Extraneous solutions predicted by the harmonic balance method

Hassan, A.; Burton, Thomas de

doi:10.1006/jsvi.1995.0214

Cited by 8 publications

(29 citation statements)

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“…For example, to analyze a resonance response, the non-linear terms, the 51 forcing and, the usually assumed linear, damping terms are multiplied by the small gauge parameter e so that these terms appear at the same time (same order) in the perturbation scheme (equations) [2]. Thus the range of system parameters and response amplitude over which the predicted perturbation solution is satisfactory is ®xed in advance by the ordering scheme; however this range is usually left unspeci®ed [6].…”

Section: A a Al-qaisia And M N Hamdanmentioning

confidence: 99%

“…Furthermore, the amplitudes of different harmonics of the predicted approximate periodic response are assumed to satisfy the established ordering scheme which determines in advance the relative importance of each of these harmonics and assumes the rapid attenuation of higher ones for the weakly nonlinear system [6]. Furthermore, the essence of the MMS perturbation method is to seek asymptotically valid, usually low order, approximations to the steady state periodic response by using a number of time scales and power series expansions for the dependent variables and parameters of the assumed weakly non-linear system in terms of a small positive gauge parameter e. These series expansions are neither unique nor convergent, and several procedural steps have been devised by various authors in order to obtain consistently ordered (asymptotically valid) ®rst and higher order MMS results.…”

Section: A a Al-qaisia And M N Hamdanmentioning

confidence: 99%

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On the Steady State Response of Oscillators With Static and Inertia Non-Linearities

Al-Qaisia

Hamdan

1999

Journal of Sound and Vibration

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Section: A a Al-qaisia And M N Hamdanmentioning

confidence: 99%

Section: A a Al-qaisia And M N Hamdanmentioning

confidence: 99%

On the Steady State Response of Oscillators With Static and Inertia Non-Linearities

Al-Qaisia

Hamdan

1999

Journal of Sound and Vibration

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“…These solutions are well known, the case having been studied in detail in [14] and [29]. Since the non-linearity is odd, the spectrum of the simplest harmonic stationary solution contains only the odd harmonics almost all over the range of parameter ω due to symmetry (3.2).…”

Section: The Symmetry Of Solutions To the Equation Of Motion With Evementioning

confidence: 99%

“…For this purpose, the harmonic balance method [3], which is widely used to study strongly non-linear systems [11][12][13][14][15][16], seems to be most effective (it is also often called by Gallerkin's method [17]). The main advantage of the harmonic balance method over time-difference methods is that it allows obtaining both stable and unstable stationary solutions [12].…”

Section: Introductionmentioning

confidence: 99%

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Cascades of subharmonic stationary states in strongly non-linear driven planar systems

Lukomsky¹,

Gandzha²

2004

Journal of Sound and Vibration

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The dynamics of a one-degree of freedom oscillator with arbitrary polynomial non-linearity subjected to an external periodic excitation is studied. The sequences (cascades) of harmonic and subharmonic stationary solutions to the equation of motion are obtained by using the harmonic balance approximation adapted for arbitrary truncation numbers, powers of non-linearity, and orders of subharmonics. A scheme for investigating the stability of the harmonic balance stationary solutions of such a general form is developed on the basis of the Floquet theorem. Besides establishing the stable/unstable nature of a stationary solution, its stability analysis allows obtaining the regions of parameters, where symmetry-breaking and period-doubling bifurcations occur. Thus, for perioddoubling cascades, each unstable stationary solution is used as a base solution for finding a subsequent stationary state in a cascade. The procedure is repeated until this stationary state becomes stable provided that a stable solution can finally be achieved. The proposed technique is applied to calculate the sequences of subharmonic stationary states in driven hardening Duffing's oscillator. The existence of stable subharmonic motions found is confirmed by solving the differential equation of motion numerically by means of a time-difference method, with initial conditions being supplied by the harmonic balance approximation.

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On the local stability analysis of the approximate harmonic balance solutions

Hassan

1996

Nonlinear Dyn

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Periodic response of nonlinear oscillators is usually determined by approximate methods. In the "steady state" type methods, first an approximate solution for the steady state periodic response is determined, and then the local stability of this solution is determined by analyzing the equation of motion linearized about this predicted "solution". An exact stability analysis of this linear variational equation can provide erroneous stability type information about the approximate solutions. It is shown that a consistent stability type information about these solutions can be obtained only when the linearized variational equation is analyzed by approximate methods, and the level of accuracy of this analysis is consistent with that of the approximate solutions. It is demonstrated that these consistent stability results do not imply that the approximate solution is qualitatively correct. It is also shown that the difference between an approximate and the next higher order stability analysis can be used to "guess" the role of higher harmonics in the periodic response. This trial and error procedure can be used to ensure the qualitatively correct and numerically accurate nature of the approximate solutions and the corresponding stability analysis.

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Extraneous solutions predicted by the harmonic balance method

Cited by 8 publications

References 0 publications

On the Steady State Response of Oscillators With Static and Inertia Non-Linearities

On the Steady State Response of Oscillators With Static and Inertia Non-Linearities

Cascades of subharmonic stationary states in strongly non-linear driven planar systems

On the local stability analysis of the approximate harmonic balance solutions

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