Nonlocal Singular Problems and Applications to MEMS (original) (raw)
Nonlocal dynamic problems with singular nonlinearities and applications to MEMS
We establish existence and regularity results for a time dependent fourth order integro-differential equation with a possibly singular nonlinearity which has applications in designing MicroElectroMechanicalSystems. The key ingredient in our approach, besides basic theory of hyperbolic equations in Hilbert spaces, exploit Near Operators Theory introduced by Campanato.
Dynamics of electrostatic microelectromechanical systems actuators
Journal of Mathematical Physics, 2012
Electrostatic actuators are simple but important switching devices for MEMS applications. Due to the difficulties associated with the electrostatic nonlinearity, precise mathematical description is often hard to obtain for the dynamics of these actuators. Here we present two sharp theorems concerning the dynamics of an undamped electrostatic actuator with one-degree of freedom, subject to linear and nonlinear elastic forces, respectively. We prove that both situations are characterized by the onset of one-stagnation-point periodic response below a well-defined pull-in voltage and a finite-time touch-down or collapse of the actuator above this pull-in voltage. In the linear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate of the movable electrode may all be determined explicitly, following the recent work of Leus and Elata based on numerics. Furthermore, in the nonlinear-force situation, the stagnation level, pull-in voltage, and pull-in coordinate may be described completely in terms of the electrostatic and mechanical parameters of the model so that they approach those in the linear-force situation monotonically in the zero nonlinear-force limit.
GLOBAL EXISTENCE FOR NONLOCAL MEMS PROBLEMS
We prove existence of solutions for a nonlocal equation arising from the mathematical modeling of MicroElectroMechanicalSystems. The existence result, obtained within a suitable Implicit Function Theorem framework, is established under rather general boundary conditions and for bounded domains whose diameter is fairly small.
Dynamics of a Canonical Electrostatic MEMS/NEMS System
Journal of Dynamics and Differential Equations, 2008
The mass-spring model of electrostatically actuated microelectromechanical systems (MEMS) or nanoelectromechanical systems (NEMS) is pervasive in the MEMS and NEMS literature. Nonetheless a rigorous analysis of this model does not exist. Here periodic solutions of the canonical mass-spring model in the viscosity dominated time harmonic regime are studied. Ranges of the dimensionless average applied voltage and dimensionless frequency of voltage variation are delineated such that periodic solutions exist. Parameter ranges where such solutions fail to exist are identified; this provides a dynamic analog to the static "pull-in" instability well known to MEMS/NEMS researchers.
Mathematical Methods in the Applied Sciences, 2021
We study periodic solutions of a one‐degree of freedom microelectromechanical system (MEMS) with a parallel‐plate capacitor under T‐periodic electrostatic forcing. We obtain analytical results concerning the existence of T‐periodic solutions of the problem in the case of arbitrary nonlinear restoring force, as well as when the moving plate is attached to a spring fabricated using graphene. We then demonstrate numerically on a T‐periodic Poincaré map of the flow that these solutions are generally locally stable with large “islands” of initial conditions around them, within which the pull‐in stability is completely avoided. We also demonstrate graphically on the Poincaré map that stable periodic solutions with higher period nT, n > 1 also exist, for wide parameter ranges, with large “islands” of bounded motion around them, within which all initial conditions avoid the pull‐in instability, thus helping us significantly increase the domain of safe operation of these MEMS models.
A Nonlinear Reduced-Order Model for Electrostatic MEMS
Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2003
We present an analytical approach and a reduced-order model (macromodel) to investigate the behavior of electrically actuated microbeam-based MEMS. The macromodel provides an effective and accurate design tool for this class of MEMS devices. The macromodel is obtained by discretizing the distributed-parameter system using a Galerkin procedure into a finite-degree-of-freedom system consisting of ordinary-differential equations in time. The macromodel accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and the nonlinear electric forces generated by the capacitors. A new technique is developed to represent the electric force in the equations of motion. The new approach allows the use of few linear-undamped mode shapes of a microbeam in its straight position as basis functions in a Galerkin procedure. The macromodel is validated by comparing its results with experimental results and finite-element solutions available in the literature. Our approach shows attractive features compared to finite-element softwares used in the literature. It is robust over the whole device operation range up to the instability limit of the device (i.e., pull-in). Moreover, it has low computational cost and allows for an easier understanding of the influence of the various design parameters. As a result, it can be of significant benefit to the development of MEMS design software.
Nonlinear non-local elliptic equation modelling electrostatic actuation
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2007
The nonlinear non-local elliptic equation governing the deflection of charged plates in electrostatic actuators is studied under the pinned and the clamped boundary conditions. Results concerning the existence, construction and approximation, and behaviour of classical and singular solutions with respect to the variation of physical parameters of the equation in various situations are presented.
On a nonlocal parabolic problem arising in electrostatic MEMS control
Discrete and Continuous Dynamical Systems, 2012
We consider a nonlocal parabolic equation associated with Dirichlet boundary and initial conditions arising in MEMS control. First, we investigate the structure of the associated steady-state problem for a general star-shaped domain. Then we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we study under which circumstances the solution of the time-dependent problem is global-in-time or quenches in finite time.
Singular Perturbation Analysis of a Regularized MEMS Model
SIAM Journal on Applied Dynamical Systems
Micro-Electro Mechanical Systems (MEMS) are defined as very small structures that combine electrical and mechanical components on a common substrate. Here, the electrostatic-elastic case is considered, where an elastic membrane is allowed to deflect above a ground plate under the action of an electric potential, whose strength is proportional to a parameter λ. Such devices are commonly described by a parabolic partial differential equation that contains a singular nonlinear source term. The singularity in that term corresponds to the so-called "touchdown" phenomenon, where the membrane establishes contact with the ground plate. Touchdown is known to imply the non-existence of steady-state solutions and blow-up of solutions in finite time. We study a recently proposed extension of that canonical model, where such singularities are avoided due to the introduction of a regularizing term involving a small "regularization" parameter ε. Methods from dynamical systems and geometric singular perturbation theory, in particular the desingularization technique known as "blow-up", allow for a precise description of steady-state solutions of the regularized model, as well as for a detailed resolution of the resulting bifurcation diagram. The interplay between the two principal model parameters ε and λ is emphasized; in particular, the focus is on the singular limit as both parameters tend to zero.
Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS
Courant Lecture Notes, 2010
Part 1. Second-Order Equations Modeling Stationary MEMS Chapter 2. Estimates for the Pull-In Voltage 2.1. Existence of the Pull-In Voltage 2.2. Lower Estimates for the Pull-In Voltage 2.3. Upper Bounds for the Pull-In Voltage 2.4. Numerics for the Pull-In Voltage Further Comments Chapter 3. The Branch of Stable Solutions 3.1. Spectral Properties of Minimal Solutions 3.2. Energy Estimates and Regularity of Solutions 3.3. Linear Instability and Compactness 3.4. Effect of an Advection on the Minimal Branch Further Comments Chapter 4. Estimates for the Pull-In Distance 4.1. Lower Estimates on the Pull-In Distance in General Domains 4.2. Upper Estimate for the Pull-In Distance in General Domains 4.3. Upper Bounds for the Pull-In Distance in the Radial Case 4.4. Effect of Power-Law Profiles on Pull-In Distances 4.5. Asymptotic Behavior of Stable Solutions near the Pull-In Voltage Further Comments Chapter 5. The First Branch of Unstable Solutions 5.1. Existence of Nonminimal Solutions 5.2. Blowup Analysis for Noncompact Sequences of Solutions 5.3. Compactness along the First Branch of Unstable Solutions 5.4. Second Bifurcation Point Further Comments vii viii CONTENTS Chapter 6. Description of the Global Set of Solutions 6.1. Compactness along the Unstable Branches 6.2. Quenching Branch of Solutions in General Domains 6.3. Uniqueness of Solutions for Small Voltage in Star-Shaped Domains 6.4. One-Dimensional Problem Further Comments Chapter 7. Power-Law Profiles on Symmetric Domains 7.1. A One-Dimensional Sobolev Inequality 7.2. Monotonicity Formula and Applications 7.3. Compactness of Higher Branches of Radial Solutions 7.4. Two-Dimensional MEMS on Symmetric Domains Further Comments Part 2. Parabolic Equations Modeling MEMS Dynamic Deflections Chapter 8. Different Modes of Dynamic Deflection 8.1. Global Convergence versus Quenching 8.2. Quenching Points and the Zero Set of the Profile 8.3. The Quenching Set on Convex Domains Further Comments Chapter 9. Estimates on Quenching Times 9.1. Comparison Results for Quenching Times 9.2. General Asymptotic Estimates for Quenching Time 9.3. Upper Estimates for Quenching Times for all > 9.4. Quenching Time Estimates in Low Dimension Further Comments Chapter 10. Refined Profile of Solutions at Quenching Time 10.1. Integral and Gradient Estimates for Quenching Solutions 10.2. Refined Quenching Profile 10.3. Refined Quenching Profiles in Dimension N D 1 10.4. Refined Quenching Profiles in the Radially Symmetric Case 10.5. More on the Location of Quenching Points Further Comments