Static Pull-in Voltage Research Articles

Pull-in instability and nonlinear vibration of microbeam with piezoelectric patch driven by electrostatic force using modified couple stress theory

The static and dynamic pull-in characteristics of electrostatically and piezo-electrically actuated microbeam are investigated. The static pull-in voltage is evaluated using analytical methods based on MCST, CCMT and validated through the use of FE based COMSOL multiphysics tools. The dynamic pull-in voltage obtained from numerical technique and verified with existing literature, lower than the static pull-in voltage. The piezoelectric patch has a significant influence on the pull-in voltage and an effect on the aspect ratio. Moreover, the pull-in band and instability zone of the microbeam with weak and hard forcing function have been investigated using multiple scale method.

Intermolecular forces and surface stress residual effects on the size-dependent performance analysis of nanocrystalline micro/nano gyroscopes

This work investigates an electrostatically-actuated vibrating beam micro/nano-gyroscope made of nanocrystalline silicon material. The comprehensive size-dependent micro/nanogyroscope model is developed by couple stress and surface elasticity theories. Then, considering the effects of intermolecular forces (Casimir force and van der Waals force) and surface stress residual, an in-depth investigation was conducted on the performance of the micro/nanogyroscope. The calculation results display that the maximum static deflection, pull-in voltage and natural frequency change obviously with the increase of surface stress residual as the scale varies from micrometer to nanoscale. However, the effect of intermolecular forces on the performance of gyroscopes is exactly the opposite, which improves the prediction of system structural softening deformation phenomenon. Considering the influence of the Casimir and van der Waals forces in micro/nano gyroscopic system will help gyroscopes to obtain reliability and functionality sensitivity, increase the accuracy of predicting static deformation, natural frequency and differential frequency. Moreover, a frequency-domain method using the difference between the two vibration natural frequencies is studied, with changing the scale of the gyroscope under the effect of the intermolecular forces. The numerical simulations exhibit that when considering smaller structure sizes of the gyroscope, strong coupling occurs between the intermolecular forces and surface stress residual, which is helpful for the design of micro/nano gyroscopes.

Exploring nonlinearity to enhance the sensitivity and bandwidth performance of a novel tapered beam micro-gyroscope

In this paper, nonlinearity is used to enhance the sensitivity and bandwidth performance of tapered beam micro-gyroscopes. Considering the electrostatic force nonlinearity and fringing field effects, the governing equations of system are derived by Euler angular transformation and Hamilton principle. The influences of parameters such as the shape factors and DC voltage on static deflection, pull-in voltage and natural frequency of tapered beam micro-gyroscope are analyzed by the Adomian decomposition method (ADM). Galerkin discretization is performed on the governing equations of the tapered beam micro-gyroscope. Frequency-response equation of the system is deduced and solved by applying the method of multi-scale (MMS). The convergence analysis of the frequency response curve is performed, and then verified by the Runge-Kutta method. Finally, the performance of the tapered beam micro-gyroscope is analyzed in terms of bandwidth, sensitivity, and calibration curve nonlinearity under the different shape factors. The maximum amplitude of response in the driving and sensing directions increases significantly with the change of shape factors of the tapered beam in the dynamics analysis. Compared with the straight beam micro-gyroscope, the tapered beam micro-gyroscope has the higher amplitude and pronounced nonlinear characteristics. Due to the electrostatic force nonlinearity, the amplitude of motion in the driving direction has a high response over a wide frequency band, while also effectively increasing the detection bandwidth. In addition, the nonlinear micro-gyroscope has less loss of sensitivity performance during damping fluctuations. Therefore, in view of the contradiction between bandwidth and sensitivity of the traditional micro-gyroscope, this paper proposes a solution from the perspective of structural design and nonlinear characteristics of the tapered beam micro-gyroscope.

Influence of Electrostatic Force Nonlinearity on the Sensitivity Performance of a Tapered Beam Micro-Gyroscope Based on Frequency Modulation.

Electrostatic force nonlinearity is widely present in MEMS systems, which could impact the system sensitivity performance. The Frequency modulation (FM) method is proposed as an ideal solution to solve the problem of environmental fluctuation stability. The effect of electrostatic force nonlinearity on the sensitivity performance of a class of FM micro-gyroscope is investigated. The micro-gyroscope consists of a tapered cantilever beam with a tip mass attached to the end. Considering the case of unequal width and thickness, the motion equations of the system are derived by applying Hamilton's principle. The differential quadrature method (DQM) was used to analyze the micro-gyroscope's static deflection, pull-in voltage, and natural frequency characteristics. We observed that from the onset of rotation, the natural frequencies of the drive and sense modes gradually split into a pair of natural frequencies that were far from each other. The FM method directly measures the angular velocity by tracking the frequency of the drive and sense modes. Then, based on the linear system, the reduced-order model was used to analyze the influence of the shape factor and DC voltage on the sensitivity performance. Most importantly, the nonlinear frequency of system was obtained using the invariant manifold method (IMM). The influence of electrostatic force nonlinearity on the performance of the FM micro-gyroscope was investigated. The results show that the different shape factors of width and thickness, as well as the different DC voltages along the drive and sense directions, break the symmetry of the micro-gyroscope and reduce the sensitivity of the system. The sensitivity has a non-linear trend with the rotation speed. The DC voltage is proportional to the electrostatic force nonlinearity coefficient. As the DC voltage gradually increases, the nonlinearity is enhanced, resulting in a significant decrease in the sensitivity of the micro-gyroscope. It is found that the negative shape factor (width and thickness gradually increase along the beam) can effectively restrain the influence of electrostatic force nonlinearity, and a larger dynamic detection range can be obtained.

Performance of multifunctional piezoelectric energy harvesting microgyroscopes with material degradation

Most microelectromechanical systems (MEMS) are made from brittle materials that are prone to cracking. Damage from the formation of cracks can occur anytime during the manufacturing process or due to mechanical loading of MEMS devices. Cracks can increase the structures flexibility and lead to changes in performance or total failure. Understanding the potential influences cracks in MEMS is necessary for the longevity of these systems. The energy harvesting microgyroscope studied in this work is susceptible to crack propagation in its silicon substrate and lead zirconate titanate piezoelectric patch. To this end, an electromechanical model is developed to capture the impacts of cracks on the performance of a damaged energy harvesting microgyroscope. Numerical methods are used to approximate the damage associated with distributed crack networks. Then, the static and dynamic responses of the damaged microgyroscope are analyzed for two possible multifunctional setups. The current study revealed that cracks can significantly affect the static and dynamic behaviors of the microgyroscope energy harvester. Combinations of cracking and piezoelectric property degradation cause a decrease in the static pull-in voltage and shift the resonance frequencies of the system. Additionally, it is shown that the systems sensitivity to changes in the base rotation and energy harvesting capabilities vary widely depending on the severity of material degradation.

Pull-in Phenomenon in the Electrostatically Micro-switch Suspended between Two Conductive Plates using the Artificial Neural Network

Artificial Neural Networks (ANN) are designed to evaluate the pull-in voltage of MEMS switches. The mathematical model of a micro-switch subjected to electrostatic force is preliminarily illustrated to get the relevant equations providing static deflection and pull-in voltage. Adopting the Step-by-Step Linearization Method together with a Galerkin-based reduced order model, numerical results in terms of pull-in voltage are obtained to be employed in the training process of ANN. Then, feed forward back propagation ANNs are designed and a learning process based on the Levenberg-Marquardt method is performed. The ability of designed neural networks to determine pull-in voltage have been compared with previous results presented in experimental and theoretical studies and it has been shown that the presented method has a good ability to approximate the threshold voltage of micro switch. Furthermore, the geometric and physical effect of the micro-switch on the pull-in voltage was also examined using these designed networks and relevant findings were provided.

Nonlinear size-dependent modeling and dynamics of nanocrystalline arc resonators

The adequate modeling of the micro/nano arc resonators' dynamics is vital for their successful implementation. Here, a size-dependent model, wherein material structure, porosity, and micro-rotation effects of the grains are considered, is derived by combining the couple stress theory, multi-phase model, and the classical Euler–Bernoulli beam model, aiming to characterize the frequency tunability of micro/nano arc resonators as monitoring either the axial load or the electrostatic force for the first time. The arc dimensions are optimized to show various phenomena in the same arc, namely snap-through, crossing, and veering. The first three natural frequencies are monitored, showing the size dependency on the frequency tuning, snap-through/back, and pull-in instability as shrinking the scale from micro- to nano-scale. Significant changes in the static snap-through and pull-in voltages and the resonance frequencies were shown as scale shrinks. A dynamic analysis of the resonator's vibration shows a dramatic effect of the size-dependency as shrinking dimensions around the veering zone.

On the Static Pull-In of Tilting Actuation in Electromagnetically Levitating Hybrid Micro-Actuator: Theory and Experiment

This work presents the results of the experimental and theoretical study of the static pull-in of tilting actuation executed by a hybrid levitation micro-actuator (HLMA) based on the combination of inductive levitation and electrostatic actuation. A semi-analytical model to study such a pull-in phenomenon is developed, for the first time, as a result of using the qualitative technique based on the Lagrangian approach to analyze inductive contactless suspensions and a recent progress in the calculation of mutual inductance and force between two circular filaments. The obtained non-linear model, accounting for two degrees of freedom of the actuator, allows us to predict accurately the static pull-in displacement and voltage. The results of modeling were verified experimentally and agree well with measurements.

Effect of the second invariant of the Cauchy–Green deformation tensor on the local dynamics of dielectric elastomers

This paper aims to investigate the local oscillations of a dielectric elastomer with emphasis on the effect of the second invariant of the deformation tensor in finite strain theory on such oscillations. Four hyperelastic constitutive models are utilized in this study, namely the classic Gent model, a modified Gent model, a modified version of the Pucci–Saccomandi model, and the Mooney–Rivlin model. We derive the governing equations for all the models through the use of the Euler–Lagrange formulation. We explore the local responses of the system in the framework of the nonlinear frequency responses by solving the equations of motion using the multiple time scales method. In addition, to provide an in-depth analysis, we analyze static and dynamic electromechanical instabilities by plotting the voltage-stretch diagram and the time signature. The numerical results for damped and undamped dielectric elastomers are simulated and compared. We analyze the system’s response for the aforementioned hyperelastic constitutive functions, and compare the differences and similarities of models. Based on the results, two saddle–node bifurcations occur in the system. Generally, increasing the second invariant parameters of the hyperelastic models, decreases the response amplitude. As contribution of the second invariant in the Mooney–Rivlin model is increased, for both damped and undamped systems, softening turns into hardening nonlinearity. We also show that the second invariant tunes the bifurcation points and instability of the system. Moreover, it can control the static and dynamic pull-in and snap-through voltages.

Pull-In Dynamics of Two MEMS Parallel-Plate Structures for Acceleration Measurement

This paper investigates the pull-in dynamics of single-sided and double-sided parallel-plate (SSPP and DSPP) structures employed in MEMS devices for acceleration measurement. A lumped electromechanical model considering a variable damping coefficient and external acceleration is developed for studying their pull-in dynamical behavior. The critical pull-in displacement and pull-in voltage for both parallel-plate structures operated with external acceleration are derived based on the static and dynamical analysis of the proposed model, which are in good agreement with the experimental results. The derived closed-form solutions are shown to be capable of predicting the pull-in behavior of both parallel-plate structures in an overdamped system. Two MEMS parallel-plate structures exhibit distinct pull-in characteristics due to their distinct configurations. The commonly used SSPP structure for MEMS devices is shown to have a relatively linear response to the external acceleration in terms of the critical pull-in displacement and pull-in voltage. However, the DSPP structure exhibits much larger yet nonlinear sensitivity of both the critical pull-in displacement and the critical pull-in voltage to a small range of external acceleration, which has the potential of developing high-sensitivity accelerometers whenever the sensitivity nonlinearity is compensated. Moreover, the effects of pressure are characterized for both DSPP and SSPP devices, which affects the difference between the critical static and dynamic pull-in voltages. The SSPP structure is shown to have the critical pull-in voltage proportional to the logarithmic magnitude of the pressure, while that of the DSPP structure do not vary linearly with respect to the logarithmic magnitude of the pressure.

Nonlinear vibration and stability of functionally graded porous microbeam under electrostatic actuation

In this work, a microbeam model based on the Euler–Bernoulli beam theory and the nonlocal strain gradient theory was developed to study nonlinear vibration and stability of a functionally graded porous microbeam under electrostatic actuation. The electrostatic force is the driving force per unit length, resulting from electrostatic excitation. The material properties of the microbeam change continuously along thickness direction according to simple power-law distribution in terms of the fractions of constituents. Two cases of even and uneven porosity distributions are considered. The Hamilton principle is used to obtain the governing equations of the motion for the microbeam. The equation of motion for the microbeams is reduced to a nonlinear ordinary differential equation with rational restoring force by applying Galerkin method. The frequency–amplitude relationship is given in the closed form by employing He’s Hamiltonian approach. The expression of the static pull-in voltage is also derived. Effects of the material length scale parameter, the nonlocal parameter, the power-law index, the porosity distribution factor, the applied voltage and the initial amplitude on nonlinear vibration and stability of the functionally graded porous microbeam are examined and discussed.

A flexible model for studying fringe field effect on parallel plate actuators

Electrostatic parallel plate actuators are common in micro-electro-mechanical systems due to their compatibility with micro-fabrication technology. Parallel plate actuators suffer from an instability problem called the pull-in phenomenon that happens when the applied DC voltage exceeds a certain value called the static pull-in voltage. The value of this critical voltage is important in many applications that depend on parallel plate actuators such as switches and static gas sensors. The fringe field around the edges of the plates could severely affect the performance of the actuator. This paper introduces a new model for the parallel plate actuator to calculate the value of static pull-in voltage. The proposed model considers the fringe field between the two plates. The static pull-in voltage of some PolyMUMPs actuators is practically measured and compared to the simulation results that involve the fringe field effect. The model is simulated using MATLAB to show the influence of that field on the static pull-in voltage. The MATLAB results of the proposed model are validated with ANSYS.

Simulation of Pull-in mechanism of plate actuator considering damping

Based on electrostatic micro-actuator, energy method is used to analyses its static and dynamic Pull-in mechanism. In addition, the model of electrostatic micro-actuator perforated-plate is used for example analysis to derive its dynamic Pull-in parameter without and with considering squeeze damping effect. By comparing the parameter values from MATLAB calculation and ANSYS simulation under the two circumstances mentioned above, the effects of damping ratio on Pull-in parameter can be obtained. The results show that, the dynamic Pull-in displacement is more than the static Pull-in displacement. And with the existence of squeeze damping effect, Pull-in voltage will increase. When the damping ratio reaches a certain extent, the dynamic Pull-in voltage value will be as large as the static Pull-in voltage value.

Investigation of pull-in behaviors of a nanoswitch tuned by piezoelectric and flexoelectric effects

In this work, the static and dynamic pull-in behaviors of an electrostatically and piezoelectrically actuated nanoswitch are investigated. The nanoswitch is based on a cantilever nanobeam with a piezoelectric actuation layer attached. In the modeling, we consider the flexoelectric effect of the piezoelectric layer, the surface effect of the unimorph as well as the Casimir effect. The governing equation of the proposed model is derived using the Hamilton's principle and discretized by employing the Galerkin method. Firstly, the results of the proposed model are compared with the numerical and experimental data, providing the validity of this model. Then, simulations are performed to show the influence of flexoelectricity on the static pull-in voltage and pull-in deflection of the nanoswitch with a lead zirconate titanate (PZT-5H) and a polyvinylidene fluoride (PVDF) actuation layer, respectively. Results indicate that the flexoelectric effect plays a significant role in determining the static pull-in behaviors of the nanoswitch and such an effect is size-dependent at the nanoscale. Moreover, we show how the flexoelectric effect influences the unstable region of dynamic pull-in instability of the proposed nanoswitch. This work suggests that the flexoelectric effect of the piezoelectric layer can be used to tune the static and dynamic pull-in behaviors of a nanoswitch, which may guide the design and development of devices in the micro- and nano-electro-mechanical-systems (M/NEMS).

Static pull-in instability and free vibration of functionally graded graphene nanoplatelet reinforced micro-sandwich beams under thermo-electrical actuation

This paper investigates the static pull-in instability and free vibration of a multilayer functionally graded graphene nanoplatelet (GPL) reinforced composite (FG-GPLRC) micro-beam sandwiched between two copper layers subjected to a combined action of an electric voltage and a uniform temperature change based on Euler–Bernoulli beam theory. The GPL nanofillers are uniformly dispersed within each individual layer while its weight fraction changes from layer to layer in the multilayer FG-GPLRC micro-beam. The modified Halpin–Tsai model is used to predict the effective Young’s modulus while the rule of mixture is used to determine the effective Poisson’s ratio, mass density and thermal expansion coefficient. The static pull-in voltage and natural frequency of clamped–clamped micro-beams are obtained by employing Galerkin and iterative method. The effects of GPL distribution pattern, weight fraction, geometry and size as well as the geometry of the beam, the temperature change and the total number of layers on the static and dynamic characteristics of the micro-beams are discussed in detail.

Qualitative Identification of the Static Pull-In and Fundamental Frequency of One-Electrode MEMS Resonators.

This paper attempts to qualitatively identify the static pull-in position, pull-in voltage, and fundamental frequency of one-electrode microresonators from a physical perspective. During theoretical derivation, a generalized one-degree-of-freedom (1-DOF) model in nondimensional form derived using the differential quadrature method (DQM) is first introduced and then transformed for frequency normalization. Based on the deduced formulas, the upper and lower bounds of the static pull-in position and pull-in voltage are both deduced through mathematical proof. To distinguish the monotonic and nonmonotonic behavior of the fundamental frequency versus direct current (DC) voltage, a critical condition decided only by cubic stiffness is then determined. For the first time, two extreme static positions, as well as the corresponding fundamental frequencies and DC voltages to identify different frequency behaviors are derived, and their variations versus cubic stiffness are then discussed and verified. During the simulation process, a high-order DQM and COMSOL 2D model are both applied for numerical analyses. Guided by nondimensional results, typical behaviors with specific physical parameters are examined in detail. Results demonstrate that the curve tendencies between all the qualitative results and quantitative numerical simulations in dimensional form agree well with each other, implying the possibility of using 1-DOF model to qualitatively discuss physical parameters effects on the system statics and dynamics.

Dynamics and vibration analysis of an electrostatically actuated FGM microresonator involving flexural and torsional modes

In this paper a nonlinear pedal like microresonator made of Functionally Graded material is introduced and modeled. Using Lagrangian formulation nonlinear equations of motion of the system are derived and the natural frequencies and mode shapes are extracted by linear analysis. The effects of various parameters such as geometry, FG material index and DC voltages on the natural frequencies and mode shapes are investigated. Static pull-in voltages of the system are obtained and it is found that depending on the type of actuation, both softening and hardening behaviors can be observed in the system. The results are compared with previous studies and finite element method where a good agreement is achieved. A two-mode reduced order model along with first-order averaging technique is implemented to derive the asymptotic dynamic equations of motion. The effects of different parameters including AC actuation level, damping and material power index on the dynamic response of the system are studied around resonance conditions. It is observed that when the system parameters are adjusted properly, the directly excited in-plane mode, in turn excites the out-of-plane torsional mode of the structure through internal resonance of 1:2 ratio within a specific frequency bandwidth. A wide range of frequency will be available by changing only the power index of the material and the system can be manipulated to be used in various MEMS applications such as mass sensors and RF filter-mixer devices.

Nonlinear analysis and characteristics of electrically-coupled microbeams under mechanical shock

Several studies have shown that MEMS devices deploying electrically-actuated vibrating beams, such as resonant sensors and RF filters may fail to operate when undergoing mechanical shocks due to the pull-in instability. To this end, we investigate the possibility to overcome or exploit this issue by considering different microsystem designs based on the application of interest. This objective is carried out through developing a nonlinear reduced-order model to simulate the dynamic response of single and dual microbeams under varying electric actuation and shock loads. The actuation of the single-beam system is made via a fixed electrode (uncoupled actuation) while the dual-beam system, composed of two movable microbeams, is actuated by applying a voltage among them (coupled actuation). We use the Galerkin method to discretize the governing equations in space and the Runge–Kutta method to integrate the resulting nonlinear ordinary differential equations. We first perform the static analysis to determine the pull-in voltage. We formulate the coupled eigenvalue problem to compute the natural frequencies of the microsystems under investigation for different applied DC voltages. Then, we introduce the AC excitation and generate the frequency-response curves. Finally, we analyze the impact of the mechanical shock (represented by an impact pulse acceleration) on the microsystems’ dynamic behavior. The present results are in good agreement with those obtained from previously-published theoretical and experimental studies. We observe a significant reduction in the static pull-in voltage and switching time when considering the dual-beam system in comparison with the single-beam case. The frequency-response curves show expanded dynamic pull-in bandwidth when operating the dual-beam system near the primary resonance. We notice that the dual-beam systems are more robust in terms of resistance to mechanical shock. This shows the suitability of such design for the operation and reliability of MEMS devices in harsh environments characterized by high mechanical shock levels. On the other hand, single-beam systems seem to be more attractive for use as microswitches which are intended to trigger a signal once receiving a mechanical shock or abrupt change in acceleration to activate safety functionalities, such as airbag systems.

An inherent nonlinear electrostatic actuator without any additional auxiliary to reduce pull-in voltage and extend travel range

This paper proposes an electrostatic actuator suspended on clamped-guided beams with inherent nonlinear behavior. Under large deflection, a flat clamped-guided beam is characterized by inherent cubic nonlinear force–displacement behavior without any additional auxiliary. We utilize the inherent nonlinearity to reduce pull-in voltage, extend travel range and simplify configurations. The theoretical analyses on pull-in voltage and travel range of the electrostatic actuator with cubic nonlinear spring were comprehensively implemented. In order to examine the proposed approach, the electrostatic actuators with a cubic suspension spring and a linear suspension spring, respectively, were fabricated. The experimental results show that the static and dynamical pull-in voltages for the cubic suspension spring, respectively, are significantly reduced by 38.6% and 36% in comparison with the linear suspension spring, and the stable and dynamic travel ranges, respectively, can be extended to 51% and 70.3% of the initial gap. These measured performances are generally slightly worse than the designed values mainly due to the subdued nonlinear degree of suspension spring caused by internal stress. The higher nonlinear level of the suspension spring is more advantageous for reduction of pull-in voltage and extension of travel range.

An improved analytical model for static pull-in voltage of a flexured MEMS switch

This paper presents the design of low-k meander based MEMS shunt capacitive switch with beam perforations. A closed form model to accurately calculate the pull-in voltage of the designed switch for two cases have been presented viz. a non-uniform meander based MEMS shunt switch and an uniform serpentine meander based MEMS shunt switch with perforated structure. The modified Mejis and Fokkema’s capacitance model have been used to propose a generalized closed form expression for the pull-in voltage which takes care of the nonlinear electrostatic force on the switch as well. The proposed model also takes into account the fringing field effect due to beam thickness and etched holes on the beam. The model is validated by calculating the pull-in voltage for both the meander designs under variation of various design parameters. The results obtained under most design specification variation has been found out to be in the range of 1.5–2.3 V for uniform meander based MEMS shunt switch and 3.2–5.2 V for the non-uniform counterpart. The model based results have been further verified by comparison with simulated results of full 3D FEM solver CoventorWare in a wide range of structural parameter variations. It has been observed that the performance of the proposed model is reasonably satisfactory with an average deviation of 4.73% for uniform serpentine flexure and 3.65% for non-uniform flexure based switch.