Flexible rod model for the rotation of a drill string in an arbitrary borehole

The behavior of an oil drill string is considered on example of a rotating flexible shaft in a rigid tube. The tube (a model of the borehole) is assumed to be an arbitrary space curve, and the shaft is considered as a nonlinear elastic Cosserat rod. The nonlinear dynamic equations for the shaft are derived and solved by means of computer mathematics. The boundary value problem for the quasi-static rotation is reduced to the ordinary differential equation (ODE). The shooting method is applied for solving the obtained nonlinear ODE. The quasi-static rotation is shown to exhibit jumps for some sets of parameters. The dynamic problem is solved by the differential-difference method. The rotation behavior, the resultant forces, and moments in the rod as well as the contact reaction of the inner surface of the tube are determined. The differences between the static and dynamic solutions are demonstrated.