Modes of vibration, stability and internal resonances on non-linear piezoelectric small-scale beams

Akhavan, Hamed; Roody, Batool Soleimani; Ribeiro, Pedro; Fotuhi, Ali Reza

doi:10.1016/j.cnsns.2018.12.006

Cited by 8 publications

(5 citation statements)

References 47 publications

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“…Changing q from 0 to 1 causes the solution to change from the initial guess to the accurate solution. Setting the right side of equation (27) to zero yields the zero-th order deformation equation [44].…”

Section: Homotopy Analysis Methodsmentioning

confidence: 99%

“…There are several studies on the strain gradient theory in nanostructures. Among these studies is the use of strain gradient in the investigating flow pulsations in nanotubes [26], vibration response of small scale beams [27,28], bending of nanoplates [29], chaotic behavior of nanoresonators [30], and instability analyses of functionally graded nanoplates and beams [31,32].…”

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confidence: 99%

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Nonlinear vibrations of Timoshenko nanobeam using stress driven nonlocal theory

Haghani¹,

Jahangiri²,

Ghaderi³

2022

Phys. Scr.

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As properties of nonlinear systems can introduce particular responses not present in linear systems, results from linear models can be inaccurate or misleading. Therefore, studying nonlinear vibrations is of extreme importance in structural vibration analysis. Nonlinear vibrations of a Timoshenko nanobeam were analyzed for the first time using the stress driven nonlocal theory (SDM) and the strain gradient theory (SGT). Nonlinear equations of motion were solved for simply supported- simply supported, clamped-clamped and clamped- simply supported beams using the homotopy analysis method. The results indicated that as the dimensionless characteristic parameter increased, the nonlinear natural frequency increased accordingly. High agreement was observed between the results from a linear frequency of Timoshenko beam model based on the SGT and those from previous research.

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Section: Homotopy Analysis Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

Nonlinear vibrations of Timoshenko nanobeam using stress driven nonlocal theory

Haghani¹,

Jahangiri²,

Ghaderi³

2022

Phys. Scr.

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“…Younis et al [47] proposed the construction of ROMs by multiplying the equation of motion by the electric load denominator. Many systems were analyzed following this strategy, such as arch resonators [28,29,[48][49][50], arches over flexible supports [51], functionally graded viscoelastic microbeams with imperfections [52], cantilever resonators [53][54][55], narrow microbeams subject to fringing fields [56][57][58] and microscale beams described by the modified couple stress theory [30][31][32][59][60][61][62][63].…”

Section: Introductionmentioning

confidence: 99%

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

Benedetti

Gonçalves

Lenci

et al. 2023

J. Micromech. Microeng.

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This work aims to study the effect of uncertainties and noise on the nonlinear global dynamics of a micro-electro-mechanical arch obtained from an imperfect microbeam under an axial load and electric excitation. An adaptative phase-space discretization strategy based on an operator approach is proposed. The Ulam method, a classical discretization of flows in phase-space, is extended here to nondeterministic cases. A unified description is formulated based on the Perron-Frobenius, Koopman, and Foias linear operators. Also, a procedure to obtain global structures in the mean sense of systems with parametric uncertainties is presented. The stochastic basins of attraction and attractors’ distributions replace the usual basin and attractor concepts. For parameter uncertainty cases, the phase-space is augmented with the corresponding probability space. The microarch is assumed to be shallow and modelled using a nonlinear Bernoulli-Euler beam theory and is discretized by the Galerkin method using as interpolating function the linear vibration modes. Then, from the discretized multi degree of freedom model (mdof) model, an accurate single degree of freedom (sdof) reduced order model, based on theory of nonlinear normal modes, is derived. Several competing attractors are observed, leading to different (acceptable or unacceptable) behaviours. Extensive numerical simulations are performed to investigate the effect of noise and uncertainties on the coexisting basins of attraction, attractors` distributions, and basins boundaries. The appearance and disappearance of attractors and stochastic bifurcation are observed, and the time-dependency of stochastic responses is demonstrated, with long-transients influencing global behaviour. To consider uncertainties and noise in design, a dynamic integrity measure is proposed via curves of constant probability, which give quantitative information about the changes in structural safety. For each attractor, the basin robustness as a function of a stochastic parameter is investigated. The weighted basin area can quantify the integrity of nondeterministic cases, being also the most natural generalization of the global integrity measure. While referring to particular MEMS, the relevance of the dynamical integrity analysis for stochastic systems to quantify tolerances and safety margins is underlined here.

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“…In order to capture the nonlinear geometric effect on a beam associated with piezoelectric layers, Monteiro et al 13 and Neto et al 14 have recently proposed a beam element whereby small strains but large rotations are incorporated in a total Lagrangian description. Akhavan et al 15 implement a nonlinear piezoelectric size-dependent Rayleigh's beam model in hinged-hinged small-scale condition for investigating its vibrations modes modification by means of couple-stress theory. In Damanpack et al 16 sandwich beams are built with piezoelectric sensor/actuator patches so to minimize vibrations via piezoelectric actuation in a closed loop feedback control.…”

Section: Introductionmentioning

confidence: 99%

A geometrically nonlinear structural formulation for analysis of beams with a new set of generalized displacements considering piezoelectric effects

Silva

Donadon

Cabral

2021

Numerical Meth Engineering

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In this article, a nonlinear structural formulation that uses a new dimensionless set of generalized displacements is proposed for solving geometrically nonlinear beam problems, being validated by several applications given in literature where a cantilever beam is likely to undergo large displacements. By this approach, useful simplifications and insights are achievable in the analysis process, as the system matrices becoming linear and the reduction of required interpolation continuity degree. The formulation is firstly developed-in a Lagrangian perspective-and the equilibrium equations are then derived using Hamilton's principle. In the sequence, using finite element method, it is substantiated by comparison to examples given in literature in static, dynamic, and finally in an application where piezoelectric effects intervene, in order to assess its multiframework capabilities and deliver a convenient approach whereby beams constituted of smart or conventional materials can be efficiently studied.

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Modes of vibration, stability and internal resonances on non-linear piezoelectric small-scale beams

Cited by 8 publications

References 47 publications

Nonlinear vibrations of Timoshenko nanobeam using stress driven nonlocal theory

Nonlinear vibrations of Timoshenko nanobeam using stress driven nonlocal theory

Parameter uncertainty and noise effects on the global dynamics of an electrically actuated microarch

A geometrically nonlinear structural formulation for analysis of beams with a new set of generalized displacements considering piezoelectric effects

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