Observing the loss of tightening torque using modal parameters is challenging due to the variability and nonlinear effects in bolted joints. Thus, this paper proposes a combined application of two probabilistic machine learning methods. First, a Gaussian mixture model (GMM) is learned using estimated natural frequencies, assuming the tightening torque in a safe situation. This probabilistic model can assuredly detect the lack of torque using indirect vibration measures in other unknown states by computing a damage index. A Gaussian process regression (GPR) is also learned considering a set of torque and damage index pairs in several conditions. The GPR model interpolates a curve to supply an estimative of the tightening torque for other conditions not used in this learning. An illustrative application is performed considering the Orion beam, an academic-scale specimen composed of a lap-joint configuration that retains the friction surface in contact patches. The structure is subjected to a random vibration with a controlled RMS level and several tightening torque conditions to identify the modal parameters. The probabilistic model learning via the GMM and GPR can detect adequately, with a low number of false diagnoses, the actual state of torque using an indirect measure of vibration, that is, without the need for a torque sensor on each bolt.
Hysteresis is a nonlinear dissipative phenomenon present in many structures, such as those assembled by bolted joints. However, the approaches for identification are still limited in the literature due to their complexity. Additionally, there are also uncertainties held in bolted structures caused to fluctuations in tightening torque and pressure distribution along the contact surface. Thus, this paper yields a methodology for identifying a stochastic Bouc-Wen model for bolted joints based on the harmonic balance method. Unfortunately, some challenges are encountered when applying conventional series approximation to hysteresis, caused mainly by non-smooth behavior, which induces abrupt transitions between different motion regimes. In this work, previous adaptations were made to split the hysteresis loop in smooth paths and then use a piecewise harmonic balance approach. In this way, it was possible to deal with a deterministic Identification problem based on minimizing the error between the Fourier amplitudes of an experimental signal and those obtained through harmonic balance applying the Cross-Entropy optimization method. So, the results were extended to a stochastic model by applying the Bayesian paradigm, in which the maximized likelihood function was also based on the harmonic balance amplitudes. This methodology was demonstrated to identify Bouc-Wen parameters capable of predicting hysteresis in the BERT benchmark, composed of two aluminum beams jointed by a bolted joint in a cantilever boundary condition. Evaluating the results in the time and frequency domain and the nonlinear behavior through the hysteresis loop, it can be concluded that the method was able to identify an accurate stochastic Bouc-Wen model in predicting the dynamics of bolted structures even taking into account the probable uncertainties of the system.
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