Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

Time-delay two-dimension mathieu equation in synchrotron dynamics

Warianty tytułu
Języki publikacji
Two dimensions Mathieu equation containing periodic terms as well as the delayed parameters has been investigated in the present work. The present system represents to a generalized form of the one-dimension delay Mathieu equation. The mathematical difficulty for delay the coupled Mathieu equation has been overcome by using the matrices method. Properties of inverse complex matrices enable us to transform the vector form of the solvability conditions to the scalar form. Small oscillation about a marginal state is introduced by using the method of multiple scales. Stability criteria for the complex matrices have been established and lead to obtain resonance curves. The analysis has been extended so that the delay 2-dimensions Mathieu equation containing weak complex damping part. Stability conditions and the transition curves that included the influence of both the delayed as well the complex damping terms has been obtained. The transition curves are analyzed using the method of harmonic balance. We note that the delayed higher dimension of the parametric excitation has a great interest and application to the design of nuclear accelerators.
Opis fizyczny
Bibliogr. 22 poz.
  • Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Egypt
  • [1] Morse, P. M. and Feshbach, H.: Methods of Theoretical Physics, New York, McGraw-Hill, 1, 5.2, 1953.
  • [2] McLachlan, N. W.:Theory and Application of Mathieu Functions, New York: Dover, 1964.
  • [3] Nayfeh, A. H.: Perturbation Methods, Wiely, New York, 1973.
  • [4] Pederson, P.: Stability of the solutions to Mathieu-Hill equations with damped, Ing. Arch., 49, 15-29, 1980.
  • [5] Hasan, S. S.: Automatic Control Systems, Delhi: Katson Publishers, 206-207, 2008.
  • [6] Choudhury, A. G. and Guha, P.: Damped equations of Mathieu type, Appl. Math. & Comp., 229, 85-93 2014.
  • [7] Bartucelli, M. V. and Gentile, G.: On a class of integrable time-dependent dynamical systems,Phys.Lett. A, 307, 5-6, 274-280, 2003.
  • [8] Turovtsev, V. V., Orlov, Y. D. and Tsirulev, A. N.: Potential and Matrix Elements of the Hamiltonian of Internal Rotation in Molecules in the Basis Set of Mathieu Functions, Opt. & Spec., 119, 2, 199-203, 2015.
  • [9] Mohamed, A. A., Elshehawey, E. F., and El-Dib, Y. O.: Electrohydrodynamic Stability of a Fluid Layer. Effect of a Tangential Periodic Field, Nuovo Cimento Soc. Ital. Fiv. , 8, 177, 1986.
  • [10] Mohamed, A. A., Elshehawey, E. F., and El-Dib, Y. O. Nonlinear Electrohydrodynamic Stability of a fluid Layer: Effect of a Tangential Electric Field, J. Phys. Soc. Jpn. , 63, 1721, 1994.
  • [11] Landa, H.,Drewsne, M. Reznik, B. and Retzker, A.: Classical and quantum modes of coupled Mathieu equations, J. Phys. A:Math. Theor., 45, 455305, 2012.
  • [12] El-Dib, Y. O. and Ghaly, A. Y.: Destabilizing effect of time-dependent oblique magnetic field on a magnetic fluids streaming in porous media, Journal of Colloid and Interface Science, 269, 224-239, 2004.
  • [13] El-Dib, Y. O. and Matoog, R. T.: Stability of Streaming in an Electrified Maxwell Fluid Sheet Influenced by a Vertical Periodic Field in the Absence of Surface Charges, Journal of Colloid and Interface Science, 229, 29-52, 2000.
  • [14] El-Dib, Y. O. and Matoog, R. T.: Electrorheological Kelvin-Helmoholtz instability of a fluid sheet, Journal of Colloid and Interface Science, 289, 223-241, 2005.
  • [15] Y. O. El-Dib,: Instability for Shearing of an Electrified Kelvin Fluid Sheet with or without Supporting Surface Charges, Journal of Colloid and Interface Science, 250, 344-363, 2002.
  • [16] Alkharashi, S. A.: Electrohydrodynamics Instability of Three Periodic Streaming Fluids through Porous Media, Open Access Library Journal, 2, e13152015.
  • [17] Al Hamdan, A. R. and Alkharashi, S. A.: Stability Characterization of Three Porous Layers Model in the Presence of Transverse Magnetic Field, Journal of Mathematics Research, 8, 2, 69-81, 2016.
  • [18] Insperger, T. and Stepan, G.: Stability of the damped Mathieu equation with time delay, Journal of Dynamic Systems, Measurement, and Control, 125, 166171, 2003.
  • [19] Garg, N. K., Mann, B. P., Kim, N. H. and Kurdi, M. H.: Stability of a time-delayed system with parametric excitation, Journal of Dynamic Systems, Measurement, and Control, 129, 125-135, 2007.
  • [20] Ahsan, Z., Sadath, A. T., Uchida, K. and Vyasarayani, C. P.: GalerkinArnoldi algorithm for stability analysis of time-periodic delay differential equations, Nonlinear Dynamics, 82, 1893-1904, 2015.
  • [21] Bernstein, A. and Rand, R. H.: Coupled Parametrically Driven Modes in Synchrotron Dynamics, Nonlinear Dynamics, V. 1: Proceedings of the 33rd IMAC, A Conference and Exposition on Structural Dynamics, G. Kerschen, Springer, 8, 107-112, 2016.
  • [22] Bernstein, A. and Rand, R. H.: Coupled Parametrically Driven Modes in Synchrotron Dynamics, Presented at Society for Experimental Mechanics (SEM) IMAC XXXIII Conference on Structural Dynamics, 2015.
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Identyfikator YADDA
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.