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.

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.

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.

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.

Analysis of the Damping of Beams With Discontinuous Viscoelastic Damping Layer Using Integral Finite Elements

Volume 2 ◽  
2004 ◽  
Author(s):  
T. Liang ◽  
Qian Chen ◽  
S. Olutunde Oyadiji ◽  
Andrew Leung
Keyword(s):  
Finite Elements ◽  
Structural Dynamics ◽  
Degrees Of Freedom ◽  
Dynamic Properties ◽  
Viscoelastic Damping ◽  
Fe Model ◽  
Dynamics Analysis ◽  
Viscoelastic Composite ◽  
Constraint Damping ◽  
Damping Treatments

The damping performance of discontinuous constrained viscoelastic damping layer using Integral Finite Elements (IFE) are investigated in this paper. The IFE involves the dynamic analysis of Elastic-Viscoelastic Composite (EVC) structures with frequency-dependant material properties. EVC structures, which incorporate constrained viscoelastic damping treatment, are modelled using IFE’s and conventional FE’s together to deal with the discontinuous treatment of the constraint damping layers. This approach dramatically reduces the number of degrees of freedom of the FE model compared with conventional FE models. By using specialised algorithms developed for EVC structural dynamics analysis, IFE makes the estimation of the dynamic properties of the EVC structures an easy task similar to the structural dynamics analysis using conventional finite element method. Using an IFE model and the special algorithms, the damping performance of various designs of viscoelastic damping treatments are investigated at extremely low computational costs compared with the use of current commercial FE software packages. A guideline is introduced based on the results of the damping performance analysis for the structural design of viscoelastic damping treatments.

scholarly journals Exact and efficient interpolation using finite elements shape functions

2009 ◽  
pp. 307-331
Author(s):  
Gustavo H.C. Silva ◽  
Rodolphe Le Riche ◽  
Jérôme Molimard ◽  
Alain Vautrin
Keyword(s):  
Experimental Data ◽  
Finite Elements ◽  
Degrees Of Freedom ◽  
Mesh Size ◽  
Large Data ◽  
Fe Model ◽  
Shape Functions ◽  
Natural Choice ◽  
Data Points ◽  
Bounding Boxes

The comparison of finite elements (FE) and experimental data fields have become ever more prevalent in numerical simulations. Since FE and experimental data fields rarely match, the interpolation of one field into the other is a fundamental step of the procedure. When one of the fields comes from FE, using the existing FE mesh and shape functions is a natural choice to determine mesh degrees of freedom at data point coordinates. This makes no assumptions beyond those already made in the FE model. In this sense, interpolation using element shape functions is exact. However, crude implementations of this technique generally display a quadratic computation complexity with respect to mesh size and number of data points, which is impractical when large data fields must be compared repeatedly. This document aims at assembling existing numerical procedures to improve the interpolation efficiency. With a combination of cross-products, bounding-boxes and indexing methods, the resulting algorithm shows linear computation cost, providing significant improvement in efficiency.

scholarly journals Simultaneous Regression and Selection in Nonlinear Modal Model Identification

Vibration ◽  
2021 ◽  
Vol 4 (1) ◽  
pp. 232-247
Author(s):  
Christopher Van Damme ◽  
Alecio Madrid ◽  
Matthew Allen ◽  
Joseph Hollkamp
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Response ◽  
Computational Cost ◽  
Model Identification ◽  
Dynamic Responses ◽  
Basis Set ◽  
Fe Model ◽  
Time Duration ◽  
Reduced Order ◽  
Polynomial Coefficients

High fidelity finite element (FE) models are widely used to simulate the dynamic responses of geometrically nonlinear structures. The high computational cost of running long time duration analyses, however, has made nonlinear reduced order models (ROMs) attractive alternatives. While there are a variety of reduced order modeling techniques, in general, their shared goal is to project the nonlinear response of the system onto a smaller number of degrees of freedom. Implicit Condensation (IC), a popular and non-intrusive technique, identifies the ROM parameters by fitting a polynomial model to static force-displacement data from FE model simulations. A notable drawback of these models, however, is that the number of polynomial coefficients increases cubically with the number of modes included within the basis set of the ROM. As a result, model correlation, updating and validation become increasingly more expensive as the size of the ROM increases. This work presents simultaneous regression and selection as a method for filtering the polynomial coefficients of a ROM based on their contributions to the nonlinear response. In particular, this work utilizes the method of least absolute shrinkage and selection (LASSO) to identify a sparse set of ROM coefficients during the IC regression step. Cross-validation is used to demonstrate accuracy of the sparse models over a range of loading conditions.

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.

Sign in / Sign up

Export Citation Format

Share Document