In this work, I propose a Caputo–Fabrizio fractional mathematical model of a drilling system (CFFMMDS) to investigate the coupled dynamics of an induction motor-driven drilling assembly within a unified fractional-order framework. The governing equations were formulated using a nonsingular Caputo–Fabrizio fractional derivative with an exponential memory kernel to characterize the temporal dependencies among the system state variables. The resulting fractional-order system was solved using the Caputo–Fabrizio q-Elzaki homotopy analysis transform method (CFq-EHATM) to derive semi-analytical solution series and to conduct a numerical investigation of the system response under varying fractional orders. The results indicated that a decrease in the fractional order 𝜇 results in attenuated growth and decay rates. This trend is consistent with the intrinsic memory structure of the Caputo–Fabrizio operator, which distributes the influence of past states over time. The proposed framework enables the analysis of the temporal evolution of coupled state variables in dynamic systems exhibiting history-dependent behavior.