Juan Berón, A. Rivera
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{"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
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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
∈
ℝ
+
,
V
∈
C
ℝ
/
T
ℤ
, and
F
D
x
,
x
̇
=
κ
x
̇
/
1
−
x
3
,
κ
∈
ℝ
+
(called squeeze film damping force), or
F
D
x
,
x
̇
=
c
x
̇
,
c
∈
ℝ
+
(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.