A theoretical model for the calculation of the quantized spectrum of spin modes frequencies in cylindrical magnetic dots of radius ranging from the nanometric to the submicrometric scale in the vortex ground state at zero applied magnetic field is presented. The effective field includes both the surface and the volume dynamic magnetostatic and exchange fields. We also show that the core energy affects the spin dynamics. The modes at lower frequencies present as radial eigenvectors Bessel functions of high orders (m greater than or equal to 1), while the axially symmetric modes at higher frequencies correspond to zero order Bessel functions.