Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Amin Gholami ◽  
Davood D. Ganji ◽  
Hadi Rezazadeh ◽  
Waleed Adel ◽  
Ahmet Bekir
Keyword(s):  
Approximate Solution ◽  
Nonlinear Vibrations ◽  
Nonlinear Problems ◽  
Iterative Approach ◽  
Mathematical Pendulum ◽  
Science And Engineering ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear System ◽  
Strong Candidate ◽  
Iteration Technique

Abstract The paper deals with the application of a strong method called the modified Mickens iteration technique which is used for solving a strongly nonlinear system. The system describes the motion of a simple mathematical pendulum with a particle attached to it through a stretched wire. This model has great applications especially in the area of nonlinear vibrations and oscillation systems. The proposed method depends on determining the frequency and amplitude of the system through the modified Mickens iterative approach which is a modification of the regular Mickens approach. The preliminaries of the proposed technique are present and the application to the model is discussed. The method depends on the Mickens iteration approach which transforms the considered equation into a linear form and then is solving this equation result in the approximate solution. Some examples are given to validate and illustrate the effectiveness and convenience of the method. These results are compared with other relative techniques from the literature in terms of finding the frequency of the two examined models. The method produces more accurate results when compared to these methods and is considered a strong candidate for solving other nonlinear problems with applications in science and engineering.

An optimal iteration method with application to the Thomas-Fermi equation

Open Physics ◽  
2011 ◽  
Vol 9 (3) ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Approximate Solution ◽  
Approximate Method ◽  
Iteration Method ◽  
Nonlinear Problems ◽  
Analytical Approximate Solution ◽  
Thomas Fermi ◽  
Strongly Nonlinear ◽  
Good Agreement

AbstractThe aim of this paper is to introduce a new approximate method, namely the Optimal Parametric Iteration Method (OPIM) to provide an analytical approximate solution to Thomas-Fermi equation. This new iteration approach provides us with a convenient way to optimally control the convergence of the approximate solution. A good agreement between the obtained solution and some well-known results has been demonstrated. The proposed technique can be easily applied to handle other strongly nonlinear problems.

Approximate Solution of Strongly Nonlinear Vibrations which Vary with Time

2018 ◽  
Vol 8 (9) ◽  
pp. 107-114
Author(s):  
Pinakee Dey ◽  
Md Asaduzzaman ◽  
Razia Pervin ◽  
M. A. Sattar
Keyword(s):  
Approximate Solution ◽  
Nonlinear Vibrations ◽  
Strongly Nonlinear

Strongly Nonlinear Vibrations of a Hyperelastic Thin-walled Cylindrical Shell Based on the Modified Lindstedt–Poincaré Method

2019 ◽  
Vol 19 (12) ◽  
pp. 1950160 ◽  
Author(s):  
Jing Zhang ◽  
Jie Xu ◽  
Xuegang Yuan ◽  
Wenzheng Zhang ◽  
Datian Niu
Keyword(s):  
Cylindrical Shell ◽  
Nonlinear System ◽  
Differential Equations ◽  
Nonlinear Vibrations ◽  
Harmonic Excitation ◽  
System Of Differential Equations ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Hardening Behavior ◽  
Thin Walled

Some significant behaviors on strongly nonlinear vibrations are examined for a thin-walled cylindrical shell composed of the classical incompressible Mooney–Rivlin material and subjected to a single radial harmonic excitation at the inner surface. First, with the aid of Donnell’s nonlinear shallow-shell theory, Lagrange’s equations and the assumption of small strains, a nonlinear system of differential equations for the large deflection vibration of a thin-walled shell is obtained. Second, based on the condensation method, the nonlinear system of differential equations is reduced to a strongly nonlinear Duffing equation with a large parameter. Finally, by the appropriate parameter transformation and modified Lindstedt–Poincar[Formula: see text] method, the response curves for the amplitude-frequency and phase-frequency relations are presented. Numerical results demonstrate that the geometrically nonlinear characteristic of the shell undergoing large vibrations shows a hardening behavior, while the nonlinearity of the hyperelastic material should weak the hardening behavior to some extent.

scholarly journals Remarks on the boundary element method for strongly nonlinear problems

1993 ◽  
Vol 34 (4) ◽  
pp. 419-438 ◽  
Author(s):  
Keijo Ruotsalainen
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Nonlinear Problems ◽  
Monotone Operators ◽  
Heat Radiation ◽  
Optimal Order ◽  
Nonlinear Boundary Value Problems ◽  
Order Error ◽  
Strongly Nonlinear ◽  
Element Method

AbstractRecently in several papers the boundary element method has been applied to non-linear problems. In this paper we extend the analysis to strongly nonlinear boundary value problems. We shall prove the convergence and the stability of the Galerkin method in Lp spaces. Optimal order error estimates in Lp space then follow. We use the theory of A-proper mappings and monotone operators to prove convergence of the method. We note that the analysis includes the u4 -nonlinearity, which is encountered in heat radiation problems.

scholarly journals A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Alvaro H. Salas S ◽  
Jairo E. Castillo H ◽  
Darin J. Mosquera P
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytical Solution ◽  
Initial Conditions ◽  
Negative Ions ◽  
Nonlinear Problems ◽  
New Approach ◽  
Weierstrass Elliptic Function ◽  
Electronegative Plasma ◽  
The Given

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.

Loudspeaker Nonlinear Distortion Signal Instantaneous Frequency Measurement and Analysis

2012 ◽  
Vol 203 ◽  
pp. 83-87
Author(s):  
Yu Tian Wang ◽  
Hui Wang ◽  
Qin Zhang
Keyword(s):  
Instantaneous Frequency ◽  
Nonlinear Distortion ◽  
Frequency Measurement ◽  
Analysis Method ◽  
Hilbert Huang Transform ◽  
Strongly Nonlinear ◽  
Measurement And Analysis ◽  
Strongly Nonlinear System ◽  
Nonlinear Signal ◽  
Signal Analysis Method

Loudspeaker is a strongly nonlinear system which is associated with several situations such as electronic, magnetic, mechanical and acoustic. Recently, most of the methods used to measure and analysis loudspeaker are based on the FFT. Unfortunately, traditional Fourier transform based signal analysis method usually causes meaningless results when it is used to analysis non-stationary and nonlinear signal. In this paper, we use Hilbert-Huang transform (HHT) to review the instantaneous frequency of loudspeaker output. Experiments demonstrate that the distortion of loudspeaker can be recognized as intrawave frequency modulation caused by wave profile deformation. Then a novel nonlinear distortion measurement method is proposed which can reveal more accurate and physical meaningful characteristic of loudspeaker.

Dynamics of Oscillator With Soft Impacts

2001 ◽  
Author(s):  
František Peterka
Keyword(s):  
Nonlinear System ◽  
Linear System ◽  
Mechanical System ◽  
Linear Model ◽  
Piecewise Linear ◽  
Impact Oscillator ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear System ◽  
New Phenomena ◽  
The Impact

Abstract The impact oscillator is the simplest mechanical system with one degree of freedom, the periodically excited mass of which can impact on the stop. The aim of this paper is to explain the dynamics of the system, when the stiffness of the stop changes from zero to infinity. It corresponds to the transition from the linear system into strongly nonlinear system with rigid impacts. The Kelvin-Voigt and piecewise linear model of soft impact was chosen for the study. New phenomena in the dynamics of motion with soft impacts in comparison with known dynamics of motion with rigid impacts are introduced in this paper.

Damped Transition of a Strongly Nonlinear System of Coupled Oscillators Into a State of Continuous Resonance Scattering

2011 ◽  
Author(s):  
David Andersen ◽  
Xingyuan Wang ◽  
Yuli Starosvetsky ◽  
Kevin Remick ◽  
Alexander Vakakis ◽  
...  
Keyword(s):  
Equations Of Motion ◽  
Resonance Scattering ◽  
Linear Oscillator ◽  
System Response ◽  
Analytical Treatment ◽  
Strongly Nonlinear ◽  
Damped System ◽  
Initial Excitation ◽  
Strongly Nonlinear System ◽  
Energy Plane

We examine analytically and experimentally a new phenomenon of ‘continuous resonance scattering’ in an impulsively excited, two-mass oscillating system. This system consists of a grounded damped linear oscillator with a light, strongly nonlinear attachment. Previous numerical simulations revealed that for certain levels of initial excitation, the system engages in a special type of response that appears to track a solution branch formed by the so-called ‘impulsive orbits’ of this system. By this term we denote the periodic (under conditions of resonance) or quasi-periodic (under conditions of non-resonance) responses of the system when a single impulse is applied to the linear oscillator with the system being initially at rest. By varying the magnitude of the impulse we obtain a manifold of impulsive orbits in the frequency-energy plane. It appears that the considered damped system is capable of entering into a state of continuous resonance scattering, whereby it tracks the impulsive orbit manifold with decreasing energy. Through analytical treatment of the equations of motion, a direct relationship is established between the frequency of the nonlinear attachment and the amplitude of the linear oscillator response, and a prediction of the system response during continuous scattering resonance is provided. Experimental results confirm the analytical predictions.

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