Analytical hierarchical Bayesian modeling framework for model updating and uncertainty propagation utilizing frequency response function data

Model updating using frequency response functions (FRFs) provides critical advantages in structural dynamics. However, existing probabilistic approaches struggle to balance computational efficiency with comprehensive uncertainty quantification. To this end, this paper introduces an analytical hierarchical Bayesian modeling (HBM) framework that overcomes these limitations through utilization of complex-valued FRF data and variational inference. In particular, the proposed approach incorporates a complex Gaussian likelihood formulation directly into the HBM framework for the FRF experimental data, which allows for a more appropriate and physically consistent treatment of FRF data, particularly when both magnitude and phase information (real and imaginary parts) are essential. Additionally, the proposed approach enables the analytical HBM solution under the complex likelihood setting, improving both the accuracy of parameter estimation and the efficiency of the computation. The framework further propagates the parameter uncertainty to the response predictions and reliability assessment. Numerical and experimental validations on a simply supported beam demonstrate the effectiveness of the proposed approach. Results indicate that the proposed framework provides a reasonable uncertainty estimate of the model parameters as well as the response predictions. Reliability computations on the numerical example also suggest that the proposed framework provides conservative and reliable failure probability estimates, compared to the classical Bayesian modeling which often leads to unsafe engineering decisions.