resonance capture; essential nonlinearity; energy transfer
Abstract :
[en] We study the dynamics of a two-degree-of-freedom (DOF) nonlinear system consisting of a grounded linear oscillator coupled to a light mass by means of an essentially nonlinear (nonlinearizable) stiffness. We consider first the undamped system and perform a numerical study based on non-smooth transformations to determine its periodic solutions in a frequency-energy plot. It is found that there is a sequence of periodic solutions bifurcating or emanating from the main backbone curve of the plot. We then study analytically the periodic orbits of the undamped system using a complexification/averaging technique in order to determine the frequency contents of the various branches of solutions, and to understand the types of oscillation performed by the system at the different regimes of the motion. The transient responses of the weakly damped system are then examined, and numerical wavelet transforms are used to study the time evolutions of their harmonic components. We show that the structure of periodic orbits of the undamped system greatly influences the damped dynamics, as it causes complicated transitions between modes in the damped transient motion. In addition, there is the possibility of strong passive energy transfer (energy pumping) from the linear oscillator to the nonlinear attachment if certain periodic orbits of the undamped dynamics are excited by the initial conditions. (c) 2005 Elsevier B.V. All rights reserved.
Disciplines :
Mechanical engineering Mathematics Physics
Author, co-author :
Lee, Young Sup
Kerschen, Gaëtan ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Laboratoire de structures et systèmes spatiaux
Vakakis, Alexander F.
Panagopoulos, Panagiotis
Bergman, Lawrence
McFarland, D. Michael
Language :
English
Title :
Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment
J. Guckenheimer, and P.J. Holmes Nonlinear Oscillators, Dynamical Systems and Bifurcations of Vector Fields 1983 Springer-Verlag Berlin/New York
S. Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos 1990 Springer-Verlag Berlin/New York
A.H. Nayfeh, and D.T. Mook Energy transfer from high-frequency to low-frequency modes in structures J. Vib. Acoustics 117 1995 186 195
P.S. Landa Nonlinear Oscillations and Waves in Dynamical Systems 1996 Kluwer Academic Publishers New York
A.F. Vakakis, and O. Gendelman Energy pumping in nonlinear mechanical oscillators II. Resonance capture J. Appl. Mech. 68 1 2001 42 48
A.F. Vakakis, L.I. Manevitch, O. Gendelman, and L.A. Bergman Dynamics of linear discrete systems connected to local essentially nonlinear attachments J. Sound Vib. 264 2003 559 577
V.I. Arnold Dynamical Systems III Encyclopaedia of Mathematical Sciences 1988 Springer-Verlag Berlin/New York
D. Quinn Resonance capture in a three degree of freedom mechanical system Nonlinear Dyn. 14 1997 309 333
V.N. Pilipchuk The calculation of strongly nonlinear systems close to vibration-impact systems PMM 49 5 1985 572 578
V.N. Pilipchuk A transformation for vibrating systems based on a non-smooth periodic pair of functions Dokl. An. Ukr. SSR Ser. 4 1988 37 40 (in Russian)
L.I. Manevitch Complex representation of dynamics of coupled oscillators Mathematical Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems 1999 Kluwer Academic/Plenum Publishers New York pp. 269-300
V.N. Pilipchuk, A.F. Vakakis, and M.A.F. Azeez Study of a class of subharmonic motions using a non-smooth temporal transformation (NSTT) Physica D 100 1997 145 164
A.F. Vakakis, L.I. Manevitch, Yu.V. Mikhlin, V.N. Pilipchuk, and A.A. Zevin Normal Modes and Localization in Nonlinear Systems 1996 Wiley-Interscience New York
A.H. Nayfeh, and D. Mook Nonlinear Oscillations 1985 Wiley-Interscience New York