The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

In this paper, the effects of silicone surfactant on the mechanical, thermal, and acoustical properties of flexible polyurethane foams are studied. At first, by introducing the reactions of the foam production and expressing the stoichiometric relationships, a method is provided to select the amounts of the basic compounds for getting the desirable properties. By knowing the OH number of polyol and the NCO content of isocyanate after selecting an amount of a blowing agent for the reaction, the quantity of each compound to produce the desirable foam can be found. Then different foams with various amounts of surfactant are produced. By preparing proper samples, the mechanical, thermal, nonacoustical, and acoustical properties of the foams are measured or calculated by using different instruments. The nonacoustical properties are obtained based on two distinct methods, i.e., a semiphenomenological method and an indirect method. The obtained properties are validated by comparing the calculated theoretical sound absorption coefficients with the measured ones. The results show that with a small increase of surfactant, the acoustical properties of foam samples improve by about 50% without any significant changes in other mechanical and thermal properties of the foams. This improvement results from the variations of reticulation rate and cell size. In addition, the initial desirable porosity and density considered in the stoichiometric relations are fairly achieved in produced samples. These results validate the accuracy of the stoichiometric relations.

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This article introduces an exact analytical method for free vibration analysis of functionally graded (FG) thin annular sector plates resting on Winkler and Pasternak elastic foundations. The annular sector plate has simply supported radial edges and arbitrary boundary conditions along the circular edges. Based on the displacement field of Kirchhoff plate theory, the governing equations of motion are obtained considering the in-plane displacements and rotary inertia. Using a set of functions, the three coupled governing equations of motion are converted into two decoupled equations. By applying the boundary conditions at inner and outer radii, an eigenvalue problem for finding the natural frequencies is obtained. The nine distinct cases are considered involve all possible combinations of boundary conditions along the circular edges. Accurate non-dimensional frequency is presented for over a wide range of sector angles, some inner to outer radii (aspect ratio) and different powers of functionally graded material. Accurate natural frequencies of FG annular sector plates resting on elastic foundations are presented for the first time and can be used as reference values for numerical analyses.

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