Dynamics of flexible shaft in rigid tube

The paper is concerned with equations and numerical methods for calculation of flexible shaft rotation in a rigid tube. In the very general statement the shaft is represented as a Cosserat rod with an arbitrary dependence of properties on the coordinate. The quasi-static motion is considered in the first. Six equations of motion are obtained for the arbitrary bent and curved rod in the tube of arbitrary geometry. The projection of the equation of moments on the tangent to the curved axis of the rod is shown to be sufficient for describing the shaft motion. This differential equation is expressed in terms of the rotation angle of the rod cross-section. The solution for the quasi-static rotation is obtained both analytically and using the shooting method for boundary-value problem for an ordinary differential equation. The closed form expression for the angles of rotation of the shaft in the rigid tube as a function of the axial coordinates is obtained. The jumps occur for some combination of the parameters and they cannot be explained in the framework of quasi-static analysis. In order to explain the instability, the dynamics statement is applied. The nonlinear dynamic problem is solved by means of differential-difference method which is tested by a comparison with a closed form solution. Solution to the dynamic problem allows one to explain the quasi-static jumps obtained. The dynamic formulation shows that instead of quasi-static jumps the initial stage of rotation is a smooth rotation which jumps are abruptly replaced by intensive vibrations. The laws of rotation at different rotational velocities are determined too. The qualitative difference in the quasi-static and dynamic solutions is exposed. The suggested approach for solving nonlinear dynamic problems of rotation of the shaft with an arbitrary geometry is promising for modeling of processes of the directional deep drilling which is vital for the problems of oil production industry.