= Bending-stiffness bias in the vibroscope area estimator
{title2=$\hat A/A=(1+r)^{-1}$}
<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 are
$$
r=\frac{\pi^2B}{TL^2},\qquad\frac{\hat A}{A}=\frac{1}{1+r}.
$$
Here $A$ 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 $A=80\,\mathrm{\mu m^2}$. The uncorrected estimate is $64\,\mathrm{\mu m^2}$, 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 $\hat A\approx75.29\,\mathrm{\mu m^2}$, 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.<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, tensioned beam and fully clamped supports; §§5–6, 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>.
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