Three-dimensional vibrations of thick, circular rings with isosceles trapezoidal and triangular cross-sections

A three-dimensional (3-D) method of analysis is presented for determining t he free vibration frequencies and mode shapes of thick, circular rings with isosceles trapezoidal and triangular cross-sections. Displacement componen ts u(s), u(z), and u(0) in the meridional, normal, and circumferential dire ctions, respectively, are taken to Be snusoidal in time, periodic in theta, and algebraic polynomials in the phi and z directions. Potential (strain) and kinetic energies of the circular ring are formulated and upper bound va lues of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased frequencies converge to the exact va lues. Novel numerical results are presented for the circular rings with iso sceles trapezoidal and equilateral triangular cross-sections having complet ely free boundaries. Convergence to four-digit exactitude is demonstrated f or the first five frequencies of the rings. The method is applicable to thi n rings, as well as thick and very thick ones.