Dynamic Stability of Transverse Oscillation of Rod with Vibrating Intermediate Support

A mathematical model of the tool behavior in deep drilling with the movable intermediate support is given. A nonlinear equation of the tool stem oscillation is derived which is solved for a system with nonstationary boundary conditions that correspond to the movable intermediate support. The numerical analysis of the dynamic stability is performed, and diagrams of the tool stability are plotted for various value ranges of parameters. It is shown that the parametrical resonance may occur depending on parameters of the intermediate support excitation.