۳D Vibration Analysis of Thick Annular Plates Using Chebyshev-Ritz Method and ۳D Elasticity Theory

Studying vibrations of plates with different shapes is significantly important for industry. It is necessary to characterize natural frequencies and mode shapes of these plates for a correct vibrational analysis. Vibrations of annular plates have been investigated in this paper. Ritz method is one energy-based which is used to solve many problems numerically. This method is adopted in conjunction with Chebyshev’s polynomials as test functions in the current problem. For this purpose, the values of kinetic and potential energies are first stated in terms of displacement for the annular plates. The displacements are defined as binary series of Chebyshev’s polynomials toward r and z directions,while boundary conditions are applied on these series. Lagrangian which is indeed the difference between potential and kinetic energies was extracted and then minimized with respect to unknown coefficients of the displacement. The system of linear equations was thus obtained which gives the values of natural frequencies by putting its determinant equal to zero. Convergence of the solution was examined after studying the solution method. In this regard, the values of natural frequencies were extracted and compared with each other for different numbers of binary series. This problem was solved using Mathematica software and an accuracy of ۱۶ decimal digits has been considered in all steps. At the end, the values of natural frequencies were presented for the annular plates with different angles and boundary conditions.

Keywords:

Thick annular plates , vibrations , ChebyshevRitz method , ۳D elasticity theory