Vibration analysis of porous metal foam plates rested on viscoelastic substrate

Abstract

In this paper, the vibration problem of a rectangular plate rested on a viscoelastic substrate and consisting of porous metal foam is solved via an analytical method with respect to the influences of various porosity distributions. Three types of porosity distribution across the thickness are covered, namely uniform, symmetric and asymmetric. The strain–displacement relations of the plate are assumed to be derived on the basis of a refined higher-order shear deformation plate theory. Then, the achieved relations will be incorporated with the Hamilton’s principle in order to reach the Euler–Lagrange equations of the structure. Next, the well-known Galerkin’s method is utilized to calculate the natural frequencies of the system. The influences of both simply supported and clamped boundary conditions are included. In order to show the accuracy of the presented method, the results of present research are compared with those reported by former published papers. The reported results show that an increase in the porosity coefficient can dramatically decrease the frequency of the plate. Also, the stiffness of the system can be lesser decreased, while a symmetrically porous metal foam is used to manufacture the plate.

Appendix

The components of stiffness and mass matrices can be calculated by:

$$\begin{aligned} k_{{11}} & = A_{{11}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{{\partial x}}Y_{n} (y)\frac{{\partial ^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + A_{{66}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{{\partial x}}Y_{n} (y)\frac{{\partial X_{m} (x)}}{{\partial x}}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ k_{{12}} & = \left( {A_{{12}} + A_{{66}} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{{\partial x}}Y_{n} (y)\frac{{\partial X_{m} (x)}}{{\partial x}}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ k_{{13}} & = - B_{{11}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{{\partial x}}Y_{n} (y)\frac{{\partial ^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x} {\text{d}}y} - \left( {B_{{12}} + 2B_{{66}} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{{\partial x}}Y_{n} (y)\frac{{\partial X_{m} (x)}}{{\partial x}}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ k_{{14}} & = - B_{{11}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{{\partial x}}Y_{n} (y)\frac{{\partial ^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x} {\text{d}}y} - \left( {B_{{12}}^{s} + 2B_{{66}}^{s} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{{\partial x}}Y_{n} (y)\frac{{\partial X_{m} (x)}}{{\partial x}}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ k_{{21}} & = \left( {A_{{12}} + A_{{66}} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{{\partial y}}\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{{\partial y}}{\text{d}}x} {\text{d}}y} , \\ k_{{22}} & = A_{{66}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{{\partial y}}\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{{\partial y}}{\text{d}}x} {\text{d}}y} + A_{{22}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{{\partial y}}X_{m} (x)\frac{{\partial ^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x} {\text{d}}y} , \\ k_{{23}} & = - \left( {B_{{12}} + 2B_{{66}} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{{\partial y}}\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{{\partial y}}{\text{d}}x} {\text{d}}y} - B_{{22}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{{\partial y}}X_{m} (x)\frac{{\partial ^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x} {\text{d}}y} , \\ k_{{24}} & = - \left( {B_{{12}}^{s} + 2B_{{66}}^{s} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{{\partial y}}\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{{\partial y}}{\text{d}}x} {\text{d}}y} - B_{{22}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{{\partial y}}X_{m} (x)\frac{{\partial ^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x} {\text{d}}y} , \\ k_{{31}} & = B_{{11}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \left( {B_{{12}} + 2B_{{66}} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ k_{{32}} & = \left( {B_{{12}} + 2B_{{66}} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} + B_{{22}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x} {\text{d}}y} , \\ k_{{33}} & = - D_{{11}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x} {\text{d}}y} - 2\left( {D_{{12}} + 2D_{{66}} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} \\ & \quad - \,D_{{22}} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x} {\text{d}}y} - k_{w} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + \,k_{p} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right), \\ k_{{34}} & = - D_{{11}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x} {\text{d}}y} - 2\left( {D_{{12}}^{s} + 2D_{{66}}^{s} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} \\ & \quad - \,D_{{22}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x} {\text{d}}y} - k_{w} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + \,k_{p} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right), \\ k_{{41}} & = B_{{11}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \left( {B_{{12}}^{s} + 2B_{{66}}^{s} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y)X_{m} (x)\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ k_{{42}} & = \left( {B_{{12}}^{s} + 2B_{{66}}^{s} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} + B_{{22}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x} {\text{d}}y} , \\ k_{{43}} & = - D_{{11}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x} {\text{d}}y} - 2\left( {D_{{12}}^{s} + 2D_{{66}}^{s} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} \\ & \quad - \,D_{{22}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x} {\text{d}}y} - k_{w} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + \,k_{p} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right), \\ k_{{44}} & = - H_{{11}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x} {\text{d}}y} - 2\left( {H_{{12}}^{s} + 2H_{{66}}^{s} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} \\ & \quad - \,H_{{22}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x} {\text{d}}y} + A_{{44}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + \,A_{{44}}^{s} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} - k_{w} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + \,k_{p} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial ^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial ^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right) \\ \end{aligned}$$

(40)

and

$$\begin{aligned} m_{11} & = - I_{0} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y){\text{d}}x} {\text{d}}y} , \\ m_{13} & = I_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y){\text{d}}x} {\text{d}}y} , \\ m_{14} & = J_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y){\text{d}}x} {\text{d}}y} , \\ m_{22} & = - I_{0} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x} {\text{d}}y} , \\ m_{23} & = I_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x} {\text{d}}y} , \\ m_{24} & = J_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}dx} dy} , \\ m_{31} & = - I_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} , \\ m_{32} & = - I_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ m_{33} & = - I_{0} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + I_{2} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right), \\ m_{34} & = - I_{0} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + J_{2} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right), \\ m_{41} & = - J_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} , \\ m_{42} & = - J_{1} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} , \\ m_{43} & = - I_{0} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + \,J_{2} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right), \\ m_{44} & = - I_{0} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x} {\text{d}}y} \\ & \quad + \,K_{2} \left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x} {\text{d}}y} + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x} {\text{d}}y} } \right) \\ \end{aligned}$$

(41)

and

$$C_{33} = C_{34} = C_{43} = C_{44} = - ic_{d} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y)dx} dy}$$

(42)