In this paper, we demonstrate the use of control-based continuation within a physical experiment: a nonlinear energy harvester, which is used to convert vibrational energy into usable electrical energy. By employing the methodology of Sieber et al. (2008, “Experimental Continuation of Periodic Orbits Through a Fold,” Phys. Rev. Lett., 100(24), p. 244101), a branch of periodic orbits is continued through a saddle-node bifurcation and along the associated branch of unstable periodic orbits using a modified time-delay controller. At each step in the continuation, the pseudo-arclength equation is appended to a set of equations that ensure that the controller is noninvasive. The resulting nonlinear system is solved using a quasi-Newton iteration, where each evaluation of the nonlinear system requires changing the excitation parameters of the experiment and measuring the response. We present the continuation results for the energy harvester in a number of different configurations.
Issue Section:
Research Papers
1.
Seydel
, R.
, 1994, Practical Bifurcation and Stability Analysis
, Springer
, New York
.2.
Doedel
, E.
, Keller
, H.
, and Kernevez
, J.
, 1991, “Numerical Analysis and Control of Bifurcation Problems, Part I
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 1
(3
), pp. 493
–520
.3.
Doedel
, E.
, Keller
, H.
, and Kernevez
, J.
, 1991, “Numerical Analysis and Control of Bifurcation Problems, Part II
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 1
(4
), pp. 745
–772
.4.
Kuznetsov
, Y.
, 1998, “Elements of Applied Bifurcation Theory
,” Applied Mathematical Sciences
, Vol. 112
, 2nd ed., Springer
, New York
.5.
Barton
, D.
, Krauskopf
, B.
, and Wilson
, R.
, 2007, “Homoclinic Bifurcations in a Neutral Delay Model of a Transmission Line Oscillator
,” Nonlinearity
0951-7715, 20
(4
), pp. 809
–829
.6.
De Feo
, O.
, Maggio
, G. M.
, and Kennedy
, M. P.
, 2000, “The Colpitts Oscillator: Families of Periodic Solutions and Their Bifurcations
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 10
(5
), pp. 935
–958
.7.
Coombes
, S.
, Lord
, G. J.
, and Owen
, M. R.
, 2003, “Waves and Bumps in Neuronal Networks With Axo-Dendritic Synaptic Interactions
,” Physica D
0167-2789, 178
(3–4
), pp. 219
–241
.8.
Luzyanina
, T.
, Roose
, D.
, and Bocharov
, G.
, 2005, “Numerical Bifurcation Analysis of Immunological Models With Time Delays
,” J. Comput. Appl. Math.
0377-0427, 184
(1
), pp. 165
–176
.9.
Lloyd
, D.
, and Champneys
, A.
, 2005, “Efficient Numerical Continuation and Stability Analysis of Spatiotemporal Quadratic Optical Solitons
,” SIAM J. Sci. Comput. (USA)
1064-8275, 27
(3
), pp. 759
–773
.10.
Lloyd
, D.
, Sandstede
, B.
, Avitabile
, D.
, and Champneys
, A.
, 2008, “Localized Hexagon Patterns of the Planar Swift–Hohenberg Equation
,” SIAM J. Appl. Dyn. Syst.
1536-0040, 7
(3
), pp. 1049
–1100
.11.
Doedel
, E.
, Champneys
, A.
, Fairgrieve
, T.
, Kuznetsov
, Y.
, Sandstede
, B.
, and Wang
, X.
, 1998, AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations.12.
Dhooge
, A.
, Govaerts
, W.
, Kuznetsov
, Y.
, Mestrom
, W.
, Riet
, A.
, and Sautois
, B.
, 2006, MATCONT and CL_MATCONT: Continuation Toolboxes in MATLAB.13.
Dankowicz
, H.
, and Schilder
, F.
, 2009, “An Extended Continuation Problem for Bifurcation Analysis in the Presence of Constraints
,” ASME Paper No. DETC2009/MSNDC-86343.14.
Engelborghs
, K.
, Luzyanina
, T.
, and Roose
, D.
, 2002, “Numerical Bifurcation Analysis of Delay Differential Equations Using DDE-BIFTOOL
,” ACM Trans. Math. Softw.
0098-3500, 28
(1
), pp. 1
–21
.15.
Barton
, D.
, Krauskopf
, B.
, and Wilson
, R.
, 2006, “Collocation Schemes for Periodic Solutions of Neutral Delay Differential Equations
,” Journal of Difference Equations and Applications
, 12
(11
), pp. 1087
–1101
.16.
Szalai
, R.
, 2005, PDDE-CONT: A Continuation and Bifurcation Software for Delay-Differential Equations.17.
Heroux
, M.
, Bartlett
, R.
, Howle
, V.
, Hoekstra
, R.
, Hu
, J.
, Kolda
, T.
, Lehoucq
, R.
, Long
, K.
, Pawlowski
, R.
, Phipps
, E.
, Salinger
, A.
, Thornquist
, H.
, Tuminaro
, R.
, Willenbring
, J.
, Williams
, A.
, and Stanley
, K.
, 2005, “An Overview of the Trilinos Project
,” ACM Trans. Math. Softw.
0098-3500, 31
(3
), pp. 397
–423
.18.
Kerschen
, G.
, Worden
, K.
, Vakakis
, A.
, and Golinval
, J. -C.
, 2006, “Past, Present and Future of Nonlinear System Identification in Structural Dynamics
,” Mech. Syst. Signal Process.
0888-3270, 20
(3
), pp. 505
–592
.19.
Pyragas
, K.
, 1992, “Continuous Control of Chaos by Self-Controlling Feedback
,” Phys. Lett. A
0375-9601, 170
(6
), pp. 421
–428
.20.
Pyragas
, K.
, 2001, “Control of Chaos via an Unstable Delayed Feedback Controller
,” Phys. Rev. Lett.
0031-9007, 86
(11
), pp. 2265
–2268
.21.
Ott
, E.
, Grebogi
, C.
, and Yorke
, J.
, 1990, “Controlling Chaos
,” Phys. Rev. Lett.
0031-9007, 64
(11
), pp. 1196
–1199
.22.
Sieber
, J.
, and Krauskopf
, B.
, 2008, “Control Based Bifurcation Analysis for Experiments
,” Nonlinear Dyn.
0924-090X, 51
(3
), pp. 365
–377
.23.
Sieber
, J.
, Gonzalez-Buelga
, A.
, Neild
, S.
, Wagg
, D.
, and Krauskopf
, B.
, 2008, “Experimental Continuation of Periodic Orbits Through a Fold
,” Phys. Rev. Lett.
0031-9007, 100
(24
), p. 244101
.24.
Misra
, S.
, Dankowicz
, H.
, and Paul
, M.
, 2008, “Event-Driven Feedback Tracking and Control of Tapping-Mode Atomic Force Microscopy
,” Proc. R. Soc. London, Ser. A
0950-1207, 464
(2096
), pp. 2113
–2133
.25.
Siettos
, C. I.
, Kevrekidis
, I. G.
, and Maroudas
, D.
, 2004, “Coarse Bifurcation Diagrams via Microscopic Simulators: A State-Feedback Control-Based Approach
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 14
(1
), pp. 207
–220
.26.
Burrow
, S.
, and Clare
, L.
, 2007, “A Resonant Generator With Non-Linear Compliance for Energy Harvesting in High Vibrational Environments
,” Proceedings of the Electric Machines and Drives Conference, IEMDC ‘07
, Antalya, Turkey, Vol. 1
, pp. 715
–720
.27.
Burrow
, S.
, Clare
, L.
, Carrella
, A.
, and Barton
, D.
, 2008, “Vibration Energy Harvesters With Non-Linear Compliance
,” Proceedings of the SPIE Smart Structures/NDE Conference
, San Diego, CA, p. 692807
.28.
Eyert
, V.
, 1996, “A Comparative Study on Methods for Convergence Acceleration of Iterative Vector Sequences
,” J. Comput. Phys.
0021-9991, 124
(2
), pp. 271
–285
.Copyright © 2011
by American Society of Mechanical Engineers
You do not currently have access to this content.
