturbance. It also raises the high-frequency absolute-motion transmission and
the damper force. A payload carried through a rough low-frequency environment may
need substantial damping to control the resonant crossing. A sensor intended to
reject a persistent high-frequency vibration often benefits from a lower damping
ratio, provided its relative travel and shock response remain safe.

Transmitted force is a second design measure. The force applied from the
isolator to the base has complex amplitude

$$
\tilde F_T=(k+i b\omega)\tilde Z.
$$

Normalize its magnitude by the base inertial scale $m\omega^2Y$. The resulting
ratio equals the absolute-motion transmissibility for this one-degree-of-freedom
model:

$$
\frac{|F_T|}{m\omega^2Y}
=\frac{\sqrt{1+(2\zeta r)^2}}
{\sqrt{(1-r^2)^2+(2\zeta r)^2}}
=T_x(r).
$$

The equality is model-specific. A flexible housing, several payload modes, or a
nonlinear damper can break it. Measuring both housing acceleration and transmitted
force is therefore valuable in a structural test; a quiet payload does not by
itself guarantee a low load on the support.

Instrument calibration also uses base excitation deliberately. An accelerometer
mounted on a shaker can be compared with a traceable reference accelerometer across
a controlled frequency and amplitude range. The mounting torque, adhesive layer,
and cable strain become part of the test article. A cable that tugs on a small
sensor adds a path for force transmission and can create a spurious mode. The
calibration report should identify the reference orientation, drive amplitude,
frequency range, phase convention, and criterion used to reject an unsettled point.

## Response reductions and model boundaries

A translating oscillator has

$$
m=0.800\,\mathrm{kg},\qquad
k=320\,\mathrm{N\,m^{-1}},\qquad
b=1.60\,\mathrm{kg\,s^{-1}},
\qquad
F_0=0.400\,\mathrm{N}.
$$

The static deflection under the stated peak force is $F_0/k=1.25\,\mathrm{mm}$.
That value is a low-frequency check. A frequency-response calculation that
predicts a much larger displacement far below resonance has likely used the wrong
force unit, sensor scale, or coordinate.

The system parameters are

$$
\omega_0=\sqrt{\frac{k}{m}}=20.0\,\mathrm{rad\,s^{-1}},
\qquad
\gamma=\frac{b}{2m}=1.00\,\mathrm{s^{-1}},
\qquad
\zeta=0.0500.
$$

The apparatus is lightly damped. Its free-decay quality factor and power
half-width are

$$
Q\approx\frac{\omega_0}{2\gamma}=10.0,
\qquad
\Delta\omega=2\gamma=2.00\,\mathrm{rad\,s^{-1}}.
$$

The half-power frequencies are obtained from the exact roots:

$$
\omega_1=\sqrt{\omega_0^2+\gamma^2}-\gamma
=19.025\,\mathrm{rad\,s^{-1}},
\qquad
\omega_2=\sqrt{\omega_0^2+\gamma^2}+\gamma
=21.025\,\mathrm{rad\,s^{-1}}.
$$

Subtracting the two frequencies gives $2.000\,\mathrm{rad\,s^{-1}}$, equal to
$b/m$. A measured
bandwidth far from this prediction should be investigated before the result is
called a material damping property.

$$
% caption: Displacement amplitude of the worked oscillator ($\omega_0=20\,\mathrm{rad\,s^{-1}}$,
% $\zeta=0.05$). The peak reaches $12.5\,\mathrm{mm}$ near $\omega_0$, ten times the
% static deflection. The marked points at $19$, $20$, and $21\,\mathrm{rad\,s^{-1}}$ show
% two off-resonance amplitudes that are similar; their phases place them on opposite sides.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.60,.56)--(6.04,.56) node[right] {source rate};
\draw[->,black] (.60,.56)--(.60,2.94) node[above] {amplitude};
\draw[acc,very thick] plot[domain=.80:5.70,samples=260]
  (\x,{.56+.2125/sqrt((1-(0.75+(\x-.90)*0.104167)^2)^2+(0.1*(0.75+(\x-.90)*0.104167))^2)});
\draw[black,dashed] (3.30,.56)--(3.30,2.86);
\draw[black,dashed] (2.82,.50)--(2.82,2.06);
\draw[black,dashed] (3.78,.50)--(3.78,1.95);
\draw[fill=white,draw=acc,thick] (2.82,2.12) circle (2.0pt);
\draw[fill=white,draw=acc,thick] (3.30,2.69) circle (2.0pt);
\draw[fill=white,draw=acc,thick] (3.78,2.01) circle (2.0pt);
\node[below] at (2.82,.40) {19};
\node[below] at (3.30,.16) {20};
\node[below] at (3.78,.40) {21};
\end{tikzpicture}
$$

At the natural frequency, elastic and inertial terms cancel. The displacement
amplitude is then determined by damping alone:

$$
A(\omega_0)=\frac{F_0}{b\omega_0}
=\frac{0.400}{(1.60)(20.0)}
=0.0125\,\mathrm{m}.
$$

The result is $12.5\,\mathrm{mm}$, ten times the static deflection. That
amplification is compatible with $Q=10$ because, for light damping, the
resonant displacement-to-static-deflection ratio is close to $Q$. The
displacement-amplitude peak is slightly below the natural frequency:

$$
\omega_A=\sqrt{\omega_0^2-2\gamma^2}
=19.950\,\mathrm{rad\,s^{-1}}.
$$

The difference is only $0.050\,\mathrm{rad\,s^{-1}}$, well below the stated
power bandwidth. Reporting $20.0\,\mathrm{rad\,s^{-1}}$ as the displacement
peak without specifying its experimental uncertainty would overstate the
resolution of this apparatus.

Evaluate the response at $19.0$, $20.0$, and
$21.0\,\mathrm{rad\,s^{-1}}$ to see how phase supplements amplitude. The
results are

$$
\begin{array}{c|c|c}
\omega\;(\mathrm{rad\,s^{-1}}) & A\;(\mathrm{mm}) & \delta\;(\mathrm{rad})\\
\hline
19.0 & 9.18 & 0.772\\
20.0 & 12.5 & 1.571\\
21.0 & 8.52 & 2.344
\end{array}
$$

The two off-resonance amplitudes are similar, but their phases occupy opposite
sides of the resonance. A data point with amplitude near $9\,\mathrm{mm}$ could
therefore belong below or above the mode; the phase resolves the ambiguity. The
phase values also test the sign convention. A recorded value near
$-2.34\,\mathrm{rad}$ may be physically equivalent after phase wrapping, while
a value near $0.80\,\mathrm{rad}$ at $21\,\mathrm{rad\,s^{-1}}$ would be
inconsistent with the stated channel orientation.

$$
% caption: Response phasors at the worked settings $19$, $20$, and $21\,\mathrm{rad\,s^{-1}}$.
% Below the mode the response leads along the positive in-phase axis; at $\omega_0$ it is
% pure quadrature and longest (largest amplitude); above the mode it has a negative
% in-phase component. Similar off-resonance amplitudes separate by phase direction.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\foreach \cx/\lab in {1.24/{19},3.22/{20},5.20/{21}} {
  \draw[->,black] (\cx-.62,1.50)--(\cx+.62,1.50);
  \draw[->,black] (\cx,.92)--(\cx,2.34);
  \node[below] at (\cx,.82) {\lab};
}
\draw[acc,very thick,->] (1.24,1.50)--(1.715,1.959);
\draw[acc,very thick,->] (3.22,1.50)--(3.22,2.40);
\draw[acc,very thick,->] (5.20,1.50)--(4.776,1.939);
\node[above] at (1.75,2.00) {low};
\node[above] at (3.22,2.42) {mode};
\node[above] at (4.74,1.98) {high};
\node[below] at (3.22,.46) {rate (rad/s)};
\end{tikzpicture}
$$

Suppose independent calibrations give

$$
u(m)=0.004\,\mathrm{kg},\qquad
u(k)=3.2\,\mathrm{N\,m^{-1}},\qquad
u(b)=0.080\,\mathrm{kg\,s^{-1}},\qquad
u(F_0)=0.010\,\mathrm{N}.
$$

Neglecting covariance, propagation for the natural frequency gives

$$
\left(\frac{u(\omega_0)}{\omega_0}\right)^2
=\frac14\left(\frac{u(k)}{k}\right)^2
+\frac14\left(\frac{u(m)}{m}\right)^2.
$$

Thus $u(\omega_0)=0.112\,\mathrm{rad\,s^{-1}}$. The damping coefficient carries
a relative uncertainty of about $5.0\%$, so
$\Delta\omega=2.00\pm0.10\,\mathrm{rad\,s^{-1}}$. For the resonant amplitude,
independent relative contributions from force, damping, and natural frequency
combine to approximately $5.6\%$, giving

$$
A(\omega_0)=12.5\pm0.7\,\mathrm{mm}.
$$

The quoted uncertainty describes propagation from separately calibrated
parameters. A response-curve fit can produce a different uncertainty because
$m$, $b$, and $k$ then become correlated fit parameters. Preserve the
covariance matrix or a parameter-correlation plot when using fitted values for
prediction.

**Model boundaries, nonlinear signatures, and reportable checks.**

The linear force law uses constant $k$ and $b$ over the observed displacement,
velocity, and frequency range. Test that assumption with controlled changes in
drive level. A simple nonlinear stiffness extension is

$$
F_s=-kx-\alpha x^3.
$$

Positive $\alpha$ produces a hardening response. The effective restoring force
grows faster than linearly with amplitude, so the resonance region moves toward
higher frequency as drive level rises. Negative $\alpha$ produces a softening
response and shifts the region downward. A strong nonlinear response can develop
multiple stable amplitudes at one drive frequency, with abrupt jumps during an
upward or downward frequency sweep. The linear $m$, $b$, $k$ fit then remains
valid only over a declared low-amplitude interval.

$$
% caption: Amplitude-dependent stiffness shifts the resonance. A hardening spring
% ($\alpha>0$) moves the peak toward higher frequency as drive level rises; a softening
% spring moves it lower. A single linear peak fitted across drive levels would hide this
% dependence and the possible multivalued response.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (.60,.56)--(6.02,.56) node[right] {frequency};
\draw[->,black] (.60,.56)--(.60,2.94) node[above] {amplitude};
\draw[black,thick] plot[domain=.80:5.70,samples=200] (\x,{.58+1.86/(1+((\x-2.02)/.22)^2)});
\draw[acc,very thick] plot[domain=.80:5.70,samples=200] (\x,{.58+2.00/(1+((\x-2.44)/.24)^2)});
\draw[black,thick] plot[domain=.80:5.70,samples=200] (\x,{.58+1.90/(1+((\x-2.94)/.27)^2)});
\draw[->,black] (2.15,2.72)--(2.85,2.62);
\node[black,anchor=west] at (2.90,2.62) {higher drive};
\node[black] at (1.30,2.66) {softening};
\node[acc] at (1.24,2.40) {linear};
\node[black] at (1.36,2.14) {hardening};
\end{tikzpicture}
$$

Frequency direction is part of the test protocol when nonlinear response is
suspected. Hold drive amplitude and thermal conditions constant, then record one
increasing-frequency scan and one decreasing-frequency scan after the same
settling rule. A gap between the two branches within the noise band indicates a
history-dependent state. The source must dwell long enough at each point for the
observed amplitude to settle; a rapid sweep can imitate a hysteresis loop in a
perfectly linear high-$Q$ oscillator by carrying energy from the previous
frequency setting.

The damping law can also change with amplitude. Three idealized loss models
illustrate distinct experimental signatures:

$$
F_d=-b\dot x,
\qquad
F_d=-F_c\,\sgn(\dot x),
\qquad
F_d=-c\dot x|\dot x|.
$$

The middle expression uses the sign function and represents dry sliding friction
over a limited range. Nearly sinusoidal motion of amplitude $A$ has mechanical
energy loss in one cycle that scales as

$$
\Delta E_{\mathrm{visc}}\approx\pi b\omega A^2,
\qquad
\Delta E_{\mathrm{dry}}=4F_cA,
\qquad
\Delta E_{\mathrm{quad}}\approx\frac{8}{3}c\omega^2A^3.
$$

Each relation leaves a different trace in ring-down data. Linear viscous loss gives
an exponential amplitude envelope. Dry friction removes nearly the same
displacement amplitude each cycle, producing a more nearly linear decay envelope.
Quadratic drag has a stronger effect at large speed, so the early decay can be much
faster than the late decay. A logarithmic decrement that changes systematically
with amplitude signals that one constant $b$ cannot represent the full record.

Harmonic content is another check. A linear oscillator driven by a single
sinusoidal force reaches a steady response at that same frequency. The measured
time record may contain noise, but the coherent spectrum should remain concentrated
at the drive frequency. A prominent second or third harmonic can arise from sensor
clipping, actuator distortion, contact nonlinearity, or nonlinear stiffness. The
source monitor must be analyzed alongside displacement; a harmonic appearing in
both channels can be created upstream of the mechanical system.

One-degree-of-freedom theory also has a limited frequency range. A real fixture,
spring, sensor arm, or support can possess additional modes. A modal compliance
over a broad band can be represented schematically as

$$
H_{xF}(\omega)
=\sum_{r}\frac{C_r}
{\omega_r^2-\omega^2+i2\gamma_r\omega}.
$$

The coefficient $C_r$ includes mode shape, force location, sensor location, and
normalization. Moving the force or sensor can strengthen one resonance and weaken
another. A single-mode fit belongs only to the isolated band where all other modal
terms vary slowly. Extending a single-mode fit through a nearby secondary peak can
return a plausible-looking damping number with no stable physical interpretation.

Digitization adds its own constraints. Sampling above twice the highest signal
frequency prevents ideal alias overlap, but phase estimation and harmonic checks
benefit from a much larger sample rate. An anti-alias low-pass filter must precede
the analog-to-digital converter when broadband sensor noise or higher mechanical
modes are present. The filter phase delay belongs in the phase calibration. A
decimated record can be adequate for plotting after the original high-rate data has
been archived; it is a poor substitute for a filtered acquisition path.

## Identification, follow-up, and dimensionless checks

A frequency-response result needs enough information for another reader to
reconstruct the physical test and test the claimed model range. Record the moving
mass, stiffness source, damping configuration, coordinate direction, drive
mechanism, force calibration, displacement sensor calibration, sampling rate,
frequency grid, dwell rule, and phase reference. Include the raw or fitted
in-phase and quadrature data together with any image of the resonance peak. The report
should state which quantity defines the quoted resonance frequency and whether it
comes from displacement, velocity, power, phase crossing, or a fitted
$\omega_0$.

Report the usable range with a positive statement of the observed conditions.
Examples include “linear fit accepted for peak displacement below
$3\,\mathrm{mm}$, $16$ to $24\,\mathrm{rad\,s^{-1}}$, and the stated
mounting torque” or “upward/downward scans separated above
$0.6\,\mathrm{N}$ drive.” Such statements preserve the measurements that
support the model while keeping amplitude dependence and extra modes visible.

A free ring-down estimates $\gamma$ from decay, a driven half-power scan
estimates $\Delta\omega$, and a phase scan identifies the in-phase zero crossing
near $\omega_0$. Agreement within uncertainty supports the stated linear viscous
model in that range. A disagreement directs a targeted follow-up measurement:
drive calibration for a power mismatch, timing calibration for a phase mismatch,
or an amplitude series for a damping mismatch.

**Asymptotic identification and sensor conversion.**

The transfer function contains parameter information away from the resonance
maximum. At low frequency,

$$
H_{xF}(\omega)\longrightarrow\frac{1}{k},
\qquad
\arg H_{xF}\longrightarrow0.
$$

A calibrated static or slowly varying force test therefore estimates stiffness
without relying on damping. At high frequency,

$$
H_{xF}(\omega)\longrightarrow-\frac{1}{m\omega^2},
\qquad
\arg H_{xF}\longrightarrow-\pi.
$$

The tail estimates the effective moving mass, including any sensor fixture,
adapter, or payload that participates in the mode. Here, effective mass includes a
rigidly attached accelerometer. A long compliant cable or a flexible sensor arm can
introduce another mode instead of increasing only the effective mass.

At the natural frequency,

$$
H_{xF}(\omega_0)=-\frac{i}{b\omega_0}.
$$

The real component crosses zero and the negative imaginary component reaches its
resonance value. Complex response data can therefore estimate $k$, $m$, and
$b$ from distinct regions of one scan. Fitting all regions jointly is usually
more stable than extracting three independent point estimates, but the asymptotes
remain valuable checks on the fitted result.

An accelerometer does not measure displacement directly. For harmonic motion,

$$
\tilde a=-\omega^2\tilde x,
\qquad
H_{aF}(\omega)=-\omega^2H_{xF}(\omega).
$$

Its low-frequency magnitude rises as $\omega^2/k$, while its high-frequency
limit approaches $1/m$ with acceleration nearly aligned with force. A
displacement sensor and an accelerometer connected to the same oscillator will
therefore show response curves with different slopes and different apparent peak
locations. Direct comparison requires conversion in the complex domain. Dividing
only the plotted magnitudes by $\omega^2$ loses phase information and magnifies
noise toward low frequency.

Numerical differentiation deserves particular caution. A finite difference
estimate of acceleration amplifies high-frequency measurement noise, while
numerical integration of an accelerometer signal accumulates offset and low-
frequency drift. A model-based sinusoidal fit avoids both operations within a
single-frequency record. Fit $X$ and $Y$ for displacement, velocity, or
acceleration in their native units, then apply multiplication by $i\omega$ or
$-\omega^2$ to the fitted complex amplitude. The conversion remains transparent
and carries the phase convention with it.

An independent mass check can use the acceleration plateau. Suppose a calibrated
force and accelerometer give $|H_{aF}|=1.19\,\mathrm{m\,s^{-2}\,N^{-1}}$ in a
frequency region above the mode and below any higher structural mode. The inferred
effective mass is

$$
m_{\mathrm{eff}}\approx\frac{1}{|H_{aF}|}
=0.840\,\mathrm{kg}.
$$

Comparing this with a weighed payload and fixture mass can expose an omitted
adapter or a compliance that invalidates the single-body approximation. The
frequency range must be demonstrated by the response data. A flat-looking
acceleration plateau that sits close to an unmeasured second mode does not provide
the same confidence as a broad, well-resolved plateau.

**Targeted follow-up measurements.**

Different inconsistencies point to different tests. A measured low-frequency
compliance that disagrees with $1/k$ calls for a force calibration and a static
stiffness check. A correct amplitude peak with a displaced phase curve calls for a
reference-delay calibration or a sensor-polarity check. Agreement between phase and
amplitude with a bandwidth wider than the ring-down prediction points toward
frequency-dependent loss, an actuator loading effect, or a sweep that retained
transient motion. A response curve whose center changes with drive level calls for
an amplitude series before any single $Q$ is reported.

Each follow-up setting isolates one hypothesis. A lower drive level tests
nonlinear stiffness and nonlinear damping without changing the geometry. A
separate static force test targets $k$. A free-decay record targets loss without
the drive chain. Swapping the sensor channel into a reference position tests sensor
calibration while leaving the oscillator unchanged. Changing every experimental
condition at once produces a new response but makes the source of a discrepancy
unidentifiable.

Keep raw time records, calibration files, fitted transfer data, residual plots, the
parameter covariance matrix, and the accepted amplitude and frequency range
together. The record then separates a tested local linear model from a numerical
fit extended beyond its measured support.

**Dimensionless checks before accepting a fitted model.**

Normalize the frequency by $\omega_0$ and the compliance by the static
compliance. The dimensionless response is

$$
G(r)=kH_{xF}(\omega)
=\frac{1}{1-r^2+i2\zeta r},
\qquad r=\frac{\omega}{\omega_0}.
$$

The form separates the horizontal scale from the response shape. Two apparatuses
with different masses and stiffnesses have the same normalized curve when they
share $\zeta$. Their physical frequency scales remain different because
$\omega_0$ differs. The normalized separation compares a prototype with a
larger machine, or when checking whether a replacement damper has changed response
shape rather than produced a pure shift of the frequency axis.

Three limiting identities provide a compact audit:

$$
G(0)=1,\qquad
\lim_{r\to\infty}r^2G(r)=-1,\qquad
G(1)=-\frac{i}{2\zeta}.
$$

The first identity tests static compliance. The second tests the inertial
high-frequency sign and slope. The third tests the quadrature response at the
natural frequency. A fitted parameter set that violates one of these relations has
usually been paired with a mismatched response variable, an inverted sensor
polarity, or an inconsistent peak/rms convention.

Parameter fitting benefits from a positive parameterization. Write

$$
m=e^\mu,\qquad b=e^\beta,\qquad k=e^\kappa
$$

when an unconstrained numerical optimizer is used. The transformed variables allow
search steps over several orders of magnitude without proposing negative mass,
negative damping, or negative stiffness. Convert the fitted covariance back to the
reported parameters before quoting uncertainties. A small residual norm alone is
insufficient evidence for a meaningful fit; the fitted values must also agree with
static stiffness, moving-mass inventory, free-decay rate, and the observed response
limits.

Each response row should preserve the quantities needed to rerun the calculation:
drive frequency, force amplitude and phase, response amplitude and phase, settling
time, acquisition length, sensor range, and a flag for clipping or rejected
transients. Store the raw two-channel waveform when practical. A later change in
the phase convention or a discovered timing delay can then be corrected from the
same record instead of reconstructed from a screenshot or a rounded amplitude
table.

Phase wrapping requires a stated rule. A phase returned in the interval
$(-\pi,\pi]$ has a discontinuity even when the physical response is smooth.
Unwrap adjacent values by adding or subtracting $2\pi$ only when the known
frequency spacing and expected response slope make that branch choice credible.
An unwrapped phase point should remain linked to the original wrapped value in the
analysis file. Large jumps can indicate a lost trigger, a poor sinusoidal fit, or a
genuine transition into a nonlinear branch; automatic unwrapping should not erase
that diagnostic evidence.

The source-force waveform deserves the same residual inspection as the mechanical
response. A force trace containing amplitude modulation or harmonic distortion
invalidates the assumed single-frequency input even if the nominal generator
setting is constant. Fit both channels over the same time window, retain their
coherence, and calculate power from the measured force and velocity rather than
from a nominal actuator specification. Those records close the connection between
the differential equation, the transfer function, and the physical energy supplied
to the oscillator.