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Global Stability Boundaries and Hidden Oscillations in Dynamical Models with Dry Friction

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Mechanics and Control of Solids and Structures

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 164))

Abstract

In this chapter, the issues of global stability, bifurcations, and emergence of nontrivial limiting dynamic regimes in systems described by differential equations with discontinuous right-hand sides are considered within the framework of the theory of hidden oscillations. Such systems are important in the problems of mechanics, engineering, and control, and arise both a priori and as a result of idealization of some characteristics included in real physical systems. Determining the boundaries of global stability, scenarios of its violation, as well as identifying all arising limiting oscillations are the key challenges in the design of real systems based on mathematical modeling. While the self-excitation of oscillations can be effectively investigated numerically, the identification of hidden oscillations requires special analytical and numerical methods. The analysis of hidden oscillations is necessary to determine the exact boundaries of global stability, to estimate the gap between the necessary and sufficient conditions of global stability, and their convergence. This work presents a number of theoretical results and engineering problems in which hidden oscillations (their absence or presence and location) play an important role.

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Notes

  1. 1.

    We use the term “global stability” for simplicity of further presentation, while in the literature there are used different terms like “globally asymptotically stable” [35, p. 137], [36, p. 144], “gradient-like” [13, p. 56], [37, p. 2], “quasi-gradient-like” [13, p. 56], [37, p. 2], and others, reflecting the features of the stationary set and the convergence of trajectories to it.

  2. 2.

    In some cases, the use of HBM makes it possible to accurately identify all periodic orbits in a system. For example, for a system described by the equation \(\ddot{x} + x - b\dot{x}\cos (x) = 0\) (see, e.g., [64, 65]), this method predicts an infinite number of periodic orbits in the form \(x^\mathrm{hbm}(t) \!=\! a^i_0 \sin (t)\), where \(\{a^i_0\}_{i=1}^\infty \) are zeros of the Bessel function: \(J_1(a^i_0) \!=\! \tfrac{1}{\pi }\int _0^\pi \cos (\tau \!-\! a^i_0 \sin \tau ) d\tau \!=\! 0\).

  3. 3.

    Without limiting the generality of the foregoing, suppose that \(\det A \ne 0\).

  4. 4.

    Further research in this field led to the onset of several conjectures on global stability of various classes of nonlinear systems (see, e.g., [75]).

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Acknowledgements

The work is supported by the Leading Scientific Schools of Russia Project NSh-2624.2020.1, NSh-4196.2022.1.1, St.Petersburg State University grant Pure ID 75207094 (section 1), and Team Finland Knowledge Programme (163/83/2021)

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  1. Institute for Problems in Mechanical Engineering RAS, V.O., Bolshoy pr., 61, St. Petersburg, 199178, Russia

    Nikolay V. Kuznetsov

  2. St. Petersburg State University, University Embankment, 7/9, St Petersburg, 199034, Russia

    Nikolay V. Kuznetsov, Elizaveta D. Akimova, Elena V. Kudryashova, Olga A. Kuznetsova, Mikhail Y. Lobachev, Ruslan N. Mokaev & Timur N. Mokaev

  3. LUT University, Lappeenranta, Yliopistonkatu 34, 53850, Lappeenranta, Finland

    Nikolay V. Kuznetsov, Elizaveta D. Akimova & Mikhail Y. Lobachev

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Kuznetsov, N.V. et al. (2022). Global Stability Boundaries and Hidden Oscillations in Dynamical Models with Dry Friction. In: Polyanskiy, V.A., K. Belyaev, A. (eds) Mechanics and Control of Solids and Structures. Advanced Structured Materials, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-030-93076-9_20

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