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

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 ◽  
Approximate Analytical Solutions ◽  
Amplitude Spectra ◽  
Stability And Bifurcations ◽  
Two Degree Of Freedom

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.

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

2015 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degrees Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Two Degrees Of Freedom ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Two Degree Of Freedom

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.

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

2015 ◽  
Vol 25 (13) ◽  
pp. 1550179 ◽  
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 Solutions of Period-1 to Period-2 Motions in a Periodically Diffused Brusselator

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Analytical Solutions ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
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.

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

2015 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Nonlinear Hardening ◽  
Two Degree Of Freedom

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.

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

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 ◽  
Two Degree Of Freedom

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.

Period-1 and Period-2 Motions in a Brusselator With a Harmonic Diffusion

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcations

In this paper, the analytical solutions of periodic evolution of Brusselator are investigated through the general harmonic balanced method. Both stable and unstable, period-1 and period-2 solutions of the Brussellator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectrums show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

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

2014 ◽  
Vol 9 (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 ◽  
Approximate Analytical Solutions ◽  
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.

On Motion Switchability in a Two Degree of Freedom, Friction-Induced Oscillator Traveling on Constant Speed Belts

2009 ◽  
Author(s):  
Albert C. J. Luo ◽  
Tingting Mao
Keyword(s):  
Degrees Of Freedom ◽  
Degree Of Freedom ◽  
Constant Speed ◽  
Periodic Motions ◽  
Mapping Structures ◽  
Two Degree Of Freedom ◽  
Motion Complexity

In this paper, all possible stick and non-stick motions in such a friction-induced oscillator are discussed and the corresponding analytical conditions for the stick and non-stick motions to the traveling belts are presented. The mapping structures are introduced and the periodic motions of the two oscillators are presented through the corresponding mapping structure. Velocity and force responses for stick and non-stick, periodic motions in the 2-DOF friction-induced system are illustrated for a better understanding of the motion complexity in such many degrees of freedom systems.

Vibration Suppression of a Harmonically Forced Oscillator Using a Passive Nonlinear Vibration Absorber

2020 ◽  
Author(s):  
Bo Yu
Keyword(s):  
Nonlinear Vibration ◽  
Analytical Solutions ◽  
Vibration Suppression ◽  
Harmonic Balance Method ◽  
Vibration Absorber ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Nonlinear Vibration Absorber ◽  
Approximate Analytical Solutions ◽  
Forced Oscillator

Abstract In this paper, the performance of a nonlinear vibration absorber with different nonlinearity is studied. The analytical solutions of periodic motions are obtained using the general harmonic balance method. As the nonlinear strength is weak, the effectiveness of the absorber is discussed. For strong nonlinearities, unstable parodic motions can be obtained and stabilities of the periodic motions are determined through the eigenvalue analysis. The Hopf and saddle bifurcations are observed. Numerical simulations are illustrated for both masses at the resonance peaks. The harmonic amplitude spectrums show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions.

Electronic Analog Computer Solutions of Nonlinear Vibratory Systems of Two Degrees of Freedom

1956 ◽  
Vol 23 (4) ◽  
pp. 629-634
Author(s):  
C. P. Atkinson
Keyword(s):  
Differential Equations ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Harmonic Approximation ◽  
Degree Of Freedom ◽  
Single Term ◽  
Approximate Analytical Solutions ◽  
Two Degree Of Freedom ◽  
Differential Analyzer ◽  
Vibrating Systems

Abstract The main contribution of this paper is its account of the use of an electronic differential analyzer for solving the exact differential equations of certain two-degree-of-freedom nonlinear vibrating systems, and the comparison of the differential-analyzer solutions with the approximate analytical solutions obtained from a single-term harmonic approximation (Ritz approximation). The results of these two methods for solving nonlinear differential equations compare favorably over much of the ranges of variables. Where discrepancies arise between the results of the two methods the value of the differential-analyzer approach can be practically appreciated. For example, in the region where superharmonics might be expected, the single-term harmonic approximation ignores them, while the analyzer solutions contain the superharmonic components of the exact solution. A secondary contribution of the paper is the account of the use of the differential analyzer for verification of suspected stability criteria for two-degree-of-freedom nonlinear vibrating systems. Analytical solutions that were reproducible on the analyzer were considered stable; those that were not reproducible were considered unstable. This paper also can be considered as a call for further application of modern computer techniques to the problems of nonlinear mechanics. Since the computer solves the exact or complete differential equations, the results are the complete solutions including transient phenomena, steady state, subharmonics, super-harmonics, and so on. These exact solutions are produced as functions of time which are analogous to the actual vibrations of the physical system studied.

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