Deflection Modelling of MEMS Cantilever Beam Through Collocation Method Taking B-Splinesas Approximating Functions

Deflection Modelling of MEMS Cantilever Beam Through Collocation Method Taking B-Splinesas Approximating Functions

Manish Kumar Mishra, P. M. Mishra, Vikas Dubey
DOI: 10.4018/IJSESD.290007
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Abstract

The Mathematical Modeling and Analysis of cantilever beam adhesion problem, under the action of electrostatic attraction force iscarried out. The model uses Euler-Bernoulli beam theory for one end fixed and other end free type beams for small deflection. A MATLAB code has prepared to predict and plot the deflection profile of MEMS cantilever beam during the action of stiction force on application of applied voltage as snap down occurs. The model predicts the cantilever behavior on occurrence of snap downvoltage. To envisage the deflection profile, A collocation method employing B-Spline as approximating functions & Gaussian quadrature point as collocation points has been utilized for solving the governing equation by keeping the four end boundary conditions of cantilever beam in mind. The numerical results reveal the deflection profile of the MEMS cantilever Beam, which are validated with the previous data & deflection profile available by numerous published research papers
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1. Introduction

Today because of the advantages of electrostatic actuators, such as, favorable scaling property, lower driving power, large deflection capacity, relative ease of fabrication, and others, have led their being more widely applied the electrostatic-actuator applications in micro-electromechanical systems (MEMS). Thundat et al. (1997) described about MEMS based micro-cantilever which has been proven as an outstanding platform for extremely sensitive sensors. In the past decade micro-cantilevers has become so popular due to its high sensitivity, selectivity, ease of fabrication and flexibility of on chip circuits. Also it has become interesting due to convenience to calibrate, readily deployable into integrated electromechanical system and does not require external detection devices (Thaysen et al. 2000; Gotszalk et al. 2000; Pinnaduwage et al. 2003; Grabiec et al. 2002). Many previous researches reported attempts made to improve the cantilever sensitivity using micro-cantilevers comprise of a poly-silicon piezo-resistor integrated with silicon / silicon-nitride cantilever (Rasmussen et al. 2003; Thaysen et al. 2000).

The electrostatic principle is common in the sensing and actuating devices of MEMS. There are two basic components of MEMS, sensors and actuators. A sensor gathers the information from the environment by utilizing mechanical, thermal, biological, chemical, optical, and magnetic properties. The actuator contains the mechanical members, which are acted upon by various mechanisms like electromagnetic, thermo actuation, use of shape memory alloys, piezo actuation, magneto static actuation and electrostatic actuation. Out of all these, electrostatic actuation is widely used. The popularity of electrostatic actuation is due to ease of fabrication, low power consumption and higher energy density is explained by Kaneria et al. (2013). There are many examples of MEMS structures depending on electrostatic deformation and/or pull-in of a beam or diaphragm. For examples, the sensing devices such as the widely used capacitive accelerometer and the capacitive pressure sensor for measuring blood pressure gradients inside the coronary artery of the heart; the actuating devices such as the comb driver, the electro-statically driven micro elastic joints, the rotary electrostatic actuator, and the electro-statically actuated MEMS power switch. The most successful commercialized examples are the static electrically actuated inkjet head applied in inkjet printers and the digital micro mirror array applied in optical scanner and digital light projector. Hu et al. (2004) explained that in addition to the sensing and actuating devices, the material parameters of microstructures, such as Young’s modulus and intrinsic stresses can also be given by the relationship between the electrostatic loads and structural deformations.

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