Equations of Motion for Nonaxisymmetric Vibrations of Prolate Spheroidal Shells

摘要

This report documents the derivation of the equations for nonaxisymmetric motion of prolate spheroidal shells of constant thickness. The equations include the effect of distributed mechanical surface forces and moments. These equations were derived from the Lagrangian of the system. The thin shell theory used in this derivation includes three displacements and two changes of curvature. Thus, the effects of membrane, bending, shear deformations, and rotatory inertias are included in this theory. The resulting five coupled partial differential equations are self-adjoint and positive definite, ensuring real and positive eigenvalues and real eigenfunctions. The frequency-wavenumber spectrum has five branches (three acoustic and two optical) representing flexural, longitudinal, torsional, and thickness-shear modes. The acoustic branches have the correct group and phase velocities, especially in the high wavenumber, short wavelength limits.