We develop a simple continuum model to analyze the vibrational modes of a nanomechanical multielement structure. In this model, arrays of submicron cantilevers located symmetrically on both sides of the central clamped-clamped nanobeam are replaced by a continuum. In this approach, the equations of motion of the structure become exactly solvable. Our analytical results capture the main features of the vibrational modes observed both numerically and experimentally and can be applied to a general class of scale-independent elasticaly coupled resonator structures.

We develop a simple continuum model to analyze the vibrational modes of a nanomechanical multielement structure. In this model, arrays of submicron cantilevers located symmetrically on both sides of the central clamped-clamped nanobeam are replaced by a continuum. In this approach, the equations of motion of the structure become exactly solvable. Our analytical results capture the main features of the vibrational modes observed both numerically and experimentally and can be applied to a general class of scale-independent elasticaly coupled resonator structures.

We study two specific features of onsite breathers in Nonlinear Schrödinger systems on d-dimensional cubic lattices with arbitrary power nonlinearity (i.e., arbitrary nonlinear exponent, n): their wavefunctions and energies close to the anti-continuum limit–small hopping limit–and their excitation thresholds. Exact results are systematically compared to the predictions of the so-called exponential ansatz (EA) and to the solution of the single nonlinear impurity model (SNI), where all nonlinearities of the lattice but the central one, where the breather is located, have been removed. In 1D, the exponential ansatz is more accurate than the SNI solution close to the anti-continuum limit, while the opposite result holds in higher dimensions. The excitation thresholds predicted by the SNI solution are in excellent agreement with the exact results but cannot be obtained analytically except in 1D. An EA approach to the SNI problem provides an approximate analytical solution that is asymptotically exact as n tends to infinity. But the EA result degrades as the dimension, d, increases. This is in contrast to the exact SNI solution which improves as n and/or d increase. Finally, in our investigation of the SNI problem we also prove a conjecture by Bustamante and Molina [C.A. Bustamante, M.I. Molina, Phys. Rev. B 62 (23) (2000) 15287] that the limiting value of the bound state energy is universal when n tends to infinity.

We propose a general one-dimensional continuous formulation to analyze the vibrational modes of antennalike nanomechanical resonators consisting of two symmetric arrays of cantilevers affixed to a central nanobeam. The cantilever arrays can have arbitrary density and length profile along the beam. We obtain the secular equation that allows for the determination of their frequency spectrum and illustrate the results on the particular examples of structures with constant or alternating cantilever length profiles. We show that our analytical results capture the vibration spectrum of such resonators and elucidate key relationships that could prove advantageous for experimental device performance. Furthermore, using a perturbative approach to treat the nonlinear and dissipative dynamics of driven structures, we analyze the anharmonic coupling between two specific widely spaced modes of the coupled-element device, with direct application to experiments