A data driven frequency-domain virtual sensing method based on cross-spectral density matrices
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Within the context of vibration-based condition monitoring, virtual sensing techniques facilitate vibration sensing at locations where sensors are not set at the time of inspection. % In this work we postulate a data-driven frequency-domain virtual sensing procedure, based on \textit{Cross Power Spectral Density} (CSD) matrices obtained from an initial dense sensor configuration. CSD matrices are used to build a conditional probability distribution for the Fourier transform of the withdrawn sensors, based on the response at the still available sensors, in a Gaussian process regression fashion. In this way, it is possible to estimate the Fourier transform of the vibration responses at the absent sensors, based on those from the available sensors. The proposed method is assessed in a wind turbine drivetrain diagnostics simulator, characterised by two speed reduction gearboxes (parallel shaft and planetary), and three shafts. The drivetrain is instrumented with accelerometers located on different bearings and on the gearboxes. The full sensor set is used to build the reference CSD based on an initial dataset at a fixed speed and load. Later, the acceleration response at some of the sensors is estimated with the proposed virtual sensing method using measured responses from a limited set of accelerometers.