Symmetry Breaking and Frequency Veering in Inclined Cables: Analytical Scaling Laws for Geometric, Elastic and Inertial Asymmetries

In the vibration analysis of cable structures, the transition from theoretically predicted frequency crossing to actual frequency veering represents a fundamental change in spectral topology. To reveal the underlying physical mechanism, this paper establishes a general operator-theoretic perturbation framework based on Hamilton’s principle. Topologically, classical eigenvalue crossings manifest as diabolic points, protected by the symmetry-induced commutativity between the structural stiffness and reflection operators. This study mathematically demonstrates that introducing conservative asymmetries namely geometric inclination, local stiffness damage, or lumped mass acts as a symmetry-breaking perturbation. This perturbation explicitly destroys the commutativity while strictly preserving the system’s self-adjointness. Consequently, the induced nonzero off-diagonal coupling between originally orthogonal subspaces lifts the eigenvalue degeneracy, generating a continuous veering region accompanied by modal hybridization. Furthermore, closed-form scaling relations are derived and numerically validated: the minimum frequency gap scales linearly with [Formula: see text] for geometric inclination, and proportionally to the modal strain product and modal displacement product for stiffness damage and lumped mass, respectively. These explicit spatial dependencies provide a unified metric for geometric, elastic, and inertial asymmetries, thereby establishing a rigorous theoretical basis for structural health monitoring and the quantitative analysis of complex cable systems.