Ashok Kumar Pandey

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.

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.

Performance of an AuPd micromechanical resonator as a temperature sensor

In this work we study the sensitivity of the primary resonance of an electrically excited microresonator for the possible usage of a temperature sensor. We find a relatively high normalized responsivity factor Rf=|TfdfdT|=0.37 with a quality factor of ∼105. To understand this outcome we perform a theoretical analysis based on experimental observation. We find that the dominant contribution to the responsivity comes from the temperature dependence of the tension in the beam. Subsequently, Rf is found to be inversely proportional to the initial tension. Corresponding to a particular temperature, the tension can be increased by applying a bias voltage.

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.

Forced and self-excited oscillations of an optomechanical cavity.

We experimentally study forced and self-excited oscillations of an optomechanical cavity, which is formed between a fiber Bragg grating that serves as a static mirror and a freely suspended metallic mechanical resonator that serves as a moving mirror. In the domain of small amplitude mechanical oscillations, we find that the optomechanical coupling is manifested as changes in the effective resonance frequency, damping rate, and cubic nonlinearity of the mechanical resonator. Moreover, self-excited oscillations of the micromechanical mirror are observed above a certain optical power threshold. A comparison between the experimental results and a theoretical model that we have recently derived and analyzed yields a good agreement. The comparison also indicates that the dominant optomechanical coupling mechanism is the heating of the metallic mirror due to optical absorption.

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.

Effect of coupled modes on pull-in voltage and frequency tuning of a NEMS device

A dynamic microelectromechanical system and a nanoelectromechanical system are designed to operate over a desired frequency range. There exist different types of frequency tuning mechanisms such as tuning due to the electrostatic softening effect and the hardening effect because of the axial tension. These two effects are discussed quite often in many studies. In this paper, we discuss about the case where both the effects can be accounted simultaneously to obtain frequency tuning. To model the effects, we take a fixed–fixed beam separated by a bottom and a side electrode. Subsequently, we solve the coupled elasticity and electrostatic equation using a Galerkin method to obtain the frequencies of two orthogonal modes, namely, the in-plane and out-of-plane modes. After validating the model with experimental results available in the literature, we study the coupled effects of these two modes on the pull-in effect and frequency tuning. It is found that the pull-in voltage can be increased by the intermodal coupling. The coupling region of the two modes can be controlled by selecting appropriate gaps between the beam and the side and bottom electrodes, respectively.

Evaluation of Mode Dependent Fluid Damping in a High Frequency Drumhead Microresonator

Design of high quality factor (Q) micromechanical resonators depends critically on our understanding of energy losses in their oscillations. The Q of such structures depends on process induced prestress in the structural geometry, interaction with the external environment, and the encapsulation method. We study the dominant fluid interaction related losses, namely, the squeeze film damping and acoustic radiation losses in a drumhead microresonator subjected to different prestress levels, operated in air, to predict its Q in various modes of oscillation. We present a detailed research of the acoustic radiation losses, associated with the 15 transverse vibration modes of the resonator using a hybrid analytical-computational approach. The prestressed squeeze film computation is based on the standard established numerical procedure. Our technique of computing acoustic damping based quality factor Qac includes calculation of the exact prestressed modes. We find that acoustic losses result in a non-monotonic variation of Qac in lower unstressed modes. Such non-monotonic variation disappears with the increase in the prestress levels. Although squeeze film damping dominates the net Q at lower frequencies, acoustic radiation losses dominate at higher frequencies. The combined computed losses correctly predict the experimentally measured Q of the resonator over a large range of resonant frequencies.

Coupling and tuning of modal frequencies in direct current biased microelectromechanical systems arrays

Understanding the coupling of different modal frequencies and their tuning mechanisms has become essential to design multi-frequency MEMS devices. In this work, we fabricate a MEMS beam with fixed boundaries separated from two side electrodes and a bottom electrode. Subsequently, we perform experiments to obtain the frequency variation of in-plane and out-of-plane mechanical modes of the microbeam with respect to both DC bias and laser heating. We show that the frequencies of the two modes coincide at a certain DC bias, which in turn can also be varied due to temperature. Subsequently, we develop a theoretical model to predict the variation of the two modes and their coupling due to a variable gap between the microbeam and electrodes, initial tension, and fringing field coefficients. Finally, we discuss the influence of frequency tuning parameters in arrays of 3, 33, and 40 microbeams, respectively. It is also found that the frequency bandwidth of a microbeam array can be increased to as high as 25 kHz for a 40 microbeam array with a DC bias of 80 V.

Capacitance and Force Computation Due to Direct and Fringing Effects in MEMS/NEMS Arrays

An accurate computation of electrical force is significant in analyzing the performance of microelectromechanical systems and nanoelectromechanical systems. Many analytical and empirical models are available for computing the forces, especially, for a single set of parallel plates. In general, these forces are computed based on the direct electric field between the overlapping areas of the plates and the fringing field effects. Most of the models, which are based on direct electric field effect, consider only the trivial cases of the fringing field effects. In this paper, we propose different models which are obtained from the numerical simulations. It is found to be useful in computing capacitance as well force in simple and complex configurations consisting of an array of beams and electrodes. For the given configurations, the analytical models are compared with the available models and numerical results. While the percentage error of the proposed model is found to be under 6% with respect to the numerical results, the error associated with the analytical model without the fringing field effects is ~50 %. The proposed model can be applied to the devices in which the fringing field effects are dominant.