Abstract
Nonlinear modal interactions have recently become the focus of intense research in micro- and nanoscale resonators for their use to improve oscillator performance and probe the frontiers of fundamental physics. However, our understanding of modal coupling is largely restricted to clamped-clamped beams, and lacking in systems with both geometric and material nonlinearities. Here we report multistable energy transfer between internally resonant modes of an electroelastic crystal plate and use a mixed analytical-numerical approach to provide new insight into these complex interactions. Our results reveal a rich bifurcation structure marked by nested regions of multistability. Even the simple case of two coupled modes generates a host of topologically distinct dynamics over the parameter space, ranging from the usual Duffing bistability to complex multistable behaviour and quasiperiodic motion.
Highlights
We derive a general model of nonlinear resonance in electroelastic crystals that accounts for both material and geometric nonlinearities and their effect on modal interactions
We use the model to explain heretofore unobserved manifestations of modal coupling in a quartz crystal resonator driven into the nonlinear regime
In addition to the well-known saddle-node bifurcations associated with Duffing bistability, we show that modal interactions can generate other saddle-node delimited, multistable regions that significantly complicate the dynamics
Summary
Our results indicate that even in the simplest case of two modes the nature of modal interactions can vary widely, and strongly depends on the system parameters: changes in the difference between eigenfrequencies of the associated linear system, the modal damping ratios, or nonlinear coupling coefficients can lead to topologically distinct bifurcation structures This explains the aforementioned difficulty of ‘pinning down’ internal resonance and related effects in real devices. Even two coupled modes can generate multiple stable solutions that are experimentally inaccessible and difficult to locate numerically without detailed knowledge of their basins of attraction Continuation methods resolve these difficulties, and once the bifurcation structure is known numerical integration can probe the dynamics of a given region of parameter space. The equations of motion for nonlinear electroelasticity, in the absence of a body force, are defined by
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