Vibration analysis of free-fixed hyperbolic cooling tower shells

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies of hyperboloidal shells free at the top edge and clamped at the bottom edge like a hyperboloidal cooling tower by the Ritz method based upon the circular cylindrical coordinate system instead of related 3- D shell coordinates which are normal and tangent to the shell midsurface. The Legendre polynomials are used as admissible displacements. Convergence to four-digit exactitude is demonstrated. Natural frequencies from the present 3-D analysis are also compared with those of straight beams with circular cross section, complete (not truncated) conical shells, and circular cylindrical shells as special cases of hyperboloidal shells from the classical beam theory, 2-D thin shell theory, and other 3-D methods.