Non-linear vibrating systems in resonance governed by hyperbolic differential equations

The asymptotic solutions of second order hyperbolic differential equations with weak non-linearities in the case of internal and external resonance are found. The method used is an extension of the Krylov-Bogoliubov-Mitropolskii method. An application is made to the longitudinal vibrations of a rod in which non-linear elastic behaviour and linear viscoelastic damping occur.