Deflection Of Microbeam Research Articles

Solving and Numerical Simulations of Fractional-Order Governing Equation for Micro-Beams

This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials are derived first. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional order governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for solution. It simplifies the calculation. MATLAB software was used to program this algorithm to simulate the numerical solution of the deflection. The effectiveness and accuracy of the algorithm are verified by the numerical example. Finally, numerical simulations are carried out on the viscoelastic micro-beams. It is found that the viscous damping coefficient is inversely proportional to the deflection, and the length scale parameter of the micro-beam is also inversely proportional to the deflection. In addition, the stress and strain of micro-beam, the change of deflection under different simple harmonic loads, and potential energy of micro-beam are discussed. The results of the study fully demonstrated that the shifted Chebyshev polynomial algorithm is effective for the numerical simulations of viscoelastic micro-beams.

Thermoelastic fractional derivative model for exciting viscoelastic microbeam resting on Winkler foundation

In the current investigation, the thermoelastic vibration of a viscoelastic microbeam resting on the Winkler foundation is studied using the fractional-order theory. To describe the damping of the viscoelastic material according to experimental results, the Kelvin–Voigt model is replaced by a new form with a fractional-order derivative. The generalized thermoelasticity model and Euler–Bernoulli beam theory are used to construct the governing equation. The microbeam is subjected to axial load, ultrafast laser heating, and varying sinusoidal heat. The governing equation is then solved using the Laplace transform technique to determine the deflection and thermoelastic interaction responses of microbeams. The effects of many parameters such as the coefficient of viscosity, axial load, fractional derivative order, laser pulse duration, and foundation parameter on the microbeam response are explained and discussed in detail.

Exact solution for frequency response of sandwich microbeams with functionally graded cores

Based on the Euler–Bernoulli beam model and the modified strain gradient theory, the size-dependent forced vibration of sandwich microbeams with a functionally graded (FG) core is presented. The equation of motion and the corresponding classical and nonclassical boundary conditions are derived using the Hamilton’s principle. An exact solution of the governing equation is developed for sandwich beams with various boundary conditions and subjected to an arbitrarily distributed harmonic transverse load. Finally, parametric studies are presented to investigate the effects of geometric ratios, length scale parameters, power index, boundary conditions, layup, and thickness of the FG layer on the frequency response of clamped and simply supported microbeams. Numerical results show that in the case of clamped microbeams, the essential and natural size-dependent boundary conditions have a significant effect on the resonance frequency and transverse deflection of microbeams. Also, it is seen that an optimal layup (without change in total volume of each material) can significantly improve the frequency characteristics of sandwich microbeams.

Chaotic Contact Dynamics of Two Microbeams under Various Kinematic Hypotheses

Abstract Different kinematic mathematical models of nonlinear dynamics of a contact interaction of two microbeams are derived and studied. Dynamics of one of the microbeams is governed by kinematic hypotheses of the first, second, and third approximation orders. The second beam is excited through a contact interaction with the first beam and is described by the kinematic hypothesis of the second-order approximation in both geometric linear and nonlinear frameworks. The derived nonlinear partial differential equations (PDEs) are transformed to the counterpart system of nonlinear ordinary differential equations (ODEs) by the finite difference method. Nonlinear contact interaction dynamics of the microbeam structure is analyzed with the help of time series (signals), Fourier spectra, and wavelet spectra based on various mother wavelets, Morlet wavelet spectra employed to study synchronization phenomena, Poincaré maps, phase portraits, and the Lyapunov exponents estimated with the Wolf, Kantz, and Rosenstein algorithms. We have illustrated that neglecting the shear function (Euler–Bernoulli model) yields erroneous numerical results. We have shown that the geometric nonlinearity cannot be neglected in the analysis even for small two-layer microbeam deflection. In addition, we have detected that the contact between two microbeams takes place in the vicinity of x ≈ 0.2 x \approx 0.2 and x ≈ 0.8 x \approx 0.8 instead of the beams central points.

Moment-curvature based elasto-plastic model for large deflection of micro-beams under combined loading

Micro-beams show size dependent structural response which is not predictable from classical beam theories. In this paper, a couple stress theory based analysis of large elasto-plastic deflection of micro-beams is carried out using an incremental moment-curvature constitutive modeling approach. An isotropic hardening model is used for illustration. The developed model is applied to solve the large deflection bending problem of a cantilever under the combined effect of distributed and concentrated load. The solution procedure adopted is based on linearization of the governing differential equation at every load step and subsequently solving the equation by numerical integration. Suitable normalization of the relevant variables have rendered generality. New results pertaining to large elasto-plastic deflections showing evident size-effects are investigated.

A hybrid solution for analyzing nonlinear dynamics of electrostatically-actuated microcantilevers

• A hybrid solution for nonlinear dynamics of microcantilevers has been introduced. • The pure odd-order nonlinear model combined with a distributed parameter system. • A comprehensive model for MEMS before, during and after switching was presented. • Two distinct model for before pull-in and after pull-in regimes are combined. This paper introduces a closed-form approximation of dynamic response of microcantilevers. The applied load on the system was linearized by Taylor series expansion and to obtain approximate solutions, model of a pure odd-order nonlinear oscillator, subjected to constant excitation was assumed. Pull-in voltage was investigated to analyze the different parameters of the examined microbeam. In order to obtain a comprehensive dynamic model for MEMS devices, before, during and after switching, the pure odd-order nonlinear model was combined with a distributed parameter system and solved after reaching the pull-in voltage. The obtained results demonstrate correct prediction of the static pull-in voltage and also the dynamic deflection of microbeams. By using the same approach, the sensitivity of the pull-in voltage to various geometrical parameters was also investigated. The obtained results indicate that excessive increase in the air gap causes substantial increase in the pull-in voltage; while increasing thickness of microcantilever has even greater effect. It was also observed that for a given thickness of microcantilever, increasing its length beyond a certain amount has no effect on the pull-in voltage.

Stability analysis of FGM microgripper subjected to nonlinear electrostatic and temperature variation loadings

AbstractConsidering the desired characteristics of functionally gradient materials (FGM), such as good thermal and wear resistance, this paper investigates the mechanical behavior of FGM microgripper under DC voltage and temperature variation. It is assumed that the two microbeams of the FGM microgripper are mounted symmetrically against each other, and the volume of ceramic varies exponentially with microbeam thickness. The nonlinear equations governing the static and dynamic deflection of microbeams due to the application of DC voltage and temperature variation are derived and solved using the step-by-step linearization method and the Galerkin method. In previous studies, the mechanical behavior of a microgripper with homogenous and the same material has been investigated; in this paper, the response and stability of the FGM microgripper against DC voltage and temperature variation are investigated. Also, the effect of geometrical dimensions, ceramic volume, and its variations with microbeam thickness on the mechanical behavior and system stability is studied.

FGM micro-gripper under electrostatic and intermolecular Van-der Waals forces using modified couple stress theory

In this paper mechanical behavior of the functional gradient materials (FGM) micro-gripper under thermal load and DC voltage is numerically investigated taking into account the effect of intermolecular forces. In contrary to the similar previous works, which have been conducted for homogenous material, here, the FGM material has been implemented. It is assumed that the FGM micro-gripper is made of metal and ceramic and that material properties are changed continuously along the beam thickness according to a given function. The nonlinear governing equations of the static and dynamic deflection of microbeams have been derived using the coupled stress theory. The equations have been solved using the Galerkin based step-by-step linearization method (SSLM). The solution procedure has been evaluated against available data of literature showing good agreement. A parametric study has been conducted, focusing on the combined effects of important parameters included DC voltage, temperature variation, geometrical dimensions and ceramic volume concentration on the dynamic response and stability of the FGM micro-gripper.

A new displacement-based framework for non-local Timoshenko beams

In this paper, a new theoretical framework is presented for modeling non-locality in shear deformable beams. The driving idea is to represent non-local effects as long-range volume forces and moments, exchanged by non-adjacent beam segments as a result of their relative motion described in terms of pure deformation modes of the beam. The use of these generalized measures of relative motion allows constructing an equivalent mechanical model of non-local effects. Specifically, long-range volume forces and moments are associated with three spring-like connections acting in parallel between couples of non-adjacent beam segments, and separately accounting for pure axial, pure bending and pure shear deformation modes. The variational consistency of the proposed non-local beam model is demonstrated by minimization of an appropriate total potential energy functional. Numerical results concerning the static behavior for different boundary and loading conditions are presented. It is shown that the proposed non-local beam model is able to capture experimental data on the static deflection of micro-beams, available in the literature.

Vibration analysis of electrostatically actuated nonlinear microbridges based on the modified couple stress theory

In this paper natural frequency of electrostatically actuated microbridges is investigated based on the modified couple stress theory. Nonlinear formulation of Euler–Bernoulli microbeam is derived using Hamilton’s principle. By considering the von-Karman strain, the nonlinearities caused by the mid-plane stretching are included in the formulation. To confirm the model, results of static deflection and natural frequency of microbeams are calculated using modified couple stress theory and compared to those evaluated based on the classical theory and experimental observations. At first, from experimental results of static deflection of a microcantilever, estimation for length scale parameter of polysilicon is presented. Using this value of length scale, natural frequency of microbridges under electrostatic actuation is calculated on the basis of the modified couple stress theory and compared with experimental results. The results of the modified couple stress theory are in very good agreement with experimental findings while the difference between the results predicted based on the classical theory and experimental findings is considerable.

Characterization of time-dependent anelastic microbeam bending mechanics

This paper presents an accurate yet straightforward methodology for characterizing time-dependent anelastic mechanics of thin metal films employed in metalic microelectromechanical systems (MEMS). The deflection of microbeams is controlled with a mechanical micro-clamp, measured with digital holographic microscopy and processed with global digital image correlation (GDIC). The GDIC processing directly incorporates kinematics into the three-dimensional correlation problem, describing drift-induced rigid body motion and the beam deflection. This yields beam curvature measurements with a resolution of <1.5 × 10−6 µm−1, or for films thinner than 5 µm, a strain resolution of <4 με. Using a simple experimental sequence, these curvature measurements are then combined with a linear multi-mode time-dependent anelastic model and a priori knowledge of the Young's modulus. This allows the characterization of the material behaviour in the absence of an additional explicit force measurement, which simplifies the experimental setup. Using this methodology we characterize the anelasticity of 5 µm-thick Al(1 wt%)-Cu microbeams of varying microstructures over relevant timescales of 1 to 1 × 105 s and adequately predict the time and amplitude response of experiments performed for various loading conditions. This demonstrates the validity of the methodology and the suitability for thin film mechanics research for MEMS development.

Assessment of mechanical properties of metallic thin-films through micro-beam testing

Microelectronic industry is driven by the continuous miniaturization process conducing to the introduction of materials with better performance. These materials are subjected to stresses mainly due to thermal mismatch, microstructural changes or process integration which can be in the origin of mechanical reliability issues. To study these phenomena and even electromigration a good mechanical characterization of the materials is needed. This work aims at developing tests to assess fracture and elastoplastic behavior of thin Cu films. The tests developed are based on the deflection of microbeams (micromachined using a focused ion beam) using a nanoindenter. Different test geometries for microbeams have been evaluated and quantitative data have been obtained combining experimental results with analytical or numerical models, depending on the property under study. Microbeam response shows a strong dependence on the orientation of the grains close to the fixed end. Grain orientation has been measured by electron backscatter diffraction and the plastic behavior has been modeled by the finite element method using an in-house crystal plasticity subroutine. The effect of film thickness on fracture energy has been determined from tests of notched beams.

Effect of mass diffusion on the damping ratio in a functionally graded micro-beam

The present paper is aimed at studying the effects of mass diffusion on the quality factor of a functionally graded cantilever micro-beam in which the material properties of the micro-beam vary continuously along the beam thickness according to a power-law. The governing equation of a micro-beam deflection is obtained using Hamilton’s principle and also the governing equations of thermo-diffusive elastic damping are established by using two dimensional non-Fourier heat conduction and non-Fickian mass diffusion models with one relaxation time based on continuum theory frame. The free vibration of the micro-beam resonators is analyzed by using Galerkin reduced order model formulation for the first mode of vibration. The mass diffusion effects on the quality factor are studied for the various micro-beam thicknesses and temperatures. Numerical computations are performed for specific materials and the results obtained are represented graphically. The effect of different power law exponent on the quality factor of the micro-beam is studied and Comparisons are made within the theory in the presence and absence of the mass diffusion effect.

Electrostatically actuated FGM micro-tweezer under the thermal moment

In previous studies, the mechanical behavior of a micro-tweezer with homogenous and the same material has been investigated. In this paper considering the desired characteristics of FGMs, as a good thermal and wear resistance, the mechanical behavior of FGM micro-tweezer under DC voltage and temperature variations is investigated. It is assumed that the two micro-beams of the FGM micro-tweezer are mounted symmetrically against each other and one side of the micro-beams is made of a pure metal the other side is a mixture of metal and ceramic, and the percentage of ceramic varies exponentially through the micro-beams thickness. The nonlinear equations governing the static deflection of micro-beams due to the application of DC voltage and the temperature changes are derived and solved using the step by step linearization method and the Galerkin method. The response and the stability of the system against the variation of DC voltage and temperature are investigated. Also, the effect of geometrical dimensions, ceramic percentage and its variations through the micro-beams thickness on the mechanical behavior and the system stability are studied.

Closed-Form Solutions for Gradient Elastic Beams with Geometric Discontinuities by Laplace Transform

The static bending solution of a gradient elastic beam with external discontinuities is presented by Laplace transform. Its utility lies in the ability to switch differential equations to algebraic forms that are more easily solved. A Laplace transformation is applied to the governing equation which is then solved for the static deflection of the microbeam. The exact static response of the gradient elastic beam with external discontinuities is obtained by applying known initial conditions when the others are derived from boundary conditions. The results are given in a series of figures and compared with their classical counterparts. The main contribution of this paper is to provide a closed-form solution for the static deflection of microbeams under geometric discontinuities.

Research on Dynamic Coupling Characteristics of Electrostatic Actuated Micro Beam Considering Gas Film Damping

It has been proved by large number of experiments that gas film damping has great influence on MEMS structures. Based on the elastic and material mechanics, this paper built a linear-vibration model for the electrostatic actuated micro-beam. According to the rarefied gas dynamics theory, a kind of modified Reynolds equation is adopted in the system modeling. In view of the study on small deflection of micro-beam has been investigated already by many researchers, this paper focus the research on large beam deflections, i.e. the mid-point deflection can be compared with the thickness of the gas film. The dynamic characteristics of micro-beam are investigated through coupling the dynamic vibration equation with the modified Reynolds equation. Through the finite difference method, the discrete form of the coupled dynamic equations has been obtained.

Measurement of static and dynamic mechanical behavior of micro and nano-scale thin metal films: using micro-cantilever beam deflection

We present the results of a novel micro-beam deflection test used to investigate the static and dynamic mechanical behavior of submicron-thick metal films. The method demonstrated in this study allows researchers to observe the motion of micro and nano-scale thin films responding to electrostatic loads, by means of laser reflection measurements at frequency rates of up to 500 Hz. Researchers fabricated a supporting frame and a novel triangular shaped “paddle” beam designed to provide uniform plane stress distribution while undergoing deflection. A simple geometric calculation, based on cantilever deflection, enabled the degree of strain to be determined, which in turn provided the Young’s modus for aluminum film of a given thickness. We also studied the dynamic behavior using the dynamic frequency response of the beam, generated by electrostatic forces under various loads and vacuum pressure conditions. Our results showed that air damping has a significant influence on the free damping behavior of specimens, and only a minor influence on damping frequency. We determined the loss angle and frequency using sweep frequency and free damping methods, which were very consistent with paddle resonant frequencies. The loss angle obtained from a simple silicon micro-beam was 2.001 × 10−4°using the free damping method and 2.23 × 10−4°using the sweep frequency method. The dynamic response loss mechanism measured in this experiment provides incentive for the further study of grain boundary motion and dislocation motion in thin films.

On the mechanical behavior of a functionally graded micro-beam subjected to a thermal moment and nonlinear electrostatic pressure

Abstract In the present work, mechanical behavior of a functionally graded cantilever micro-beam subjected to a nonlinear electrostatic pressure and temperature changes has been studied. It has been assumed that the top surface is made of pure metal and the bottom surface from a metal–ceramic mixture. The ceramic constituent percent of the bottom surface varies from 0% to 100%. In addition to the Volume Fractional Rule of material, exponential function has been used for representation of continuous gradation of the material properties through micro-beam thickness. Attention being paid to the ceramic constituent percent of the bottom surface, five different types of FGM micro-beams have been investigated. Nonlinear integro-differential thermo-electro mechanical equation based on Euler–Bernoulli beam theory has been derived and solved using Step-by-Step Linearization Method and Finite Difference Method. The effects of temperature changes and the electrostatic pressure on the deflection and stability of FGM micro-beams having various amounts of the ceramic constituent have been studied and normal stress distributions in the cross section along the beam thickness have been given and compared with a classic metal beam.

SiGe based grating light valves: A leap towards monolithic integration of MOEMS

The Grating Light Valve (GLV) is a microelectromechanical reflection grating. It operates on the principle of controlled diffraction of incident light due to electrostatic deflection of microbeams, thus producing bright and dark pixels in a display system. This unique approach offers significant advantages compared to other display systems in terms of speed, high optical efficiency, accuracy and ease of manufacturing. At the same time monolithic integration of MEMS on top of CMOS is getting more popular because CMOS-integrated MEMS exhibit less parasitics and have a reduced assembly and packaging cost over hybrid approaches. The relatively low deposition temperature (∼450 °C) of poly-SiGe compared to poly-Si (∼800 °C) makes poly-SiGe suitable for back end processing. Hence, in this work we report the fabrication and functioning of CVD poly-SiGe GLVs in terms of their response to the amplitude and frequency of the applied actuation voltage.

Low cycle fatigue of single crystal silicon thin films

Silicon will continue to be the critical structure material for micromechanical components for years to come so that reliability must be a key concern. Consequently, in order to ensure reliability design approaches must account for fatigue behavior. This work is aimed at studying the fatigue of single crystal silicon (SCS) thin films by a specially designed method. The films were tested using cantilever microbeam deflection with respect to the effect of loading conditions. To conduct a fatigue test under cyclic strain would be more realistic because many MEMS applications, such as micro-accelerometers and micro-filters, consist of beams vibrating in the same mode. A micro-force testing machine with a micro-probe and specially designed fixtures is used to contact and load the beams under the cyclic strain. Before the fatigue test, the failure strain ɛ f of beams in the flexural test is achieved as the testing criterion. In fatigue testing, various percentages of failure strain ɛ f, two times of the strain amplitude, are adopted. First of all, fatigue strain–life cycle (S/N) curve is achieved. Further, the curve of fatigue force detected on the SCS versus fatigue life is measured. SEM is also employed to observe the fracture modes of films under fatigue. Based on the SEM observation and force/life curve, the failure mechanism of the fatigued SCS films under the cyclic strain is proposed. This provides a viable method to evaluate the reliability of the SCS.