Nonlinear polarization coupling in freestanding nanowire/nanotube resonators

In this work, we study the nonlinear coupling between the transverse modes of nanoresonators such as nanotubes or nanowires in a singly clamped configuration. We previously showed that at high driving, this coupling could result in a transition from independent planar modes to a locked elliptical motion, with important modifications of the resonance curves. Here, we clarify the physical origins, associated with a 1: 1 internal resonance, and study in depth this transition as a function of the relevant parameters. We present simple formulae that permit to predict the emergence of this transition as a function of the frequency difference between the polarizations and the nonlinear coefficients and give the "backbone curves" corresponding to the elliptical regime. We also show that the elliptical regime is associated with the emergence of a new set of solutions of which one branch is stable. Finally, we compare single and double clamped configurations and explain why the elliptical transition appears on different polarizations.