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

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 ◽  
The Stability

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.

On Period-1 and Period-2 Motions of a Jeffcott Rotor System

2019 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Mapping Method ◽  
Rotor System ◽  
Periodic Motions ◽  
Period 2 ◽  
Jeffcott Rotor ◽  
Mapping Structures ◽  
Point Scheme ◽  
Stability And Bifurcations ◽  
Specific Mapping

Abstract In this paper, the semi-analytical solutions of period-1 and period-2 motions in a nonlinear Jeffcott rotor system are presented through the discrete mapping method. The periodic motions in the nonlinear Jeffcott rotor system are obtained through specific mapping structures with a certain accuracy. A bifurcation tree of period-1 to period-2 motion is achieved, and the corresponding stability and bifurcations of periodic motions are analyzed. For verification of semi-analytical solutions, numerical simulations are carried out by the mid-point scheme.

Period Motions and Stability in a Nonlinear Spring Pendulum

2018 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yaoguang Yuan
Keyword(s):  
Fourier Series ◽  
Excitation Frequency ◽  
Analytic Method ◽  
Analytical Prediction ◽  
Discrete Fourier Series ◽  
Periodic Motions ◽  
Harmonic Frequency ◽  
Pendulum System ◽  
Nonlinear Spring ◽  
The Stability

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.

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

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.

Analytical Prediction of Period-1 Motions in a Time-Delayed, Softening Duffing Oscillator

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Duffing Oscillator ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Implicit Mapping ◽  
Stability And Bifurcations ◽  
The Stability

In this paper, symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, softening Duffing oscillator is analytically predicted through a discrete implicit mapping method. Such a method is based on the discretization of the corresponding differential equation. The stability and bifurcations of the symmetric and asymmetric period-1 motions are determined through eigenvalue analysis. Numerical simulation of the period-1 motions in the time-delayed softening Duffing oscillator is presented for verification of the analytical prediction.

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.

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

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 ◽  
The Stability

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.

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.

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

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 ◽  
Stability And Bifurcation Analysis ◽  
The Stability

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.

Vibration Suppression of a Four-Degrees-of-Freedom Nonlinear Spring Pendulum via Longitudinal and Transverse Absorbers

2011 ◽  
Vol 79 (1) ◽  
Author(s):  
M. Eissa ◽  
M. Kamel ◽  
A. T. El-Sayed
Keyword(s):  
Degrees Of Freedom ◽  
Vibration Suppression ◽  
Perturbation Technique ◽  
Time History ◽  
Multiple Scale ◽  
Threshold Value ◽  
Response Curves ◽  
Pendulum System ◽  
Nonlinear Spring ◽  
The Stability

An investigation into the passive vibration reduction of the nonlinear spring pendulum system, simulating the ship roll motion is presented. This leads to a four-degree-of-freedom (4-DOF) system subjected to multiparametric excitation forces. The two absorbers in the longitudinal and transverse directions are usually designed to control the vibration near the simultaneous subharmonic and internal resonance where system damage is probable. The theoretical results are obtained by applying the multiple scale perturbation technique (MSPT). The stability of the obtained nonlinear solution is studied and solved numerically. The obtained results from the frequency response curves confirmed the numerical results which were obtained using time history. For validity, the numerical solution is compared with the analytical solution. Effectiveness of the absorbers (Ea) are about 13 000 for the first mode (x) and 10 000 for the second mode (ϕ). A threshold value of linear damping coefficient can be used directly for vibration suppression of both vibration modes. Comparison with the available published work is reported.

Period-1 to Period-2 Motions in a Discontinuous Oscillator

2020 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
System Design ◽  
Spring Stiffness ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Period 2 ◽  
Stability And Bifurcations ◽  
The Stability ◽  
And Control ◽  
Grazing Bifurcations

Abstract In this paper, grazing bifurcations on bifurcation trees in a discontinuous dynamical oscillator are discussed. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Thus, grazing bifurcations on a bifurcation tree of period-1 to period-2 motions varying spring stiffness are presented in a discontinuous oscillator with three domains divided by circular boundaries. The stability and bifurcations of period-1 and period-2 motions are discussed. From analytical predictions, periodic motions are simulated numerically. Stiffness effects on the periodic motions are discussed. Such studies will help one understand parameter effects in discontinuous dynamical systems, which can be applied for system design and control.

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