= 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 (materials scientist)>, <Rajnish Kumar (materials scientist)> and <Hans Lilholt> compared area-analysis procedures using measured frequencies of tensioned glass and carbon strands.<Vibroscope method for determination of cross-sectional area of glass and carbon fibres – Experiments and analyses by Bo Madsen and others>[Public abstract, frequency measurements and simple/direct/iterative area analyses]{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 estimator>
$$
f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}},\qquad
\hat\mu=\frac{T}{4L^2f_1^2},\qquad
\hat A=\frac{T}{4\rho L^2f_1^2}.
$$
The string wave-speed and fixed-end standing-wave relations supply this model.<University Physics Volume 1 by William Moebs and others>[§16.3, wave speed on a stretched string; §16.6, fixed-end standing-wave modes]{ref}
Choose $L=20\,\mathrm{mm}$, $T=1.00\,\mathrm{mN}$, $f_1=2000\,\mathrm{Hz}$ and $\rho=2500\,\mathrm{kg\,m^{-3}}$. The model gives $\hat\mu=1.5625\times10^{-7}\,\mathrm{kg\,m^{-1}}$ and $\hat A=62.5\,\mathrm{\mu m^2}$. Only if a circular area is assumed does this correspond to diameter $\sqrt{4\hat A/\pi}\approx8.92\,\mathrm{\mu m}$.
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.
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