On the Usage of Berger’s Model for Electrostatically Actuated Circular Curved Micro Plates

2017 ◽  
Author(s):  
Lior Medina ◽  
Rivka Gilat ◽  
Slava Krylov
Keyword(s):  
Flat Plate ◽  
Continuous Method ◽  
Structural Elements ◽  
Buckling Modes ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Significant Interest ◽  
Base Functions ◽  
Galerkin Models ◽  
Micro Plates

Micro- and nanolectromechanical systems (MEMS/NEMS) incorporating two-dimensional structural elements such as plates attracted significant interest in recent years. In this work, we explore implementation of a model based on Berger’s approximation, which significantly simplifies the formulation of a curved plate and describes it by a single governing equation. The solution of this equation is based on the Galerkin decomposition with buckling modes of an initially flat plate used as the base functions. To track the unstable branches of the equilibrium curve, a continuous method based on the Riks algorithm is implemented. The validation of the models is conducted for two loading cases, “mechanical” deflection-independent load, and electrostatic displacement-dependent load. In the case of an initially flat plate, results provided by the reduced order (RO) Galerkin models were compared to results available in the literature. In the case of a curved plate undergoing “mechanical” loading, results of a direct finite elements (FE) analysis, as well as of a finite differences (FD) analysis, were used as a reference. We show that the DOF Berger RO model can be conveniently used for analysis of plates with small curvature, as it provides satisfactory accuracy. Further more, a single DOF model can be used for the development of a bistability criterion.

Latching Criteria in Bistable Prestressed Curved Micro Beams

2015 ◽  
Author(s):  
Lior Medina ◽  
Rivka Gilat ◽  
Slava Krylov
Keyword(s):  
Response Curve ◽  
Transverse Force ◽  
Curved Beams ◽  
Buckling Modes ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Straight Beam ◽  
Base Functions ◽  
Symmetric Response ◽  
Good Agreement

Curved beams subjected to transverse force may exhibit a latching phenomena, namely remain in their buckled configuration under zero force such that an opposite force is required for their release. In this study, we investigate the latching in bistable electrostatically actuated prestressed curved beams. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition with buckling modes of a straight beam as base functions. Criteria for the existence of latching are derived in terms of the beam geometric parameters and the axial load. Two conditions are formulated: A necessary criterion establishes the appearance of latching on the symmetric response curve and a sufficient condition which assures the existence of latching in the presence of bifurcations. A comparison between the model results and those obtained by numerical analysis shows good agreement up to a certain elevation. It is noted that as the latching is not affected by the nonlinear electrostatic load, the obtained criteria stand for all types of loading.

Tristable Properties in Electrostatically Actuated Initially Curved Coupled Micro Beams

2021 ◽  
Author(s):  
Lior Medina ◽  
Ashwin A. Seshia
Keyword(s):  
Degrees Of Freedom ◽  
Geometrical Parameters ◽  
Curved Beams ◽  
Buckling Modes ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Critical Thresholds ◽  
All Solutions ◽  
Base Functions

Abstract A limit point behaviour analysis of a metastructure, composed of two double clamped, initially curved beams, coupled via a rigid truss at their respective centres, is carried out when subjected to a distributed electrostatic load. The analysis is based on a reduced order (RO) model resulting from Galerkin’s decomposition, with symmetric buckling modes taken as the base functions, for either beam. All solutions employed the implicit arc-length “Riks” method to accommodate for winding equilibrium paths, while validation of the said results were carried out against finite differences (FD) direct solutions. In addition, local stability analysis via the energy method, conducted on the primary beam was instrumental in clarifying the role of the various extremum points by characterising which branches are stable, and which are not. The combined analysis has shown that the driving beam, which directly encounters the load, is able to possess bistable as well as tristable properties, provided that the metastructure meets certain geometrical parameters. Several variations of tristability are disclosed in the study. The analysis indicates that a model with at least three degrees of freedom (DOF) is needed to predict such configurations, as well as the various critical thresholds, with reasonable errors of around one percent when compared against FD. In so doing, the model can be used to provide static characterisation of the structure.

Limit Points Behavior in Electrostatically Actuated Initially Curved Micro Plate

2016 ◽  
Author(s):  
Lior Medina ◽  
Rivka Gilat ◽  
Slava Krylov
Keyword(s):  
Degrees Of Freedom ◽  
Mechanical Load ◽  
Electrostatic Force ◽  
Equilibrium Path ◽  
Buckling Modes ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Base Functions ◽  
Three Degrees Of Freedom ◽  
Good Agreement

The axisymmetric snap-through of an initially curved circular micro plate, subjected to a transversal distributed electrostatic force is studied. The analysis is based on a reduced order (RO) model resulting from the Galerkin decomposition, with buckling modes of a flat plate used as the base functions. In order to check the validity of the RO model, the corresponding problem for a displacement-independent (“mechanical”) load is solved, and a comparison between the RO model and those obtained using finite elements (FE) analysis is carried out. It is shown, that the two are in good agreement, indicating that the RO model can be used for a plate undergoing electrostatic loading. However, the study shows that at least three degrees of freedom (DOF) are required for an accurate prediction of the equilibrium path and bistability. The coupled electromechanical analysis shows that due to the nonlinearity of the electrostatic load, the snap-through occurs at a lower displacement than in the case of the “mechanical” load. Moreover, the study concludes that actuation of plates of realistic dimensions can be achieved by reasonably low voltages.

The Effect of Membrane Load on the Usage of Berger’s Model in Electrostatically Actuated Pre-Stressed Circular Curved Micro Plates

2021 ◽  
Author(s):  
Lior Medina ◽  
Rami Eliasi ◽  
Rivka Gilat ◽  
Slava Krylov
Keyword(s):  
Finite Elements ◽  
Degrees Of Freedom ◽  
Mechanical Load ◽  
Fe Model ◽  
Buckling Modes ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
All Solutions ◽  
Convergence Study ◽  
Micro Plates

Abstract The effect of membrane load on the behaviour of axisymmetric bistable circular curved microplates on Berger’s based axisymmetric reduced order (RO) model, incorporating radial prestress, is studied. The model is first validated for a “mechanical” load, against a Föppl-von-Kármán’s RO model with twenty degrees of freedom (DOF), a finite differences (FD) solution and a finite elements (FE) model, serving as the reference. All solutions implement the “Riks” method to track possible unstable branches, which can swerve in due to the presence of higher buckling modes. A convergence study is carried out for the snap-through location and load, as well as for the critical elevation and prestress required for bistability. Based on validated results of the analysis, the reliability of the model for predicting the effect of prestress on the plate behaviour under nonlinear electrostatic load is then investigated while using FD solutions as the reference. The study furnishes a reliable expended RO model, which includes prestress on the as-fabricated curved plate. The resulting model can further be used to estimate the value of residual prestress, present in an electrostatically actuated curved plate, based on its response.

Dynamic Trapping in Bi-Stable Electrostatically Actuated Curved Micro Beams

2014 ◽  
Author(s):  
Lior Medina ◽  
Rivka Gilat ◽  
Slava Krylov
Keyword(s):  
Stable State ◽  
Stable Configuration ◽  
Structural Elements ◽  
Dynamic Phenomenon ◽  
Order Model ◽  
Reduced Order ◽  
Actuation Force ◽  
Numerical Investigations ◽  
Electrostatically Actuated ◽  
Beam Characteristics

Micro and nano devices incorporating bi-stable structural elements such as micro beams are designed to exploit the fact that the latter possess two stable configurations at the same actuation force. Generally, the transition of a micro beam from one table state to another, namely the snap-through which is essentially dynamic phenomenon, can be initiated by either static or dynamic activations. In this work, results of theoretical and numerical investigations of the transient dynamics of a pre-stressed initially curved double clamped micro beams actuated by a time dependent electrostatic load are presented. We show by means of a reduced order model of a shallow beam, derived using the Galerkin procedure, that the beam may exhibit various types of responses. For certain beam characteristics, the second stable state is inaccessible under a static loading but is attainable only by means of a specially tailored dynamic actuation. This gives way to the possibility of trapping the dynamically bi-stable beam at a stable configuration which is close to the electrode by applying special loading sequences.

Symmetry Breaking in an Initially Curved Micro Beam Loaded by a Distributed Electrostatic Force

2012 ◽  
Author(s):  
Lior Medina ◽  
Rivka Gilat ◽  
Slava Krylov
Keyword(s):  
Symmetry Breaking ◽  
Degrees Of Freedom ◽  
Electrostatic Force ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Straight Beam ◽  
Base Functions ◽  
Approximate Expressions ◽  
Symmetric Response ◽  
Good Agreement

The asymmetric buckling of a shallow initially curved stress-free micro beam subjected to distributed nonlinear deflection-dependent electrostatic force is studied. The analysis is based on a two degrees of freedom reduced order (RO) model, resulting from the Galerkin decomposition with linear undamped eigen-modes of a straight beam used as the base functions. Simple approximate expressions are derived defining the geometric parameters of beams for which an asymmetric response bifurcates from the symmetric one. The necessary criterion establishes the conditions for the appearance of bifurcation points on the unstable branch of the symmetric response curve; the sufficient criterion assures a realistic asymmetric buckling bifurcating from the stable branches of the curve. It is shown that while the symmetry breaking conditions are affected by the nonlinearity of the electrostatic force, its influence is less pronounced than in the case of the symmetric snap-through criterion. A comparison between the RO model results and those obtained by direct numerical analysis shows good agreement between the two and indicates that the obtained criteria can be used to predict non-symmetric buckling in electrostatically actuated bistable micro beams.

The Static and Dynamic Behavior of MEMS Arches Under Electrostatic Actuation

2009 ◽  
Author(s):  
Hassen M. Ouakad ◽  
Mohammad I. Younis ◽  
Fadi M. Alsaleem ◽  
Ronald Miles ◽  
Weili Cui
Keyword(s):  
Multiple Scales ◽  
Perturbation Technique ◽  
Dynamical Behavior ◽  
Primary Resonance ◽  
Reduced Order Model ◽  
Harmonic Load ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Model Equations

In this paper, we investigate theoretically and experimentally the static and dynamic behaviors of electrostatically actuated clamped-clamped micromachined arches when excited by a DC load superimposed to an AC harmonic load. A Galerkin based reduced-order model is used to discretize the distributed-parameter model of the considered shallow arch. The natural frequencies of the arch are calculated for various values of DC voltages and initial rises of the arch. The forced vibration response of the arch to a combined DC and AC harmonic load is determined when excited near its fundamental natural frequency. For small DC and AC loads, a perturbation technique (the method of multiple scales) is also used. For large DC and AC, the reduced-order model equations are integrated numerically with time to get the arch dynamic response. The results show various nonlinear scenarios of transitions to snap-through and dynamic pull-in. The effect of rise is shown to have significant effect on the dynamical behavior of the MEMS arch. Experimental work is conducted to test polysilicon curved microbeam when excited by DC and AC loads. Experimental results on primary resonance and dynamic pull-in are shown and compared with the theoretical results.

Reduced Order Model of MEMS Sensors Near Natural Frequency

2011 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Jose C. Solis Silva
Keyword(s):  
Natural Frequency ◽  
Nonlinear Response ◽  
Casimir Effect ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Mass Detection ◽  
Order Model ◽  
Reduced Order ◽  
First Order ◽  
Electrostatically Actuated

The nonlinear response of an electrostatically actuated cantilever beam microresonator sensor for mass detection is investigated. The excitation is near the natural frequency. A first order fringe correction of the electrostatic force, viscous damping, and Casimir effect are included in the model. The dynamics of the resonator is investigated using the Reduced Order Model (ROM) method, based on Galerkin procedure. Steady-state motions are found. Numerical results for uniform microresonators with mass deposition and without are reported.

ROM of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Resonators

2015 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Second Order ◽  
Voltage Response ◽  
Order Model ◽  
Reduced Order ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Method Effects ◽  
Ground Plate

This paper deals with the voltage-amplitude response (or voltage response) of superharmonic resonance of second order of MEMS resonator sensors under electrostatic actuation. The system consists of a MEMS flexible cantilever above a parallel ground plate. The AC frequency of actuation is near one fourth the natural frequency. The voltage response of the superharmonic resonance of second order of the structure is investigated using the Reduced Order Model (ROM) method. Effects of voltage and damping voltage response are reported.

The Motion Formalism for Flexible Multibody Systems

2021 ◽  
Author(s):  
Olivier Bauchau ◽  
Valentin Sonneville
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Solution Process ◽  
Flexible Multibody Systems ◽  
Structural Elements ◽  
Governing Equations ◽  
Euclidean Group ◽  
Reduced Order ◽  
Flexible Multibody ◽  
Element Approach

Abstract This paper describes a finite element approach to the analysis of flexible multibody systems. It is based on the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) recognizes that these are members of the Special Euclidean group thereby coupling their displacement and rotation components, and (3) resolves all tensors components in local frames. The goal of this review paper is not to provide an in-depth derivation of all the elements found in typical multibody codes but rather to demonstrate how the motion formalism (1) provides a theoretical framework that unifies the formulation of all structural elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, and (4) prevents the occurrence of singularities.

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