Rudra Pratap

Effect of flexural modes on squeeze film damping in MEMS cantilever resonators

We present an analytical model that gives the values of squeeze film damping and spring coefficients for MEMS cantilever resonators taking into account the effect of flexural modes of the resonator. We use the exact mode shapes of a 2D cantilever plate to solve for pressure in the squeeze film and then derive the equivalent damping and spring coefficient relations from the back force calculations. The relations thus obtained can be used for any flexural mode of vibration of the resonators. We validate the analytical formulae by comparing the results with numerical simulations carried out using coupled finite element analysis in ANSYS, as well as experimentally measured values from MEMS cantilever resonators of various sizes vibrating in different modes. The analytically predicted values of damping are, in the worst case, within less than 10% of the values obtained experimentally or numerically. We also compare the results with previously reported analytical formulae based on approximate flexural mode shapes and show that the current results give much better estimates of the squeeze film damping. From the analytical model presented here, we find that the squeeze film damping drops by 84% from the first mode to the second mode in a cantilever resonator, thus improving the quality factor by a factor of 6 to 7. This result has significant implications in using cantilever resonators for mass detection where a significant increase in the quality factor is obtained by using a vacuum.

Analytical solutions for the stiffness and damping coefficients of squeeze films in MEMS devices with perforated back plates

Closed-form expressions for the stiffness and the damping coefficients of a squeeze film are derived for MEMS devices with perforated back plates. Two kinds of perforation configurations are considered—staggered and matrix or non-staggered configuration. The analytical solutions are motivated from the observation of repetitive pressure patterns obtained fromnumerical (FEM) solutions of the compressible Reynolds equation for the two configurations using ANSYS. A single pressure pattern is isolated and further subdivided into circular pressure cells. Circular geometry is used based on observed symmetry. Using suitable boundary conditions, the Reynolds equation is analytically solved over the pressure cells. The complex pressure obtained is used to identify the stiffness and damping offered by the pressure cells. The stiffness and damping forces due to pressure cells within a pattern are added up separately. In turn, the stiffness and damping due to all the patterns are summed up resulting in the stiffness and damping forces due to the entire squeeze film. The damping and spring forces thus obtained analytically are compared with those obtained from the FEM simulations in ANSYS. The match is found to be very good. The regime of validity and limitations of the analytical solutions are assessed in terms of design parameters such as pitch to air gap, hole length to diameter and pitch to hole radius ratios. The analysis neglects inertial effects. Hence, the results are presented for low values of Reynolds number.

A Compact Squeeze-Film Model Including Inertia, Compressibility, and Rarefaction Effects for Perforated 3-D MEMS Structures

We present a comprehensive analytical model of squeeze-film damping in perforated 3D microelectromechanical system structures. The model includes effects of compressibility, inertia, and rarefaction in the flow between two parallel plates forming the squeeze region, as well as the flow through perforations. The two flows are coupled through a nontrivial frequency-dependent pressure boundary condition at the flow entry in the hole. This intermediate pressure is obtained by solving the fluid flow equations in the two regions using the frequency-dependent fluid velocity as the input velocity for the hole. The governing equations are derived by considering an approximate circular pressure cell around a hole, which is representative of the spatially invariant pressure pattern over the interior of the flow domain. A modified Reynolds equation that includes the unsteady inertial term is derived from the Navier-Stokes equation to model the flow in the circular cell. Rarefaction effects in the flow through the air gap and the hole are accounted for by considering the slip boundary conditions. The analytical solution for the net force on a single cell is obtained by solving the Reynolds equation over the annular region of the air gap and supplementing the resulting force with a term corresponding to the loss through the hole. The solution thus obtained is valid over a range of air gap and perforation geometries, as well as a wide range of operating frequencies. We compare the analytical results with extensive simulations carried out using the full 3D Navier-Stokes equation solver in a commercial simulation package (ANSYS-CFX). We show that the analytical solution performs very well in tracking the net force and the damping force up to a frequency f = 0.8fn where fn is the first resonance frequency) with a maximum error within 20% for thick perforated cells and within 30% for thin perforated cells. The error increases considerably beyond this frequency. The prediction of the first resonance frequency is within 21 % error for various perforation geometries.

Studies on the dynamics of a supercavitating projectile

Due to imperfect water entry, a high speed supercavitating projectile, while moving in the forward direction, rotates inside the cavity. This rotation leads to a series of impacts between the projectile tail and the cavity wall. The impacts affiect the trajectory as well as the stability of motion of the projectile. The present paper discusses the in-fiight dynamics of such a projectile. Equations of motion of the projectile are developed for two distinct phases of motion - Phase I: the projectile moves in the cavity without interaction with the cavity wall, and Phase II: the tail impacts on the cavity wall.The equations are found to be coupled and nonlinear. A simple model based on the concepts of flow planes is used to determine the forces acting on the projectile during impact. The effect of the mass distribution on the projectile dynamics is also studied. The results show that despite the impacts with the cavity wall, the projectile nearly follows a straight line path. The frequency of the impacts between the projectile tail and cavity boundary increases initially,reaches a maximum, and then decreases gradually. The results also indicate that the frequency of impacts decreases with the projectiles moment of inertia. It is also shown that the impact of the projectile with the cavity wall can be modelled as an impact with a rigid barrier with variable coefficient of restitution. A functional form of the coefficient of restitution is proposed, and it is shown that the proposed form predicts the impact behaviour quite well.

A semi-analytical model for squeeze-film damping including rarefaction in a MEMS torsion mirror with complex geometry

A semi-analytical approach is presented to model the effects of complicated boundary conditions and rarefaction on the squeeze-film damping dependent quality factor in a double-gimballed MEMS torsion mirror. To compute squeeze-film damping in a rectangular torsion mirror with simple boundaries, compact models derived by solving the conventional Reynolds equation with zero pressure boundary conditions on the edges of the plate are generally used. These models are not applicable if the air-gap thickness is comparable to the length of the plate. To extend the validity of the existing models in devices with large air-gap thickness and complicated boundaries, we present a procedure that requires the computation of the effective length of the structure and uses this length for the computation of damping in all flow regimes using a modified effective viscosity model. The effective length is computed by comparing the damping obtained from a numerical solution of Navier–Stokes equations with that obtained from a Reynolds-equation-based compact model. To capture the effect of rarefaction in different flow regimes, we use two different approaches: the effective viscosity approach which is valid for continuum, slip, transition and molecular flow regimes, and an approach based on the free molecular model which is valid only in a molecular flow regime. We show that the effective length obtained for complicated structures in the continuum regime may still be used to capture the rarefaction effect in the slip, transition and molecular regimes. On comparing different empirical models based on the effective viscosity approach with experimental results, we find some anomaly in the region between the molecular regime and the intrinsic regime where non-fluid damping dominates. To improve modelling in the rarified regimes, we modify the best model among the existing models by minimizing error obtained with respect to the experimental results. We find that the proposed model captures the rarefaction effect not only in the slip, transition and molecular regimes but also couples well with the non-fluid damping in the intrinsic regime and captures the transition to purely intrinsic losses.

Graphene on Paper: A Simple, Low-Cost Chemical Sensing Platform

Graphene layers have been transferred directly on to paper without any intermediate layers to yield G-paper. Resistive gas sensors have been fabricated using strips of G-paper. These sensors achieved a remarkable lower limit of detection of ∼300 parts per trillion (ppt) for NO2, which is comparable to or better than those from other paper-based sensors. Ultraviolet exposure was found to dramatically reduce the recovery time and improve response times. G-paper sensors are also found to be robust against minor strain, which was also found to increase sensitivity. G-paper is expected to enable a simple and inexpensive low-cost flexible graphene platform.

Graphene field emission devices

Graphene field emission devices are fabricated using a scalable process. The field enhancement factors, determined from the Fowler-Nordheim plots, are within few hundreds and match the theoretical predictions. The devices show high emission current density of ∼10 nA μm−1 at modest voltages of tens of volts. The emission is stable with time and repeatable over long term, whereas the noise in the emission current is comparable to that from individual carbon nanotubes emitting under similar conditions. We demonstrate a power law dependence of emission current on pressure which can be utilized for sensing. The excellent characteristics and relative ease of making the devices promise their great potential for sensing and electronic applications.

Elasto-Electrostatic Analysis of Circular Microplates Used in Capacitive Micromachined Ultrasonic Transducers

The active structural component of a capacitive micromachined ultrasonic transducer (CMUT) is the top plate which vibrates under the influence of a time-varying electrostatic force thereby producing ultrasound waves of the desired frequency in the surrounding medium. Analysis of MEMS devices which rely on electrostatic actuation is complicated due to the fact that the structural deformations alter the electrostatic forces, which redistribute and modify the applied loads. Hence, it becomes imperative to consider the electrostatics-structure coupling aspect in the design of these devices. This paper presents an approximate analytical solution for the static deflection of a thin, clamped circular plate caused by electrostatic forces which are inherently nonlinear. Traditionally, finite element simulations using some commercial software such as ANSYS are employed to determine the structural deflections caused by electrostatic forces. Since the structural deformation alters the electrostatic field, a coupled-field simulation is required wherein the electrostatic mesh is continuously updated to coincide with the deflection of the structure. Such simulations are extremely time consuming, in addition to being nontransparent and somewhat hard to implement. We employ the classical thin-plate theory which is adequate when the ratio of the diameter to thickness of the plate is very large, a situation commonly prevalent in many MEMS devices, especially the CMUTs. We solve the thin-plate electrostatic-elastic equation using the Galerkin-weighted residual technique, under the assumption that the deflections are small in comparison to the thickness of the plate. The evaluation of the electrostatic force between the two plates is simplified due to the fact that the electrostatic gap is much smaller than the lateral dimensions of the device. The results obtained are compared to those found from ANSYS simulations and an excellent agreement is observed between the two. The pull-in voltage predicted by our model is close to the value predicted by ANSYS simulations.

Influence of Boundary Conditions on the Dynamic Characteristics of Squeeze Films in MEMS Devices

Micromechanical structures that have squeeze-film damping as the dominant energy dissipation mechanism are of interest in this paper. For such structures with narrow air gap, the Reynolds equation is used for calculating squeeze-film damping, which is generally solved with trivial pressure boundary conditions on the side walls. This procedure, however, fails to give satisfactory results for structures under two important conditions: 1) for an air gap thickness comparable to the lateral dimensions of the microstructure and 2) for nontrivial pressure boundary conditions such as fully open boundaries on an extended substrate or partially blocked boundaries that provide side clearance to the fluid flow. Several formulas exist to account for simple boundary conditions. In practice, however, there are many micromechanical structures such as torsional microelectromechanical system (MEMS) structures that have nontrivial boundary conditions arising from partially blocked boundaries. Such boundaries usually have clearance parameters that can vary due to fabrication. These parameters, however, can also be used as design parameters if we understand their role on the dynamics of the structure. We take a MEMS torsion mirror as an example device that has large air gap and partially blocked boundaries due to static frames. We actuate the device and experimentally determine the quality factor Q from the response measurements. Next, we model the same structure in ANSYS and carry out computational fluid dynamics analysis to evaluate the stiffness constant K, the damping constant D, and the quality factor Q due to the squeeze film. We compare the computational results with experimental results and show that without taking care of the partially blocked boundaries properly in the computational model, we get unacceptably large errors.

Coupled nonlinear effects of surface roughness and rarefaction on squeeze film damping in MEMS structures

Many MEMS devices employ parallel plates for capacitive sensing and actuation. The desire to get a significant change in capacitance has been pushing the need to reduce the gap between the moving plate and the fixed plate. With fabrication processes making rapid strides, it is now possible to push the gap to be so small that it becomes comparable to the mean free path of gas or air molecules present in the gap. In all MEMS devices, where the essential function of the device depends on the dynamics of the mechanical components, the presence of air or a gas in such narrow gaps leads to energy dissipation if the gas is squeezed between the two plates due to transverse motion of the movable plate. This energy dissipation, known as squeeze film damping, plays a critical role in determining the quality factor of such devices. For many devices, simple approximation of squeeze film damping based on the linearized Reynolds equation is sufficient. However, under moderate vacuum and very narrow gaps, the linearized Reynolds equation does not give satisfactory results, especially if large amplitude motions of the movable plate are desired. In this paper, we carry out an analysis of the fluid flow in the narrow gap taking rarefaction and surface roughness into account and show that both these factors have a significant effect on the squeeze film damping of the devices.