An approximate analytical solution of a 4-DOF variable-length pendulum model

• Autorzy: Yakubu Godiya , Olejnik Paweł

Identyfikatory

DOI

10.15632/jtam-pl/188882

• Języki publikacji: EN

Abstrakty

EN

In this work, we employ the multiple scale method to introduce a novel analytical solution for an extended four-degrees-of-freedom dynamical system modeled on a swinging Atwood machine. We provide a methodology for obtaining the asymptotic solution up to the second-order approximation for both the swinging and modified swinging Atwood machine, demonstrating its solvability through the multiple scale approach. Subsequently, we present a comparative analysis of time histories between numerical and analytical solutions. These analytical solutions are of particular significance in applied mechanics, given their practical applications in parametric dynamical models grounded in the pendulum concept.

Słowa kluczowe

EN

analytical solutions; asymptotic solutions; swinging Atwood machine; multiple scale method

• Wydawca: Polskie Towarzystwo Mechaniki Teoretycznej i Stosowanej
• Czasopismo: Journal of Theoretical and Applied Mechanics
• Rocznik: 2024
• Tom: Vol. 62 nr 3
• Strony: 461--476
• Opis fizyczny: Bibliogr. 21 poz., rys.

Twórcy

autor

Yakubu Godiya

• Lodz University of Technology, Department of Automation, Biomechanics and Mechatronics, Lodz, Poland

autor

Olejnik Paweł

• Lodz University of Technology, Department of Automation, Biomechanics and Mechatronics, Lodz, Poland

Bibliografia

• 1. Abady I.M., Amer T.S., Gad H.M., Bek M.A., 2022, The asymptotic analysis and stability of 3DOF non-linear damped rigid body pendulum near resonance, Ain Shams Engineering Journal, 13, 2, 101554.
• 2. Abohamer M.K., Awrejcewicz J., Amer T.S., 2023a, Modeling and analysis of a piezoelectric transducer embedded in a nonlinear damped dynamical system, Nonlinear Dynamics, 31, 1, 1-16.
• 3. Abohamer M.K., Awrejcewicz J., Amer T.S., 2023b, Modeling of the vibration and stability of a dynamical system coupled with an energy harvesting device, Alexandria Engineering Journal, 63, 377-397.
• 4. Awrejcewicz J., Starosta R., Sypniewska-Kamińska G. (Edit.), 2022, Asymptotic Multiple Scale Method in Time Domain: Multi-Degree-of-Freedom Stationary and Nonstationary Dynamics, 1st ed., Boca Raton, United States: Taylor and Francis Group.
• 5. Awrejcewicz J., Starosta R., Sypniewska-Kamińska G., 2014, Asymptotic analysis and limiting phase trajectories in the dynamics of spring pendulum, [In:] Applied Non-Linear Dynamical Systems, Awrejcewicz J. (Ed.), Springer Proceedings in Mathematics and Statistics, Springer, Cham, vol. 93.
• 6. Casasayas J., Nunes A., Tufillaro N.B., 1990, Swinging Atwood’s machine: integrability and dynamics, Journal de Physique II, 51, 1693-1702.
• 7. Elmandouh A.A., 2016, On the integrability of the motion of 3D-Swinging Atwood machine and related problems, Physics Letters A, 380, 9, 989-991.
• 8. Manafian J., Allahverdiyeva N., 2022, An analytical analysis to solve the fractional differential equations, Advanced Mathematical Models and Applications, 6, 2, 128-161.
• 9. Nayfeh A.H., 2005, Resolving controversies in the application of the method of multiple scales and the generalized method of averaging, Nonlinear Dynamics, 40, 61-102.
• 10. Nunes A., Casasayas J., Tufillaro N.B., 1995, Periodic orbits of the integrable swinging Atwood’s machine, American Journal of Physics, 63, 2, 121-126.
• 11. Prokopenya A.N., 2017, Motion of a swinging Atwood’s machine: simulation and analysis with Mathematica, Mathematics in Computer Science, 11, 4, 417-425.
• 12. Prokopenya A.N., 2021, Searching for equilibrium states of Atwood’s machine with two oscillating bodies by means of computer algebra, Programming and Computer Software, 47, 1, 43-49.
• 13. Pujol O., Pérez J.P., Ramis J.P., Simó C., Simon S., Weil J.A., 2010, Swinging Atwood machine: experimental and numerical results, and a theoretical study, Physica D: Nonlinear Phenomena, 239, 12, 1067-1081.
• 14. Seadawy A.R., Manafian J., 2018, New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod, Results in Physics, 8, 1158-1167.
• 15. Starosta R., Sypniewska-Kamińska G., Awrejcewicz J., 2017, Quantifying non-linear dynamics of mass-springs in series oscillators via asymptotic approach, Mechanical Systems and Signal Processing, 89, 149-158.
• 16. Tufillaro N., 1986, Integrable motion of a swinging Atwood’s machine, American Journal of Physics, 54, 142-143.
• 17. Tufillaro N.B., 1985, Motions of a swinging Atwood’s machine, Journal de Physique, 46, 1495-1500.
• 18. Tufillaro N.B., 1994, Teardrop and heart orbits of a swinging Atwood’s machine, American Journal of Physics, 62, 3, 231-233.
• 19. Tufillaro N.B., Nunes A., Casasayas J., 1988, Unbounded orbits of a swinging Atwood’s machine, American Journal of Physics, 56, 1117-1120.
• 20. Yakubu G., Olejnik P., Awrejcewicz J., 2022, On the modeling and simulation of variable-length pendulum systems: A review, Archives of Computational Methods in Engineering, 29, 4, 2397-2415.
• 21. Yehia H.M., 2006, On the integrability of the motion of a heavy particle on a tilted cone and the swinging Atwood’s machine, Mechanics Research Communications, 33, 5, 711-716.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).