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

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.

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.

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.

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.

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.

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.

Volume 7: Dynamic Systems and Control; Mechatronics and Intelligent Machines, Parts A and B ◽

Stability and Bifurcation Analysis of Periodic Motions in a Periodically Forced Duffing Oscillator

10.1115/imece2011-62949 ◽

2011 ◽

Author(s):

Albert C. J. Luo ◽

Jianzhe Huang

Keyword(s):

Approximate Solution ◽

Bifurcation Analysis ◽

Duffing Oscillator ◽

Approximate Solutions ◽

Nonlinear Damping ◽

Periodic Motions ◽

Stability And Bifurcation Analysis

In this paper, the analytical, approximate solutions of period-1 motions in the nonlinear damping, periodically forced, Duffing oscillator is obtained. The corresponding stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out. Numerical illustrations of period-1 motions are presented.

On Time-Delay Effects on Period-1 Motions in a Periodically Forced, Time-Delayed, Duffing Oscillator

Volume 4B: Dynamics, Vibration, and Control ◽

10.1115/imece2016-66198 ◽

2016 ◽

Author(s):

Albert C. J. Luo ◽

Siyuan Xing

Keyword(s):

Time Delay ◽

Analytical Solutions ◽

Analytical Method ◽

Duffing Oscillator ◽

Delay Systems ◽

Harmonic Amplitude ◽

Time Delay Systems ◽

Periodic Motions ◽

Delay Effects ◽

In recent decades, nonlinear time-delay systems were often applied in controlling nonlinear systems, and the stability of such time-delay systems was very actively discussed. Recently, one was very interested in periodic motions in nonlinear time-delay systems. Especially, the semi-analytical solutions of periodic motions in time-delay systems are of great interest. From the semi-analytical solutions, the nonlinearity and complexity of periodic motions in the time-delay systems can be discussed. In this paper, time-delay effects on periodic motions of a periodically forced, damped, hardening, Duffing oscillator are analytically discussed through a semi-analytical method. The semi-analytical method is based on discretization of the differential equation of such a Duffing oscillator to obtain the corresponding implicit discrete mappings. Through such implicit mappings and mapping structures of periodic motions, period-1 motions varying with time-delay are discussed, and the corresponding stability and bifurcation analysis of periodic motions are carried out through eigenvalue analysis. Numerical results of periodic motions are illustrated to verify analytical predictions. The corresponding harmonic amplitude spectrums and harmonic phases are presented for a better understanding of periodic motions in such a nonlinear oscillator.

An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4026425 ◽

2014 ◽

Vol 9 (3) ◽

Cited By ~ 3

Author(s):

Albert C. J. Luo ◽

Arash Baghaei Lakeh

Keyword(s):

Approximate Solution ◽

Fourier Series ◽

Analytical Solutions ◽

Amplitude Spectrum ◽

Van Der Pol Oscillator ◽

Harmonic Amplitude ◽

Periodic Motions ◽

Van Der Pol ◽

Series Expression

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 balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms (N) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., AN≤ɛ and ɛ=10-8). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.

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.