Analysis of the dynamic behaviour of a torsional micro-mirror

In this paper, the results of the dynamic pull-in voltage characteristics of a micro-mirror using electrostatic actuation are analyzed. Based on torsional dynamic theory, appropriate equations are developed that allowed to give insight into the actuating voltage, switching time and other dynamic parameters. The analytical results are discussed in detail without and with considering air squeeze film damping, respectively. This is equivalent to assuming the mirror is operated in vacuum or at ambient pressure. When the effect of the damping is considered, the movement trajectory of the cantilever beam is changed, and the calculated results of the pull-in voltage and switching time are considerably different compared to those without considering damping. Therefore, the effect of the air squeeze film damping is an important factor in the design and fabrication of micro-electro-mechanical systems. Finally, the experimental results in the air environment are discussed and compared to the theoretical analysis.     

虚拟环境中微加速度传感器建模及拟实运行

对虚拟环境中微加速度传感器建模及拟实运行进行了研究.此微加速度传感器采用悬臂梁式结构.论文分析了其静态特性和动态特性,建立了悬臂梁变形模型和微加速度传感器动态行为模型;讨论了在修改结构尺寸过程中,梁长、梁厚对微加速度传感器的影响,并优化了结构参数;最后建立虚拟环境,进行拟实运行,验证设计结果.     

Design, fabrication and characterization of a high fill-factor micromirror array for wavelength selective switch applications

This paper presents the design, fabrication and characterization of a high fill-factor micromirror array in application of wavelength selective switch (WSS). The micromirror array consists of 52 independent micromirrors. Each micromirror is composed of a cantilever-type micromirror plate (800 μm × 120 μm) with a bumper and an eight-terraced bottom electrode with a limiting plane. A cantilever beam is designed to obtain the rotation angle of micromirror plate and achieve a high fill-factor for the micromirror array. Meanwhile, the bumper and limiting plane are used to prevent the damage possibly caused by the pull-in effect or some vibration instance. An eight-terraced electrode is utilized for reducing the driving voltage. The micromirror array with a high fill-factor in excess of 97% has been successfully achieved using the bulk micromachining technologies. The measured static and dynamic characteristics show that the micromirror can achieve a maximal rotation angle of 0.87° with a Direct Current (DC) driving voltage of 156 V. The turn-on responding time is 0.57 ms, and the turn-off responding time is 4.36 ms. Furthermore micromirror plate can be easily released from the pull-in state without damaged due to the novel bumper design. The switching function between the two output ports of a WSS optical system has also been demonstrated.     

Dynamic response of a torsional micromirror to electrostatic force and mechanical shock

In this paper dynamic characteristics of a capacitive torsional micromirror under electrostatic forces and mechanical shocks have been investigated. A 2DOF model considering the torsion and bending stiffness of the micromirror structure has been presented. A set of nonlinear equations have been derived and solved by Runge–Kutta method. The Static pull-in voltage has been calculated by frequency analyzing method, and the dynamic pull-in voltage of the micromirror imposed to a step DC voltage has been derived for different damping ratios. It has been shown that by increasing the damping ratio the dynamic pull-in voltage converges to static one. The effects of linear and torsional shock forces on the mechanical behavior of the electrostatically deflected and undeflected micromirror have been studied. The results have shown that the combined effect of a shock load and an electrostatic actuation makes the instability threshold much lower than the threshold predicted, considering the effect of shock force or electrostatic actuation alone. It has been shown that the torsional shock force has negligible influence on dynamic response of the micromirror in comparison with the linear one. The results have been calculated for linear shocks with different durations, amplitudes, and input times.     

Dynamic characteristics and forced response of an electrostatically-actuated microbeam subjected to fluid loading

In this paper flexural vibrations of an electrostatically actuated cantilever microbeam in an incompressible inviscid stationary fluid have been studied. By applying “Three dimensional aerodynamic theory” pressure jump across the microbeam has been investigated and the inertial effects of fluid on microbeam dynamics have been modeled as a mass added to microbeam mass. Magnitude of the added mass has been calculated for various aspect ratios of cantilever microbeams and compared with those of clamped-clamped microbeams. To investigate the dynamic characteristics, it has been considered that the microbeam has been deflected by a DC voltage, V _DC and then the dynamic characteristics and forced response of the system have been considered about these conditions. Galerkin-based step by step linearization method (SSLM) and Galerkin-based reduced order model have been applied to solve the nonlinear static and dynamic governing equations, respectively. Water by neglecting viscidity effects, as an instant has been considered as a surrounding fluid and the frequency response of the microbeam has been compared with that of vacuum conditions. It has been shown that because of the added mass effects in watery environment, the natural frequencies of the microbeam decrease. Because of the higher dielectric coefficient and increasing electrical stiffness and decreasing total stiffness consequently, maximum amplitude of the microbeam vibrations increases in watery environment, compared with vacuum. Moreover, it has been shown that increasing the DC voltage, increases the electrical stiffness and maximum amplitude of the microbeam vibrations, consequently, It has been shown that in higher voltages (near pull-in voltage), the rate of variation of resonance frequency and maximum amplitude is stronger than lower voltages.     

Influence of van der Waals force on the pull-in parameters of cantilever type nanoscale electrostatic actuators

In this paper, the influence of the van der Waals force on two main parameters describing an instability point of cantilever type nanomechanical switches, which are the pull-in voltage and deflection are investigated by using a distributed parameter model. The fringing field effect is also taken into account. The nonlinear differential equation of the model is transformed into the integral form by using the Green’s function of the cantilever beam. The integral equation is solved analytically by assuming an appropriate shape function for the beam deflection. The detachment length and the minimum initial gap of the cantilever type switches are given, which are the basic design parameters for NEMS switches. The pull-in parameters of micromechanical electrostatic actuators are also investigated as a special case of our study by neglecting the van der Waals force.     

A comprehensive model to study nonlinear behavior of multilayered micro beam switches

A novel model to study the pull-in behavior of nonlinear electromechanically coupled systems has been developed. The proposed model is based on the multilayered cantilever and fixed–fixed micro beam type MEMS switches. Due to the complexity of the nonlinear beam mechanics, exact analytical solutions are not generally available; therefore, the derived nonlinear equation has been numerically solved fully using the nonlinear finite difference method. Furthermore, the results obtained are summarized and compared with the other existing empirical and analytical models. These results can be useful in the optimization of MEMS switch designs or other actuators. In addition, the method developed in this paper has a good potential for analyzing other types of complex MEMS devices. An erratum to this article can be found at     

High-speed imaging by electromagnetic alloy actuated probe withdual spring

This paper describes a magnetically actuated cantilever with dual spring (cantilevered actuator and torsional cantilever) for a high-speed imaging of atomic force microscopy (AFM). A fabricated cantilever beam with a high resonant frequency is successfully actuated by electromagnetic force. A planar coil is placed on the free end of the cantilever beam and embedded in a groove formed on the silicon cantilever to get a large deflection. Static and dynamic mechanical characteristics of the fabricated probes have been measured. The experimental results of the mechanical properties are compared with the calculation results obtained from a finite element method. When flowing a current of ±10 mA, a static deflection of ±2 can be achieved by a cantilever with a length of 400 μm. The scanning speed of AFM is increased up to 1 mm/s by actuating the high resonant frequency cantilever in constant force mode     

Application of the Generalized Differential Quadrature Method to the Study of Pull-In Phenomena of MEMS Switches

This paper reports on the pull-in behavior of nonlinear microelectromechanical coupled systems. The generalized differential quadrature method has been used as a high-order approximation to discretize the governing nonlinear integro-differential equation, yielding more accurate results with a considerably smaller number of grid points. Various electrostatically actuated microstructures such as cantilever beam-type and fixed-fixed beam-type microelectromechanical systems (MEMS) switches are studied. The proposed models capture the following effects: (1) the intrinsic residual stress from fabrication processes; (2) the fringing effects of the electrical field; and (3) the nonlinear stiffening or axial stress due to beam stretching. The effects of important parameters on the mechanical performance have been studied in detail. These results are expected to be useful in the optimum design of MEMS switches or other actuators. Further, the results obtained are summarized and compared with other existing empirical and analytical models.     

Effects of the slip boundary condition on dynamics and pull-in instability of carbon nanotubes conveying fluid

This paper addresses the effects of the slip boundary condition on dynamics and pull-in instability of carbon nanotubes (CNTs) containing internal fluid flow. Both the clamped–clamped and the cantilever boundary conditions are considered. The structure of CNTs is modelled using the size-dependent strain gradient theory (SGT) of continuum mechanics. It is shown that the Knudsen number (Kn) has a significant effect on the static and dynamic CNT response due to pull-in voltage loading and the existence of the instability region.     

On the tunability of a MEMS based variable capacitor with a novel structure

In this work a novel MEMS based variable capacitor has been presented. To increase the tunability and decrease the applied voltage, the conventional fixed-fixed beam used in CPW lines has been changed to a fixed-simple supported beam. The proposed structure is a simple cantilever micro-beam in the first step of deflection and is changed to a fixed-simple supported micro-beam in the second step of motion. In the capacitive micro-structures increasing the applied voltage decreases the equivalent stiffness of the structure and leads the system to an unstable condition by undergoing to a saddle node bifurcation. In the proposed structure to avoid pull-in instability and increase the capacitance tuning range, mechanical stiffness of the structure is increased by changing boundary conditions by locating a pedestal in the end of the cantilever beam. The governing nonlinear equation for static deflection of the micro-beam, based on Euler–Bernoulli micro-beam theory has been presented. The results show that the proposed structure increases the capacitance tuning range and decreases the applied voltage. The results also show that the position of the pedestal affects the tunability and the threshold voltage of the structure.     

Pull-in voltage study of electrostatically actuated fixed-fixed beams using a VLSI on-chip interconnect capacitance model

A highly accurate computationally efficient closed-form model has been developed to determine the pull-in voltage of an electrostatically actuated fixed-fixed beam. The approach includes the electrostatic spring softening effects due to the fringing field capacitances along with the nonlinear spring hardening effects associated with the load-deflection characteristics of a uniformly loaded fixed-fixed beam. Meijs and Fokkema's highly accurate empirical formula for the capacitance of a VLSI on-chip interconnect has been used to determine the spring softening effects due to the fringing field capacitances. The developed model has been verified by comparing the results with published experimentally verified three-dimensional (3-D) finite element analysis (FEA) results and with those from other published representative closed-form models. The developed model can determine the pull-in voltage with a maximum deviation of 1.27% from the FEA results for small deflections and for large deflections (airgap-beam thickness ratio =12), the deviation from the FEA results is 2.0%. A maximum deviation of 0.5% from the FEA results has been observed for extreme fringing field cases (beamwidth-airgap ratio /spl les/0.5). The model's accuracy range is better compared to the other published models.     

Dynamic pull-in of parallel-plate and torsional electrostatic MEMS actuators

An analysis of the dynamic characteristics of pull-in for parallel-plate and torsional electrostatic actuators is presented. Traditionally, the analysis for pull-in has been done using quasi-static assumptions. However, it was recently shown experimentally that a step input can cause a decrease in the voltage required for pull-in to occur. We propose an energy-based solution for the step voltage required for pull-in that predicts the experimentally observed decrease in the pull-in voltage. We then use similar energy techniques to explore pull-in due to an actuation signal that is modulated depending on the sign of the velocity of the plate (i.e., modulated at the instantaneous mechanical resonant frequency). For this type of actuation signal, significant reductions in the pull-in voltage can theoretically be achieved without changing the stiffness of the structure. This analysis is significant to both parallel-plate and torsional electrostatic microelectromechanical systems (MEMS) switching structures where a reduced operating voltage without sacrificing stiffness is desired, as well as electrostatic MEMS oscillators where pull-in due to dynamic effects needs to be avoided.     

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.

     

MEMS薄膜平均应力梯度及等效弹性模量在线提取方法

薄膜沿厚度方向的平均应力梯度及薄膜的弹性模量对器件性能有重要影响.提出了一种利用静电作用下悬臂梁的吸合电压提取薄膜沿厚度方向的平均应力梯度及等效弹性模量的方法,该方法的关键在于实现悬臂梁吸合电压的快速精确计算.考虑了悬臂梁由应力梯度引起的沿宽度方向的弯曲及实现其固定端接近理想固支的方法,提高了吸合电压的计算精度.实际模拟表明该测量方法计算速度快、精度高,能够应用于实际工艺过程中材料参数的在线测量.     

A bidirectional magnetic microactuator using electroplatedpermanent magnet arrays

A novel bidirectional magnetic microactuator using electroplated permanent magnet arrays has been designed, fabricated and characterized. To realize a bidirectional microactuator, CoNiMnP-based permanent magnet arrays have been fabricated first on a silicon cantilever beam using a new electroplating technique. In the fabricated permanent magnets, the vertical coercivity and retentivity have been achieved up to 87.6 kA/m (1100 Oe) and 190 mT (1900 G), respectively by applying magnetic field during electroplating. A prototype bidirectional magnetic microactuator has been realized by integrating an electromagnet with a silicon cantilever beam, which has permanent magnet arrays on its tip. By applying a do current of 100 mA and altering its polarity, bidirectional motion on the tip of the cantilever beam has been successfully achieved in the deflection range of ±80 μm     

Shape optimization of electrostatically actuated microbeams for extending static and dynamic operating ranges

We present the systematic development and application of a generic shape optimization methodology to enhance the static and dynamic pull-in ranges of electrostatically actuated microbeams. Energy-based techniques are used to extract static and dynamic pull-in parameters of the distributed electromechanical model that accounts for the fringing field capacitance, moderately large deflections, and coupling between the mechanical and electric forces. Versatile parametric width functions are used to characterize the nonprismatic geometries of cantilever and fixed–fixed microbeams, and parameters of the proposed width functions are optimized using the Nelder–Mead method of function minimization together with a penalty method to enforce the constraints. We consider several test cases in order to fully demonstrate the utility of the proposed methodology. Results indicate that an increase in the travel range of as much as 20% can be obtained using our optimization approach. In case of fixed–fixed microbeams, this enhancement in the travel range is found to be dependent on the extent of geometric nonlinearity. We present the optimal shapes of these microbeams, that easily lend themselves to microfabrication, which exhibit the improved pull-in response in both static and dynamic regimes. For a set of representative cases, the enhanced travel range in both static and dynamic modes of actuation is positively verified by 3D finite element analyses performed on the referential and optimized geometries.     

Analysis of a novel RF MEMS switch using different meander techniques

In this paper, two types of RF MEMS switches namely step structure and Normal beam structure are designed and analyzed using different meander techniques. Three techniques namely plus, zigzag and three-square meander were used to lower the pull-in voltage. The actuating beam is designed with the rectangular perforations affects the performance of a switch by lowering the pull-in voltage, switching speed and results in better isolation. In this paper a comparative analysis is done for all three meander techniques with and without perforations on the beam. Total six structures have been designed with the combination three meanders and two different beam structures. The proposed stepdown structure exhibits high performance characteristics with a very low pull-in voltage of 1.2 V having an airgap of 0.8 µm between the actuation electrodes. The gold is used as beam material and HfO₂ as the dielectric material such that the upstate and downstate capacitance is seen as 1.02 fF and 49 fF. The FEM analysis is done to calculate the spring constant and thereby the pull-in voltage and behavior of the switch is studied with various parameters. The switch with a step structure and three-square meander configuration has shown best performance of all by requiring a pull-in voltage of 1.2 V and lower switching time of 0.2 µs. The proposed switch also exhibits good RF performance characteristics with an insertion loss below − 0.07 dB and return loss below − 60 dB over the frequency range of 1–40 GHz. At 28 GHz a high isolation of − 68 dB is exhibited.