531.3

nonlinear dynamics, oscillations, space problem, spherical pendulum, Lagrange equations of the 2nd kind, mathematical model, numerical experiment.

Using the Lagrange equation of the second kind, a mathematical model in the form of spatial equations of a spherical pendulum motion on an elastic suspension under the action of a variable side load is obtained. The system has three degrees of freedom. The relations between the angular and Cartesian coordinates are determined. Software is compiled and a numerical experiment is performed. The model and software make it possible to obtain the time dependences of linear and angular displacements, as well as linear and angular velocities, and to construct the corresponding graphs, phase portraits, and spatial trajectory. The solution found in general form allows further research to be performed by setting specific parameter values. The study was conducted for a nonlinear model without the use of asymptotic methods, which allowed us to exclude the methodological error of the solution.

1. Krasilnikov P. S. O nelinejnykh kolebaniyakh mayatnika  peremennoj dliny na vibriruyushhem osnovanii. PMM. 2012. T. 76. Vol. 1, рр. 36–51. [In Russian].
2. Markeev A. P. Nelinejnye kolebaniya simpaticheskikh mayatnikov. Nelinejnaya dinamika. 2010. T. 6. No. 3, рр. 605–622. [In Russian].
3. Shvecz A. Yu. Determinirovannyj khaos sfericheskogo mayatnika pri ogranichennom vozbuzhdenii. Ukr. mat. zhurn., 2007, t. 59, no. 4, рр. 534–548. [In Russian].
4. Chelombit'ko V. F. Heometrychne modelyuvannya kolyvannya sferychnoho mayatnyka. Byonyka yntellekta. 2016. No. 1 (86), рр. 43–46. [In Ukrainian].
5. Kochetkov A. P., Fedotov P. V. Novy`e metodicheskie podkhody resheniya sfericheskogo mayatnika v elementarnykh funkcziyakh. Vvedenie v topologicheskuyu mekhaniku. Vestnik Evrazijskoj nauki. 2019, no. 2, Tom 11. URL: https://esj.today/PDF/46SAVN219.pdf. [In Russian].
6. Korytov M. S., Shherbakov V. S., Titenko V. V., Belyakov V. E. Model sfericheskogo mayatnika s podvizhnoj tochkoj podvesa v zadache prostranstvennogo peremeshheniya gruza gruzopod`emny`m kranom pri ogranichenii kolebanij. Dinamika sistem, mekhanizmov i mashin. 2019. Tom 7, No. 1,
   рр. 104–110. [In Russian].
7. Korytov M. S., Shherbakov V. S., Titenko V. V. Ispolzovanie splajnov e`rmita pri reshenii zadachi peremeshheniya gruza na nezhestkom kranovom podvese po krivolinejnoj traektorii. Dinamika sistem, mekhanizmov i mashin. 2019. Tom 7. No. 1, рр. 95–104. [In Russian].
8. Nespirnyj V. N., Korolev V. A. Staczionarnye rezhimy sfericheskogo mayatnika s podvizhnoj tochkoj podvesa. Mekhanika tverdogo tela. 2011. Vol. 41, рр. 225–232. [In Russian].
9. Goldobina L. A., Vlasov A. V., Bochkov A. L. Teoreticheskoe obosnovanie snizheniya raskachivaniya gruza na kanate stroitel`nogo krana. Tekhniko-tekhnologicheskie problemy` servisa, no. 2 (16), 2011,
   рр. 52–60. [In Russian].
10. Perig A. V., Stadnik A. N., Deriglazov A. I., and Podlesny S. V., 3 DOF spherical pendulum oscillations with a uniform slewing pivot center and a small angle assumption, Shock and Vibration, vol. 2014, Article ID 203709, 32 p., 2014. URL: https://www.researchgate.net/publication/265385700.
11. Perig A. V., Stadnik A. N., A. A. Kostikov, S. V. Podlesny Research into 2D dynamics and control of small oscillations of a cross-beam during transportation by two overhead cranes. Shock and Vibration, 2017. URL: http://downloads.hindawi.com/journals/sv/2017/9605657.pdf.
12. Selyuczkij Yu. D., Andronov P. R. O modelirovanii povedeniya mayatnika v potoke sredy. Vestnik Nizhegorodskogo universiteta im. N. I. Lobachevskogo, 2011, No. 4 (2), рр. 307–309. [In Russian].
13. Zaika V. V., Maslennikov A. L. Matematicheskoe modelirovanie odnozvennogo sfericheskogo mayatnika v sfericheskoj sisteme koordinat. Politekhnicheskij molodezhny`j zhurnal. 2019. No. 09, рр. 1–12. [In Russian].
14. Malafayev M. T. Obertannya molekul vody yak rukh sferychnoho mayatnyka v neodnoridnomu poli syl. Prohresyvni tekhnika ta tekhnolohiyi kharchovykh vyrobnytstv restorannoho hospodarstva i torhivli. 2014. Vol. 1, рр. 291–298. URL: http://nbuv.gov.ua/UJRN/Pt_2014_1_36. [In Ukrainian].
15. Loveykin V., Lymar P. Dynamic analysis of movement of carriage hoisting crane with a displaced center of mass cargo for grips. Bulletin of TNTU. Ternopil: TNTU, 2014. Volume 73. No. 1. P. 102–109. (engineering, factory automation and processes of mechanical treatment).
16. Iurchenko M. (2016) Rozviazok obernenoi zadachi kolyvan neodnoridnoho sterzhnia [Solution of the inverse problem of vibrations of a heterogeneous rod]. Scientific Journal of TNTU (Tern.), vol. 83, no. 3,
    pp. 43–50. [In Ukrainian].
17. Yasniy P., Pyndus Y., Hud M. (2017) Methodology for the experimental research of reinforced cylindrical shell forced oscillations. Scientific Journal of TNTU (Tern.), vol. 86, no. 2, pp. 7–13. [In English].

1. Krasilnikov P. S. O nelinejnykh kolebaniyakh mayatnika  peremennoj dliny na vibriruyushhem osnovanii. PMM. 2012. T. 76. Vol. 1, рр. 36–51. [In Russian].
2. Markeev A. P. Nelinejnye kolebaniya simpaticheskikh mayatnikov. Nelinejnaya dinamika. 2010. T. 6. No. 3, рр. 605–622. [In Russian].
3. Shvecz A. Yu. Determinirovannyj khaos sfericheskogo mayatnika pri ogranichennom vozbuzhdenii. Ukr. mat. zhurn., 2007, t. 59, no. 4, рр. 534–548. [In Russian].
4. Chelombit'ko V. F. Heometrychne modelyuvannya kolyvannya sferychnoho mayatnyka. Byonyka yntellekta. 2016. No. 1 (86), рр. 43–46. [In Ukrainian].
5. Kochetkov A. P., Fedotov P. V. Novy`e metodicheskie podkhody resheniya sfericheskogo mayatnika v elementarnykh funkcziyakh. Vvedenie v topologicheskuyu mekhaniku. Vestnik Evrazijskoj nauki. 2019, no. 2, Tom 11. URL: https://esj.today/PDF/46SAVN219.pdf. [In Russian].
6. Korytov M. S., Shherbakov V. S., Titenko V. V., Belyakov V. E. Model sfericheskogo mayatnika s podvizhnoj tochkoj podvesa v zadache prostranstvennogo peremeshheniya gruza gruzopod`emny`m kranom pri ogranichenii kolebanij. Dinamika sistem, mekhanizmov i mashin. 2019. Tom 7, No. 1,
   рр. 104–110. [In Russian].
7. Korytov M. S., Shherbakov V. S., Titenko V. V. Ispolzovanie splajnov e`rmita pri reshenii zadachi peremeshheniya gruza na nezhestkom kranovom podvese po krivolinejnoj traektorii. Dinamika sistem, mekhanizmov i mashin. 2019. Tom 7. No. 1, рр. 95–104. [In Russian].
8. Nespirnyj V. N., Korolev V. A. Staczionarnye rezhimy sfericheskogo mayatnika s podvizhnoj tochkoj podvesa. Mekhanika tverdogo tela. 2011. Vol. 41, рр. 225–232. [In Russian].
9. Goldobina L. A., Vlasov A. V., Bochkov A. L. Teoreticheskoe obosnovanie snizheniya raskachivaniya gruza na kanate stroitel`nogo krana. Tekhniko-tekhnologicheskie problemy` servisa, no. 2 (16), 2011,
   рр. 52–60. [In Russian].
10. Perig A. V., Stadnik A. N., Deriglazov A. I., and Podlesny S. V., 3 DOF spherical pendulum oscillations with a uniform slewing pivot center and a small angle assumption, Shock and Vibration, vol. 2014, Article ID 203709, 32 p., 2014. URL: https://www.researchgate.net/publication/265385700.
11. Perig A. V., Stadnik A. N., A. A. Kostikov, S. V. Podlesny Research into 2D dynamics and control of small oscillations of a cross-beam during transportation by two overhead cranes. Shock and Vibration, 2017. URL: http://downloads.hindawi.com/journals/sv/2017/9605657.pdf.
12. Selyuczkij Yu. D., Andronov P. R. O modelirovanii povedeniya mayatnika v potoke sredy. Vestnik Nizhegorodskogo universiteta im. N. I. Lobachevskogo, 2011, No. 4 (2), рр. 307–309. [In Russian].
13. Zaika V. V., Maslennikov A. L. Matematicheskoe modelirovanie odnozvennogo sfericheskogo mayatnika v sfericheskoj sisteme koordinat. Politekhnicheskij molodezhny`j zhurnal. 2019. No. 09, рр. 1–12. [In Russian].
14. Malafayev M. T. Obertannya molekul vody yak rukh sferychnoho mayatnyka v neodnoridnomu poli syl. Prohresyvni tekhnika ta tekhnolohiyi kharchovykh vyrobnytstv restorannoho hospodarstva i torhivli. 2014. Vol. 1, рр. 291–298. URL: http://nbuv.gov.ua/UJRN/Pt_2014_1_36. [In Ukrainian].
15. Loveykin V., Lymar P. Dynamic analysis of movement of carriage hoisting crane with a displaced center of mass cargo for grips. Bulletin of TNTU. Ternopil: TNTU, 2014. Volume 73. No. 1. P. 102–109. (engineering, factory automation and processes of mechanical treatment).
16. Iurchenko M. (2016) Rozviazok obernenoi zadachi kolyvan neodnoridnoho sterzhnia [Solution of the inverse problem of vibrations of a heterogeneous rod]. Scientific Journal of TNTU (Tern.), vol. 83, no. 3,
    pp. 43–50. [In Ukrainian].
17. Yasniy P., Pyndus Y., Hud M. (2017) Methodology for the experimental research of reinforced cylindrical shell forced oscillations. Scientific Journal of TNTU (Tern.), vol. 86, no. 2, pp. 7–13. [In English].