In this paper, a mathematical representation is proposed to further understand the dynamic behavior and risk factors associated with vibration-assisted drilling (VAD) technology. The proposed Timoshenko beam model, which characterizes VAD technology, consists of two passive, counter-rotating coaxial rotors operating simultaneously, subjected to a stochastic excitation. In this regard, a finite element technique was incorporated to determine the physical parameters of the governing equation of motion, where the shear and rotary effects, as well as the gyroscopic couples generated perpendicular to the axis of rotation, were accounted for. Further, the relative velocity between the coaxial rotors was accounted in the equation of motion, which induced a modification on the stiffness and damping parameters. A fourth-order Runge-Kutta method was utilized to estimate the response and power spectra for soft and compact formations. Results indicated that resonance conditions due to gyroscopic effects manifest at soft and compact formations during drilling, however, at soft formations, the natural frequencies separated at lower RPMs, while at compact formations, higher RPMs were required to induce instabilities.

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