Clamped Microbeam Research Articles

The effect of capillary and intermolecular forces on instability of the electrostatically actuated microbeam with T-shaped paddle in the presence of fringing field

• The mechanical behavior of a clamped–clamped microbeam with middle paddle is analyzed. • The pull in voltage declines and the deflection enlarges as the intermolecular forces increase. • Presence of the electrostatic force accelerated the instability resulting from the capillary force . • Fringing field effect considerably affected determination of the bifurcation points . The present study is aimed to investigate pull-in instability, static deformation, natural frequency, primary and subharmonic resonances in the clamped–clamped microbeam with middle T-shaped paddle in presence of electrostatic force, Casimir force, van der Waals force and fringing field effect; besides, it is focused on investigating the effect of capillary force on instability and adhesion considering the effects of middle layer elongation and axial force. The microbeam is modeled as an Euler Bernoulli beam. In order to solve the nonlinear equations, the Galerkin-based rank reduction method was used. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances. The obtained results showed that the van der Waals and Casimir forces had attractive nature and led to the increased static deformation as well as occurrence of the pull-in at lower voltages; besides, the effects of attraction in Casimir was more than that in van der Waals. Results showed that with increase in the dimensionless length, which is influenced by the capillary force, the maximum static deformation was increased as well, leading to instability in capillary coefficients. Furthermore, presence of the electrostatic force also accelerated the instability resulting from the capillary force. Effects of the axial force and elongation in the clamped state resulted in the reduced static deformation as well as delayed instability. It is shown that changing the Casimir and Van der Waals forces as well as the fringing field effect, in addition to causing changes in the voltage and range, considerably affected determination of the bifurcation points. To validate the analytical results, numerical simulation is performed.

Stability analysis of a capacitive micro-resonator with embedded pre-strained SMA wires

This paper presents a study on the effects of the SMA wires’ characteristics on tuning the stability of a capacitive micro-resonator. In the proposed model, pre-strained SMA wires have been embedded in a double clamped resonant microbeam which is actuated electrostatically. The governing equations of the system have been introduced and then an eigen-value problem has been adopted to investigate stability. Galerkin-based numerical methods have been used to solve the governing equation of motion for obtaining the motion trajectories of the beam. The effects of the number of SMA wires, their diameter, pre-strain and temperature on the pull-in instability and frequency response of the resonator have been shown. Critical values of recovery stress and SMA temperature for avoiding instability, with and without applying DC voltage have been obtained. The results have shown that by changing each of the SMA parameters, one can reach a needed magnitude of recovery stress as well as transmitted longitudinal force to the host beam, and consequently control and tune the stability and resonance frequency of the micro-resonator.

On the response of MEMS resonators under generic electrostatic loadings: experiments and applications

We present an investigation of the dynamic behavior of an electrostatically actuated clamped–clamped microbeam, under the simultaneous excitation of primary and subharmonic resonance. The simultaneous excitation of primary and subharmonic resonances of similar strength is experimentally investigated by using different combinations of AC and DC voltages. It is observed that the response of the resonator is governed by a mixed effect of both excitations. Subharmonic-dominated response shows sharp amplitude transitions and smaller monostable regime, while primary-dominated response shows gradual amplitude transition and larger monostable regime. Two potential applications are experimentally demonstrated. The first is a resonator-based MEMS AND logic gate based on AC only subharmonic excitation. The second is a charge sensor based on the transition from subharmonic-dominated response to primary-dominated response, which is potentially capable of detecting a small amount of electric charges.

Stochastic dynamic behavior of electrostatically actuated clamped–clamped microbeams with consideration of thermal field

Environment perturbation and structure vibration can lead to the change of the thermal field, which may have certain effects on intrinsic frequency and dissipation mechanism for small-scale mechanical resonators. This article aims to theoretically investigate static and stochastic dynamic behavior of electrostatically actuated microbeams via a low dimensional model. Considering thermoelastic damping and thermal residual stress, an improved single degree of freedom model to describe microbeam-based resonators is obtained by using Hamilton’s principle and Galerkin method. Through static behavior analysis, the influence of temperature field and electric field on the natural frequency and thermoelastic damping of the system is theoretically derived. The results show that the perturbation of environment temperature can significantly change the natural frequency and quality factor of the system. The following, a random frequency perturbation parameter is introduced to describe the perturbation of the environment temperature. An efficient approximation method is proposed to qualitatively study nonlinear stochastic dynamic behavior under small perturbation of stochastic parameter. It is found that the randomness of environment temperature suppresses nonlinear behavior and reduces the possibility of large amplitude vibration. Typically, the environment temperature disturbance reduces the resonance frequency of the system. Finally, the numerical method is put forward to not only verify the validity of the theoretical method but also quantitatively give the effects of noise strength and correlation rate on the nonlinear dynamic behavior. The present work provides theoretical framework for analyzing the effects of the perturbation of temperature field on nonlinear system response.

Dynamic evolution of a primary resonance MEMS resonator under prebuckling pattern

This paper analytically investigates the dynamic evolution of the primary frequency response of a prebuckling microbeam-based resonator with $$\hbox {Z}_{2}$$ symmetry. A doubly clamped straight microbeam actuated by two symmetric stationary electrodes is simplified as a time-varying capacitor model for qualitative analysis purpose. Nonlinearities induced by the midplane stretching of the microbeam and the electrostatic force are considered. During solution procedure, electrostatic force holds its original form without any Taylor series expansion, and only one assumption with a small ratio of AC to DC voltage is introduced. The average equation, frequency response, backbone curve and stability condition are determined, respectively, based on the method of multiple scales combined with homotopy concept. Results demonstrate for the first time that the frequency response includes two types of branches, namely low- energy branch and high-energy branch. As the increase ion AC excitation amplitude, both branches close to each other along the backbone curve until they intersect. Further analyses are then performed to investigate the details of the backbone curve and the frequency response equation. Analytical formulas to determine the hardening and softening switches of the frequency response and the intersection condition of the low- and high-energy branches are both deduced and examined in depth. Primary frequency response properties in pull-in and secondary pull-in case are classified and depicted through theoretical predictions via the method of multiple scales and then verified through numerical results via the finite difference method combined with Floquet theory. Finally, a specific case study based on equivalent lumped parameters via Galerkin method is presented. Excellent agreements between theoretical predictions and simulation results illustrate the effectiveness of the whole analyses.

Electrostatic nonlinear bending, buckling and free vibrations of viscoelastic microbeams based on the modified couple stress theory

This paper presents a size-dependent, nonlinear beam model for bending, buckling and free vibrations of electrically actuated viscoelastic clamped–clamped microbeams. The modified couple stress theory is used to model the size effect. The Euler–Bernoulli beam model including von Karman geometric nonlinearity and intermolecular van der Waals and Casimir forces is adopted. The Boltzmann superposition viscoelastic model is used to simulate the linear behavior of the viscoelastic material. The governing integral–differential equation and corresponding boundary conditions are obtained using Hamilton’s principle. Galerkin’s discretization is used to obtain a reduced-order model of the problem. The quasi-elastic method is used to approximate the viscoelastic behavior. The model has been verified by comparing some results with those available in the literature, and a good agreement is obtained. The influence of the small size, creep modulus, relaxation time and intermolecular forces on the electrostatic bending, buckling and free vibrations are investigated and found significant.

Resonant Gas Sensor and Switch Operating in Air With Metal-Organic Frameworks Coating

We demonstrate a resonant gas sensor uniformly coated with a metal-organic framework (MOF) and excited near the second vibration mode for enhanced sensitivity. The possibility of realizing a smart switch triggered upon exceeding a threshold mass is demonstrated when operating the resonator near the dynamic pull-in instability. The resonator is based on an electrostatically excited clamped–clamped microbeam. The microbeam is fabricated from a polyimide layer coated from top with Cr/Au and from bottom with Cr/Au/Cr layer. The geometry of the resonator is optimized to reduce the effect of squeeze film damping, thereby allowing operation under atmospheric pressure. The electrostatic electrode is designed to enhance the excitation of the second mode of vibration with the minimum power required. Significant frequency shift in kilohertz is demonstrated for the first time upon water, acetone, and ethanol vapor exposure due to the MOF functionalization and the higher order mode excitation. The adsorption dynamics and MOF selectivity are investigated by studying the decaying time constants of the response upon gas exposure. [2017–0218]

A MEMS coupled resonator for frequency filtering in air

We present design, fabrication, and characterization of a mechanically coupled MEMS H resonator capable of performing simultaneous mechanical amplification and filtering in air. The device comprises of two doubly clamped polyimide microbeams joined through the middle by a coupling beam of the same size. The resonator is fabricated via a multi-layer surface micromachining process. A special fabrication process and device design is employed to enable operation in air and to achieve mechanical amplification of the output response. Moreover, mixed-frequency excitation is used to demonstrate a tunable wide band filter for low frequency applications. It is demonstrated that through the multi-source harmonic excitation and the operation in air, an improved band-pass filter with flat response and minimal ripples can be achieved.

A Microbeam Resonator With Partial Electrodes for Logic and Memory Elements

We demonstrate logic and memory elements based on an in-plane clamped–clamped microbeam resonator. The microresonator is electrostatically actuated through a drive electrode and the motional signal is capacitively sensed at a sense electrode, while the resonance characteristics are modulated by dc voltage pulses provided at two separate partial electrodes, independent of the drive/sense electrodes. For the logic applications, we use two separate electrodes to provide dc voltages defined as the logic inputs. The high (low) motional signal at on-resonance (off-resonance) state is defined as the logic output state “1” (“0”). For the memory operation, two stable vibrational states, high and low, within the hysteretic regime are defined as the memory states, “1” and “0,” respectively. We take advantage of the split electrode configuration to provide positive and negative dc voltage pulses selectively to set/reset the memory states (“1”/“0”) without affecting the driving and sensing terminals. Excluding the energy cost for supporting electronics, these devices consume energy in tens of picojoules per logic/memory operation. Furthermore, the devices are fabricated using silicon-on-insulator wafers, have the potential for on-chip integration, and operate at moderate pressure (~1 Torr) and room temperature.

Response of thermoelastic microbeams to a periodic external transverse excitation based on MCS theory

In this paper, we investigated the dynamics vibration of clamped–clamped microbeams exposed to associate exterior cross excitation. The theory of modified couple stress (MCS) and thermoelasticity model based on replacing Fourier’s law (LS) are applied to the constructed model. In sight of on Laplace transform technique, an analytical method is utilized to get the deflection and therefore the temperature fields. The inverse of Laplace transforms are calculated numerically utilizing Fourier development methods. The effects of angular frequency of the external force and couple stress term on temperature, deflection, displacement and bending moment are taken into consideration. Numerical results we obtained are displayed in graphs and discussed.

Micromachined microbeams made from porous silicon for dynamic and static mode sensing

Abstract Through a controlled variation of the applied current during porous silicon formation, newly developed processes enable previously unattainable structural integrity of all-mesoporous silicon microelectromechanical systems (MEMS) structures. Such structures are desirable for applications such as sensing where the large surface area and low Young’s modulus of the high porosity layer enable ultra-high sensitivity detection of adsorbed species. In this work, micromachined all-mesoporous silicon microbeams were released, allowing both the dynamic and static sensing modes to be studied using such porous structures. Resonant frequencies (50–250 kHz) of released doubly clamped porous silicon microbeams were measured, allowing mechanical properties to be extracted. Static mode sensing of vapour at the 1100 ppm level was also performed, with the released porous silicon cantilevers showing a significant 6.5 μm (3.7% of a 175 μm beam length) and repeatable deflection after exposure.

Vibrational Response of Initially Deformed Bistable Microbeams Under the Combined Effect of Mechanical Shock Loads and Electrostatic Forces

This paper focuses on the influence of sudden drop tests on the nonlinear structural behavior of electrically actuated bi-table shallow microelectromechanical system (MEMS) arches. The assumed structure consists of an initially bell-shaped doubly clamped microbeam with a rectangular cross section. The Euler–Bernoulli beam theory is assumed to model the nonlinear structural behavior of the bistable system under the combined effect of both the direct current (DC) actuating load and the shaking waves. Moreover, the structural model takes into account both geometric midplane stretching and electric actuation nonlinear terms. A multimode Galerkin-based decomposition is used to discretize the beam equations to extract a reduced-order model (ROM). The convergence of the ROM simulations are first verified and furthermore compared to published experimental data. A thorough ROM parametric study showed that the effect of increasing the shallow arch initial rise alter drastically the system behavior from undergoing a uninterrupted snap-through motion to a sudden snap-through instability. Moreover, the arch rise relationship with its shock spectrum response (SSR) is investigated and it was concluded that as increasing the rise value can cause the system to collapse under the combined DC and shock wave loadings if the shock wave duration is lower or near the system fundamental natural period. All the presented graphs in this investigation represent some robust numerical approaches and design tools to help MEMS designers in improving both the reliability and efficiency of these bistable-based microdevices under shaking dynamic environments.

Thermoelastic damping in microbeam resonators based on modified strain gradient elasticity and generalized thermoelasticity theories

The present study aimed to investigate the size-dependent quality factor of thermoelastic damping in a microbeam resonator based on modified strain gradient elasticity theory. The governing differential equation of motion is derived by using the principle of virtual displacements. The generalized thermoelasticity theory of the well-known Lord–Shulman model is also utilized to derive the equation of coupled thermoelasticity. The quality factor of thermoelastic damping in the microbeam is obtained based on both the complex frequency and the entropy generation methods. Further, by applying the generalized thermoelasticity theory and entropy generation method, explicit formulae for the quality factor of thermoelastic damping in the microbeam resonators are developed in the frameworks of both classical and modified couple stress elasticity theories. The effects of several parameters including size effect are investigated on thermoelastic damping in cantilever and clamped–clamped silicon microbeams as case studies. The results are validated by observing an excellent agreement regarding the comparison of the results of the present study and with previously published results.

Nonlinear coupled vibration of electrostatically actuated clamped–clamped microbeams under higher-order modes excitation

Nonlinear modal interactions have recently become the focus of intense research in micro-resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. Understanding and controlling nonlinear coupling between vibrational modes is critical for the development of advanced micromechanical devices. This article aims to theoretically investigate the influence of antisymmetry mode on nonlinear dynamic characteristics of electrically actuated microbeam via considering nonlinear modal interactions. Under higher-order modes excitation, two nonlinear coupled flexural modes to describe microbeam-based resonators are obtained by using Hamilton’s principle and Galerkin method. Then, the Method of Multiple Scales is applied to determine the response and stability of the system for small amplitude vibration. Through Hopf bifurcation analysis, the bifurcation sets for antisymmetry mode vibration are theoretically derived, and the mechanism of energy transfer between antisymmetry mode and symmetry mode is detailed studied. The pseudo-trajectory processing method is introduced to investigate the influence of external drive on amplitude and bifurcation behavior. Results show that nonlinear modal interactions can transit vibration energy from one mode to nearby mode. In what follows, an effective way is proposed to suppress midpoint displacement of the microbeam and to reduce the possibility of large deflection. The quantitative relationship between vibrational modes is also obtained. The displacement of one mode can be predicted by detecting another mode, which shows great potential of developing parameter design in MEMS. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.

A Coupled Resonator for Highly Tunable and Amplified Mixer/Filter

We present an H-shaped resonator made of two clamped–clamped microbeams mechanically coupled at the middle with a strong coupler to achieve, in a single device, mechanical amplification of the response signal, filtering, and frequency conversion simultaneously. Usingmechanicalamplification combinedwith combination resonances generated from a mixed-frequency excitation, a wideband tunable filter, and a simultaneous frequency up and down convertors at multiple bands is demonstrated. The proposed coupled structure, when combined with the easy-to-implement technique of frequency mixing, is promising for applications in an RF chain.

On the Nonlinear Dynamics of a Doubly Clamped Microbeam Near Primary Resonance

This work aims to investigate theoretically and experimentally various nonlinear dynamic behaviors of a doubly clamped microbeam near its primary resonance. Mainly, we investigate the transition behavior from hardening, mixed, and then softening behavior. We show in a single frequency–response curve, under a constant voltage load, the transition from hardening to softening behavior demonstrating the dominance of the quadratic electrostatic nonlinearity over the cubic geometric nonlinearity of the beam as the motion amplitudes becomes large, which may lead eventually to dynamic pull-in. The microbeam is fabricated using polyimide as a structural layer coated with nickel from top and chromium and gold layers from the bottom. Frequency sweep tests are conducted for different values of direct current (DC) bias revealing hardening, mixed, and softening behavior of the microbeam. A multimode Galerkin model combined with a shooting technique are implemented to generate the frequency–response curves and to analyze the stability of the periodic motions using the Floquet theory. The simulated curves show a good agreement with the experimental data.

Nonlinear vibration analysis of pre-twisted functionally graded microbeams in thermal environment

The large amplitude free vibration behavior of pre-twisted functionally graded (FG) microbeams in thermal environment is investigated based on the modified strain gradient theory (MSGT) in conjunction with the first-order shear deformation theory (FSDT). The geometrical nonlinearity effects are taken into account in the sense of von Kármán nonlinear kinematic assumptions. The Chebyshev–Ritz method combined with harmonic balance method is employed to derive the nonlinear algebraic eigenfrequency equations of the microbeams subjected to different boundary conditions. The material properties are assumed to be temperature-dependent and graded in the thickness direction. After numerically demonstrating the fast rate of convergence and accuracy of the method, the effects of different geometrical and material parameters on the nonlinear free vibration behavior of pre-twisted FG microbeams are investigated. It is found that the effect of twist angle on the hardening or softening of the microbeams depend on the boundary conditions, and the largest and smallest values of nonlinear to linear frequency ratios belong to simply supported and fully clamped microbeams, respectively.

Closed-form relation to predict static pull-in voltage of an electrostatically actuated clamped–clamped microbeam under the effect of Casimir force

The current work presents an accurate closed-form model to Microelectro Mechanical System designers for computing static pull-in voltage of electrostatically actuated microbeams with clamped–clamped end condition. The model incorporates the effect of Casimir force including correction for finite conductivity. The Euler–Bernoulli beam equation is adapted considering the effects of mid-plane stretching, residual stress, fringing field and Casimir force to derive the governing differential equation for electrostatically actuated microbeams. The Galerkin method is used with a multimodes reduced-order model to solve the governing differential equation of microbeams. The results obtained using the reduced-order model are further compared with the solution of the boundary value problem and are validated with published numerical and experimental results. The results of the current work indicate that at least three modes in reduced-order model are essential for the prediction of pull-in voltage of microbeams which have a large value of mid-plane stretching parameter. In order to develop a closed-form relation, dimensionless parameters are used to plot the curves of pull-in voltage versus various parameters such as axial force due to residual stress, Casimir force, fringing field, Casimir force including finite conductivity correction, and mid-plane stretching. Based on the relationship observed in the plotted curves for the independent effect and interaction effects of these parameters on static pull-in voltage, a closed-form model is proposed for the computation of static pull-in voltage. Optimised coefficients of the proposed model are determined using nonlinear regression analysis. An adjusted R $$^2$$ value equal to 0.99909, a P value equal to zero, and $${\chi }^2$$ tolerance equal to $$1\times 10^{-9}$$ obtained by statistical analysis exhibit the precision of fitted data, significance of model, and convergence of the fit, respectively. The proposed model is validated by comparing the results of the model with results of boundary value problem solutions, results predicted from reduced-order model and other several reported numerical and experimental results. The proposed model is robust enough for calculating the static pull-in voltage under different conditions with maximum error of 3% when compared to reported experimental and numerical results.

Vibrations of fluid conveying microbeams under non-ideal boundary conditions

This study aimed to present the effects of non-ideal boundary conditions (BCs) on fundamental parametric resonance behavior of fluid conveying clamped microbeams. Non-ideal BCs are modelled by using the weighting factor (k). Equations of motion are obtained by using the Hamilton’s Principle. A perturbation technique, method of multiple scales, is applied to solve the non-linear equations of motions. In this study, frequency-response curves of fundamental parametric resonance are plotted and the effects of non-ideal BCs are shown. Besides, instability areas of microbeams under ideal and non-ideal BCs are investigated by considering different system parameters. Numerical results show that instability areas significantly changed by the effect of non-ideal BCs.

Dynamic analysis and design of electrically actuated viscoelastic microbeams considering the scale effect

Viscoelastic phenomena widely exist in MEMS materials, which may have certain effects on quasi-static behaviors and transition mechanism of nonlinear jumping phenomena. The static and dynamic behaviors of a doubly clamped viscoelastic microbeam actuated by one sided electrode are investigated in detail, based on a modified couple stress theory. The governing equation of motion is introduced here, which is essentially nonlinear due to its midplane stretching effect and electrostatic force. Through quasi-static analysis, the equilibrium position, pull-in voltage and pull-in location of the system are obtained with differential quadrature method and finite element method. The equivalent geometric nonlinear parameter is presented to explain the influence of the scale effect on the pull-in location. Different from elastic material, there are two kinds of pull-in voltages called as instantaneous pull-in voltage and the durable pull-in voltage in viscoelastic system. Then, Galerkin discretization and the method of multiple scales are applied to determine the response and stability of the system for small vibration amplitude. A new perturbation method to deal with viscoelastic term is presented. Theoretical expressions about the parameter spaces of linear-like vibration, hardening-type vibration and softening-type vibration are then deduced. The influence of viscoelasticity and scale effect on nonlinear dynamic behavior is studied. Results show that the viscoelasticity can reduce the effective elastic modulus and make the system tend to softening-type vibration; the scale effect can increase effective elastic modulus and make the system tend to hardening-type vibration. And most of all, simulation results of case studies are used to realize parameter optimization. Then parameter conditions of linear-like vibration, which is desired for many applications, are obtained. In this paper, the results of multi-physical field coupling simulation are used to verify the theoretical analysis.