Abstract
The development of the theory of discontinuous dynamical systems and differential inclusions was not only due to research in the field of abstract mathematics but also a result of studies of particular problems in mechanics. One of the first methods, used for the analysis of dynamics in discontinuous mechanical systems, was the harmonic balance method developed in the thirties of the twentieth century. In our work, the results of analysis obtained by the method of harmonic balance, which is an approximate method, are compared with the results obtained by rigorous mathematical methods and numerical simulation.
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Notes
- 1.
The history of the dry friction law can be found, e.g., in [14].
References
Hartog, J.D.: Lond. Edinb. Dubl. Philos. Mag. J. Sci. 9(59), 801 (1930)
Andronov, N., Bautin, A.A.: Dokl. Akad. Nauk SSSR (in Russian) 43(5), 197 (1944)
Keldysh, M.: TsAGI Tr. (in Russian) 557, 26 (1944)
Krylov, N., Bogolyubov, N.: Introduction to Non-linear Mechanics (in Russian). AN USSR, Kiev (1937) (English transl: Princeton Univ. Press, 1947)
Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer, Dordrecht (1988)
Gelig, A., Leonov, G., Yakubovich, V.: Stability of Nonlinear Systems with Nonunique Equilibrium (in Russian). Nauka, Moscow (1978) (English transl: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, 2004, World Scientific)
Orlov, Y.: Discontinuous Systems: Lyapunov Analysis and Robust Synthesis Under Uncertainty Conditions. Communications and Control Engineering. Springer, New York (2008)
Boiko, I.: Discontinuous Control Systems: Frequency-Domain Analysis and Design. Springer, London (2008)
Adly, S.: A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics. SpringerBriefs in Mathematics. Springer International Publishing, Cham (2018)
Leonov, G., Ponomarenko, D., Smirnova, V.: Frequency-Domain Methods for Nonlinear Analysis. Theory and Applications. World Scientific, Singapore (1996)
Aizerman, M., Pyatnitskiy, E.: Autom. Remote Control (in Russian) 7 8, 33 (1974)
Dontchev, A., Lempio, F.: SIAM Rev. 34(2), 263 (1992)
Piiroinen, P.T., Kuznetsov, Y.A.: ACM Trans. Math. Softw. 34(3), 13 (2008)
Zhuravlev, V.: Herald of the Bauman Moscow State Technical University. Series Natural Sciences (2(53)), 21 (2014)
Leonov, G.A., Kuznetsov, N.V.: On flutter suppression in the Keldysh model. Dokl. Phys. 63(9), 366–370 (2018). https://doi.org/10.1134/S1028335818090021
Leonov, G., Kuznetsov, N.: Int. J. Bifurcation Chaos 23(1) (2013) art. no. 1330002. https://doi.org/10.1142/S0218127413300024
Leonov, G., Kuznetsov, N., Mokaev, T.: Eur. Phys. J. Spec. Top. 224(8), 1421 (2015). https://doi.org/10.1140/epjst/e2015-02470-3
Kuznetsov, N.: Lect. Notes Electr. Eng. 371, 13 (2016). https://doi.org/10.1007/978-3-319-27247-4_2. Plenary lecture at International Conference on Advanced Engineering Theory and Applications 2015
Kiseleva, M., Kudryashova, E., Kuznetsov, N., Kuznetsova, O., Leonov, G., Yuldashev, M., Yuldashev, R.: Int. J. Parall. Emerg. Distrib. Syst. (2018). https://doi.org/10.1080/17445760.2017.1334776
Stankevich, N., Kuznetsov, N., Leonov, G., Chua, L.: Int. J. Bifurcation Chaos 27(12) (2017). Art. num. 1730038
Chen, G., Kuznetsov, N., Leonov, G., Mokaev, T.: Int. J. Bifurcation Chaos 27(8) (2017). Art. num. 1750115
Kuznetsov, N., Leonov, G., Mokaev, T., Prasad, A., Shrimali, M.: Nonlinear Dyn. (2018). https://doi.org/10.1007/s11071-018-4054-z
Danca, M.F., Fečkan, M., Kuznetsov, N., Chen, G.: Nonlinear Dyn. 91(4), 2523 (2018)
Pliss, V.A.: Some Problems in the Theory of the Stability of Motion (in Russian). Izd LGU, Leningrad (1958)
Fitts, R.E.: Trans. IEEE AC-11(3), 553 (1966)
Barabanov, N.E.: Sib. Math. J. 29(3), 333 (1988)
Bernat, J., Llibre, J.: Dyn. Contin. Discrete Impuls. Syst. 2(3), 337 (1996)
Leonov, G., Bragin, V., Kuznetsov, N.: Dokl. Math. 82(1), 540 (2010). https://doi.org/10.1134/S1064562410040101
Bragin, V., Vagaitsev, V., Kuznetsov, N., Leonov, G.: J. Comput. Syst. Sci. Int. 50(4), 511 (2011). https://doi.org/10.1134/S106423071104006X
Leonov, G., Kuznetsov, N.: Dokl. Math. 84(1), 475 (2011). https://doi.org/10.1134/S1064562411040120
Alli-Oke, R., Carrasco, J., Heath, W., Lanzon, A.: IFAC Proceedings Volumes (IFAC-PapersOnline), vol. 7, p. 27 (2012). https://doi.org/10.3182/20120620-3-DK-2025.00161
Heath, W.P., Carrasco, J., de la Sen, M.: Automatica 60, 140 (2015)
Leonov, G., Kuznetsov, N., Kiseleva, M., Mokaev, R.: Differ. Equ. 53(13), 1671 (2017)
Leonov, G., Mokaev, R.: Dokl. Math. 96(1), 1 (2017). https://doi.org/10.1134/S1064562417040111
Acknowledgement
This work was supported by the grant NSh-2858.2018.1 of the President of Russian Federation for the Leading Scientific Schools of Russia (2018–2019).
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Kudryashova, E.V., Kuznetsov, N.V., Kuznetsova, O.A., Leonov, G.A., Mokaev, R.N. (2019). Harmonic Balance Method and Stability of Discontinuous Systems. In: Matveenko, V., Krommer, M., Belyaev, A., Irschik, H. (eds) Dynamics and Control of Advanced Structures and Machines. Springer, Cham. https://doi.org/10.1007/978-3-319-90884-7_11
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