Vibration analysis of cantilever FG-CNTRC trapezoidal plates

Abstract

In this paper, a numerical solution is presented for free vibration analysis of cantilever functionally graded carbon nanotube-reinforced trapezoidal plates. The plate is modeled based on the first-order shear deformation theory, effective mechanical properties are estimated according to extended rule of mixture, and the set of governing equations and boundary conditions are derived using Hamilton’s principle. Generalized differential quadrature method is employed, and natural frequencies and corresponding mode shapes are derived numerically. Convergence and accuracy of the solution are confirmed, and effect of various parameters on the natural frequencies is investigated including geometrical characteristics, volume fraction and distribution of carbon nanotubes. Because of similarity of the studied model with the wing, tail and fin of aircrafts and missiles, results of this paper can be useful in design and analysis of aeronautic vehicles in the near future. It is worth mentioning that results of this paper may serve as benchmarks for future studies.

Appendices

Appendix 1

$$ \begin{array}{*{20}l} {S_{11} = A_{11} \frac{{\partial^{2} }}{{\partial x^{2} }} + A_{66} \frac{{\partial^{2} }}{{\partial y^{2} }} - I_{0} \frac{{\partial^{2} }}{{\partial t^{2} }}} \hfill & {S_{12} = \left( {A_{12} + A_{66} } \right)\frac{{\partial^{2} }}{\partial x\partial y}} \hfill & {S_{13} = 0} \hfill \\ {S_{14} = B_{11} \frac{{\partial^{2} }}{{\partial x^{2} }} + B_{66} \frac{{\partial^{2} }}{{\partial y^{2} }} - I_{1} \frac{{\partial^{2} }}{{\partial t^{2} }}} \hfill & {S_{15} = \left( {B_{12} + B_{66} } \right)\frac{{\partial^{2} }}{\partial x\partial y}} \hfill & {S_{22} = A_{66} \frac{{\partial^{2} }}{{\partial x^{2} }} + A_{22} \frac{{\partial^{2} }}{{\partial y^{2} }} - I_{0} \frac{{\partial^{2} }}{{\partial t^{2} }}} \hfill \\ {S_{23} = 0} \hfill & {S_{24} = \left( {B_{12} + B_{66} } \right)\frac{{\partial^{2} }}{\partial x\partial y}} \hfill & {S_{25} = B_{66} \frac{{\partial^{2} }}{{\partial x^{2} }} + B_{22} \frac{{\partial^{2} }}{{\partial y^{2} }} - I_{1} \frac{{\partial^{2} }}{{\partial t^{2} }}} \hfill \\ {S_{33} = - A_{55} \frac{{\partial^{2} }}{{\partial x^{2} }} - A_{44} \frac{{\partial^{2} }}{{\partial y^{2} }} + I_{0} \frac{{\partial^{2} }}{{\partial t^{2} }}} \hfill & {S_{34} = - A_{55} \frac{\partial }{\partial x}} \hfill & {S_{35} = - A_{44} \frac{\partial }{\partial y}} \hfill \\ {S_{44} = D_{11} \frac{{\partial^{2} }}{{\partial x^{2} }} + D_{66} \frac{{\partial^{2} }}{{\partial y^{2} }} - A_{55} - I_{2} \frac{{\partial^{2} }}{{\partial t^{2} }}} \hfill & {S_{45} = \left( {D_{12} + D_{66} } \right)\frac{{\partial^{2} }}{\partial x\partial y}} \hfill & {S_{55} = D_{66} \frac{{\partial^{2} }}{{\partial x^{2} }} + D_{22} \frac{{\partial^{2} }}{{\partial y^{2} }} - A_{44} - I_{2} \frac{{\partial^{2} }}{{\partial t^{2} }}} \hfill \\ \end{array} $$

(42)

Appendix 2

$$ \begin{array}{*{20}l} {P_{11} = \left( {n_{x}^{2} A_{11} + n_{y}^{2} A_{12} } \right)\left( {} \right)_{,x} + 2n_{x} n_{y} A_{66} \left( {} \right)_{,y} } \hfill & {P_{21} = n_{x} n_{y} \left( {A_{12} - A_{11} } \right)\left( {} \right)_{,x} + \left( {n_{x}^{2} - n_{y}^{2} } \right)A_{66} \left( {} \right)_{,y} } \hfill \\ {P_{12} = 2n_{x} n_{y} A_{66} \left( {} \right)_{,x} + \left( {n_{x}^{2} A_{12} + n_{y}^{2} A_{22} } \right)\left( {} \right)_{,y} } \hfill & {P_{22} = \left( {n_{x}^{2} - n_{y}^{2} } \right)A_{66} \left( {} \right)_{,x} + n_{x} n_{y} \left( {A_{22} - A_{12} } \right)\left( {} \right)_{,y} } \hfill \\ {P_{13} = 0} \hfill & {P_{23} = 0} \hfill \\ {P_{14} = \left( {n_{x}^{2} B_{11} + n_{y}^{2} B_{12} } \right)\left( {} \right)_{,x} + 2n_{x} n_{y} B_{66} \left( {} \right)_{,y} } \hfill & {P_{24} = n_{x} n_{y} \left( {B_{12} - B_{11} } \right)\left( {} \right)_{,x} + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} \left( {} \right)_{,y} } \hfill \\ {P_{15} = 2n_{x} n_{y} B_{66} \left( {} \right)_{,x} + \left( {n_{x}^{2} B_{12} + n_{y}^{2} B_{22} } \right)\left( {} \right)_{,y} } \hfill & {P_{25} = \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} \left( {} \right)_{,x} + n_{x} n_{y} \left( {B_{22} - B_{12} } \right)\left( {} \right)_{,y} } \hfill \\ {P_{31} = \left( {n_{x}^{2} B_{11} + n_{y}^{2} B_{12} } \right)\left( {} \right)_{,x} + 2n_{x} n_{y} B_{66} \left( {} \right)_{,y} } \hfill & {P_{41} = n_{x} n_{y} \left( {B_{12} - B_{11} } \right)\left( {} \right)_{,x} + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} \left( {} \right)_{,y} } \hfill \\ {P_{32} = 2n_{x} n_{y} B_{66} \left( {} \right)_{,x} + \left( {n_{x}^{2} B_{12} + n_{y}^{2} B_{22} } \right)\left( {} \right)_{,y} } \hfill & {P_{42} = \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} \left( {} \right)_{,x} + n_{x} n_{y} \left( {B_{22} - B_{12} } \right)\left( {} \right)_{,y} } \hfill \\ {P_{33} = 0} \hfill & {P_{43} = 0} \hfill \\ {P_{34} = \left( {n_{x}^{2} D_{11} + n_{y}^{2} D_{12} } \right)\left( {} \right)_{,x} + 2n_{x} n_{y} D_{66} \left( {} \right)_{,y} } \hfill & {P_{44} = n_{x} n_{y} \left( {D_{12} - D_{11} } \right)\left( {} \right)_{,x} + \left( {n_{x}^{2} - n_{y}^{2} } \right)D_{66} \left( {} \right)_{,y} } \hfill \\ {P_{35} = 2n_{x} n_{y} D_{66} \left( {} \right)_{,x} + \left( {n_{x}^{2} D_{12} + n_{y}^{2} D_{22} } \right)\left( {} \right)_{,y} } \hfill & {P_{45} = \left( {n_{x}^{2} - n_{y}^{2} } \right)D_{66} \left( {} \right)_{,x} + n_{x} n_{y} \left( {D_{22} - D_{12} } \right)\left( {} \right)_{,y} } \hfill \\ {P_{51} = 0\quad P_{52} = 0\quad P_{53} = n_{x} A_{55} \left( {} \right)_{,x} + n_{y} A_{44} \left( {} \right)_{,y} } \hfill & {P_{54} = n_{x} A_{55} \left( {} \right)_{,x} \quad P_{55} = n_{y} A_{44} \left( {} \right)_{,y} } \hfill \\ \end{array} $$

(43)

Appendix 3

$$ \begin{aligned} R_{11} & = \left( {A_{11} + A_{66} F^{2} } \right)E^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + A_{66} \left( {2FE\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + \frac{{\partial^{2} }}{{\partial \eta^{2} }} + 2GFE^{2} \frac{\partial }{\partial \zeta }} \right) + \varOmega^{2} I_{0} L^{2} \\\\ R_{12} & = \left( {A_{12} + A_{66} } \right)\left( {FE^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + E\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + GE^{2} \frac{\partial }{\partial \zeta }} \right)\quad R_{13} = 0 \\\\ R_{14} & = \left( {B_{11} + B_{66} F^{2} } \right)E^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + B_{66} \left( {2FE\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + \frac{{\partial^{2} }}{{\partial \eta^{2} }} + 2GFE^{2} \frac{\partial }{\partial \zeta }} \right) + \varOmega^{2} I_{1} L^{2} \\\\ R_{15} & = \left( {B_{12} + B_{66} } \right)\left( {FE^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + E\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + GE^{2} \frac{\partial }{\partial \zeta }} \right) \\\\ R_{22} & = \left( {A_{66} + A_{22} F^{2} } \right)E^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + A_{22} \left( {2FE\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + \frac{{\partial^{2} }}{{\partial \eta^{2} }} + 2GFE^{2} \frac{\partial }{\partial \zeta }} \right) + \varOmega^{2} I_{0} L^{2} \\\\ R_{23} & = 0\quad R_{24} = \left( {B_{12} + B_{66} } \right)\left( {FE^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + E\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + GE^{2} \frac{\partial }{\partial \zeta }} \right) \\\\ R_{25} & = \left( {B_{66} + B_{22} F^{2} } \right)E^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + B_{22} \left( {2FE\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + \frac{{\partial^{2} }}{{\partial \eta^{2} }} + 2GFE^{2} \frac{\partial }{\partial \zeta }} \right) + \varOmega^{2} I_{1} L^{2} \\\\ R_{33} & = - \left( {A_{55} + A_{44} F^{2} } \right)E^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} - A_{44} \left( {2FE\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + \frac{{\partial^{2} }}{{\partial \eta^{2} }} + 2GFE^{2} \frac{\partial }{\partial \zeta }} \right) - \varOmega^{2} I_{0} L^{2} \\\\ R_{34} & = - A_{55} LE\frac{\partial }{\partial \zeta }\quad R_{35} = - A_{44} L\left( {FE\frac{\partial }{\partial \zeta } + \frac{\partial }{\partial \eta }} \right) \\\\ R_{44} & = \left( {D_{11} + D_{66} F^{2} } \right)E^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + D_{66} \left( {2FE\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + \frac{{\partial^{2} }}{{\partial \eta^{2} }} + 2GFE^{2} \frac{\partial }{\partial \zeta }} \right) - A_{55} L^{2} + \varOmega^{2} I_{2} L^{2} \\\\ R_{45} & = \left( {D_{12} + D_{66} } \right)\left( {FE^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + E\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + GE^{2} \frac{\partial }{\partial \zeta }} \right) \\\\ R_{55} & = \left( {D_{66} + D_{22} F^{2} } \right)E^{2} \frac{{\partial^{2} }}{{\partial \zeta^{2} }} + D_{22} \left( {2FE\frac{{\partial^{2} }}{\partial \zeta \partial \eta } + \frac{{\partial^{2} }}{{\partial \eta^{2} }} + 2GFE^{2} \frac{\partial }{\partial \zeta }} \right) - A_{44} L^{2} + \varOmega^{2} I_{2} L^{2} \\\\ \end{aligned} $$

(44)

Appendix 4

$$ \begin{aligned} J_{11} & = \left( {n_{x}^{2} A_{11} + n_{y}^{2} A_{12} + 2n_{x} n_{y} A_{66} F} \right)E\left( {} \right)_{,\zeta } + 2n_{x} n_{y} A_{66} \left( {} \right)_{,\eta } \\ J_{12} & = \left[ {\left( {n_{x}^{2} A_{12} + n_{y}^{2} A_{22} } \right)F + 2n_{x} n_{y} A_{66} } \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} A_{12} + n_{y}^{2} A_{22} } \right)\left( {} \right)_{,\eta } \\ J_{13} & = 0 \\ J_{14} & = \left( {n_{x}^{2} B_{11} + n_{y}^{2} B_{12} + 2n_{x} n_{y} B_{66} F} \right)E\left( {} \right)_{,\zeta } + 2n_{x} n_{y} B_{66} \left( {} \right)_{,\eta } \\ J_{15} & = \left[ {\left( {n_{x}^{2} B_{12} + n_{y}^{2} B_{22} } \right)F + 2n_{x} n_{y} B_{66} } \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} B_{12} + n_{y}^{2} B_{22} } \right)\left( {} \right)_{,\eta } \\ J_{21} & = \left[ {n_{x} n_{y} \left( {A_{12} - A_{11} } \right) + \left( {n_{x}^{2} - n_{y}^{2} } \right)A_{66} F} \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} - n_{y}^{2} } \right)A_{66} \left( {} \right)_{,\eta } \\ J_{22} & = \left[ {n_{x} n_{y} \left( {A_{22} - A_{12} } \right)F + \left( {n_{x}^{2} - n_{y}^{2} } \right)A_{66} } \right]E\left( {} \right)_{,\zeta } + n_{x} n_{y} \left( {A_{22} - A_{12} } \right)\left( {} \right)_{,\eta } \\ J_{23} & = 0 \\ J_{24} & = \left[ {n_{x} n_{y} \left( {B_{12} - B_{11} } \right) + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} F} \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} \left( {} \right)_{,\eta } \\ J_{25} & = \left[ {n_{x} n_{y} \left( {B_{22} - B_{12} } \right)F + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} } \right]E\left( {} \right)_{,\zeta } + n_{x} n_{y} \left( {B_{22} - B_{12} } \right)\left( {} \right)_{,\eta } \\ J_{31} & = \left( {n_{x}^{2} B_{11} + n_{y}^{2} B_{12} + 2n_{x} n_{y} B_{66} F} \right)E\left( {} \right)_{,\zeta } + 2n_{x} n_{y} B_{66} \left( {} \right)_{,\eta } \\ J_{32} & = \left[ {\left( {n_{x}^{2} B_{12} + n_{y}^{2} B_{22} } \right)F + 2n_{x} n_{y} B_{66} } \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} B_{12} + n_{y}^{2} B_{22} } \right)\left( {} \right)_{,\eta } \\ J_{33} & = 0 \\ J_{34} & = \left( {n_{x}^{2} D_{11} + n_{y}^{2} D_{12} + 2n_{x} n_{y} D_{66} F} \right)E\left( {} \right)_{,\zeta } + 2n_{x} n_{y} D_{66} \left( {} \right)_{,\eta } \\ J_{35} & = \left[ {\left( {n_{x}^{2} D_{12} + n_{y}^{2} D_{22} } \right)F + 2n_{x} n_{y} D_{66} } \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} D_{12} + n_{y}^{2} D_{22} } \right)\left( {} \right)_{,\eta } \\ J_{41} & = \left[ {n_{x} n_{y} \left( {B_{12} - B_{11} } \right) + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} F} \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} \left( {} \right)_{,\eta } \\ J_{42} & = \left[ {n_{x} n_{y} \left( {B_{22} - B_{12} } \right)F + \left( {n_{x}^{2} - n_{y}^{2} } \right)B_{66} } \right]E\left( {} \right)_{,\zeta } + n_{x} n_{y} \left( {B_{22} - B_{12} } \right)\left( {} \right)_{,\eta } \\ J_{43} & = 0 \\ J_{44} & = \left[ {n_{x} n_{y} \left( {D_{12} - D_{11} } \right) + \left( {n_{x}^{2} - n_{y}^{2} } \right)D_{66} F} \right]E\left( {} \right)_{,\zeta } + \left( {n_{x}^{2} - n_{y}^{2} } \right)D_{66} \left( {} \right)_{,\eta } \\ J_{45} & = \left[ {n_{x} n_{y} \left( {D_{22} - D_{12} } \right)F + \left( {n_{x}^{2} - n_{y}^{2} } \right)D_{66} } \right]E\left( {} \right)_{,\zeta } + n_{x} n_{y} \left( {D_{22} - D_{12} } \right)\left( {} \right)_{,\eta } \\ J_{51} & = 0\quad J_{52} = 0\quad J_{53} = \left( {n_{x} A_{55} + n_{y} A_{44} F} \right)E\left( {} \right)_{,\zeta } + n_{y} A_{44} \left( {} \right)_{,\eta } \quad J_{54} = n_{x} A_{55} L\quad J_{55} = n_{y} A_{44} L \\ \left( {} \right)_{,\zeta } & = \frac{\partial }{\partial \zeta }\left( {} \right)_{,\eta } = \frac{\partial }{\partial \eta }C \\ \end{aligned} $$

(45)