Asymptotic behavior of resonant frequencies in anisotropic cylindrical geometries

This study presents a theoretical analysis of the asymptotic behavior of resonant frequencies in anisotropic elastic cylinders. We use the Rayleigh–Ritz method with a normalized basis of Zernike and Legendre polynomials to derive analytical relationships governing the eigenfrequency spectrum. A central proof shows that for a sample with defined geometry, the ratio of the eigenvalue to the eigenvalue number converges to a constant determined solely by the diagonal elements of the material’s stiffness tensor. This directly links the asymptotic vibrational spectrum to longitudinal and shear stiffness constants, independent of the cross-stiffness constants. To facilitate this analytical derivation, we also present recursive formulas for the partial derivatives of Zernike polynomials, expressing them as linear combinations of lower-order terms. Numerical verification compares anisotropic iron with a geometrically scaled test material possessing proportionally adjusted elastic constants, demonstrating that both converge to the same asymptotic limit.