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...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
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Elsevier
2021-01-01
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| Series: | Heliyon |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2405844020327183 |
| _version_ | 1818414903508074496 |
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| author | Jiarong Gan Hong Yuan Shanqing Li Qifeng Peng Huanliang Zhang |
| author_facet | Jiarong Gan Hong Yuan Shanqing Li Qifeng Peng Huanliang Zhang |
| author_sort | Jiarong Gan |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-12-14T11:26:30Z |
| format | Article |
| id | doaj.art-0b0f6b2dc1ed45bab95ead9336ac442a |
| institution | Directory Open Access Journal |
| issn | 2405-8440 |
| language | English |
| last_indexed | 2024-12-14T11:26:30Z |
| publishDate | 2021-01-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Heliyon |
| spelling | doaj.art-0b0f6b2dc1ed45bab95ead9336ac442a2022-12-21T23:03:30ZengElsevierHeliyon2405-84402021-01-0171e05876An analytical method for shallow spherical shell free vibration on two-parameter foundationJiarong Gan0Hong Yuan1Shanqing Li2Qifeng Peng3Huanliang Zhang4MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, ChinaCorresponding author.; MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, ChinaCorresponding author.; MOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, ChinaMOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, ChinaMOE Key Laboratory of Disaster Forecast and Control in Engineering, School of Mechanics and Construction Engineering, Jinan University, Guangzhou 510632, ChinaThe 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.http://www.sciencedirect.com/science/article/pii/S2405844020327183Shallow spherical shellTwo-parameter foundationHelmholtz equationLaplace equationR-functionGreen's function |
| spellingShingle | Jiarong Gan Hong Yuan Shanqing Li Qifeng Peng Huanliang Zhang An analytical method for shallow spherical shell free vibration on two-parameter foundation Heliyon Shallow spherical shell Two-parameter foundation Helmholtz equation Laplace equation R-function Green's function |
| 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 | Shallow spherical shell Two-parameter foundation Helmholtz equation Laplace equation R-function Green's function |
| url | http://www.sciencedirect.com/science/article/pii/S2405844020327183 |
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