An Interpretable Frequency–Domain Volterra Nonlinear Feature for Bolt Damage Detection and Preload Identification in Bolted Joints

Vibration–based identification methods provide a feasible way to evaluate bolt loosening, but measured responses of bolted joints are often affected by environmental and measurement noise, making robust nonlinear feature extraction challenging. Although Volterra–series–based methods have been used to characterize nonlinear structural behavior, existing studies generally employ different–order kernels or diagonal kernel vectors directly as preload–sensitive indicators. Such treatments provide limited explanation of the physical mechanisms underlying Volterra–based nonlinear separation, especially from the perspective of frequency–domain coupling. To address this limitation, this study proposes a frequency–domain Volterra analysis framework for nonlinear feature extraction and bolt preload identification. The bolted joint is first modeled as a nonlinear dynamic system, and a Kautz–expanded Volterra model is established to identify the input–output relationship under dynamic excitation. The identified Volterra responses are then transformed into the frequency domain to reveal the harmonic composition of different nonlinear orders. Based on the generalized frequency response function (GFRF), a nonlinear frequency component associated with high–order harmonic generation is extracted to construct an interpretable Volterra nonlinear frequency–domain feature (VNFF). Furthermore, an unsupervised preload–sensitive indicator is developed based on the VNFF using the initial–state feature center, and supervised preload identification is performed using machine learning models. Experimental results show that the identified Volterra model can accurately reconstruct the measured responses. The proposed preload–sensitive indicator exhibits a consistent increasing trend with preload degradation across different datasets, demonstrating its monotonicity and sensitivity to bolt loosening. In addition, the proposed VNFF achieves the best preload identification performance, with optimal accuracies of 97.73%, 95.45%, and 93.18% on three datasets, respectively. These results demonstrate that the VNFF effectively highlights nonlinear contact information and supports reliable tightening–state assessment.