Modal stability of sagged cables

Abstract

The conservative problem of the stability of symmetric nonlinear normal modes (NNMs) of sagged cables is analysed. Based on harmonic shape functions, the equations of motion for a conservative sagged cable are derived and nonlinear normal modes are calculated as a continuation of the linear modes, via the harmonic balance approach. Leveraging symmetry, we decouple the equations of motion, obtaining equations in the form d²y/dt² = A(t)y. The harmonic balance approach reduces the equations of motion to a nonlinear eigenproblem, in which the matrix of coefficients depends nonlinearly on a single parameter (energy level or amplitude). The relevant eigensolutions provide the stability boundaries. Floquet theory provides a complementary tool for individuating stable and unstable regions. Numerical examples illustrate how the insurgence of modal instability is related to increasing energy levels and demonstrate the mechanism of energy transfer from a mode to a specific harmonic. The first five symmetric modes are analysed in detail, and the recipient harmonics are individuated, providing a complete stability mapping. The proposed approach provides a tool for a systematic understanding of modal instability and energy transfer between modes in cable-suspended structures, thus enabling more engineering-oriented analyses.