Two mode sub-harmonic and harmonic vibrations of a non-linear beam forced by a two mode harmonic load

The non-linear transverse vibrations of a uniform beam with ends restrained to remain a fixed distance apart and forced by a two mode function which is harmonic in time, are studied by a corresponding two mode approach. The existence of sub-harmonic response of order 1/3 and harmonic response in the sub-harmonic resonance region of the forcing frequency is proved. Approximate solutions are found by Urabe's numerical method applied to Galerkin's procedure and by an analytical harmonic balance-perturbation method. Error bounds of the Galerkin approximations are given.