Identification of nonlinear modes using phase-locked-loop experimental continuation and normal form

Denis, Vivien; Jossic, Marguerite; Giraud-Audine, Christophe; Chomette, Baptiste; Renault, Alexandre; Thomas, Olivier

doi:10.1016/j.ymssp.2018.01.014

Cited by 72 publications

(98 citation statements)

References 62 publications

(123 reference statements)

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“…Although similar results can be obtained using normal form theory, see e.g. Denis et al [11], a clear advantage of the LSM reduction method is that it requires no nonlinear change of coordinates. As a consequence, the reduced model (17) keeps the original modeling coordinate x as a physically meaningful modal coordinate.…”

Section: Explicit Form Of the Lsm-reduced Model When The Non-modelingsupporting

confidence: 69%

“…Consequently, for a conservative system with symmetric cubic nonlinearities, a third-order reduced model can immediately be sought as a Duffing-oscillator. For such systems, therefore, the numerical and experimental procedure of Olivier et al [11] is justified when they define an equivalent Duffing-oscillator for reduced models obtained from a normal form analysis.…”

Section: Explicit Form Of the Lsm-reduced Model When The Non-modelingmentioning

confidence: 99%

“…A more recent application of the method of normal forms for model-order reduction purposes can be found in the work of Denis et al [11]. They observe that, in the absence of internal resonances, a reduced-order model for each mode can be approximated by a Duffing oscillator.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Explicit third-order model reduction formulas for general nonlinear mechanical systems

Veraszto

Ponsioen

Haller

2020

Journal of Sound and Vibration

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For general nonlinear mechanical systems, we derive closed-form, reduced-order models up to cubic order based on rigorous invariant manifold results. For conservative systems, the reduction is based on Lyapunov Subcenter Manifold (LSM) theory, whereas for damped-forced systems, we use Spectral Submanifold (SSM) theory. To evaluate our explicit formulas for the reduced model, no coordinate changes are required beyond an initial linear one. The reduced-order models we derive are simple and depend only on physical and modal parameters, allowing us to extract fundamental characteristics, such as backbone curves and forcedresponse curves, of multi-degree-of-freedom mechanical systems. To numerically verify the accuracy of the reduced models, we test the reduction formulas on several mechanical systems, including a higher-dimensional nonlinear Timoshenko beam.

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Section: Explicit Form Of the Lsm-reduced Model When The Non-modelingsupporting

confidence: 69%

Section: Explicit Form Of the Lsm-reduced Model When The Non-modelingmentioning

confidence: 99%

See 1 more Smart Citation

Explicit third-order model reduction formulas for general nonlinear mechanical systems

Veraszto

Ponsioen

Haller

2020

Journal of Sound and Vibration

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“…In the second continuation technique, which is used in this study, the tracking of the backbone is carried out in forced regime by setting the system at phase resonance using a phase-locked-loop (PLL) controller. [22][23][24][25] The forcing amplitude is set incrementally step by step and the forcing frequency is adjusted by the PLL in order to achieve the nonlinear phase resonance. The main advantage of continuation techniques is to directly obtain the backbone curve from forced regime rather than rely on the free vibration frequency-time analysis, which is performed with the NPR method.…”

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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“…Following the same general principles as CBC, phase-locked loops were exploited to trace the nonlinear frequency response of a nonlinear oscillator in [27]. A similar method was used to measure the backbone curves of a beam with nonlinear boundary conditions [28] and of a circular plate and a Chinese gong [29].The experimental demonstration of CBC remains largely limited to systems whose dynamics can be approximated by SDOF oscillators. The objective of this paper is to demonstrate ex-…”

mentioning

confidence: 99%

Application of control-based continuation to a nonlinear structure with harmonically coupled modes

Renson

Shaw

Barton

et al. 2019

Mechanical Systems and Signal Processing

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This paper presents a systematic method for exploring the nonlinear dynamics of multi-degree-of-freedom (MDOF) physical experiments. To illustrate the power of this method, known as control-based continuation (CBC), it is applied to a nonlinear beam structure that exhibits a strong 3:1 modal coupling between its first two bending modes. CBC is able to extract a range of dynamical features, including an isola, directly from the experiment without recourse to model fitting or other indirect data-processing methods.Previously, CBC has only been applied to (essentially) single-degree-of-freedom experiments; in this paper we show that the required feedback-control methods and pathfollowing techniques can equally be applied to MDOF systems. A low-level broadband excitation is initially applied to the experiment to obtain the requisite information for controller design and, subsequently, the physical experiment is treated as a black box that is probed using CBC. The invasiveness of the controller used is analysed and experimental results are validated with open-loop measurements. Good agreement between open-and closed-loop results is achieved, though it is found that care needs to be taken in dealing with the presence of higher-harmonics in the force applied to the structure. when compared with linear systems which can be characterised very efficiently with broadband modal analysis techniques. The presence, in nonlinear systems, of multiple coexisting steadystate responses to identical inputs requires repeated tests to capture the possible behaviours. Determining, precisely and with confidence, the regions where these multiple responses exist further multiplies the testing effort as experimental errors such as noise can lead to untimely transitions between these responses even before the region boundaries are reached.Nonlinear systems can also exhibit a wide variety of other complex phenomena not seen in linear systems such as modal interactions [5], quasi-periodic oscillations [6] and isola [7]. The latter correspond to branches of stable responses that are isolated from the main peak of the nonlinear frequency response of the system. Isola can arise through a number of mechanisms such as sub-and super-harmonic resonances on simple single-degree-of-freedom (SDOF) nonlinear systems [6,8,9] as well as through modal interactions. As such, isola have been observed across a wide range of nonlinear mechanical systems, for instance, suspended cables [10], pedestrian bridges [11], vibration absorbers [12,13] and a satellite structure [14]. The reader is referred to [15] for a more detailed review on this topic.The aforementioned challenges associated with nonlinear MDOF structures mean that a robust systematic method for experimental testing is highly desirable. This paper explores the use of control-based continuation (CBC) for this purpose. CBC exploits feedback control to provide a number of advantages compared with classical open-loop approaches to experimental testing. The controller maintains the experiment around a pres...

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Identification of nonlinear modes using phase-locked-loop experimental continuation and normal form

Cited by 72 publications

References 62 publications

Explicit third-order model reduction formulas for general nonlinear mechanical systems

Explicit third-order model reduction formulas for general nonlinear mechanical systems

Effects of internal resonances in the pitch glide of Chinese gongs

Application of control-based continuation to a nonlinear structure with harmonically coupled modes

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