Analytical Periodic Motions of a Parametric Oscillator With Quadratic Nonlinearity

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.

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 ◽

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: 12th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

Analytical Solutions for Periodic Motions in a Twin-Well Potential Mathieu-Duffing Oscillator

10.1115/detc2016-60280 ◽

2016 ◽

Author(s):

Dennis M. O’Connor ◽

Albert C. J. Luo

Keyword(s):

Fourier Series ◽

Bifurcation Analysis ◽

Analytical Solutions ◽

Duffing Oscillator ◽

Series Solutions ◽

Periodic Motions ◽

Unstable Solutions ◽

Analytical solutions for periodic motion in a twin-well potential Mathieu-Duffing oscillator with damping 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 twin-well solutions are presented for Period-1 asymmetric motions. Further, simulation of the unstable solutions is considered to demonstrate the resonant bands separating the twin-wells.

On Periodic Motions in a Parametric Hardening Duffing Oscillator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414300043 ◽

2014 ◽

Vol 24 (01) ◽

pp. 1430004 ◽

Cited By ~ 5

Author(s):

Albert C. J. Luo ◽

Dennis M. O'Connor

Keyword(s):

Fourier Series ◽

Bifurcation Analysis ◽

Analytical Solutions ◽

Duffing Oscillator ◽

Periodic Motions ◽

Period 2 ◽

Series Expression ◽

Amplitude Characteristics

In this paper, analytical solutions for periodic motions in a parametric hardening Duffing oscillator are presented using the finite Fourier series expression, and the corresponding stability and bifurcation analysis for such periodic motions are carried out. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions are discussed. The hardening Mathieu–Duffing oscillator is also numerically simulated to verify the approximate analytical solutions of periodic motions. Period-1 asymmetric and period-2 symmetric motions are illustrated for a better understanding of periodic motions in the hardening Mathieu–Duffing oscillator.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500758 ◽

2014 ◽

Vol 24 (05) ◽

pp. 1450075 ◽

Cited By ~ 1

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Fourier Series ◽

Nonlinear Systems ◽

Analytical Solutions ◽

Nonlinear Oscillator ◽

Series Solution ◽

Parametric Oscillator ◽

Periodic Motions ◽

Parametric Oscillators ◽

Harmonic Term ◽

Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413300097 ◽

2013 ◽

Vol 23 (03) ◽

pp. 1330009 ◽

Cited By ~ 5

Author(s):

ALBERT C. J. LUO ◽

MOZHDEH S. FARAJI MOSADMAN

Keyword(s):

Dynamical Systems ◽

Bifurcation Analysis ◽

Degrees Of Freedom ◽

Eigenvalue Analysis ◽

Analytical Dynamics ◽

Periodic Motions ◽

Mapping Structures ◽

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

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 ◽

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.

Analytical Solutions of Period-1 Motions in a Periodically Forced van der Pol Oscillator

Volume 4: Dynamics, Control and Uncertainty, Parts A and B ◽

10.1115/imece2012-86589 ◽

2012 ◽

Cited By ~ 2

Author(s):

Albert C. J. Luo ◽

Arash Baghaei Lakeh

Keyword(s):

Bifurcation Analysis ◽

Analytical Solutions ◽

Eigenvalue Analysis ◽

Van Der Pol Oscillator ◽

Comprehensive Understanding ◽

Van Der Pol ◽

In this paper, the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balanced method. The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis, and numerical illustrations of periodic-1 solutions are given to verify the approximate motion. This investigation provides more accurate solutions of period-1 motions in the van der pol oscillator for a better and comprehensive understanding of motions in such an oscillator.

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 ◽

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.