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An analytical method for shallow spherical shell free vibration on two-parameter foundation
The free vibration control differential equation of shallow spherical shell on two-parameter foundation is a four order differential equation. Using the intermediate variable, the four order differential equation is reduced to two lower order differential equations. The first lower order differentia...
Autores principales: | , , , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
Elsevier
2021
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Materias: | |
Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7855341/ https://www.ncbi.nlm.nih.gov/pubmed/33553716 http://dx.doi.org/10.1016/j.heliyon.2020.e05876 |
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author | Gan, Jiarong Yuan, Hong Li, Shanqing Peng, Qifeng Zhang, Huanliang |
author_facet | Gan, Jiarong Yuan, Hong Li, Shanqing Peng, Qifeng Zhang, Huanliang |
author_sort | Gan, Jiarong |
collection | PubMed |
description | The free vibration control differential equation of shallow spherical shell on two-parameter foundation is a four order differential equation. Using the intermediate variable, the four order differential equation is reduced to two lower order differential equations. The first lower order differential equation is a Helmholtz equation. A new method of two-dimensional Helmholtz operator is proposed as shown in the paper in which the Bessel function included in Helmholtz equation needs to be treated appropriately to eliminate singularity. The first lower order differential equation is transformed into the integral equation using the proposed method in the paper. The second lower order differential equation which is a Laplace equation is transformed into the integral equation by existing methods. Then the two integral equations are discretized according to the middle rectangle formula, and the corresponding solutions can be obtained by MATLAB programming. In this paper, the R-function theory is used to select the appropriate boundary equation to eliminate the singularity. Based on the properties of R-function, the combined method of Helmholtz equation and Laplace equation can solve the free vibration problem of irregular shallow spherical shell on two-parameter foundation. Five examples are given to verify the feasibility of the method. |
format | Online Article Text |
id | pubmed-7855341 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | Elsevier |
record_format | MEDLINE/PubMed |
spelling | pubmed-78553412021-02-05 An analytical method for shallow spherical shell free vibration on two-parameter foundation Gan, Jiarong Yuan, Hong Li, Shanqing Peng, Qifeng Zhang, Huanliang Heliyon Research Article The free vibration control differential equation of shallow spherical shell on two-parameter foundation is a four order differential equation. Using the intermediate variable, the four order differential equation is reduced to two lower order differential equations. The first lower order differential equation is a Helmholtz equation. A new method of two-dimensional Helmholtz operator is proposed as shown in the paper in which the Bessel function included in Helmholtz equation needs to be treated appropriately to eliminate singularity. The first lower order differential equation is transformed into the integral equation using the proposed method in the paper. The second lower order differential equation which is a Laplace equation is transformed into the integral equation by existing methods. Then the two integral equations are discretized according to the middle rectangle formula, and the corresponding solutions can be obtained by MATLAB programming. In this paper, the R-function theory is used to select the appropriate boundary equation to eliminate the singularity. Based on the properties of R-function, the combined method of Helmholtz equation and Laplace equation can solve the free vibration problem of irregular shallow spherical shell on two-parameter foundation. Five examples are given to verify the feasibility of the method. Elsevier 2021-01-06 /pmc/articles/PMC7855341/ /pubmed/33553716 http://dx.doi.org/10.1016/j.heliyon.2020.e05876 Text en © 2020 The Authors http://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). |
spellingShingle | Research Article Gan, Jiarong Yuan, Hong Li, Shanqing Peng, Qifeng Zhang, Huanliang An analytical method for shallow spherical shell free vibration on two-parameter foundation |
title | An analytical method for shallow spherical shell free vibration on two-parameter foundation |
title_full | An analytical method for shallow spherical shell free vibration on two-parameter foundation |
title_fullStr | An analytical method for shallow spherical shell free vibration on two-parameter foundation |
title_full_unstemmed | An analytical method for shallow spherical shell free vibration on two-parameter foundation |
title_short | An analytical method for shallow spherical shell free vibration on two-parameter foundation |
title_sort | analytical method for shallow spherical shell free vibration on two-parameter foundation |
topic | Research Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7855341/ https://www.ncbi.nlm.nih.gov/pubmed/33553716 http://dx.doi.org/10.1016/j.heliyon.2020.e05876 |
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