Schauder fixed point theorem based existence of periodic solution for the response of Duffing’s oscillator
- 172 Downloads
An initial-boundary value problem that is Duffing’s oscillator with time varying coefficients will be studied. Using Banach’s fixed-point theorem, the existence of periodic solution of the equation will be predicted. The method applied in this paper is the Schauder second fixed point theorem, which includes the response of structures under vibratory force systems. As an example, the dynamics of nonlinear simply supported rectangular thin plate under influence of a relatively moving mass is studied. By expansion of the solution as a series of mode functions, the governing equations of motion are reduced to an ordinary differential equation for time development vibration amplitude, which is Duffing’s oscillator. Finally, a parametric study is developed, after that some numerical examples are solved, and the validity of the present analysis is clearly shown.
KeywordsBanach’s theorem Large deformation of thin plates Moving loads Non-linear vibration Schauder fixed point theorem
Unable to display preview. Download preview PDF.
- HABIB MÂAGLI, FATEN TOUMI, & MALEK ZRIBI, EXISTENCE OF POSITIVE SOLUTIONS FOR SOME POLYHARMONIC NONLINEAR BOUNDARY-VALUE PROBLEMS, Electronic Journal of Differential Equations, 2003(58) (2003) 58 1–19.Google Scholar
- P. J. Holmes, New Approaches to Nonlinear Problems in Dynamics, SIAM, Philadelphia, PA, 1981.Google Scholar
- C. AVRAMESCU, A fixed point theorem for multivalued mappings, Electronic Journal of Qualitative Theory of Differential Equations.Google Scholar
- J. CHU and P. J. TORRES, APPLICATIONS OF SCHAUDER’s FIXED POINT THEOREM TO SINGULAR DIFFERENTIAL EQUATIONS, Submitted exclusively to the London Mathematical Society, 2000 Mathematics Subject Classification, 34B15, 34D20.Google Scholar
- D. H. Griffle, Applied Functional Analysis, Wiley, New York, 1985.Google Scholar
- A. C. Ugural, Stresses in Plates and Shells, Wiley, New York, 1990.Google Scholar