Derivative-based statistical analysis of energy dissipation and dynamic stability in ultrasonic-assisted deep drilling

Ultrasonic-assisted deep drilling (UAD) behaves as a nonlinear, energy-dissipative system driven by high-frequency excitation, where torque fluctuations directly reflect transient mechanical energy within the cutting zone. This study introduces a derivative-based statistical framework that utilizes the first-order torque gradient with respect to drilling depth ([Formula: see text]) to quantify the instantaneous rate, polarity and fluctuation intensity of energy dissipation. High-resolution torque–depth data were collected for both conventional drilling (CD) and UAD of AISI 304 stainless steel under identical cutting conditions. Eight quantitative metrics — including mean derivative, standard deviation, extreme derivative amplitudes ([Formula: see text], [Formula: see text]), spike frequency (F_spike), sign-change frequency (F_sign), cumulative dissipation activity [Formula: see text] and composite dynamic-stability index [Formula: see text] — were extracted to characterize system dynamics. Experimental results indicate that UAD significantly mitigates high-magnitude derivative spikes, with F_spike reduced by approximately 65%, and decreases cumulative dissipation activity (A) by 78.2% compared to CD. Simultaneously, the dynamic-stability index [Formula: see text] increased by 127%, reflecting a more coherent and energy-efficient cutting regime. Furthermore, UAD decreased the mean derivative [Formula: see text] by 64% and the standard deviation StD[Formula: see text] by 47.8%, while extreme transient energy bursts [Formula: see text] and [Formula: see text] increased by 173% and 176%, respectively, illustrating controlled and reversible energy transfer induced by ultrasonic excitation. These findings demonstrate that UAD stabilizes energy flow, reduces stochastic, entropy-generating irregularities at the tool–material interface and promotes a self-regulated oscillatory drilling state. The proposed derivative-based framework provides a novel and physically interpretable methodology for quantifying dynamic stability and energy dissipation in complex nonlinear machining systems.