= Vibroscope area uncertainty
{title2=$d\log\hat A=d\log T-d\log\rho-2d\log L-2d\log f_1$}
<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.<Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results by Barry N. Taylor and Chris E. Kuyatt>[§5.1; AppendixA, equationA-3, sensitivity coefficients and covariance terms]{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 gives
$$
\frac{u_A}{\hat A}\simeq\sqrt{1^2+2^2+1^2+0.2^2}\%=\sqrt{6.04}\%\approx2.46\%.
$$
For a stipulated estimate $\hat A=80\,\mathrm{\mu m^2}$, this is $u_A\approx1.97\,\mathrm{\mu m^2}$. Density contributes $4/6.04\approx66.2\%$ 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 $\sqrt{3.04}\%\approx1.74\%$ 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.
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