Model Problems on Oscillations of Mechanical and Biological Membranes
Various models of membrane oscillations emerging in the theory of elasticity of mechanical systems, biomechanics of the internal ear of vertebrata, and in the theory of electrical circuits are discussed in the article. The considered oscillations have different natures, but their mathematical models...
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MDPI AG
2023-10-01
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| Series: | Inventions |
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| Online Access: | https://www.mdpi.com/2411-5134/8/6/139 |
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| author | Yury Kostikov Aleksandr Romanenkov |
| author_facet | Yury Kostikov Aleksandr Romanenkov |
| author_sort | Yury Kostikov |
| collection | DOAJ |
| description | Various models of membrane oscillations emerging in the theory of elasticity of mechanical systems, biomechanics of the internal ear of vertebrata, and in the theory of electrical circuits are discussed in the article. The considered oscillations have different natures, but their mathematical models are described using similar initial boundary value problems for the second-order hyperbolic equation with the nontrivial boundary condition. The differential equations in these problems are the same. Thus, for example, the model of voltage distribution in the telegraph line emerges for the one-dimensional equation of oscillations. The model of oscillations of a circular homogeneous solid membrane, a membrane with a hole, and the model of gas oscillations in a sphere and spherical region emerge for the two-dimensional and three-dimensional operators, but take into account the radial symmetry of oscillations. The model problem on membrane oscillation can be considered as the problem on ear drum membrane oscillations. The unified approach to reducing the corresponding problems to the initial boundary value problem with zero boundary conditions is suggested. The technique of formulating the solution in the form of a Fourier series using eigenfunctions of the corresponding Sturm–Liouville problem is described. |
| first_indexed | 2024-03-08T20:40:11Z |
| format | Article |
| id | doaj.art-c636507b2aa94751b1a47810d2f3a9e9 |
| institution | Directory Open Access Journal |
| issn | 2411-5134 |
| language | English |
| last_indexed | 2024-03-08T20:40:11Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Inventions |
| spelling | doaj.art-c636507b2aa94751b1a47810d2f3a9e92023-12-22T14:16:28ZengMDPI AGInventions2411-51342023-10-018613910.3390/inventions8060139Model Problems on Oscillations of Mechanical and Biological MembranesYury Kostikov0Aleksandr Romanenkov1Department 916 Mathematics, Moscow Aviation Institute; Volokolamskoye Shosse, 4, Moscow 125993, RussiaDepartment 916 Mathematics, Moscow Aviation Institute; Volokolamskoye Shosse, 4, Moscow 125993, RussiaVarious models of membrane oscillations emerging in the theory of elasticity of mechanical systems, biomechanics of the internal ear of vertebrata, and in the theory of electrical circuits are discussed in the article. The considered oscillations have different natures, but their mathematical models are described using similar initial boundary value problems for the second-order hyperbolic equation with the nontrivial boundary condition. The differential equations in these problems are the same. Thus, for example, the model of voltage distribution in the telegraph line emerges for the one-dimensional equation of oscillations. The model of oscillations of a circular homogeneous solid membrane, a membrane with a hole, and the model of gas oscillations in a sphere and spherical region emerge for the two-dimensional and three-dimensional operators, but take into account the radial symmetry of oscillations. The model problem on membrane oscillation can be considered as the problem on ear drum membrane oscillations. The unified approach to reducing the corresponding problems to the initial boundary value problem with zero boundary conditions is suggested. The technique of formulating the solution in the form of a Fourier series using eigenfunctions of the corresponding Sturm–Liouville problem is described.https://www.mdpi.com/2411-5134/8/6/139hyperbolic equationmodeling of oscillations with attenuationboundary value problemexact solutions |
| spellingShingle | Yury Kostikov Aleksandr Romanenkov Model Problems on Oscillations of Mechanical and Biological Membranes Inventions hyperbolic equation modeling of oscillations with attenuation boundary value problem exact solutions |
| title | Model Problems on Oscillations of Mechanical and Biological Membranes |
| title_full | Model Problems on Oscillations of Mechanical and Biological Membranes |
| title_fullStr | Model Problems on Oscillations of Mechanical and Biological Membranes |
| title_full_unstemmed | Model Problems on Oscillations of Mechanical and Biological Membranes |
| title_short | Model Problems on Oscillations of Mechanical and Biological Membranes |
| title_sort | model problems on oscillations of mechanical and biological membranes |
| topic | hyperbolic equation modeling of oscillations with attenuation boundary value problem exact solutions |
| url | https://www.mdpi.com/2411-5134/8/6/139 |
| work_keys_str_mv | AT yurykostikov modelproblemsonoscillationsofmechanicalandbiologicalmembranes AT aleksandrromanenkov modelproblemsonoscillationsofmechanicalandbiologicalmembranes |