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Introduction To Perturbation Techniques

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 discretised with the finite element method is addressed. Implicit condensation, giving rise to a projection onto a stress manifold, and modal derivatives, used in the framework of the quadratic manifold, are first reviewed. Finally, recent developments allowing direct computation of reduced-order models relying on invariant manifolds theory are detailed. Applicative examples are shown and the extension of the methods to deal with further complications are reviewed. Finally, open problems and future directions are highlighted.

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

Dimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in operating directly from the physical space, hence avoiding the computation of the complete eigenfunctions spectrum. Explicit solutions are given, thus enabling a fully non-intrusive version of the reduction method. The reduced dynamics is obtained from the normal form of the geometrically nonlinear mechanical problem, free of non-resonant monomials, and truncated to the selected master coordinates, thus making a direct link with the parametrisation of invariant manifolds. The method is fully expressed with a complex-valued formalism by detailing the homological equations in a systematic manner, and the link with real-valued expressions is established. A special emphasis is put on the treatment of second-order internal resonances and the specific case of a 1:2 resonance is made explicit. Finally, applications to large-scale models of micro-electro-mechanical structures featuring 1:2 and 1:3 resonances are reported, along with considerations on computational efficiency.

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Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance

This article critically analyzes the trends and limits of integrated technologies for sensing, computing, and data storage when devices and systems originally developed for consumer electronics (CE) are used for self-driving cars. Some hints, supported by theoretical analysis and experimental work, are provided to overcome the issues of inertial sensors, micromirrors for lidar scanners, car data/program memory, and computing platforms.

/pdf/is-consumer-electronics-redesigning-our-cars-challenges-of-43ftau318w.pdf

Is Consumer Electronics Redesigning Our Cars?: Challenges of Integrated Technologies for Sensing, Computing, and Storage

In this work, we address the modelling of piezo-micromirrors with large opening angles. We focus on various sources of nonlinearites induced by the interaction with the surrounding fluid and by occurrence of geometric large transformations. Specific attention is also devoted to the piezoelectric coupling coefficient. Piezoelectric thin films have gained attention as key materials for the actuation of micro devices since they provide high drive forces, enabling devices with higher performances compared to electrostatic actuation. However they also induce material nonlinearities that cannot be neglected. The model is validated using experimental data obtained for various excitation levels. [2020-0229]

Nonlinear Response of PZT-Actuated Resonant Micromirrors

We consider an electrostatically actuated torsional micromirror, a key element of recent optical microdevices. The mechanical response is analyzed with specific emphasis on its nonlinear features. We show that the mirror motion is an example of parametric resonance, activated when the drive frequency is twice the natural frequency of the system. The numerical model, solved with a continuation approach, is validated with very good accuracy through an extensive experimental campaign.

/pdf/parametric-resonance-in-electrostatically-actuated-9wx0q5sgv7.pdf

Parametric Resonance in Electrostatically Actuated Micromirrors

In this work, we address the simulation and testing of MEMS micromirrors with hardening and softening behaviour excited with patches of piezoelectric materials. The forces exerted by the piezoelectric patches are modelled by means of the theory of ferroelectrics developed by Landau–Devonshire and are based on the experimentally measured polarisation hysteresis loops. The large rotations experienced by the mirrors also induce geometrical nonlinearities in the formulation up to cubic order. The solution of the proposed model is performed by discretising the device geometry using the Finite Element Method, and the resulting large system of coupled differential equations is solved by means of the Harmonic Balance Method. Numerical results were validated with experimental data collected on the devices.

https://www.mdpi.com/2076-0825/10/2/21/pdf

Analysis of the Nonlinear Response of Piezo-Micromirrors with the Harmonic Balance Method

The main torsional mode of electrostatically actuated micromirrors is known to be dominated by parametric resonance when the actuation is performed via in-plane comb fingers. Here, we show that, for specific geometrical features of the mirror, parametric resonance simultaneously activates a spurious yaw mode. Due to the large torsional rotations, the two modes are nonlinearly coupled, inducing mutual stiffness variations and an unexpected temperature dependence of the main mode. After presenting an experimental evidence of the coupling, we develop and discuss a numerical model capable of capturing the key phenomena and of providing guidelines for a robust design.

Mode Coupling and Parametric Resonance in Electrostatically Actuated Micromirrors

This work presents a long-term characterization of a newly designed microelectromechanical-system (MEMS) micromirror capable of achieving a field-of-view (FOV) of almost 94°. The process used to fabricate the device allows to combine piezoelectric (PZT) actuation and piezoresistive (PZR) sensing of the tilt angle, enabling testing under closed loop control. Consistently with the adopted design guidelines, which considered fatigue limits on the silicon springs, the tested devices are able to withstand up to 3.6 billion continuous operating cycles in an uncontrolled laboratory environment when operated in resonant mode at 500 Hz. [2020-0366]

Roberto Carminati

Papers

Parametric Resonance in Electrostatically Actuated Micromirrors

Nonlinear Response of PZT-Actuated Resonant Micromirrors

Analysis of the Nonlinear Response of Piezo-Micromirrors with the Harmonic Balance Method

Mode Coupling and Parametric Resonance in Electrostatically Actuated Micromirrors

Long-Term Characterization of a New Wide-Angle Micromirror With PZT Actuation and PZR Sensing