A split-frequency harmonic balance method for nonlinear oscillators with multi-harmonic forcing

2006 ◽  
Vol 295 (3-5) ◽  
pp. 939-963 ◽  
Author(s):  
J.F. Dunne ◽  
P. Hayward
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Balance Method ◽  
Harmonic Forcing ◽  
Split Frequency ◽  
Frequency Harmonic

Subharmonic-Response Computation and Stability Analysis for a Nonlinear Oscillator Using a Split-Frequency Harmonic Balance Method

2006 ◽  
Vol 1 (3) ◽  
pp. 221-229 ◽  
Author(s):  
J. F. Dunne
Keyword(s):  
Stability Analysis ◽  
High Frequency ◽  
Harmonic Balance ◽  
Low Frequency ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Equation Error ◽  
Subharmonic Response ◽  
Split Frequency ◽  
Frequency Harmonic

A split-frequency harmonic balance method (SF-HBM) is developed to obtain subharmonic responses of a nonlinear single-degree-of-freedom oscillator driven by periodic excitation. This method is capable of generating highly accurate periodic solutions involving a large number of solution harmonics. Responses at the excitation period, or corresponding multiples (such as period 2 and period 3), can be readily obtained with this method, either in isolation or as combinations. To achieve this, the oscillator equation error is first expressed in terms of two Mickens functions, where the assumed Fourier series solution is split into two groups, nominally associated with low-frequency or high-frequency harmonics. The number of low-frequency harmonics remains small compared to the number of high-frequency harmonics. By exploiting a convergence property of the equation-error functions, accurate low-frequency harmonics can be obtained in a new iterative scheme using a conventional harmonic balance method, in a separate step from obtaining the high-frequency harmonics. The algebraic equations (needed in the HBM part of the method) are generated wholly numerically via a fast Fourier transform, using a discrete-time formulation to include inexpansible nonlinearities. A nonlinear forced-response stability analysis is adapted for use with solutions obtained with this SF-HBM. Period-3 subharmonic responses are obtained for an oscillator with power-law nonlinear stiffness. The paper shows that for this type of oscillator, two qualitatively different period-3 subharmonic response branches can be obtained across a broad frequency range. Stability analysis reveals, however, that for an increasingly stiff model, neither of these subharmonic branches are stable.

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.

scholarly journals Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

2008 ◽  
Vol 372 (39) ◽  
pp. 6047-6052 ◽  
Author(s):  
A. Beléndez ◽  
E. Gimeno ◽  
M.L. Álvarez ◽  
D.I. Méndez ◽  
A. Hernández
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Balance Method ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

scholarly journals Continuation of quasi-periodic solutions with two-frequency Harmonic Balance Method

2017 ◽  
Vol 394 ◽  
pp. 434-450 ◽  
Author(s):  
Louis Guillot ◽  
Pierre Vigué ◽  
Christophe Vergez ◽  
Bruno Cochelin
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Frequency Harmonic
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