Analysis of drilling vibrations: A time-delay system approach

The main purpose of this study is the description of the qualitative dynamical response of a rotary drilling system with a drag bit, using a model that takes into consideration the axial and the torsional vibration modes of the bit. The studied model, based on the interface bit-rock, contains a couple of wave equations with boundary conditions consisting of the angular speed and the axial speed at the top additionally to the angular and axial acceleration at the bit whose contain a realistic frictional torque. Our analysis is based on the center manifold Theorem and Normal forms theory whose allow us to simplify the model.

I Introduction

Interconnected oscillatory systems often display what is called propagation phenomena, [13]. In general by Lossless propagation it is understood the phenomenon associated with long transmission lines for physical signals. In engineering, this problem is strongly related to electric and electronic applications, e.g. circuit structures consisting of multipoles connected through LC transmission lines; this can also be seen in steam or gaz flows or pressures and water pipes [21], [9], [22]. The mathematical model is described in all these cases by a mixed initial and boundary value problem for hyperbolic partial differential equations modeling the lossless propagation. The boundary conditions are of special type, being in feedback connection with some system described by ordinary differential equations. This leads to the so called derivative boundary conditions considered by Cooke & Krumme [6], but also to the even more general boundary conditions of Abolina & Myshkis described by Volterra operators, see [22]. Integration along characteristics of the hyperbolic partial differential equations (here d'Alembert method) allows the association of certain system of functional equations to the mixed problem.