Three-dimensional vibrations of thick spherical shell segments with variable thickness. (English) Zbl 0999.74059

Kang, Jae-Hoon; Leissa, Arthur W.

Int. J. Solids Struct. 37, No. 35, 4811-4823 (2000).

Summary: We present a three-dimensional method for determining free vibration frequencies and mode shapes of spherical shell segments with variable thickness. Displacement components \(u_\varphi\), \(u_z\), and \(u_\theta\) in the meridional, normal, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in \(\theta\), and algebraic polynomials in \(\varphi\) and \(z\). Potential (strain) and kinetic energies of the spherical shell segment are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Numerical results are presented for thick spherical shell segments with constant or linearly varying thickness and completely free boundaries. Convergence to four-digit accuracy is demonstrated for the first five frequencies of spherical shell segments.

MSC:

74H45	Vibrations in dynamical problems in solid mechanics
74K25	Shells
74H15	Numerical approximation of solutions of dynamical problems in solid mechanics
74S30	Other numerical methods in solid mechanics (MSC2010)

Keywords:

Ritz method; convergence; free vibration frequencies; mode shapes; spherical shell segments; variable thickness