Subharmonic Resonance Cascades in a Class of Coupled Resonators

We consider a chain of N nonlinear resonators with natural frequency ratios of approximately 2:1 along the chain and weak nonlinear coupling that allows energy to flow between resonators. Specifically, the coupling is such that the response of one resonator parametrically excites the next resonator in the chain, and also creates a resonant back-action on the previous resonator in the chain. This class of systems, which is a generic model for passive frequency dividers, is shown to have rich dynamical behavior. Of particular interest in applications is the case when the high frequency end of the chain is resonantly excited, and coupling results in a cascade of subharmonic bifurcations down the chain. When the entire chain is activated, that is, when all N resonators have nonzero amplitudes, if the input frequency on the first resonator is Ω, the terminal resonator responds with frequency Ω/2^N. The details of the activation depend on the strength and frequency of the input, the level of resonator dissipation, and the frequency mistuning in the chain. In this paper we present analytical results, based on perturbation methods, which provide useful predictions about these responses in terms of system and input parameters. Parameter conditions for activation of the entire chain are derived, along with results about other phenomena, such as the period doubling accumulation to full activation, and regions of multistability. We demonstrate the utility of the predictive results by direct comparison with simulations of the equations of motion, and we also present a sample mechanical system that embodies the desired properties. These results are useful for the design and operation of mechanical frequency dividers that are based on subharmonic resonances.