Uncertainty in frequency response function estimates in experimental modal analysis

In experimental modal analysis, the estimation of frequency response functions is based on input and output measurements, which are prone to uncertainty due to noise. Different estimators exist, such as $H_1$, $H_2$, etc. Which estimator to use depends solely on the experimental setup, as the estimators have different statistical properties in different setups. This work presents covariance expressions for real and imaginary parts of different frequency response function estimators: $H_1$, $H_2$, and $H_v$. The covariance expressions are obtained by propagating the uncertainty from measurements. Different techniques to identify the uncertainty from measurements are briefly presented before the covariance expressions for each estimator are developed. These covariance expressions for the considered estimators are structured identically, and a general expression to propagate the uncertainty from the real and imaginary parts to the amplitude and phase is developed and presented. The covariance expressions are developed using the Delta-method, where the Jacobians are obtained using first-order perturbation theory. The expressions are validated using a simple 6-DoF simulated mass–spring system and Monte-Carlo simulation. The validation shows good agreement between the estimated uncertainties using the developed covariance expressions and the uncertainties from the Monte-Carlo simulations when the measurement uncertainty is adequately identified. The application of the derived uncertainty expressions is illustrated using excitation and response data from an experiment conducted on a beam-like structure.