Structural optimization of rotating tapered laminated thick composite plates with ply drop-offs

Abstract

In this study, structural optimization of rotating tapered thick laminated composite plates with ply drop-offs has been investigated numerically. The governing differential equations of motion of the tapered composite plate have been presented including the energy associated with the inertia force, coriolis force, displacement dependent centrifugal force and initial stress resultants due to steady state rotation. Four noded quadrilateral finite element has been formulated based on the first order shear deformation theory. Finite element analysis results are validated with experimental results for natural frequencies of the tapered plate with various configurations. Various cases of optimization problems are formulated with different objective functions in terms of maximization of natural frequencies and damping factors (individually and combined) and solved using genetic algorithm in order to obtain optimal ply sequence and ply orientation. It is shown that the optimization problem with maximization of fundamental modal damping factor without rotating condition yields the optimal layout as 90° for all the layers in the plate. It is also observed that maximization of the fundamental modal damping factor yields identical optimal orientation for uniform and all the configurations of a tapered composite plate.

Appendix

1.1 Strain displacement matrix

$$ \left[ {B(x,y)} \right] = \left[ \partial \right]\left\{ u \right\} $$

where,

$$ \begin{aligned} \left[ \partial \right] & = \left[ {\begin{array}{*{20}c} {\frac{\partial }{\partial x}} & 0 & 0 & 0 & 0 \\ 0 & {\frac{\partial }{\partial y}} & 0 & 0 & 0 \\ {\frac{\partial }{\partial y}} & {\frac{\partial }{\partial x}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{\partial }{\partial x}} & 0 \\ 0 & 0 & 0 & 0 & {\frac{\partial }{\partial y}} \\ 0 & 0 & 0 & {\frac{\partial }{\partial y}} & {\frac{\partial }{\partial x}} \\ 0 & 0 & {\frac{\partial }{\partial y}} & 0 & 1 \\ 0 & 0 & {\frac{\partial }{\partial x}} & 1 & 0 \\ \end{array} } \right] \\ \left\{ \chi \right\} & = \left[ {\bar{B}(x,y)} \right]\left\{ d \right\} \\ [\bar{B}_{i} (x,y)] & = \left[ {\begin{array}{*{20}c} {\frac{{\partial N_{1} }}{\partial x}} & 0 & 0 & 0 & 0 \\ 0 & {\frac{{\partial N_{1} }}{\partial y}} & 0 & 0 & 0 \\ {\frac{{\partial N_{1} }}{\partial y}} & {\frac{{\partial N_{1} }}{\partial x}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{{\partial N_{1} }}{\partial x}} & 0 \\ 0 & 0 & 0 & 0 & {\frac{{\partial N_{1} }}{\partial y}} \\ 0 & 0 & 0 & {\frac{{\partial N_{1} }}{\partial y}} & {\frac{{\partial N_{1} }}{\partial x}} \\ 0 & 0 & {\frac{{\partial N_{1} }}{\partial y}} & 0 & {N_{1} } \\ 0 & 0 & {\frac{{\partial N_{1} }}{\partial x}} & {N_{1} } & 0 \\ \end{array} \,{\kern 1pt} \;\ldots\begin{array}{*{20}c} {\frac{{\partial N_{4} }}{\partial x}} & 0 & 0 & 0 & 0 \\ 0 & {\frac{{\partial N_{4} }}{\partial y}} & 0 & 0 & 0 \\ {\frac{{\partial N_{4} }}{\partial y}} & {\frac{{\partial N_{4} }}{\partial x}} & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{{\partial N_{4} }}{\partial x}} & 0 \\ 0 & 0 & 0 & 0 & {\frac{{\partial N_{4} }}{\partial y}} \\ 0 & 0 & 0 & {\frac{{\partial N_{4} }}{\partial y}} & {\frac{{\partial N_{4} }}{\partial x}} \\ 0 & 0 & {\frac{{\partial N_{4} }}{\partial y}} & 0 & {N_{4} } \\ 0 & 0 & {\frac{{\partial N_{4} }}{\partial x}} & {N_{4} } & 0 \\ \end{array} } \right] \\ [F_{i} (x,y)] & = \left[ {\begin{array}{*{20}c} 0 & 0 & {\frac{{\partial N_{w01} }}{\partial x}} & 0 & 0 \\ 0 & 0 & {\frac{{\partial N_{w01} }}{\partial y}} & 0 & 0 \\ \end{array} \;\;\ldots\begin{array}{*{20}c} 0 & 0 & {\frac{{\partial N_{w04} }}{\partial x}} & 0 & 0 \\ 0 & 0 & {\frac{{\partial N_{w04} }}{\partial y}} & 0 & 0 \\ \end{array} } \right] \\ N_{x} & = I_{T} (x)\;\varOmega^{2} \;\left( { - r_{0} (x + a) - q\;Le\;(x + a) - \frac{{(x + a)^{2} }}{2} + r_{0} \;Le\;(n_{x} - q) + \frac{{Le^{2} \;(n_{x}^{2} - q^{2} )}}{2}} \right),\quad \quad q = 0,1,2,3,\ldots\ldots,\left( {n_{x} - 1} \right) \\ N_{y} & = \;\frac{{I_{T} (x)}}{2}\;\varOmega^{2} \cos^{2} \phi \;\left( {\frac{{b^{2} }}{4} + y^{2} } \right) \\ \end{aligned} $$

where, r ₀, Le, and n _x are the hub radius, length of the finite element, total number of elements in the longitudinal direction of the tapered laminated composite plate respectively.

1.2 Inertia matrix

$$ \begin{aligned} \left[ I \right] & = \left[ {\begin{array}{*{20}c} {I_{T} (x)} & 0 & 0 & {I_{C} (x)} & 0 \\ 0 & {I_{T} (x)} & 0 & 0 & {I_{C} (x)} \\ 0 & 0 & {I_{T} (x)} & 0 & 0 \\ {I_{C} (x)} & 0 & 0 & {I_{R} (x)} & 0 \\ 0 & {I_{C} (x)} & 0 & 0 & {I_{R} (x)} \\ \end{array} } \right] \\ \left[ {I_{1} } \right] & = \left[ {\begin{array}{*{20}c} 0 & { - 2I_{T} (x)\varOmega_{z} } & {2I_{T} (x)\varOmega_{y} } & 0 & 0 \\ {2I_{T} (x)\varOmega_{z} } & 0 & { - 2I_{T} (x)\varOmega_{x} } & 0 & 0 \\ { - 2I_{T} (x)\varOmega_{y} } & {2I_{T} (x)\varOmega_{x} } & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \left[ {I_{2} } \right] & = \left[ {\begin{array}{*{20}c} {I_{T} (x)\left( {\varOmega_{y}^{2} + \varOmega_{z}^{2} } \right)} & { - I_{T} (x)\varOmega_{x} \varOmega_{y} } & { - I_{T} (x)\varOmega_{x} \varOmega_{z} } & 0 & 0 \\ { - I_{T} (x)\varOmega_{x} \varOmega_{y} } & {I_{T} (x)\left( {\varOmega_{x}^{2} + \varOmega_{z}^{2} } \right)} & { - I_{T} (x)\varOmega_{z} \varOmega_{y} } & 0 & 0 \\ { - I_{T} (x)\varOmega_{x} \varOmega_{z} } & { - I_{T} (x)\varOmega_{z} \varOmega_{y} } & {I_{T} (x)\left( {\varOmega_{x}^{2} + \varOmega_{y}^{2} } \right)} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \end{array} } \right] \\ \end{aligned} $$