Nonresonant Modal Interactions

1995 ◽  
Author(s):  
Zaichun Feng
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Mechanical System ◽  
Modal Interactions ◽  
Chaotic Motions ◽  
Work Mechanism ◽  
One To One ◽  
Energy Interchange

Abstract Modal interactions in nonlinear systems provide a means by which energy can be transferred between modes. This energy interchange may give rise to chaotic motions in dynamical systems. Extensive research has been focused on the resonant modal interactions when the frequencies of the interacting modes are commensurate such as one-to-one and one-to-two resonances. It is recently realized that modal interactions can also occur even if the frequencies of the interacting modes are non-commensurate. In this work, mechanism for these nonresonant modal interactions is identified and illustrated through a simple mechanical system.

Modal Interactions in Dynamical and Structural Systems

1989 ◽  
Vol 42 (11S) ◽  
pp. S175-S201 ◽  
Author(s):  
A. H. Nayfeh ◽  
B. Balachandran
Keyword(s):  
Dynamical Systems ◽  
Surface Waves ◽  
Nonlinear Response ◽  
Experimental Studies ◽  
Structural Systems ◽  
Qualitative Behavior ◽  
Modal Interactions ◽  
Chaotic Motions ◽  
Beam Structures ◽  
Quadratic Nonlinearities

We review theoretical and experimental studies of the influence of modal interactions on the nonlinear response of harmonically excited structural and dynamical systems. In particular, we discuss the response of pendulums, ships, rings, shells, arches, beam structures, surface waves, and the similarities in the qualitative behavior of these systems. The systems are characterized by quadratic nonlinearities which may lead to two-to-one and combination autoparametric resonances. These resonances give rise to a coupling between the modes involved in the resonance leading to nonlinear periodic, quasi-periodic, and chaotic motions.

A Wavelet Based Procedure to Study Vibrations and Stability

1999 ◽  
Author(s):  
S. Pernot ◽  
C. H. Lamarque
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Periodic Solutions ◽  
Time Varying ◽  
Differential Systems ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential

Abstract A Wavelet-Galerkin procedure is introduced in order to obtain periodic solutions of multidegrees-of-freedom dynamical systems with periodic time-varying coefficients. The procedure is then used to study the vibrations of parametrically excited mechanical systems. As problems of stability analysis of nonlinear systems are often reduced after linearization to problems involving linear differential systems with time-varying coefficients, we demonstrate the method provides efficient practical computations of Floquet exponents and consequently allows to give estimators for stability/instability levels. A few academic examples illustrate the relevance of the method.

Identification of Nonlinear Systems Parameters Using Polyspectral Measurements and Analysis

1995 ◽  
Author(s):  
Muhammad R. Hajj ◽  
Ali H. Nayfeh ◽  
Pavol Popovic
Keyword(s):  
Experimental Study ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Analytical Techniques ◽  
Modal Interactions ◽  
Unknown Parameters ◽  
Nonlinear Parameters ◽  
The Subject ◽  
Beam Frame ◽  
Subharmonic Resonances

Abstract Experimental and analytical techniques that characterize nonlinear modal interactions in structures are used to quantify parameters in representative nonlinear systems. The subject of the experimental study is a three-beam frame. Subharmonic resonances and interaction between widely spaced modes are exploited to determine nonlinear parameters in models that represent these interactions. The phases of the auto-bispectra of the response of this structure appear in the analytical solutions of the representative models. Values of these phases could thus aid in determining other unknown parameters of nonlinear systems.

scholarly journals A Dynamical Systems Approach to Nonlinear Stellar Pulsations

1993 ◽  
Vol 134 ◽  
pp. 9-31
Author(s):  
J. R. Buchler
Keyword(s):  
Dynamical Systems ◽  
Ad Hoc ◽  
Systems Approach ◽  
Chaotic Behavior ◽  
Theoretical Explanation ◽  
Modal Interactions ◽  
Reduction Techniques ◽  
Numerical Hydrodynamics ◽  
Stellar Pulsations ◽  
The One

AbstractOver the last decade we have seen the application of novel techniques to the old problem of nonlinear stellar pulsations. Together with numerical hydrodynamics this approach provides a more fundamental understanding of the systematics of the pulsational behavior. For weakly nonadiabatic pulsations, whether regular or multi-periodic, dimensional reduction techniques lead to amplitude equations and to a description in terms of modal interactions and resonances. In particular they shed new light on the bump progression in the classical Cepheids. In more dissipative stars numerical hydrodynamical modelling has uncovered the existence of irregular variability, both in radiative and in convective models. An application of modern dynamical systems techniques has shown that this behavior occurs according to well understood routes from regular to chaotic behavior. The mechanism is very robust and represents the first non ad hoc theoretical explanation of irregular stellar variability. Finally, we discuss how a comparison with observations of irregular variability shows the need for more suitable observations, on the one hand, and of better techniques of signal processing, on the other.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

Bifurcation Analysis of a Fermi-Acceleration Oscillator Under Different Excitations

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions

In this paper, switchability and bifurcation of motions in a double excited Fermi acceleration oscillator is discussed using the theory of discontinuous dynamical systems. The two oscillators are chosen to have different excitation and parameters. The analytical conditions for motion switching in such a Fermi-oscillator are presented. Bifurcation scenario for periodic and chaotic motions is presented, and the analytical predictions of periodic motions are presented. Finally, different motions in such an oscillator are illustrated.

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