SYMMETRY EQUIVALENCE OF NON-UNIFORM BEAMS WITH INTERMEDIATE MASSES

Abstract

This paper investigates the spectral properties of non-uniform Euler--Bernoulli beams resting on a Winkler--type elastic foundation and subjected to an axial load, in the presence of two concentrated masses placed symmetrically with respect to the beam midpoint. The governing equation includes variable bending stiffness, foundation modulus, and distributed mass, while the point masses are modeled through classical continuity and jump conditions. Under an agreed symmetry of the variable coefficients, we prove that the spectral problem admits a symmetry-based decomposition: the eigenvalues and eigenfunctions of the full beam split into symmetric and antisymmetric families. This is achieved through two auxiliary problems on the half-interval, equipped with sliding-hinged and hinged-hinged (or sliding--clamped and hinged--clamped) midpoint conditions. The resulting factorization reduces the complexity of eigenvalue computation and extends known symmetry equivalence results to beams with intermediate masses. Numerical examples confirm the theoretical findings and illustrate the loss of symmetry when coefficient symmetry is violated. The proposed framework significantly reduces the computational complexity of spectral analysis for non-uniform beams with discrete masses and extends existing symmetry-based results to a broader class of structurally heterogeneous systems.