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Self-sustained Oscillations and Limit Cycles in Rayleigh System with Cubic Return Force
- Authors: Kumakshev S.A.1
-
Affiliations:
- Ishlinsky Institute for Problems in Mechanics RAS
- Issue: Vol 87, No 5 (2023)
- Pages: 765-772
- Section: Articles
- URL: https://journals.rcsi.science/0032-8235/article/view/232508
- DOI: https://doi.org/10.31857/S0032823523050090
- EDN: https://elibrary.ru/QIZOZG
- ID: 232508
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Abstract
An oscillatory system with an excitation mechanism as in a Rayleigh oscillator, but with a nonlinear (cubic) returning force, is investigated. Using the accelerated convergence method and the continuation procedure for the parameter, limit cycles are constructed and the amplitudes and periods of self-oscillations are calculated. This is done for a wide range of feedback coefficient values, in which this coefficient is not asymptotically small or large. The proposed iterative procedure allows to achieve the specified accuracy of calculations. The analysis of the features of the limit cycle caused by an increase in the self-excitation coefficient is carried out. The results obtained are compared with the self-oscillations of a classical Rayleigh oscillator with a linear returning force.
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About the authors
S. A. Kumakshev
Ishlinsky Institute for Problems in Mechanics RAS
Author for correspondence.
Email: kumak@ipmnet.ru
Russia, Moscow
References
- Kharkevich A.A. Self-Induced Vibrations. Moscow: Gostekhizdat, 1953. 171 p. (in Russian)
- Andronov A.A., Vitt A.A., Khaikin S.E. Theory of Oscillations. Moscow: Fizmatgiz, 1959. 915 p. (in Russian)
- Lefschetz S. Differential Equations: Geometric Theory. N.Y.: Wiley, 1957. 390 p.
- Bogoliubov N.N., Mitropolsky Y.A. Asymptotic Methods in the Theory of Nonlinear Oscillations. Мoscow: Nauka, 1958. 462 p.
- Blaquiere A. Nonlinear System Analysis. N.Y.: Acad. Press, 1966. 408 pp.
- Zhuravlev V.F., Klimov D.M. Applied Methods in the Vibration Theory. Мoscow: Nayka, 1988. 326 p. (in Russian)
- Mishchenko E.F., Rozov N.Kh. Differential Equations with Small Parameters and Relaxation Oscillations. N. Y.: Plenum Press, 1980. 228 p.
- Volosov V.M., Morgunov B.I. Method of Averaging in the Theory of Nonlinear Oscillatory Systems. Moscow: Izd-vo MGU, 1971. 507 p. (in Russian)
- Akulenko L.D. Asymptotic Methods of Optimal Control. Moscow: Nauka, 1987. 365 p. (in Russian)
- Malkin I.G. Some Problems of the Theory of Nonlinear Oscillations. Moscow: Gostekhizdat, 1956. 491 p. (in Russian)
- Dorodnicyn A.A. Asymptotic solution of Van Der Pol’s equation // PMM, 1947, vol. 11, no. 3, pp. 313–328.
- Cartwright M.L. Van der Pol’s equation for relaxation oscillations // Contribut. to Theory Nonlin. Oscill. Ann. Math. Studies, 1952, no. 29, pp. 3–18.
- Krogdahl W.S. Numerical solutions of the Van der Pol equation // Z. Angew. Math. Phys., 1960, vol. 2, no. 1, pp. 59–63.
- Urabe M. Numerical study of periodic solutions of van der Pol’s equation. // in: Tr. Mezhdunarod. Simpoz. Nelinejnym Kolebaniyam, Mezhdunarod. Soyuz. Teor. Prikl. Mekh., Kiev, 1961, 1963, no. 2, pp. 367–375.
- Akulenko L.D., Korovina L.I., Nesterov S.V. Self-induced vibrations in an essentially nonlinear system // Mech. Solids, 2002, vol. 37, no. 3, pp. 36–41.
- Akulenko L.D., Kumakshev S.A., Nesterov S.V. Effective numerical-analytical solution of isoperimetric variational problems of mechanics by an accelerated convergence method // JAMM, 2002, vol. 66, no. 5, pp. 693–708.
- Akulenko L.D., Korovina L.I., Kumakshev S.A., Nesterov S.V. Self-sustained oscillations of Rayleigh and Van der Pol oscillators with moderately large feedback factors // JAMM, 2004, vol. 68, no. 2, pp. 241–248.
