Period Motions and Stability in a Nonlinear Spring Pendulum

In this paper, period-1 motions varying with excitation frequency in a periodically forced, nonlinear spring pendulum system are predicted by a semi-analytic method. The harmonic frequency-amplitude for periodical motions are analyzed from the finite discrete Fourier series. The stability of the periodical solutions are shown on the bifurcation trees as well. From the analytical prediction, numerical illustrations of periodic motions are given, the comparison of numerical solution and analytical solution are given.

Period-1 to Period-2 Motions in a Periodically Forced Nonlinear Spring Pendulum

Volume 8: 31st Conference on Mechanical Vibration and Noise ◽

10.1115/detc2019-97292 ◽

2019 ◽

Author(s):

Albert C. J. Luo ◽

Yaoguang Yuan

Keyword(s):

Analytical Solutions ◽

Perturbation Analysis ◽

Mapping Method ◽

Analytical Prediction ◽

Pendulum System ◽

Nonlinear Spring ◽

Period 2 ◽

Stability And Bifurcations ◽

Abstract In this paper, bifurcation trees of period-1 to period-2 motions in a periodically forced, nonlinear spring pendulum system are predicted analytically through the discrete mapping method. The stability and bifurcations of period-1 to period-2 motions on the bifurcation trees are presented as well. From the analytical prediction, numerical illustrations of period-1 and period-2 motions are completed for comparison of numerical and analytical solutions. The results presented in this paper is totally different from the traditional perturbation analysis.

Periodic Motions and Bifurcations of a Periodically Forced Spring Pendulum Varying With Excitation Amplitude

Volume 7B: Dynamics, Vibration, and Control ◽

10.1115/imece2020-23384 ◽

2020 ◽

Author(s):

Yu Guo ◽

Albert C. J. Luo

Keyword(s):

Dynamical Behavior ◽

Excitation Amplitude ◽

Periodic Motions ◽

Pendulum System ◽

Bifurcation Tree ◽

Nonlinear Spring ◽

Nonlinear Dynamical ◽

Discrete Maps ◽

Period 2 ◽

Stability And Bifurcations

Abstract In this paper, the semi-analytical method of implicit discrete maps is employed to investigate the nonlinear dynamical behavior of a nonlinear spring pendulum. The implicit discrete maps are developed through the midpoint scheme of the corresponding differential equations of a nonlinear spring pendulum system. Using discrete mapping structures, different periodic motions are obtained for the bifurcation trees. With varying excitation amplitude, a bifurcation tree of period-1 motion to chaos is achieved through the bifurcation tree of period-1 to period-2 motions. The corresponding stability and bifurcations are studied through eigenvalue analysis. Finally, numerical illustrations of periodic motions are obtained numerically and analytically.

Periodic Motions in the Periodically Forced Oscillator With Multiple Discontinuities

Design Engineering and Computers and Information in Engineering, Parts A and B ◽

10.1115/imece2006-13827 ◽

2006 ◽

Author(s):

Albert C. J. Luo ◽

Mehul T. Patel

Keyword(s):

Dynamical System ◽

Bifurcation Analysis ◽

Analytical Prediction ◽

Periodic Motions ◽

Stability And Bifurcation ◽

Mapping Structures ◽

Forced Oscillator ◽

In this paper, the stability and bifurcation of periodic motions in periodically forced oscillator with multiple discontinuities is investigated. The generic mappings are introduced for the analytical prediction of periodic motions. Owing to the multiple discontinuous boundaries, the mapping structures for periodic motions are very complicated, which causes more difficulty to obtain periodic motions in such a dynamical system. The analytical prediction of complex periodic motions is carried out and verified numerically, and the corresponding stability and bifurcation analysis are performed. Due to page limitation, grazing and stick motions and chaos in this system will be investigated further.

Period-m Motions in a Periodically Forced van der Pol Oscillator

Volume 4B: Dynamics, Vibration and Control ◽

10.1115/imece2013-62465 ◽

2013 ◽

Author(s):

Albert C. J. Luo ◽

Arash Baghaei Lakeh

Keyword(s):

Fourier Series ◽

Periodic Motion ◽

Approximate Solutions ◽

Van Der Pol Oscillator ◽

Periodic Motions ◽

Van Der Pol ◽

Stable Period ◽

Series Expression ◽

Period-m motions in a periodically forced, van der Pol oscillator are investigated through the Fourier series expression, and the stability and bifurcation analysis of such periodic motions are carried out. To verify the approximate solutions of period-m motions, numerical illustrations are given. Period-m motions are separated by quasi-periodic motion or chaos, and the stable period-m motions are in independent periodic motion windows.

FEED FORWARD RESIDUE HARMONIC BALANCE METHOD FOR A QUADRATIC NONLINEAR OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411029434 ◽

2011 ◽

Vol 21 (06) ◽

pp. 1783-1794 ◽

Cited By ~ 13

Author(s):

AYT LEUNG ◽

ZHONGJIN GUO

Keyword(s):

Fourier Series ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

High Order ◽

Balance Method ◽

Quadratic Nonlinearity ◽

Analytical Prediction ◽

Original Solution ◽

Feed Forward ◽

The harmonic balance method truncates the Fourier series in a finite number of terms. In this paper we show that the truncated residues may be important to determine the stability of the approximated solution and that the truncated residues in the stability analysis can fully be considered without increasing the number of equations in the original solution. Therefore, the high order superharmonic and subharmonic responses and the cascade of bifurcations to irregular attractor can be accurately approximated by just the first few terms of the Fourier series so that analytical prediction is possible. A harmonically driven oscillator with quadratic nonlinearity is taken as examples. The explicitly analytical solutions are obtained for the steady state solutions and for the high order superharmonic approximation. The stabilities of the solutions are determined by the Floquet theory. It is shown that the predicted stability of the solution can be qualitatively different with and without the consideration of the feed forward residues. The second-, fourth- and eighth-order subharmonic analytical bifurcation solutions are calculated to obtain the cascades of bifurcations to irregular attractor. The improved analytical harmonic approximations are compared with other results and with numerical solutions. It is proved that a two superharmonic expansion with appropriate subharmonic is sufficient for determining the characteristics of the solutions of a harmonically driven oscillator with quadratic nonlinearity.

Analytical Periodic Motions of a Parametric Oscillator With Quadratic Nonlinearity

Volume 8: 26th Conference on Mechanical Vibration and Noise ◽

10.1115/detc2014-35467 ◽

2014 ◽

Cited By ~ 1

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Fourier Series ◽

Series Expansion ◽

Bifurcation Analysis ◽

Analytical Solutions ◽

Parametric Oscillator ◽

Quadratic Nonlinearity ◽

Fourier Series Expansion ◽

Periodic Motions ◽

In this paper, the analytical solutions of periodic motions in a parametric oscillator are presented by the finite Fourier series expansion, and the stability and bifurcation analysis of periodic motions are performed. Numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum.

Positive Position Feedback Controllers for Reduction the Vibration of a Nonlinear Spring Pendulum

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v12i11.7 ◽

2016 ◽

Vol 12 (11) ◽

pp. 6758-6772

Author(s):

Y a Amer

Keyword(s):

Perturbation Technique ◽

Multiple Scale ◽

Resonance Case ◽

Feedback Controllers ◽

Pendulum System ◽

Nonlinear Spring ◽

Positive Position Feedback ◽

Position Feedback ◽

Scale Perturbation ◽

In this paper, the two positive position feedback controllers (PPF) are proposed to reduce the longitudinal and angular vibrations of the nonlinear spring pendulum system which simulated the ship roll motion. This described by a four-degreeof- freedom system (4-DOF) which subjected to the external excitation force at simultaneous primary and internal resonance case. The method of multiple scale perturbation technique (MSPT) is applied to study the approximate solution of the given system. The stability of the system is investigated near the resonance case applying the frequency-response equations. Numerically, the effects of different controllers parameters on the basic system behavior are studied.

Analytical Solutions for Periodic Motions in a Hardening Mathieu-Duffing Oscillator

Volume 4B: Dynamics, Vibration and Control ◽

10.1115/imece2013-62417 ◽

2013 ◽

Author(s):

Albert C. J. Luo ◽

Dennis M. O’Connor

Keyword(s):

Fourier Series ◽

Bifurcation Analysis ◽

Analytical Solutions ◽

Duffing Oscillator ◽

Series Solutions ◽

Periodic Motions ◽

Period 2 ◽

Approximate Analytical Solutions ◽

Analytical solutions for period-m motions in a hardening Mathieu-Duffing oscillator are obtained using the finite Fourier series solutions, and the stability and bifurcation analysis of such periodic motions are completed. To verify the approximate analytical solutions of periodic motions, numerical simulations of the hardening Mathieu-Duffing oscillator are presented. Period-1 asymmetric and period-2 symmetric motions are illustrated.

Volume 6: 10th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

Period-1 Motions in a Time-Delayed Duffing Oscillitor With Periodic Excitation

10.1115/detc2014-35473 ◽

2014 ◽

Author(s):

Albert C. J. Luo ◽

Hanxiang Jin

Keyword(s):

Fourier Series ◽

Analytical Solutions ◽

Duffing Oscillator ◽

Eigenvalue Analysis ◽

Periodic Excitation ◽

Periodic Motions ◽

Stability And Bifurcation ◽

The Stability ◽

Amplitude Characteristics

In this paper, analytical solutions of period-1 motions in a time-delayed Duffing oscillator with a periodic excitation are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed by eigenvalue analysis. The symmetric and asymmetric period-1 motions in such time-delayed Duffing oscillator are obtained analytically, and the frequency-amplitude characteristics of period-1 motions in such a time-delayed Duffing oscillator are investigated. Numerical illustrations of period-1 motions are given by numerical and analytical solutions.

Periodic Motions to Chaos in Pendulum

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416501595 ◽

2016 ◽

Vol 26 (09) ◽

pp. 1650159 ◽

Cited By ~ 5

Author(s):

Albert C. J. Luo ◽

Yu Guo

Keyword(s):

Complex Dynamics ◽

Traditional Approach ◽

Analytical Prediction ◽

Periodic Motions ◽

Pendulum System ◽

Linear Dynamical Systems ◽

Mapping Structures ◽

Forced Pendulum ◽

And Mathematics ◽

Analytical Bifurcation Trees

It is not easy to find periodic motions to chaos in a pendulum system even though the periodically forced pendulum is one of the simplest nonlinear systems. However, the inherent complex dynamics of the periodically forced pendulum is much beyond our imaginations through the traditional approach of linear dynamical systems. Until now, we did not know complex motions of pendulum yet. What are the mechanism and mathematics of such complex motions in the pendulum? The results presented herein give a new view of complex motions in the periodically forced pendulum. Thus, in this paper, periodic motions to chaos in a periodically forced pendulum are predicted analytically by a semi-analytical method. The method is based on discretization of differential equations of the dynamical system to obtain implicit maps. Using the implicit maps, mapping structures for specific periodic motions are developed, and the corresponding periodic motions can be predicted analytically through such mapping structures. Analytical bifurcation trees of periodic motions to chaos are obtained, and the corresponding stability and bifurcation analysis of periodic motions to chaos are carried out by eigenvalue analysis. From the analytical prediction of periodic motions to chaos, the corresponding frequency-amplitude characteristics are obtained for a better understanding of motions’ complexity in the periodically forced pendulum. Finally, numerical simulations of selected periodic motions are illustrated. The nontravelable and travelable periodic motions on the bifurcation trees are discovered. Through this investigation, the periodic motions to chaos in the periodically forced pendulums can be understood further. Based on the perturbation method, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the periodically forced pendulum.