Vibration Analysis of Orthotropic Functionally Graded Composite Plates in Thermal Environment Using High-Order Shear Deformation Theory: Frequency Suppression by Tuning the In-Plane Forces

Abstract

The vibration absorption of rectangular functionally graded cross-ply thick laminates is achieved analytically. The high-order shear deformation theory containing von Kármán strains is employed to study the plate. The laminate structure is assumed to function in a thermal environment subjected to in-plane loadings. Mechanical properties are assumed to vary continuously through the thickness, while each layer of the plate represent an orthotropic composite lamina. The governing equations are derived using Hamilton’s principle and the Levy solution is utilized for the direct implementation of various boundary conditions, including simply support, free, and clamped on the edges of the structure. A particular solution is derived that takes into account the in-plane thermal stresses. Effect of the temperature variation, in-plane loadings, and boundary conditions on the natural frequency of the plate was quantified. Finally, the magnitude of the in-plane forces has been designed in such a way that it can suppress the vibration of the plate under each thermal and boundary condition.

Appendices

Appendix A

The elements of matrices $\varvec{\varGamma}$ and R given in Eq. (20)

$$ \begin{aligned} & \varGamma_{11} = A_{66} \frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} + A_{11} \frac{{\partial^{3} }}{{\partial x_{1}^{3} }} \\ & \varGamma_{12} = \left( {A_{12} + A_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{13} = \left( {B_{66} - \gamma E_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {B_{11} - \gamma E_{11} } \right)\frac{{\partial^{3} }}{{\partial x_{1}^{3} }} \\ & \varGamma_{14} = \left( {B_{12} + B_{66} - \gamma \left( {E_{12} + E_{66} } \right)} \right)\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{15} = - \gamma \left( {\left( {E_{12} + 2E_{66} } \right)\frac{{\partial^{4} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + E_{11} \frac{{\partial^{4} }}{{\partial x_{1}^{4} }}} \right) \\ & \varGamma_{21} = \left( {A_{12} + A_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} \\ & \varGamma_{22} = A_{22} \frac{{\partial^{3} }}{{\partial x_{2}^{3} }} + A_{66} \frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{23} = \left( {B_{12} + B_{66} - \gamma \left( {E_{12} + E_{66} } \right)} \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} \\ & \varGamma_{24} = \left( {B_{22} - \gamma E_{22} } \right)\frac{{\partial^{3} }}{{\partial x_{2}^{3} }} + \left( {B_{66} - \gamma E_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{25} = - \gamma \left( {E_{22} \frac{{\partial^{4} }}{{\partial x_{2}^{4} }} + \left( {E_{12} + 2E_{66} } \right)\frac{{\partial^{4} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }}} \right) \\ & \varGamma_{31} = \gamma \left( {\left( {E_{12} + 2E_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} + E_{11} \frac{{\partial^{3} }}{{\partial x_{1}^{3} }}} \right) \\ & \varGamma_{32} = \gamma \left( {E_{22} \frac{{\partial^{3} }}{{\partial x_{2}^{3} }} + \left( {E_{12} + 2E_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }}} \right) \\ & \varGamma_{33} = \left( {9\gamma^{2} F_{55} - 6\gamma D_{55} + A_{55} } \right)\frac{\partial }{{\partial x_{1} }} + (\gamma F_{12} + 2\gamma F_{66} - \gamma^{2} H_{12} - 2\gamma^{2} H_{66} )\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {\gamma F_{11} - \gamma^{2} H_{11} } \right)\frac{{\partial^{3} }}{{\partial x_{1}^{3} }} \\ & \varGamma_{34} = \left( {9\gamma^{2} F_{44} - 6\gamma D_{44} + A_{44} } \right)\frac{\partial }{{\partial x_{2} }} + \left( {\gamma F_{22} - \gamma^{2} H_{22} } \right)\frac{{\partial^{3} }}{{\partial x_{2}^{3} }} + (\gamma F_{12} + 2\gamma F_{66} - \gamma^{2} H_{12} - 2\gamma^{2} H_{66} )\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{35} = \left( {9\gamma^{2} F_{44} - 6\gamma D_{44} + A_{44} } \right)\frac{{\partial^{2} }}{{\partial x_{2}^{2} }} + \left( {9\gamma^{2} F_{55} - 6\gamma D_{55} + A_{55} } \right)\frac{{\partial^{2} }}{{\partial x_{1}^{2} }} - 2\gamma^{2} (H_{12} + 2H_{66} )\frac{{\partial^{4} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }}\\ & \qquad \qquad - \gamma^{2} \left( {H_{11} \frac{{\partial^{4} }}{{\partial x_{1}^{4} }} + H_{22} \frac{{\partial^{4} }}{{\partial x_{2}^{4} }}} \right) \\ & \varGamma_{41} = \left( {B_{66} - \gamma E_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} + (B_{11} - \gamma E_{11} )\frac{{\partial^{3} }}{{\partial x_{1}^{3} }} \\ & \varGamma_{42} = \left( {B_{12} + B_{66} - \gamma \left( {E_{12} + E_{66} } \right)} \right)\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{43} = \left( {6\gamma D_{55} - 9\gamma^{2} F_{55} - A_{55} } \right)\frac{\partial }{{\partial x_{1} }} + \left( {D_{66} - 2\gamma F_{66} + \gamma^{2} H_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} + \left( {D_{11} - 2\gamma F_{11} + \gamma^{2} H_{11} } \right)\frac{{\partial^{3} }}{{\partial x_{1}^{3} }} \\ & \varGamma_{44} = \left( {D_{12} + D_{66} - 2\gamma \left( {F_{12} + F_{66} } \right) + \gamma^{2} \left( {H_{12} + H_{66} } \right)} \right)\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{45} = \left( {6\gamma D_{55} - 9\gamma^{2} F_{55} - A_{55} } \right)\frac{{\partial^{2} }}{{\partial x_{1}^{2} }} + (\gamma^{2} \left( {H_{12} + 2H_{66} } \right) - \gamma (F_{12} + 2F_{66} ))\frac{{\partial^{4} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} + \left( {\gamma^{2} H_{11} - \gamma F_{11} } \right)\frac{{\partial^{4} }}{{\partial x_{1}^{4} }} \\ & \varGamma_{51} = \left( {B_{12} + B_{66} - \gamma \left( {E_{12} + E_{66} } \right)} \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} \\ & \varGamma_{52} = \left( {B_{22} - \gamma E_{22} } \right)\frac{{\partial^{3} }}{{\partial x_{2}^{3} }} + \left( {B_{66} - \gamma E_{66} } \right)\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{53} = \left( {D_{12} + D_{66} - 2\gamma \left( {F_{12} + F_{66} } \right) + \gamma^{2} \left( {H_{12} + H_{66} } \right)} \right)\frac{{\partial^{3} }}{{\partial x_{1} \partial x_{2}^{2} }} \\ & \varGamma_{54} = \left( {6\gamma D_{44} - A_{44} - 9\gamma^{2} F_{44} } \right)\frac{\partial }{{\partial x_{2} }} + \left( {D_{22} - 2\gamma F_{22} + \gamma^{2} H_{22} } \right)\frac{{\partial^{3} }}{{\partial x_{2}^{3} }} + (D_{66} - 2\gamma F_{66} + \gamma^{2} H_{66} )\frac{{\partial^{3} }}{{\partial x_{1}^{2} \partial x_{2} }} \\ & \varGamma_{55} = \left( {6\gamma D_{44} - 9\gamma^{2} F_{44} - A_{44} } \right)\frac{\partial^2}{{\partial x_{2}^2 }} + \left( {\gamma^2 H_{22} - \gamma F_{22} } \right)\frac{{\partial^{4} }}{{\partial x_{2}^{4} }} + (\gamma^2 (H_{12} +2 H_{66} )- \gamma(F_{12} + 2F_{66}))\frac{{\partial^{4} }}{{\partial x_{1}^{2} \partial x_{2}^{2} }} \\ & R_{11} = - I_{1} \frac{\partial }{{\partial x_{1} }}, R_{12} = 0, R_{13} = - \left( {I_{2} - \gamma I_{4} } \right)\frac{\partial }{{\partial x_{1} }}, R_{14} = 0, \\ & R_{15} = \gamma I_{4} \frac{{\partial^{2} }}{{\partial x_{1}^{2} }}, \\ & R_{21} = 0, R_{22} = - I_{1} \frac{\partial }{{\partial x_{2} }}, R_{23} = 0, R_{24} = - \left( {I_{2} - \gamma I_{4} } \right)\frac{\partial }{{\partial x_{2} }}, \\ & R_{25} = \gamma I_{4} \frac{{\partial^{2} }}{{\partial x_{2}^{2} }}, \\ & R_{31} = - \gamma I_{4} \frac{\partial }{{\partial x_{1} }}, R_{32} = - \gamma I_{4} \frac{\partial }{{\partial x_{2} }}, \\ & R_{33} = - \gamma \left( {I_{5} - \gamma I_{7} } \right)\frac{\partial }{{\partial x_{1} }}, \\ & R_{34} = - \gamma \left( {I_{5} - \gamma I_{7} } \right)\frac{\partial }{{\partial x_{2} }}, R_{35} = - I_{1} + \gamma^{2} I_{7} \left( {\frac{{\partial^{2} }}{{\partial x_{1}^{2} }} + \frac{{\partial^{2} }}{{\partial x_{2}^{2} }}} \right), \\ & R_{41} = - (I_{2} - \gamma I_{4} )\frac{\partial }{{\partial x_{1} }}, R_{42} = 0, \\ & R_{43} = - \left( {I_{3} - 2\gamma I_{5} + \gamma^{2} I_{7} } \right)\frac{\partial }{{\partial x_{1} }}, R_{44} = 0, \\ & R_{45} = \gamma (I_{5} - \gamma I_{7} )\frac{{\partial^{2} }}{{\partial x_{1}^{2} }} \\ & R_{51} = 0, R_{52} = - (I_{2} - \gamma I_{4} )\frac{\partial }{{\partial x_{2} }}, R_{53} = 0, \\ & R_{54} = - \left( {I_{3} - 2\gamma I_{5} + \gamma^{2} I_{7} } \right)\frac{\partial }{{\partial x_{2} }}, R_{55} = \gamma (I_{5} - \gamma I_{7} )\frac{{\partial^{2} }}{{\partial x_{2}^{2} }} \\ \end{aligned} $$

(A.1)

Appendix B

The coefficients $\alpha_{i} ,\beta_{j} ,\gamma_{i} , i = 1, \ldots .,6, j = 1, \ldots ,7$ in Eq. (34)

$$ \begin{aligned} & \alpha_{1} = I_{1} \omega^{2} , \alpha_{2} = A_{22} , \alpha_{3} = \left( {I_{2} - \gamma I_{4} } \right)\omega^{2} , \\ & \alpha_{4} = \left( {B_{22} - \gamma E_{22} } \right), \alpha_{5} = - \gamma I_{4} \omega^{2} , \alpha_{6} = - \gamma E_{22} \\ & \beta_{1} = \gamma I_{4} \omega^{2} , \beta_{2} = \gamma E_{22} , \\ & \beta_{3} = A_{44} + 9\gamma^{2} F_{44} - 6\gamma D_{44} + \gamma \left( {I_{5} - \gamma I_{7} } \right)\omega^{2} , \\ & \beta_{4} = \gamma F_{22} - \gamma^{2} H_{22} , \beta_{5} = I_{1} \omega^{2} , \\ & \beta_{6} = A_{44} + 9\gamma^{2} F_{44} - 6\gamma D_{44} - \gamma^{2} I_{7} \omega^{2} - \psi_{22}^{1} , \beta_{7} = - \gamma^{2} H_{22} \\ & \gamma_{1} = \left( {I_{2} - \gamma I_{4} } \right)\omega^{2} , \gamma_{2} = \left( {B_{22} - \gamma E_{22} } \right), \\ & \gamma_{3} = - A_{44} - 6\gamma D_{44} + 9\gamma^{2} F_{44} + \left( {I_{3} - 2\gamma I_{5} + \gamma^{2} I_{7} } \right)\omega^{2} , \\ & \gamma_{4} = D_{44} - 2\gamma F_{22} + \gamma^{2} H_{22} , \\ & \gamma_{5} = - A_{44} - 9\gamma^{2} F_{44} + 6\gamma D_{44} - \gamma \left( {I_{5} - \gamma I_{7} } \right)\omega^{2} , \\ & \gamma_{6} = - \gamma F_{22} + \gamma^{2} H_{22} \\ \end{aligned} $$

(B.1)

Appendix C

The coefficients $\delta_{4} , \delta_{3} ,\delta_{2} ,\delta_{1}$ and $\delta_{0}$ used in Eq. (35)

$$ \begin{aligned} & \delta_{4} = \frac{{\alpha_{7} \gamma_{10} - \gamma_{7} \alpha_{10} }}{{\alpha_{7} - \beta_{8} \gamma_{7} }}, \delta_{3} = \frac{{\alpha_{10} - \beta_{8} \gamma_{10} - \alpha_{9} \gamma_{7} + \beta_{11} \gamma_{7} + \alpha_{7} \gamma_{9} }}{{\alpha_{7} - \beta_{8} \gamma_{7} }} \\ & \delta_{2} = \frac{{\alpha_{9} - \beta_{11} - \alpha_{8} \gamma_{7} + \beta_{10} \gamma_{7} + \alpha_{7} \gamma_{8} - \beta_{8} \gamma_{9} }}{{\alpha_{7} - \beta_{8} \gamma_{7} }} \\ & \delta_{1} = \frac{{\alpha_{8} - \beta_{10} + \beta_{9} \gamma_{7} - \beta_{8} \gamma_{8} }}{{\alpha_{7} - \beta_{8} \gamma_{7} }}, \delta_{0} = \frac{{ - \beta_{9} }}{{\alpha_{7} - \beta_{8} \gamma_{7} }} \\ & \alpha_{7} = \frac{1}{d}\left( {\alpha_{4} \beta_{3} \gamma_{2} - \alpha_{3} \beta_{4} \gamma_{2} - \alpha_{4} \beta_{2} \gamma_{3} + \alpha_{2} \beta_{4} \gamma_{3} + \alpha_{3} \beta_{2} \gamma_{4} - \alpha_{2} \beta_{3} \gamma_{4} } \right) \\ & \alpha_{8} = \frac{{\beta_{5} \left( { - \alpha_{3} \gamma_{2} + \alpha_{2} \gamma_{3} } \right)}}{d } \\ & \alpha_{9} = \frac{ - 1}{d}\left( { - \alpha_{5} \beta_{3} \gamma_{2} + \alpha_{3} \beta_{6} \gamma_{2} + \alpha_{5} \beta_{2} \gamma_{3} - \alpha_{2} \beta_{6} \gamma_{3} - \alpha_{3} \beta_{2} \gamma_{5} + \alpha_{2} \beta_{3} \gamma_{5} } \right) \\ & \alpha_{10} = \frac{ - 1}{d}\left( { - \alpha_{6} \beta_{3} \gamma_{2} + \alpha_{3} \beta_{7} \gamma_{2} + \alpha_{6} \beta_{2} \gamma_{3} - \alpha_{2} \beta_{7} \gamma_{3} - \alpha_{3} \beta_{2} \gamma_{6} + \alpha_{2} \beta_{3} \gamma_{6} } \right) \\ & \beta_{8} = \frac{1}{d}\left( { - \alpha_{4} \beta_{3} \gamma_{1} + \alpha_{3} \beta_{4} \gamma_{1} + \alpha_{4} \beta_{1} \gamma_{3} - \alpha_{1} \beta_{4} \gamma_{3} - \alpha_{3} \beta_{1} \gamma_{4} + \alpha_{1} \beta_{3} \gamma_{4} } \right) \\ & \beta_{9} = \frac{{\beta_{5} \left( {\alpha_{3} \gamma_{1} - \alpha_{1} \gamma_{3} } \right)}}{d} \\ & \beta_{10} = \frac{1}{d}\left( { - \alpha_{5} \beta_{3} \gamma_{1} + \alpha_{3} \beta_{6} \gamma_{1} + \alpha_{5} \beta_{1} \gamma_{3} - \alpha_{1} \beta_{6} \gamma_{3} - \alpha_{3} \beta_{1} \gamma_{5} + \alpha_{1} \beta_{3} \gamma_{5} } \right) \\ & \beta_{11} = \frac{1}{d}\left( { - \alpha_{6} \beta_{3} \gamma_{1} + \alpha_{3} \beta_{7} \gamma_{1} + \alpha_{6} \beta_{1} \gamma_{3} - \alpha_{1} \beta_{7} \gamma_{3} - \alpha_{3} \beta_{1} \gamma_{6} + \alpha_{1} \beta_{3} \gamma_{6} } \right) \\ & \gamma_{7} = \frac{ - 1}{d}\left( { - \alpha_{4} \beta_{2} \gamma_{1} + \alpha_{2} \beta_{4} \gamma_{1} + \alpha_{4} \beta_{1} \gamma_{2} - \alpha_{1} \beta_{4} \gamma_{2} - \alpha_{2} \beta_{1} \gamma_{4} + \alpha_{1} \beta_{2} \gamma_{4} } \right) \\ & \gamma_{8} = \frac{{ - \beta_{5} \left( {\alpha_{2} \gamma_{1} - \alpha_{1} \gamma_{2} } \right)}}{d} \\ & \gamma_{9} = \frac{ - 1}{d}\left( { - \alpha_{5} \beta_{2} \gamma_{1} + \alpha_{2} \beta_{6} \gamma_{1} + \alpha_{5} \beta_{1} \gamma_{2} - \alpha_{1} \beta_{6} \gamma_{2} - \alpha_{2} \beta_{1} \gamma_{5} + \alpha_{1} \beta_{2} \gamma_{5} } \right) \\ & \gamma_{10} = \frac{ - 1}{d}\left( { - \alpha_{6} \beta_{2} \gamma_{1} + \alpha_{2} \beta_{7} \gamma_{1} + \alpha_{6} \beta_{1} \gamma_{2} - \alpha_{1} \beta_{7} \gamma_{2} - \alpha_{2} \beta_{1} \gamma_{6} + \alpha_{1} \beta_{2} \gamma_{6} } \right) \\ & d = - \alpha_{3} \beta_{2} \gamma_{1} + \alpha_{2} \beta_{3} \gamma_{1} + \alpha_{3} \beta_{1} \gamma_{2} - \alpha_{1} \beta_{3} \gamma_{2} - \alpha_{2} \beta_{1} \gamma_{3} + \alpha_{1} \beta_{2} \gamma_{3} \\ \end{aligned} $$

(C.1)

Appendix D

The coefficients in Eq. (36)

$$ \begin{aligned} & \bar{C}_{i} = C_{i} \left( {\frac{{\alpha_{8} }}{{\tau_{i} }} + \alpha_{9} \tau_{i} + \alpha_{10} \tau_{i}^{3} + \frac{{\alpha_{7} \left( {A_{1} + A_{2} \tau_{i}^{2} + A_{3} \tau_{i}^{4} + A_{4} \tau_{i}^{6} } \right)}}{{\tau_{i} }}} \right)e^{{\tau_{i} x_{2} }} \\ & \tilde{C}_{i} = C_{i} \left( {\frac{{A_{1} }}{{\tau_{i}^{3} }} + \frac{{A_{2} }}{{\tau_{i} }} + A_{3} \tau_{i} + A_{4} \tau_{i}^{3} } \right)e^{{\tau_{i} x_{2} }} ,\quad i = 1, \ldots ,8,\quad \tau_{2i - 1} = - \tau_{2i} \\ & A_{1} = - \frac{{\beta_{9} \gamma_{7} }}{{(\beta_{8} \gamma_{7} - \alpha_{7} )}}, A_{2} = \frac{{\alpha_{8} \gamma_{7} - \beta_{10} \gamma_{7} - \alpha_{7} \gamma_{8} }}{{(\beta_{8} \gamma_{7} - \alpha_{7} )}}, \\ & A_{3} = \frac{{\alpha_{9} \gamma_{7} - \beta_{11} \gamma_{7} - \alpha_{7} \gamma_{9} }}{{(\beta_{8} \gamma_{7} - \alpha_{7} )}}, A_{4} = \frac{{\alpha_{10} \gamma_{7} - \alpha_{7} \gamma_{10} }}{{(\beta_{8} \gamma_{7} - \alpha_{7} )}} \\ \end{aligned} $$

(D.1)