scholarly journals On the approximate solution of a piecewise nonlinear oscillator under super-harmonic resonance

2005 ◽  
Vol 283 (1-2) ◽  
pp. 467-474 ◽  
Author(s):  
J.C. Ji ◽  
Colin H. Hansen
Keyword(s):  
Approximate Solution ◽  
Nonlinear Oscillator ◽  
Harmonic Resonance ◽  
Super Harmonic Resonance

scholarly journals The modulation of response caused by the fractional derivative in the Duffing system under super-harmonic resonance

2021 ◽  
pp. 126-126
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Qixun Lan ◽  
Yujie Cai ◽  
Huafeng Xu ◽  
...  
Keyword(s):  
Approximate Solution ◽  
Fractional Derivative ◽  
Frequency Response ◽  
Duffing Oscillator ◽  
Time History ◽  
Singularity Theory ◽  
Critical Parameters ◽  
Amplitude Frequency Response ◽  
Harmonic Resonance ◽  
Super Harmonic Resonance

The dynamic characteristics of the 3:1 super-harmonic resonance response of the Duffing oscillator with the fractional derivative are studied. Firstly, the approximate solution of the amplitude-frequency response of the system is obtained by using the periodic characteristic of the response. Secondly, a set of critical parameters for the qualitative change of amplitude-frequency response of the system is derived according to the singularity theory and the two types of the responses are obtained. Finally, the components of the 1X and 3X frequencies of the system?s time history are extracted by the spectrum analysis, and then the correctness of the theoretical analysis is verified by comparing them with the approximate solution. It is found that the amplitude-frequency responses of the system can be changed essentially by changing the order and coefficient of the fractional derivative. The method used in this paper can be used to design a fractional order controller for adjusting the amplitude-frequency response of the fractional dynamical system.

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

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 ◽  
Approximate Analytical Solutions ◽  
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.

Frequency–Amplitude Relationship of Coupled Anharmonic Oscillators

2011 ◽  
Vol 105-107 ◽  
pp. 271-274
Author(s):  
Zheng Biao Li ◽  
Wei Hong Zhu
Keyword(s):  
Approximate Solution ◽  
Nonlinear Oscillator ◽  
Nonlinear Problems ◽  
Solution Procedure ◽  
Parameter Method ◽  
Anharmonic Oscillators ◽  
First Order ◽  
Linear And Nonlinear ◽  
Relationship Of

The frequency–amplitude relationship of coupled anharmonic oscillators is an important problem. Many powerful methods for solving this problem have been proposed. He’s parameter-expanding method is an important one. It holds the advantages of modified Lindstedt–Poincare parameter method and bookkeeping parameter method. The first iteration is enough. It is very effective and convenient and quite accurate to both linear and nonlinear problems. In this paper, He’s parameter-expanding method is applied to coupled anharmonic oscillators. The frequency-amplitude relationship and the first-order approximate solution of the oscillators are obtained respectively. The solution procedure shows that the method is very powerful and convenient to nonlinear oscillator. This method has great potential and can be applied to other types of nonlinear oscillator problems.

scholarly journals Higher-Order Approximate Periodic Solutions of a Nonlinear Oscillator with Discontinuity by Variational Approach

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
M. Orhan Kaya ◽  
S. Altay Demirbağ ◽  
F. Özen Zengin
Keyword(s):  
Approximate Solution ◽  
Periodic Solutions ◽  
Relative Error ◽  
Variational Approach ◽  
Nonlinear Oscillator ◽  
Higher Order ◽  
Second Order ◽  
Elastic Force ◽  
Force Term ◽  
The Third

He's variational approach is modified for nonlinear oscillator with discontinuity for which the elastic force term is proportional to sgn(u). Three levels of approximation have been used. We obtained 1.6% relative error for the first approximate period, 0.3% relative error for the second-order approximate period. The third approximate solution has the accuracy as high as 0.1%.

Super-Harmonic Resonance Analysis on Torsional Vibration of Misaligned Rotor Driven by Universal Joint

2010 ◽  
Vol 26-28 ◽  
pp. 1226-1231 ◽  
Author(s):  
Chang Lin Feng ◽  
Yong Yong Zhu ◽  
De Shi Wang
Keyword(s):  
Periodic Solution ◽  
Torsional Vibration ◽  
Stable Region ◽  
Universal Joint ◽  
Weakly Nonlinear ◽  
Resonance Analysis ◽  
Actual Error ◽  
The Stability ◽  
Harmonic Resonance ◽  
Super Harmonic Resonance

The super-harmonic resonance of the nonlinear torsional vibration of misaligned rotor system driven by universal joint was studied considering both natural structure misalignment and actual error misalignment. Utilizing multi-scale method, the periodic solution of weakly nonlinear torsional vibration equation was obtained corresponding to super-harmonic resonance, including amplitude-frequency and phase-frequency characteristic expressions of steady periodic solution. The stability of equilibrium point was investigated using Lyapunov first approximate stability theory, then the stable region and unstable region on the amplitude of the super-harmonic resonance periodic solution, which varied with the detuning parameter. At last, the driving shaft’s steady periodic motion of the first approximation and its calculation simulation were carried out according to the kinetic relation about driven shaft and driving shaft. It is found that jumping phenomenon and dynamic bifurcation occur when the rotating angular velocity of driving shaft is half of the natural frequency of the deriving system. The results above indicate the fundamental characteristic of the nonlinear dynamic on the misaligned rotor, also applying the foundation for advanced bifurcation and singularities analysis.

Approximate solution of a piecewise linear–nonlinear oscillator using the homotopy analysis method

2017 ◽  
Vol 24 (19) ◽  
pp. 4551-4562 ◽  
Author(s):  
Jixiong Fei ◽  
Bin Lin ◽  
Shuai Yan ◽  
Xiaofeng Zhang
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Homotopy Analysis Method ◽  
Nonlinear Oscillator ◽  
Piecewise Linear ◽  
Dynamical Behavior ◽  
Nonlinear Damping ◽  
Analysis Method ◽  
Bifurcation Diagrams ◽  
Homotopy Analysis

Most of the piecewise oscillators in engineering fields include nonlinear damping or stiffness and the contained damping or stiffness is strongly nonlinear, but to the authors’ knowledge little attention has been paid to those systems. Thus, in the present paper, a sinusoidal excited piecewise linear–nonlinear oscillator is analyzed. The mathematical model of the oscillator is described by a combination of a linear and a nonlinear differential equation which contains strong nonlinear terms of stiffness. An approximate solution for the oscillator is proposed by using the homotopy analysis method and matching method. The validity of the proposed solution is verified by comparing it with the exact solution. It is found that the approximate solution is in good agreement with the exact solution. The influence of some system parameters on the dynamical behavior of the oscillator is also investigated by the bifurcation diagrams of these parameters. From these bifurcation diagrams, one can observe the motion of the oscillator directly.

scholarly journals Super-harmonic resonance of fractional-order van der Pol oscillator

2014 ◽  
Vol 63 (1) ◽  
pp. 010503
Author(s):  
Wei Peng ◽  
Shen Yong-Jun ◽  
Yang Shao-Pu
Keyword(s):  
Fractional Order ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Harmonic Resonance ◽  
Super Harmonic Resonance
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