By E. T. Whittaker
AC ourse of contemporary A nalysis: An advent to the overall conception of countless sequence and of Analytic services; With an Account of the vital Transcendental capabilities used to be written via E. T. Whittaker in 1902. it is a 393 web page publication, containing 101264 phrases and 10 images. seek within is enabled for this title.
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Additional resources for A Course of Modern Analysis. An Introduction to the General Theory of Infinite Series and of Analytic Functions; with an account of the Principal Transcendental Functions
To properly describe the string behavior at the upper end, it is essential to consider the actuators’ dynamics; induction and DC motor models are then recalled. Furthermore, top boundary conditions must be defined to characterize axial-torsional drilling trajectories at this level. • Drillstring sections. , the series of drillpipes of length Lp and the BHA of length Lb , so that L = Lp + Lb , where L is the total length of the rod, as illustrated in Fig. 1. • Bottom extremity. The system trajectories at the bit level are characterized by the bottom boundary conditions involving a nonlinear term to represent the bit-rock contact from which undesirable vibrations arise.
This experimentally obtained model is validated in ; a specially designed test rig with a braking device was constructed for this purpose. 4 Notes and References 43 Besides the friction force arising from the contact between the cutting surface and the bit, the regenerative phenomena have influence on the occurrence of vibrations. In drilling operations, the depth of cut is determined by both the current and previous position of the bit. The regenerative cutting effect, which refers to this dependence on the current and previous states, is analyzed in  through a PDE model with switching boundary conditions.
The modeling of the torque on the bit constitutes a crucial aspect of the system description since it allows to reproduce the vibrational phenomena; this subject will be discussed in Chap. 3. 3 Neutral-Type Time-Delay Models The wave equation model provides a realistic description of the distributed system variables; however, under certain circumstances, it is convenient to deal with a relatively simpler model involving just the primary interest variables. This section presents a direct procedure to derive, from the wave equations, equivalent input-output 2 The inertia moment is such that I B = M B r 2 , where r is taken as the averaged radius of drillpipe.
A Course of Modern Analysis. An Introduction to the General Theory of Infinite Series and of Analytic Functions; with an account of the Principal Transcendental Functions by E. T. Whittaker