Combining ARX Models, Latent Force Representation, and Kalman Filters for Nonlinear Vibration Analysis

Resumo

The identification of nonlinear mechanical systems has wide application in the study of structural dynamics, both during the design and optimization phases and in the monitoring of complex systems in operation. In the context of structural health monitoring (SHM), identifying nonlinearities allows not only the detection of damage and anomalies but also their quantification and characterization. Although several methods have been proposed for identifying and characterizing nonlinear behavior, new techniques continue to emerge, aiming to simplify the methodologies and broaden the applicability of nonlinear dynamics identification in structural engineering. Recently, approaches that combine data analysis and machine learning models with physical modeling have gained significant attention due to their flexibility, accuracy, extrapolation capabilities, and physical interpretability. In this work, a novel methodology is proposed for characterizing nonlinearities in structural dynamic systems. The approach combines a classical linear AutoRegressive model with eXogenous inputs (ARX) with a Bayesian filtering method — the classical Kalman filter — to extract nonlinear contributions from vibration signals. These contributions are modeled in the state equations using a Latent Force representation based on a Gaussian Process machine learning model. The extracted nonlinear component can be used for damage detection, characterization, and quantification. Moreover, once the nonlinear contribution is isolated, a machine learning model can be employed to represent this component, enabling the prediction of the dynamic behavior under varying operating conditions, including different vibration levels. The proposed methodology is applied to a classical nonlinear dynamic system — the Duffing oscillator — showing promising results both in extracting the nonlinear restoring force and in predicting the system response under previously unseen conditions.