Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a three-dimensional theory

• Published in: Computers & structures Vol. 83; no. 31; pp. 2594 - 2608
• Main Authors: Kang, Jae-Hoon, Leissa, Arthur W.
• Format: Journal Article
• Language: English
• Published: Oxford Elsevier Ltd 01.12.2005; Elsevier Science
• Subjects: Complete paraboloidal shells; Computational techniques; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical methods in physics; Physics; Ritz method; Shells of revolution; Solid mechanics; Solid paraboloids; Static elasticity (thermoelasticity...); Structural and continuum mechanics; Thick shell; Three-dimensional analysis; Variable thickness; Vibration; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Variable thickness; Vibration; Complete paraboloidal shells; Three-dimensional analysis; Shells of revolution; Ritz method; Thick shell; Solid paraboloids; Variable thickness shell; Modeling; Dimensional analysis; Tridimensional elasticity; Eigenvalue problem; Kinetic energy; Potential energy; Strain energy; Free vibration; Paraboloidal shell; Revolution shell; Wall thickness; Axial symmetry; Variable section; Upper bound; Vibrational mode; Structural analysis
• ISSN: 0045-7949; 1879-2243
• DOI: 10.1016/j.compstruc.2005.03.018

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloids and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components u r , u θ and u z in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.