In this paper, a new model is proposed to study the coupled axial–torsional vibration of the drill string. It is assumed that rotary table angular speed is constant and equals to the nominal angular speed of the drill string. In addition, axial displacement of any point on the drill string is considered to be as the sum of rigid-body motion and elastic vibrations. The depth of cut is defined using instantaneous dynamic states instead of using the delayed model as presented in previous researches. A velocity-weakening function is introduced for modeling the behavior of the frictional component of the torque-on-bit (TOB) with respect to the bit angular speed. After discretizing vibration equations, stability analysis of the system is investigated by linearizing the nonlinear system around its steady-state response point. Considering nominal weight-on-bit (WOB) ($W0$) and nominal rotational speed ($Ω$) as the input parameters of the drilling, variation of maximum allowable value of ($W0$) is presented with respect to variation of $Ω$ . It is shown that the maximum allowable value of $W0$ has an increasing–decreasing behavior with respect to $Ω$. The effect of drill string upper and lower part lengths is studied on the stability of the system, and practical results are presented both in the condition that $W0$ is constant and in the condition that the hook upward force is constant. It is shown that by increasing the drill string length, the system is more exposed to instability, and this must be considered in regulating the input parameters of drilling.

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