Large Deflection Analysis of Peripherally Fixed Circular Membranes Subjected to Liquid Weight Loading: A Refined Design Theory of Membrane Deflection-Based Rain Gauges

The anticipated use of elastic membranes for deflection-based rain gauges has provided an impetus for this paper to revisit the large deflection problem of a peripherally fixed circular membrane subjected to liquid weight loading, a statics problem when the fluid–structure interaction of membrane and liquid reaches static equilibrium. The closed-form solution of this statics problem of fluid–structure interaction is necessary for the design of such membrane deflection-based rain gauges, while the existing closed-form solution, due to the use of the small rotation angle assumption of the membrane, cannot meet the design requirements for computational accuracy. In this paper, the problem under consideration is reformulated by giving up the small rotation angle assumption, which gives rise to a new and somewhat intractable nonlinear integro-differential equation of the governing out-of-plane equilibrium. The power series method has played an irreplaceable role in analytically solving membrane equations involving both integral and differential operations, and a new and more refined closed-form solution without the small rotation angle assumption is finally presented. Numerical examples conducted show that the new and more refined closed-form solution presented has satisfactory convergence, and the effect of giving up the small rotation angle assumption is also investigated numerically. The application of the closed-form solution presented in designing such membrane deflection-based rain gauges is illustrated, and the reliability of the new and more refined closed-form solution presented was confirmed by conducting a confirmatory experiment.

Development of High-Sensitivity Piezoresistive Pressure Sensors for -0.5…+0.5 kPa

A mathematical model of an ultrahigh sensitivity piezoresistive chip of a pressure sensor with a range from -0.5 to 0.5 kPa has been developed. The optimum geometrical dimensions of a specific silicon membrane with a combination of rigid islands to ensure a trade-off relationship between sensitivity (Ssamples = 34.5 mV/kPa/V) and nonlinearity (2KNL samples = 0.81 %FS) have been determined. The paper also studies the range of the membrane deflection and makes recommendations on position of stops limiting diaphragm deflection in both directions; the stops allow for increasing burst pressure Pburst up to 450 кPa. The simulated data has been related to that of experimental samples and their comparative analysis showed the relevance of the mathematical model (estimated sensitivity and nonlinearity errors calculated on the basis of average values are 1.5% and 19%, respectively).

Development of High-Sensitivity Piezoresistive Pressure Sensors for -0.5…+0.5 kPa

<p>A mathematical model of an ultrahigh sensitivity piezoresistive chip of a pressure sensor with a range from -0.5 to 0.5 kPa has been developed. The optimum geometrical dimensions of a specific silicon membrane with a combination of rigid islands to ensure a trade-off relationship between sensitivity (S_samples = 34.5 mV/kPa/V) and nonlinearity (2K_NL samples = 0.81 %FS) have been determined. The paper also studies the range of the membrane deflection and makes recommendations on position of stops limiting diaphragm deflection in both directions; the stops allow for increasing burst pressure P_burst up to 450 кPa. The simulated data has been related to that of experimental samples and their comparative analysis showed the relevance of the mathematical model (estimated sensitivity and nonlinearity errors calculated on the basis of average values are 1.5% and 19%, respectively).</p>

Development of High-Sensitivity Piezoresistive Pressure Sensors for -0.5…+0.5 kPa

<p>A mathematical model of an ultrahigh sensitivity piezoresistive chip of a pressure sensor with a range from -0.5 to 0.5 kPa has been developed. The optimum geometrical dimensions of a specific silicon membrane with a combination of rigid islands to ensure a trade-off relationship between sensitivity (S_samples = 34.5 mV/kPa/V) and nonlinearity (2K_NL samples = 0.81 %FS) have been determined. The paper also studies the range of the membrane deflection and makes recommendations on position of stops limiting diaphragm deflection in both directions; the stops allow for increasing burst pressure P_burst up to 450 кPa. The simulated data has been related to that of experimental samples and their comparative analysis showed the relevance of the mathematical model (estimated sensitivity and nonlinearity errors calculated on the basis of average values are 1.5% and 19%, respectively).</p>

Development of high-sensitivity piezoresistive pressure sensors for −0.5...+0.5 kPa

A mathematical model of an ultrahigh sensitivity piezoresistive chip of a pressure sensor with a range from −0.5 to 0.5 kPa has been developed. The optimum geometrical dimensions of a specific silicon membrane with a combination of rigid islands to ensure a trade-off relationship between sensitivity (Ssamples = 34.5 mV kPa−1 V−1) and nonlinearity (2KNL samples = 0.81%FS) have been determined. The paper also studies the range of the membrane deflection and makes recommendations on position of stops limiting diaphragm deflection in both directions; the stops allow for increasing burst pressure Pburst up to 450 kPa. The simulated data has been related to that of experimental samples and their comparative analysis showed the relevance of the mathematical model (estimated sensitivity and nonlinearity errors calculated on the basis of average values are 1.5% and 19%, respectively).

APPLICATION OF FAST EXPANSIONS TO OBTAIN EXACT SOLUTIONS TO A PROBLEM ON RECTANGULAR MEMBRANE DEFLECTION UNDER ALTERNATING LOAD

The problem of rectangular membrane deflection under alternating loads is solved in general terms by means of the method of fast expansions. The exact solution is represented by the finite expression borrowed from the theory of fast expansions as a sum of the boundary function and Fourier sine series with two Fourier coefficients taken into account. The obtained exact solution includes free parameters. Changing the values of these parameters, one can derive many new exact solutions. Obtaining of exact solutions to a problem of the rigidly fixed membrane under two types of loads (dome-shaped and sinusoidal) is shown as an example. Graphs of the dome-shaped and sinusoidal loads on the membrane and the curves of the corresponding deflections and stress components are presented in the paper. From the analysis of the exact solutions, it is obvious that only when a symmetrical alternating load is used, the membrane maximum deflection is attained in the center of the membrane, and the stresses reach the highest values in the middle of both long sides. In the case of a non-symmetrical load, the maximum stress occurs in the middle of either one of two long sides of the rectangular membrane, and the maximum deflection is found in the central region.

Explicit solution of the scattering problem involving a vertical flexible membrane

&lt;p&gt;The problem of normally incident water wave scattering by a flexible membrane is completely solved. The physical problem in a half-plane is reduced to a couple of equivalent quarter-plane problems by allowing incident waves from either direction of the membrane. In the same way, quarter-plane boundary value problems are posed for solid wave potentials that are solutions of the scattering problem involving a rigid structure of the same geometric configuration. Then, two novel integral relations are introduced to establish a link between the required solution wave potentials and few resolvable solid wave potentials. Explicit expressions for the scattering quantities such as the reflection and the transmission wave amplitudes are obtained. Also, the deflection of the flexible vertical membrane and the solution potentials are determined analytically. Numerical results for the scattering quantities and the membrane deflection are presented.&lt;/p&gt;

Membrane Micro Electro-Mechanical Systems for Industrial Applications

The objective of this chapter is to provide the analytical-numerical tools for the simplified rewriting of the most important mathematical models of MEMS membrane devices for Mechatronics, exploiting advanced concepts and results in the theory of curves and surfaces. Moreover, when the solution in closed form could not be obtained (that is, it is impossible to obtain the membrane deflection analytically), some consolidated techniques will be described both to obtain conditions ensuring existence/uniqueness of the solution, and the most suitable approaches for obtaining numerical solutions in the absence of ghost solutions. Finally, some practical examples will illustrate the approaches presented.

Membrane Deflection and Stress in Thermal Flow Sensors

The effect of membrane deflection has been investigated for thermal flow sensors. Catastrophic membrane breakage is a common occurrence in membrane based thermal flow sensors due to thermal expansion and internal stresses. This work analyses three sensors comprising a tungsten heater embedded in buried oxide membrane with a silicon nitride passivation layer, the use of back etching creates a cavity underneath to reduce the thermal conduction. The investigation is done using interferometry to measure the membrane shape at room and operating temperature for three membranes of different sizes. As expected, the deflection increases with temperature up to 15 µm at operating temperature and with the reduction of membrane size the deflection is reduced to a minimum of 3 µm for the smallest membrane. The lower deflection measured in devices with a smaller cavity can be related to a reduced internal stress, improving the long term stability.

Actuation of Flexible Membranes via Capillary Force: Single-Active-Surface Experiments

Conventional approaches to microscale actuation, such as electrostatic, have difficulty in achieving large motion at moderate voltages. Recently, actuators relying on the active control of capillary pressure have been demonstrated, with the pressure change caused by electrowetting on a pair of opposing surfaces. In this work, experimental results are presented from five prototype devices in which only a single active surface is used. The results demonstrate that pressure changes induced in a liquid bridge in this manner can produce large deflections (15 μm) of a flexible membrane. Voltages employed in the tests were moderate (≤25 V). The influence of several design variables, such as membrane diameter and thickness, on the membrane deflection are examined. Theoretical predictions are also presented and generally follow the experimental values. Potential sources for the discrepancies between theory and experimental results are discussed. While deflections obtained using a single active surface are not as large as those obtained with two active surfaces, single-active-surface configurations offer a simple route to achieving adequate deflections for lab-on-a-chip microsystems.