Energy Method for Determining Natural Frequencies and Associated Mode Shapes of Truncated Conical and Parabolic Shells

Abstract

An energy method is presented for calculating natural frequencies of thin, truncated conical shells or parabolic shells of revolution. Extensional and inextensional vibrations are treated separately, and the corresponding (approximate) mode shapes determined. For the inextensional case, the vanishing of the middle surface strains yield solutions for the displacement field in the three principal directions. The displacement expressions are used in the appropriate energy integral, together with Lagrange's method, to obtain an algebraic equation for the oscillatory frequencies for each mode. For the extensional vibrations, the displacements are written as orthonormal series of functions, selected to match the boundary conditions imposed. The (unknown) series coefficients are obtained by minimization of the energy integral (Ritz procedure), and a set of algebraic equations for the frequencies is obtained. Convergence of the orthonormal series is discussed. Finally, the square root of the sum of squares of corresponding inextensional and extensional frequencies, regarded as a measure of the correct natural frequencies for the shell, is examined.