This review provides a summary of the work completed to date on the nonlinear dynamics of resonant micro- and nanoelectromechanical systems (MEMS/NEMS). This research area, which has been active for approximately a decade, involves the study of nonlinear behaviors arising in small scale, vibratory, mechanical devices that are typically integrated with electronics for use in signal processing, actuation, and sensing applications. The inherent nature of these devices, which includes low damping, desired resonant operation, and the presence of nonlinear potential fields, sets an ideal stage for the appearance of nonlinear behavior, and this allows engineers to beneficially leverage nonlinear dynamics in the course of device design. This work provides an overview of the fundamental research on nonlinear behaviors arising in micro/nanoresonators, including direct and parametric resonances, parametric amplification, impacts, selfexcited oscillations, and collective behaviors, such as localization and synchronization, which arise in coupled resonator arrays. In addition, the work describes the active exploitation of nonlinear dynamics in the development of resonant mass sensors, inertial sensors, and electromechanical signal processing systems. The paper closes with some brief remarks about important ongoing developments in the field.

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Nonlinear Dynamics and Its Applications in Micro- and Nanoresonators

https://imechanica.org/files/Resonant%20behavior%20of%20a%20membrane%20of%20a%20dielectric%20elastomer.pdf

Resonant behavior of a membrane of a dielectric elastomer

This review provides a summary of the work completed to date on the nonlinear dynamics of resonant micro- and nanoelectromechanical systems (MEMS/NEMS). This research area, which has been active for approximately a decade, involves the study of nonlinear behaviors arising in small scale, vibratory, mechanical devices that are typically integrated with electronics for use in signal processing, actuation, and sensing applications. The inherent nature of these devices, which includes low damping, desired resonant operation, and the presence of nonlinear potential fields, sets an ideal stage for the appearance of nonlinear behavior, and this allows engineers to beneficially leverage nonlinear dynamics in the course of device design. This work provides an overview of the fundamental research on nonlinear behaviors arising in micro/nanoresonators, including direct and parametric resonances, parametric amplification, impacts, selfexcited oscillations, and collective behaviors, such as localization and synchronization, which arise in coupled resonator arrays. In addition, the work describes the active exploitation of nonlinear dynamics in the development of resonant mass sensors, inertial sensors, and electromechanical signal processing systems. The paper closes with some brief remarks about important ongoing developments in the field.Copyright © 2008 by ASME

The geometrically nonlinear governing differential equations of motion and the corresponding boundary conditions are derived for the mechanical analysis of Timoshenko microbeams with large deflections, based on the strain gradient theory. The variational approach is employed to achieve the formulation. Hinged-hinged beams are considered as an important practical case, and their nonlinear static and free-vibration behaviors are investigated based on the derived formulation.

A size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory

In this paper the radial deformation and the corresponding stresses in a non-homogeneous hollow elastic cylinder rotating about its axis with a constant angular velocity is investigated. The material of the cylinder is assumed to the non-homogeneous and cylindrically orthotropic. The system of fundamental equations is solved by means of a finite difference method and the numerical calculations are carried out for the temperature, the components of displacement and the components of stress with the time t and through the thickness of the cylinder. The results indicate that the effect of inhomogeneity is very pronounced.

Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder

This paper studies parametric resonance of coupled micromechanical oscillators under periodically varying nonlinear coupling forces. Different from most of previous related works in which the periodically varying coupling forces between adjacent oscillators are linearized, our work focuses on new physical phenomena caused by the periodically varying nonlinear coupling. Harmonic balance method (HBM) combined with Newton iteration method is employed to find steady-state periodic solutions. Similar to linearly coupled oscillators studied previously, the present model predicts superharmonic parametric resonance and the lower-order subharmonic parametric resonance. On the other hand, the present analysis shows that periodically varying nonlinear coupling considered in the present model does lead to the appearance of high-order subharmonic parametric resonance when the external excitation frequency is a multiple or nearly a multiple (≥3) of one of the natural frequencies of the oscillator system. This remarkable new phenomenon does not appear in the linearly coupled micromechanical oscillators studied previously, and makes the range of exciting resonance frequencies expanded to infinity. In addition, the effect of a linear damping on parametric resonance is studied in detail, and the conditions for the occurrence of the high-order subharmonics with a linear damping are discussed.

High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators

A novel method is developed to study the structural instability of a parallel array of identical microbeams each of which interacts with two neighboring microbeams through distance-dependent surface attractive forces (such as electrostatic, van der Waals or Casimir forces). First, based on a simplified spring model, it is verified that the equilibrium deflections of intermediate beams (except the beams at the two ends of the parallel array) would be negligibly small. Thus, when the end-effect of the beams at the ends of the parallel array is neglected, exact analysis of parallel beams shows that the critical value of the beam–beam interaction coefficient for instability of the parallel array, defined on the initial distance between adjacent beams, is exactly a half of the critical value of the interaction coefficient for instability of two mutually attracting beams, or equivalently a quarter of the critical value of the interaction coefficient for instability of a single beam attracted by a rigid body. Furthermore, the end-effect is studied by examining the dependence of the critical value on the amplified interaction coefficient for the two end beams due to their non-negligible deflection caused by one-side attraction from the adjacent beam. The results show that the end-effect leads to a 20%–35% reduction of the critical interaction coefficient for instability of the parallel array, and therefore the actual critical value can be given, conservatively, by 60% of the critical value without the end-effect. These results provide a simple design criterion for the pull-in instability of interacting parallel microbeams. In particular, the results obtained by the present method are found to be in good agreement with known results for a few special cases and the exact data obtained by an iteration numerical method for the spring model.

Structural instability of a parallel array of mutually attracting identical microbeams

High-order subharmonic parametric resonance of coupled microbeams is studied based on a model of multiple nonlinearly coupled micromechanical nonlinear oscillators. Of particular interest are the effects of nonlinear coupling between adjacent oscillators and elastic nonlinearity of individual oscillators. First, we analyze parametric resonance of a nonlinearly coupled nonlinear oscillator, and conduct instability analysis of steady-state solutions. Then we study parametric resonance of three nonlinearly coupled nonlinear oscillators. It is found that the nonlinear coupling between adjacent oscillators leads to the appearance of high-order subharmonic parametric resonance which, to the best of our knowledge, has received little attention in previous related works. In the present analysis, the effects of elastic nonlinearity of individual oscillators, damping and some loading parameters (such as dc and ac voltages) on high-order subharmonics are also examined in detail. Our results suggest that the range of excitation frequency for parametric resonance of coupled oscillators can be much larger than what the previous studies predicted, due to the appearance of high-order subharmonic parametric resonance. These results are believed to offer new and interesting insights into the ongoing research on nonlinear dynamics of coupled microbeams or nanobeams in MEMS or NEMS.

High-order subharmonic parametric resonance of multiple nonlinearly coupled micromechanical nonlinear oscillators

Comb-drive structures consisting of two opposing arrays of microcantilevers have wide-spread use in MEMS In this paper, a novel method is developed to study jump-to-together instability of such comb-drive microcantilevers, each of which is attracted by two neighboring microcantilevers through surface forces Based on a representative spring model given in the Appendix, it is verified that equilibrium deflections of all intermediate microcantilevers (except those at the ends of the array) would be negligibly small because two attractions from two opposing sides are almost equal but opposite Thus, when the end-effect of the microcantilevers at the ends of the array is neglected, the critical value of the beam–beam interaction coefficient for instability of the array is found to be exactly a half of the critical value of the interaction coefficient for instability of a pair of opposing microcantilevers The dependence of the critical value on the overlap length of opposing microcantilevers is demonstrated 

Surface-forces-driven instability of comb-drive microcantilevers in MEMS

This paper is concerned with the finite spherically symmetric motion of a compressible hyperelastic spherical shell, subjected to a spatially uniform step funtion application of pressure at its inner surface. A method, given in a previous paper [1], for the determination of the field of characteristics, for expansion of a spherical cavity in an unbounded solid is adapted to consider the spherical shell problem. Results are presented graphically for a particular strain energy function and are compared with results obtained for an incompressible material and from linear elasticity theory.

A. Mioduchowski

Papers

High-order subharmonic parametric resonance of nonlinearly coupled micromechanical oscillators

Structural instability of a parallel array of mutually attracting identical microbeams

High-order subharmonic parametric resonance of multiple nonlinearly coupled micromechanical nonlinear oscillators

Surface-forces-driven instability of comb-drive microcantilevers in MEMS

Dynamic expansion of a compressible hyperelastic spherical shell