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

Prokopenya, Alexander N.

doi:10.1007/s11786-017-0301-9

Cited by 10 publications

(4 citation statements)

References 7 publications

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“…One of the areas of the variable-length pendulum is the Swinging Atwood's Machine (SAM). In this case, the pendulum's mass swings in a two-dimensional plane, producing a chaotic behavior without colliding with the other mass known as the counterweight [50][51][52]. The inextensible massless string suspended on two frictionless pulleys connects the two groups (see also [53]), as presented in Fig.…”

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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“…To clarify the physical reasons of such influence of oscillation on the system motion in the previous paper [10] we considered the simplest generalization of the Atwood machine when only one body is allowed to swing in a plane while the other body can move only along a vertical. We have shown that oscillation results in increasing of an averaged tension of the thread which depends on the amplitude of oscillation.…”

Section: A N Prokopenya (B)mentioning

confidence: 99%

“…In case of the two oscillating bodies the equations of motion (2.3)-(2.5) are much more complicated in comparison to the case of one body oscillation (see [10]). And to demonstrate some new peculiarities of the system motion we have to look for numerical solutions of these equations for different initial conditions and different values of the system parameter μ.…”

Section: Motion Of Atwood's Machine With Two Oscillating Bodiesmentioning

confidence: 99%

Modelling Atwood’s Machine with Three Degrees of Freedom

Prokopenya

2018

Math.Comput.Sci.

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A generalized model of the Atwood machine when two bodies 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 oscillations can completely modify motion of the system while the simple Atwood machine demonstrates only the uniformly accelerated motion of the bodies. In particular, a quasi-periodic motion of the system can take place even in case of equal masses of the bodies. We have also obtained a differential equation determining an averaged translational motion of the system and have shown that its solution corresponds completely to the numerical solution of the exact differential equations of motion. 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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“…The investigation of the proposed scenario outlined in the abstract stems from questioning, "For two identical blocks which one reaches the vertical leg of the table first?" [1] [2]. Intuitively, it is an easy task without quantifying… Claim the block on the table!…”

Section: Motivation and Goalsmentioning

confidence: 99%

Characteristics of a Two-Body Holonomic Constraint Mechanical System

Sarafian¹,

Hickey²

2017

WJM

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Two massive blocks are connected with a massless unstretchable line of 2ℓ. One of the masses is placed on a horizontal frictionless table, ℓ distance away from the edge of the table the other one is held horizontally equidistance from the edge along the extension of the line. The latter is released from rest. As it falls under gravity's pull, it drags the one on the table. It is the interest of this investigation to analyze the kinematics of the system. Because of the holonomic constraint of the system, analysis of the problem encounters complicated super nonlinear coupled differential equations. Utilizing Mathematica we solve the equations numerically. Applying the solutions we quantify numerous kinematic quantities; most interestingly we evaluate the run-time, and the trajectory of the falling block. Analysis is robust allowing us to address the "what if" scenarios.

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

Cited by 10 publications

References 7 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

Modelling Atwood’s Machine with Three Degrees of Freedom

Characteristics of a Two-Body Holonomic Constraint Mechanical System

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