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Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor

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Abstract

This work is devoted to the investigation of mathematical models of drilling systems described by ordinary differential equations. Here, we continue the study done by the researchers from Eindhoven where the two-mass mathematical model of a drilling system has been investigated (Mihajlovic et al. J. Dyn. Syst. Meas. Control 126(4): 709–720, 2004; de Bruin et al. Automatica 45(2): 405–415, 2009). The modified version of this model, which takes into account a full description of an induction motor, is studied. It is shown that such complex effects as hidden oscillations may appear in these kinds of systems. These effects may lead to drill string failures and breakdowns.

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Notes

  1. Usually both stator and rotor are made of laminated electrical steel.

  2. Without this assumption it is necessary to consider a stator, what leads to more complicated derivation of equations and more complicated equations themselves, which are difficult for analytical and numerical analyzing.

  3. An oscillation in a dynamical system can be easily localized numerically if the initial conditions from its open neighborhood lead to the long-time behavior that approaches the oscillation. Such an oscillation (or a set of oscillations) is called an attractor, and its attracting set is called a basin of attraction. Thus, from the point of view of the numerical analysis of nonlinear dynamical models in applied problems, it is essential to classify an attractor as self-excited or hidden attractor depending on simplicity of finding its basin of attraction [13, 35, 46–48, 50, 51]. For a self-excited attractor, its basin of attraction is connected with an unstable equilibrium: self-excited attractors can be localized numerically by standard computational procedure, in which after a transient process a trajectory, started from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation and, therefore, it can be easily identified. In contrast, for a hidden attractor, its basin of attraction does not intersect with small neighborhoods of equilibria. While many classical attractors are self-exited attractors and therefore can be obtained numerically by standard computational procedure, for localization of hidden attractors, it is necessary to develop special procedures since there are no similar transient processes leading to such attractors. See, e.g., localization of hidden oscillations in various modifications of electrical Chua circuits [33, 34, 36, 50–52], counterexamples to Aizerman conjecture and Kalman conjecture on absolute stability of nonlinear control systems [35, 44, 45], phase-locked loops [48, 49], and aircraft control systems [5, 6].

  4. Nowadays the length of drill string in drilling systems, used in oil and gas industry, is varied from 1 to 8 km, while the diameter of drill string is several tens of centimeters [57].

  5. The models studied are described by equations with discontinuous right-hand sides and therefore a special method for numerical computation of their solutions is required. Here, the method based on Filippov definition [63] is used for computer modeling.

  6. Parameters of a drilling system corresponding to the lower disk were taken from [15]. Parameters of the driving part of drilling system were chosen in such a way as to show the presence of hidden oscillations which may lead to damaging vibrations of the drill string. These parameters are in agreement with the parameters of the 4A series induction motors with wound rotor considered in [32].

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Acknowledgments

This study was supported by President Grants for Government Support of the Leading Scientific Schools, the Ministry of Education and Science, Saint Petersburg State University, Russian Foundation of Basic Research (Russia), and the Academy of Finland.

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  1. Saint-Petersburg State University, University of Jyväskylä, Jyväskylä, Finland

    G. A. Leonov, N. V. Kuznetsov, M. A. Kiseleva, E. P. Solovyeva & A. M. Zaretskiy

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  1. G. A. Leonov
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  2. N. V. Kuznetsov
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Leonov, G.A., Kuznetsov, N.V., Kiseleva, M.A. et al. Hidden oscillations in mathematical model of drilling system actuated by induction motor with a wound rotor. Nonlinear Dyn 77, 277–288 (2014). https://doi.org/10.1007/s11071-014-1292-6

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