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Fractional Euler-Lagrange Equations Applied to Oscillatory Systems

Author

Listed:
  • Sergio Adriani David

    (University of São Paulo at Pirassununga, Av. Duque de Caxias Norte, 225-13635-900, Pirassununga-SP, Brazil)

  • Carlos Alberto Valentim

    (University of São Paulo at Pirassununga, Av. Duque de Caxias Norte, 225-13635-900, Pirassununga-SP, Brazil)

Abstract

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction.

Suggested Citation

  • Sergio Adriani David & Carlos Alberto Valentim, 2015. "Fractional Euler-Lagrange Equations Applied to Oscillatory Systems," Mathematics, MDPI, vol. 3(2), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:3:y:2015:i:2:p:258-272:d:48455
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    References listed on IDEAS

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    1. Edmundo Capelas de Oliveira & José António Tenreiro Machado, 2014. "A Review of Definitions for Fractional Derivatives and Integral," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-6, June.
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    Cited by:

    1. Tirumalasetty Chiranjeevi & Raj Kumar Biswas, 2017. "Discrete-Time Fractional Optimal Control," Mathematics, MDPI, vol. 5(2), pages 1-12, April.

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