Velocity corrected theory of laminated plates applied to free plate strip vibrations

Abstract

Simple thickness frequencies for the C.T. Sun “effective stiffness” theory for flexural and extensional vibrations of laminated plates are compared to exact simple thickness frequency results previously obtained from the “effective stiffness” continuum theory and are found to differ by a factor of 12 π 2 . By following Mindlin, Sun's plate theory is rederived and modified with three velocity correction factors introduced through the kinetic energy to make the plate theory consistent with the continuum theory for simple thickness modes. A comparison of simple thickness frequencies for both theories gives the correction factors as K i = √12 π , i = 1, 2, 3 . The velocity corrected flexural equations are reduced for the case of a plate strip and frequency equations are derived for simply supported edges by the method of M. Levy, where solutions harmonic in plate width are assumed and the boundary conditions for simple supports are automatically satisfied. For comparison purposes, the velocity corrected Sun “effective stiffness” plate equations of motion and boundary conditions are reduced to a corresponding “effective modulus” theory by a suitable elimination of microstructure terms. Also, the anisotropic Mindlin plate equations for flexure, with both shear and velocity correction factors included, are reduced to the transversely isotropic case and the Postma elastic constants are introduced; by setting either the shear or velocity correction factors equal to one, the velocity and shear corrected Mindlin plate theories are respectively obtained. The variation of dimensionless frequency versus such dimensionless quantities as width-to-thickness ratio, number of layer pairs, and elastic ratio for the “effective stiffness” theory and the three “effective modulus” theories leads to a number of significant trends, which are discussed in detail.