In this research, a new approach for stability analysis of drill string is investigated. To this goal, the potential energy of drill string for axial and lateral vibrations is written in an integral equation. In this equation, the effect of geometrical shortening, which causes nonlinear coupling between axial and lateral vibration, is considered. The work done by WOB force and weight of drill string is calculated. The finite element method is employed to convert the integral potential energy of the continuous system to a discrete one. The effects of stabilizers are modeled by dropping the nodes coincided with them. Dropping and considering third and fourth order tensor of potential energy lead to linear and nonlinear stability analysis, respectively. Taking the first order derivative of discrete potential energy, the equilibrium position of drill string can be found. Taking second order derivative, the stability of the equilibrium position can be analyzed. Illustrated results demonstrate that as the length of hole is increased, the differences between linear and nonlinear cases become larger. This analysis shows the working condition of drill string is stable or not. These results can be used to obtain safe working conditions in drilling progress.

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