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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


An Impact Oscillator with A Grazing Cycle

Discontinuity, Nonlinearity, and Complexity 6(2) (2017) 105--111 | DOI:10.5890/DNC.2017.06.001

M. U. Akhmet; A. Kıvılcım

Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

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An oscillator which impacts against a rigid barrier is taken into account. A cycle with zero impact velocity is discussed. The main result of this article concerns stability of the grazing cycle. A significant attention to a model with the variable coefficient of restitution depending on velocity is paid. The mechanical reasons for that are provided as well as new theoretical advantages have been discovered for the investigation of dynamics near a grazing cycle. The W-map which reduces the system with variable moments of impacts to that with fixed moments and simplifies the analysis, is defined. A new type of linearization system with two compounds is applied to investigate the stability of the grazing cycle whose existence is easily examined. A new approach to suppress a singularity, caused by the tangency, in linearization has been developed. Simulations are provided to visualize the stability of the grazing cycle.


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