Investigation of Resonance Modes in Iced Transmission Lines Using Two Discrete Methods

To investigate the oscillation modes of iced transmission lines, this study introduces a forcing term into the galloping equation and applies two discretization approaches: Discrete Method I (DMI), which directly transforms the partial differential equation into an ordinary differential form, and Discrete Method II (DMII), which first averages dynamic tension along the span. The finite element method is employed to validate the analytical solutions. Using a multiscale approach, amplitude-frequency responses under primary, harmonic, and internal resonance are derived. Results show that DMII yields larger galloping amplitudes and trajectories than DMI, with lower resonant frequencies and weaker geometric nonlinearities. In harmonic resonance, superharmonic and subharmonic modes (notably 1/2) are more easily excited. Under 2:1:2 internal resonance, amplitude differences in the vertical ( z ) direction are more sensitive to the discretization method, whereas the 1:1:1 case shows minimal variation across directions. These findings suggest that the choice of discretization significantly influences galloping behavior, with DMII offering a more conservative prediction.