Galerkin method, ansatz method, and He’s frequency formulation for modeling the forced damped parametric driven pendulum oscillators

The (un)forced (un)damped parametric pendulum oscillator is analyzed analytically and numerically using some simple, effective, and more accurate techniques. In the first technique, the ansatz method is employed for analyzing the unforced damped parametric pendulum oscillator (PPO) and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions. In the second approach, some approximations to (un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method. In the third approach, He's Frequency-Amplitude principle (He's-FAP) is applied for deriving some approximations to the (un)damped PPO. In the forth approach, He's Homotopy technique (He's-HT) is employed for analyzing the forced (un)damped PPO numerically. In the fifth approach, the p-solution Method which is constructed based on Krylov-Bogoliúbov Mitropolsky method (KBMM) is introduced for deriving an approximation to the forced damped PPO. In the final approach, the hybrid Padé-finite difference method (Padé-FDM) is carried out for analyzing the damped PPO numerically. All proposed techniques are compared to the fourth-order Runge-Kutta (RK4) numerical solution. Moreover, the global maximum residual distance error is estimated for checking the accuracy of the obtained approximations. The proposed methodologies and approximations can help many researchers in studying and investigating several nonlinear phenomena related to the oscillations that can arise in various branches of science, e.g. waves and oscillations in plasma physics.

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“…(35) with the i.v.p. j w g j j j j j j j The following homotopy is considered [46][47][48][49] r j w j r g j f j .…”

Section: He's-fap For Analyzing the (Un)damped Pendulum Oscillatormentioning

confidence: 99%

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Alyousef

Alharthi²,

Salas³

et al. 2022

Commun. Theor. Phys.

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“…He's frequency formulation [10][11][12] not only can quickly unveil the relationship between the frequency and amplitude of various vibrators but also ensures high accuracy. Elías-Zúñiga, et al [19] extended the formulation by coupling Jacobi elliptic function, Ma [20] suggested a Hamiltonian-based modification, Alyousef, et al [21] used the formulation to solve a complex nonlinear system with great success. He and Liu [22] gave a mathematical proof of the formulation and suggested a powerful modification.…”

Section: Introductionmentioning

confidence: 99%

Application of He’s Frequency Formula to Nonlinear Oscillators With Generalized Initial Conditions

Zhang,

Song,

Zhang

et al. 2023

FU Mech Eng

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This paper focuses on the vibration periodic property of Duffing oscillator with generalized initial conditions. Firstly, the undamped case is solved by Ji-Huan He’s frequency formulation; Secondly, the formulation is extended to the damped case. Numerical verification shows that the frequency formulation is mathematically simple and physically insightful and practically applicable. This paper paves a novel way for engineers to use the formulation to study nonlinear vibration system with ease and reliability.

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“…For more recent applications of He's frequency formulation, Ma [21] suggested an alternative modification for He's frequency formulation to solve a quintic/cubic Duffing equation. For modeling a damped forced parametric pendulum model, Alyousef et al [22] used He's frequency formulation. Feng and Niu [23] also estimated a new analytical solution for a fractal Toda model.…”

Section: Introductionmentioning

confidence: 99%

An innovative technique to solve a fractal damping Duffing-jerk oscillator

El‐Dib

Elgazery

Khattab

et al. 2023

Commun. Theor. Phys.

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The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative approach. Using a new analytical technique, namely, the modification of He's a fractal derivative that converts the fractal derivative to the traditional derivative in continuous space, this study provides an effective and easy-to-apply procedure that is dependent on He's fractal derivative approach. The analytic approximate solution has significant match with the results of the numerical simulation as the fractal parameter is very closer to unity, which proves the reliability of the method. Stability behavior is discussed and illustrated graphically. These new powerful analytical tools are developed in an attempt to obtain effective analytical tools, valid for any fractal nonlinear problems.

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Galerkin method, ansatz method, and He’s frequency formulation for modeling the forced damped parametric driven pendulum oscillators

Cited by 14 publications

References 32 publications

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Optimal analytical and numerical approximations to the (un)forced (un)damped parametric pendulum oscillator

Application of He’s Frequency Formula to Nonlinear Oscillators With Generalized Initial Conditions

An innovative technique to solve a fractal damping Duffing-jerk oscillator

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