Skip to main content
SpringerLink
Log in

Parametric stability analysis for planar bicircular restricted four-body problem

  • Research Article
  • Published:
Astrodynamics Aims and scope Submit manuscript

A Correction to this article was published on 11 February 2022

This article has been updated

Abstract

Stability for the non-autonomous bicircular four-body model is analytically investigated in this study. The governing equation is derived from Newton's law of gravity. When the distance between the infinitesimal mass and the third primary is expanded as Taylor expansions, the governing equation can be regarded as two parts: the unperturbed conservative system and the small periodically parametric excitations. The unperturbed system's natural frequency and parametric frequency are analyzed for the possibility of principal parametric resonances. The method of multiple scales is applied directly to the governing equation. The stability conditions are obtained analytically for the principal parametric resonance. Numerical method demonstrates the efficiency of the analytical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Change history

References

  1. Papadakis, K. E. Asymptotic orbits in the restricted four-body problem. Planetary and Space Science, 2007, 55(10): 1368–1379.

    Article  Google Scholar 

  2. Kumari, R., Kushvah, B. S. Stability regions of equilibrium points in restricted four-body problem with oblateness effects. Astrophysics and Space Science, 2014, 349(2): 693–704.

    Article  Google Scholar 

  3. Falaye, B. J. Effect of oblateness, radiation and a circular cluster of material points on the stability of equilibrium points in the restricted four-body problem. Few-Body Systems, 2015, 56(1): 29–40.

    Article  Google Scholar 

  4. Lhotka, C., Celletti, A. The effect of Poynting-Robertson drag on the triangular Lagrangian points. Icarus, 2015, 250: 249–261.

    Article  Google Scholar 

  5. Baltagiannis, A. N., Papadakis, K. E. Equilibrium points and their stability in the restricted four-body problem. International Journal of Bifurcation and Chaos, 2011, 21(8): 2179–2193.

    Article  MathSciNet  MATH  Google Scholar 

  6. Arribas, M., Abad, A., Elipe, A., Palacios, M. Equilibria of the symmetric collinear restricted fourbody problem with radiation pressure. Astrophysics and Space Science, 2016, 361(2): 84.

    Article  Google Scholar 

  7. Singh, J. The equilibrium points in the perturbed R3BP with triaxial and luminous primaries. Astrophysics and Space Science, 2013, 346(1): 41–50.

    Article  MATH  Google Scholar 

  8. Papadouris, J. P., Papadakis, K. E. Equilibrium points in the photogravitational restricted four-body problem. Astrophysics and Space Science, 2013, 344(1): 21–38.

    Article  MATH  Google Scholar 

  9. Singh, J., Vincent, A. E. Equilibrium points in the restricted four-body problem with radiation pressure. Few-Body Systems, 2016, 57(1): 83–91.

    Article  Google Scholar 

  10. Kalvouridis, T. J., Arribas, M., Elipe, A. Parametric evolution of periodic orbits in the restricted four-body problem with radiation pressure. Planetary and Space Science, 2007, 55(4): 475–493.

    Article  Google Scholar 

  11. Ceccaroni, M., Biggs, J. Low-thrust propulsion in a coplanar circular restricted four body problem. Celestial Mechanics and Dynamical Astronomy, 2012, 112(2): 191–219.

    Article  MathSciNet  MATH  Google Scholar 

  12. Alvarez-Ramíez, M., Skea, J. E. F., Stuchi, T. J. Nonlinear stability analysis in a equilateral restricted four-body problem. Astrophysics and Space Science, 2015, 358(1): 3.

    Article  Google Scholar 

  13. Budzko, D. A., Prokopenya, A. N. On the stability of equilibrium positions in the circular restricted fourbody problem. In: Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, Vol. 6885. Gerdt, V. P., Koepf, W., Mayr, E. W., Vorozhtsov, E. V. Eds. Springer Berlin Heidelberg, 2011: 88–100.

    Article  MATH  Google Scholar 

  14. Hou, X. Y., Liu, L. On quasi-periodic motions around the triangular libration points of the real Earth-Moon system. Celestial Mechanics and Dynamical Astronomy, 2010, 108(3): 301–313.

    Article  MathSciNet  MATH  Google Scholar 

  15. Hou, X. Y., Liu, L. On quasi-periodic motions around the collinear libration points in the real Earth-Moon system. Celestial Mechanics and Dynamical Astronomy, 2011, 110(1): 71–98.

    Article  MathSciNet  MATH  Google Scholar 

  16. Salmani, M., Buskens, C. Real-time control of optimal low-thrust transfer to the Sun-Earth L1 halo orbit in the bicircular four-body problem. Acta Astronautica, 2011, 69(9-10): 882–891.

    Article  Google Scholar 

  17. Machuy, A. L., Prado, A. F. B. D., Stuchi, T. D. Numerical study of the gravitational capture in the bi-circular four-body problem. In: Proceedings of the 11th WSEAS International Conference on Automatic Control, Modelling and Simulation, 2009: 456–461.

    Google Scholar 

  18. Masago, B. Y. P. L., Prado, A. F. B. D., Chiaradia, A. P. M., Gomes, V. M. Developing the Precessing Inclined Bi-Elliptical Four-Body Problem with Radiation Pressure" to search for orbits in the triple asteroid 2001SN263. Advances in Space Research, 2016, 57(4): 962–982.

    Article  Google Scholar 

  19. Assaidian, N., Pourtakdoust, S. H. On the quasiequilibria of the BiElliptic four-body problem with noncoplanar motion of primaries. Acta Astronautica, 2010, 66(1-2): 45–58.

    Article  Google Scholar 

  20. Oshima, K., Topputo, F., Campagnola, S., Yanao, T. Analysis of medium-energy transfers to the Moon. Celestial Mechanics and Dynamical Astronomy, 2017, 127(3): 285–300.

    Article  MathSciNet  Google Scholar 

  21. Yagasaki, K. Sun-perturbed earth-to-moon transfers with low energy and moderate fiight time. Celestial Mechanics and Dynamical Astronomy, 2004, 90(3-4): 197–212.

    Article  MathSciNet  MATH  Google Scholar 

  22. Simó, C., Góez, G., Jorba, À., Masdemont, J. The bicircular model near the triangular libration points of the RTBP. In: From Newton to Chaos. NATO ASI Series (Series B: Physics), Vol. 336. Roy, A. E., Steves, B. A. Eds. Springer, 1995, 336: 343–370.

    Article  MathSciNet  MATH  Google Scholar 

  23. Nayfeh, A. H., Mook, D. T. Nonlinear Oscillations. John Wiley & Sons, 2008.

    MATH  Google Scholar 

  24. Huang, J. L., Su, R. K. L., Li, W. H., Chen, S. H. Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. Journal of Sound and Vibration, 2011, 330(3): 471–485.

    Article  Google Scholar 

  25. Kesimli, A., Ozkaya, E., Bagdatli, S. M. Nonlinear vibrations of spring-supported axially moving string. Nonlinear Dynamics, 2015, 81(3): 1523–1534.

    Article  MathSciNet  MATH  Google Scholar 

  26. Malookani, R. A., van Horssen, W. T. On resonances and the applicability of Galerkin's truncation method for an axially moving string with time-varying velocity. Journal of Sound and Vibration, 2015, 344: 1–17.

    Article  Google Scholar 

  27. Chen, L.-Q., Tang, Y.-Q., Zu, J.-W. Nonlinear transverse vibration of axially accelerating strings with exact internal resonances and longitudinally varying tensions. Nonlinear Dynamics, 2014, 76(2): 1443–1468.

    Article  MATH  Google Scholar 

  28. Sahoo, B., Panda, L. N., Pohit, G. Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation. International Journal of Non-Linear Mechanics, 2016, 78: 35–44.

    Article  Google Scholar 

  29. Kholostova, O. Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances. International Journal of Non-Linear Mechanics, 2015, 73: 64–68.

    Article  Google Scholar 

  30. Alfriend, K. T., Rand, R. H. A perturbation solution for the stability of the triangluar points in the elliptic restricted problem. The Journal of the Asrtonautical Science, 1968, 15(3): 105–109.

    Google Scholar 

  31. Gong, S., Liu, C. Hill stability of the satellites in coplanar four-body problem. Monthly Notices of the Royal Astronomical Society, 2016, 462(1): 547–553.

    Article  Google Scholar 

  32. Liu, C., Gong, S. The Hill stability of triple planets in the Solar system. Astrophysics and Space Science, 2017, 362(7): 127.

    Article  MathSciNet  Google Scholar 

  33. Szebehely, V. Theory of Orbit: The Restricted Problem of Three Bodies. Elsevier, 2012.

    MATH  Google Scholar 

  34. Chen, L. Q., Yang, X.-D. Stability in parametric resonance of axially moving viscoelastic beams with time-dependent speed. Journal of Sound and Vibration, 2005, 284(3-5): 879–891.

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge the support of the National Natural Science Foundation of China through Grant Nos. 11402007, 11772009, and 11672007.

Author information

Authors and Affiliations

  1. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing, 100124, China

    Ying-Jing Qian, Lei-Yu Yang, Xiao-Dong Yang & Wei Zhang

Authors
  1. Ying-Jing Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lei-Yu Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiao-Dong Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Wei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ying-Jing Qian.

Additional information

Ying-Jing Qian received her Ph.D. degree in aeronautical and astronautical science and technology from Harbin Institute of Technology, China, in 2013, and was a visiting scholar of Purdue University in 2010 and 2011. After two years as a research assistant, she joined Beijing University of Technology in 2015. Her current research interests are astrodynamics and interplanetary trajectory design.

Lei-Yu Yang is a master student in Beijing University of Technology. He received his bachelor degree in industry design from Jilin Institute of Chemical Technology in 2015. His current research interest is astrodynamics.

Xiao-Dong Yang is the Distinguished Professor of dynamics and control in Beijing University of Technology. He received his Ph.D. degree in mechanics from Shanghai University, China, in 2004, and was a visiting scholar of Wilfred Laurier University in 2005 and 2006. His area of expertise is nonlinear vibrations and gyroscopic dynamics.

Wei Zhang is the Distinguished Professor of dynamics and control in Beijing University of Technology. He received his Ph.D. degree in mechanics from Tianjin University, China, in 1997, and was a visiting scholar of University of Western Ontario, University of Toronto, and Hong Kong University. His area of expertise is nonlinear dynamics.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qian, YJ., Yang, LY., Yang, XD. et al. Parametric stability analysis for planar bicircular restricted four-body problem. Astrodyn 2, 147–159 (2018). https://doi.org/10.1007/s42064-017-0017-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42064-017-0017-2

Keywords

Search

Navigation