A Time Marching Integration for Semi-Analytical Solutions of Nonlinear Oscillators Based On Synchronization

2021 ◽  
Author(s):  
R.Z. Yao ◽  
Y.M. Chen ◽  
Q.X. Liu
Keyword(s):  
Fourier Series ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Numerical Scheme ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Harmonic Oscillators ◽  
Time Marching ◽  
Novel Method ◽  
Basic Ideas

Abstract A novel method is proposed in this study for solving the semi-analytical solutions of periodic responses of nonlinear oscillators. The basic ideas comes from the fact that any periodic response can be described by Fourier series. By transforming the Fourier series into a system of harmonic oscillators, we thus establish a novel numerical scheme for tracking the periodic responses, as long as a synchronized motion can be achieved between the system of harmonic oscillators and the nonlinear oscillators considered. The presented method can be implemented by conducting time marching integration only, but it is capable of providing semi-analytical solutions straightforwardly. Different from some widely used methods such as harmonic balance method and its improved forms, this method can solve solutions involving high order harmonics without incorporating any tedious derivations as it is totally a numerical scheme. Several typical oscillators with smooth as well as non-smooth nonlinearities are taken as numerical examples to test the validity and efficiency.

Analytical Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2012 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Balance Method ◽  
Periodic Motions ◽  
Conditions Of Stability ◽  
Generalized Harmonic Balance Method

The analytical solutions of the period-1 motions for a hardening Duffing oscillator are presented through the generalized harmonic balance method. The conditions of stability and bifurcation of the approximate solutions in the oscillator are discussed. Numerical simulations for period-1 motions for the damped Duffing oscillator are carried out.

Experiment on a novel method for fundamental ferroresonance suppression

2014 ◽  
Vol 28 (05) ◽  
pp. 1450035 ◽  
Author(s):  
Wen-Xia Sima ◽  
Ming Yang ◽  
Qing Yang ◽  
Tao Yuan ◽  
Mi Zou
Keyword(s):  
High Frequency ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Suppression Mechanism ◽  
Novel Method

A novel method for fundamental ferroresonance suppression is proposed in this paper. The suppression mechanism is analyzed based on the harmonic balance method and a novel method for fundamental ferroresonance suppression is proposed. Experiments show that the fundamental ferroresonance with different saturation degrees can be suppressed by a single damping resistor controlled by the high frequency controllable switches located in the suppression module.

scholarly journals Use and Limitations of the Harmonic Balance Method for Rub-Impact Phenomena in Rotor-Stator Dynamics

2012 ◽  
Author(s):  
Loïc Peletan ◽  
Sebastien Baguet ◽  
Georges Jacquet-Richardet ◽  
Mohamed Torkhani
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Current Form ◽  
Transient Simulations ◽  
Future Extension ◽  
Periodic Behaviour ◽  
State Behaviour ◽  
Impact Phenomena ◽  
Time Marching

In the present paper, a Harmonic Balance Method (HBM) coupled with a pseudo-arc length continuation algorithm is presented for the prediction of the steady state behaviour of a rotor-stator contact problem. The ability of the HBM to reproduce the four most common phenomena encountered during rotor to stator contact situations (i.e. ‘no-rub’, ‘full annular rub’, ‘partial rub’ and ‘backward whirl/whip’) is investigated. A modified Jeffcott rotor model is used and results of the proposed algorithm are compared with traditional time marching solutions and analytical predictions. The advantages and limitations of the HBM for this kind of problem are analyzed. It is shown that the HBM is orders of magnitude faster than transient simulations, and provides very accurate results. However, in its current form it is unable to predict quasi-periodic behaviour. Detailed analysis of the transient solutions yields valuable information for the future extension of the HBM to efficient quasi-periodic simulations.

Period Motions and Limit Cycle in a Periodically Forced, Plunged Galloping Oscillator

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Limit Cycle ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Hopf Bifurcations ◽  
Periodic Motions ◽  
Stability And Bifurcations ◽  
Generalized Harmonic Balance Method

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

Harmonic Balance Vibration Analysis of Turbine Blades With Friction Dampers

1997 ◽  
Vol 119 (1) ◽  
pp. 96-103 ◽  
Author(s):  
K. Y. Sanliturk ◽  
M. Imregun ◽  
D. J. Ewins
Keyword(s):  
Harmonic Balance ◽  
Turbine Blades ◽  
Numerical Test ◽  
Harmonic Balance Method ◽  
Nonlinear Behavior ◽  
Friction Damper ◽  
Friction Dampers ◽  
Equivalent Time ◽  
Time Marching ◽  
Detailed Assessment

Although considerable effort has been devoted to the formulation of predictive models of friction damper behavior in turbomachinery applications, especially for turbine blades, the problem is far from being solved due to the complex nonlinear behavior of the contact surfaces. This paper primarily focuses on analytical and numerical aspects of the problem and addresses the problem in the frequency domain while exploring the viability of equivalent time-domain alternatives. The distinct features of this work are: (i) the modelling of nonlinear friction damper behavior as an equivalent amplitude-dependent complex stiffness via a first-order harmonic balance method (HBM), (ii) the use of sine sweep excitation in time-marching analysis, (iii) the application of the methodology to numerical test cases, including an idealised 3D turbine blade model with several friction dampers, (iv) the verification of the numerical findings using experimental data, and (v) a detailed assessment of the suitability of HBM for the analysis of structures with friction dampers.

RATIONAL ENERGY BALANCE METHOD TO NONLINEAR OSCILLATORS WITH CUBIC TERM

2013 ◽  
Vol 06 (02) ◽  
pp. 1350019 ◽  
Author(s):  
M. Daeichin ◽  
M. A. Ahmadpoor ◽  
H. Askari ◽  
A. Yildirim
Keyword(s):  
Energy Balance ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Problems ◽  
Nonlinear Oscillators ◽  
Initial Guess ◽  
Second Order ◽  
Balance Method ◽  
Strong Nonlinearity ◽  
Novel Approach

In this paper, a novel approach is proposed for solving the nonlinear problems based on the collocation and energy balance methods (EBMs). Rational approximation is employed as an initial guess and then it is combined with EBM and collocation method for solving nonlinear oscillators with cubic term. Obtained frequency amplitude relationship is compared with exact numerical solution and subsequently, a very excellent accuracy will be revealed. According to the numerical comparisons, this method provides high accuracy with 0.03% relative error for Duffing equation with strong nonlinearity in the second-order of approximation. Furthermore, achieved results are compared with other types of modified EBMs and the second-order of harmonic balance method. It is demonstrated that the new proposed method has the highest accuracy in comparison with different approaches such as modified EBMs and the second-order of harmonic balance method.

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