 Vibroscope area uncertainty

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.