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

We propose and experimentally validate a first-principles based model for the nonlinear piezoelectric response of an electroelastic energy harvester The analysis herein highlights the importance of modeling inherent piezoelectric nonlinearities that are not limited to higher order elastic effects but also include nonlinear coupling to a power harvesting circuit Furthermore, a nonlinear damping mechanism is shown to accurately restrict the amplitude and bandwidth of the frequency response The linear piezoelectric modeling framework widely accepted for theoretical investigations is demonstrated to be a weak presumption for near-resonant excitation amplitudes as low as 05 g in a prefabricated bimorph whose oscillation amplitudes remain geometrically linear for the full range of experimental tests performed (never exceeding 025% of the cantilever overhang length) Nonlinear coefficients are identified via a nonlinear least-squares optimization algorithm that utilizes an approximate analytic solution obta

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Nonlinear piezoelectricity in electroelastic energy harvesters: Modeling and experimental identification

This paper presents an experimental piezoelectric energy harvester exhibiting strong mechanical nonlinear behavior. Vibration energy harvesters are usually resonant mechanical systems working at resonance. The resulting mechanical amplification gives an output power multiplied by the mechanical quality factor Q when compared to non-resonant systems, provided that the electromechanical coupling k2 is high as well as the mechanical quality factor Q. However, increasing the Q value results in a narrowband energy harvester, and the main drawback is the difficulty of matching a given vibration frequency range to the energy harvester's resonance frequency. Mechanical nonlinear stiffness results in a distortion of the resonance peak that may lead to a broadband energy harvesting capability while keeping a large output power as for high Q systems. This paper is devoted to an experimental study of a Duffing oscillator exhibiting piezoelectric electromechanical coupling. A nonlinear electromechanical model is first presented including piezoelectric coupling, a nonlinear stiffness as for a Duffing oscillator, and an additional nonlinear loss term. Under harmonic excitation, it is shown that for a particular excitation range, the power frequency bandwidth is multiplied by a factor of 5.45 whereas the output power is decreased by a factor of 2.4. In addition, when compared to a linear system exhibiting the same power bandwidth as for the nonlinear one (which is here 7.75%), the output power is increased by a factor of 16.5. Harmonic study is, however, partially irrelevant, because Duffing oscillators exhibit a frequency range where two stable harmonic solutions are possible. When excited with sine bursts or colored noise, the oscillator remains most of the time at the lowest solution. In this paper, we present a technique—called fast burst perturbation—which consists of a fast voltage burst applied to the piezoelectric element. It is then shown that the resonator may jump from the low solution to the high solution at a very small energy cost. Time-domain solution of the model is presented to support experimental data.

https://iopscience.iop.org/article/10.1088/0964-1726/20/10/102001/pdf

Experimental Duffing oscillator for broadband piezoelectric energy harvesting

We present a novel hysteresis compensation method for piezoelectric actuators. We consider the hysteresis nonlinearity of the actuator as a disturbance over a linear system. A disturbance observer (DOB) is then utilized to estimate and compensate for the hysteresis nonlinearity. In contrast to the existing inverse-model-based approach, the DOB-based hysteresis compensation does not rely on any particular hysteresis model, and therefore provides a simple and effective compensation mechanism. We design and fabricate a lead magnesium niobate-lead titanate (PMN-PT) piezoelectric actuator for microscale tip-based power sintering process. Experimental validation of the proposed hysteresis compensation is performed on the PMN-PT cantilever piezoelectric actuator. The experimental results demonstrate the effectiveness and efficiency of the approach.

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Disturbance-Observer-Based Hysteresis Compensation for Piezoelectric Actuators

Inherent nonlinearities of piezoelectric materials are pronounced in various engineering applications such as sensing, actuation, combined applications for vibration control, and energy harvesting from dynamical systems The existing literature focusing on the dynamics of electroelastic structures made of piezoelectric materials has explored such nonlinearities separately for the problems of mechanical and electrical excitation Similar manifestations of softening nonlinearities have been attributed to purely elastic nonlinear terms, coupling nonlinearities, hysteresis alone, or a combination of these effects by various authors In order to develop a unified nonlinear nonconservative framework with two-way coupling, the present work investigates the nonlinear dynamic behavior of a bimorph piezoelectric cantilever under low to moderately high mechanical and electrical excitation levels in energy harvesting, sensing, and actuation The highest voltage levels, for near resonance investigation, are well below the coercive field A distributed parameter electroelastic model is developed by accounting for softening and dissipative nonlinearities to analyze the primary resonance of a soft (eg, PZT-5A, PZT-5H) piezoelectric cantilever for the fundamental bending mode using the method of harmonic balance Excellent agreement between the model and experimental investigation is found, providing evidence that quadratic stiffness, damping, and electromechanical coupling effects accurately model predominantly observed nonlinear effects in geometrically linear vibration of piezoelectric cantilever beams The backbone curves of both energy harvesting and actuation frequency responses for a PZT-5A cantilever are experimentally found to be dominantly of first order and specifically governed by ferroelastic quadratic softening for a broad range of mechanical and electrical excitation levels Cubic and higher-order nonlinearities become effective only near the physical limits of the brittle and stiff (geometrically linear) bimorph cantilever when excited near resonance

Unified nonlinear electroelastic dynamics of a bimorph piezoelectric cantilever for energy harvesting, sensing, and actuation

Nanomechanical cantilever sensors (NMCSs) have recently emerged as an effective means for label-free chemical and biological species detection. They operate through the adsorption of species on the functionalized surface of mechanical cantilevers. Through this functionalization, molecular recognition is directly transduced into a micromechanical response. In order to effectively utilize these sensors in practice and correctly relate the micromechanical response to the associated adsorbed species, the chief technical issues related to modeling must be resolved. Along these lines, this paper presents a general nonlinear-comprehensive modeling framework for piezoelectrically actuated microcantilevers and validates it experimentally. The proposed model considers both longitudinal and flexural vibrations and their coupling in addition to the ever-present geometric and material nonlinearities. Utilizing Euler-Bernoulli beam theory and employing the inextensibility conditions, the coupled longitudinal-flexural equations of motion are reduced to one nonlinear partial differential equation describing the flexural vibrations of the sensor. Using a Galerkian expansion, the resulting equation is discretized into a set of nonlinear ordinary differential equations. The method of multiple scales is then implemented to analytically construct the nonlinear response of the sensor near the first modal frequency (primary resonance of the first vibration mode). These solutions are compared to experimental results demonstrating that the sensor exhibits a softening-type nonlinear response. Such behavior can be attributed to the presence of quadratic material nonlinearities in the piezoelectric layer. This observation is critical, as it suggests that unlike macrocantilevers where the geometric hardening nonlinearities dominate the response behavior, material nonlinearities dominate the response of microcantilevers yielding a softening-type response. This behavior should be accounted for when designing and employing such sensors for practical applications.

Modeling, Nonlinear Dynamics, and Identification of a Piezoelectrically Actuated Microcantilever Sensor

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

This paper undertakes the passive nonlinear vibration analysis of a directly excited composite cantilever beam modeled as an inextensible Euler-Bernoulli beam. The composite beam consists of two elastic layers of high-carbon steel sandwiched together through a viscoelastic layer of carbon nanotube (CNT)-epoxy mixture. The resulting viscoelastic damping is modeled as Kelvin-Voigt damping model, with the nonlinearities present due to inextensibility assumption. In order to study the system response characteristics, the method of multiple scales is employed to arrive at the modulation equations and the closed-loop frequency response function. Such analytically-derived frequency response is experimentally verified through harmonic force excitation of samples of CNT-reinforced composite beams. The results demonstrate that increasing the excitation amplitude or decreasing damping ratio can cause a minor decrease in the nonlinear resonance frequency despite the significant increase in the amplitude of vibration due to reduced damping.Copyright © 2005 by ASME

S.N. Mahmoodi

Papers

Modeling, Nonlinear Dynamics, and Identification of a Piezoelectrically Actuated Microcantilever Sensor

Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

Passive Nonlinear Vibrations of a Directly Excited Nanotube-Reinforced Composite Cantilevered Beam