Subharmonic-response computation and stability analysis for a nonlinear oscillator using a split-frequency harmonic balance method

Dunne, Julian F (2006) Subharmonic-response computation and stability analysis for a nonlinear oscillator using a split-frequency harmonic balance method. Journal of Computational and Nonlinear Dynamics, 1 (3). pp. 221-228. ISSN 1555-1415

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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.

Item Type: Article
Additional Information: Stability analysis of nonlinear systems such as turbine rotors, requires very accurate solutions, especially for sub-harmonic responses. This new method accurately generates period-3 solutions in the frequency domain. The existence of multiple solutions is important, where stability analysis can assess whether they are sustainable. Using this method, an example is shown how varying the strength of nonlinearity can cause an otherwise stable period-3 solution to become unstable (such as increasing nonlinearity of a jet engine rotor bearing stiffness). The method is being explored by CDH-AG. Contact: Dr D. F. Bella, Technical Manager, Special Applications Department, CDH-AG. Email:
Schools and Departments: School of Engineering and Informatics > Engineering and Design
Depositing User: Julian Dunne
Date Deposited: 06 Feb 2012 20:57
Last Modified: 11 Jul 2012 12:48
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