Periodic solutions of the Nathanson’s and the Comb-drive models

Llibre, Jaume; Núñez, Daniel; Rivera, Andrés

doi:10.1016/j.ijnonlinmec.2018.05.009

Cited by 8 publications

(8 citation statements)

References 16 publications

(28 reference statements)

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“…We also revisited the Nathanson model under linear damping. Although this problem is considered in [7][8][9], Theorem 5 improves the existence and stability results as a result of appropriate conditions over the voltage load and the viscous damping coefficient c, providing new and significant knowledge of the dynamics of this model.…”

Section: Discussionmentioning

confidence: 95%

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Periodic Oscillations in MEMS under Squeeze Film Damping Force

Beron

Rivera

2022

Journal of Applied Mathematics

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We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Liénard-type equation x ¨ + F D x , x ̇ + x = β V 2 t / 1 − x 2 , x ∈ − ∞ , 1 with β ∈ ℝ + , V ∈ C ℝ / T ℤ , and F D x , x ̇ = κ x ̇ / 1 − x 3 , κ ∈ ℝ + (called squeeze film damping force), or F D x , x ̇ = c x ̇ , c ∈ ℝ + (called linear damping force). If F D is of squeeze film type, we have proven that there exists at least two positive periodic solutions, one of them locally asymptotically stable. Meanwhile, if F D is a linear damping force, we have proven that there are only two positive periodic solutions. One is unstable, and the other is locally exponentially asymptotically stable with rate of decay of c / 2 . Our technique can be applied to a class of Liénard equations that model several microelectromechanical system devices, including the comb-drive finger model and torsional actuators.

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Section: Discussionmentioning

confidence: 95%

“…After more than 50 years, the Nathanson model continues to draw a lot of attention. Many researchers have been devoted to its analytical and numerical study, mainly to understanding and characterizing the pull-in phenomenon through different techniques and mathematical formulations (see for instance [1,2,[6][7][8][9][10]).…”

Section: Introductionmentioning

confidence: 99%

Periodic Oscillations in MEMS under Squeeze Film Damping Force

Beron

Rivera

2022

Journal of Applied Mathematics

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“…Let us compare the results of calculations of displacements, velocities, and phase portraits for the original (2) and approximate Equation (15) (Figures [3][4][5][6][7][8][9][10][11]. The following notation is used in the figures: 𝑊 0 is the initial displacement relative to point 𝑊 = 0, 𝑊 𝑠𝑡 is the position of a minimal stable point of equilibrium in rest, 𝑊 𝑢𝑛𝑠𝑡 is the position of an unstable point of equilibrium in rest, and 𝐴 0 = 𝑊 0 − 𝑊 𝑠𝑡 is the initial displacement relative to the stable equilibrium point.…”

Section: Numerical Results For Nathanson Equationmentioning

confidence: 99%

“…One of the first models describing the phenomenon under study was the Nathanson one DoF model [5,6] (Figure 1).…”

Section: Nathanson Equation: Analytical Estimationsmentioning

confidence: 99%

Nonlinear oscillation of a microbeam due to an electric actuation—Comparison of approximate models

Andrianov,

Awrejcewicz,

Koblik

et al. 2023

Z Angew Math Mech

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In this paper, oscillations of the electrically actuated microbeam are considered, described by strongly nonlinear differential equations. One of the essentially nonlinear effects is the pull‐in phenomenon, that is, the transition of the oscillatory regime to the attraction regime. This effect is considered in the paper on the basis of various approximate models. In particular, the error of replacing the original nonlinearity by its expansion in the Maclaurin series is estimated. It is shown that such an approximation can only be used for voltage values that are far from critical. Estimates are also given for initial displacements and velocities, which guarantee an oscillating character of solutions. The electrically activated oscillations of the microbeam are considered taking into account the geometric nonlinearity within the framework of the Kirchhoff model. The dependence of pull‐in value on geometric nonlinearity is investigated.

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“…In [14], some sufficient conditions for existence and stability properties of periodic solutions for two periodically forced canonical MEMS models: the Nathanson's model and the Combdrive finger model (see [22] chapter 3.5.1 and 3.5.3 respectively) are provided using Averaging Theory and Lower and Upper Solutions Method.…”

Section: Introductionmentioning

confidence: 99%

Periodic oscillations for a graphene-based electrostatic micro actuator

Núñez

Murcia

Galán

2021

Preprint

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We study the mechanical oscillations for a novel model of a graphene-based electrostatic parallel plates micro actuator introduced by Wei et al.(2017), considering damping effects when a periodic voltage with alternating current is applied. Our analysis starts from recent results about this MEMS model with constant voltage, and provides new insights on the periodic mechanical responses for a variable input voltage. We derive sufficient conditions on the system physical components for which periodic oscillations with constant sign exist together with their stability properties. Specifically, under some conditions, the existence of three periodic solutions is established, one of them is negative and the others are positive in sign. The positive one nearby the origin is asymptotically locally stable, whilst the other two are unstable. Additionally, we prove that no further constant sign periodic solutions can be found. The existence of periodic solutions is approached from direct and reverse order Lower and Upper Solutions Method, and the stability assertions are derived from the Liapounoff-Zukovskii criteria for Hill's equations and the linearization principle. Theoretical results are complemented by numerical simulations and numerical continuation results. Furthermore, these numerical simulations evidence the robustness of the graphene-based MEMS model over the traditional ones.

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Periodic solutions of the Nathanson’s and the Comb-drive models

Cited by 8 publications

References 16 publications

Periodic Oscillations in MEMS under Squeeze Film Damping Force

Periodic Oscillations in MEMS under Squeeze Film Damping Force

Nonlinear oscillation of a microbeam due to an electric actuation—Comparison of approximate models

Periodic oscillations for a graphene-based electrostatic micro actuator

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