Global Bifurcations in Externally Excited Two-Degree-of-Freedom Nonlinear Systems

We consider the response of a linear structural system when coupled to an attachment containing strong or even essential nonlinearities. For this system, the attachment is designed as a nonlinear vibration absorber, serving to dissipate energy from the structural system. Moreover, the attachment not only leads to a reduction in the total energy of the system, but also couples together the vibration modes of the linear structural system, thereby allowing for energy to also be redistributed among these structural modes of the system. The effect of the nonlinear attachment on the linear primary system can be quantified in terms of equivalent measures for the damping and frequency of each mode, derived through consideration of the energy in each mode. The identification of these equivalent measures is illustrated on a two degree-of-freedom primary system. Moreover, this procedure depends only on the time history of the response and is therefore suited to both simulation and experimental results.

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Two-degree-of-freedom output feedback controllers for discrete-time nonlinear systems

Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148) ◽

10.1109/acc.2001.946411 ◽

2001 ◽

Cited By ~ 1

Author(s):

R.A. Wright ◽

C. Kravaris

Keyword(s):

Nonlinear Systems ◽

Discrete Time ◽

Output Feedback ◽

Degree Of Freedom ◽

Feedback Controllers ◽

Two Degree Of Freedom

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Free Periodic Motions of an Undamped Two-Degree-of-Freedom Oscillatory System With Nonlinear Unsymmetrical Elasticity

Journal of Applied Mechanics ◽

10.1115/1.4010098 ◽

1950 ◽

Vol 17 (2) ◽

pp. 185-190

Author(s):

Walter W. Soroka

Keyword(s):

Nonlinear Systems ◽

Linear Systems ◽

Oscillatory System ◽

Degree Of Freedom ◽

The Other ◽

Periodic Motions ◽

Deflection Curve ◽

Approximate Treatment ◽

Load Deflection Curve ◽

Two Degree Of Freedom

Abstract Precise solutions are presented for a two-degree-of-freedom oscillatory system containing a preset spring. Such a system is characteristic of an aircraft propeller-engine-supercharger installation. The periodic free motions obtained indicate the possibility of highly unconventional motions when the nonlinearity is pronounced, motions which may be easily overlooked in the usual approximate treatment of nonlinear systems. The results presented in this paper show that one mass may oscillate several times while the other mass is going through one oscillation. The ratio of oscillations of one mass with respect to the other changes with amplitude. By considering the load-deflection curve for the nonlinear spring to be a broken line, periodic motions may be obtained to any desired degree of precision by combining conventional general solutions for two-degree-of-freedom linear systems.

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HIGHER-ORDER AVERAGING FOR PERIODICALLY FORCED WEAKLY NONLINEAR SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127499000353 ◽

1999 ◽

Vol 09 (03) ◽

pp. 519-531 ◽

Cited By ~ 9

Author(s):

K. YAGASAKI ◽

T. ICHIKAWA

Keyword(s):

Nonlinear Systems ◽

Computer Algebra ◽

Computer Algebra System ◽

Higher Order ◽

Subharmonic Resonance ◽

Degree Of Freedom ◽

Weakly Nonlinear ◽

Lie Transforms ◽

Two Degree Of Freedom

We consider periodically forced, weakly nonlinear systems and perform higher-order averaging analyses. Especially, we describe an algorithm for computing the higher-order averaging terms by the Lie transforms. The necessary computations can be implemented on a developed package of the computer algebra system, Mathematica. We also give three examples for two Duffing-type oscillators with the primary or ultra-subharmonic resonance and a two-degree-of-freedom system with internal and external resonances, to demonstrate our results.

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