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A purely frequency based Floquet-Hill formulation for the efficient stability computation of periodic solutions of ordinary differential systems
- Source :
- Journal of Computational Physics, Journal of Computational Physics, Elsevier, 2020, 416, pp.109477. ⟨10.1016/j.jcp.2020.109477⟩, Journal of Computational Physics, 2020, 416, pp.109477. ⟨10.1016/j.jcp.2020.109477⟩
- Publication Year :
- 2020
- Publisher :
- Elsevier BV, 2020.
-
Abstract
- International audience; Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined , either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.
- Subjects :
- Floquet theory
Physics and Astronomy (miscellaneous)
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS]
010103 numerical & computational mathematics
01 natural sciences
Mécanique: Vibrations [Sciences de l'ingénieur]
symbols.namesake
Harmonic balance
Bifurcations
Quadratic equation
Floquet multipliers
Taylor series
Applied mathematics
Time domain
0101 mathematics
Mécanique: Mécanique des structures [Sciences de l'ingénieur]
Fourier series
Stability analysis of periodic solutions
Mathematics
Numerical Analysis
Numerical analysis
Applied Mathematics
Hill's method
[SPI.MECA]Engineering Sciences [physics]/Mechanics [physics.med-ph]
Computer Science Applications
Asymptotic Numerical Method
010101 applied mathematics
Computational Mathematics
Frequency domain
Modeling and Simulation
Quadratic nonlinearities recast
symbols
Backbone curve
Harmonic Balance Method
Subjects
Details
- Language :
- English
- ISSN :
- 00219991 and 10902716
- Database :
- OpenAIRE
- Journal :
- Journal of Computational Physics, Journal of Computational Physics, Elsevier, 2020, 416, pp.109477. ⟨10.1016/j.jcp.2020.109477⟩, Journal of Computational Physics, 2020, 416, pp.109477. ⟨10.1016/j.jcp.2020.109477⟩
- Accession number :
- edsair.doi.dedup.....4856d8d093cee4332723f2e968125a5c