A problem on longitudinal vibration of a bar with elastic fixing
In this paper, we study longitudinal vibration in a thick short bar fixed by point forces and springs. For mathematical model we consider a boundary value problem with dynamical boundary conditions for a forth order partial differential equation. The choice of this model depends on a necessity to ta...
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| Format: | Article |
| Language: | English |
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Samara State Technical University
2016-06-01
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| Series: | Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki |
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| Online Access: | https://journals.eco-vector.com/1991-8615/article/viewFile/20495/16742 |
| _version_ | 1828311402366894080 |
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| author | Alexander B Beylin |
| author_facet | Alexander B Beylin |
| author_sort | Alexander B Beylin |
| collection | DOAJ |
| description | In this paper, we study longitudinal vibration in a thick short bar fixed by point forces and springs. For mathematical model we consider a boundary value problem with dynamical boundary conditions for a forth order partial differential equation. The choice of this model depends on a necessity to take into account the result of a transverse strain. It was shown by Rayleigh that neglect of a transverse strain leads to an error. This is confirmed by modern nonlocal theory of vibration. We prove existence of orthogonal with load eigenfunctions and derive representation of them. Established properties of eigenfunctions make possible using the separation of variables method and finding a unique solution of the problem. |
| first_indexed | 2024-04-13T15:58:43Z |
| format | Article |
| id | doaj.art-65aced90afc84cd9a884f9dbbadf2752 |
| institution | Directory Open Access Journal |
| issn | 1991-8615 2310-7081 |
| language | English |
| last_indexed | 2024-04-13T15:58:43Z |
| publishDate | 2016-06-01 |
| publisher | Samara State Technical University |
| record_format | Article |
| series | Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki |
| spelling | doaj.art-65aced90afc84cd9a884f9dbbadf27522022-12-22T02:40:36ZengSamara State Technical UniversityVestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki1991-86152310-70812016-06-0120224925810.14498/vsgtu147417915A problem on longitudinal vibration of a bar with elastic fixingAlexander B Beylin0Samara State Technical UniversityIn this paper, we study longitudinal vibration in a thick short bar fixed by point forces and springs. For mathematical model we consider a boundary value problem with dynamical boundary conditions for a forth order partial differential equation. The choice of this model depends on a necessity to take into account the result of a transverse strain. It was shown by Rayleigh that neglect of a transverse strain leads to an error. This is confirmed by modern nonlocal theory of vibration. We prove existence of orthogonal with load eigenfunctions and derive representation of them. Established properties of eigenfunctions make possible using the separation of variables method and finding a unique solution of the problem.https://journals.eco-vector.com/1991-8615/article/viewFile/20495/16742dynamic boundary conditionslongitudinal vibrationloaded orthogonalityrayleigh's model |
| spellingShingle | Alexander B Beylin A problem on longitudinal vibration of a bar with elastic fixing Vestnik Samarskogo Gosudarstvennogo Tehničeskogo Universiteta. Seriâ: Fiziko-Matematičeskie Nauki dynamic boundary conditions longitudinal vibration loaded orthogonality rayleigh's model |
| title | A problem on longitudinal vibration of a bar with elastic fixing |
| title_full | A problem on longitudinal vibration of a bar with elastic fixing |
| title_fullStr | A problem on longitudinal vibration of a bar with elastic fixing |
| title_full_unstemmed | A problem on longitudinal vibration of a bar with elastic fixing |
| title_short | A problem on longitudinal vibration of a bar with elastic fixing |
| title_sort | problem on longitudinal vibration of a bar with elastic fixing |
| topic | dynamic boundary conditions longitudinal vibration loaded orthogonality rayleigh's model |
| url | https://journals.eco-vector.com/1991-8615/article/viewFile/20495/16742 |
| work_keys_str_mv | AT alexanderbbeylin aproblemonlongitudinalvibrationofabarwithelasticfixing AT alexanderbbeylin problemonlongitudinalvibrationofabarwithelasticfixing |