Analytical solution for the motion of a pendulum with rolling wheel: stability analysis

Moatimid, Galal M.; Amer, T. S.

doi:10.1038/s41598-022-15121-w

Cited by 22 publications

(16 citation statements)

References 18 publications

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“…Time delays of position and velocity are used to reduce the nonlinear oscillation of the existing model. A modified HPM is employed to obtain a significantly more precise approximate solution [30]. For various amounts of the used factors, the temporal difference of this solution is graphed.…”

Section: Discussionmentioning

confidence: 99%

Dynamical Analysis of a Damped Harmonic Forced Duffing Oscillator with Time Delay

Moatimid

Amer

2022

Preprint

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This paper is concerned with a time-delayed controller of a damped nonlinear excited Duffing oscillator (DO). Since time-delayed techniques have recently been the focus of numerous studies, the topic of this investigation is quite contemporary. Therefore, time delays of position and velocity are utilized to reduce the nonlinear oscillation of the model under consideration. A much supplementary precise approximate solution is achieved using an advanced Homotopy perturbation method (HPM). The temporal variation of this solution is graphed for different amounts of the employed factors. The organization of the model is verified through a comparison between the plots of the estimated solution and the numerical one which is obtained utilizing the fourth order Runge-Kutta technique (RK4). The outcomes show that the improved HPM is appropriate for a variety of damped nonlinear oscillators since it minimizes the error of the solution while increasing the validation variety. Furthermore, it presents a potential model that deals with a diversity of nonlinear problems. The multiple scales homotopy technique is used to achieve an estimated formula for the suggested time-delayed structure. The controlling nonlinear algebraic equation for the amplitude oscillation at the steady state is gained. The effectiveness of the proposed controller, the time delays impact, controller gains, and feedback gains have been investigated. The realized outcomes show that the controller performance is influenced by the total of the product of the control and feedback gains, in addition to the time delays in the control loop. The analytical and numerical calculations reveal that for certain amounts of the control and feedback signal improvement, the suggested controller could completely reduce the system vibrations.

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Section: Discussionmentioning

confidence: 99%

Dynamical Analysis of a Damped Harmonic Forced Duffing Oscillator with Time Delay

Moatimid

Amer

2022

Preprint

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“…Moatimid 19 and 20 used an extended frequency concept combined with the HPM and Laplace transform to analyze a parameterized Duffing equation to arrive at a valid constricted formula of solution. There have been recent efforts that were connected to the current manuscript 21 – 23 .…”

Section: Introductionmentioning

confidence: 99%

Dynamical system of a time-delayed of rigid rocking rod: analytical approximate solution

Moatimid

Amer

2023

Sci Rep

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The stability analysis of a rocking rigid rod is investigated in this paper using a time-delayed square position and velocity. The time delay is an additional safety against the nonlinearly vibrating system under consideration. Because time-delayed technologies have lately been the core of several investigations, the subject of this inquiry is extremely relevant. The Homotopy perturbation method (HPM) is modified to produce a more precise approximate outcome. Therefore, the novelty of the exciting paper arises from the coupling of the time delay and its correlation with the modified HPM. A comparison with the fourth-order Runge–Kutta (RK4) technique is employed to evaluate the precision between the analytical as well as the numerical solutions. The study allows for a comprehensive examination of the recognition of the outcome of the realistic approximation analytical methodology. For different amounts of the physical frequency and time delay factors, the time histories of the found solutions are depicted in various plots. These graphs are discussed in the context of the shown curves according to the relevant parameter values. The organized nonlinear prototype approach is examined by the multiple-time scale method up to the first approximation. The obtained results have periodic behavior and a stable manner. The current study makes it possible to carefully examine the findings arrived at by employing the analytical technique of practicable estimation. Additionally, the time delay performs as extra protection as opposed to the system potential for nonlinear oscillation.

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“…Many numerical methods have been applied for solving linear and non-linear differential equations [13][14][15][16]. One of the most popular physical models encountered in undergraduate courses is the simple pendulum and the differential equation describing its motion [14,15,[17][18][19][20][21][22][23][24][25]. Historically, the equation arises when studying the oscillations of a pendulum clock, but also appears in various other areas of physics, since problems often can be reduced to a differential equation similar to that describing the pendulum [16,21,22,26,27].…”

Section: Introductionmentioning

confidence: 99%

“…He et al, presents a Periodic property and instability of a rotating pendulum system [27], Moatimid and Amer presents an analytical solution and stability analysis for pendulum in [25]. Li et al, studies Theoretical, numerical, and experimental in a vibrator-pendulum coupling system [28] Simple pendulum is a simple mechanical system in terms of setup, but it is difficult to calculate the factors that act on its motion, such as time period, amplitude, angle of oscillation, acting forces, and energy [23].…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution for Time Period of Simple Pendulum Under Magnetic Field

et al. 2023

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In the present study, the numerical solution of the time period of a Simple Pendulum under a magnetic field investigated. The analytical solution presented for the given problem. The numerical solution for the problem achieved by using two numerical quadrature methods, namely, Simpson’s 3/8 and Boole’s method. The period of a simple pendulum with a large angle is presented. The results of the numerical quadrature have been compared to the exact solution. Absolute and relative mistakes of the problem have been presented. The Matlab program 2013R has created a numerical method to analyze the outcome. Moreover, it is established that the comparison results guarantee the present method's ability and accuracy.

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Analytical solution for the motion of a pendulum with rolling wheel: stability analysis

Cited by 22 publications

References 18 publications

Dynamical Analysis of a Damped Harmonic Forced Duffing Oscillator with Time Delay

Dynamical Analysis of a Damped Harmonic Forced Duffing Oscillator with Time Delay

Dynamical system of a time-delayed of rigid rocking rod: analytical approximate solution

Numerical Solution for Time Period of Simple Pendulum Under Magnetic Field

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