Random vibrations of three-dimensional functionally graded plates

Due to the practical importance of functionally graded materials (FGMs) in various branches of engineering, Koizumi (1997), an increasing amount of attention has been devoted to different problems associated with the mechanical behavior of plates made of these materials, see, among the others, Reddy (2004). However, up to now the aspect of vibrations of such plates due to random excitation has not been investigated. Recently, Ferrante and Graham-Brady (2004) take into account the effect of random variations in both the component volume fractions and porosity for FGM plates subjected to deterministic thermal boundary conditions. Analysis of random vibrations is essential for engineers in order to ensure that the design of FGM components is reliable and safe. In fact, since FGM plates are widely employed in aerospace structures, they are often subjected to dynamic excitations of random nature in complex environmental conditions. Based on the above remarks, in this work a preliminary study on the response characteristics of FGM plates to random loading in the three-dimensional setting is presented. Material properties, namely, modulus of elasticity and mass density, are assumed to follow a through the thickness variation according to a simple power law distribution in terms of the volume fraction of the constituents. The plate is assumed to be all round simply supported. A three-dimensional Ritz energy approach is employed in the free vibration analysis. Once the natural frequencies and mode shapes have been determined, the transient response is studied by the modal superposition method. It is assumed that the plate under study is subjected to a stationary random excitation with given mean and correlation function. Displacements and external forces are expressed in terms of series of approximate modes of vibration, which satisfy the boundary conditions, Elishakoff (1999). Analytical expressions for the distribution of the mean-square displacement and autocorrelation function of the response are found. Preliminary numerical results are presented in tabular and graphical format for the natural frequencies and dynamic response of aluminum/zirconia square plates. The approach seems to be suitable for the computationally intensive task of three-dimensional modeling due to ease of generation and numerical implementation. Three-dimensional solutions for functionally graded plates are useful because they can be used as benchmarks to assess the accuracy of various two-dimensional theories.