= Vibroscope area estimator
{title2=$\hat A=T/(4\rho L^2f_1^2)$}
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 gives
$$
f_1=\frac{1}{2L}\sqrt{\frac{T}{\mu}},\qquad
\boxed{\hat A=\frac{T}{4\rho L^2f_1^2}.}
$$
The 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.<University Physics Volume 1 by William Moebs and others>[§16.3, equation 16.8, string wave speed; §16.6, fixed-end standing-wave modes]{ref}
The experimental study by <Bo Madsen (materials scientist)>, <Rajnish Kumar (materials scientist)> 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 $1\,\mathrm{\mu N}$ and $0.1\,\mathrm{\mu m}$. Those readout resolutions do not themselves specify the estimator's full <measurement uncertainty>.<Vibroscope method for determination of cross-sectional area of glass and carbon fibres – Experiments and analyses by Bo Madsen and others>[§3.2, apparatus, acoustic excitation, swept-sine/polynomial peak determination and measurement resolutions]{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.<Vibroscope method for determination of cross-sectional area of glass and carbon fibres - Theory and further development by Bo Madsen and others>[§§5.1–5.2, volume and area changes; §6, complete vibroscope equation]{ref}
The <vibroscope area uncertainty> follows locally by <logarithmic differentiation>:
$$
\frac{d\hat A}{\hat A}=\frac{dT}{T}-\frac{d\rho}{\rho}-2\frac{dL}{L}-2\frac{df_1}{f_1}.
$$
Treat the small input errors as <independent random variables>, and let each $u$ denote the corresponding <standard uncertainty>. First-order <uncertainty propagation> then gives
$$
\left(\frac{u_A}{\hat A}\right)^2\simeq
\left(\frac{u_T}{T}\right)^2+
\left(\frac{u_\rho}{\rho}\right)^2+
4\left(\frac{u_L}{L}\right)^2+
4\left(\frac{u_f}{f_1}\right)^2.
$$
Correlated inputs require their <covariances> as well. Four independent 1% relative standard uncertainties give $\sqrt{10}\%\approx3.16\%$ for the area. A systematic 1% overestimate of frequency alone instead multiplies the area estimate by $(1.01)^{-2}\approx0.9803$: 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.<Vibroscope method for determination of cross-sectional area of glass and carbon fibres - Theory and further development by Bo Madsen and others>[§§2.1–2.2, beam equation and clamped-end conditions; §§5–6, corrections for the tensioned fibre]{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 $y=\sin(n\pi x/L)\cos(2\pi f_nt)$ into the tensioned <Euler-Bernoulli beam equation> $\mu y_{tt}+B y_{xxxx}-T y_{xx}=0$ gives
$$
\frac{f_n^2}{n^2}=a+bn^2,\qquad
a=\frac{T}{4\mu L^2},\qquad
b=\frac{\pi^2B}{4\mu L^4}.
$$
With $r=b/a=\pi^2B/(TL^2)$, 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>:
$$
b=\frac{f_2^2/4-f_1^2}{3},\qquad
a=\frac{4f_1^2-f_2^2/4}{3},\qquad
A=\frac{T}{4\rho L^2a}.
$$
For $T>0$ and $B\geq0$, 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.
\Image[/vibroscope-bending-bias.png]
{title=Original simply supported fibre model: bending resistance lowers the uncorrected area estimate and changes the ratio of the first two frequencies.}
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