Extending the functionality of a symbolic computational dynamic solver by using a novel term-tracking method

Symbolic computational dynamic solvers are currently under development in order to provide new and powerful tools for modelling nonlinear dynamical systems. Such solvers consist of two parts; the core solver, which comprises an approximate analytical method based on perturbation, averaging, or harmonic balance, and a specialised term-tracker. A term-tracking approach has been introduced to provide a powerful new feature into computational approximate analytical solutions by highlighting the many mathematical connections that exist, but which are invariably lost through processing, between the physical model of the system, the solution procedure itself, and the final result which is usually expressed in equation form. This is achieved by a highly robust process of term-tracking, recording, and identification of all the symbolic mathematical information within the problem. In this paper, the novel source and evolution encoding method is introduced for the first time and an implementation in Mathematica is described through the development of a specialised algorithm

Drillstring-borehole interaction: backward whirl instabilities and axial loading

A major concern within the oil drilling industry remains the interaction between the drillstring and borehole. The interaction between the drillstring and borehole wall involves nonlinearities in the form of friction and contact. The drillstring borehole interaction induces whirling behaviour of the drillstring causing forward whirl, backward whirl or intermittent bouncing behaviour depending on the system parameters. The purpose of this study is to analyse the steady backward whirl behaviour within the system which reduces the fatigue life of the drillstring. Initially a two discs model was developed to analyse the behaviour of the system. The theoretical model was tuned by altering the phase of the eccentric mass. This excites each lateral modes of the system in isolation. The effects of impact, friction and mass unbalance are included in the model. For the tuned system the backward whirl behaviour was analysed by carrying out a rotor speed sweep spanning the lateral natural frequencies. The influence of rotor speed on the system dynamics is explored using a run up and run down and is analysed using a waterfall plot. The waterfall plot indicated the frequency of maximum response corresponding to each rotor speed. Depending on the whirling behaviour the dominant frequency was observed at the lateral natural frequency, the rotational speed or the backward whirl frequency. The influence of variation in whirling behaviour due to altering of the axial load was analysed for a multiple disc case consisting of five discs. A transition in behaviour along the length of the drillstring was observed due to the axial load and bending moment interactions. Depending on the mode excited impact and sustained contact initiation with the borehole varied across the different stabilizer locations