The objective of the proposed analytical research is to develop a technical basis for predicting the limit of dynamic stability of drill bits used in various earth boring operations. The investigation will incorporate the combined effects of axial, torsional and lateral modes in the analysis of bit motion and in the determination of the limit of dynamic stability. A study of the effects of mode coupling will show which modes of vibration are critical and lead to practical methods of changing the structural stability of the drilling system. The knowledge gained from this research will provide a technical guidance for reducing drillstring failures and increasing penetration rate.