Mathematical Modeling of Drill Strings Nonlinear Vibrations taking into account a Friction Forces

In this work the nonlinear transverse vibrations of a drill string taking into account a frictional forces was studied. This forces are arising between the string and the borehole, between the drilling tool and the borehole at longitudinal feed during drilling. Solution of the model is obtained by the Bubnov-Galerkin method in the third approximation. Comparative analysis of the models with and without consideration the friction force is carried out, and the significance of application of the governing equations taking into account damping forces is shown. Drill strings with various parameters of length, material, external compressive force, and coefficient of friction are investigated. The resonant phenomenon is also investigated. The amplitude-frequency characteristics of the first and the third harmonic resonance are constructed. As a result of the research, the energy of the basic resonance is transferred to the third harmonic resonance. This fact confirms the appearance of resonance at higher frequencies in nonlinear systems with a stiff characteristic.