Asymptotics of the Spectrum of One-Dimensional Natural Vibrations in a Layered Medium Consisting of Viscoelastic Material and Viscous Fluid
The spectrum of one-dimensional natural vibrations propagating through a two-phase layered medium in the direction of the normal to the layers is investigated. The medium considered consists of many periodically alternating layers of isotropic viscoelastic material and viscous compressible fluid. It...
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Published in | Fluid dynamics Vol. 54; no. 6; pp. 749 - 760 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Moscow
Pleiades Publishing
01.11.2019
Springer Springer Nature B.V |
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Abstract | The spectrum of one-dimensional natural vibrations propagating through a two-phase layered medium in the direction of the normal to the layers is investigated. The medium considered consists of many periodically alternating layers of isotropic viscoelastic material and viscous compressible fluid. It is found that the spectrum mentioned above consists of the roots of transcendental equations whose number is proportional to the number of the layers of original medium. As the initial approximations of the roots for solving these equations numerically, it is proposed to use the points of the spectrum of one-dimensional natural vibrations of the corresponding homogenized medium. These points represent the roots of linear fractional equations. It is shown that the points at which the denominators of fractions in the linear fractional equations vanish should be also taken as the initial approximations. The accuracy of the initial approximations is proved to increase when the number of layers of the original medium increases and the layer thickness decreases simultaneously. |
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AbstractList | The spectrum of one-dimensional natural vibrations propagating through a two-phase layered medium in the direction of the normal to the layers is investigated. The medium considered consists of many periodically alternating layers of isotropic viscoelastic material and viscous compressible fluid. It is found that the spectrum mentioned above consists of the roots of transcendental equations whose number is proportional to the number of the layers of original medium. As the initial approximations of the roots for solving these equations numerically, it is proposed to use the points of the spectrum of one-dimensional natural vibrations of the corresponding homogenized medium. These points represent the roots of linear fractional equations. It is shown that the points at which the denominators of fractions in the linear fractional equations vanish should be also taken as the initial approximations. The accuracy of the initial approximations is proved to increase when the number of layers of the original medium increases and the layer thickness decreases simultaneously. The spectrum of one-dimensional natural vibrations propagating through a two-phase layered medium in the direction of the normal to the layers is investigated. The medium considered consists of many periodically alternating layers of isotropic viscoelastic material and viscous compressible fluid. It is found that the spectrum mentioned above consists of the roots of transcendental equations whose number is proportional to the number of the layers of original medium. As the initial approximations of the roots for solving these equations numerically, it is proposed to use the points of the spectrum of one-dimensional natural vibrations of the corresponding homogenized medium. These points represent the roots of linear fractional equations. It is shown that the points at which the denominators of fractions in the linear fractional equations vanish should be also taken as the initial approximations. The accuracy of the initial approximations is proved to increase when the number of layers of the original medium increases and the layer thickness decreases simultaneously. Key words: spectrum, viscous fluid, viscoelastic material, two-phase medium, homogenized model DOI: 10.1134/S0015462819060107 |
Audience | Academic |
Author | Shamaev, A. S. Shumilova, V. V. |
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References | Shamaev, Shumilova (CR2) 2013; 448 Il’yushin, Pobedrya (CR8) 1970 Vlasov, Wu, Kabiriva (CR14) 2010; 35 Shumilova (CR13) 2013; 73 Akulenko, Nesterov (CR1) 2005; 40 CR7 Kosmodem’yanskii, Shamaev (CR5) 2009; 44 Zhikov (CR3) 2000; 191 Il’yushin (CR10) 1990 Oleinik, Shamaev, Yosifian (CR4) 1992 Shamaev, Shumilova (CR12) 2012; 48 Shamaev, Shumilova (CR6) 2016; 295 Sanchez-Palencia (CR11) 1980 Shamaev, Shumilova (CR9) 2012; 15 AS Shamaev (7007_CR6) 2016; 295 VV Zhikov (7007_CR3) 2000; 191 VV Vlasov (7007_CR14) 2010; 35 E Sanchez-Palencia (7007_CR11) 1980 LD Akulenko (7007_CR1) 2005; 40 AA Il’yushin (7007_CR10) 1990 DA Kosmodem’yanskii (7007_CR5) 2009; 44 7007_CR7 AS Shamaev (7007_CR9) 2012; 15 AS Shamaev (7007_CR2) 2013; 448 AS Shamaev (7007_CR12) 2012; 48 OA Oleinik (7007_CR4) 1992 AA Il’yushin (7007_CR8) 1970 VV Shumilova (7007_CR13) 2013; 73 |
References_xml | – volume: 15 start-page: 124 issue: 4 year: 2012 end-page: 134 ident: CR9 publication-title: Spectrum of one-dimensional vibrations of a composite consisting of layers of elastic and viscoelastic materials contributor: fullname: Shumilova – volume: 44 start-page: 75 issue: 6 year: 2009 end-page: 114 ident: CR5 article-title: Spectral properties of some problems in mechanics of strongly inhomogeneous media publication-title: Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela contributor: fullname: Shamaev – year: 1990 ident: CR10 publication-title: Mekhanika sploshnoi sredy contributor: fullname: Il’yushin – volume: 448 start-page: 43 issue: 1 year: 2013 end-page: 46 ident: CR2 article-title: Spectrum of natural vibrations in a layered medium consisting of an elastic material and a viscous fluid publication-title: Dokl. Ross. Akad. Nauk contributor: fullname: Shumilova – year: 1980 ident: CR11 publication-title: Non-Homogeneous Media and Vibration Theory contributor: fullname: Sanchez-Palencia – volume: 191 start-page: 31 issue: 7 year: 2000 end-page: 72 ident: CR3 article-title: On an extension of the method of two-scale convergence and its applications publication-title: Mat. Sb. doi: 10.4213/sm491 contributor: fullname: Zhikov – volume: 73 start-page: 167 year: 2013 end-page: 172 ident: CR13 article-title: Spectral analysis of a class of integro-differential equations in viscoelasticity theory publication-title: Probl. Mat. Anal. contributor: fullname: Shumilova – year: 1970 ident: CR8 publication-title: Osnovy matematicheskoi teorii termovyazkouprugosti contributor: fullname: Pobedrya – volume: 40 start-page: 109 issue: 1 year: 2005 end-page: 119 ident: CR1 article-title: Inertial and dissipative properties of a porous medium occupied by a viscous fluid publication-title: Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela contributor: fullname: Nesterov – year: 1992 ident: CR4 publication-title: Mathematical Problems in Elasticity and Homogenization contributor: fullname: Yosifian – volume: 295 start-page: 202 issue: 1 year: 2016 end-page: 212 ident: CR6 article-title: Asymptotic behavior of the spectrum of one-dimensional vibrations in a layered medium consisting of elastic and Kelvin—Voigt viscoelastic materials publication-title: Tr. Mat. Inst. RAN im. V.A. Steklova doi: 10.1134/S0081543816080137 contributor: fullname: Shumilova – ident: CR7 – volume: 35 start-page: 44 year: 2010 end-page: 59 ident: CR14 article-title: Well-posed solvability and the spectral properties of abstract hyperbolic equations with residual effect publication-title: Sovr. Matem. Fundam. Napravleniya contributor: fullname: Kabiriva – volume: 48 start-page: 1174 issue: 8 year: 2012 end-page: 1186 ident: CR12 article-title: Averaging of acoustic equations for a porous viscoelastic material with long-time memory filled with a viscous fluid publication-title: Dif. Uravneniya contributor: fullname: Shumilova – ident: 7007_CR7 doi: 10.1016/j.ifacol.2018.03.022 – volume: 35 start-page: 44 year: 2010 ident: 7007_CR14 publication-title: Sovr. Matem. Fundam. Napravleniya contributor: fullname: VV Vlasov – volume: 48 start-page: 1174 issue: 8 year: 2012 ident: 7007_CR12 publication-title: Dif. Uravneniya contributor: fullname: AS Shamaev – volume: 73 start-page: 167 year: 2013 ident: 7007_CR13 publication-title: Probl. Mat. Anal. contributor: fullname: VV Shumilova – volume: 191 start-page: 31 issue: 7 year: 2000 ident: 7007_CR3 publication-title: Mat. Sb. doi: 10.4213/sm491 contributor: fullname: VV Zhikov – volume: 295 start-page: 202 issue: 1 year: 2016 ident: 7007_CR6 publication-title: Tr. Mat. Inst. RAN im. V.A. Steklova doi: 10.1134/S0081543816080137 contributor: fullname: AS Shamaev – volume-title: Non-Homogeneous Media and Vibration Theory year: 1980 ident: 7007_CR11 contributor: fullname: E Sanchez-Palencia – volume: 44 start-page: 75 issue: 6 year: 2009 ident: 7007_CR5 publication-title: Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela contributor: fullname: DA Kosmodem’yanskii – volume: 40 start-page: 109 issue: 1 year: 2005 ident: 7007_CR1 publication-title: Izv. Ross. Akad. Nauk, Mekh. Tverd. Tela contributor: fullname: LD Akulenko – volume-title: Mekhanika sploshnoi sredy year: 1990 ident: 7007_CR10 contributor: fullname: AA Il’yushin – volume-title: Osnovy matematicheskoi teorii termovyazkouprugosti year: 1970 ident: 7007_CR8 contributor: fullname: AA Il’yushin – volume: 448 start-page: 43 issue: 1 year: 2013 ident: 7007_CR2 publication-title: Dokl. Ross. Akad. Nauk contributor: fullname: AS Shamaev – volume: 15 start-page: 124 issue: 4 year: 2012 ident: 7007_CR9 publication-title: Spectrum of one-dimensional vibrations of a composite consisting of layers of elastic and viscoelastic materials contributor: fullname: AS Shamaev – volume-title: Mathematical Problems in Elasticity and Homogenization year: 1992 ident: 7007_CR4 contributor: fullname: OA Oleinik |
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SubjectTerms | Classical and Continuum Physics Classical Mechanics Compressible fluids Engineering Fluid Dynamics Fluid- and Aerodynamics Isotropic material Mathematical analysis Physics Physics and Astronomy Roots Thickness Viscoelasticity Viscous fluids |
Title | Asymptotics of the Spectrum of One-Dimensional Natural Vibrations in a Layered Medium Consisting of Viscoelastic Material and Viscous Fluid |
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