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Journal of Computational and Nonlinear Dynamics | Volume 1 | Issue 3 | Research Paper
RESEARCH PAPERS

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

[+] Author and Article Information
J. F. Dunne

Department of Engineering and Design, School of Science and Technology,  The University of Sussex, Falmer, Brighton, BN1 9QT, UKj.f.dunne@sussex.ac.uk

J. Comput. Nonlinear Dynam 1(3), 221-229 (Feb 20, 2006) (9 pages) doi:10.1115/1.2198875 History: Received September 05, 2005; Revised February 20, 2006
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Citing Articles

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.
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Copyright © 2006 by American Society of Mechanical Engineers
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References

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Figures

Grahic Jump Location
+(a) Phase-plane trajectory for the stable subharmonic response of the Duffing oscillator (m=3); (b) equation error rms as a function of iteration number (shown as •) with four iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ●); (c) absolute magnitudes of the total 64 response harmonics (low- plus high-frequency); and (d) complex eigenvalues associated with the (stable) subharmonic response
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(a) Phase-plane trajectory for the stable subharmonic response of the Duffing oscillator (m=3); (b) equation error rms as a function of iteration number (shown as •) with four iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ●); (c) absolute magnitudes of the total 64 response harmonics (low- plus high-frequency); and (d) complex eigenvalues associated with the (stable) subharmonic response
Figure 1

(a) Phase-plane trajectory for the stable subharmonic response of the Duffing oscillator (m=3); (b) equation error rms as a function of iteration number (shown as •) with four iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ●); (c) absolute magnitudes of the total 64 response harmonics (low- plus high-frequency); and (d) complex eigenvalues associated with the (stable) subharmonic response

Grahic Jump Location
+(a) Phase-plane trajectory for the unstable subharmonic response of the Duffing oscillator; (b) equation error rms as a function of iteration number (shown as • symbol) with four iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ●); (c) absolute magnitudes of the total 64 response harmonics (low- plus high-frequency); and (d) complex eigenvalues associated with the (unstable) subharmonic
View Large  |  Save Figure  |  Download Slide (.ppt)
(a) Phase-plane trajectory for the unstable subharmonic response of the Duffing oscillator; (b) equation error rms as a function of iteration number (shown as • symbol) with four iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ●); (c) absolute magnitudes of the total 64 response harmonics (low- plus high-frequency); and (d) complex eigenvalues associated with the (unstable) subharmonic
Figure 2

(a) Phase-plane trajectory for the unstable subharmonic response of the Duffing oscillator; (b) equation error rms as a function of iteration number (shown as • symbol) with four iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ●); (c) absolute magnitudes of the total 64 response harmonics (low- plus high-frequency); and (d) complex eigenvalues associated with the (unstable) subharmonic

Grahic Jump Location
+Contour plot of the initial-final-state-error surface for the Duffing oscillator for Ω∕ωn=5 computed via time-domain integration showing initial conditions for periodic solutions obtained with a systematic grid search (highlighted with an asterisk)
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Contour plot of the initial-final-state-error surface for the Duffing oscillator for Ω∕ωn=5 computed via time-domain integration showing initial conditions for periodic solutions obtained with a systematic grid search (highlighted with an asterisk)
Figure 3

Contour plot of the initial-final-state-error surface for the Duffing oscillator for Ω∕ωn=5 computed via time-domain integration showing initial conditions for periodic solutions obtained with a systematic grid search (highlighted with an asterisk)

Grahic Jump Location
+Phase-plane trajectories obtained by time domain integration for the Duffing oscillator with Ω∕ωn=5 associated with the initial conditions found in Fig. 3
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Phase-plane trajectories obtained by time domain integration for the Duffing oscillator with Ω∕ωn=5 associated with the initial conditions found in Fig. 3
Figure 4

Phase-plane trajectories obtained by time domain integration for the Duffing oscillator with Ω∕ωn=5 associated with the initial conditions found in Fig. 3

Grahic Jump Location
+Phase-plane trajectories obtained with the split-frequency HBM for the oscillator model with power-law stiffness for Ω∕ωn=5 with m=5, m=9, and m=13
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Phase-plane trajectories obtained with the split-frequency HBM for the oscillator model with power-law stiffness for Ω∕ωn=5 with m=5, m=9, and m=13
Figure 5

Phase-plane trajectories obtained with the split-frequency HBM for the oscillator model with power-law stiffness for Ω∕ωn=5 with m=5, m=9, and m=13

Grahic Jump Location
+Equation error rms as a function of iteration number (shown as •) with 4 iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ∘): (a) m=5, (b) m=9, (c) m=13; (d), (e), and (f), show corresponding absolute magnitudes of the total 64 response harmonics (low-plus high-frequency)
View Large  |  Save Figure  |  Download Slide (.ppt)
Equation error rms as a function of iteration number (shown as •) with 4 iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ∘): (a) m=5, (b) m=9, (c) m=13; (d), (e), and (f), show corresponding absolute magnitudes of the total 64 response harmonics (low-plus high-frequency)
Figure 6

Equation error rms as a function of iteration number (shown as •) with 4 iterations for each new low-frequency harmonic (total number=8), fourth iteration shown overlaid with ∘): (a) m=5, (b) m=9, (c) m=13; (d), (e), and (f), show corresponding absolute magnitudes of the total 64 response harmonics (low-plus high-frequency)

Grahic Jump Location
+Complex eigenvalues associated with the subharmonic response of the oscillator with power-law stiffness for Ω∕ωn=5 (a) m=7 and (b) m=9
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Complex eigenvalues associated with the subharmonic response of the oscillator with power-law stiffness for Ω∕ωn=5 (a) m=7 and (b) m=9
Figure 7

Complex eigenvalues associated with the subharmonic response of the oscillator with power-law stiffness for Ω∕ωn=5 (a) m=7 and (b) m=9

Grahic Jump Location
+Subharmonic frequency response amplitudes as a function of frequency ratio Ω∕ωn for an oscillator with power-law stiffness parameter m=7 showing different solution branches
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Subharmonic frequency response amplitudes as a function of frequency ratio Ω∕ωn for an oscillator with power-law stiffness parameter m=7 showing different solution branches
Figure 8

Subharmonic frequency response amplitudes as a function of frequency ratio Ω∕ωn for an oscillator with power-law stiffness parameter m=7 showing different solution branches

Grahic Jump Location
+Subharmonic frequency response amplitudes as a function of frequency ratio Ω∕ωn for an oscillator with power-law stiffness parameter m=11 showing different solution branches
View Large  |  Save Figure  |  Download Slide (.ppt)
Subharmonic frequency response amplitudes as a function of frequency ratio Ω∕ωn for an oscillator with power-law stiffness parameter m=11 showing different solution branches
Figure 9

Subharmonic frequency response amplitudes as a function of frequency ratio Ω∕ωn for an oscillator with power-law stiffness parameter m=11 showing different solution branches

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