This paper contains the results of an approximate analysis of the axially symmetric vibrations of deep spherical elastic shells. The approximation is based on the asymptotic formulas for Legendre functions of large degree. The results are relatively simple approximate formulas for the natural frequencies and mode shapes under a variety of boundary conditions at the shell edge. The results agree very well with values obtained by Kalnins, using numerical methods.
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Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells
E. W. Ross, Jr.
U. S. Army Materials Research Agency, Watertown Arsenal, Watertown, Mass.
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Ross, E. W., Jr. (September 1, 1965). "Natural Frequencies and Mode Shapes for Axisymmetric Vibration of Deep Spherical Shells." ASME. J. Appl. Mech. September 1965; 32(3): 553–561. https://doi.org/10.1115/1.3627258
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