Driven Oscillators

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 \overset{x}{¨} + b \overset{x}{˙} + k x = F_{0} cos (ω 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: $m \overset{x}{¨}$ has units of $kg m s^{- 2}$ , $b \overset{x}{˙}$ has units of $kg s^{- 1} m s^{- 1}$ , and $k x$ has units of $N m^{- 1} m$ . That unit check catches a common error in which a force amplitude is treated as a displacement amplitude.

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 ω t

along the

x

direction. Displacement is measured relative to the laboratory frame.

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

ω_{0} = \frac{k}{m}, γ = \frac{b}{2 m}, ζ = \frac{γ}{ω _{0}} = \frac{b}{2 mk} .

The compact driven equation is

\overset{x}{¨} + 2 γ \overset{x}{˙} + ω_{0}^{2} x = \frac{F _{0}}{m} cos (ω t) .

The source frequency $ω$ 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 $r (t) = cos ω t$ and write the measured displacement as

x (t) = X cos ω t + Y sin ω t .

Comparison with $A cos (ω t - δ)$ gives

X = A cos δ, Y = A sin δ, A = X^{2} + Y^{2}, δ = atan2 (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 $π$ 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 $r = ω / ω_{0}$ :

δ (r) = atan2 (2 ζ r, 1 - r^{2}) .

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 $π$ . Those sign changes diagnose the response when an amplitude calibration is uncertain. A phase trace that falls from zero to minus $π$ 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.

Steady-state phase lag

δ (r) = atan2 (2 ζ r, 1 - r^{2})

for three damping ratios. Each curve climbs from

0

at low frequency toward

π

well above resonance and passes through the quarter-cycle value

π /2

exactly at

r = 1

, independent of damping. Lighter damping makes the crossing steeper.

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 $ω$ , fit

x_{j} = C + D t_{j} + X cos ω t_{j} + Y sin ω t_{j} + ε_{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 $Δ t$ produces

Δ δ = ω Δ t .

An uncorrected $10 μ s$ channel delay is only $0.00063 rad$ at $10 Hz$ , but it becomes $0.063 rad$ at $1 kHz$ . 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 $σ_{x}$ , a well-distributed record of $N$ samples has a rough phase standard uncertainty

u (δ) \approx \frac{2 σ _{x}}{A 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 (ω) = (ω_{0}^{2} - ω^{2})^{2} + 4 γ^{2} ω^{2} .

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

\frac{d D}{d ω} = 4 ω (ω^{2} - ω_{0}^{2} + 2 γ^{2}) .

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

ω_{A} = ω_{0}^{2} - 2 γ^{2} = ω_{0} 1 - 2 ζ^{2} .

The expression is real only when $ζ < 1/ 2$ . 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.

Normalized displacement amplitude

A / A_{static} = 1/ (1 - r^{2})^{2} + (2 ζ 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

ζ = 1/ 2

the interior peak disappears though the same mode is present. All curves start at the static value

1

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

V^{2} (ω) = \frac{( F _{0} / m ) ^{2} ω ^{2}}{( ω _{0}^{2} - ω ^{2} ) ^{2} + 4 γ ^{2} ω ^{2}} .

Differentiating with respect to $ω^{2}$ places the maximum at $ω = ω_{0}$ for any positive $b$ . That result has a mechanical interpretation. At $ω_{0}$ , 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.

Response peaks depend on the measured observable. Displacement peaks slightly below

ω_{0}

, velocity peaks exactly at

ω_{0}

, and acceleration peaks above it. The dashed guide marks

ω_{0}

; the three maxima separate once damping is appreciable.

Instantaneous mechanical power from the source is

p_{in} (t) = F_{0} cos (ω t) \overset{x}{˙} (t) .

Average this product over a complete cycle in the steady state. Using $\overset{x}{˙} = - ω A sin (ω t - δ)$ gives

\overline{P} = \frac{1}{2} F_{0} ω A sin δ = \frac{1}{2} b ω^{2} A^{2} = \frac{1}{2} b V^{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 $ω_{0}$ , the force is in phase with velocity and

\overline{P}_{max} = \frac{F _{0}^{2}}{2 b} .

The factor one-half arises because $F_{0}$ and $V$ are peak amplitudes. Values entered as rms quantities require the corresponding rms identity $\overline{P} = F_{rms} V_{rms}$ only when force and velocity are in phase. Units remain a strong check. A force times a velocity has units of $N m s^{- 1} = W$ .

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 $U = \frac{1}{2} k x^{2}$ , the mass holds $K = \frac{1}{2} m \overset{x}{˙}^{2}$ , and the damper converts energy at rate $b \overset{x}{˙}^{2}$ . Over a complete steady cycle,

t \int t + T F \overset{x}{˙} d t = t \int t + T b \overset{x}{˙}^{2} d t, T = \frac{2 π}{ω} .

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 $ω_{0}$ :

\frac{P ( ω )}{P _{max}} = \frac{4 γ ^{2} ω ^{2}}{( ω _{0}^{2} - ω ^{2} ) ^{2} + 4 γ ^{2} ω ^{2}} .

Half-power frequencies $ω_{1}$ and $ω_{2}$ satisfy

\overline{P} (ω_{1}) = \overline{P} (ω_{2}) = \frac{1}{2} \overline{P}_{max}, ω_{1} < ω_{0} < ω_{2} .

Substitution reduces the half-power condition to

∣ ω_{0}^{2} - ω^{2} ∣ = 2 γ ω .

Solving the two signs gives the exact roots

ω_{1} = ω_{0}^{2} + γ^{2} - γ, ω_{2} = ω_{0}^{2} + γ^{2} + γ .

Their separation is exact for this linear viscous model:

Δ ω = ω_{2} - ω_{1} = 2 γ = \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.

Half-power bandwidth of the average-power response. The curve crosses

\frac{1}{2} P_{max}

ω_{1}

and

ω_{2}

; their separation is the bandwidth

Δ ω = 2 γ = b / m

. The peak sits at

ω_{0}

in the linear model.

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

Q = 2 π \frac{E _{stored}}{∣Δ E ∣ _{cycle}} \approx \frac{ω _{0}}{2 γ} = \frac{m ω _{0}}{b} = \frac{1}{2 ζ} .

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

Q \approx \frac{ω _{0}}{Δ ω} .

The approximation replaces the resonant frequency scale in the numerator with $ω_{0}$ 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 $E (t) = E_{0} e^{- 2 γ t}$ . Over one near-resonant period, the fractional loss is approximately $2 γ (2 π / ω_{0})$ , giving the same $Q \approx ω_{0} / (2 γ)$ . 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 $y (t) = Y cos ω t$ , absolute mass motion be $x (t)$ , and spring extension be $z = x - y$ . The force balance becomes

m \overset{x}{¨} + b (\overset{x}{˙} - \overset{y}{˙}) + k (x - y) = 0,

or, in the relative coordinate,

m \overset{z}{¨} + b \overset{z}{˙} + k z = - m \overset{y}{¨} .

The right side has amplitude $m ω^{2} Y$ . Base excitation thus has the same mathematical form as force drive after the drive amplitude is replaced by $m ω^{2} Y$ , 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.

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

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 $e^{- γ t} cos (ω_{d} t + ϕ_{t})$ , where

ω_{d} = ω_{0}^{2} - γ^{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 $ω_{0}$ . 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.

Starting the drive from rest, the response builds up to steady amplitude: the homogeneous transient decays as

e^{- γ 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.

The amplitude of a transient decays by $e^{- γ t_{s}}$ 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 $γ t_{s} > ln 100 \approx 4.61$ . 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 $ω_{0}$ and $γ$ 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 $F_{0} e^{iω t}$ and seek a steady response $x = Re {\tilde{x} e^{iω t}}$ . Substitution gives

(k - m ω^{2} + ibω) \tilde{x} = F_{0} .

The complex dynamic stiffness is

\tilde{K} (ω) = k - m ω^{2} + ibω, \tilde{x} = \frac{F _{0}}{K ~ ( ω )} .

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

Dynamic stiffness

\tilde{K} = k - m ω^{2} + ibω

in the complex plane. The horizontal leg is the elastic-minus-inertial part

k - m ω^{2}

, the vertical leg is the loss part

bω

, the hypotenuse is

∣ \tilde{K} ∣ = F_{0} / A

, and its angle at the origin is the displacement lag

δ

Taking the magnitude gives the displacement amplitude:

A (ω) = \frac{F _{0}}{( k - m ω ^{2} ) ^{2} + b ^{2} ω ^{2}} = \frac{F _{0} / m}{( ω _{0}^{2} - ω ^{2} ) ^{2} + ( 2 γ ω ) ^{2}} .

With $F (t) = F_{0} cos ω t$ , write the response as

x (t) = A cos (ω t - δ), δ = atan2 (bω, k - m ω^{2}) .

The two-argument angle is essential. A one-argument inverse tangent cannot distinguish $k - m ω^{2} > 0$ from $k - m ω^{2} < 0$ when the ratio has the same value. The physical lag increases continuously from $0$ toward $π$ as the source frequency crosses the resonance region; $atan2$ returns the correct quadrant.

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

A \approx \frac{F _{0}}{k}, δ \approx 0.

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

A \approx \frac{F _{0}}{m ω ^{2}}, δ \approx π .

The mass then moves almost opposite to the applied force, with an amplitude that falls as $ω^{- 2}$ . A response curve whose high-frequency tail falls only as $ω^{- 1}$ 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 $2$ .

The complex compliance, or displacement transfer function, is

H_{x F} (ω) = \frac{x ~ ( ω )}{F ~ ( ω )} = \frac{1}{k - m ω ^{2} + ibω} .

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

H_{v F} (ω) = iω H_{x F} (ω),

which has units of $m s^{- 1} N^{- 1}$ . 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

ρ_{j} = \tilde{x}_{j} - H_{x F} (ω_{j}; m, b, k) \tilde{F}_{j} .

When the channel uncertainties are known, minimize a weighted sum of $∣ ρ_{j} ∣^{2}$ . 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 $y = Y cos ω t$ describe the support, $x$ describe the absolute mass coordinate, and $z = x - y$ describe spring extension. The relative equation is

m \overset{z}{¨} + b \overset{z}{˙} + k z = - m \overset{y}{¨} = m ω^{2} Y cos ω t .

The relative displacement is therefore driven by an effective force with amplitude $m ω^{2} Y$ . Introducing $r = ω / ω_{0}$ gives the relative-motion ratio

\frac{Z}{Y} = \frac{r ^{2}}{( 1 - r ^{2} ) ^{2} + ( 2 ζ 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{X ~}{Y ~} = \frac{1 + i 2 ζ r}{1 - r ^{2} + i 2 ζ r} .

Its magnitude is

T_{x} (r) = \frac{X}{Y} = \frac{1 + ( 2 ζ r ) ^{2}}{( 1 - r ^{2} ) ^{2} + ( 2 ζ 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 $r > 2$ . 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.

Absolute-motion transmissibility

T_{x} (r) = 1 + (2 ζ r)^{2} / (1 - r^{2})^{2} + (2 ζ r)^{2}

for two damping ratios. Both curves cross unity exactly at

r = 2

; 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.

The high-frequency asymptote makes the damping compromise explicit:

T_{x} (r) \approx \frac{2 ζ}{r}, \frac{Z}{Y} \approx 1 (r ≫ 1) .

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 + ibω) \tilde{Z} .

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

\frac{∣ F _{T} ∣}{m ω ^{2} Y} = \frac{1 + ( 2 ζ r ) ^{2}}{( 1 - r ^{2} ) ^{2} + ( 2 ζ 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 kg, k = 320 N m^{- 1}, b = 1.60 kg s^{- 1}, F_{0} = 0.400 N .

The static deflection under the stated peak force is $F_{0} / k = 1.25 mm$ . 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

ω_{0} = \frac{k}{m} = 20.0 rad s^{- 1}, γ = \frac{b}{2 m} = 1.00 s^{- 1}, ζ = 0.0500.

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

Q \approx \frac{ω _{0}}{2 γ} = 10.0, Δ ω = 2 γ = 2.00 rad s^{- 1} .

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

ω_{1} = ω_{0}^{2} + γ^{2} - γ = 19.025 rad s^{- 1}, ω_{2} = ω_{0}^{2} + γ^{2} + γ = 21.025 rad s^{- 1} .

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

Displacement amplitude of the worked oscillator (

ω_{0} = 20 rad s^{- 1}

ζ = 0.05

). The peak reaches

12.5 mm

near

ω_{0}

, ten times the static deflection. The marked points at

19

20

, and

21 rad s^{- 1}

show two off-resonance amplitudes that are similar; their phases place them on opposite sides.

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

A (ω_{0}) = \frac{F _{0}}{b ω _{0}} = \frac{0.400}{( 1.60 ) ( 20.0 )} = 0.0125 m .

The result is $12.5 mm$ , 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:

ω_{A} = ω_{0}^{2} - 2 γ^{2} = 19.950 rad s^{- 1} .

The difference is only $0.050 rad s^{- 1}$ , well below the stated power bandwidth. Reporting $20.0 rad s^{- 1}$ 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 $21.0 rad s^{- 1}$ to see how phase supplements amplitude. The results are

ω (rad s^{- 1}) 19.0 20.0 21.0 A (mm) 9.18 12.5 8.52 δ (rad) 0.772 1.571 2.344

The two off-resonance amplitudes are similar, but their phases occupy opposite sides of the resonance. A data point with amplitude near $9 mm$ 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 $- 2.34 rad$ may be physically equivalent after phase wrapping, while a value near $0.80 rad$ at $21 rad s^{- 1}$ would be inconsistent with the stated channel orientation.

Response phasors at the worked settings

19

20

, and

21 rad s^{- 1}

. Below the mode the response leads along the positive in-phase axis; at

ω_{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.

Suppose independent calibrations give

u (m) = 0.004 kg, u (k) = 3.2 N m^{- 1}, u (b) = 0.080 kg s^{- 1}, u (F_{0}) = 0.010 N .

Neglecting covariance, propagation for the natural frequency gives

(\frac{u ( ω _{0} )}{ω _{0}})^{2} = \frac{1}{4} (\frac{u ( k )}{k})^{2} + \frac{1}{4} (\frac{u ( m )}{m})^{2} .

Thus $u (ω_{0}) = 0.112 rad s^{- 1}$ . The damping coefficient carries a relative uncertainty of about $5.0%$ , so $Δ ω = 2.00 \pm 0.10 rad s^{- 1}$ . For the resonant amplitude, independent relative contributions from force, damping, and natural frequency combine to approximately $5.6%$ , giving

A (ω_{0}) = 12.5 \pm 0.7 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} = - k x - α x^{3} .

Positive $α$ 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 $α$ 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.

Amplitude-dependent stiffness shifts the resonance. A hardening spring (

α > 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.

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 \overset{x}{˙}, F_{d} = - F_{c} sgn (\overset{x}{˙}), F_{d} = - c \overset{x}{˙} ∣ \overset{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

Δ E_{visc} \approx π bω A^{2}, Δ E_{dry} = 4 F_{c} A, Δ E_{quad} \approx \frac{8}{3} c ω^{2} A^{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_{x F} (ω) = r \sum \frac{C _{r}}{ω _{r}^{2} - ω ^{2} + i 2 γ _{r} ω} .

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 $ω_{0}$ .

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

A free ring-down estimates $γ$ from decay, a driven half-power scan estimates $Δ ω$ , and a phase scan identifies the in-phase zero crossing near $ω_{0}$ . 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_{x F} (ω) ⟶ \frac{1}{k}, ar g H_{x F} ⟶ 0.

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

H_{x F} (ω) ⟶ - \frac{1}{m ω ^{2}}, ar g H_{x F} ⟶ - π .

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_{x F} (ω_{0}) = - \frac{i}{b ω _{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} = - ω^{2} \tilde{x}, H_{a F} (ω) = - ω^{2} H_{x F} (ω) .

Its low-frequency magnitude rises as $ω^{2} / k$ , 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 $ω^{2}$ 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 $iω$ or $- ω^{2}$ 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 $∣ H_{a F} ∣ = 1.19 m s^{- 2} N^{- 1}$ in a frequency region above the mode and below any higher structural mode. The inferred effective mass is

m_{eff} \approx \frac{1}{∣ H _{a F} ∣} = 0.840 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 $ω_{0}$ and the compliance by the static compliance. The dimensionless response is

G (r) = k H_{x F} (ω) = \frac{1}{1 - r ^{2} + i 2 ζ r}, r = \frac{ω}{ω _{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 $ζ$ . Their physical frequency scales remain different because $ω_{0}$ 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, r \to \infty lim r^{2} G (r) = - 1, G (1) = - \frac{i}{2 ζ} .

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^{μ}, b = e^{β}, k = e^{κ}

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 $(- π, π]$ has a discontinuity even when the physical response is smooth. Unwrap adjacent values by adding or subtracting $2 π$ 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.