Analytical approach and numerical /spl alpha/-lines method for pull-in hyper-surface extraction of electrostatic actuators with multiple uncoupled voltage sources

This work presents a systematic analysis of electrostatic actuators driven by multiple uncoupled voltage sources. The use of multiple uncoupled voltage sources has the potential of enriching the electromechanical response of electrostatically actuated deformable elements. This in turn may enable novel MEMS devices with improved and even new capabilities. It is therefore important to develop methods for analyzing this class of actuators. Pull-in is an inherent instability phenomenon that emanates from the nonlinear nature of the electromechanical coupling in electrostatic actuators. The character of pull-in in actuators with multiple uncoupled voltage sources is studied, and new insights regarding pull-in are presented. An analytical method for extracting the pull-in hyper-surface by directly solving the voltage-free K-N pull-in equations derived here, is proposed. Solving simple but interesting example problems illustrate these new insights. In addition, a novel /spl alpha/-lines numerical method for extracting the pull-in hyper-surface of general electrostatic actuators is presented and illustrated. This /spl alpha/-lines method is motivated by new features of pull-in, that are exhibited only in electrostatic actuators with multiple uncoupled voltage sources. This numerical method permits the analysis of electrostatic actuators that could not have been analyzed by using current methods.     

Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage

In this paper, a novel method has been developed to control the pull-in voltage of the fixed-fixed and cantilever MEM actuators and measure the residual stress in the fixed-fixed model using of the piezoelectric layers that have been located on the upper and lower surfaces of actuator. In the developed model, the tensile or compressive residual stresses, fringing-field and axial stress effects in the fixed-fixed end type micro-electro-mechanical systems actuator have been considered. The non-linear governing differential equations of the MEM actuators have been derived by considering the piezoelectric layers and mentioned effects. The results show that due to different applied voltage to the piezoelectric layers, the pull-in voltage can be controlled and in the fixed-fixed type the unknown value of the residual stress can be obtained.     

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.     

Shape optimizations and static/dynamic characterizations of deformable microplate structures with multiple electrostatic actuators

Micromirrors used in many optoelectronic devices can be considered as microplates. The functions and accuracy of the micromirrors depend on the static and dynamic deflection shapes of microplates. The objective of this study is to develop an efficient method for predicting the shapes of the microplates subjected to unsymmetrical electrostatic forces produced by electrostatic actuators such that micromirrors can be effectively optimized and controlled in their real time operation. The non-classical boundary conditions which result from the microfabrication process were modeled with artificial springs at the edges. A classical energy method using boundary characteristic orthogonal polynomials was applied to formulate the equations of motion of the microsystem. Based on this method, influence functions were built and least squares method was used to optimize the desired deflection under electrostatic forces from the electrostatic actuators. Softening effect of the electrostatic stiffness was also evaluated and considered in the simulation. Static deflections and dynamic responses were compared with those from finite element analysis (FEA) using Reduced Order Modeling (ROM) method. This study found that the static and dynamic responses of microplates predicted from the proposed method were highly consistent with those calculated from FEA. However, the proposed method is simpler and more efficient than FEA and can be conveniently used for any non-classical boundary condition situations. These features make the proposed method useful to effectively control and optimize the shape of a microplate with multiple electrostatic actuators.     

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.     

On degrees of freedom of the cellular network

In this paper, we consider the capacity limits of the cellular network by modeling it as consisting of two mutually interfering multiple access channels with multiple antennas at each receiver (e.g., base station). By developing a tight outerbound as well as an achievable scheme which exploits the idea of interference alignment, we are able to exactly characterize the sum degrees of freedom (DoF) of the network when the channel coefficients are timeor frequency-varying, which equals KM K+min(M,K) (where M a...     

A methodology and model for the pull-in parameters of magnetostatic actuators

Magnetostatic actuators exhibit bistability similarly to the pull-in phenomena of electrostatic actuators. In this paper a methodology and model for the extraction of the magnetic Pull-In parameters of magnetostatic actuators are derived. The flux-controlled magnetostatic actuator is analyzed based on the energy representation and the magnetomotive force-controlled magnetostatic actuator is analyzed in the thermodynamic potential energy (or co-energy) representation. An algebraic equation, referred to as the magnetic Pull-In equation, for each of the two cases (flux-controlled and magnetomotive force-controlled actuators) is derived. By solving these Pull-In equations either analytically or numerically, the magnetic Pull-In parameters are obtained. Several case studies, covering displacement and torsion magnetic actuators, are presented and analyzed, illustrating the usefulness of the proposed methodology, its relative simplicity as well as the adaptability and practical usage in wide spectrum of magnetic actuators. [1425].     

On large displacement-small strain analysis of structures with rotational degrees of freedom

The matrix displacement analysis of geometrically nonlinear structures becomes an intricate task as soon as finite elements in space with rotational degrees of freedom are considered. The fundamental reason for these difficulties lies in the non-commutativity of successive finite rotations about fixed axes with different directions. In order to circumvent this difficulty, a new definition of rotations — the so-called semitangential rotations — is introduced in this paper. Our new definition leads to a reformulation of the theory of [1,2]which in itself is clearly consistent and correct.In contrast to rotations about fixed axes these semitangential rotations which correspond to the semitangential torques of Ziegler [3]possess the most important property of being commutative. In this manner, all complexities involved in the standard definition of rotations are avoided ab initio.A specific aspect of this paper is a careful exposition of semitangential torques and rotations, as well as the consequences of the semitangential definitions for the geometrical stiffness of finite elements. In fact, these new definitions permit a very simple and consistent derivation of the geometrical stiffness matrices. Moreover, the semitangential definition automatically leads to a symmetric geometrical stiffness which clearly expresses that the nonlinear strain-displacement relations must satisfy the condition of conservativity of the structure itself — independently of any loading.The general theory of geometrical stiffness matrices as evolved in this paper is applied to beams in space. The consistency of the theory is demonstrated by a large number of numerical examples not only of straight beams but also of the lateral and torsional buckling and post-buckling behaviour of stiff-joined frames. Most of the former developments appear to be inadequate.     

Extending the travel range of analog-tuned electrostatic actuators

The pull-in instability limits the travel distance of elastically suspended parallel-plate electrostatic microactuators to about 1/3 of the undeflected gap distance. In this paper, we examine the “leveraged bending” and “strain-stiffening” methods for extending the travel range of electrostatic actuators. The leveraged bending effect can be used to achieve full gap travel at the cost of increased actuation voltage. The strain-stiffening effect can be used to minimize actuation voltage for a given travel range. An analytical approximation shows that the strain-stiffening effect can be used to achieve a stable travel distance up to about 3/5 of the gap. A tunable reflective diffraction grating known as the polychromator has been designed using these actuation techniques, and selected designs have been fabricated and tested for actuation behavior. Gratings with 1024 flat, closely packed grating-element actuators have been fabricated with over 1-cm-long mirrors, achieving stable vertical travel distances of more than 1.75 μm out of a 2-μm gap     

Control performance with three translational degrees of freedom

For multiple degree-of-freedom (DOF) systems, it is important to determine how accurately operators can control each DOF and what influence perceptual, information processing, and psychomotor components have on performance. Sixteen right-handed male students participated in 2 experiments: 1 involving positioning and 1 involving tracking with 3 translational DOFs. To separate perceptual and psychomotor effects, we used 2 control-display mappings that differed in the coupling of vertical and depth dimensions to the up-down and fore-aft control axes. We observed information processing effects in the positioning task: Initial error correction on the vertical dimension lagged in time behind the horizontal dimension. The depth dimension error correction lagged behind both, which was ascribed to the poorer perceptual information. We observed this perceptual effect also in the tracking experiment: Tracking error along the depth dimension was 3.8 times larger than along the other dimensions. Motor effects were also present, with tracking errors along the up-down axis of the hand controller being 1.1 times larger than along the fore-aft axis. These results indicate that all 3 components contribute to control performance. Actual applications of this research include interface design for remote control and virtual reality.     

Robotic welding system with parallel degrees of freedom

This paper considers some problems concerning robotic welding of large structures. If spot welding is applied, then the task consists of fast motions between welding points, and at each point the end effector stops and maintains position for a short time. Similar examples can be found in arc welding applications. Since large structures and, accordingly, large robots are considered, a significant problem appears when extremely nonuniform motion is required. One way to solve this problem is to introduce parallel degrees of freedom. Two methods for this distribution of motion to parallel degrees of freedom are discussed. The theory resulted in the design of a large portal robot with high dynamic capabilities.     

Self-calibration of wireless cameras with restricted degrees of freedom

This paper presents an approach for the automatic calibration of low-cost cameras which are assumed to be restricted in their freedom of movement to either pan or tilt movements. Camera parameters, including focal length, principal point, lens distortion parameter and the angle and axis of rotation, can be recovered from a minimum set of two images of the camera, provided that the axis of rotation between the two images goes through the camera’s optical center and is parallel to either the vertical (panning) or horizontal (tilting) axis of the image. Previous methods for auto-calibration of cameras based on pure rotations fail to work in these two degenerate cases. In addition, our approach includes a modified RANdom SAmple Consensus (RANSAC) algorithm, as well as improved integration of the radial distortion coefficient in the computation of inter-image homographies. We show that these modifications are able to increase the overall efficiency, reliability and accuracy of the homography computation and calibration procedure using both synthetic and real image sequences.     

Charge control of parallel-plate, electrostatic actuators and the tip-in instability

Controlling the charge, rather than the voltage, on a parallel-plate, electrostatic actuator theoretically permits stable operation for all deflections. Practically, we show that, using charge control, the maximum stable deflection is limited by 1) charge pull-in, in which the actuator snaps due to the presence of parasitic capacitance and 2) tip-in, in which the rotation mode becomes unstable. This work presents a circuit that controls the amount of charge on a parallel-plate, electrostatic actuator. This circuit reduces the sensitivity to parasitic capacitance, so that tip-in is the limiting instability. A small-signal model of the actuator is developed and used to determine the circuit bandwidth and gain requirements for stable deflections. Four different parallel-plate actuators have been designed and tested to verify the charge control technique as well as to verify charge pull-in, tip-in, and the bandwidth requirements. One design travels 83% of the gap before tip-in. Another design can only travel 20% of the gap before tip-in, regardless of whether voltage control or charge control is used.     

Analytical behavior of rectangular electrostatic torsion actuators with nonlinear spring bending

In this paper, we study the pull-in effect for rectangular electrostatic torsion actuators by using analytical calculations that include the higher order effects of nonlinear spring bending. The calculation approach speeds the design of such systems. The method is found to be suitable for actuators with single long beam springs where the ratio of the resonant frequencies for the torsion and bending modes is up to at least 3.5, in the region where bending dominates torsion. After fitting the theory in this paper to Coventor simulation results with three nonphysical coefficients, the fractional differences between Coventor simulation and analytical calculation results are smaller than 6%. The method is also suitable for at least one class of folded spring designs, with greatly decreased bending mode displacement. The main results are also verified by comparing them with published experimental results.     

On the vectorial nature of the rotational degrees of freedom in spatial beam elements

In this note, it is shown that the rotational degrees of freedom at a node of a spatial finite beam element are the components of a vector regardless of their magnitudes, as long as the structure to be analysed is composed of slender beams, and what may prevent them from being considered as the components of a vector is the presence of significant transverse shear strains, which usually appear in short beams, rather than the magnitudes of the rotations.     

Design and simulation of RF MEMS comb drive with ultra-low pull-in voltage and maximum displacement

Microsystem Technologies - Comb drive actuators are very important microelectromechanical systems structures that are widely used in many applications. They are currently used in force balanced...     

Out-of-plane deformation and pull-in voltage of cantilevers with residual stress gradient: experiment and modelling

The out-of-plane deformation and the pull-in voltage of electrostatically actuated cantilevers with a residual stress gradient, is investigated in the length range 100–300 µm. Measured pull-in voltages are compared with calculations, which are obtained using previously proposed analytical expressions and a finite element method (FEM) modelling. In particular, a simplified model of the residual stress distribution inside cantilevers is formulated that enables FEM simulation of measured out-of-plane deformations and pull-in voltages for all lengths of fabricated cantilevers. The presented experimental results and FEM model are exploitable in the design of cantilever-based microelectromechanical systems, in order to provide a reliable prediction of the influence of residual stress gradient on device shape and pull-in voltage.

     

Computation of static shapes and voltages for micromachineddeformable mirrors with nonlinear electrostatic actuators

In modeling micromachined deformable mirrors with electrostatic actuators whose gap spacings are of the same order of magnitude as those of the surface deformations, it is necessary to use nonlinear models for the actuators. In this paper, we consider micromachined deformable mirrors modeled by a membrane or plate equation with nonlinear electrostatic actuator characteristics. Numerical methods for computing the mirror deformation due to given actuator voltages and the actuator voltages required for producing the desired deformations at the actuator locations are presented. The application of the proposed methods to circular deformable mirrors whose surfaces are modeled by elastic membranes is discussed in detail. Numerical results are obtained for a typical circular micromachined mirror with electrostatic actuators