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Abstrakty
In this paper, we presented the fractional-order dynamics of a double pendulum, at a small oscillation, with a non-singular derivative kernel. The equation of motion has been derived from the fractional Lagrangian of the system and the considered fractional Euler-Lagrange equation. The generalized force has also been presented in studying the different cases of force, such as horizontal and vertical forcing. The source term is described by the imposed periodic force, and the memory effect gives an additional damping factor described by the fractional order. The integer and fractional orders of the sample phase diagrams were obtained and presented to visualize the effect of the presented fractional order on the system. Also, since the motion of the system dissipates in the fractional regime, the applied force will drive the system out of equilibrium.
Rocznik
Tom
Strony
95--106
Opis fizyczny
Bibliogr. 19 poz., rys.
Twórcy
autor
- Department of Physics, Mindanao State University - Main Campus, 9700 Marawi City, Philippines
autor
- Department of Physics, Mindanao State University - Main Campus, 9700 Marawi City, Philippines
- Department of Physics, Mindanao State University - Main Campus, 9700 Marawi City, Philippines
Bibliografia
- [1] Qureshi, S., Rangaig, N.A., & Baleanu, D. (2019). New numerical aspects of Caputo-Fabrizio fractional derivative operator. Mathematics, 7(4), 374.
- [2] Frederico, G.S.F., & Torres, D.F.M. (2008). Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int. Math. Forum, 3, 9-12, 479–493. arXiv preprint arXiv:0712.1844.
- [3] Rangaig, N.A., & Convicto, V.C. (2019). On fractional modelling of dye removal using fractional derivative with non-singular kernel. Journal of King Saud University-Science, 31(4), 525-527.
- [4] Rangaig, N. (2018). Finite di_erence approximation for Caputo-Fabrizio time fractional derivative on non-uniform mesh and some applications. Phys. J, 1, 255-263.
- [5] Riewe, F. (1997). Mechanics with fractional derivatives. Physical Review E, 55(3), 3581.
- [6] Agrawal, O.P. (2010). Generalized variational problems and Euler-Lagrange equations. Computers & Mathematics with Applications, 59(5), 1852-1864.
- [7] Sene, N. (2020). Second-grade fluid model with Caputo-Liouville generalized fractional derivative. Chaos, Solitons & Fractals, 133, 109631.
- [8] Thiao, A., & Sene, N. (2019l). Fractional optimal economic control problem described by the generalized fractional order derivative. International Conference on Computational Mathematics and Engineering Sciences. Cham: Springer, 36-48.
- [9] Sene, N., & Atangana, A. (2019). Integral-balance methods for the fractional diffusion equation described by the Caputo-generalized fractional derivative. In: Methods of Mathematical Modelling: Fractional Differential Equations, 83.
- [10] Elmas, A., & Ibrahim, O. (2010). Classical and fractional-order analysis of the free and forced double pendulum. Engineering, 2, 935-943.
- [11] Baleanu, D., Jajarmi, A., & Asad, J.H. (2019). Classical and fractional aspects of two coupled pendulums. Romanian Reports in Physics, 71(1).
- [12] Lopes, A.M., & Tenreiro Machado, J.A. (2017). Dynamics of the N-link pendulum: a fractional perspective. International Journal of Control, 90(6), 1192-1200.
- [13] Caputo, M., & Fabrizio, M. (2016). Applications of new time and spatial fractional derivatives with exponential kernels. Progr. Fract. Di_er. Appl., 2(2), 1-11.
- [14] Rangaig, N.A., & Conding R.M. (2020). Numerics of fractional Langevin equation driven by fractional Brownian motion using non-singular fractional derivative. Prog. Fract. Di_er. Appl., In Press.
- [15] Rangaig, N.A., & Pido, A.A.G. (2019). Finite di_erence approximation method for twodimensional space-time fractional di_usion equation using nonsingular fractional derivative. Prog. Fract. Di_er. Appl., 5(4), 1-11.
- [16] Losada, J., & Nieto, J.J. (2015). Properties of a new fractional derivative without singular kernel. Progr. Fract. Di_er. Appl., 1(2), 87-92.
- [17] Al-Refai, M., & Pal, K. (2019). New aspects of Caputo–Fabrizio fractional derivative. Prog. Fract. Di_er. Appl., 5(2), 157-166.
- [18] Greiner, W. (2009). Classical Mechanics: Systems of Particles and Hamiltonian Dynamics. Springer Science & Business Media.
- [19] Atangana, A., & Owolabi, K.M. (2018). New numerical approach for fractional differential equations. Mathematical Modelling of Natural Phenomena, 13(1), 3.
Uwagi
Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).
Typ dokumentu
Bibliografia
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