The modified KdV equation for a nonlinear evolution problem with perturbation technique

Self Cite

The dynamic analysis of complex mechanical systems often requires the application of advanced mathematical techniques. In this study, we present a variation iteration-based solution for a pendulum system coupled with a rolling wheel, forming a combined translational and rotational system. Furthermore, the Lagrange multiplier is calculated using the Elzaki transform. The system under investigation consists of a pendulum attached to a wheel that rolls without slipping on a horizontal surface. The coupled motion of the pendulum and the rolling wheel creates a complex system with both translational and rotational degrees of freedom. To solve the governing equations of motion, we employ the variation iteration method, a powerful numerical technique that combines the advantages of both variational principles and iteration schemes. The Lagrange multiplier plays a crucial role in incorporating the constraints of the system into the equations of motion. In this study, we determine the Lagrange multiplier using the Elzaki transform, which provides an effective means to calculate Lagrange multipliers for constrained mechanical systems. The proposed solution technique is applied to analyse the dynamics of a pendulum with a rolling wheel system. The effects of various system parameters, such as the pendulum length, wheel radius and initial conditions, are investigated to understand their influence on the system dynamics. The results demonstrate the effectiveness of the variation iteration method combined with the Elzaki transform in capturing the complex behaviour of a combined translational and rotational system. The proposed approach serves as a valuable tool for analysing and understanding the dynamics of similar mechanical systems encountered in various engineering applications.

Section: Combined Translational and Rotational Systemmentioning

confidence: 99%

The Variational Iteration Method for a Pendulum with a Combined Translational and Rotational System

Amir,

Ashraf,

Haider

2024

Self Cite

“…The goal of the ideal homotopy equation is to minimize an objective function, which may have to do with deflection, stress, power usage, or any other relevant performance indicator. Engineers and researchers can improve the design and performance of MEMS devices by finding the ideal values for system parameters by solving the optimal homotopy equation [29][30][31][32][33][34]. Through the examination of different design configurations and optimization methodologies made possible by this methodology, the field of MEMS is ultimately advanced, and the functionality and efficiency of MEMSs in a variety of applications are improved.…”

Section: Introductionmentioning

confidence: 99%

The Homotopy Perturbation Method for Electrically Actuated Microbeams in Mems Systems Subjected to Van Der Waals Force and Multiwalled Carbon Nanotubes

Amir,

Haider,

Ashraf

2024

Self Cite

This paper presents a summary of a study that uses the Aboodh transformation and homotopy perturbation approach to analyze the behavior of electrically actuated microbeams in microelectromechanical systems that incorporate multiwalled carbon nanotubes and are subjected to the van der Waals force. All of the equations were transformed into linear form using the HPM approach. Electrically operated microbeams, a popular structure in MEMS, are the subject of this work. Because of their interaction with a nearby surface, these microbeams are sensitive to a variety of forces, such as the van der Waals force and body forces. MWCNTs are also incorporated into the MEMSs in this study because of their special mechanical, thermal, and electrical characteristics. The suggested method uses the HPM to model how electrically activated microbeams behave when MWCNTs and the van der Waals force are present. The nonlinear equations controlling the dynamics of the system can be roughly solved thanks to the HPM. The HPM offers a precise and effective way to analyze the microbeam’s reaction to these outside stimuli by converting the nonlinear equations into linear forms. The study’s findings shed important light on how electrically activated microbeams behave in MEMSs. A more thorough examination of the system’s performance is made possible with the addition of MWCNTs and the van der Waals force. With its ability to approximate solutions and characterize system behavior, the HPM is a potent instrument that improves comprehension of the physics at play and facilitates the design and optimization of MEMS devices. The aforementioned method’s accuracy is verified by comparing it with published data that directly aligns with Anjum et al.’s findings. We have faith in this method’s accuracy and its current application.

“…In several subfields within physics and engineering, the notion of soliton is an important concept. The solitons are used in modelling of optics, hydrodynamics, nuclear physics, biomechanics, plasma physics and many other fields [1][2][3]. While in most cases, the solution to nonlinear governing equations is associated with a soliton, this equation describes a wave that maintains its form across time [4].…”

Section: Introductionmentioning

confidence: 99%

Travelling Wave Solutions of the Non-Linear Wave Equations

Haider

Gul

Rahman

et al. 2023

Self Cite

This article focuses on the exact periodic solutions of nonlinear wave equations using the well-known Jacobi elliptic function expansion method. This method is more general than the hyperbolic tangent function expansion method. The periodic solutions are found using this method which contains both solitary wave and shock wave solutions. In this paper, the new results are computed using the closed-form solution including solitary or shock wave solutions which are obtained using Jacobi elliptic function method. The corresponding solitary or shock wave solutions are compared with the actual results. The results are visualised and the periodic behaviour of the solution is described in detail. The shock waves are found to break with time, whereas, solitary waves are found to be improved continuously with time.