An iterative method is proposed for finding periodic orbits of strongly nonlinear oscillators. The method combines the strength of analytical approaches, where the candidate solution is assumed in the form of a Fourier series, and the convenience of numerical methods that can be applied to larger systems with strong nonlinearity. The proposed method does not require integration of the vector field over any period of time and examples presented here illustrate that it is faster than traditional collocation algorithms, has a large radius of convergence, and is capable of finding several periodic orbits in each solution.
Issue Section:
Research Papers
Topics:
Algorithms
References
1.
R. J.
Kuether
, B.
Deaner
, M. S.
Allen
, and J. J.
Hollkamp
, “Evaluation of Geometrically Nonlinear Reduced Order Models with Nonlinear Normal Modes
,” AIAA.2.
Kerschen
, G.
, Peeters
, M.
, Golinval
, J. C.
, and Vakakis
, A. F.
, 2009
, “Nonlinear Normal Modes, Part I: A Useful Framework for the Structural Dynamicist
,” Mech. Syst. Signal Process.
, 23
(1
), pp. 170
–194
.3.
Nayfeh
, A. H.
, 2011
, Introduction to Perturbation Techniques
, Wiley-VCH
, New York
.4.
Blanchard
, P.
, Brüning
, E.
, and Hayes
, G. M.
, 1992
, Variational Methods in Mathematical Physics: A Unified Approach
, Springer
, New York/Berlin
.5.
Roberts
, S. M.
, and Shipman
, J. S.
, 1972
, Two-Point Boundary Value Problems: Shooting Methods
, American Elsevier Publishing Company
, New York
.6.
Allgower
, E. L.
, and Georg
, K.
, 1990
, Numerical Continuation Methods
, Vol. 33
, Springer-Verlag
, Berlin
.7.
Krauskopf
, B.
, Osinga
, H. M.
, and Galán-Vioque
, J.
, 2007
, Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value Problems
, Springer
, Dordrecht
.8.
Peeters
, M.
, Viguié
, R.
, Sérandour
, G.
, Kerschen
, G.
, and Golinval
, J. C.
, 2009
, “Nonlinear Normal Modes, Part II: Toward a Practical Computation Using Numerical Continuation Techniques
,” Mech. Syst. Signal Process.
, 23
(1
), pp. 195
–216
.9.
Champneys
, A. R.
, Fairgrieve
, T. F.
, Kuznetsov
, Y. A.
, Sandstede
, B.
, and Wang
, X. J.
, 1997
, “auto97: Continuation and Bifurcation Software for Ordinary Differential Equations
,” http://indy.cs.concordia.ca/auto/10.
Dhooge
, A.
, Govaerts
, W.
, and Kuznetsov
, Y. A.
, 2003
, “matcont: A MATLAB Package for Numerical Bifurcation Analysis of ODEs
,” ACM Trans. Math. Software
, 29
(2
), pp. 141
–164
.11.
Barbu
, V.
, 2010
, Nonlinear Differential Equations of Monotone Types in Banach Spaces
, Springer Science & Business Media
, New York
.12.
Bauschke
, H. H.
, and Combettes
, P. L.
, 2011
, Convex Analysis and Monotone Operator Theory in Hilbert Spaces
, Springer Science & Business Media
, New York
.13.
Simons
, S.
, 1998
, Minimax and Monotonicity
(Lectures Notes in Mathematics), Vol. 1963
, Springer
, New York/Berlin
.14.
Fletcher
, R.
, and Powell
, M. J. D.
, 1963
, “A Rapidly Convergent Descent Method for Minimization
,” Comput. J.
, 6
(2
), pp. 163
–168
.15.
Fletcher
, R.
, and Reeves
, C. M.
, 1964
, “Function Minimization by Conjugate Gradients
,” Comput. J.
, 7
(2
), pp. 149
–154
.16.
Fletcher
, R.
, 2013
, Practical Methods of Optimization
, Wiley
, Hoboken, NJ
.17.
Powell
, M. J. D.
, 1984
, “Nonconvex Minimization Calculations and the Conjugate Gradient Method
,” Numerical Analysis
, Springer
, Berlin
, pp. 122
–141
.18.
Rosenberg
, R. M.
, and Atkinson
, C. P.
, 1959
, “On the Natural Modes and Their Stability in Nonlinear Two-Degree-of-Freedom Systems
,” ASME J. Appl. Mech.
, 26
(3
), pp. 377
–385
.19.
Shaw
, S. W.
, and Pierre
, C.
, 1991
, “Non-Linear Normal Modes and Invariant Manifolds
,” J. Sound Vib.
, 150
(1
), pp. 170
–173
.20.
Ardeh
, H. A.
, and Allen
, M. S.
, 2012
, “Instantaneous Center Manifolds and Nonlinear Modes of Vibrations
,” ASME 2012 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference
.21.
Vakakis
, A. F.
, 1990
, “Analysis and Identification of Linear and Nonlinear Normal Modes in Vibrating Systems
,” Ph.D. thesis, California Institute of Technology, Pasadena, CA.22.
Ardeh
, H. A.
, and Allen
, M. S.
, 2013
, “Investigating Cases of Jump Phenomenon in A Nonlinear Oscillatory System
,” 31st IMAC, A Conference on Structural Dynamics
, Anaheim, CA, pp. 299
–318
.23.
Keller
, H. B.
, 1977
, “Numerical Solution of Bifurcation and Nonlinear Eigenvalue Problems
,” Applications of Bifurcation Theory
, Academic Press
, New York/London
, pp. 359
–384
.24.
Butler
, G.
, Freedman
, H. I.
, and Waltman
, P.
, 1986
, “Uniformly Persistent Systems
,” American Mathematical Society, pp. 425
–430
.25.
Hale
, J. K.
, and Waltman
, P.
, 1989
, “Persistence in Infinite-Dimensional Systems
,” SIAM J. Math. Anal.
, 20
(2
), pp. 388
–395
.26.
Shi
, J.
, 1999
, “Persistence and Bifurcation of Degenerate Solutions
,” J. Funct. Anal.
, 169
(2
), pp. 494
–531
.27.
Doedel
, E. J.
, 2010
, “Lecture Notes on Numerical Analysis of Nonlinear Equations
,” http://indy.cs.concordia.ca/auto/notes.pdf28.
Kuether
, R. J.
, and Allen
, M. S.
, 2013
, “Structural Modification of Nonlinear FEA Subcomponents Using Nonlinear Normal Modes
,” 31st International Modal Analysis Conference (IMAC XXXI) Garden Grove, CA.Copyright © 2016 by ASME
You do not currently have access to this content.
