  The harmonic oscillations of a spring-ball model of benzene-like nanosystems
with Hooke's law interactions between nearest, second, and third neighbors are
explored. We show that in the cylindrical coordinates the dynamics of this
cyclic hexagonal system is described by the Lagrange equations similar to those
of the one-dimensional two-component crystal model. We expose that the
vibration frequencies of the hexagonal model lie on the branches of the
dispersion law of the associated lattice model, and their positions are
determined by the cyclic Born-Von Karman condition. The hexagonal model is
generalized to one describing the benzene molecule and the fully deuterated and
halogenated benzenes. The effect of hybridization of vibration modes and the
pushing apart of spectral branches in the crossover situation is revealed. All
the discrete frequency spectrum and normal modes of oscillations and their
explicit dependencies on all the constants of elastic interactions are exactly
found.
