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Article
Computer Science, Interdisciplinary Applications
Ji-Huan He et al.
Summary: This paper reveals the effectiveness of the homotopy perturbation method for strongly nonlinear oscillators. A generalized Duffing oscillator is adopted to illustrate the solving process step by step. A nonlinear frequency-amplitude relationship with a relative error of 0.91% is obtained, and the solution morphology is discussed along with the improvement of accuracy.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Interdisciplinary Applications
Chun-Hui He et al.
Summary: The simplest frequency formulation for nonlinear oscillators is presented and proved in this article, along with a suggested modification. Additionally, a fractal vibration in a porous medium is introduced and its low-frequency characteristics are elucidated using the frequency formulation. It is revealed that the inertia force in a fractal space is equivalent to a combination of the inertia force and the damping force in the traditional differential model.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2022)
Article
Engineering, Multidisciplinary
T. S. Amer et al.
Summary: This article investigates the motion of a spring pendulum system with the suspension point moving in an elliptic trajectory. Lagrange's equations are used to derive the equations of motion, and asymptotic solutions up to third approximation are obtained. Different resonance cases are classified and steady-state solutions are verified.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Engineering, Mechanical
F. M. El-Sabaa et al.
Summary: This study investigated the planar dynamical motion of a double-rigid-body pendulum near resonance, where its pivot point moves in a Lissajous curve. The controlling equations of motion were constructed using Lagrange equations. New approximate analytical solutions were obtained using the multiple scales method, and their accuracy was confirmed through comparison with numerical solutions. The stability of these solutions was analyzed, and the results provide a generalization of previous works.
JOURNAL OF VIBRATION ENGINEERING & TECHNOLOGIES
(2022)
Article
Engineering, Mechanical
T. S. Amer et al.
Summary: This article studies the motion of a three-degree-of-freedom dynamical system consisting of a triple rigid body pendulum under the influence of three harmonically external moments. The governing equations of motion are obtained using Lagrange's equations based on the generalized coordinates of the system. Approximate solutions up to the third approximation are obtained using the multiple scales approach, and the solvability conditions are determined by eliminating secular terms. The importance of this work lies in its applications in engineering vibrational control.
NONLINEAR DYNAMICS
(2022)
Article
Multidisciplinary Sciences
Galal M. Moatimid et al.
Summary: This study focuses on the motion of a simple pendulum connected to a wheel and a lightweight spring. A complicated nonlinear ordinary differential equation is obtained from the fundamental equation of motion under restricted conditions. The combination of the Homotopy perturbation method (HPM) and Laplace transforms is used to obtain an approximate regular solution. The validity of the solution is verified using the fourth-order Runge-Kutta method (RK4). The influence of parameters on the motion behavior is displayed through the graphs of the obtained solutions over time and their related phase plane plots.
SCIENTIFIC REPORTS
(2022)
Article
Mechanics
T. S. Amer et al.
Summary: The main goal of this study is to investigate the motion of a damped two degrees-of-freedom auto-parametric dynamical system. Lagrange's equations are used to derive the equations of motion, and the method of multiple scales is employed to obtain approximate solutions. The study examines two cases of resonance and explores the solvability conditions and modulation equations. The stability of fixed points is estimated, and plots are used to depict the motion of the system. Routh-Hurwitz conditions are used to analyze stability zones. The outcomes of this study are considered novel and original, with applications in various disciplines.
ARCHIVE OF APPLIED MECHANICS
(2022)
Article
Engineering, Multidisciplinary
S. A. Abdelhfeez et al.
Summary: This article investigates the planar motion of a novel three degrees of freedom dynamic system and derives analytic solutions using the multiple scales method. The accuracy of the solutions is demonstrated by comparing them with numerical results. Resonance cases, modulation equations, and steady-state solutions are characterized and analyzed. The significance of the studied dynamical system in engineering and physics, as well as the impact of different parameters on dynamic behavior, is discussed.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Acoustics
Chun-Hui He et al.
Summary: This work focuses on vibration alleviation and energy harvesting in a dynamical system of a spring-pendulum. The structure of the pendulum is modified using an independent electromagnetic harvesting system, and efforts have been made to achieve both energy harvesting and vibration mitigation efficacy.
JOURNAL OF LOW FREQUENCY NOISE VIBRATION AND ACTIVE CONTROL
(2022)
Review
Acoustics
Chun-Hui He et al.
Summary: The article discusses the application of the homotopy perturbation method in nonlinear vibration theory, focusing on the treatment of non-conservative oscillators, including models like Duffing oscillators and fractional oscillators. The authors heuristically explain the method's specific applications through examples, demonstrating its potential in solving practical problems.
JOURNAL OF LOW FREQUENCY NOISE VIBRATION AND ACTIVE CONTROL
(2022)
Article
Multidisciplinary Sciences
A. I. Ismail
Summary: This paper represents a pendulum model as a mechanical system with a simple pendulum suspended on a spring, exploring the motion of the pendulum and spring. Equations of motion are obtained using Lagrange's equation, and the influence of system parameters on motion is studied using a computerized program, showing the accuracy of the methods through graphical representations.
SCIENTIFIC REPORTS
(2021)
Article
Multidisciplinary Sciences
T. S. Amer et al.
Summary: This article investigates a nonlinear dynamical system with three degrees of freedom involving multiple pendulums, using perturbation methods to obtain approximate solutions and examining stability through nonlinear analysis approach. Solution diagrams and resonance curves illustrate the impact of parameters on the solutions.
KUWAIT JOURNAL OF SCIENCE
(2021)
Article
Materials Science, Multidisciplinary
T. S. Amer et al.
Summary: This article investigates the motion of a double nonlinear damped spring pendulum system with three degrees of freedom, using Lagrange's equations and multiple scales technique. The study categorizes resonance cases, examines steady-state solutions for stability, and analyzes nonlinear stability to identify stability and instability regions. The obtained high accuracy solutions have a significant impact on the system's dynamic behavior.
RESULTS IN PHYSICS
(2021)
Article
Mathematics, Applied
Ji-Huan He et al.
Summary: This paper investigates the dynamical properties of a pendulum attached to a rotating rigid frame with a constant angular velocity. The analytic solution of the governing nonlinear differential equation of motion is obtained using He's homotopy perturbation method. The obtained solution is verified for high accuracy using the fourth-order Runge-Kutta method and He's frequency formulation. The stability condition of the motion is examined and discussed, and the impact of different parameters on the dynamical motion is revealed through graphical representations of time histories.
Article
Engineering, Aerospace
A. I. Ismail
INTERNATIONAL JOURNAL OF AEROSPACE ENGINEERING
(2020)
Article
Materials Science, Multidisciplinary
F. M. El-Sabaa et al.
RESULTS IN PHYSICS
(2020)
Article
Acoustics
Zhong-Fu Ren et al.
JOURNAL OF LOW FREQUENCY NOISE VIBRATION AND ACTIVE CONTROL
(2019)
Article
Acoustics
Mateusz Wojna et al.
JOURNAL OF SOUND AND VIBRATION
(2018)
Article
Materials Science, Multidisciplinary
W. S. Amer et al.
RESULTS IN PHYSICS
(2018)
Article
Physics, Mathematical
T. S. Amer et al.
ADVANCES IN MATHEMATICAL PHYSICS
(2016)
Proceedings Paper
Mechanics
Jan Awrejcewicz et al.
IUTAM SYMPOSIUM ANALYTICAL METHODS IN NONLINEAR DYNAMICS
(2016)
Article
Mathematics, Applied
Jan Awrejcewicz et al.
DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
(2013)
Article
Mathematics, Interdisciplinary Applications
Roman Starosta et al.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2011)
Article
Physics, Multidisciplinary
M. Gitterman
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
(2010)
Review
Mathematics, Interdisciplinary Applications
Jan Awrejcewicz et al.
INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
(2008)
Article
Engineering, Mechanical
Jan Awrejcewicz et al.
NONLINEAR DYNAMICS
(2007)
Article
Mechanics
J Awrejcewicz et al.
ARCHIVE OF APPLIED MECHANICS
(2005)
