Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

Monteil, Mélodie; Touzé, Cyril; Thomas, Olivier; Benacchio, Simon

doi:10.1007/s11071-013-1057-7

Cited by 22 publications

(26 citation statements)

References 35 publications

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“…Thus the Hopf bifurcation for the modulated solutions indicates the occurrence of quasi-periodic solutions for the original system, which can result in important physical consequences [54] , such as a behaviour from transition to chaos. The saddle-node bifurcation is another bifurcation type analyzed in this paper, which indicates the stability changing of the responses, resulting in the occurrences for the jump phenomenon and multiple solutions.…”

Section: Numerical Resultsmentioning

confidence: 99%

Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

Chen

Wang

Wiercigroch

et al. 2016

Appl. Math. Mech.-Engl. Ed.

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A reduced model simulating vortex-induced vibrations (VIVs) for turbine blades is proposed and analyzed. The rotating blade is modeled as a uniform cantilever beam while a van der Pol oscillator is used to represent the time-varying characteristics of the vortex shedding, which interacts with equations of motion for a blade to simulate the fluid-structure interaction. The action for the structural motion on the fluid is considered as a linear inertia coupling. Nonlinear characteristics for the dynamic responses are investigated with the multiple scale method and the modulation equations are derived. The transition set consisting of the bifurcation and hysteresis sets is constructed by using the singularity theory and the effects of system parameters including the van der Pol damping and the coupling parameter on the equilibrium solutions are analyzed. Frequency-response curves are obtained and the stabilities are determined by using the Routh-Hurwitz criterion. The phenomena including the saddle-node and Hopf bifurcations are found to occur under certain parameter values. A direct numerical method is used to analyze the dynamic characteristics for the original system and verify the validity of the multiple scale method. The results indicate that the new coupled model can be useful to explain the rich dynamic response characteristics including possible bifurcation phenomena in the VIVs.

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Section: Numerical Resultsmentioning

confidence: 99%

Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

Chen

Wang

Wiercigroch

et al. 2016

Appl. Math. Mech.-Engl. Ed.

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“…As the steelpan's shell is particularly thin (of the order of 0.5 mm in the center of the notes), geometrically nonlinear vibrations are excited in normal conditions of playing. easily fulfilled via quadratic nonlinear terms of the model [20,[25][26][27]. Hence energy exchanges between modes will be more easily activated, favoring the transfer of energy to higher frequencies.…”

Section: Modal Analysis Of the Steelpanmentioning

confidence: 99%

“…The measurements obviously reveal the complexity of the vibrational patterns and the number of excited modes, even for very small levels of vibration amplitudes. Simple models displaying 1:2:2 and 1:2:4 internal resonance, available in [20], are then used in order to fit some experimental frequency response curves. Finally, time-domain simulations of the models identified from forced vibrations, are used to compare the time response of the first four harmonics of the signal of an impacted note, showing a perfect agreement.…”

Section: Introductionmentioning

confidence: 99%

Identification of mode couplings in nonlinear vibrations of the steelpan

2015

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The authors are grateful to Bertrand David (Telecom-ParisTech) for computing the code allowing the STFT filtering procedure used in Section 5.1. The filter has been designed in the framework of the PAFI project (Plateforme d’Aide la facture Instrumentale, www.pafi.fr) which is also thanked.The vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics.International audienceThe vibrations and sounds produced by two notes of a double second steelpan are investigated, the main objective being to quantify the nonlinear energy exchanges occurring between vibration modes that are responsible of the peculiar sound of the instrument. A modal analysis first reveals the particular tuning of the modes and the systematic occurence of degenerate modes, from the second one, this feature being a consequence of the tuning and the mode localization. Forced vibrations experiments are then performed to follow precisely the energy exchange between harmonics of the vibration and thus quantify properly the mode couplings. In particular, it is found that energy exchanges are numerous, resulting in complicated frequency response curves even for very small levels of vibration amplitude. Simple models displaying1:2:2 and 1:2:4 internal resonance are then fitted to the measurements, allowing to identify the values of the nonlinear quadratic coupling coefficients resulting from the geometric nonlinearity. The identified 1:2:4 model is finally used to recover the time domain variations of an impacted note in normal playing condition, resulting in an excellent agreement for the temporal behaviour of the first four harmonics

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“…This small interest is all the more surprising since, as it will be shown, a single Duffing oscillator is sufficient to describe the frequencyamplitude dependence; in contrast, many degree-of-freedom models are required to describe internal resonances and/or chaos phenomena, which have received considerable interest in the case of gongs and cymbals [7][8][9][10] or the steelpan. 11,12 Many previous studies on plates and shells (see, e.g., Refs. 1, 2, and 13) have highlighted that geometrical nonlinearities lead to quadratically and/or cubically coupled modal equations; conversely, the linear case is characterized by uncoupled modal equations.…”

Section: Introductionmentioning

confidence: 99%

Effects of internal resonances in the pitch glide of Chinese gongs

Jossic

Thomas

Denis

et al. 2018

The Journal of the Acoustical Society of America

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The framework of nonlinear normal modes gives a remarkable insight into the dynamics of nonlinear vibratory systems exhibiting distributed nonlinearities. In the case of Chinese opera gongs, geometrical nonlinearities lead to a pitch glide of several vibration modes in playing situation. This study investigates the relationship between the nonlinear normal modes formalism and the ascendant pitch glide of the fundamental mode of a xiaoluo gong. In particular, the limits of a single nonlinear mode modeling for describing the pitch glide in playing situation are examined. For this purpose, the amplitude-frequency relationship (backbone curve) and the frequency-time dependency (pitch glide) of the fundamental nonlinear mode is measured with two excitation types, in free vibration regime: first, only the fundamental nonlinear mode is excited by an experimental appropriation method resorting to a phase-locked loop; second, all the nonlinear modes of the instrument are excited with a mallet impact (playing situation). The results show that a single nonlinear mode modeling fails at describing the pitch glide of the instrument when played because of the presence of 1:2 internal resonances implying the nonlinear fundamental mode and other nonlinear modes. Simulations of two nonlinear modes in 1:2 internal resonance confirm qualitatively the experimental results.

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Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances

Cited by 22 publications

References 35 publications

Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

Bifurcation and dynamic response analysis of rotating blade excited by upstream vortices

Identification of mode couplings in nonlinear vibrations of the steelpan

Effects of internal resonances in the pitch glide of Chinese gongs

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