Incoprea research · leads batch N · 3 November 2026

Acoustics, vibration and mechanical wave measurement: 200 leads, three computations, one proof

0 · The structure being generalised

A mechanical measurement reports a quantity carried by a transducer, through a fixture, over a finite bandwidth, at a finite number of points, in a stated unit. Every one of those four carriers can fail independently, and the failure appears as a property of the object.

The four sections are the four carriers. Modal analysis is the space and bandwidth carrier: which modes exist in the data at all. Damping is the mechanism carrier: which physical process the single number stands for. Nonlinearity is the amplitude carrier: whether the response scales. Transduction is the reference and unit carrier: whether the number is the quantity it claims to be.

Three quantities recur, and each is dimensionless:

• The modal overlap 2ζ/Δf — the half-power bandwidth divided by the mode spacing. It sets what the first section can see, and it is the only parameter in the first computation's band table.
• The loss-factor ratio η_max/η_min across a band — it sets how badly one constant represents a curve, and it is the only parameter that matters in the second computation's bias table.
• The drive exponent d ln H_k / d ln F — the order of the nonlinearity as measured, which is the one number that separates a material from a fixture in the third computation.

The proposition closes the damping section: a partition of a loss factor among mechanisms with the same frequency dependence is not identifiable from the loss factor, at any number of frequencies, and the numerical signature of that non-identifiability is a condition number of order 10¹⁵.

Y · Modal analysis: identifiability of modes from a finite measurement set

A mode list is a list of what a particular sensor array, in a particular band, with a particular damping, could see. The leads below ask what bounds that list, and the first computation measures two of the bounds — one in frequency and one in space.

Z · Damping: which mechanism a loss factor can and cannot distinguish

A loss factor is one number per frequency. A mechanism is a curve over frequency, amplitude and temperature. The leads below ask when one number can stand for a curve, and the second computation puts a number on what a single loss factor does to a two-mechanism curve.

AA · Nonlinearity: when a nonlinear signature is real and when it is a fixture

Every nonlinear measurement is a measurement of the whole rig. The leads below separate the material from the fixture, and the third computation shows that at a single drive the two are indistinguishable from their harmonic ratio, while the drive-scaling exponent separates them.

AB · Transduction and calibration: reciprocity, reference standards and unit traps

The transducer is the reference the whole measurement rests on, and the unit is the reference the number rests on. The leads below are claims about both, and each is falsifiable by a calibration record or a substitution.

Tests run

Three computations, all in incoprea_nov3/code/acoustics_tests.py, writing incoprea_nov3/results/acoustics_vibration.json. Tests 1 and 3 are simulations with stipulated parameters; test 2 is arithmetic on a stipulated mechanism model plus a seeded noise propagation. No measured data are used anywhere on this page, and no number in the three test tables was typed by hand: each is read from the JSON the script wrote.

1 · How many modes a finite measurement set can distinguish

Three sub-computations. (a) Two modes at 1000 Hz and 1000(1+δ) with equal damping, unit residues (or a residue ratio of 0.3), summed as a modal receptance and weighted by f² to give accelerance. δ is bisected until the peak count of |H| falls from two to one. A peak counts only if its prominence exceeds a stated fraction of its height, which is the same as requiring the saddle beside it to be at least that many dB down; three thresholds are reported, because the threshold is a choice and the answer depends on it.

κ = δ_crit/(2ζ) is the separation in units of one half-power bandwidth. At the 1 dB threshold κ = 0.41 for equal residues and 0.67 for a residue ratio of 0.3, and it varies by about 10% across three decades of ζ. So the working criterion is not "one full bandwidth apart" but 0.41 of a bandwidth, and a mode 30% weaker must be 1.6× as far out to be seen at all.

(b) A band from 100 to 1000 Hz containing a stipulated number of modes per decade, log-spaced, all present in the FRF. The count is of interior peaks standing at least 1 dB above their saddle, divided by the number of modes whose peaks lie strictly inside the window. The last column is the prediction from (a) applied pairwise to adjacent modes.

What the table says. The identifiable fraction is not a smooth function of damping. It is one until the overlap 2ζ/Δf reaches about 0.8, partial between 0.8 and 1.7, and zero above 2.5: at 30 modes per decade and ζ = 0.1 the response is a single envelope with no interior maximum at all, so the count is 0 and not "some". And the pairwise prediction from (a) is optimistic wherever it matters — at 50 modes/decade and ζ = 0.05 it predicts all 49 peaks and the computation finds 14 — because a mode has a neighbour on each side, and the two neighbours fill the saddle that part (a) measured with only one.

(c) P sensors on a uniform grid, sensor j at x_j = jL/(P+1), mode n with shape sin(nπx/L). The sensor matrix's rank is the number of independent mode-shape measurements, and two modes are indistinguishable when their sensor vectors agree up to sign.

The identities sin((n+2(P+1))πj/(P+1)) = sin(nπj/(P+1)) and sin((2(P+1)−n)πj/(P+1)) = −sin(nπj/(P+1)) hold exactly, and the computation verifies them to 6×10⁻¹⁴. So of every 2(P+1) consecutive modes at most P+1 sensor signatures exist, and one of those — the mode with a node at every sensor — carries no signal at all. With 8 sensors the 54-mode set collapses to 8 independent signatures plus one all-zero class: 46 of the 54 modes are confounded with another, 40 as duplicates and 6 as invisible. Half the mode list is unidentifiable in space before any question of bandwidth arises, and the mechanism is a periodicity of the mode shapes, not a limit of the analyser.

The expectation this contradicted. The working rule was that two modes are separable once their separation exceeds one half-power bandwidth, and that a dense mode list is a question of frequency resolution. Both are wrong in opposite directions. The separation threshold is about 0.41 of a bandwidth, not one, so more modes are separable than the rule allows; but the pairwise criterion then fails by a factor of 3.5 at realistic overlap, so fewer are separable than the corrected rule allows. And the binding limit is spatial: with P sensors, no bandwidth and no analyser recovers the other half of the modes.

2 · What one loss factor does to a two-mechanism curve

Two mechanisms, hysteretic and viscous: η(f) = η_h + η_v(f/f₀) with η_h = 0.010 and η_v = 0.004 at f₀ = 1 kHz. One constant is fitted as the arithmetic mean over log-spaced points, which is the unweighted least-squares constant. The bias is reported at both band edges, where it is largest, together with the error it produces in Q = 1/η.

The bias is antisymmetric and grows with the band. At a 2:1 band it is ±10%, which is inside most damping uncertainties; at 100:1 it is +79% at one edge and −63% at the other, and the Q inferred at the high edge is wrong by 169%. A single loss factor is not a wrong number — it is a number for a band, and it is meaningless without the band.

The next table asks whether a better two-parameter model helps. The curve is now three mechanisms, η = η_h + η_vf/f₀ + η_te·2(f/f_te)/(1+(f/f_te)²) with 0.010, 0.004 and 0.006 at f₀ = 1 kHz and f_te = 1.5 kHz, over 200–5000 Hz, where η runs from 0.0124 to 0.0333, a ratio of 2.69. Bias is the maximum relative deviation over the band; RMS is in log η.

A Rayleigh model is the standard two-parameter answer and it still leaves 42% — it cannot represent a Debye peak, whose curvature has the wrong sign. Increasing the parameter count is not the solution; matching the mechanism is. The three-mechanism fit recovers all three coefficients to better than 0.05%, which is a check on the arithmetic and not a result: the data were generated from that basis.

The identifiability question. If two mechanisms are both flat in frequency, no number of frequencies separates them. The condition number of the design matrix, computed after normalising every column to unit norm, is the numerical statement of that, and the last column asks whether a temperature axis with different activation energies rescues the partition.

Hysteretic against friction: 1.7×10¹⁵ from frequency alone — two identical columns, a rank-one map — and 9.7 once four temperatures are added. Every other pair is already identifiable from frequency, with a condition number below 7. So a loss-factor curve distinguishes almost every pair of mechanisms except the one pair that matters most: the two that are both constant in frequency are the two a single loss factor can never separate, and the axis that separates them is temperature, not frequency.

The noise propagation says how much this costs. A 0.1% standard deviation in η, propagated through the three-mechanism fit over 4000 seeded draws, gives coefficient standard deviations of 0.05σ, 0.01σ and 0.07σ — the fit averages the measurement noise down, which is why the partition is worth attempting at all when the basis is right and hopeless when the basis is degenerate.

The expectation this contradicted. I expected the two-parameter Rayleigh model to repair most of the single-constant bias, since it contains both a 1/f and an f term and the true curve contains both. It repairs less than half: 56% against 71%. The bias is not caused by the model having too few parameters but by the thermoelastic term having the wrong shape, and a wrong shape is not a parameter-count problem. The second surprise is that the degenerate pair is not viscous against hysteretic — those are separable at a condition number of 2.7 — but hysteretic against friction, which no frequency-only measurement can touch.

3 · The harmonic signature of a quadratic and of a cubic nonlinearity at fixed drive

RK4 on x″ + 2ζx′ + g(x) = F cos(ωt) with ω₀ = 1, ζ = 0.03, ω/ω₀ = 0.98, 160 cycles of which the last 32 are analysed, 512 points per cycle, so the harmonics fall on exact FFT bins. g(x) = x + αx², x + βx³, or x + γ|x|, with α = 0.05, β = 0.1, γ = 0.05. The |x| model is the idealisation of a one-sided contact: piecewise linear, hence linear in the drive, which is the point of it.

The quadratic's second harmonic follows F^2.004 and the contact's follows F^1.000; the cubic's third follows F^2.807, not F³, and the next table says why. At the lowest drive the contact's second harmonic is 12.0× the quadratic's, and at the highest drive only 1.5×: a fixture does not weaken with drive, and a material nonlinearity does.

The exponent that is not three. The cubic's third harmonic should follow F³ and follows F^2.81. Repeating the sweep with the drive point moved away from resonance recovers the exponent: 2.98 at ω/ω₀ = 0.90, 2.94 at ω/ω₀ = 0.94, 2.81 at ω/ω₀ = 0.98. The cause is the backbone: a cubic nonlinearity shifts the resonance upward with amplitude, so at a fixed drive frequency the operating point walks down the resonance flank as the drive rises, and part of the "harmonic growth" is really the linear response at a changing detuning. The standard amplitude-scaling test is therefore biased toward low exponents exactly where the nonlinearity is strongest, which is where it is usually measured.

Numerical checks. The linear control gives H₂/H₁ between 1.2e-13 and 1.2e-13 at every drive, which is the arithmetic floor. The contact model's step-size convergence, from 256 to 1024 points per cycle, moves H₂/H₁ by 0.0007% and H₃/H₁ by 0.0016%, so the kink in |x| is resolved.

The expectation this contradicted. The expectation was that the harmonic pattern at a single drive identifies the nonlinearity: even harmonics mean quadratic, odd mean cubic. It does not, because a one-sided contact also produces a large second harmonic and at low drive produces a larger one than a quadratic material nonlinearity does. At one drive the two differ by a factor of 12 in size and not at all in kind. What separates them is the exponent — 2 against 1 — and the exponent is the one thing a single-drive measurement cannot see. The second surprise is that the cubic's own exponent is not 3: the standard test is biased by the backbone shift, and the bias is largest at the amplitude where the test is most sensitive.

The proof: a loss factor cannot partition itself

proved Everything in this section is a derivation. No number in it is computed, and the numerical corroboration that follows is labelled as such.

Proposition

Let the loss factor of a linear viscoelastic solid be a superposition of mechanism contributions, η(ω) = Σ_k c_k φ_k(ω), with stipulated basis functions φ_k and unknown non-negative coefficients c_k. Let η be measured at any finite set of frequencies ω₁…ω_m in a band Ω. If two basis functions are proportional on Ω, then the coefficients are not identifiable from those measurements: the data determine one linear combination of the two and nothing more.

Assumptions

1. A1 — superposition. The mechanisms contribute additively to the loss factor at fixed amplitude, temperature and frequency. This is the standard linear viscoelastic assumption; it fails for a mechanism whose loss depends on the strain at which another mechanism is measured.
2. A2 — stipulated basis. The functions φ_k are known. The proposition is about the coefficients, not about discovering the basis; a misspecified basis makes the problem worse, not better.
3. A3 — one measured quantity. Only η(ω) is measured. Amplitude, temperature and static preload are held fixed and are not varied, so they supply no additional equations.
4. A4 — exact arithmetic. The measurements are noiseless. Noise only widens the non-identifiable set; it cannot shrink it.

Proof

Write the measured vector as η = Φc, where Φ is the m×K matrix with entries Φ_ij = φ_j(ω_i) and c ∈ ℝ^K is the coefficient vector. The map c ↦ Φc is linear, so it is injective exactly when Φ has full column rank K. Suppose φ₁|_Ω = λφ₂|_Ω for some λ ≠ 0. Then for any c,

c₁φ₁ + c₂φ₂ = (c₁ + λc₂) φ₁ ,

so the data depend on c₁ and c₂ only through the single combination c₁ + λc₂. The vector v = (λ, −1, 0, …, 0)^T satisfies Φv = λφ₁ − φ₂ = 0, so v lies in the null space of Φ and c and c + tv give identical measurements for every real t. The null space has dimension at least one, rank Φ ≤ K − 1, and the identifiable quantities are the projections of c onto the row space of Φ. In particular, adding frequencies inside Ω does not help: the proportionality is a property of the functions on Ω, and Φ gains rows but not rank. ∎

Corollary 1 (the pair that matters)

Structural (hysteretic) damping has η independent of frequency. Coulomb friction, in the equivalent linearisation used in modal analysis, also has η independent of frequency. So φ_h = φ_f = 1 on every band, λ = 1, and no measurement of η(ω), at any number of frequencies and any bandwidth, can separate them. They are separated only on an axis where their basis functions differ: amplitude, because friction is nonlinear and structural damping is not, or temperature, because their activation energies differ. This is the reason the third column of the condition-number table is the only column that rescues the pair.

Corollary 2 (noise amplification)

Let σ be the standard deviation of an independent error on each η_i and let κ(Φ) = ‖Φ‖‖Φ⁺‖ be the condition number. The standard deviation of the recovered coefficients along the worst direction is σ‖Φ⁺‖ = σ/σ_min(Φ), and since σ_min ≤ σ_max/κ = ‖Φ‖/κ, that error is at least σκ/‖Φ‖ and grows at least as fast as κ. A condition number of 10¹⁵ therefore does not mean "poorly determined"; it means "determined to zero digits", which is the numerical way of saying the coefficient does not exist as a measurable quantity.

Numerical corroboration computed

The proposition is proved, but the claim that floating point exhibits it is computed. A two-column design matrix of two frequency-flat mechanisms over 200–5000 Hz has singular values 31.65 and 2.1e-14, a null vector of (-0.707, 0.707), and two coefficient vectors summing to 0.010 and 0.010 whose predicted measurements differ by 3.8e-16. The machine reproduces the null direction to three digits and the indistinguishability to the arithmetic floor.

If one sentence survives this section: a loss factor is a sum, and a sum cannot be decomposed on an axis where its terms are equal — the pair of mechanisms that a loss-factor curve can never separate is exactly the pair, hysteretic and frictional, that a single loss factor is usually asked to represent.

Status of the claims

Counts

Tags are editorial: test marks a lead that can be attacked with a computation, a dataset, or a re-analysis already to hand; note marks one that needs a literature sample, a new measurement, or an institutional change. A "test" tag is not evidence that the lead is true.

Limits

• Test 1 uses unit residues and equal damping. Real structures have residues that vary by an order of magnitude between modes and damping that varies with mode. The κ table covers one residue ratio (0.3) and no damping ratio; the band table covers neither. The qualitative result — a threshold near 0.4 of a bandwidth, and a pairwise criterion that fails at overlap above about 1 — is robust to these choices, but the numbers are not.
• The band count's zero is an artefact of the model, not of the physics. At high overlap the modal sum becomes a single envelope whose maximum sits at a band edge, so the count of interior maxima is 0 rather than small. The identifiable fraction is therefore a step, and the step location is what the table is for.
• Test 2's mechanism model is stipulated, and the thermoelastic term is a single Debye peak. Real thermoelastic loss in a plate is a spectrum of Debye peaks over thickness modes. A single peak gives the cleanest possible case for the fit, so the bias figures are lower bounds.
• The condition numbers are not uncertainties. They measure collinearity of a stipulated basis at stipulated activation energies. A pair whose activation energies were closer than 20 and 35 kJ/mol would be less separable than the table shows, and one further apart more so.
• Test 3 is a single-degree-of-freedom model. No mode coupling, no modal interaction, no quadratic damping. Real fixtures produce combination tones and subharmonics that a 1-DOF model cannot represent, so the fixture result is a lower bound on how much a rig can imitate a material.
• The |x| contact model is an idealisation. A real one-sided contact has a gap, a Hertzian stiffness and a restitution, and it produces odd harmonics as well as even. The model was chosen because it is exactly linear in the drive, which makes the exponent argument exact; a real contact would have an exponent between 1 and 2.
• No literature search. Several leads are certainly known under other names — the Rayleigh separation criterion, modal overlap, reciprocity calibration, the backbone curve — and are stated here as falsifiable claims rather than as discoveries.
• All 200 leads are untested. The tags describe intent, not status. Nothing on this page is a measurement; the only numbers are closed-form evaluations, seeded simulations and one proof.

If one sentence survives: a mechanical measurement identifies at most half the modes its sensor count allows, represents a mechanism only on an axis where that mechanism differs from its neighbours, distinguishes a material nonlinearity from a fixture only through the drive exponent and not through the harmonic, and is only as traceable as the transducer and the unit it was reported in.

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