Vibroacoutic Behavior of Finite Composite Sandwich Plates with Foam Core

Abstract

In this study, based on a modal superposition method and Biot’s theory, an analytical model on vibroacoustic behavior of clamped and simply supported orthotropic rectangular composite sandwich plates with foam core has been derived. Theoretical predictions of sound transmission loss (STL) across finite composite sandwich plates with poroelastic material agree well with the experimental results in most frequency ranges of interest. Basing on the numerical results obtained, the influence of different parameters of the two thin laminated composite sheets and polyurethane foam core layer on STL of a sandwich plate is quantitatively evaluated and discussed in detail.

Appendix A. Expressions of the Matrix Equation (58)

The expressions of the M² x M² submatrices $\Delta _{1}^{{*i}} \left( {i = 1, 2, 3} \right)$, $\Delta _{2}^{{*i}} \left( {i = 1, 2, 3, 4} \right)$ and $\Delta _{1}^{{**i}} \left( {i = 1, 2, 3} \right)$, as appearing in the matrix elements T_ij,mn (i, j = 1, 2; m, n = 1, 2, …, M), are as flows:

$$ \begin{gathered} \Delta _{1}^{{*1}} = \left[ {\begin{array}{*{20}c} {C_{1} } & {} & {} & {} \\ {} & {C_{2} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {C_{M} } \\ \end{array} } \right]_{{M^{2} xM^{2} }} ;{\text{ }}C_{i} = \left[ {\begin{array}{*{20}c} {\lambda _{{1,1i}}^{{*1}} } & {} & {} & {} \\ {} & {\lambda _{{1,2i}}^{{*1}} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {\lambda _{{1,Mi}}^{{*1}} } \\ \end{array} } \right]_{{MxM}} {\text{ (i = 1,2,}}...{\text{,M); }} \hfill \\ \lambda _{{1,mn}}^{{*1}} = 3D_{{11}} \left( {\frac{m}{a}} \right)^{4} + 3D_{{22}} \left( {\frac{n}{b}} \right)^{4} + 4\left( {D_{{12}} + 2D_{{66}} } \right)\left( {\frac{m}{a}} \right)^{2} \left( {\frac{n}{b}} \right)^{2} \hfill \\ \end{gathered} $$

(A.1)

$$ \Delta _{1}^{{*2}} = \left[ {\begin{array}{*{20}c} {\lambda _{{1,1}}^{{*2}} } & {} & {} & {} \\ {} & {\lambda _{{1,2}}^{{*2}} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {\lambda _{{1,M}}^{{*2}} } \\ \end{array} } \right]_{{M^{2} xM^{2} }} {\text{ ; }}\lambda _{{1,n}}^{{*2}} = 2D_{{22}} \left( {\frac{n}{b}} \right)^{4} \left[ {\begin{array}{*{20}c} 0 & 1 & \cdots & \cdots & 1 \\ 1 & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & 1 \\ 1 & \cdots & \cdots & 1 & 0 \\ \end{array} } \right]_{{MxM}} $$

(A.2)

$$ \Delta _{1}^{{*3}} = \left[ {\begin{array}{*{20}c} 0 & {\lambda _{1}^{{*3}} } & \cdots & \cdots & {\lambda _{1}^{{*3}} } \\ {\lambda _{1}^{{*3}} } & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & {\lambda _{1}^{{*3}} } \\ {\lambda _{1}^{{*3}} } & \cdots & \cdots & {\lambda _{1}^{{*3}} } & 0 \\ \end{array} } \right]_{{M^{2} xM^{2} }} ;{\text{ }}\lambda _{1}^{{*3}} = 2D_{{11}} \left( {\frac{m}{a}} \right)^{4} \left[ {\begin{array}{*{20}c} {1^{4} } & {} & {} & {} \\ {} & {2^{4} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {M^{4} } \\ \end{array} } \right]_{{MxM}} $$

(A.3)

$$ \Delta _{2}^{{*1}} = \frac{{9ab}}{4}\left[ {\begin{array}{*{20}c} 1 & {} & {} & {} \\ {} & 1 & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & 1 \\ \end{array} } \right]_{{M^{2} xM^{2} }} ;{\text{ }} $$

(A.4)

$$ \Delta _{2}^{{*2}} = \frac{{9ab}}{4}\left[ {\begin{array}{*{20}c} {\lambda _{2}^{{*2}} } & {} & {} & {} \\ {} & {\lambda _{2}^{{*2}} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {\lambda _{2}^{{*2}} } \\ \end{array} } \right]_{{M^{2} xM^{2} }} ,\lambda _{2}^{{*2}} = \frac{{3ab}}{2}\left[ {\begin{array}{*{20}c} 0 & 1 & \cdots & \cdots & 1 \\ 1 & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & 1 \\ 1 & \cdots & \cdots & 1 & 0 \\ \end{array} } \right]_{{MxM}} $$

(A.5)

$$ \Delta _{2}^{{*3}} = \left[ {\begin{array}{*{20}c} 0 & {\lambda _{2}^{{*3}} } & \cdots & \cdots & {\lambda _{2}^{{*3}} } \\ {\lambda _{2}^{{*3}} } & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & {\lambda _{2}^{{*3}} } \\ {\lambda _{2}^{{*3}} } & \cdots & \cdots & {\lambda _{2}^{{*3}} } & 0 \\ \end{array} } \right]_{{M^{2} xM^{2} }} ;{\text{ }}\lambda _{2}^{{*3}} = \frac{{3ab}}{2}\left[ {\begin{array}{*{20}c} 1 & {} & {} & {} \\ {} & 1 & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & 1 \\ \end{array} } \right]_{{MxM}} $$

(A.6)

$$ \Delta _{2}^{{*4}} = \left[ {\begin{array}{*{20}c} 0 & {\lambda _{2}^{{*4}} } & \cdots & \cdots & {\lambda _{2}^{{*4}} } \\ {\lambda _{2}^{{*4}} } & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & {\lambda _{2}^{{*4}} } \\ {\lambda _{2}^{{*4}} } & \cdots & \cdots & {\lambda _{2}^{{*4}} } & 0 \\ \end{array} } \right]_{{M^{2} xM^{2} }} ;{\text{ }}\lambda _{2}^{{*4}} = ab\left[ {\begin{array}{*{20}c} 0 & 1 & \cdots & \cdots & 1 \\ 1 & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & 1 \\ 1 & \cdots & \cdots & 1 & 0 \\ \end{array} } \right]_{{MxM}} $$

(A.7)

$$ \begin{gathered} \Delta _{1}^{{**1}} = \left[ {\begin{array}{*{20}c} {C_{1}^{*} } & {} & {} & {} \\ {} & {C_{2}^{*} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {C_{M}^{*} } \\ \end{array} } \right]_{{M^{2} xM^{2} }} ;{\text{ }}C_{i}^{*} = \left[ {\begin{array}{*{20}c} {\lambda _{{1,1i}}^{{**1}} } & {} & {} & {} \\ {} & {\lambda _{{1,2i}}^{{**1}} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {\lambda _{{1,Mi}}^{{**1}} } \\ \end{array} } \right]_{{MxM}} {\text{ (i = 1,2,}}...{\text{,M); }} \hfill \\ \lambda _{{1,mn}}^{{**1}} = 3D_{{11}}^{*} \left( {\frac{m}{a}} \right)^{4} + 3D_{{22}}^{*} \left( {\frac{n}{b}} \right)^{4} + 4\left( {D_{{12}}^{*} + 2D_{{66}}^{*} } \right)\left( {\frac{m}{a}} \right)^{2} \left( {\frac{n}{b}} \right)^{2} \hfill \\ \end{gathered} $$

(A.8)

$$ \Delta _{1}^{{**2}} = \left[ {\begin{array}{*{20}c} {\lambda _{{1,1}}^{{**2}} } & {} & {} & {} \\ {} & {\lambda _{{1,2}}^{{**2}} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {\lambda _{{1,M}}^{{**2}} } \\ \end{array} } \right]_{{M^{2} xM^{2} }} {\text{ ; }}\lambda _{{1,n}}^{{**2}} = 2D_{{22}}^{*} \left( {\frac{n}{b}} \right)^{4} \left[ {\begin{array}{*{20}c} 0 & 1 & \cdots & \cdots & 1 \\ 1 & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & 1 \\ 1 & \cdots & \cdots & 1 & 0 \\ \end{array} } \right]_{{MxM}} $$

(A.9)

$$ \Delta _{1}^{{**3}} = \left[ {\begin{array}{*{20}c} 0 & {\lambda _{1}^{{*3}} } & \cdots & \cdots & {\lambda _{1}^{{*3}} } \\ {\lambda _{1}^{{*3}} } & 0 & \ddots & {} & \vdots \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & {} & \ddots & 0 & {\lambda _{1}^{{*3}} } \\ {\lambda _{1}^{{*3}} } & \cdots & \cdots & {\lambda _{1}^{{*3}} } & 0 \\ \end{array} } \right]_{{M^{2} xM^{2} }} ;{\text{ }}\lambda _{1}^{{**3}} = 2D_{{11}} \left( {\frac{m}{a}} \right)^{4} \left[ {\begin{array}{*{20}c} {1^{4} } & {} & {} & {} \\ {} & {2^{4} } & {} & {} \\ {} & {} & \ddots & {} \\ {} & {} & {} & {M^{4} } \\ \end{array} } \right]_{{MxM}} $$

(A.10)

The expression of the constants f_mn (k_x, k_y) is in the form:

$$ \begin{array}{*{20}l} \begin{gathered} f_{{mn}} \left( {k_{x} ,k_{y} } \right) = \int\limits_{0}^{b} {\int\limits_{0}^{a} {\varphi _{{mn}} \left( {x,y} \right)e^{{ - j\left( {k_{x} x + k_{y} y} \right)}} dxdy} } \hfill \\ = \int\limits_{0}^{b} {\int\limits_{0}^{a} {\left( {1 - \cos \frac{{2m\pi x}}{a}} \right)\left( {1 - \cos \frac{{2n\pi y}}{b}} \right)e^{{ - j\left( {k_{x} x + k_{y} y} \right)}} dxdy} } \hfill \\ \end{gathered} \hfill \\ {{\text{ = }}\left\{ {\begin{array}{*{20}c} {ab{\text{ for k}}_{x} = 0,{\text{ }}k_{y} = 0} \\ {\frac{{4in^{2} \pi ^{2} a\left( {1 - e^{{ - jbk_{y} }} } \right)}}{{k_{y} \left( {k_{y}^{2} b^{2} - 4n^{2} \pi ^{2} } \right)}}{\text{ for k}}_{x} = 0,{\text{ }}k_{y} \ne 0{\text{ }}} \\ {\frac{{4im^{2} \pi ^{2} b\left( {1 - e^{{ - jak_{x} }} } \right)}}{{k_{x} \left( {k_{x}^{2} a^{2} - 4m^{2} \pi ^{2} } \right)}}{\text{ for k}}_{x} \ne 0,{\text{ }}k_{y} = 0} \\ { - \frac{{16m^{2} n^{2} \pi ^{4} \left( {1 - e^{{ - jak_{x} }} } \right)\left( {1 - e^{{ - jbk_{y} }} } \right)}}{{k_{x} k_{y} \left( {k_{x}^{2} a^{2} - 4m^{2} \pi ^{2} } \right)\left( {k_{y}^{2} b^{2} - 4n^{2} \pi ^{2} } \right)}}{\text{ for k}}_{x} \ne 0,{\text{ }}k_{y} \ne 0\,{\text{ }}} \\ \end{array} } \right.} \hfill \\ \end{array} $$

$$ \begin{aligned} f_{{mn}} \left( {k_{x} ,k_{y} } \right) & = \int\limits_{0}^{b} {\int\limits_{0}^{a} {\varphi _{{mn}} \left( {x,y} \right)e^{{ - j\left( {k_{x} x + k_{y} y} \right)}} } } \\ & = \int\limits_{0}^{b} {\int\limits_{0}^{a} {\sin \left( {\frac{{m\pi x}}{a}} \right)\sin \left( {\frac{{n\pi y}}{b}} \right)e^{{ - j\left( {k_{x} x + k_{y} y} \right)}} dxdy} } \\ \end{aligned} $$

$$ { = \left\{ {\begin{array}{*{20}c} {\frac{{ab{\text{ }}\left\{ {{\text{ }}1 - ( - 1)^{m} - \left( { - 1} \right)^{n} + \left( { - 1} \right)^{{m + n}} } \right\}}}{{mn\pi ^{2} }}{\text{ }}if{\text{ }}k_{x} = 0{\text{ }}and{\text{ }}k_{y} = 0} \\ {\frac{{mab\left\{ {{\text{ }}1 - ( - 1)^{m} e^{{ - jak_{x} }} - \left( { - 1} \right)^{n} + \left( { - 1} \right)^{{m + n}} e^{{ - jak_{x} }} } \right\}}}{{mn\pi ^{2} }}{\text{ }}if{\text{ }}k_{x} \ne 0{\text{ }}and{\text{ }}k_{y} = 0} \\ {\frac{{nab\left\{ {{\text{ }}1 - ( - 1)^{m} - \left( { - 1} \right)^{n} e^{{ - jbk_{y} }} + \left( { - 1} \right)^{{m + n}} e^{{ - jbk_{y} }} } \right\}}}{{mn\pi ^{2} }}{\text{ }}if{\text{ }}k_{x} = 0{\text{ }}and{\text{ }}k_{y} \ne 0} \\ {\frac{{mnab\pi ^{2} \left\{ {{\text{ }}1 - ( - 1)^{m} e^{{ - jak_{x} }} - \left( { - 1} \right)^{n} e^{{ - jbk_{y} }} + \left( { - 1} \right)^{{m + n}} e^{{ - j\left( {ak_{x} + bk_{y} } \right)}} } \right\}}}{{\left( {k_{x}^{2} a^{2} - m^{2} \pi ^{2} } \right)\left( {k_{y}^{2} b^{2} - n^{2} \pi ^{2} } \right)}}{\text{ }}if{\text{ }}k_{x} \ne 0{\text{ }}and{\text{ }}k_{y} \ne 0} \\ \end{array} } \right.}$$