Single-joint spherical pendulum mathematical modeling in cartesian coordinate system

Authors: Zaika V.V., Maslennikov A.L.
Published in issue: #1(42)/2020
DOI: 10.18698/2541-8009-2020-1-566
Category: Mechanics | Chapter: Biomechanics
Keywords: spherical pendulum, mathematical spherical pendulum, physical spherical pendulum, dissipation, modeling, mathematical model, numerical ODE integration, Runge-Kutta method
Published: 24.01.2020

Mathematical model of the n-link spherical pendulum could be used as a mathematical model of different technical systems, including biotechnical systems. Mathematical or physical spherical pendulum model could be used in different problems. In this paper the derivation of mathematical models of the mathematical and the physical single-joint spherical pendulum in spherical coordinate system is shown. Two cases are considered for both systems: without and with dissipation. Numerical simulation is realized utilizing explicit Runge-Kutta 4-th order method.

References

[1] Ryabina K.E., Isaev A.P. Biomechanics of erect posture maintaining (review of equilibrium control models). Vestnik YuUrGU. Seriya Obrazovanie, zdravookhranenie, fizicheskaya kul’tura [Bulletin of the South Ural State University. Ser. Education, health, physical culture], 2015, vol. 15, no. 4, pp. 93–98. DOI: https://doi.org/10.14529/ozfk150417 (in Russ.).

[2] Dubrovskiy V.I., Fedorova V.N. Biomekhanika [Biomechanics]. Moscow, Vlados-press Publ., 2003 (in Russ.).

[3] Olsson M.G. Spherical pendulum revisited. Am. J. Phys., 1981, vol. 49, no. 6, pp. 531–534. DOI: https://doi.org/10.1119/1.12666

[4] Miles J.W. Resonant motion of a spherical pendulum. Physica D, 1984, vol. 11, no. 3, pp. 309–323. DOI: https://doi.org/10.1016/0167-2789(84)90013-7

[5] Awrejcewicz J. Classical mechanics. Dynamics. New York, Springer, 2012.

[6] Zommerfel’d A. Mekhanika [Mechanics]. Moscow, NITs “Regulyarnaya i khaoticheskaya dinamika” Publ., 2001 (in Russ.).

[7] Zaika V.V., Maslennikov A.L. Single-joint spherical pendulum mathematical modeling in spherical coordinate system. Politekhnicheskiy molodezhnyy zhurnal [Politechnical student journal], 2019, no. 9. DOI: http://dx.doi.org/10.18698/2541-8009-2019-9-522 (in Russ.).

[8] Filimonov A.B., Filimonov N.B. Concerning the problems of synthesis of coordinated systems of automatic control. Izvestiya YuFU. Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2012, no. 3, pp. 172–180 (in Russ.).

[9] Farkhatdinov I.G., Poduraev Yu.V., Dzh.-Kh. Yu. Experimental study of position, speed and combined position-speed control in the teleoperation of mobile robot. Mekhatronika, avtomatizatsiya, upravlenie, 2010, no. 1, pp. 70–78 (in Russ.).

[10] Amosov A.A., Dubinskiy Yu.A. Kopchenova N.V. Vychislitel’nye metody [Computational methods]. Moscow, Lan’ Publ., 2014 (in Russ.).

[11] Bakhvalov N.S., Zhidkov N.P., Kobel’kov G.M. Chislennye metody [Numerical methods]. Moscow, Binom. Laboratoriya znaniy Publ., 2017 (in Russ.).

[12] Kil’chevskiy N.A. Kurs teoreticheskoy mekhaniki. T. 1 [Theoretical mechanics course. Vol. 1]. Moscow, Nauka Publ., 1977 (in Russ.).

[13] Prokhorov A.M., ed. Fizicheskaya entsiklopediya. T. 4 [Physical encyclopedia. Vol. 4]. Moscow, Bol’shaya rossiyskaya entsiklopediya Publ., 1994 (in Russ.).