Equivalent Finite Element Formulations for the Calculation of Eigenvalues Using Higher-Order Polynomials

This paper investigates higher-order approximations in order to extract Sturm-Liouville eigenvalues in one-dimensional vibration problems in continuum mechanics. Several alternative global approximations of polynomial form such as Lagrange, Bernstein, Legendre as well as Chebyshev of first and second kind are discussed. In an instructive way, closed form analytical formulas are derived for the stiffness and mass matrices up to the quartic degree. A rigorous proof for the transformation of the matrices, when the basis changes, is given. Also, a theoretical explanation is provided for the fact that all the aforementioned alternative pairs of matrices lead to identical eigenvalues. The theory is sustained by one numerical example under three types of boundary conditions.