Swinging Atwood Machine: Experimental and numerical results, and a theoretical study

Pujol, Olivier; Pérez, José Miguel Sánchez; Ramis, Jean-Pierre; Simó, Carles; Simon, Sergi; Weil, Jacques-Arthur

doi:10.1016/j.physd.2010.02.017

Cited by 26 publications

(12 citation statements)

References 24 publications

(35 reference statements)

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“…When the compensation is less than the pendulum mass, the system will have a large parameter space, which leads to rich dynamical behaviour [50,54]. These categories are terminating or non-terminating, chaotic or quasi-periodic, bounded or unbounded, singular or non-singular, which depends on the pendulum's reactive centrifugal force counteracting the counterweight [55][56][57][58][59].…”

Section: A Variable-length Pendulum System: Swinging Atwood's Machinementioning

confidence: 99%

On the Modeling and Simulation of Variable-Length Pendulum Systems: A Review

Yakubu

Olejnik

Awrejcewicz

2022

Arch Computat Methods Eng

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A comprehensive review of variable-length pendulums is presented. An attempt at a unique evaluation of current trends in this field is carried out in accordance with mathematical modeling, dynamical analysis, and original computer simulations. Perspectives of future trends are also noted on the basis of various concepts and possible theoretical and engineering applications. Some important physical concepts are verified using dedicated numerical procedures and assessed based on dynamical analysis. At the end of the review, it is concluded that many variable-length pendulums are very demanding in the modeling and analysis of parametric dynamical systems, but basic knowledge about constant-length pendulums can be used as a good starting point in providing much accurate mathematical description of physical processes. Finally, an extended model for a variable-length pendulum’s mechanical application being derived from the Swinging Atwood Machine is proposed. The extended SAM presents a novel SAM concept being derived from a variable-length double pendulum with a suspension between the two pendulums. The results of original numerical simulations show that the extended SAM’s nonlinear dynamics presented in the current work can be thoroughly studied, and more modifications can be achieved. The new technique can reduce residual vibrations through damping when the desired level of the crane is reached. It can also be applied in simple mechatronic and robotic systems.

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Section: A Variable-length Pendulum System: Swinging Atwood's Machinementioning

confidence: 99%

On the Modeling and Simulation of Variable-Length Pendulum Systems: A Review

Yakubu

Olejnik

Awrejcewicz

2022

Arch Computat Methods Eng

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“…This approach of finding necessary conditions for integrability in a framework of differential Galois theory was mostly used for Hamiltonian systems, and it is frequently called Morales-Ramis theory. Thanks to this approach, new integrable cases have been found; see for instance [63][64][65].…”

Section: For Periodic Motionmentioning

confidence: 99%

Integrability analysis of chaotic and hyperchaotic finance systems

Szumiński

2018

Nonlinear Dyn

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In this paper, we consider two chaotic finance models recently studied in the literature. The first one, introduced by Huang and Li, has a form of three first-order nonlinear differential equationṡ x = z + (y − a)x,ẏ = 1 − by − x 2 ,ż = − x − cz. The second system, called a hyperchaotic finance model, is defined bẏ x = z + (y − a)x + u, z = − x − cz,ẏ = 1 − by − x 2 , u = − dx y − ku. In both models, (a, b, c, d, k) are real positive parameters. In order to present the complexity of these systems Poincaré cross sections, bifurcation diagrams, Lyapunov exponents spectrum and the Kaplan-Yorke dimension have been calculated. Moreover, we show that the Huang-Li system is not integrable in a class of functions meromorphic in variables (x, y, z), for all real values of parameters (a, b, c), while the hyperchaotic system is not integrable in the case when k = c and := 1 + d(a + d − c) > 0. We give analytic proofs of these facts analyzing properties of the differential Galois groups of variational equations along cer-The research has been supported by the National Science Centre of Poland under Grants DEC

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“…It should be noted that swinging Atwood's machine has been a subject of a number of papers (see [2][3][4][5][6][7][8]) and its mechanical behaviour has been studied quite well. In particular, it has been proven that the system of differential equations describing dynamics of swinging Atwood's machine is not integrable, in general.…”

Section: Introductionmentioning

confidence: 99%

Motion of a Swinging Atwood’s Machine: Simulation and Analysis with Mathematica

Prokopenya

2017

Math.Comput.Sci.

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A generalized model of the Atwood machine when one body is constrained to move along a vertical axis while the other one can swing in a plane is considered. Combining symbolic and numerical calculations, we have obtained equations of motion of the system and analyzed their solutions. We have shown that oscillation can completely modify a motion of the system while the simple Atwood machine demonstrates only the uniformly accelerated motion of the bodies. The validity of the results obtained is demonstrated by means of the simulation of motion of swinging Atwood's machine with the computer algebra system Wolfram Mathematica.

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Swinging Atwood Machine: Experimental and numerical results, and a theoretical study

Cited by 26 publications

References 24 publications

On the Modeling and Simulation of Variable-Length Pendulum Systems: A Review

On the Modeling and Simulation of Variable-Length Pendulum Systems: A Review

Integrability analysis of chaotic and hyperchaotic finance systems

Motion of a Swinging Atwood’s Machine: Simulation and Analysis with Mathematica

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