Fundamental and subharmonic resonances in a nonlinear oscillator with bifurcating modes

The fundamental and subharmonic resonances of a discrete oscillator with cubic stiffness nonlinearities and linear viscous damping are examined using a multiple-scales averaging analysis. The system is in a '1-1' internal resonance, i.e., it has two equal linearized eigenfrequencies, and possesses nonlinear normal modes. For weak coupling stiffnesses the internal resonance gives rise to a Hamiltonian Pitchfork bifurcation of normal modes which in turn affects the topology of the fundamental and subharmonic resonance curves. It is shown that the number of resonance branches differs before and after the mode bifurcation, and that jump phenomena are possible between forced modes. Some of the steady state solutions were found to be very sensitive to damping: a whole branch of fundamental resonances was eliminated even for small amounts of viscous damping, and subharmonic steady state solutions were shifted by damping to higher frequencies.