A Hilbert Space-Based Approach for Vibration Signal Representation in Diagnostics and Classification of Rock Disintegration in Rotary Drilling

Rotary drilling represents a complex dynamic process in which vibration signals contain significant information regarding the interaction between rock material, drilling tools, and mechanical subsystems of the drilling apparatus. Despite considerable advances in vibration-based diagnostics, there remains a need for methodologies that integrate signal processing, geometric interpretation, and operational-state classification. This paper proposes an approach based on the representation of vibration signals in Hilbert space, where signals are interpreted as vectors in an infinite-dimensional functional space. The proposed methodology utilizes geometric metrics, such as distances, norms, and angular similarities, for the classification of rotary-drilling operational regimes and the analysis of dynamic system behavior. Experimental vibration data were obtained from a laboratory drilling stand during drilling of limestone, granite, and concrete, while auxiliary operational modes of the mechanical system were also analyzed. The results demonstrate that the Hilbert-space representation can distinguish the investigated operating states. The granite-drilling regime exhibited distinct amplitude, energetic, and impulsive vibration characteristics, whereas auxiliary mechanical regimes formed compact geometrical clusters with high mutual similarity. Among the evaluated symptoms, the power spectrum showed the greatest sensitivity to frequency-energy differences across the investigated operating regimes. The evaluated symptoms provided complementary information for distinguishing between the investigated operating regimes. The proposed methodology provides a mathematically consistent framework that may serve as a basis for future vibration-based condition monitoring and predictive maintenance systems.