Tables of integrals

The paper presents an overview and reports the recent progress of research on squeeze film air damping in MEMS. The review starts with the governing equations of squeeze film air damping: the nonlinear isothermal Reynolds equation and various reduced forms of the equation for different conditions. After the basic effects of squeeze film damping on the dynamic performances of micro-structures are discussed based on the analytical solutions to parallel plate problems, recent research on various aspects of squeeze film air damping are reviewed, including the squeeze film air damping of perforated and slotted plate, the squeeze film air damping in rarefied air and the squeeze film air damping of torsion mirrors. Finally, the simulation of squeeze film air damping is reviewed. For quick reference, important equations and curves are included.

Squeeze film air damping in MEMS

In this paper some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m or = 1 Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations A mixed boundary value problem from the theory of elasticity is considered as an example Particularly for integral equations where the kernel contains, in addition to the dominant term (t,x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results

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On the solution of integral equations with strong ly singular kernels

The purpose of this chapter is to present the basic ideas of the theory of distributions. Distributions or generalized functions, as they are also known, have proved to be very useful in many branches of pure and applied mathematics. Many textbooks, monographs and articles have been written on their theory and their applications [12], [23], [53], [63], [64], [69], [71], [80], [97], [111]. Our present aim is to give a brief but solid introduction to the theory of distributions, particularly to those aspects that are important in the theory of asymptotic expansions.

Introduction to the Theory of Distributions

Most heart diseases are associated with and reflected by the sounds that the heart produces. Heart auscultation, defined as
 listening to the heart sound, has been a very important method for the early diagnosis of cardiac dysfunction. Traditional auscultation requires substantial clinical experience and good listening skills. The emergence of the electronic stethoscope has paved the way for a new field of computer-aided auscultation. This article provides an in-depth study of (1) the electronic stethoscope technology, and (2) the methodology for diagnosis of cardiac disorders based on computer-aided auscultation. The paper is based on a comprehensive review of (1) literature articles, (2) market (state-of-the-art) products, and (3) smartphone stethoscope apps. It covers in depth every key component of the computer-aided system with electronic stethoscope, from sensor design, front-end circuitry, denoising algorithm, heart sound segmentation, to the final machine learning techniques. Our intent is to provide an informative and illustrative presentation of the electronic stethoscope, which is valuable and beneficial to academics, researchers and engineers in the technical field, as well as to medical professionals to facilitate its use clinically. The paper provides the technological and medical basis for the development and commercialization of a real-time integrated heart sound detection, acquisition and quantification system.

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The electronic stethoscope

The paper contains an analysis of the viscous damping in perforated planar microstructures that often serve as backplates or protecting surfaces in capacitive microsensors. The focus of this work is on planar surfaces containing an offset system of periodic oval holes or its limit cases: a system of circular holes or of slits. The viscous damping is calculated as the sum of squeeze film and the holes’ resistances. The optimum number of holes is determined which minimizes the total viscous damping for a given percentage of open area. Graphs and formulas are provided for designing these devices. In the case the open area is higher than 15% the numerical results show that the influence of the holes’ geometry (circular or oval) has a slight influence on viscous damping. As the planar structures containing oval holes assure a better protection against dust particles and water drops, they should be preferred in designing protective surfaces for microphones working in a natural environment. The obtained results ...

Modeling of viscous damping of perforated planar microstructures. Applications in acoustics

The paper gives an analytical approximation to the viscous damping coefficient due to the motion of a gas between a pair of closely spaced fluctuating plates in which one of the plates contains a regular system of circular holes. These types of structures are important parts of many microelectromechanical devices realized in MEMS technology as microphones, microaccelerometers, resonators, etc.

The pressure satisfies a Reynolds’ type equation with coefficients accounting for all the important effects: compressibility of the gas, inertia and possibly slip of the gas on the plates. An analytical expression for the optimum number of circular holes which assure a minimum value of the total damping coefficient is given. This value realizes an equilibrium between the squeeze-film damping and the viscous resistance of the holes.

The paper also provides analytical design formulas to be used in the case of regular circular perforated plates.

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Viscous damping of perforated planar micromechanical structures.

A silicon microelectromechanical systems microphone is described that detects sound pressure gradients. The diaphragm consists of a stiffened plate that rotates around a central axis in response to sound pressure gradients. The motion of the diaphragm is converted into an electronic signal through the use of interdigitated comb fins that enable capacitive sensing. Measured results show that the microphone achieves a substantially lower low-frequency sound pressure-referred noise floor than can be achieved using existing dual miniature microphone systems. Measured directivity patterns are shown to be very close to what is expected for sound pressure gradient receivers over a broad range of frequencies.

A MEMS Low-Noise Sound Pressure Gradient Microphone With Capacitive Sensing

Conventional approaches to measuring animal vibrational signals on plant stems use a single transducer to measure the amplitude of vibrations. Such an approach, however, will often underestimate the amplitude of bending waves traveling along the stem. This occurs because vibration transducers are maximally sensitive along a single axis, which may not correspond to the major axis of stem motion. Furthermore, stem motion may be more complex than that of a bending wave propagating along a single axis, and such motion cannot be described using a single transducer. Here, we describe a method for characterizing stem motion in two dimensions by processing the signals from two orthogonally positioned transducers. Viewed relative to a cross-sectional plane, a point on the stem surface moves in an ellipse at any one frequency, with the ellipse’s major axis corresponding to the maximum amplitude of vibration. The method outlined here measures the ellipse’s major and minor axes, and its angle of rotation relative to one of the transducers. We illustrate this method with measurements of stem motion during insect vibrational communication. It is likely the two-dimensional nature of stem motion is relevant to insect vibration perception, making this method a promising avenue for studies of plant-borne transmission.

A method for two-dimensional characterization of animal vibrational signals transmitted along plant stems

In the free surface flow of dense suspensions in confined domains, spontaneous particle enrichment occurs in the region adjacent to the free surface with an attendant increase in the effective viscosity. We present experimental evidence that this particle accretion introduces stability considerations that would otherwise not exist if the suspension remained uniform. For conditions favoring global stability, a local fingering instability can occur in the accretion band. For conditions favoring global instability, accretion can suppress the global growth of viscous fingers at the interface.

Dorel Homentcovschi

Papers

Modeling of viscous damping of perforated planar microstructures. Applications in acoustics

Viscous damping of perforated planar micromechanical structures.

A MEMS Low-Noise Sound Pressure Gradient Microphone With Capacitive Sensing

A method for two-dimensional characterization of animal vibrational signals transmitted along plant stems

Stability considerations associated with the meniscoid particle band at advancing interfaces in hele-shaw suspension flows