A three-dimensional nonlinear finite element model for coupled vibration analysis of rotary steerable systems

Reliable operation of intelligent rotary steerable systems (RSSs) critically depends on the dynamic stability of the bottom-hole assembly (BHA), where high-frequency vibrations can lead to tool malfunction, accelerating structural fatigue, and degrade steering performance under harsh downhole conditions. This study establishes a three-dimensional nonlinear coupled vibration model for a static push-the-bit RSS-BHA based on finite-element dynamics, energy-based methods, and Hamilton's principle, enabling a unified analysis of axial, lateral, and torsional vibrations and their coupled responses under field-relevant high-frequency conditions. The results show that the proposed model can identify seven representative vibration modes commonly observed in drilling operations, including single-mode vibrations, pairwise coupled vibrations, and fully coupled vibration behaviors, and its predictive capability is validated through field-based case studies. Distinct dynamic characteristics clearly differentiate torsion-dominated stick-slip vibration from lateral-torsional coupled whirl vibration. Sensitivity analyses further demonstrate that stick-slip vibration is primarily governed by torsional excitation, whereas whirl vibration is dominated by lateral-torsional coupling and exhibits strong sensitivity to rotational speed and drill-collar geometry. Overall, this work provides a quantitative, physics-based framework for vibration-mode identification, interpretation of coupling mechanisms, and mode-specific vibration control and parameter optimization of RSS-BHA systems.