Circular Microplate Research Articles

Frequency characteristics of a GPL-reinforced composite microdisk coupled with a piezoelectric layer

This is the first research on the frequency analysis of a graphene nanoplatelet composite (GPLRC) microdisk in the framework of a numerical-based generalized differential quadrature method. The stresses and strains are obtained using the higher-order shear deformable theory. Rule of mixture is employed to obtain varying mass density, thermal expansion, and Poisson’s ratio, while module of elasticity is computed by modified Halpin–Tsai model. Governing equations and boundary conditions of the GPLRC microdisk covered with piezoelectric layer are obtained by implementing Hamilton’s principle. Regarding perfect bonding between the piezoelectric layer and core, the compatibility conditions are derived. In addition, due to the existence of piezoelectric layer, Maxwell’s equation is derived. The results show that outer-to-inner ratio of radius ($$ R_{\text{o}} /R_{\text{i}} $$), ratios of length scale and nonlocal to thickness (l/h and $$ \mu $$/h), ratio of piezoelectric to core thickness (hp/h), applied voltage, and GPL weight fraction (gGPL) have significant influence on the frequency characteristics of the GPLRC microdisk. Another important consequence is that in addition to the nonlinear indirect effects of applied voltage on the natural frequency of the GPLRC microdisk covered with piezoelectric for each specific value of $$ R_{\text{o}} /R_{\text{i}} $$, the impact of the $$ R_{\text{o}} /R_{\text{i}} $$ on the natural frequency is indirect. A useful suggestion of this research is that, for designing the GPLRC circular microplate at the low value of the $$ R_{\text{o}} /R_{\text{i}} $$ should be more attention to the gGPL and $$ R_{\text{o}} /R_{\text{i}} $$, simultaneously.

Vibration of circular micro-ceramic (Si3N4) plate resonators in the context of the generalized viscothermoelastic dual-phase-lagging theory

In this article, the analysis and numerical results are represented for the thermoelastic of an isotropic homogeneous, thermally conducting, Kelvin–Voigt-type circular micro-plate in the context of Kirchhoff’s Love plate theory of generalized viscothermoelasticity based on the dual-phase-lagging model. The governing equations are obtained for the generalized dual-phase-lagging model and coupled viscothermoelastic plates. The scaled viscothermoelasticity has been illustrated in the case of the circular plate and the axisymmetric circular plate for an aspect ratio for clamped boundary conditions. Laplace transform has been applied, and its inversions have been calculated numerically by using the Tzou method. The results have been carried out for the ceramic (Si3N4). It is noted that the temperature increment and lateral deflection are significantly affected by the time, the width, the thickness, and the mechanical relaxation times of the material.

Size-Dependent Green’s Function for Bending of Circular Micro Plates Under Eccentric Load

In this paper, a Green’s function is developed for bending analysis of micro plates under an asymmetric load. In order to consider the length scale effect, the modified couple stress theory is used. This theory can accurately predict the behavior of micro structures. A thin micro plate is considered and therefore the classical plate theory is utilized. The size dependent governing equilibrium equation of a circular micro plate under an eccentric load is obtained by using the minimum total potential energy principle. This equation is a partial differential equation and it is hard to solve it for an arbitrary loading. A transformation of the coordinate system is introduced to obtain the asymmetric exact solution for deflection of circular micro-plates. By using the obtained size dependent Green’s function, the bending behavior of microplates under arbitrary loads can be easily defined. The results are presented for different asymmetric loads. Also, it is concluded that the length scale has a significant effect on bending of micro plates.

Modeling and experimental characterization of squeeze film effects in nonlinear capacitive circular microplates

Abstract In this article, an original approach to model the squeeze film effects in capacitive circular microplates is developed. The nonlinear von Karman plate theory is used while taking into consideration the electrostatic and geometric nonlinearities of the clamped edge microplate. The fluid underneath the plate is modeled using the nonlinear Reynolds equation with a corrected effective dynamic viscosity due to size effect. The strongly coupled system of equations is solved using the Differential Quadrature Method (DQM) by discretizing the structural and the fluid domains into a set of grid points. The linear effects of the squeeze film on the microplate have been investigated based on the complex eigenfrequencies of the multiphysical problem. It is shown that the air film can alter the resonance frequencies by adding stiffness as well as damping to the system. The model has been validated numerically with respect to a Finite Element Model (FEM) implemented in ANSYS and experimentally on a fabricated circular microplates. The nonlinear effects of the squeeze film have been studied by determining the steady state solution of the system using the finite difference method (FDM) coupled with the arclength continuation technique. It is shown that the decrease of the static pressure shifts the resonance frequency and leads to an increase of the vibration amplitude due to the reduction of the damping coefficient, while the increase in the pressure enlarges the bistability domain. The developed model can be exploited as an effective tool to predict the nonlinear dynamic behavior of microplates under the effect of air film for the design of Capacitive Micromachined Ultrasonic Transducers (CMUTs).

Free vibration analysis of bilayered circular micro-plate including surface effects

• A new size-dependent Kirchhoff bilayered micro-plate model is developed. • A displacement field is proposed for the bilayered micro-plate. • Natural frequency is larger or smaller than those predicted by existing models. • Larger normalized effective flexural rigidity is predicted. • Transition region is expanded for stiffened surface and reduced for softened surface. Based on the Kirchhoff plate theory and the continuum surface elasticity theory, a novel model of bilayered circular micro-plates including surface effects is developed. Furthermore, the governing equation is derived, and the Galerkin method is employed to obtain approximate results. The influence of surface effects on the natural frequency of bilayered circular micro-plate is investigated. Besides, the effect of residual stresses on the transition behavior of the bilayered circular micro-plate from plate to membrane is predicted. The results are compared with the single-layer model and the modified laminated plate model. It is found that the natural frequency predicted by the present model is obviously different from that of the existing models. For different thickness ratios, the natural frequency obtained by the new model will be larger or smaller than the results of the single-layer model. The transition region from plate to membrane is expanded for stiffened surface and reduced for softened surface. The results of the novel model can provide effective guidance for the design of bilayered circular micro-plate in MEMS devices.

Nonlinear vibration analysis of partially coated circular microplate under electrostatic actuation

This paper investigates the optimal configuration for a partially two-layered circular capacitive microplate subjected to AC-DC electrostatic actuation. To this end, the static deflection due to DC electrostatic actuation, natural frequency of vibration about static position and primary resonance response due to AC electrostatic actuation are studied. Primarily, the nonlinear equations of motion are derived through classical laminated plate theory (CLPT). Then, the static position and natural frequency of vibration around static position are obtained using Galerkin approach. The linear mode shapes of non-uniform microplate i.e. a microplate coated as partial by a second layer are used as comparison functions. The forced vibration equations around static position are separated using Galerkin method, and solved by the multiple scale perturbation theory. Firstly, the impact of changes in the second layer radius on the variations of static and dynamic response of the system is studied while its thickness remains constant. Then, the effect of changes in the second layer thickness is studied while its radius remains constant. Finally, the impact of simultaneous change in the radius and thickness of the second layer is studied while its volume remains constant.

Multi-objective optimization of a circular dual back-plate MEMS microphone: tradeoff between pull-in voltage, sensitivity and resonance frequency

In this study, the optimization of a circular dual back-plate condenser microphone has been done in order to increase the pull-in voltage, sensitivity and resonance frequency simultaneously. Microphone’s diaphragm is assumed as a circular micro-plate subjected to symmetric two-sided electrostatic force. An accurate eighth order polynomial function is determined as the first mode shape of the circular micro-plate and Galerkin decomposition method is employed to find the analytical formulations for the microphone metrics performance. The analytical relations are validated by comparing them with finite element results. Next, the applied voltage, gap size, diaphragm radius and thickness are assumed as the design variables and modified non-dominated sorting genetic algorithm is utilized for multi-objective optimization procedure. Considering low standard deviations and high mean values, a new design point is chosen among the suggested optimal points so that there is an increment in each of three objective functions. Compare to a fabricated sample of a dual back-plate microphone, for the new design point, pull-in voltage has increased more than 3.6 times, sensitivity has improved 4% and resonance frequency has extended 24%.

Investigations of the Effects of Geometric Imperfections on the Nonlinear Static and Dynamic Behavior of Capacitive Micomachined Ultrasonic Transducers.

In order to investigate the effects of geometric imperfections on the static and dynamic behavior of capacitive micomachined ultrasonic transducers (CMUTs), the governing equations of motion of a circular microplate with initial defection have been derived using the von Kármán plate theory while taking into account the mechanical and electrostatic nonlinearities. The partial differential equations are discretized using the differential quadrature method (DQM) and the resulting coupled nonlinear ordinary differential equations (ODEs) are solved using the harmonic balance method (HBM) coupled with the asymptotic numerical method (ANM). It is shown that the initial deflection has an impact on the static behavior of the CMUT by increasing its pull-in voltage up to 45%. Moreover, the dynamic behavior is affected by the initial deflection, enabling an increase in the resonance frequencies and the bistability domain and leading to a change of the frequency response from softening to hardening. This model allows MEMS designers to predict the nonlinear behavior of imperfect CMUT and tune its bifurcation topology in order to enhance its performances in terms of bandwidth and generated acoustic power while driving the microplate up to 80% beyond its critical amplitude.

Global dynamics and integrity of a micro-plate pressure sensor

This paper investigates the complex bifurcation scenario of electrically-actuated circular micro-plates subjected to differential pressure. Our analysis deals with the primary, secondary and ultimate saddle-node bifurcation points responsible for the device snap-through and addresses the pressure range in which the robustness of the two main stable configurations is undermined by minor coexisting attractors. By making use of basins of attraction and integrity profiles, safe dynamical regions of motion are evaluated with respect to the applied DC voltage. It is found that period-doubling bifurcations are accountable for a sensible reduction in the dynamical integrity, small variations of the DC voltage largely modify the response of the pressure sensor.

Optimal Design of a Stable Fuzzy Controller for Beyond Pull-In Stabilization of Electrostatically Actuated Circular Microplates

The current paper aims to provide an optimal stable fuzzy controller to extend the travel range of a pair of flexible electrostatically actuated circular microplates beyond their pull-in limit. The single mode assumption is utilized to derive the equation of motion of the system based on a Lagrangian approach. The static behavior of the system is studied using the proposed model, and the utilized assumption and the relevant results are closely verified by nonlinear finite element simulations. The open-loop dynamic analysis is also performed to derive the linguistic rules governing the voltage-deflection behavior of the system. The mentioned rules are then employed for designing a fuzzy controller, which controls the deflection of the microplates. The controller is then optimized to provide better response specifications. The performance of the optimal fuzzy controller is compared with that of the optimal proportional–integral–derivative (PID) controller and obvious superiorities in terms of noise suppression and stability enhancement are observed. To guarantee the stability of the closed-loop system, another higher level controller is designed to oversee the behavior of the fuzzy controller. Simulation results reveal that the superintended fuzzy controller can prevent instability, while fairly extending the travel range of system and providing it with a better transient response. The suggested design approach proposed in this paper may be used to improve the performance of many nano/micro devices and nano/micro positioning systems.

Nonlinear vibration analysis of a circular micro-plate in two-sided NEMS/MEMS capacitive system by using harmonic balance method

In this study, forced nonlinear vibration of a circular micro-plate under two-sided electrostatic, two-sided Casimir and external harmonic forces is investigated analytically. For this purpose, at first, von Karman plate theory including geometrical nonlinearity is used to obtain the deflection of the micro-plate. Galerkin decomposition method is then employed, and nonlinear ordinary differential equations (ODEs) of motion are determined. A harmonic balance method (HBM) is applied to equations and analytical relation for nonlinear frequency response (F–R) curves are derived for two categories (including and neglecting Casimir force) separately. The analytical results for three cases: (1) semi-linear vibration; (2) weakly nonlinear vibration; (3) highly nonlinear vibration, are validated by comparing with the numerical solutions. After validation, the effects of the voltage and Casimir force on the natural frequency of two-sided capacitor system are investigated. It is shown that by assuming Casimir force in small gap distances, reduction of the natural frequency is considerable. The influences of the applied voltage, damping, micro-plate thickness and Casimir force on the frequency response curves have been presented too. The results of this study can be useful for modeling circular parallel-plates in nano/microelectromechanical transducers such as microphones and pressure sensors.

Study of Volumetric Flow Rate of a Micropump Using a Non-classical Elasticity Theory

The purpose of this research is to study the mechanical behavior of a micropump with clamped circular diaphragm which is the main component of drug delivery systems. In this paper, the non-linear governing equations of the circular microplate using Kirchhoff thin plate theory was been extracted based on the modified couple stress (MCST) and classical (CT) theories. Then, the non-linear equation of static deflection is solved using Step-by-Step Linearization Method (SSLM) in order to escape the nonlinearity of the differential equation and Galerkin-based reduced-order model is applied to investigate the dynamic motion of the microplate. Afterwards, static and dynamic stabilities of the micropump have been studied based on both MCST and CT, then compared. Also, volumetric flow rate of the micropump was been delved based on both theories and in entire research, presence of the length scale parameter in modified couple stress theory brings this opportunity to study the size effect on the mechanical behavior of the micropump.

Generalized viscothermoelasticity theory of dual-phase-lagging model for damping analysis in circular micro-plate resonators

Analysis and numerical results are presented for the thermoelastic dissipation of a homogeneous isotropic, thermally conducting, Kelvin–Voigt type circular micro-plate based on Kirchhoff’s Love plate theory utilizing generalized viscothermoelasticity theory of dual-phase-lagging model. The analytical expressions for thermoelastic damping of vibration and frequency shift are obtained for generalized dual-phase-lagging model and coupled viscothermoelastic plates. The scaled thermoelastic damping has been illustrated in case of circular plate and axisymmetric circular plate for fixed aspect ratio for clamped and simply supported boundary conditions. It is observed that the damping of vibrations significantly depend on time delay and mechanical relaxation times in addition to thermo-mechanical coupling in circular plate under resonance conditions and plate dimensions.

Modeling of the flexoelectric annular microplate based on strain gradient elasticity theory

ABSTRACTA size-dependent annular microplate model is established based on the isotropic flexoelectric theory, which considering the effects of strain gradient, polarization gradient, and microscopic electrical field gradient. The displacement mode of the Kirchhoff plate theory is used to obtain the detailed expressions of the electric enthalpy, dynamic energy, and the work done by external forces. The governing equations, boundary conditions, and initial conditions are derived using the Hamilton's principle. By solving the governing equations with only mechanical and only electrical boundary conditions, the analytical solutions of deflection and polarization in the static direct and converse flexoelectric responses are obtained, respectively. Meanwhile, the free vibration problem is also solved. Finally, based on the analytical solutions, some numerical results are given to illustrate the unique flexoelectric properties of the annular microplate compared with the solid circular microplate and influence of the strain gradient elastic effect on the static direct and converse flexoelectric responses as well as the natural frequency.

Effect of pressure on nonlinear dynamics and instability of electrically actuated circular micro-plates

Characterization of nonlinear behavior of micro-mechanical components in MEMS applications plays an important role in their design process. In this paper, nonlinear dynamics, stability and pull-in mechanisms of an electrically actuated circular micro-plate subjected to a differential pressure are studied. For this purpose, a reduced-order model based on an energy approach is formulated. It has been shown that nonlinear dynamics of an electrically actuated micro-plate, in the presence of differential pressure, significantly differs from those under purely electrostatic loads. The micro-plate may lose stability upon either saddle-node or period-doubling bifurcations. It has also been found that in the presence of a differential pressure, increasing the DC or AC voltages may surprisingly help to stabilize the motion of the micro-plate.

Free vibration and critical angular velocity of a rotating variable thickness two-directional FG circular microplate

In this paper, the free vibration of a rotating variable thickness two-directional FG circular microplate is studied. The governing equations of motion for the microplate are extracted utilizing the Hamiltonian’s principle in conjunction with the first shear deformation theory as well as the modified couple stress theory. The solution of equations is presented utilizing the differential quadrature method. In special cases, the natural frequency results obtained by the reduced form of the proposed formulation are compared with those available in the literature, indicating a very good accuracy. The results reveal that there is a non-proportional relation between the natural frequencies of the microplate and the thickness-variations of the section. In contrast, the critical angular velocity of that is not much sensitive with respect to the thickness variation. Moreover, the analyses indicate the significant impact of the two-directionality-variation of the graded material on the natural frequencies as well as the critical angular velocities. A map on the effects of the two-directionality-variation of the material property on the free vibration of the microplate is presented. The results show that the increase of the size dependency would lead to the reduction of the non-dimensional natural frequency as well as the critical angular velocity.

Size dependent thermal buckling and postbuckling of functionally graded circular microplates based on modified couple stress theory

ABSTRACTIn this article, size-dependent thermal buckling and postbuckling behavior of a functionally graded circular microplate under uniform temperature rise field and clamped boundary conditions is investigated. Material properties are assumed to gradually vary through the thickness according to a simple power law. Equilibrium equations and associated boundary conditions are derived using variational method and based on modified couple stress theory, classical plate theory and von Kármán geometric nonlinearity. The differential quadrature method is used to discretize the governing equations. This technique is accompanied by an iterative method to determine the thermal postbuckling behavior of microplate. Finally, effects of length scale parameter, power law index and ratio of thickness to radius on the thermal buckling and postbuckling behavior of FG circular microplate are investigated.

The Stability and Non-linear Vibration Analysis of a Circular Clamped Microplate under Electrostatic Actuation

A study was made of the dynamic response of a circular clamped micro-plate actuated by a DC/AC voltage. The analysis considers not only the non-linear electrostatic coupling force, residual stress effect, and hydrostatic pressure acting on the upper surface, but also the squeeze-film damping effect generated by the air gap between the vibrating micro-plate and the fixed substrate. The non-linear governing equation of motion of the circular micro-plate was solved using a hybrid numerical method comprising the differential transformation and the finite difference. It was shown that the numerical results obtained for the pure DC pull-in voltage deviate by no more than 1.6% from the results presented in the literature. The effects of the actuating voltage, hydrostatic pressure, squeeze-film damping, and residual stress on the dynamic response of the clamped circular micro-plate were systematically examined. In addition, the stability of the vibrating micro-plate was investigated examined by reference to phase portraits.

A comparison of strain gradient theories with applications to the functionally graded circular micro-plate

• The strain gradient elasticity theory of Zhou et al. is re-expressed. • The formulations of present theory in cylindrical coordinates are derived. • The differences of strain gradient theories are investigated. • A FGM circular plate model is established. • The present model can degenerate to the corresponding models of other theories. The strain gradient theory of Zhou et al. is re-expressed in a more direct form and the differences with other strain gradient theories are investigated by an application on static and dynamic analyses of FGM circular micro-plate. To facilitate the modeling, the strain gradient theory of Zhou et al. is re-expressed in cylindrical coordinates, and then the governing equation, boundary conditions and initial condition for circular plate are derived with the help of the Hamilton's principle. The present model can degenerate into other models based on the strain gradient theory of Lam et al., the couple stress theory, the modified couple stress theory or even the classical theory, respectively. The static bending and free vibration problems of a simply supported circular plate are solved. The results indicate that the consideration of strain gradients results in an increase in plate stiffness, and leads to a reduction of deflection and an increase in natural frequency. Compared with the reduced models, the present model can predict a stronger size effect since the contribution from all strain gradient components is considered, and the differences of results from all these models are diminishing when the plate thickness is far greater than the material length-sale parameter.

Size-dependent axisymmetric vibration of functionally graded circular plates in bifurcation/limit point instability

In the present paper, vibration behavior of size-dependent functionally graded (FG) circular microplates subjected to thermal loading are carried out in pre/post-buckling of bifurcation/limit-load instability for the first time. Two kinds of frequently used thermal loading, i.e., uniform temperature rise and heat conduction across the thickness direction are considered. Thermo-mechanical material properties of FG plate are supposed to vary smoothly and continuously throughout the thickness based on power law model. Modified couple stress theory is exploited to describe the size dependency of microplate. The nonlinear governing equations of motion and associated boundary conditions are extracted through generalized form of Hamilton’s principle and von-Karman geometric nonlinearity for the vibration analysis of circular FG plates including size effects. Ritz finite element method is then employed to construct the matrix representation of governing equations which are solved by two different strategies including Newton–Raphson scheme and cylindrical arc-length method. Moreover, in the following a parametric study is accompanied to examine the effects of the several parameters such as material length scale parameter, temperature distributions, type of buckling, thickness to radius ratio, boundary conditions and power law index on the dimensionless frequency of post-buckled/snapped size-dependent FG plates in detail. It is found that the material length scale parameter and thermal loading have a significant effect on vibration characteristics of size-dependent circular FG plates.