Influences of angular velocity and periodic axial load on the dynamic instability of functionally graded porous cylindrical panels

Abstract

This work presents an analytical study on the dynamic instability of rotating functionally graded porous cylindrical panels subjected to periodic axial load under simply supported boundary conditions. Considering the Coriolis acceleration, centrifugal component, and hoop stress, the partial general equations of motion have been derived based on first-order shear deformation theory (FSDT) and by using Hamilton's principle. The excitation frequencies of rotating functionally graded porous cylindrical panels were calculated by substituting all motion equations for the transverse displacement and then applying the Galerkin method to the resulting equation. Numerical results are employed to validate the analytical results and show good agreement. The influences of the porosity distribution method, porosity coefficient, static and dynamic load factors, angular velocity, Coriolis acceleration, initial hoop stress, circumferential wave number, angle of the cylindrical panel, and cylindrical panel dimensions on the dynamic instability of rotating functionally graded porous cylindrical panels were investigated. Some of the important outputs that were reached in this study are that static and dynamic loads increase the width of the instability zone and that the structure is more stable when the porosity moves away from the surface and when the angular velocity increases. In addition, the cylindrical panel has a large frequency when the panel's angle is small (i.e., θ₀ = 30°), but it is less sensitive to dynamic load when the angle is large (i.e., θ₀ ≥ 100°).Please confirm if the author names are presented accurately and in the correct sequence (given name, middle name/initial, family name). Author 1 Given name: [specify authors given name] Last name [specify authors last name]. Also, kindly confirm the details in the metadata are correct.Thank you for this inquiry, author names have been given accurately and in the correct sequence.

Appendices

Appendix 1

$\Gamma_{i,j} \;(i,j = 1,2,3,4,5)$ are parameters and given by:

$$\begin{aligned} \Gamma_{11} & = - \left( {I_{10} \alpha^{2} + I_{30} \frac{{\beta^{2} }}{{R^{2} }} + \frac{1}{4}I_{0} \Omega^{2} \beta^{2} } \right), \\ \Gamma_{12} & = I_{20} \frac{\beta }{R}\alpha + I_{30} \frac{\beta }{R}\alpha - \frac{1}{4}I_{0} \Omega^{2} R\beta \alpha \\ \Gamma_{13} & = I_{20} \frac{\alpha }{R},\quad \Gamma_{14} = - \left( {I_{11} \alpha^{2} + I_{31} \frac{{\beta^{2} }}{{R^{2} }}} \right),\;\;\Gamma_{15} = \left( {I_{21} + I_{31} } \right)\frac{\beta }{R}\alpha \\ \Gamma_{21} & = \Gamma_{12} ,\quad \Gamma_{22} = - \left( {I_{30} \alpha^{2} + I_{10} \frac{{\beta^{2} }}{{R^{2} }} + \frac{1}{4}I_{0} \Omega^{2} R^{2} \alpha^{2} + kI_{30} \frac{1}{{R^{2} }}} \right), \\ \Gamma_{23} & = - \left( {(I_{10} + kI_{30} )\frac{1}{{R^{2} }} + I_{0} \Omega^{2} } \right)\beta \\ \Gamma_{24} & = \left( {I_{21} + I_{31} } \right)\frac{\beta }{R}\alpha ,\quad \Gamma_{25} = - \left( {I_{31} \alpha^{2} + I_{11} \frac{{\beta^{2} }}{{R^{2} }} - I_{1} \Omega^{2} - kI_{30} \frac{1}{{R^{2} }}} \right), \\ \Gamma_{31} & = \Gamma_{13} ,\quad \Gamma_{32} = \Gamma_{23} \\ \Gamma_{33} & = - \left( {I_{10} \frac{1}{{R^{2} }} + kI_{30} \left( {\frac{{\beta^{2} }}{{R^{2} }} + \alpha^{2} } \right) + I_{0} \Omega^{2} (\beta^{2} - 1)} \right), \\ \Gamma_{34} & = \left( {I_{21} \frac{1}{R} - kI_{30} } \right)\alpha ,\quad \Gamma_{35} = \left( { - I_{11} \frac{1}{{R^{2} }} + k\frac{{I_{30} }}{R}} \right)\beta \\ \Gamma_{41} & = \Gamma_{14} ,\quad \Gamma_{42} = \Gamma_{24} ,\quad \Gamma_{43} = \Gamma_{34} , \\ \Gamma_{44} & = - \left( {I_{12} \alpha^{2} + I_{32} \frac{{\beta^{2} }}{{R^{2} }} + kI_{30} } \right),\quad \Gamma_{45} = \left( {I_{22} + I_{32} } \right)\frac{\beta }{R}\alpha \\ \Gamma_{51} & = \Gamma_{15} ,\quad \Gamma_{52} = \Gamma_{25} ,\quad \Gamma_{53} = \Gamma_{35} , \\ \Gamma_{54} & = \Gamma_{45} ,\quad \Gamma_{55} = - \left( {I_{32} \alpha^{2} + I_{12} \frac{{\beta^{2} }}{{R^{2} }} - I_{2} \Omega^{2} + kI_{30} } \right) \\ \end{aligned}$$

Appendix 2

$D_{i} ,K_{i} ,T_{i} ,H_{i} \;(i = 1,2)$ are defined as:

$$\begin{aligned} D_{1} & = - \frac{1}{{\Gamma_{11} }}\left( {\Gamma_{12} K_{1} + \Gamma_{13} + \Gamma_{14} T_{1} + \Gamma_{15} H_{1} } \right), \\ D_{2} & = - \frac{1}{{\Gamma_{11} }}\left( {\Gamma_{12} K_{2} + \Gamma_{14} T_{2} + \Gamma_{15} H_{2} } \right) \\ K_{1} & = a_{1} + a_{2} T_{1} + a_{3} H_{1} ,\quad K_{2} = a_{4} + a_{2} T_{2} + a_{3} H_{2} , \\ T_{1} & = j_{1} + j_{2} H_{1} ,\quad T_{2} = j_{3} + j_{2} H_{2} \\ H_{1} & = - \frac{{(a_{1} + a_{2} j_{1} )(\Gamma_{11} \Gamma_{52} - \Gamma_{51} \Gamma_{12} ) + j_{1} (\Gamma_{11} \Gamma_{54} - \Gamma_{51} \Gamma_{14} ) + (\Gamma_{11} \Gamma_{53} - \Gamma_{51} \Gamma_{13} )}}{{(a_{3} + a_{2} j_{2} )(\Gamma_{11} \Gamma_{52} - \Gamma_{51} \Gamma_{12} ) + j_{2} (\Gamma_{11} \Gamma_{54} - \Gamma_{51} \Gamma_{14} ) + (\Gamma_{11} \Gamma_{55} - \Gamma_{51} \Gamma_{15} )}} \\ H_{2} & = - \frac{{(a_{4} + a_{2} j_{3} )(\Gamma_{11} \Gamma_{52} - \Gamma_{51} \Gamma_{12} ) + j_{3} (\Gamma_{11} \Gamma_{54} - \Gamma_{51} \Gamma_{14} ) + \Gamma_{11} I_{1} }}{{(a_{3} + a_{2} j_{2} )(\Gamma_{11} \Gamma_{52} - \Gamma_{51} \Gamma_{12} ) + j_{2} (\Gamma_{11} \Gamma_{54} - \Gamma_{51} \Gamma_{14} ) + (\Gamma_{11} \Gamma_{55} - \Gamma_{51} \Gamma_{15} )}} \\ \end{aligned}$$

where

$$\begin{aligned} j_{1} & = - \frac{{a_{1} (\Gamma_{11} \Gamma_{42} - \Gamma_{41} \Gamma_{12} ) + (\Gamma_{11} \Gamma_{43} - \Gamma_{41} \Gamma_{13} )}}{{a_{2} (\Gamma_{11} \Gamma_{42} - \Gamma_{41} \Gamma_{12} ) + (\Gamma_{11} \Gamma_{44} - \Gamma_{41} \Gamma_{14} )}}, \\ j_{2} & = - \frac{{a_{3} (\Gamma_{11} \Gamma_{42} - \Gamma_{41} \Gamma_{12} ) + (\Gamma_{11} \Gamma_{45} - \Gamma_{41} \Gamma_{15} )}}{{a_{2} (\Gamma_{11} \Gamma_{42} - \Gamma_{41} \Gamma_{12} ) + (\Gamma_{11} \Gamma_{44} - \Gamma_{41} \Gamma_{14} )}} \\ j_{3} & = - \frac{{a_{4} (\Gamma_{11} \Gamma_{42} - \Gamma_{41} \Gamma_{12} )}}{{a_{2} (\Gamma_{11} \Gamma_{42} - \Gamma_{41} \Gamma_{12} ) + (\Gamma_{11} \Gamma_{44} - \Gamma_{41} \Gamma_{14} )}}, \\ a_{1} & = - \frac{{\Gamma_{11} \Gamma_{23} - \Gamma_{21} \Gamma_{13} }}{{\Gamma_{11} \Gamma_{22} - \Gamma_{21} \Gamma_{12} }} \\ a_{2} & = - \frac{{\Gamma_{11} \Gamma_{24} - \Gamma_{21} \Gamma_{14} }}{{\Gamma_{11} \Gamma_{22} - \Gamma_{21} \Gamma_{12} }}, \\ a_{3} & = - \frac{{\Gamma_{11} \Gamma_{25} - \Gamma_{21} \Gamma_{15} }}{{\Gamma_{11} \Gamma_{22} - \Gamma_{21} \Gamma_{12} }},\quad a_{4} = - \frac{{\Gamma_{11} I_{0} }}{{\Gamma_{11} \Gamma_{22} - \Gamma_{21} \Gamma_{12} }} \\ \end{aligned}$$

Appendix 3

$$\begin{gathered} C_{1} = I_{0} + 4\Omega^{2} (I_{0} K_{2} + I_{1} H_{2} ) \\ C_{2} = - 2\Omega \left( {I_{0} K_{1} + I_{1} H_{1} + \Gamma_{31} L_{2} + \Gamma_{32} K_{2} + \Gamma_{34} T_{2} + \Gamma_{35} H_{2} } \right) \\ F = - \left( {\Gamma_{31} L_{1} + \Gamma_{32} K_{1} + \Gamma_{33} + \Gamma_{34} T_{1} + \Gamma_{35} H_{1} } \right) \\ \end{gathered}$$