Vibroscope

A vibroscope measures the vibration of a tensioned material fibre. In fibre testing, the measured fundamental frequency, suspended length and applied tension can be used to estimate linear mass density; known mass density then permits an area estimate. The physical measurement complements the single-fibre tensile test, whose conversion from force to engineering stress requires cross-sectional area.

The 2024 study by Bo Madsen, Rajnish Kumar and Hans Lilholt compared area-analysis procedures using measured frequencies of tensioned glass and carbon strands.^[ref] A practical arrangement holds a known strand length, applies a known tension, excites small transverse oscillations and detects the frequency of its response. The supports and identification of the vibration mode enter the interpretation.

For an original idealization, treat the strand as a uniform flexible string with fixed endpoints, length $L$, tension $T$ and linear mass density $\mu$. Its wave speed is $\sqrt{T/\mu }$ and its fundamental wavelength is $2L$, giving the vibroscope area estimatorThe string wave-speed and fixed-end standing-wave relations supply this model.^[ref]

Choose $L=20\mathrm{mm}$, $T=1.00\mathrm{mN}$, $f_1=2000\mathrm{Hz}$ and $\rho =2500\mathrm{kg}{\mathrm{m}}^{-3}$. The model gives $\hat{\mu }=1.5625\times 10^{-7}\mathrm{kg}{\mathrm{m}}^{-1}$ and $\hat{A}=62.5\mu {\mathrm{m}}^2$. Only if a circular area is assumed does this correspond to diameter $\sqrt{4\hat{A}/\pi }\approx 8.92\mu \mathrm{m}$.

These are stipulated model values, not published fibre data. Checking several tensions tests the predicted proportionality of $f_1^2$ to $T$. Finite bending resistance, uncertain effective length and changes in strand dimensions under tension can invalidate the flexible-string estimate. A resonance peak alone does not establish that all of its assumptions hold.

The vibroscope area estimator converts the fundamental frequency of a tensioned material fibre into a cross-sectional area through a mechanical model. For a uniform flexible strand of length $L$, positive tension $T$, linear mass density $\mu$ and mass density $\rho$, the fixed-end string relation givesThe second equation uses $\mu =\rho A$: the frequency first identifies mass per length under the stated tension, and a chosen mass density then converts it into area. Circular geometry is needed only to infer a diameter from that area.^[ref]

The experimental study by Bo Madsen, Rajnish Kumar and Hans Lilholt used a Favimat+ fibre testing machine. Acoustic waves excited the strand, and a frequency sweep plus a polynomial fit located the resonance peak. A load cell measured tension, while grip separation supplied $L$. The reported frequency range was 500–4000 Hz, with an estimated resolution of 0.1 Hz; the stated force and length readout resolutions were $1\mu \mathrm{N}$ and $0.1\mu \mathrm{m}$. Those readout resolutions do not themselves specify the estimator's full measurement uncertainty.^[ref]

The area and mass density must refer to the same loaded or reference configuration. The companion theory derives volume and area corrections for recovering an untensioned reference area from the tensioned measurement.^[ref]

The vibroscope area uncertainty follows locally by logarithmic differentiation:Treat the small input errors as independent random variables, and let each $u$ denote the corresponding standard uncertainty. First-order uncertainty propagation then givesCorrelated inputs require their covariances as well. Four independent 1% relative standard uncertainties give $\sqrt{10}\%\approx 3.16\%$ for the area. A systematic 1% overestimate of frequency alone instead multiplies the area estimate by $(1.01{)}^{-2}\approx 0.9803$: it produces about a 1.97% underestimate.

Bending resistance can also increase the observed frequency. The companion theory paper models fully clamped supports and derives the corresponding frequency equation and deformation corrections.^[ref] An independently derived, simply supported example isolates the bending-stiffness bias in the vibroscope area estimator. Let $B=EI$ be the filament bending modulus and impose zero deflection and zero bending moment at each endpoint. Substituting $y=\sin (n\pi x/L)\cos (2\pi f_{n}t)$ into the tensioned Euler-Bernoulli beam equation $\mu y_{tt}+B{y}_{xxxx}-T{y}_{xx}=0$ givesWith $r=b/a={\pi }^{2}B/(T{L}^2)$, the uncorrected flexible-string estimate satisfies $\hat{A}/A=1/(1+r)$. At $r=0.10$, it is 9.09% too small. The same model yields two-mode separation of tension and bending stiffness:For $T>0$ and $B\ge 0$, the predicted mode ratio is $f_2/f_1=2\sqrt{(1+4r)/(1+r)}$, ranging from 2 toward 4. These identities belong to the simply supported model; clamped grips require their own boundary conditions. Neither a single resonance peak nor an assumed diameter establishes the appropriate support, density or bending correction.

Bending-stiffness bias in the vibroscope area estimator occurs when bending resistance raises a strand's frequency but its area is inferred with a flexible-string model. In the independently derived simply supported beam model in vibroscope area estimator, let $B=EI$ be the bending rigidity, $T>0$ the tension and $L$ the length. The dimensionless ratio and area error areHere $A$ is the actual area in the model's measurement configuration. The parent article derives this relation; no circular-area assumption is needed for the ratio once $B$ is specified.

For an original example, choose $r=0.25$ and $A=80\mu {\mathrm{m}}^2$. The uncorrected estimate is $64\mu {\mathrm{m}}^2$, twenty percent below the model area. Doubling only $L$, with the same $B$, $T$ and material configuration, divides $r$ by four. The new ratio is $1/1.0625=16/17$, giving $\hat{A}\approx 75.29\mu {\mathrm{m}}^2$, about 5.88 percent low. A length-dependent discrepancy is therefore a useful diagnostic to investigate, rather than evidence that the material area itself changed.

The boundary conditions are essential. Simply supported endpoints impose zero displacement and zero bending moment. The companion experimental theory instead treats fully clamped endpoints, with zero displacement and slope, and derives its own frequency and deformation corrections.^[ref] The displayed correction cannot be transferred unchanged to that apparatus.

Two-mode separation of tension and bending stiffness supplies another check within the same simply supported model. A model correction for bending bias also remains distinct from the standard input uncertainties treated in vibroscope area uncertainty.

Two-mode separation of tension and bending stiffness uses two identified transverse eigenmodes to separate coefficients that a single resonance cannot determine independently. For the uniform simply supported beam model derived in vibroscope area estimator,Thus plotting $f_n^2/n^2$ against $n^2$ gives an intercept associated with tension and a slope associated with bending. Frequencies alone identify these ratios; determining dimensional $T$ and $B$ also requires the remaining scale quantities.

An original exact example takes $f_1=800\mathrm{Hz}$ and $f_2=\sqrt{3328000}\mathrm{Hz}\approx 1824.28\mathrm{Hz}$. The two transformed values are $640000$ and $832000\mathrm{H}{\mathrm{z}}^2$ at $n^2=1$ and $4$. Their difference givesHence $r=b/a=1/9$. Using the first mode as if $b=0$ would use $640000$ instead of $576000$ for the tension coefficient; the corresponding area correction is a factor $10/9$ under this model.

Two observations fit two free coefficients, so that fit alone does not validate the model. A third identified mode should satisfy $f_3^2=9(a+9b)=10368000\mathrm{H}{\mathrm{z}}^2$, or $f_3\approx 3219.94\mathrm{Hz}$, in this example. Repeated measurements and another mode can test consistency.

For $T>0$ and $B\ge 0$, the same model requires $2\le f_2/f_1<4$. A ratio outside that range contradicts at least one assumption or the mode assignment. Actual clamped grips require the clamped-end frequency model described in the companion theory.^[ref] Do not label arbitrary neighbouring resonance peaks as modes one and two without additional evidence.

Vibroscope area uncertainty quantifies uncertainty in an area inferred from a vibration model and its measured inputs. For the flexible-string vibroscope area estimator, length and frequency enter with inverse-square dependence, whereas tension and density enter to the first power. The parent article gives the full first-order propagation formula. A measurement uncertainty budget records the input standard uncertainties, their sensitivities and any correlations.^[ref]

For an original independent-input example, stipulate relative standard uncertainties of $1\%$ in tension, $2\%$ in density, $0.5\%$ in length and $0.1\%$ in frequency. After applying the sensitivity coefficients, their relative area contributions are $1\%,2\%,1\%,0.2\%$. Combining their squares givesFor a stipulated estimate $\hat{A}=80\mu {\mathrm{m}}^2$, this is $u_A\approx 1.97\mu {\mathrm{m}}^2$. Density contributes $4/6.04\approx 66.2\%$ of the combined variance. Improving the frequency readout alone would therefore make little difference to this budget. Reducing the density contribution from $2\%$ to $1\%$ instead gives $\sqrt{3.04}\%\approx 1.74\%$ overall.

These input uncertainties are assigned model values, not the readout resolutions quoted for the Favimat+ fibre testing machine. A resolution increment does not determine a standard uncertainty without a model for the associated error. Correlated inputs require covariance terms; their signed sensitivities can either reinforce or cancel contributions.

The budget also presumes an adequate measurement model. Bending-stiffness bias in the vibroscope area estimator, uncertain supports or deformation conventions may require a correction and additional uncertainty terms. Increasing the numerical precision of the frequency does not repair a wrong physical model.