Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates
The free vibration of isotropic gradient elastic thick non-rectangular microplates is analyzed in this paper. To capture the microstructure-dependent effects of microplates, a negative second-order gradient elastic theory with symmetry is utilized. The related equations of motion and boundary condit...
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MDPI AG
2022-12-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/14/12/2592 |
| _version_ | 1797455155973586944 |
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| author | Bo Zhang Cheng Li Limin Zhang Feng Xie |
| author_facet | Bo Zhang Cheng Li Limin Zhang Feng Xie |
| author_sort | Bo Zhang |
| collection | DOAJ |
| description | The free vibration of isotropic gradient elastic thick non-rectangular microplates is analyzed in this paper. To capture the microstructure-dependent effects of microplates, a negative second-order gradient elastic theory with symmetry is utilized. The related equations of motion and boundary conditions are obtained using the energy variational principle. A closed-form solution is presented for simply supported free-vibrational rectangular microplates with four edges. A <i>C</i><sup>1</sup>-type differential quadrature finite element (DQFE) is applied to solve the free vibration of thick microplates. The DQ rule is extended to the straight-sided quadrilateral domain through a coordinate transformation between the natural and Cartesian coordinate systems. The Gauss–Lobato quadrature rule and DQ rule are jointly used to discretize the strain and kinetic energies of a generic straight-sided quadrilateral plate element. Selective numerical examples are validated against those available in the literature. Finally, the impact of various parameters on the free vibration characteristics of annular sectorial and triangular microplates is shown. It indicates that the strain gradient and inertia gradient effects can result in distinct changes in both vibration frequencies and mode shapes. |
| first_indexed | 2024-03-09T15:47:29Z |
| format | Article |
| id | doaj.art-314c6149cac0409985c8d2c85a9afadc |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T15:47:29Z |
| publishDate | 2022-12-01 |
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| series | Symmetry |
| spelling | doaj.art-314c6149cac0409985c8d2c85a9afadc2023-11-24T18:19:42ZengMDPI AGSymmetry2073-89942022-12-011412259210.3390/sym14122592Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick MicroplatesBo Zhang0Cheng Li1Limin Zhang2Feng Xie3Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 611756, ChinaApplied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 611756, ChinaApplied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 611756, ChinaSchool of Rail Transportation, Soochow University, Suzhou 215131, ChinaThe free vibration of isotropic gradient elastic thick non-rectangular microplates is analyzed in this paper. To capture the microstructure-dependent effects of microplates, a negative second-order gradient elastic theory with symmetry is utilized. The related equations of motion and boundary conditions are obtained using the energy variational principle. A closed-form solution is presented for simply supported free-vibrational rectangular microplates with four edges. A <i>C</i><sup>1</sup>-type differential quadrature finite element (DQFE) is applied to solve the free vibration of thick microplates. The DQ rule is extended to the straight-sided quadrilateral domain through a coordinate transformation between the natural and Cartesian coordinate systems. The Gauss–Lobato quadrature rule and DQ rule are jointly used to discretize the strain and kinetic energies of a generic straight-sided quadrilateral plate element. Selective numerical examples are validated against those available in the literature. Finally, the impact of various parameters on the free vibration characteristics of annular sectorial and triangular microplates is shown. It indicates that the strain gradient and inertia gradient effects can result in distinct changes in both vibration frequencies and mode shapes.https://www.mdpi.com/2073-8994/14/12/2592free vibrationnon-rectangular microplatesgradient elastic theory with symmetrymicrostructure-dependent effectsdifferential quadrature finite element |
| spellingShingle | Bo Zhang Cheng Li Limin Zhang Feng Xie Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates Symmetry free vibration non-rectangular microplates gradient elastic theory with symmetry microstructure-dependent effects differential quadrature finite element |
| title | Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates |
| title_full | Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates |
| title_fullStr | Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates |
| title_full_unstemmed | Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates |
| title_short | Size-Dependent Free Vibration of Non-Rectangular Gradient Elastic Thick Microplates |
| title_sort | size dependent free vibration of non rectangular gradient elastic thick microplates |
| topic | free vibration non-rectangular microplates gradient elastic theory with symmetry microstructure-dependent effects differential quadrature finite element |
| url | https://www.mdpi.com/2073-8994/14/12/2592 |
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