A non-uniform, axially loaded Euler-Bernoulli beam having complex ends

An operator-based formulation is used to show the completeness of the eigenfunctions of a non-uniform, axially-loaded, transversely-vibrating Euler-Bernoulli beam having eccentric masses and supported by offset linear springs. This result generalizes the classical expansion theorem for a beam having conventional end conditions. Furthermore, the effect of truncating a series approximation of the initial deflection is investigated for the first time. New asymptotic forms of the eigenvalues and eigenfunctions are determined which are themselves often sufficiently accurate for high-frequency calculations.