A dynamic model of the drilling action of percussive tools
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A phenomenological model is proposed to explain the existence of sweet spot in the relationship between the rate of penetration (ROP) and the weight-on-bit (WOB), i.e., the existence of an optimal WOB for which the ROP is maximized. The reasons that have been previously proposed to explain the existence of a sweet spot include increasing wear of the bit, reduced indexing, and poor cleaning of debris. Instead, motivated by experimental evidence, we consider an increase of the pseudo-stiffness of the bit/rock interface (BRI) with the WOB, and investigate its consequence to the impact energy transmitted to rock. The model approximates the dynamics underlying the drilling process by assuming that the impact of the hammer generates a longitudinal wave in the bit. It is shown that the BRI pseudo-stiffness influences the stress and associated energy transmitted from the bit to the rock. As a consequence, the drilling efficiency is affected by the dependence of the BRI stiffness on the WOB. According to this model, there exist optimal conditions for the energy transfer from the bit to the rock in terms of the impedance ratio and the BRI stiffness. The model confirms that there is a sweet spot as seen in practice, thus suggesting that the root cause of the existence of a sweet spot in the ROP-WOB relationship lies in the nature of the BRI laws, rather than with issues related to indexing of the bit and/or cleaning of the debris.