Dynamical Behavior of Fractional-Order Delayed Feedback Control on the Mathieu Equation by Incremental Harmonic Balance Method
Open Access
- 19 July 2022
- journal article
- research article
- Published by Hindawi Limited in Shock and Vibration
- Vol. 2022, 1-13
- https://doi.org/10.1155/2022/7515080
Abstract
In this study, the dynamical analysis of the Mathieu equation with multifrequency excitation under fractional-order delayed feedback control is investigated by the incremental harmonic balance method (IHBM). IHBM is applied to the fractional-order delayed feedback control system, and the general formulas of the first-order approximate periodic solution for the Mathieu equation are derived. Caputos definition is adopted to process the fractional-order delayed feedback term. The general formulas of this system are suitable for not only the weakly but also the strongly nonlinear fractional-order system. Through the analysis of the general formulas of this system, it shows that fractional-order delayed feedback control has two functions, which are velocity delayed feedback control and displacement delayed feedback control. Next, the numerical simulation of the system is carried out. The comparison between the approximate analytical solution and the numerical iterative result is made, and the accuracy of the approximate analytical result by IHBM is proved to be high. At last, the effects of the time delay, feedback coefficient, and fractional order are investigated, respectively. It is generally known that time delay is common and inevitable in the control system. But the fractional order can be used to adjust the influence caused by time delay in fractional-order delayed feedback control. Those new system characteristics will provide theoretical guidance to the design and the control of this kind system.Keywords
Funding Information
- National Natural Science Foundation of China (12072206, U1934201, 11802183, ZD2020310)
This publication has 34 references indexed in Scilit:
- Strongly Nonlinear Subharmonic Resonance and Chaotic Motion of Axially Moving Thin Plate in Magnetic FieldJournal of Computational and Nonlinear Dynamics, 2015
- An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomialsMathematical Methods in the Applied Sciences, 2014
- Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative dampingFractional Calculus and Applied Analysis, 2013
- On the stochastic response of a fractionally-damped Duffing oscillatorCommunications in Nonlinear Science and Numerical Simulation, 2012
- Numerical approaches to fractional calculus and fractional ordinary differential equationJournal of Computational Physics, 2011
- A Lyapunov approach to the stability of fractional differential equationsSignal Processing, 2011
- Stability of a linear oscillator with damping force of the fractional-order derivativeScience China Physics Mechanics and Astronomy, 2010
- Nonlinear vibrations of dynamical systems with a general form of piecewise-linear viscous damping by incremental harmonic balance methodPhysics Letters A, 2002
- Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance methodJournal of Sound and Vibration, 1995
- Nonlinear Vibrations of Piecewise-Linear Systems by Incremental Harmonic Balance MethodJournal of Applied Mechanics, 1992