A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution
In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential eq...
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| Format: | Article |
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/649 |
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| author | Stanislav Papkov Jnan Ranjan Banerjee |
| author_facet | Stanislav Papkov Jnan Ranjan Banerjee |
| author_sort | Stanislav Papkov |
| collection | DOAJ |
| description | In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions. |
| first_indexed | 2024-03-11T09:34:13Z |
| format | Article |
| id | doaj.art-45545db6cae2428e9b663758a4450194 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:34:13Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-45545db6cae2428e9b663758a44501942023-11-16T17:22:36ZengMDPI AGMathematics2227-73902023-01-0111364910.3390/math11030649A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series SolutionStanislav Papkov0Jnan Ranjan Banerjee1Department of Mathematics, Sevastopol State University, 299046 Sevastopol, RussiaSchool of Science and Technology, City, University of London, London EC1V 0HB, UKIn this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions.https://www.mdpi.com/2227-7390/11/3/649isotropic and orthotropic triangular platesfree vibrationnatural modessuperposition methodinfinite system of linear equations |
| spellingShingle | Stanislav Papkov Jnan Ranjan Banerjee A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution Mathematics isotropic and orthotropic triangular plates free vibration natural modes superposition method infinite system of linear equations |
| title | A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution |
| title_full | A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution |
| title_fullStr | A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution |
| title_full_unstemmed | A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution |
| title_short | A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution |
| title_sort | new method for free vibration analysis of triangular isotropic and orthotropic plates of isosceles type using an accurate series solution |
| topic | isotropic and orthotropic triangular plates free vibration natural modes superposition method infinite system of linear equations |
| url | https://www.mdpi.com/2227-7390/11/3/649 |
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