挤压膜阻尼力作用下MEMS的周期振荡

Juan Berón, A. Rivera
{"title":"挤压膜阻尼力作用下MEMS的周期振荡","authors":"Juan Berón, A. 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":"{\"title\":\"Periodic Oscillations in MEMS under Squeeze Film Damping Force\",\"authors\":\"Juan Berón, A. 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}","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

摘要

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。
Periodic Oscillations in MEMS under Squeeze Film Damping Force
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.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信