Dynamic Analysis of Nonlinear Oscillator Equation Arising in Double-Sided Driven Clamped Microbeam-Based Electromechanical Resonator

2012 ◽  
Vol 67 (8-9) ◽  
pp. 435-440 ◽  
Author(s):  
Yasir Khan ◽  
Mehdi Akbarzade
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Variational Approach ◽  
Nonlinear Oscillator ◽  
Analytical Form ◽  
Angular Frequency ◽  
Initial Amplitude ◽  
Hamiltonian Approach ◽  
The Galerkin Method ◽  
The Relationship

In this paper, three different analytical methods have been successfully used to study a nonlinear oscillator equation arising in the microbeam-based electromechanical resonator. These methods are: variational approach, Hamiltonian approach, and amplitude-frequency formulation. The governing equation is based on the Euler-Bernoulli hypothesis and the partial differential equation (PDE) is simplified into an ordinary differential equartion (ODE) by using the Galerkin method. A frequency analysis is carried out, and the relationship between the angular frequency and the initial amplitude is obtained in closed analytical form. A comparison of the present solutions is made with the existing solutions and excellent agreement is noted.

scholarly journals On the Integration of the Canonic Equations and the Principle of Duality

1909 ◽  
Vol 28 ◽  
pp. 6-13
Author(s):  
G. D. C. Stokes
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Standard Form ◽  
Analytical Form ◽  
System Of Differential Equations ◽  
Analytical Treatment ◽  
Change Of Variables ◽  
Principle Of Duality ◽  
The Subject ◽  
Two Bodies

The object of this paper is to essay an analytical statement of the reduction of the integration of a canonic system of differential equations (into which time does not enter explicitly) to that of the partial differential equation of Jacobi and Hamilton; and to illustrate the principle of duality by an outline of the solution for the problem of two bodies both by the standard form of the equation referred to and by the analogous form which that principle involves. Most statements of the reduction are verifications and somewhat obscure the symmetry of the canonic form. The shortest procedure, of course, is by means of the well known theorem of Jacobi, and this verificatory method is followed by Tisserand, Charlier and Appell. Poincaré gives a proof depending on a simple form given by him to the conditions for a canonical change of variables, but again the statement lacks analytical form. The essentials of this proof will be given here, but in an entirely different way. An analytical treatment of the subject has been given by Professor L. Becker in his class lectures at Glasgow, but it has not been published.

Analysis for transverse shake vibration of high-rise buildings based on partial differential equation

2021 ◽  
Vol 64 (1) ◽  
pp. 103-111
Author(s):  
Li Li ◽  
◽  
Raquel Martínez ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Energy Consumption ◽  
Vibration Analysis ◽  
High Energy ◽  
Lateral Vibration ◽  
Partial Differential ◽  
High Rise ◽  
The Galerkin Method ◽  
High Energy Consumption

In order to overcome the problems of long analysis time, low accuracy and high energy consumption in traditional lateral vibration analysis methods of high-rise buildings, a new method of lateral vibration analysis of high-rise buildings based on partial differential equation is proposed. Based on Hamilton's principle, the partial differential equation of lateral vibration of high-rise buildings is established, and the Galerkin method is used to solve the partial differential equation until the discrete solution is obtained, and then the displacement response of high-rise buildings under different excitation frequencies is obtained. The experimental results show that compared with the traditional method, the proposed method has the advantages of short calculation time, high accuracy and low energy consumption.

Modelling MEMS Resonators Past Pull-In

2008 ◽  
Author(s):  
Chandrika P. Vyasarayani ◽  
Eihab M. Abdel-Rahman ◽  
John McPhee ◽  
Stephen Birkett
Keyword(s):  
Differential Equation ◽  
Mathematical Model ◽  
Partial Differential Equation ◽  
Numerical Solution ◽  
Nonlinear Ordinary Differential Equations ◽  
Mems Resonators ◽  
Electrostatic Mems ◽  
Stationary Electrode ◽  
External Forces ◽  
The Galerkin Method

In this paper, we develop a mathematical model of an electrostatic MEMS beam undergoing impact with a stationary electrode subsequent to pull-in. We model the contact between the beam and the substrate using a nonlinear foundation of springs and dampers. The system partial differential equation (PDE) is converted into coupled nonlinear ordinary differential equations (ODEs) using the Galerkin method. A numerical solution is obtained by treating all nonlinear terms as external forces.

scholarly journals Investigation of dynamics of elastic element of vibration device

2021 ◽  
pp. 67-83
Author(s):  
П.А. Вельмисов ◽  
А.В. Анкилов ◽  
Г.А. Анкилов
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Complex Variable ◽  
Elastic Element ◽  
Numerical Experiments ◽  
Original Problem ◽  
Fourier Method ◽  
Partial Differential ◽  
The Galerkin Method ◽  
Liquid Flows

ва подхода к решению аэрогидродинамической части задачи, основанные на методах теории функций комплексного переменного и методе Фурье. В результате применения каждого подхода решение исходной задачи сведено к исследованию дифференциального уравнения с частными производными для деформации элемента, позволяющего изучать его динамику. На основе метода Галеркина произведены численные эксперименты для конкретных примеров механической системы, подтверждающие идентичность решений, найденных для каждого дифференциального уравнения с частными производными. The dynamics of an elastic element of a vibration device, simulated by a channel, inside which a stream of a liquid flows, is investigated. Two approaches to solving the aerohydrodynamic part of the problem, based on the methods of the theory of functions of a complex variable and the Fourier method, are given. As a result of applying each approach, the solution to the original problem is reduced to the study of a partial differential equation for the deformation of an element, which makes it possible to study its dynamics. Based on the Galerkin method, the numerical experiments were carried out for specific examples of mechanical system, confirming the identity of the solutions found for each partial differential equation.

scholarly journals Analysis of nonlinear oscillator arising in the microelectromechanical system by using the parameter expansion and equivalent linearization methods

2018 ◽  
Vol 7 (2) ◽  
pp. 597 ◽  
Author(s):  
D. V. Hieu ◽  
N. T. K. Thoa ◽  
L. Q. Duy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Nonlinear Ordinary Differential Equation ◽  
Equivalent Linearization ◽  
Parameter Expansion ◽  
Linearization Methods ◽  
The Galerkin Method ◽  
Micro Electromechanical System

In this paper, the nonlinear oscillator arising in the microbeam-based micro-electromechanical system (MEMS) is described. The motion equation of a microbeam is simplified into an ordinary differential equation by using the Galerkin method. The nonlinear ordinary differential equation is solved by using two methods including the Parameter-Expansion and Equivalent Linearization Methods. To verify the accuracy of the present methods, illustrative examples are provided and compared with other analytical, exact and numerical solutions.

scholarly journals Disparity Computation Through PDE and Data-Driven CeNN Technique

2021 ◽  
Vol 38 (4) ◽  
pp. 1051-1059
Author(s):  
Mahima Lakra ◽  
Sanjeev Kumar
Keyword(s):  
Neural Network ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Experimental Study ◽  
Variational Approach ◽  
Cellular Neural Network ◽  
Energy Functional ◽  
Data Driven ◽  
Pde Model ◽  
Distance Regularization

This paper proposes a variational approach by minimizing the energy functional to compute the disparity from a given pair of consecutive images. The partial differential equation (PDE) is modeled from the energy function to address the minimization problem. We incorporate a distance regularization term in the PDE model to preserve the boundaries' discontinuities. The proposed PDE is numerically solved by a cellular neural network (CeNN) algorithm. This CeNN based scheme is stable and consistent. The effectiveness of the proposed algorithm is shown by a detailed experimental study along with its superiority over some of the existing algorithms.

Integral operator methods for generalized axially symmetric potentials in (n+1) variables

1965 ◽  
Vol 5 (3) ◽  
pp. 331-348 ◽  
Author(s):  
R. P. Gilbert ◽  
H. C. Howard
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Integral Operator ◽  
Laplace Equation ◽  
Operator Method ◽  
Axially Symmetric ◽  
Partial Differential ◽  
Image Position ◽  
The Relationship ◽  
Generalized Axially Symmetric Potentials

In this paper we shall use the integral operator method of Bergman, B[1–6], to investigate solutions of the partial differential equation where s > −1. In particular, information concerning the growth, and location of singularities, of solutions of (1.1) will be obtained. Equations of the form (1.1) with s = 1, 2, arise from the (n+k+1)-dimensional Laplace equation Δn+k+1u = 0 in the “axially symmetric” coordinates x1, …xn, p where the relationship between cartesian and “axially symmetric” coordinates is given by

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