Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Gobat, Giorgio; Guillot, Louis; Frangi, Attilio; Cochelin, Bruno; Touzé, Cyril

doi:10.1007/s11012-021-01351-1

Cited by 37 publications

(40 citation statements)

References 64 publications

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“…Important consequences are in the field of identification methods, where minimal nonlinear models can be used reliably, see, e.g. circular plates with 1:1 internal resonances [68,70,272], shallow shells with 1:1:2 and 1:2:2:4 internal resonances [106,184,274], MEMS structure with 1:2 and 1:3 resonance [42,71], and the identification of the hardening/softening behaviour of particular modes of a structure [45].…”

Section: Applicationsmentioning

confidence: 99%

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures dis-C. Touzé (B)

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Section: Applicationsmentioning

confidence: 99%

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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“…The low-order internal resonances are the most well known and have given rise to a vast literature investigating their solutions, see e.g. [64,65,66,67,68] to cite only a few entries. Second-order internal resonances are related to quadratic nonlinearities, and give rise to the 1:2 case where ω j 2ω p , as well as combination of resonance such as ω p ω k ± ω l .…”

Section: Resonancesmentioning

confidence: 99%

High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

Vizzaccaro¹,

Opreni²,

Salles³

et al. 2021

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This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems expressed in the physical basis, so that the technique is directly applicable to problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion. The same development is performed on the reduced-order dynamics which is computed at generic order following the different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clampedclamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that the invariant manifold of the first mode shows a folding point at large amplitudes which is not connected to an internal resonance. This exemplifies the failure of the graph style due to the folding point, whereas the normal form style is able to pass over the folding. Finally, A MEMS micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example.

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“…Internal resonances (IRs) are associated to energy transfer between modes, and are frequently experienced by MEMS structures mainly due to the very low dissipation. Often IRs are strongly linked to the stability of the associated periodic response, and quasi-periodic regimes might arise as a consequence of Neimark-Sacker (NS) bifurcations [61]. The numerical prediction of such phenomena has been tackled only recently with ad-hoc approaches like the Implicit Condensation [6] or the DPIM [62].…”

Section: Internal Resonance In a Shallow Archmentioning

confidence: 99%

“…satisfying the necessary condition to induce a 1:2 internal resonance, and leads to qualitative and quantitative changes in the dynamics. An in-depth analysis has been developed in Gobat et al [61], where 1:2 IR systems are analysed starting their normal form and the existence of the so-called parabolic modes between the coupled oscillators is demonstrated. Such modes exist on the two branches of the FRF (see Figure 11) and are associated to the two backbones that onset from ω 1 and ω 6 /2, respectively.…”

Section: Internal Resonance In a Shallow Archmentioning

confidence: 99%

Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches

Gobat¹,

Fresca²,

Manzoni³

et al. 2022

Preprint

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Micro-Electro-Mechanical-Systems are complex structures, often involving nonlinearites of geometric and multiphysics nature, that are used as sensors and actuators in countless applications. Starting from fullorder representations, we apply deep learning techniques to generate accurate, efficient and real-time reduced order models to be used as virtual twin for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches and gyroscopes, also displaying intricate dynamical evolutions like internal resonances. In particular, we discuss the accuracy of the deep learning technique and its ability to replicate and converge to the invariant manifolds predicted using the recently developed direct parametrization approach that allows extracting the nonlinear normal modes of large finite element models. Finally, by addressing an electromechanical gyroscope, we show that the non-intrusive deep learning approach generalizes easily to complex multiphysics problems.

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Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS

Cited by 37 publications

References 64 publications

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point

Virtual twins of nonlinear vibrating multiphysics microstructures: physics-based versus deep learning-based approaches

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