The extended periodic motion concept for fast limit cycle detection of self-excited systems

Research output: Contribution to journal › Article › Research › peer review

Authors

  • Martin Jahn
  • Merten Stender
  • Sebastian Tatzko
  • Norbert Hoffmann
  • Aurélien Grolet
  • Jörg Wallaschek

Research Organisations

External Research Organisations

  • Hamburg University of Technology (TUHH)
  • Imperial College London
  • Arts et Métiers ParisTech (ENSAM)
View graph of relations

Details

Original languageEnglish
Article number106139
JournalComputers and Structures
Volume227
Early online date11 Nov 2019
Publication statusPublished - 15 Jan 2020

Abstract

Limit cycle solutions of self-excited dynamic systems can be determined by continuation of solutions along a system parameter variation or by brute-force testing. While the brute-force search for basins of attraction is computationally intractable, continuation methods compute only those branches that are connected to others, thus neglecting a-priori unknown solutions and detached branches, such as isolas. In this work, a method is proposed for finding limit cycles of self-excited dynamic systems. The method is based on the continuation of nonlinear modes for non-conservative systems, for which the Extended Periodic Motion Concept (E-PMC) is applied. The E-PMC allows for finding stable and unstable periodic solutions along the nonlinear mode and is especially helpful for determining solutions that are detached from other solution branches. Hence, the a-priori selection of proper initial conditions for the limit cycle computation is no longer necessary. A self-excited frictional oscillator with cubic stiffness terms is studied. The proposed technique proves to be robust and finds all isolated periodic solutions that were published previously by other authors. In an extended model configuration, the E-PMC finds co-existing stable limit cycles and unstable periodic orbits, one of which gives rise to hyper-chaotic motion with multiple positive Lyapunov exponents.

Keywords

    Bifurcations, Chaos, Nonlinear modes, Self-excited vibrations, Stability, Unstable periodic orbits

ASJC Scopus subject areas

Cite this

The extended periodic motion concept for fast limit cycle detection of self-excited systems. / Jahn, Martin; Stender, Merten; Tatzko, Sebastian et al.
In: Computers and Structures, Vol. 227, 106139, 15.01.2020.

Research output: Contribution to journal › Article › Research › peer review

Jahn, M., Stender, M., Tatzko, S., Hoffmann, N., Grolet, A., & Wallaschek, J. (2020). The extended periodic motion concept for fast limit cycle detection of self-excited systems. Computers and Structures, 227, Article 106139. https://doi.org/10.1016/j.compstruc.2019.106139
Jahn M, Stender M, Tatzko S, Hoffmann N, Grolet A, Wallaschek J. The extended periodic motion concept for fast limit cycle detection of self-excited systems. Computers and Structures. 2020 Jan 15;227:106139. Epub 2019 Nov 11. doi: 10.1016/j.compstruc.2019.106139
Jahn, Martin ; Stender, Merten ; Tatzko, Sebastian et al. / The extended periodic motion concept for fast limit cycle detection of self-excited systems. In: Computers and Structures. 2020 ; Vol. 227.
Download
@article{5d9ee20a58be417299d023162406a2d2,
title = "The extended periodic motion concept for fast limit cycle detection of self-excited systems",
abstract = "Limit cycle solutions of self-excited dynamic systems can be determined by continuation of solutions along a system parameter variation or by brute-force testing. While the brute-force search for basins of attraction is computationally intractable, continuation methods compute only those branches that are connected to others, thus neglecting a-priori unknown solutions and detached branches, such as isolas. In this work, a method is proposed for finding limit cycles of self-excited dynamic systems. The method is based on the continuation of nonlinear modes for non-conservative systems, for which the Extended Periodic Motion Concept (E-PMC) is applied. The E-PMC allows for finding stable and unstable periodic solutions along the nonlinear mode and is especially helpful for determining solutions that are detached from other solution branches. Hence, the a-priori selection of proper initial conditions for the limit cycle computation is no longer necessary. A self-excited frictional oscillator with cubic stiffness terms is studied. The proposed technique proves to be robust and finds all isolated periodic solutions that were published previously by other authors. In an extended model configuration, the E-PMC finds co-existing stable limit cycles and unstable periodic orbits, one of which gives rise to hyper-chaotic motion with multiple positive Lyapunov exponents.",
keywords = "Bifurcations, Chaos, Nonlinear modes, Self-excited vibrations, Stability, Unstable periodic orbits",
author = "Martin Jahn and Merten Stender and Sebastian Tatzko and Norbert Hoffmann and Aur{\'e}lien Grolet and J{\"o}rg Wallaschek",
note = "Funding information: The authors M.J. and M.S. thank the German Research Foundation (DFG) for the support within the priority program SPP 1897 ”Calm, Smooth and Smart” Appendix A",
year = "2020",
month = jan,
day = "15",
doi = "10.1016/j.compstruc.2019.106139",
language = "English",
volume = "227",
journal = "Computers and Structures",
issn = "0045-7949",
publisher = "Elsevier Ltd.",

}

Download

TY - JOUR

T1 - The extended periodic motion concept for fast limit cycle detection of self-excited systems

AU - Jahn, Martin

AU - Stender, Merten

AU - Tatzko, Sebastian

AU - Hoffmann, Norbert

AU - Grolet, Aurélien

AU - Wallaschek, Jörg

N1 - Funding information: The authors M.J. and M.S. thank the German Research Foundation (DFG) for the support within the priority program SPP 1897 ”Calm, Smooth and Smart” Appendix A

PY - 2020/1/15

Y1 - 2020/1/15

N2 - Limit cycle solutions of self-excited dynamic systems can be determined by continuation of solutions along a system parameter variation or by brute-force testing. While the brute-force search for basins of attraction is computationally intractable, continuation methods compute only those branches that are connected to others, thus neglecting a-priori unknown solutions and detached branches, such as isolas. In this work, a method is proposed for finding limit cycles of self-excited dynamic systems. The method is based on the continuation of nonlinear modes for non-conservative systems, for which the Extended Periodic Motion Concept (E-PMC) is applied. The E-PMC allows for finding stable and unstable periodic solutions along the nonlinear mode and is especially helpful for determining solutions that are detached from other solution branches. Hence, the a-priori selection of proper initial conditions for the limit cycle computation is no longer necessary. A self-excited frictional oscillator with cubic stiffness terms is studied. The proposed technique proves to be robust and finds all isolated periodic solutions that were published previously by other authors. In an extended model configuration, the E-PMC finds co-existing stable limit cycles and unstable periodic orbits, one of which gives rise to hyper-chaotic motion with multiple positive Lyapunov exponents.

AB - Limit cycle solutions of self-excited dynamic systems can be determined by continuation of solutions along a system parameter variation or by brute-force testing. While the brute-force search for basins of attraction is computationally intractable, continuation methods compute only those branches that are connected to others, thus neglecting a-priori unknown solutions and detached branches, such as isolas. In this work, a method is proposed for finding limit cycles of self-excited dynamic systems. The method is based on the continuation of nonlinear modes for non-conservative systems, for which the Extended Periodic Motion Concept (E-PMC) is applied. The E-PMC allows for finding stable and unstable periodic solutions along the nonlinear mode and is especially helpful for determining solutions that are detached from other solution branches. Hence, the a-priori selection of proper initial conditions for the limit cycle computation is no longer necessary. A self-excited frictional oscillator with cubic stiffness terms is studied. The proposed technique proves to be robust and finds all isolated periodic solutions that were published previously by other authors. In an extended model configuration, the E-PMC finds co-existing stable limit cycles and unstable periodic orbits, one of which gives rise to hyper-chaotic motion with multiple positive Lyapunov exponents.

KW - Bifurcations

KW - Chaos

KW - Nonlinear modes

KW - Self-excited vibrations

KW - Stability

KW - Unstable periodic orbits

UR - http://www.scopus.com/inward/record.url?scp=85075292914&partnerID=8YFLogxK

U2 - 10.1016/j.compstruc.2019.106139

DO - 10.1016/j.compstruc.2019.106139

M3 - Article

AN - SCOPUS:85075292914

VL - 227

JO - Computers and Structures

JF - Computers and Structures

SN - 0045-7949

M1 - 106139

ER -