Irrational elliptic functions and the analytical solutions of SD oscillator

• Autorzy: Cao Q. , Wang D. , Chen Y. , Wiercigroch M.

Warianty tytułu

PL

Niewymierne funkcje eliptyczne i rozwiązania analityczne dla oscylatora typu SD

• Języki publikacji: EN

Abstrakty

EN

The smooth and discontinuous (SD) oscillator is a strongly nonlinear system with an irrational restoring force proposed in P.R.E (2006), which leads to barriers for the conventional methods to investigate the dynamical behaviour directly. In this paper, two kinds of irrational elliptic functions and a kind of hyperbolic functions are defined in the real domain to formulate the analytical solutions of the system. The properties of the functions are obtained including differentiability, periodicity and parity. As the application of the defined irrational functions, the chaotic thresholds of the oscillator are also depicted by using the Melnikov method. Numerical analysis shows the efficiency of the proposed procedure.

PL

Oscylator typu gładkiego i nieciągłego (smooth and discontinuous— SD) jest silnie nieliniowym układem mechanicznym z niewymierną siłą restytucyjną opisaną przez P.R.E. w 2006 r. Jej charakter stanowi barierę dla konwencjonalnych metod badania dynamiki oscylatorów SD w sposób bezpośredni. W pracy zdefiniowano dwa rodzaje niewymiernych funkcji eliptycznych i jeden typ hiperbolicznych w dziedzinie liczb rzeczywistych do wyznaczenia rozwiązań analitycznych rozważanego układu. Właściwości tych funkcji obejmują różniczkowalność, okresowość i parzystość. Tak sformułowanych funkcji, jako przykład ich zastosowania, uzyto do określenia zakresów występowania drgań chaotycznych oscylatora przy wykorzystaniu metody Mielnikowa. Symulacje numeryczne potwierdziły efektywność zaproponowanej metody.

Słowa kluczowe

EN

SD oscillator; irrational nonlinearity; irrational elliptic functions; threshold of chaos

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2012
• Tom: Vol. 50 nr 3
• Strony: 701--715
• Opis fizyczny: Bibliogr. 27 poz., rys., tab.

Twórcy

autor

Cao Q.

autor

Wang D.

autor

Chen Y.

autor

Wiercigroch M.

• Centre for Nonlinear Dynamics Research, School of Astronautics, Harbin Institute of Technology, Harbin, China, and Centre for Nonlinear Dynamics Research, Shijiazhuang Tiedao University, Shijiazhuang, China,

Bibliografia

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