Amplitude modulated motions in a two degree-of-freedom system with quadratic nonlinearities under parametric excitation: experimental investigation

General two-degree-of-freedom dynamical systems with weak quadratic nonlinearities are studied. With the aid of an asymptotic method of analysis a classification of these systems is made and the more interesting subclasses are studied in detail. The study includes an examination of the stability of the solutions. Depending on the values of the system parameters, several different physical phenomena are shown to occur. Among these is the phenomenon of amplitude-modulated motions with modulation periods that are much larger than the periods of the excitation forces.

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Experimental Determination of the Complete Dynamical Properties of a Two-Degree-Of-Freedom Model Having Nearly Coincident Natural Frequencies

Journal of Mechanical Engineering Science ◽

10.1243/jmes_jour_1967_009_061_02 ◽

1967 ◽

Vol 9 (5) ◽

pp. 402-413 ◽

Cited By ~ 6

Author(s):

R. W. Traill-Nash ◽

G. Long ◽

C. M. Bailey

Keyword(s):

Experimental Investigation ◽

Experimental Determination ◽

Natural Frequencies ◽

Degree Of Freedom ◽

Influence Coefficients ◽

Stiffness Properties ◽

Modes Of Vibration ◽

Dynamical Properties ◽

Two Degree Of Freedom

Existing techniques of resonance testing have shown a marked inability to find the principal modes, natural frequencies and levels of damping in a structure which possesses two or more close natural frequencies (1)§. This paper describes an experimental investigation on a two-degree-of-freedom model of a technique which makes use of dynamical influence coefficients (or receptances) measured at a number of stations on the structure (2) (3) (4) (5). The measured coefficients are used to calculate natural frequencies and modes of vibration, and the mass, damping and stiffness properties of the system. Several model configurations having different natural frequency separations were tested and no special difficulty resulted when natural frequencies were close or even coincident.

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Parametrically Excited Nonlinear Two-Degree-of-Freedom Systems With Repeated Natural Frequencies

14th Biennial Conference on Mechanical Vibration and Noise: Nonlinear Vibrations ◽

10.1115/detc1993-0047 ◽

1993 ◽

Author(s):

A. H. Nayfeh ◽

C. Chin ◽

D. T. Mook

Keyword(s):

Parametric Excitation ◽

Nonlinear Response ◽

Normal Forms ◽

Natural Frequencies ◽

Linear Part ◽

Degree Of Freedom ◽

Cubic Nonlinearity ◽

Simply Supported ◽

The Stability ◽

Two Degree Of Freedom

Abstract The method of normal forms is used to study the nonlinear response of two-degree-of-freedom systems with repeated natural frequencies and cubic nonlinearity to a principal parametric excitation. The linear part of the system has a nonsemisimple one-to-one resonance. The character of the stability and various types of bifurcation are analyzed. The results are applied to the flutter of a simply-supported panel in a supersonic airstream.

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PARAMETRICALLY EXCITED NONLINEAR TWO-DEGREE-OF-FREEDOM SYSTEMS WITH NONSEMISIMPLE ONE-TO-ONE RESONANCE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127495000545 ◽

1995 ◽

Vol 05 (03) ◽

pp. 725-740 ◽

Cited By ~ 5

Author(s):

C. CHIN ◽

A.H. NAYFEH

Keyword(s):

Parametric Resonance ◽

Multiple Scales ◽

Period Doubling ◽

Pitchfork Bifurcation ◽

Degree Of Freedom ◽

Parametrically Excited ◽

Parametric Resonances ◽

Quadratic Nonlinearities ◽

One To One ◽

Two Degree Of Freedom

The response of a parametrically excited two-degree-of-freedom system with quadratic and cubic nonlinearities and a nonsemisimple one-to-one internal resonance is investigated. The method of multiple scales is used to derive four first-order differential equations governing the modulation of the amplitudes and phases of the two modes for the cases of fundamental and principal parametric resonances. Bifurcation analysis of the case of fundamental parametric resonance reveals that the quadratic nonlinearities qualitatively change the response of the system. They change the pitchfork bifurcation to a transcritical bifurcation. Cyclic-fold, Hopf bifurcations of the nontrivial constant solutions, and period-doubling sequences leading to chaos are induced by these quadratic terms. The effects of quadratic nonlinearities for the case of principal parametric resonance are discussed.

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