Period-1 Motions in a Two-Degree-of-Freedom Nonlinear Oscillator With Periodic Excitation

The Stability ◽

In this paper, analytical solutions for period-1 motions in a periodically forced, two-degrees-of-freedom system with a nonlinear spring are developed. The stability and bifurcation of the periodic motions are completed by the eigenvalue analysis. Both symmetric and asymmetric periodic motions are found in the system. Analytical solutions of both stable and unstable period-1 are presented. Finally, numerical simulations of stable and unstable motions in the two degrees of freedom systems are presented. The harmonic amplitude spectrums show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be observed.

Volume 6: 12th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

Period-3 Motions in a Two Degree-of-Freedom Nonlinear Oscillator

10.1115/detc2016-59397 ◽

2016 ◽

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Analytical Solutions ◽

Nonlinear Oscillator ◽

Degrees Of Freedom ◽

Degree Of Freedom ◽

Harmonic Amplitude ◽

Periodic Motions ◽

Amplitude Spectra ◽

Stability And Bifurcations ◽

In this paper, periodic motions of a two-degree-of-freedom nonlinear oscillator are studied by using general harmonic balanced method. Stable and unstable period-3 motions are obtained. The corresponding stability and bifurcations of the period-3 motions are determined through the eigenvalue analysis. Both symmetric and asymmetric period-3 motions are found in the system with a certain set of parameter. Numerical simulations of both stable and unstable period-3 motions in the two degrees of freedom systems are illustrated. The harmonic amplitude spectra show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be observed.

Bifurcation Trees of a Periodically Forced, Two-Degree-of-Freedom Oscillator With a Nonlinear Hardening Spring

Volume 4B: Dynamics, Vibration, and Control ◽

10.1115/imece2015-50028 ◽

2015 ◽

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Analytical Solutions ◽

Degree Of Freedom ◽

Harmonic Amplitude ◽

Periodic Motions ◽

Stable Period ◽

Period 2 ◽

The Stability ◽

Nonlinear Hardening ◽

In this paper, analytical solutions of periodic motions in a periodically forced, damped, two-degree-of-freedom oscillator with a nonlinear hardening spring are obtained. The bifurcation trees of periodic motions are presented, and the stability and bifurcation of the periodic motion are determined through the eigenvalue analysis. Numerical simulations of stable period-1 and period-2 motions in the two-degree-of-freedom systems are presented, and the harmonic amplitude spectrums are presented to show the harmonic effects on periodic motions, and the accuracy of approximate analytical solutions can be estimated through the harmonic amplitudes.

Bifurcation Trees of Period-1 Motions to Chaos in a Two-Degree-of-Freedom, Nonlinear Oscillator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415501795 ◽

2015 ◽

Vol 25 (13) ◽

pp. 1550179 ◽

Cited By ~ 8

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Dynamical System ◽

Fourier Series ◽

Analytical Solutions ◽

Nonlinear Oscillator ◽

Degree Of Freedom ◽

Harmonic Amplitude ◽

Periodic Motions ◽

Series Transformation ◽

Two Degree Of Freedom ◽

Analytical Bifurcation Trees

In this paper, analytical solutions for period-[Formula: see text] motions in a two-degree-of-freedom (2-DOF) nonlinear oscillator are developed through the finite Fourier series. From the finite Fourier series transformation, the dynamical system of coefficients of the finite Fourier series is developed. From such a dynamical system, the solutions of period-[Formula: see text] motions are obtained and the corresponding stability and bifurcation analyses of period-[Formula: see text] motions are carried out. Analytical bifurcation trees of period-1 motions to chaos are presented. Displacements, velocities and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator.

Analytical Period-1 Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

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

10.1115/imece2012-86592 ◽

2012 ◽

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Analytical Solutions ◽

Harmonic Balance ◽

Nonlinear Oscillator ◽

Harmonic Balance Method ◽

Series Expansions ◽

Periodic Motions ◽

Generalized Harmonic Balance Method

Forced Oscillator ◽

The Stability ◽

In this paper, period-1 motions in a quadratic nonlinear oscillator under excitation are investigated by the generalized harmonic balance method. The analytical solutions of period-1 motion for such an oscillator are presented by the Fourier series expansions. The stability and bifurcation analysis of period-1 motion is carried out via eigenvalue analysis. To verify the approximate analytical solutions, numerical simulations are performed for a better understanding of the parameter characteristics of the period-1 solutions, and the stable and unstable periodic motions are illustrated. The analytical period-1 solutions are different from the perturbation analysis.

Galloping Vibration of Nonlinear Cables Through a Two Degree-of-Freedom Oscillator

Volume 4B: Dynamics, Vibration, and Control ◽

10.1115/imece2016-66173 ◽

2016 ◽

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Transmission Line ◽

Analytical Solutions ◽

Transmission Lines ◽

Nonlinear Oscillator ◽

Numerical Solutions ◽

Degree Of Freedom ◽

Periodic Motions ◽

Stability And Bifurcation ◽

In this paper, galloping vibrations of a lightly iced transmission line are investigated through a two-degree-of-freedom (2-DOF) nonlinear oscillator. The 2-DOF nonlinear oscillator is used to describe the transverse and torsional motions of the galloping cables. The analytical solutions of periodic motions of galloping cables are presented through generalized harmonic balanced method. The analytical solutions of periodic motions for the galloping cable are compared with the numerical solutions, and the corresponding stability and bifurcation of periodic motions are analyzed by the eigenvalues analysis. To demonstrate the accuracy of the analytical solutions of periodic motions, the harmonic amplitudes are presented. This investigation will help one better understand galloping mechanism of iced transmission lines.

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 ◽

The Stability

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.

Asymptotic trajectories and the stability of the periodic motions of an autonomous hamiltonian system with two degrees of freedom

Journal of Applied Mathematics and Mechanics ◽

10.1016/0021-8928(88)90079-2 ◽

1988 ◽

Vol 52 (3) ◽

pp. 283-289

Author(s):

A.P. Markeyev

Keyword(s):

Hamiltonian System ◽

Degrees Of Freedom ◽

Periodic Motions ◽

Two Degrees Of Freedom ◽

Autonomous Hamiltonian System ◽

The Stability

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.

An Approximate Solution for Period-m Motions in a Periodically Forced Oscillator With Quadratic Nonlinearity

Volume 4B: Dynamics, Vibration and Control ◽

10.1115/imece2013-62418 ◽

2013 ◽

Author(s):

Albert C. J. Luo ◽

Bo Yu

Keyword(s):

Approximate Solution ◽

Numerical Simulations ◽

Analytical Solutions ◽

Periodic Motion ◽

Nonlinear Oscillator ◽

Approximate Solutions ◽

Bifurcation Tree ◽

Forced Oscillator ◽

The Stability

The approximate analytical solutions of the period-m motions for a periodically forced, quadratic nonlinear oscillator are presented. The stability and bifurcation of such approximate solutions in the quadratic nonlinear oscillator are discussed. The bifurcation tree of period-1 to chaos is presented. Numerical simulations for period-1 to period-4 motions in such quadratic oscillator are carried out for comparison of approximate analytical solutions. Such an investigation provides how to analytically determine bifurcation of periodic motion to chaos.

Analytical Solutions of Period-1 to Period-2 Motions in a Periodically Diffused Brusselator

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4038204 ◽

2018 ◽

Vol 13 (9) ◽

Cited By ~ 1

Author(s):

Albert C. J. Luo ◽

Siyu Guo

Keyword(s):

Analytical Solutions ◽

Harmonic Amplitude ◽

Periodic Motions ◽

Bifurcation Tree ◽

Stable Period ◽

Period 2 ◽

Amplitude Spectra ◽

Unstable Solutions ◽

Stability And Bifurcations

In this paper, the analytical solutions of periodic evolutions of the periodically diffused Brusselator are obtained through the generalized harmonic balanced method. Stable and unstable solutions of period-1 and period-2 evolutions in the Brusselator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis, and the corresponding Hopf bifurcations are presented on the analytical bifurcation tree of the periodic motions. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectra show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.