A Modal State-Space Formulation for Operational Modal Analysis

Operational Modal Analysis (OMA) is widely used for the identification of the dynamic properties of civil and mechanical structures under ambient excitation. In this work, a state-space formulation expressed directly in modal coordinates is proposed for stochastic vibration systems subjected to unknown excitation forces. Starting from the second-order equations of motion, the structural dynamics are transformed into modal coordinates, leading to a decoupled representation where each vibration mode is described by a low-dimensional state-space model parameterized by the natural frequencies, damping ratios, and modal vectors. The unknown excitation forces are modeled as Gaussian stochastic processes, allowing the formulation of the problem within a probabilistic framework suitable for Operational Modal Analysis. The resulting discrete-time stochastic state-space model is estimated using the maximum likelihood method combined with Kalman filtering. The likelihood function is optimized numerically using a BFGS algorithm in order to estimate the modal parameters directly from measured acceleration responses. A numerical example involving a simulated four-degree-of-freedom system is presented to evaluate the performance of the proposed methodology. The results show accurate estimation of natural frequencies, damping ratios, and modal shapes, with high Modal Assurance Criterion (MAC) values and reasonable standard errors. The proposed formulation provides a physically interpretable and computationally convenient framework for modal parameter estimation in Operational Modal Analysis applications.