A general closed-form second-moment framework for probabilistic updating of damped MDOF systems

Probabilistic updating of damped structural systems lacks closed-form expressions for operator statistics, leaving iterative numerical methods or implicit auxiliary variable formulations that obscure analytical uncertainty propagation. This work develops a closed-form second-moment framework for simultaneous updating of mass, damping, and stiffness matrices under combined eigenvalue, eigenvector, and modal damping uncertainty, a combination for which explicit operator statistics were previously unavailable. The formulation exploits a sequential property of weighted Frobenius norm minimisation subject to modal orthogonality constraints, whereby the three updates are obtained in three successive convex steps. Stiffness updating depends only on measured frequencies, damping updating depends only on measured modal damping, and mass updating is deduced from the measured modal mass matrix through Berman’s incomplete-mode correction. Gaussian moment closures give exact mean operators, and Fréchet derivative-based perturbation expansions give Kronecker-structured sensitivity matrices that map modal uncertainty to physical-space operator covariance without auxiliary variables or nonlinear solvers. The cross-covariances between the identified modal quantities are retained throughout. Numerical verification on a non-proportionally damped benchmark confirms close analytical-Monte Carlo agreement for mean and covariance predictions, with cross-correlation inclusion yielding order-of-magnitude improvement over independence assumptions. The methodology exchanges finite-element sparsity for analytical tractability, positioning it as a component of digital twin applications, in which the updated operators are interpreted as reduced models condensed onto the instrumented degrees of freedom and the measured modes are taken to span that reduced space. In that setting operator-level uncertainty quantification supports probabilistic forward prediction, data assimilation, and lifecycle prognostics that sampling-based alternatives render prohibitive. The scope of the paper is the operator statistics themselves, and their embedding in a particular assimilation architecture is left open. The framework requires as many sensors as identified modes, with the analytical operators condensed onto those sensors, and it recovers the true mass exactly on the span of the identified modes.