Periodic Oscillations in MEMS under Squeeze Film Damping Force

Juan Berón, A. Rivera
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Rivera","doi":"10.1155/2022/1498981","DOIUrl":null,"url":null,"abstract":"<jats:p>We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Liénard-type equation <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <mover accent=\"true\">\n <mi>x</mi>\n <mo>¨</mo>\n </mover>\n <mo>+</mo>\n <msub>\n <mrow>\n <mi>F</mi>\n </mrow>\n <mrow>\n <mi>D</mi>\n </mrow>\n </msub>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mover accent=\"true\">\n <mi>x</mi>\n <mo>̇</mo>\n </mover>\n </mrow>\n </mfenced>\n <mo>+</mo>\n <mi>x</mi>\n <mo>=</mo>\n <mi>β</mi>\n <msup>\n <mrow>\n <mi mathvariant=\"script\">V</mi>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>t</mi>\n </mrow>\n </mfenced>\n <mo>/</mo>\n <msup>\n <mrow>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mi>x</mi>\n </mrow>\n </mfenced>\n </mrow>\n <mrow>\n <mn>2</mn>\n </mrow>\n </msup>\n <mo>,</mo>\n <mi>x</mi>\n <mo>∈</mo>\n <mfenced open=\"]\" close=\"[\">\n <mrow>\n <mo>−</mo>\n <mrow>\n <mo>∞</mo>\n </mrow>\n <mrow>\n <mo>,</mo>\n </mrow>\n <mn>1</mn>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> with <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <mi>β</mi>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mo>+</mo>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <mi mathvariant=\"script\">V</mi>\n <mo>∈</mo>\n <mi>C</mi>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>ℝ</mi>\n <mo>/</mo>\n <mi>T</mi>\n <mi>ℤ</mi>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula>, and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <msub>\n <mrow>\n <mi>F</mi>\n </mrow>\n <mrow>\n <mi>D</mi>\n </mrow>\n </msub>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mover accent=\"true\">\n <mi>x</mi>\n <mo>̇</mo>\n </mover>\n </mrow>\n </mfenced>\n <mo>=</mo>\n <mi>κ</mi>\n <mover accent=\"true\">\n <mi>x</mi>\n <mo>̇</mo>\n </mover>\n <mo>/</mo>\n <msup>\n <mrow>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mi>x</mi>\n </mrow>\n </mfenced>\n </mrow>\n <mrow>\n <mn>3</mn>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\n <mi>κ</mi>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mo>+</mo>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> (called squeeze film damping force), or <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\n <msub>\n <mrow>\n <mi>F</mi>\n </mrow>\n <mrow>\n <mi>D</mi>\n </mrow>\n </msub>\n <mfenced open=\"(\" close=\")\">\n <mrow>\n <mi>x</mi>\n <mo>,</mo>\n <mover accent=\"true\">\n <mi>x</mi>\n <mo>̇</mo>\n </mover>\n </mrow>\n </mfenced>\n <mo>=</mo>\n <mi>c</mi>\n <mover accent=\"true\">\n <mi>x</mi>\n <mo>̇</mo>\n </mover>\n </math>\n </jats:inline-formula>, <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\n <mi>c</mi>\n <mo>∈</mo>\n <msup>\n <mrow>\n <mi>ℝ</mi>\n </mrow>\n <mrow>\n <mo>+</mo>\n </mrow>\n </msup>\n </math>\n </jats:inline-formula> (called linear damping force). If <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M8\">\n <msub>\n <mrow>\n <mi>F</mi>\n </mrow>\n <mrow>\n <mi>D</mi>\n </mrow>\n </msub>\n </math>\n </jats:inline-formula> is of squeeze film type, we have proven that there exists at least two positive periodic solutions, one of them locally asymptotically stable. Meanwhile, if <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M9\">\n <msub>\n <mrow>\n <mi>F</mi>\n </mrow>\n <mrow>\n <mi>D</mi>\n </mrow>\n </msub>\n </math>\n </jats:inline-formula> is a linear damping force, we have proven that there are only two positive periodic solutions. One is unstable, and the other is locally exponentially asymptotically stable with rate of decay of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M10\">\n <mi>c</mi>\n <mo>/</mo>\n <mn>2</mn>\n </math>\n </jats:inline-formula>. Our technique can be applied to a class of Liénard equations that model several microelectromechanical system devices, including the comb-drive finger model and torsional actuators.</jats:p>","PeriodicalId":14766,"journal":{"name":"J. Appl. Math.","volume":"2013 1","pages":"1498981:1-1498981:15"},"PeriodicalIF":0.0000,"publicationDate":"2022-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"J. Appl. Math.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1155/2022/1498981","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2

Abstract

We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Liénard-type equation x ¨ + F D x , x ̇ + x = β V 2 t / 1 x 2 , x , 1 with β + , V C / T , and F D x , x ̇ = κ x ̇ / 1 x 3 , κ + (called squeeze film damping force), or F D x , x ̇ = c x ̇ , c + (called linear damping force). If F D is of squeeze film type, we have proven that there exists at least two positive periodic solutions, one of them locally asymptotically stable. Meanwhile, if F D is a linear damping force, we have proven that there are only two positive periodic solutions. One is unstable, and the other is locally exponentially asymptotically stable with rate of decay of c / 2 . Our technique can be applied to a class of Liénard equations that model several microelectromechanical system devices, including the comb-drive finger model and torsional actuators.
挤压膜阻尼力作用下MEMS的周期振荡
本文给出了用li<s:1> + fd型方程建模的理想静电致动器周期解存在的充分条件x ,x * + x = β V 2T / 1 - x2, x∈−∞;1与β∈h +,V∈c,和fdx,X氧= κ X氧/1−× 3;κ∈λ +(称为挤膜阻尼力),或者fdx,X =
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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