Electrostatically Actuated M/NEMS With Casimir Effect: Primary Resonance — Comparison Between Three Methods

2017 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Julio S. Beatriz ◽  
Christian Reyes
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Multiple Scales ◽  
Casimir Effect ◽  
Boundary Value ◽  
Primary Resonance ◽  
Initial Value ◽  
Electrostatically Actuated ◽  
Mems Resonator ◽  
Modes Of Vibration

This paper deals with electrostatically actuated micro- and nano-electromechanical systems (M/NEMS) cantilever resonator under electrostatic actuation. The model includes Casimir effect. Three different methods are used to investigate the primary resonance of the MEMS resonator. The first two methods are based on a Galerkin approach in which the initial value and boundary value problem, given by the partial differential equation (PDE) of motion and the initial and boundary conditions, is transformed into an initial value problem of one ordinary differential equation (ODE) or a system of ODEs depending on how many modes of vibrations are considered in the model. The first method used is the Method of Multiple Scales (MMS) which is an approximate analytical method used to solve the model using one mode of vibration. The second method referred to as Reduced Order Model (ROM) solves the model using two to five modes of vibration using numerical integration. The third method is different than the first two in the sense that the initial value and boundary value problem describing the MEMS resonator is transformed into a boundary value problem (BVP) by using finite differences tor time derivatives. For this Matlab built-in function bvp4c is used to solve the problem. This built-in function is used for two different versions of the same equation. One which involves Taylor expansions of the nonlinear terms, and the other which does not. Results between the methods are in agreement. Thus any of these methods can be used to accurately predict the behavior of the MEMS resonator. For the ROM, two equations are also used, one for Casimir and one for without. The influence of damping, Casimir, and voltage parameter are also shown.

Voltage Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonators: Comparison Between Boundary Value Problem Model and Reduced Order Model

2018 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value ◽  
Mode Shapes ◽  
Voltage Response ◽  
Order Model ◽  
Partial Differential ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

This paper deals with the voltage response of electrostatically actuated NEMS resonators at superharmonic resonance. In this work a comparison between Boundary Value Problem (BVP) model, and Reduced Order Model (ROM) is conducted for this type of resonance. BVP model is developed from the partial differential equation by replacing the time derivatives with finite differences. So, the partial differential equation is replaced by a sequence of boundary value problems, one for each step in time. Matlab’s function bvp4c is used to numerically integrate the BVPs. ROMs are based on Galerkin procedure and use the mode shapes of the resonator as a basis of functions. Therefore, the partial differential equation is replaced by a system of differential equations in time. The number of the equations in the system is equal to the number of mode shapes (or modes of vibration) used in the ROM. One mode of vibration ROM is solved using the method of multiple scales. Two modes of vibration ROM is numerically integrated using Matlab’s function ode15s in order to obtain time responses, and a continuation and bifurcation analysis is conducted using AUTO 07P. The effects of different nonlinearities in the system on the voltage response are reported. This work shows that BVP model is a valid method to predict the voltage response of a micro/nano cantilevers.

Influence of Timestep and Mesh Sizes on Numerical Simulations of Boundary Value Problem Model of Electrostatically Actuated MEMS Resonators

2019 ◽  
Author(s):  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Boundary Value Problem ◽  
Numerical Simulations ◽  
Multiple Scales ◽  
Boundary Value ◽  
Mesh Size ◽  
Problem Solver ◽  
Time Step ◽  
Problem Model ◽  
Electrostatically Actuated ◽  
Mems Resonators

Abstract This paper deals with the effects of mesh size and time step on the numerical simulations using bvp4c, a Matlab Boundary Value Problem solver, on the time response of electrostatically actuated MEMS resonators. These results are compared to the reduced order model as well as the method of multiple scales to test how accurate these results are at lower amplitudes. The refinement of mesh size leads to more accurate results to a certain extent, as it eventually reaches a convergence. It should be said that the larger the mesh size, the longer the calculations take. A similar result occurs with timestep size. The smaller the timestep the more accurate the results. However, the CPU time increases significantly. However, beyond a certain timestep, any smaller time step would not yield any noticeable differences. Thus it can be said convergence has been reached.

ANALYTIC SOLUTIONS TO THE OSCILLATORY BEHAVIOR AND PRIMARY RESONANCE OF ELECTROSTATICALLY ACTUATED MICROBRIDGES

2011 ◽  
Vol 11 (06) ◽  
pp. 1119-1137 ◽  
Author(s):  
M. MOJAHEDI ◽  
M. MOGHIMI ZAND ◽  
M. T. AHMADIAN ◽  
M. BABAEI
Keyword(s):  
Differential Equation ◽  
Natural Frequencies ◽  
Multiple Scales ◽  
Axial Stress ◽  
Nonlinear Partial Differential Equation ◽  
Primary Resonance ◽  
Dynamic Responses ◽  
Design Parameters ◽  
Electrostatically Actuated ◽  
Homotopy Perturbation

In this paper, the vibration and primary resonance of electrostatically actuated microbridges are investigated, with the effects of electrostatic actuation, axial stress, and mid-plane stretching considered. Galerkin's decomposition method is adopted to convert the governing nonlinear partial differential equation to a nonlinear ordinary differential equation. The homotopy perturbation method (a special case of homotopy analysis method) is then employed to find the analytic expressions for the natural frequencies of predeformed microbridges, by which the effects of the voltage, mid-plane stretching, axial force, and higher mode contribution on the natural frequencies are studied. The primary resonance of the microbridges is also investigated, where the microbridges are predeformed by a DC voltage and driven to vibrate by an AC harmonic voltage. The methods of homotopy perturbation and multiple scales are combined to find the analytic solution for the steady-state motion of the microbeam. In addition, the effects of the design parameters and damping on the dynamic responses are discussed. The results are shown to be in good agreement with the existing ones.

Casimir Effect on Simultaneous Resonances of Electrostatically Actuated MEMS Resonators

2015 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christian Reyes
Keyword(s):  
Natural Frequency ◽  
Bottom Electrode ◽  
Multiple Scales ◽  
Casimir Effect ◽  
Parallel Plates ◽  
Casimir Forces ◽  
Electrostatically Actuated ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Simultaneous Resonances

This paper deals with the influence of Casimir effect on MEMS resonator sensors under double electrostatic actuation and simultaneous resonances. The MEMS cantilever is between two parallel plates (electrodes) under soft AC double actuation. The AC voltage with the bottom electrode is of frequency near half natural frequency of the resonator and the AC voltage with the top electrode is of frequency near natural frequency of the resonator. The method of multiple scales is used to model the behavior of the system. In the model the damping, fringing, voltage, and Casimir forces are taken into consideration and the effects of these parameters on the frequency response are reported. Designing MEMS resonators for applications in fields such as automotive and biomedical can benefit from this work.

scholarly journals Second-order neutrosophic boundary-value problem

2021 ◽  
Author(s):  
Sandip Moi ◽  
Suvankar Biswas ◽  
Smita Pal(Sarkar)
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Differential Calculus ◽  
Boundary Value ◽  
Second Order ◽  
Numerical Examples ◽  
Different Types ◽  
First Time

AbstractIn this article, some properties of neutrosophic derivative and neutrosophic numbers have been presented. This properties have been used to develop the neutrosophic differential calculus. By considering different types of first- and second-order derivatives, different kind of systems of derivatives have been developed. This is the first time where a second-order neutrosophic boundary-value problem has been introduced with different types of first- and second-order derivatives. Some numerical examples have been examined to explain different systems of neutrosophic differential equation.

On a Singular Two-Point Boundary Value Problem for the Nonlinear mth-Order Differential Equation with Deviating Arguments

1997 ◽  
Vol 4 (6) ◽  
pp. 557-566
Author(s):  
B. Půža
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Vector Function ◽  
Unique Solvability ◽  
Order Differential Equation ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Point Boundary ◽  
Deviating Arguments ◽  
Measurable Functions

Abstract Sufficient conditions of solvability and unique solvability of the boundary value problem u (m)(t) = f(t, u(τ 11(t)), . . . , u(τ 1k (t)), . . . , u (m–1)(τ m1(t)), . . . . . . , u (m–1)(τ mk (t))), u(t) = 0, for t ∉ [a, b], u (i–1)(a) = 0 (i = 1, . . . , m – 1), u (m–1)(b) = 0, are established, where τ ij : [a, b] → R (i = 1, . . . , m; j = 1, . . . , k) are measurable functions and the vector function f : ]a, b[×Rkmn → Rn is measurable in the first and continuous in the last kmn arguments; moreover, this function may have nonintegrable singularities with respect to the first argument.

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