A Computer Presentation of the Analytical and Numerical Study of Nonlinear Vibration Response for Porous Functionally Graded Cylindrical Panel

Abstract

The current study uses a new analytical model and numerical method to present a study of free vibration carried out on a cylindrical shell panel that is simply supported and functionally graded. It is anticipated that the FG thickness attributes will be dependent on the porosity level and will change along the thickness axis in accordance with a distribution that follows a power-law. This work makes a contribution by analyzing the performance of porous FGMs, which are employed in a particularly wide variety of biomedical applications. For the purpose of determining the free vibration characteristics as well as the nonlinear vibration response, the governing equations are constructed on a first-order shear deformation theory by utilizing the Galerkin technique with the fourth-order Runge Kutta a close encounter with an incomplete FGM cylindrical shell panel and include different parameters. Parameters included are the power-law index, graded distributions of porosity, and FG thickness. With the help of both the ANSYS 2021-R1 software, a numerical investigation was carried out making use of the finite element approach, and a modal investigation was carried out. This was done in order to verify the analytical strategy.

Appendix

$$ \begin{aligned} & I_{10} = \frac{{E_{1} }}{{1 - \upsilon^{2} }},I_{20} = \frac{{\upsilon E_{1} }}{{1 - \upsilon^{2} }},I_{30} = \frac{{E_{1} }}{{2\left( {1 + \upsilon } \right)}},I_{11} = \frac{{E_{2} }}{{1 - \upsilon^{2} }},I_{21} = \frac{{\upsilon E_{2} }}{{1 - \upsilon^{2} }},I_{31} = \frac{{E_{2} }}{{2\left( {1 + \upsilon } \right)}}, \\ & I_{12} = \frac{{E_{3} }}{{1 - \upsilon^{2} }},I_{22} = \frac{{\upsilon E_{3} }}{{1 - \upsilon^{2} }},I_{32} = \frac{{E_{3} }}{{2\left( {1 + \upsilon } \right)}}, \\ & A_{11} = \frac{1}{\Delta }I_{10} ,A_{22} = \frac{1}{\Delta }I_{10} ,A_{12} = \frac{{I_{20} }}{\Delta },A_{66} = \frac{1}{{I_{30} }},\Delta = I_{10}^{2} - I_{20}^{2} ,B_{11} = A_{22} I_{11} - A_{12} I_{21} ,B_{22} = A_{11} I_{11} - A_{12} I_{21} , \\ & B_{12} = A_{22} I_{21} - A_{12} I_{11} ,B_{21} = A_{11} I_{21} - A_{12} I_{11} ,B_{66} = \frac{{I_{31} }}{{I_{30} }},D_{11} = I_{12} - B_{11} B_{12} - I_{21} B_{21} ,D_{22} = I_{22} - B_{22} I_{11} - I_{21} B_{12} , \\ & D_{12} = I_{22} - B_{12} I_{11} - I_{21} B_{22} ,D_{21} = I_{22} - B_{21} I_{11} - I_{21} B_{11} ,D_{66} = I_{32} - I_{31} B_{66} , \\ & {\text{ T}}_{11} \left( w \right) = K_{s} I_{30} \frac{{\partial^{2} w}}{{\partial x^{2} }} + K_{s} I_{30} \frac{{\partial^{2} w}}{{\partial y^{2} }},{ T}_{12} \left( {\phi_{x} } \right) = K_{s} I_{30} \frac{{\partial \phi_{x} }}{\partial x}, \\ & {\text{ T}}_{13} \left( {\phi_{y} } \right) = K_{s} I_{30} \frac{{\partial \phi_{y} }}{\partial y},R_{1} \left( {w,f} \right) = \frac{{\partial^{2} f}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial x^{2} }} - 2\frac{{\partial^{2} f}}{\partial x\partial y}\frac{{\partial^{2} w}}{\partial x\partial y} + \frac{{\partial^{2} f}}{{\partial x^{2} }}\frac{{\partial^{2} w}}{{\partial y^{2} }} + \frac{1}{R}\frac{{\partial^{2} f}}{{\partial x^{2} }}, \\ & {\text{ T}}_{21} \left( w \right) = - K_{s} I_{30} \frac{\partial w}{{\partial x}},{ T}_{22} \left( {\phi_{x} } \right) = D_{11} \frac{{\partial^{2} \phi_{x} }}{{\partial x^{2} }} + D_{66} \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} - K_{s} I_{30} \phi_{x} ,{\text{ T}}_{23} \left( {\phi_{y} } \right) = \left( {D_{12} + D_{66} } \right)\frac{{\partial^{2} \phi_{y} }}{\partial x\partial y}, \\ & R_{2} \left( f \right) = B_{21} \frac{{\partial^{3} f}}{{\partial x^{3} }} + \left( {B_{11} - B_{66} } \right)\frac{{\partial^{3} f}}{{\partial x\partial y^{2} }},{\text{ T}}_{31} \left( w \right) = - K_{s} I_{30} \frac{\partial w}{{\partial y}},{\text{ T}}_{32} \left( {\phi_{x} } \right) = \left( {D_{21} + D_{66} } \right)\frac{{\partial^{2} \phi_{x} }}{\partial x\partial y}, \\ & {\text{ T}}_{33} \left( {\phi_{y} } \right) = D_{22} \frac{{\partial^{2} \phi_{y} }}{{\partial y^{2} }} + D_{66} \frac{{\partial^{2} \phi_{y} }}{{\partial x^{2} }} - K_{s} I_{30} \phi_{y} ,R_{3} \left( f \right) = B_{12} \frac{{\partial^{3} f}}{{\partial y^{3} }} + \left( {B_{22} - B_{66} } \right)\frac{{\partial^{3} f}}{{\partial x^{2} \partial y}}, \\ \end{aligned} $$