Interaction Between Slow and Fast Oscillations in an Infinite Degree-of-Freedom Linear System Coupled to a Nonlinear Subsystem: Theory and Experiment

J. Appl. Mech 66(2), 448-459 (Oct 25, 1999) (12 pages) doi:10.1115/1.2791069 History: Received December 23, 1997; Received February 10, 1998; Revised October 30, 1998; Revised February 02, 1999


The interaction dynamics of a cantilever linear beam coupled to a nonlinear pendulum, a prototype for linear/nonlinear coupled structures of infinite degrees-of-freedom, has been studied analytically and experimentally. The spatio-temporal characteristics of the dynamics is analyzed by using tools from geometric singular perturbation theory and proper orthogonal decompositions. Over a wide range of coupling between the linear beam and the nonlinear pendulum, the coupled dynamics is dominated by three proper orthogonal (PO) modes. The first two dominant PO modes stem from those characterizing the reduced slow free dynamics of the stiff/soft (weakly coupled) system. The third mode appears in all interactions and stems from the reduced fast free dynamics. The interaction creates periodic and quasi-periodic motions that reduce dramatically the forced resonant dynamics in the linear substructure. These regular motions are characterized by four PO modes. The irregular interaction dynamics consists of low-dimensional and high-dimensional chaotic motions characterized by three PO modes and six to seven PO modes, respectively. Experimental tests are also carried out and there is satisfactory agreement with theoretical predictions.

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FenichelN1979Geometric singular perturbation theory for ordinary differential equationsJ. of Diff. Equat6625398 [CrossRef]
BoothbyWM1985An Introduction to Differential Manifolds and Riemanian Geometry Academic Press London
CusumanoJPSharkadyMTKimpleBW1994Experimental measurements of dimensionality and spatial coherence in the dynamics of a flexible-beam impact oscillatorPhil. Trans. R. Soc. Lond662421438 [CrossRef]
CuvalciOErtasA1996Pendulum as Vibration Absorber for Flexible Structures: Experiments and TheoryJournal of Vibration and Acoustics662558566 [CrossRef]
GeorgiouIT1993Nonlinear dynamics and chaotic motions of a singularly perturbed nonlinear viscoelastic beamPh.D. dissertation Purdue University West Lafayette, IN
GeorgiouITBajajAKCorlessM1998aSlow and fast invariant manifolds, and normal modes in a two-degree-of-freedom structural dynamical system with multiple equilibrium statesInt. J. of Non-Linear Mech662275300 [CrossRef]
GeorgiouITCorlessMBajajAK1996aDynamics of nonlinear structures with multiple equilibria: A singular perturbation-invariant manifold approachJ. Appl. Math. Phys. (ZAMP)to appear
GeorgiouITSchwartzIB1999aDynamics of Large Scale Coupled Structural/Mechanical Systems: A Singular Perturbation/Proper Orthogonal Decomposition ApproachJ. Appl. Math. (SIAM)66211781207
GeorgiouITSchwartzIB1997aSlaving the In-Plane Motions of a Nonlinear Plate to its Flexural Motions: An Invariant Manifold ApproachASME JOURNAL OF APPLIED MECHANICS662175182 [CrossRef]
GeorgiouITSansourJSansourCSchwartzIB1997bNonlinear dynamics of geometrically exact in-plane rods: the finite element/proper orthogonal decomposition approachActa Mechanicato appear
GeorgiouITSchwartzIB1999bThe Slow Nonlinear Normal Mode and Stochasticity in the Dynamics of a Conservative Flexible Rod/Pendulum ConfigurationJournal of Sound and Vibration662383411 [CrossRef]
GeorgiouITSchwartzIBEmaciEVakakisAF1997cChaos and Hysteresis in Dynamics of Coupled Structures: Experiment and TheoryProceedings of the Fourth Experimental Chaos ConferenceDingMDittoWPecoraLSpanoMVohraS World Scientific Singapore
GeorgiouITSansourJ1998bAnalyzing the finite element dynamics of nonlinear in-plane rods by the method of proper orthogonal decompositionComputational Mechanics, New Trends and ApplicationsIdelsohnSOnateEDvorkinE CIMNE Barcelona, Spain
JonesC1995Geometric singular perturbation theory in dynamical systems(Springer Lecture Notes Math. 1609)44120
KnoblochH. WAulbachB1984Singular perturbations and integral manifoldsJ. Math. Phys. Sci.662415423
NippK1985Invariant manifolds of singularly perturbed ordinary differential equationsJ. Appl. Math. Phys. (ZAMP)662311320
MiraC1997Some Historical Aspects of Nonlinear Dynamics-Possible Trends for the FutureJ. Franklin Inst.66210751113 [CrossRef]
PontrjaginLS1957Asymptotic behavior of solutions of differential equations when the higher derivatives contain a small parameter as a factorIzv. Akad. Nauk (Ser. mat.)662605621(in Russian)
ScheckF1990Mechanics Springer-Verlag New York
TikhonovAN1952Systems of differential equations containing a small parameter with higher order derivativesMat. Sbornik662575584(in Russian)
SirovichL1987Turbulence and the Dynamics of Coherent Structures, Pt. I, Coherent StructuresQuart. Appl. Math.662561571
SchwartzIBTriandafI1996Chaos and intermittent bursting in a reaction-diffusion processChaos662229237 [CrossRef]
SchwartzIBGeorgiouIT1998Instant chaos and hysteresis in coupled linear-nonlinear oscillatorsPhysics Letters, A662307312 [CrossRef]
WigginsS1990Introduction to Applied Nonlinear Dynamical Systems and Chaos Springer-Verlag New York





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