Application of Nonlocal Theory in Dynamic Pull-In Analysis of Electrostatically Actuated Micro and Nano Beams

2011 ◽  
Author(s):  
M. T. Ahmadian ◽  
Abdolreza Pasharavesh ◽  
Ali Fallah
Keyword(s):  
Differential Equation ◽  
Differential Quadrature Method ◽  
Classical Mechanics ◽  
Nonlocal Theory ◽  
Equation Of Motion ◽  
Nonlocal Effect ◽  
Size Change ◽  
Restoring Force ◽  
Electrostatically Actuated ◽  
The Difference

One of the most important phenomena related to electrically actuated micro and nano electromechanical systems (MEMS\NEMS) is dynamic pull-in instability which occurs when the electrical attraction and beam inertia forces are more than elastic restoring force of the beam. According to failure of classical mechanics constitutive equations in prediction of dynamic behavior of small size systems, nonlocal theory is implemented here to analyze the dynamic pull-in behavior. Equation of motion of an electrostatically actuated micro to nano scale doubly clamped beam is rewritten using differential form of nonlocal theory constitutive equation. To analyze the nonlocal effect equation of motion is nondimentionalized. Governing partial differential equation is transformed to an ordinary differential equation using the Galerkin decomposition method and then is solved implementing differential quadrature method (DQM). Change of dynamic pull-in voltage with respect to size change is investigated. Results indicate as the beam length decreases dynamic pull-in voltage increases due to nonlocal effect and the difference with clasical mechanics results is up to 20% for nano beams.

Boundary Value Problem Method to Investigate Superharmonic Resonance of MEMS Resonators

2017 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Christian Reyes ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Equation Of Motion ◽  
The Other ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Amplitude Frequency Response ◽  
Problem Method ◽  
Mems Resonators

This paper deals with two different methods to analyze the amplitude frequency response of an electrostatically actuated micro resonator. The methods used in this paper are the method of multiple scales, which is an analytical method with one mode of vibration. The other method is based on system of odes which is derived using the partial differential equation of motion, as well as the boundary conditions. This system is then solved using a built in matlab function known as BVP4C. Results are then shown comparing the two methods, under a variety of parameters, including the influence of damping, voltage, and fringe.

Vibration Analysis of Axially Compressed Nanobeams and its Critical Pressure Using a New Nonlocal Stress Theory

2011 ◽  
Vol 105-107 ◽  
pp. 1788-1792 ◽  
Author(s):  
Cheng Li ◽  
C.W. Lim ◽  
Zhong Kui Zhu
Keyword(s):  
Differential Equation ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Separation Of Variables ◽  
Classical Mechanics ◽  
Bending Moment ◽  
Nonlocal Elasticity Theory ◽  
Equation Of Motion ◽  
Basic Element ◽  
Mode Shapes

The transverse vibration of a nanobeam subject to initial axial compressive forces based on nonlocal elasticity theory is investigated. The effects of a small nanoscale parameter at molecular level unavailable in classical mechanics theory are presented and analyzed. Explicit solutions for natural frequency, vibration mode shapes are derived through two different methods: separation of variables and multiple scales. The respective numerical solutions are in close agreement. Validity of the models and approaches presented in the work are verified. Unlike the previous studies for a nonlocal nanostructure, this paper adopts the effective nonlocal bending moment instead of the pure traditional nonlocal bending moment. The analysis yields an infinite-order differential equation of motion which governs the vibrational behaviors. For practical analysis and as examples, an eight-order governing differential equation of motion is solved and the results are discussed. The paper presents a complete nonlocal nanobeam model and the results may be helpful for the application and design of various nano-electro-mechanical devices, e.g. nano-drivers, nano-oscillators, nano-sensors, etc., where a nanobeam acts as a basic element.

Equivalent Linearization for Hysteretic Systems Under Random Excitation

1980 ◽  
Vol 47 (1) ◽  
pp. 150-154 ◽  
Author(s):  
Y. K. Wen
Keyword(s):  
Differential Equation ◽  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Closed Form ◽  
Present Method ◽  
Equation Of Motion ◽  
Random Excitation ◽  
Equivalent Linearization ◽  
Restoring Force ◽  
Hysteretic Systems

A method of equivalent linearization for smooth hysteretic systems under random excitation is proposed. The hysteretic restoring force is modeled by a nonlinear differential equation and the equation of motion is linearized directly in closed form without recourse to Krylov-Bogoliubov technique. Compared with previously proposed similar methods, the formulation of the present method is versatile and considerably simpler. The accuracy of this method is verified against Monte-Carlo simulation for all response levels. It has a great potential in the analysis of multidegree-of-freedom and degrading systems.

Dynamic Stability of Cracked Viscoelastic Rectangular Plate Subjected to Tangential Follower Force

2008 ◽  
Vol 75 (6) ◽  
Author(s):  
Zhong-min Wang ◽  
Yan Wang ◽  
Yin-feng Zhou
Keyword(s):  
Differential Equation ◽  
Plate Theory ◽  
Differential Quadrature Method ◽  
Viscoelastic Plate ◽  
Equation Of Motion ◽  
Follower Force ◽  
Inverse Transformation ◽  
Complex Eigenvalue ◽  
The Time Domain ◽  
Viscoelastic Rectangular Plate

Based on the thin plate theory and the two-dimensional viscoelastic differential type constitutive relation, the differential equation of motion of a viscoelastic plate containing an all-over part-through crack and subjected to uniformly distributed tangential follower force is established in Laplace domain. Then, by performing the Laplace inverse transformation, the differential equation of motion of the plate in the time domain is obtained. The expression of the additional rotation induced by the crack is given. The complex eigenvalue equations of the cracked viscoelastic plate subjected to uniformly distributed tangential follower force are obtained by the differential quadrature method, and the δ method is adopted at the crack continuity conditions. The general eigenvalue equations of the cracked viscoelastic plate subjected to uniformly distributed tangential follower force under the different boundary conditions are calculated. The transverse vibration characteristics, type of instability, and corresponding critical loads of the cracked viscoelastic plate subjected to uniformly distributed tangential follower force are analyzed.

scholarly journals Differential Quadrature and Differential Transformation Methods in Buckling Analysis of Nanobeams

2019 ◽  
Vol 6 (1) ◽  
pp. 68-76 ◽  
Author(s):  
Subrat Kumar Jena ◽  
S. Chakraverty
Keyword(s):  
Boundary Conditions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Transformation Method ◽  
Nonlocal Theory ◽  
Buckling Analysis ◽  
Differential Quadrature ◽  
Load Parameter ◽  
Computationally Efficient ◽  
Differential Transformation

AbstractIn this paper, two computationally efficient techniques viz. Differential Quadrature Method (DQM) and Differential Transformation Method (DTM) have been used for buckling analysis of Euler-Bernoulli nanobeam incorporation with the nonlocal theory of Eringen. Complete procedures of both the methods along with their mathematical formulations are discussed, and MATLAB codes have been developed for both the methods to handle the boundary conditions. Various classical boundary conditions such as SS, CS, and CC have been considered for investigation. A comparative study for the convergence of DQM and DTM approaches are carried out, and the obtained results are also illustrated to demonstrate the effects of the nonlocal parameter, aspect ratio (L/h) and the boundary condition on the critical buckling load parameter.

The Numerical Methods of Peridynamics Theory Used in Failure Analysis of Materials

2014 ◽  
Vol 1030-1032 ◽  
pp. 223-227
Author(s):  
Lin Fan ◽  
Song Rong Qian ◽  
Teng Fei Ma
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Numerical Integration ◽  
Integration Method ◽  
Equation Of Motion ◽  
Numerical Integration Method ◽  
Volume Integral ◽  
Volume Integral Equation ◽  
Method Of Molecular Dynamics ◽  
Classic Theory

In order to analysis the force situation of the material which is discontinuity,we can used the new theory called peridynamics to slove it.Peridynamics theory is a new method of molecular dynamics that develops very quickly.Peridynamics theory used the volume integral equation to constructed the model,used the volume integral equation to calculated the PD force in the horizon.So It doesn’t need to assumed the material’s continuity which must assumed that use partial differential equation to formulates the equation of motion. Destruction and the expend of crack which have been included in the peridynamics’ equation of motion.Do not need other additional conditions.In this paper,we introduce the peridynamics theory modeling method and introduce the relations between peridynamics and classic theory of mechanics.We also introduce the numerical integration method of peridynamics.Finally implementation the numerical integration in prototype microelastic brittle material.Through these work to show the advantage of peridynamics to analysis the force situation of the material.

Study on Silo Pressure by Consider Storage Materials Density Changed along the Height

2012 ◽  
Vol 517 ◽  
pp. 797-800
Author(s):  
Zhi Yong Yang ◽  
Shun Hu Liu ◽  
Song Zhao ◽  
Jun Hu ◽  
Zeng Chan Lu
Keyword(s):  
Differential Equation ◽  
Density Stratification ◽  
Maximum Pressure ◽  
Actual Case ◽  
Storage Materials ◽  
Pressure Calculation ◽  
The Difference

The difference existed between results of silos pressure calculation and the actual case, because the influence of density stratification was not taken into consideration. The aim of this paper was to obtain silo pressure calculating formula by consider of storage materials density stratified. To this end, we assume that the density was continuous changed along the height and differential equation of the storage materials pressure was established. By compared the results calculated from the equation with the results calculated from the code, it is found that the maximum pressure increased. The results showed density stratified is an import factor for silo pressure calculation and the equation obtained in this paper can provide references for the theory of silo pressure calculation.

scholarly journals Matching loci surveyed to questions asked in phylogeography

2016 ◽  
Vol 283 (1826) ◽  
pp. 20152340 ◽  
Author(s):  
Chih-Ming Hung ◽  
Sergei V. Drovetski ◽  
Robert M. Zink
Keyword(s):  
Population Diversity ◽  
Single Locus ◽  
Population Divergence ◽  
Summary Statistics ◽  
Recent Past ◽  
Size Change ◽  
The Caucasus ◽  
Nuclear Loci ◽  
The Difference ◽  
Coalescent Time

Although mitochondrial DNA (mtDNA) has long been used for assessing genetic variation within and between populations, its workhorse role in phylogeography has been criticized owing to its single-locus nature. The only choice for testing mtDNA results is to survey nuclear loci, which brings into contrast the difference in locus effective size and coalescence times. Thus, it remains unclear how erroneous mtDNA-based estimates of species history might be, especially for evolutionary events in the recent past. To test the robustness of mtDNA and nuclear sequences in phylogeography, we provide one of the largest paired comparisons of summary statistics and demographic parameters estimated from mitochondrial, five Z-linked and 10 autosomal genes of 30 avian species co-distributed in the Caucasus and Europe. The results suggest that mtDNA is robust in estimating inter-population divergence but not in intra-population diversity, which is sensitive to population size change. Here, we provide empirical evidence showing that mtDNA was more likely to detect population divergence than any other single locus owing to its smaller N e and thus faster coalescent time. Therefore, at least in birds, numerous studies that have based their inferences of phylogeographic patterns solely on mtDNA should not be readily dismissed.

A reduction theory of second order meromorphic differential equations, I

1987 ◽  
Vol 101 (2) ◽  
pp. 323-342
Author(s):  
W. B. Jurkat ◽  
H. J. Zwiesler
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Standard Form ◽  
Reduction Theory ◽  
Fundamental Solution Matrix ◽  
Equivalent Equation ◽  
The Difference ◽  
Solution Matrix ◽  
Meromorphic Differential Equation ◽  
Meromorphic Differential

In this article we investigate the meromorphic differential equation X′(z) = A(z) X(z), often abbreviated by [A], where A(z) is a matrix (all matrices we consider have dimensions 2 × 2) meromorphic at infinity, i.e. holomorphic in a punctured neighbourhood of infinity with at most a pole there. Moreover, X(z) denotes a fundamental solution matrix. Given a matrix T(z) which together with its inverse is meromorphic at infinity (a meromorphic transformation), then the function Y(z) = T−1(z) X(z) solves the differential equation [B] with B = T−1AT − T−1T [1,5]. This introduces an equivalence relation among meromorphic differential equations and leads to the question of finding a simple representative for each equivalence class, which, for example, is of importance for further function-theoretic examinations of the solutions. The first major achievement in this direction is marked by Birkhoff's reduction which shows that it is always possible to obtain an equivalent equation [B] where B(z) is holomorphic in ℂ ¬ {0} (throughout this article A ¬ B denotes the difference of these sets) with at most a singularity of the first kind at 0 [1, 2, 5, 6]. We call this the standard form. The question of how many further simplifications can be made will be answered in the framework of our reduction theory. For this purpose we introduce the notion of a normalized standard equation [A] (NSE) which is defined by the following conditions:(i) , where r ∈ ℕ and Ak are constant matrices, (notation: )(ii) A(z) has trace tr for some c ∈ ℂ,(iii) Ar−1 has different eigenvalues,(iv) the eigenvalues of A−1 are either incongruent modulo 1 or equal,(v) if A−1 = μI, then Ar−1 is diagonal,(vi) Ar−1 and A−1 are triangular in opposite ways,(vii) a12(z) is monic (leading coefficient equals 1) unless a12 ≡ 0; furthermore a21(z) is monic in case that a12 ≡ 0 but a21 ≢ 0.

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