Random vibration response of composite–sandwich laminates

Abstract

This paper presents the random free vibration response of laminated composite and sandwich plates using inverse hyperbolic zigzag theory. Recently developed inverse hyperbolic zigzag plate model by authors satisfies the inter-laminar continuity of transverse shear stresses at layer interfaces. The uncertainties induced in the system parameters are accounted by employing first-order perturbation technique in conjunction with the deterministic finite element method. The second-order statistics of natural frequency of the laminated composite and sandwich structures is obtained considering the material properties of the structures to be random in nature. The random natural frequency results are obtained in terms of mean and standard deviations of the responses for different values of span-to-thickness ratios, geometric parameters, stacking sequence, and boundary conditions. The evaluated results are ensured by comparing with those existing in the literature and with independent Monte Carlo simulation results obtained in the framework of random free vibration analysis of laminated composite and sandwich plates using inverse hyperbolic zigzag theory.

Appendices

Appendix A.1

Coefficients of kinematic variables:

$r_{1} = s_{2}$ = 1,$r_{2} = r_{3} = r_{5} = r_{7} = s_{1} = s_{3} = s_{4} = s_{6}$ = 0,$r_{4} = s_{5} = - z$

$$r_{6} = \left[ {\left[ {g\left( z \right) + \Omega z} \right] + \sum\limits_{i = 1}^{{n_{u - 1} }} {\left( {z - z_{i}^{u} } \right)H\left( {z - z_{i}^{u} } \right)} R_{x}^{i} \left[ {g^{\prime}\left( {z_{i}^{u} } \right) + \Omega^{i} } \right] + \sum\limits_{j = 1}^{{n_{l - 1} }} {\left( {z - z_{j}^{l} } \right)H\left( { - z + z_{j}^{l} } \right)} R_{x}^{j} \left[ {g^{\prime}(z_{j}^{l} ) + \Omega^{j} } \right]} \right]$$

$$s_{7} = \left[ {\left[ {g\left( z \right) + \Omega z} \right] + \sum\limits_{i = 1}^{{n_{u - 1} }} {\left( {z - z_{i}^{u} } \right)H\left( {z - z_{i}^{u} } \right)} R_{y}^{i} \left[ {g^{\prime}\left( {z_{i}^{u} } \right) + \Omega^{i} } \right] + \sum\limits_{j = 1}^{{n_{l - 1} }} {\left( {z - z_{j}^{l} } \right)H\left( { - z + z_{j}^{l} } \right)} R_{y}^{j} \left[ {g^{\prime}(z_{j}^{l} ) + \Omega^{j} } \right]} \right]$$

$${\text{Where }}R_{x}^{i} = \frac{{Q_{55}^{i + 1} }}{{Q_{55}^{i} }} - 1{\text{ and }}R_{y}^{i} = \frac{{Q_{44}^{i + 1} }}{{Q_{44}^{i} }} - 1$$

Appendix A.2

Elements of matrix $\left[ H \right]$

$$\left[ H \right] = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & z & 0 & 0 & {p_{1} } & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & z & 0 & 0 & {p_{2} } & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & z & 0 & 0 & {p_{1} } & {p_{2} } & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & {q_{1} } & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & {q_{2} } \\ \end{array} } \right]$$

where

$$\begin{gathered} p_{1} = g(z) + \sum\limits_{i = 1}^{{n_{u - 1} }} {(\Omega - \Omega_{x}^{u} )(z - z_{i}^{u} )H(z - z_{i}^{u} )} + \sum\limits_{j = 1}^{{n_{l - 1} }} {(\Omega - \Omega_{x}^{l} )(z - z_{j}^{l} )H( - z + z_{j}^{l} )} \hfill \\ p_{2} = g(z) + \sum\limits_{i = 1}^{{n_{u - 1} }} {(\Omega - \Omega_{y}^{u} )(z - z_{i}^{u} )H(z - z_{i}^{u} )} + \sum\limits_{j = 1}^{{n_{l - 1} }} {(\Omega - \Omega_{y}^{l} )(z - z_{j}^{l} )H( - z + z_{j}^{l} )} \hfill \\ q_{1} = g^{\prime}(z) + \sum\limits_{i = 1}^{{n_{u - 1} }} {(\Omega - \Omega_{y}^{u} )H(z - z_{i}^{u} )} + \sum\limits_{j = 1}^{{n_{l - 1} }} {(\Omega - \Omega_{y}^{l} )H( - z + z_{j}^{l} )} \hfill \\ q_{2} = g^{\prime}(z) + \sum\limits_{i = 1}^{{n_{u - 1} }} {(\Omega - \Omega_{x}^{u} )H(z - z_{i}^{u} )} + \sum\limits_{j = 1}^{{n_{l - 1} }} {(\Omega - \Omega_{x}^{l} )H( - z + z_{j}^{l} )} \hfill \\ \end{gathered}$$

Appendix A.3

Linear strain terms

$\varepsilon_{1}^{0} = \frac{\partial u}{{\partial x}}$,$\varepsilon_{2}^{0} = \frac{\partial v}{{\partial y}}$,$\varepsilon_{6}^{0} = \left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} \right)$,$k_{1}^{1} = \left( { - \frac{{\partial \theta_{x} }}{\partial x} + \Omega \frac{{\partial \beta_{x} }}{\partial x}} \right)$,$k_{2}^{1} = \left( { - \frac{{\partial \theta_{y} }}{\partial y} + \Omega \frac{{\partial \beta_{y} }}{\partial y}} \right)$,$k_{6}^{1} = \left[ { - \left( {\frac{{\partial \theta_{x} }}{\partial y} + \frac{{\partial \theta_{y} }}{\partial x}} \right) + \Omega \left( {\frac{{\partial \beta_{x} }}{\partial y} + \frac{{\partial \beta_{y} }}{\partial x}} \right)} \right]$,$k_{1}^{2} = \frac{{\partial \beta_{x} }}{\partial x}$,$k_{2}^{3} = \frac{{\partial \beta_{y} }}{\partial y}$,$k_{6}^{4} = \frac{{\partial \beta_{x} }}{\partial y}$,$k_{6}^{5} = \frac{{\partial \beta_{y} }}{\partial x}$,$\varepsilon_{4}^{0} = \left( {\frac{\partial w}{{\partial y}} + \Omega \beta_{y} - \theta_{y} } \right)$,$\varepsilon_{5}^{0} = \left( {\frac{\partial w}{{\partial x}} + \Omega \beta_{x} - \theta_{x} } \right)$,$k_{4}^{6} = \beta_{y}$,$k_{5}^{7} = \beta_{x}$.

Appendix A.4

Elements of matrix $\left[ Q \right]$

$$Q = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & {M_{x} } & 0 & { - z} & 0 \\ 0 & 1 & 0 & 0 & {M_{y} } & 0 & { - z} \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ \end{array} } \right]$$

where $M_{x} = \bigg[ \left[ {g\left( z \right) + \Omega z} \right] + \sum\limits_{i = 1}^{{n_{u - 1} }} {\left( {z - z_{i}^{u} } \right)H\left( {z - z_{i}^{u} } \right)} R_{x}^{i} \left[ {g^{\prime}\left( {z_{i}^{u} } \right) + \Omega^{i} } \right] + \sum\limits_{j = 1}^{{n_{l - 1} }} {\left( {z - z_{j}^{l} } \right)H\left( { - z + z_{j}^{l} } \right)} R_{x}^{j} \left[ {g^{\prime}(z_{j}^{l} ) + \Omega^{j} } \right] \bigg]$

$$M_{y} = \left[ {\left[ {g\left( z \right) + \Omega z} \right] + \sum\limits_{i = 1}^{{n_{u - 1} }} {\left( {z - z_{i}^{u} } \right)H\left( {z - z_{i}^{u} } \right)} R_{y}^{i} \left[ {g^{\prime}\left( {z_{i}^{u} } \right) + \Omega^{i} } \right] + \sum\limits_{j = 1}^{{n_{l - 1} }} {\left( {z - z_{j}^{l} } \right)H\left( { - z + z_{j}^{l} } \right)} R_{y}^{j} \left[ {g^{\prime}(z_{j}^{l} ) + \Omega^{j} } \right]} \right]$$