Keywords: transversely isotropic plates, annular, Mindlin, Bessel's function.

As a result of the wide variety of engineering applications, the study of plate vibrations and in particular transversely isotropic plates have been of great interest to researchers. As structural elements, circular annular plates are extensively used in the construction of aircrafts, ships, automobiles and other vehicles. The turbine disc is a well-known example for industrial application of the annular plate. The analysis becomes more complex when the transversely isotropic material is considered and before finalizing the design of a structure first few frequencies of such structure should be known.

Very few results are found related to vibrations of transversely isotropic circular plates with transversely isotropic materials. Irie et al. [1] have tabulated natural frequencies of annular isotropic plates based on the Mindlin plate theory. Wang et. al. [2] have presented the mode shapes and stress resultants of isotropic circular plates with free edge condition. Mirsky [3] solved the three-dimensional equations of elastodynamics for transversely isotropic media in terms of three displacement potentials, each satisfying a partial differential equation of the second order. In the present work, analysis of axisymmetric vibrations of transversely isotropic annular plates has been carried out. The method used by [3] has been extended to compute vibration characteristics of transversely isotropic annular circular plates using Mindlin's Plate Theory with the shear deformation consideration. To the best of our knowledge no work related to transversely isotropic Mindlin annular plates giving exact vibration frequencies for the boundary conditions discussed in this paper exists.

The numerical solution is found assuming auxiliary variables using a Bessel function. The frequency determinant equations are derived in explicit form for various sets of boundary conditions. The results have been computed for three different isotropic materials possessing transverse isotropy. Numerical results for the frequency parameters of annular plates of various materials, different ratio of thickness of plate to radius of plate and different ratio of inner to outer radii have been obtained. As a special case, the results are also obtained for isotropic plates and compared with the known results. Few results that are available in case of transversely isotropic circular plates are also compared with the present and are found to be in good agreement. Here, the exact vibration solutions of annular circular transversely isotropic Mindlin plates have been presented. Numerical results have been obtained by considering outer edge clamped and all the possible boundary conditions (clamped, simply supported and free) have been considered for the inner edge.

Thus, exact vibration frequencies of annular transversely isotropic Mindlin plates have been presented here. Results are obtained for various boundary conditions and different materials possessing transverse isotropy. The obtained results may prove to be as benchmark for other investigations.

1

T. Irie, G. Yamada, K. Takagi, "Natural frequencies of thick annular plates", Journal of Applied Mechanics, 49, 633-638, 1982.

2

C.M. Wang, Y. Xiang, E. Watanabe, T. Utsunomiya, "Mode shapes and stress-resultants of circular Mindlin plates with free edges", Journal of Sound and Vibration, 276, 511-525, 2004. doi:10.1016/j.jsv.2003.08.010

3

I. Mirsky, "Wave Propagation in Transversely Isotropic Circular Cylinders Part I: Theory", Journal of Acoustical Society of America, 36, 41-51, 1964. doi:10.1121/1.1918910

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