Abstract
In this paper, we discuss the problem of searching for equilibrium states of Atwood’s machine in which a pulley of a finite radius is replaced by two separate small pulleys and both masses can oscillate in a vertical plane. Differential equations of motion for the system are derived, and their solutions are computed in the form of power series in a small parameter. It is shown that, in the case of equal masses, the equilibrium position \(r = {\text{const}}\) of the system exists only if the oscillation amplitudes and frequencies of the masses are the same and the phase shift is α = 0 or \(\alpha = \pi \). In addition, there is a dynamic equilibrium state when both masses oscillate with the same amplitudes and frequencies and the phase shift is \(\alpha = \pm \pi {\text{/}}2\). In this case, the lengths of the pendulums also oscillate around a certain equilibrium value. Comparison of the results with the corresponding numerical solutions of the motion equations confirms their correctness. All necessary computations are carried out using the Wolfram Mathematica computer algebra system.
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Atwood, G., A Treatisa on the Rectilinear Motion and Rotation of Bodies, Cambridge University Press, 1784.
Tufillaro, N.B., Abbott, T.A., and Griffiths, D.J., Swinging Atwood’s machine, Am. J. Phys., 1984, vol. 52, no. 10, pp. 895–903.
Tufillaro, N.B., Motions of a swinging Atwood’s machine, J. Phys., 1985, vol. 46, pp. 1495–1500.
Tufillaro, N.B., Integrable motion of a swinging Atwood’s machine, Am. J. Phys., 1986, vol. 54, pp. 142–143.
Casasayas, J., Nunes, T.A., and Tufillaro, N.B., Swinging Atwood’s machine: Integrability and dynamics, J. Phys., 1990, vol. 51, pp. 1693–1702.
Yehia, H.M., On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood’s machine, Mech. R. Comm., 2006, vol. 33, no. 5, pp. 711–716.
Pujol, O., Pérez, J.P., Ramis, J.P., Simo, C., Simon, S., and Weil, J.A., Swinging Atwood machine: Experimental and numerical results, and a theoretical study, Phys. D, 2010, vol. 239, no. 12, pp. 1067–1081.
Prokopenya, A.N., Motion of a swinging Atwood’s machine: Simulation and analysis with mathematica, Math. Comput. Sci., 2017, vol. 11, no. 3–4, pp. 417–425.
Prokopenya, A.N., Construction of a periodic solution to the equations of motion of generalized Atwood’s machine using computer algebra, Program. Comput. Software, 2020, vol. 46, pp. 120–125.
Prokopenya, A.N., Modelling Atwood’s machine with three degrees of freedom, Math. Comput. Sci., 2019, vol. 13, nos. 1–2, pp. 247–257.
Abramov, S.A., Zima, E.B., and Rostovtsev, V.A., Computer algebra, Programmirovanie, 1992, no. 5, pp. 4–25.
Vasil’ev, N.N. and Edneral, V.F., Computer algebra in physical and mathematical applications, Programmirovanie, 1994, no. 1, pp. 70–82.
Prokopenya, A.N., Some symbolic computation algorithms in cosmic dynamics problems, Program. Comput. Software, 2006, vol. 32, pp. 71–76.
Prokopenya, A.N., Symbolic computation in studying stability of solutions of linear differential equations with periodic coefficients, Program. Comput. Software, 2007, vol. 33, pp. 60–66.
Prokopenya, A.N., Hamiltonian normalization in the restricted many-body problem by computer algebra methods, Program. Comput. Software, 2012, vol. 38, pp. 156–166.
Wolfram, S., An Elementary Introduction to the Wolfram Language, Wolfram Media, 2017, 2nd ed.
Markeev, A.P., Teoreticheskaya mekhanika (Theoretical Mechanics), Izhevsk: NITs “Regulyarnaya Khaoticheskaya Din.,” 2001.
Landau, L.D. and Lifshits, E.M., Mekhanika (Mechanics), Moscow: Nauka, 1988, 4th ed.
Nayfeh, A.H., Introduction to Perturbation Techniques, New York: Wiley, 1981.
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Prokopenya, A.N. Searching for Equilibrium States of Atwood’s Machine with Two Oscillating Bodies by Means of Computer Algebra. Program Comput Soft 47, 43–49 (2021). https://doi.org/10.1134/S0361768821010084
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DOI: https://doi.org/10.1134/S0361768821010084