Display on the Phase Plane of the Slow Processes in Conservative Chains with One Degree of Freedom at Nonlinear Resonance

The weakly nonlinear electric chains without resistors with a single source of harmonic excitation are discussed. Differential equations of the chains are identical to the typical nonlinear equation of second order. The amplitude of the excitation and its frequency detuning are small. The equations for the amplitude and phase of resonant oscillation are obtained by the method of averaging. A simple and effective method (applicable only in the case of conservative chains) is proposed for the analysis of amplitude-phase plane. Using it amplitude-frequency characteristic is presented and modes of one- and bistability are selected. The phase portrait corresponds to each mode: in the mode of one-stability it has one particular point, in the mode of bistability-three points. The coordinates of special points, the nature of their stability and phase trajectory in their surroundings are determined. In the mode of bistability the full phase portrait is built qualitatively and it enables to present transient processes at various initial conditions.

References

[1] Bessonov L.A. Teoreticheskie osnovy elektrotekhniki. Elektricheskie tsepi [Theoretical basis of Electrical Engineering]. Moscow, Nauka Publ., 2007. 701 p.

[2] Danilov L.V., Matkhanov P.N. Filippov E.S. Teoriya nelineynykh elektricheskikh tsepey [The theory of nonlinear electrical circuits]. Moscow, Energoizdat Publ., 1990. 256 p.

[3] Rozo M. Nelineynye kolebaniya i teoriya ustoychivosti [Nonlinear vibration and stability theory]. Moscow, Nauka Publ., 1971. 228 p.

[4] Bogolyubov N.N., Mitropol’skiy Yu.A. Asimptoticheskie metody v teorii nelineynykh kolebaniy [Asymptotic methods in the theory of nonlinear vibration]. Moscow, Nauka Publ., 1974. 504 p.

[5] Moiseev N.N. Asimptoticheskie metody nelineynoy mekhaniki [Asymptotic methods of nonlinear mechanics]. Moscow, Nauka Publ., 1969. 381 p.

[6] Butenin N.V., Neymark Yu.I., Fufaev N.L. Vvedenie v teoriyu nelineynykh kolebaniy [Introduction to the theory of nonlinear vibration]. Moscow, Nauka Publ., 1987.382 p.

[7] Landau L.D., Lifshits E.M. Teoreticheskaya fizika. V 10 t. T. 1. Mekhanika [Theoretical physics. Ten-volume set. Vol. 1. Mechanics]. Moscow, Nauka Publ., 1988. 214 p. (Eng. Ed.: Landau L.D., Lifshitz E.M. Mechanics. Volume 1 (Course of Theoretical Physics). Second Ed. Oxford, New York, Pergamon Press, 1969.)

[8] Korn G.A., Korn T.M. Mathematical handbook for scientists and engineers: definitions, theorems, and formulas for reference and review. USA, McGraw-Hill, 1968. 1130 p. (Russ. ed.: Korn G., Korn T. Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov. Moscow, Nauka Publ., 1984. 831 p.)

[9] Andronov A.A., Vitt A.A., Khaykin S.E. Teoriya kolebaniy [Theory of oscillations]. Moscow, Nauka Publ., 1981. 918 p. (Eng. ed.: Andronov A.A., Vitt A.A. and Khaykin S.E. Theory of Oscillations. Translated from the Russian by F. Immirzi. Oxford-N.Y.-Toronto, Ont., Pergamon Press, 1966).