Micron-sized ferromagnetic Permalloy disks exhibiting an in-plane ferromagnetic vortex structure are excited by a fast rise time perpendicular magnetic field pulse and their modal structure is analyzed. We find azimuthal and axial modes. By a Fourier filtering technique we can separate and analyze the time dependence of individual modes. Analysis of the experimental data demonstrates that the azimuthal modes damp more quickly than the axial modes. We interpret these results as mode conversion from low-frequency azimuthal modes to the fundamental mode which is higher in frequency, i.e., mode-mode coupling in a system with a single Landau-Lifshitz-Gilbert phenomenological damping constant alpha.