Mathematical modelling of the partial differential equations in microelectromechanical systems (MEMS) and its applications

Author:

Khan Muhammad Naveed1,Haider Jamil Abbas2,Wang Zhentao1,Gul Sana2,Lone Showkat Ahmad3,Elkotb Mohamed Abdelghany4

Affiliation:

1. School of Energy and Power Engineering, Jiangsu University, Zhenjiang, Jiangsu 212013, China

2. Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54600, Pakistan

3. Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia

4. Mechanical Engineering Department, College of Engineering, King Khalid University, Abha 61421, Saudi Arabia

Abstract

This paper presents a model of a doubly clamped electrically actuated microbeam, which is a commonly used structure in microelectromechanical systems (MEMS). The model is based on Euler–Bernoulli beam theory and includes the effect of electrostatic forces on the beam’s deflection. The electric field is modeled using the parallel plate capacitor model, and the deflection of the beam is calculated using the Galerkin method. The behavior of a microbeam subjected to the van der Waals force, which is a weak intermolecular force that arises from the interaction between the beam and a nearby surface. The microbeam is modeled using the Euler–Bernoulli beam theory, and the van der Waals force is modeled using the Lennard–Jones potential. At the last we study the model of MEMS based on multi-walled carbon nanotubes (MWCNTs). MWCNTs have unique mechanical, thermal, and electrical properties, which make them ideal for use in MEMS applications. The approximate solution of the developed models is found by using homotopy perturbation based Aboodh transformation (HPATM). HPATM is a mathematical method used to solve nonlinear equations by converting them into linear forms. This approach involves introducing a small parameter and applying perturbation theory to obtain a solution in a series form. The method’s accuracy is defined based on the existing literature because its solution matched the variation iteration-based Laplace method. Also, we compared its results with the finite difference method. The validity of the stability analysis is further established by examining the stability in the vicinity of the fixed points. Sketches are made of the phase portraits close to the equilibrium points.

Funder

the Deanship of Scientific Research at King Khalid University

Publisher

World Scientific Pub Co Pte Ltd

Subject

Condensed Matter Physics,Statistical and Nonlinear Physics

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