scholarly journals A 2D Non-Linear Second-Order Differential Model for Electrostatic Circular Membrane MEMS Devices: A Result of Existence and Uniqueness

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1193 ◽  
Author(s):  
Paolo Di Barba ◽  
Luisa Fattorusso ◽  
Mario Versaci
Keyword(s):  
Steady State ◽  
Mean Curvature ◽  
Existence And Uniqueness ◽  
Second Order ◽  
Circular Membrane ◽  
Mems Devices ◽  
Differential Model ◽  
Non Linear ◽  
The Mean ◽  
State Case

In the framework of 2D circular membrane Micro-Electric-Mechanical-Systems (MEMS), a new non-linear second-order differential model with singularity in the steady-state case is presented in this paper. In particular, starting from the fact that the electric field magnitude is locally proportional to the curvature of the membrane, the problem is formalized in terms of the mean curvature. Then, a result of the existence of at least one solution is achieved. Finally, two different approaches prove that the uniqueness of the solutions is not ensured.

scholarly journals Recovering of the Membrane Profile of an Electrostatic Circular MEMS by a Three-Stage Lobatto Procedure: A Convergence Analysis in the Absence of Ghost Solutions

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 487 ◽  
Author(s):  
Mario Versaci ◽  
Giovanni Angiulli ◽  
Alessandra Jannelli
Keyword(s):  
Convergence Analysis ◽  
Electrostatic Field ◽  
Mean Curvature ◽  
Mechanical Systems ◽  
Numerical Approach ◽  
Second Order ◽  
Circular Membrane ◽  
Differential Model ◽  
The Mean

In this paper, a stable numerical approach for recovering the membrane profile of a 2D Micro-Electric-Mechanical-Systems (MEMS) is presented. Starting from a well-known 2D nonlinear second-order differential model for electrostatic circular membrane MEMS, where the amplitude of the electrostatic field is considered proportional to the mean curvature of the membrane, a collocation procedure, based on the three-stage Lobatto formula, is derived. The convergence is studied, thus obtaining the parameters operative ranges determining the areas of applicability of the device under analysis.

scholarly journals Electrostatic field in terms of geometric curvature in membrane MEMS devices

2017 ◽  
Vol 8 (1) ◽  
pp. 165-184 ◽  
Author(s):  
Paolo Di Barba ◽  
Luisa Fattorusso ◽  
Mario Versaci
Keyword(s):  
Fixed Point ◽  
Steady State ◽  
Existence And Uniqueness ◽  
Mems Devices ◽  
Micro Electro Mechanical Systems ◽  
Membrane Deformation ◽  
Numerical Tests ◽  
Geometric Curvature ◽  
State Case ◽  
Uniqueness Of A Solution

Abstract In this paper we present, in a framework of 1D-membrane Micro-Electro-Mechanical- Systems (MEMS) theory, a formalization of the problem of existence and uniqueness of a solution related to the membrane deformation u for electrostatic actuation in the steady- state case. In particular, we propose a new model in which the electric field magnitude E is proportional to the curvature of the membrane and, for it, we obtain results of existence by Schauder-Tychono's fixed point application and subsequently we establish conditions of uniqueness. Finally, some numerical tests have been carried out to further support the analytical results.

scholarly journals Stationary surfaces in Lorentz–Minkowski space

2008 ◽  
Vol 138 (5) ◽  
pp. 1067-1096 ◽  
Author(s):  
Rafael López
Keyword(s):  
Minkowski Space ◽  
Critical Point ◽  
Linear Function ◽  
Mean Curvature ◽  
Existence And Uniqueness ◽  
Support Plane ◽  
Stationary Surface ◽  
The Mean

Consider a space-like plane Π in Minkowski space. Under the presence of a uniform time-like potential directed towards Π, this paper analyses the configurations of shapes that show a space-like surface supported in Π with prescribed volume and show that it is a critical point of the energy of this system. Such a surface is called stationary and it is determined by the condition that the mean curvature is a linear function of the distance from Π and the fact that the angle of contact with the plate Π is constant. We prove that the surface must be rotational symmetric with respect to an axis orthogonal to Π. Next, we show existence and uniqueness of symmetric solutions for a prescribed angle of contact with Π. Finally, we study the shapes that a stationary surface can adopt in terms of its size. We thus derive estimates of its height and the enclosed volume by surface with the support plane.

scholarly journals Electrostatic Circular Membrane MEMS: An Approach to the Optimal Control

Computation ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 41
Author(s):  
Mario Versaci ◽  
Francesco Carlo Morabito
Keyword(s):  
Optimal Control ◽  
Stable Equilibrium ◽  
Equilibrium Configuration ◽  
Micro Electro Mechanical System ◽  
Circular Membrane ◽  
Differential Model ◽  
Equilibrium Configurations ◽  
Electrostatic Effect ◽  
The Mean ◽  
The Stability

The recovery of the membrane profile of an electrostatic micro-electro-mechanical system (MEMS) is an important issue, because, when an external electrical voltage is applied, the membrane deforms with the risk of touching the upper plate of the device producing an unwanted electrostatic effect. Therefore, it is important to know whether the movement admits stable equilibrium configurations especially when the membrane is closed to the upper plate. In this framework, this work analyzes the behavior of a two-dimensional (2D) electrostatic circular membrane MEMS device subjected to an external voltage. Specifically, starting from a well-known 2D non-linear second-order differential model in which the electrostatic field in the device is proportional to the mean curvature of the membrane, the stability of the only possible equilibrium configuration is studied. Furthermore, when considering that the membrane is equipped with mechanical inertia and that it must not touch the upper plate of the device, a useful range of possible values has been obtained for the applied voltage. Finally, the paper concludes with some computations regarding the variation of potential energy, identifying some optimal control conditions.

Helicoidal surfaces with prescribed curvatures in a conformally flat 3-space

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Chul Woo Lee ◽  
Jae Won Lee ◽  
Dae Won Yoon
Keyword(s):  
Mean Curvature ◽  
Extrinsic Curvature ◽  
Conformal Factor ◽  
Second Order ◽  
Conformally Flat ◽  
Non Linear ◽  
Linear Ode ◽  
Helicoidal Surfaces ◽  
Conformally Flat Metric

Abstract In this paper, we study a conformally flat 3-space 𝔽 3 {\mathbb{F}_{3}} which is an Euclidean 3-space with a conformally flat metric with the conformal factor 1 F 2 {\frac{1}{F^{2}}} , where F ⁢ ( x ) = e - x 1 2 - x 2 2 {F(x)=e^{-x_{1}^{2}-x_{2}^{2}}} for x = ( x 1 , x 2 , x 3 ) ∈ ℝ 3 {x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}} . In particular, we construct all helicoidal surfaces in 𝔽 3 {\mathbb{F}_{3}} by solving the second-order non-linear ODE with extrinsic curvature and mean curvature functions. As a result, we give classification of minimal helicoidal surfaces as well as examples for helicoidal surfaces with some extrinsic curvature and mean curvature functions in 𝔽 3 {\mathbb{F}_{3}} .

scholarly journals On the reconstruction of a star-shaped body from its “half-volumes”

1984 ◽  
Vol 37 (2) ◽  
pp. 243-257 ◽  
Author(s):  
Stefano Campi
Keyword(s):  
Mean Curvature ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
A Priori ◽  
Symmetry Condition ◽  
Shaped Body ◽  
A Priori Bound ◽  
Uniqueness Of The Solution ◽  
The Mean

AbstractThe problem is the reconstruction of the shape of an object, whose shell is a surface star-shaped with respect to a point 0, from the knowledge of the volume of every “half-object” obtained by taking any plane through 0. Conditions for the existence and uniqueness of the solution are given. The main result consists in showing that any uniform a-priori bound on the mean curvature of the shell reestablishes continuous dependence on the data for bodies satisfying a certain symmetry condition.

Weighted minimal translation surfaces in Minkowski 3-space with density

2017 ◽  
Vol 14 (12) ◽  
pp. 1750178 ◽  
Author(s):  
Dae Won Yoon
Keyword(s):  
Mean Curvature ◽  
Second Order ◽  
Translation Surfaces ◽  
Weighted Mean ◽  
Smooth Functions ◽  
Weighted Mean Curvature ◽  
Straight Line ◽  
Non Linear ◽  
Linear Ode

The aim of this work is to study translation surfaces in a Minkowski 3-space [Formula: see text] with density. Translation surfaces in [Formula: see text] are defined as the two generating curves which lie in orthogonal planes. They have actually three different possible parametrizations according to the intersecting straight line of the two planes. We completely classify all translation surfaces with zero weighted mean curvature in [Formula: see text] with density [Formula: see text] by solving the second-order non-linear ODE with some smooth functions.

Dynamics of a stochastically perturbed two-body problem

2007 ◽  
Vol 463 (2080) ◽  
pp. 979-1003 ◽  
Author(s):  
Shambhu N Sharma ◽  
H Parthasarathy
Keyword(s):  
Body Problem ◽  
Random Force ◽  
Second Order ◽  
Differential Model ◽  
Two Body Problem ◽  
System Nonlinearity ◽  
The Mean ◽  
Perturbed Model ◽  
Perturbation Effect ◽  
Stochastically Perturbed

In classical mechanics, the two-body problem has been well studied. The governing equations form a system of two-coupled second-order nonlinear differential equations for the radial and angular coordinates. The perturbation induced by the astronomical disturbance like ‘dust’ is normally not considered in the orbit dynamics. Distributed dust produces an additional random force on the orbiting particle, which can be modelled as a random force having ‘Gaussian statistics’. The estimation of accurate positioning of the orbiting particle is not possible without accounting for the stochastic perturbation felt by the orbiting particle. The objective of this paper is to use the stochastic differential equation (SDE) formalism to study the effect of such disturbances on the orbiting body. Specifically, in this paper, we linearize SDEs about the mean of the state vector. The linearization operation performed above, transforms the system of SDEs into another system of SDEs that resembles a bilinear system, as described in signal processing and control literature. However, the mean trajectory of the resulting bilinear stochastic differential model does not preserve the perturbation effect felt by the orbiting particle; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the dust-perturbed model is examined on the basis of the bilinear and second-order approximations of the system nonlinearity . The bilinear and second-order approximations of the system nonlinearity allow substantial simplifications for the numerical implementation and preserve some of the properties of the original stochastically perturbed model. Most notably, this paper reveals that the Brownian motion process is accurate to model and study the effect of dust perturbation on the orbiting particle. In addition, analytical findings are supported with finite difference method-based numerical simulations.

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