The Journal of the Astronautical Sciences

, Volume 66, Issue 4, pp 383–403 | Cite as

On Nonlinear Dynamic Modeling for a Rotating Paraboloidal Thin-Shell Appendage Attached to a Spacecraft Mechanism with Paraboloidal Curved Shell Element

  • Bindi YouEmail author
  • Yiming Sun
  • Peibo Hao
  • Dong Liang
  • Zhihui Gao


A methodology for modeling a rotating paraboloidal thin-shell structure attached to a spacecraft body is proposed considering its geometric nonlinear effect. Instead of using conventional shell elements to discretize the paraboloidal thin-shell in the Cartesian coordinate, the paraboloidal coordinates in the meridional, circumferential and normal directions are employed to completely express the deformations of the curved thin-shell. Then, explicit expressions for the generalized elastic forces and stiffness varying matrices are deduced on the basis of exact strain-displacement relations. And, the rigid-flexible coupled dynamic model for the flexible multibody system is derived by the principle of virtual work. In contrast with the previous modeling approaches, the present method shows an advantage in avoiding large calculation quantity of nonlinear stiffness matrix due to more formalized. Furthermore, a full analysis with specific numerical simulation is achieved by using the present model and conventional shell model, respectively. All simulation results obtained by the two modeling methods verify the correctness and better convergence of the proposed methodology.


Dynamic modeling Rotating paraboloidal thin-shell Spacecraft mechanism Curved shell element Geometric nonlinearity 



This material is partially based upon Project (No. 51575126) supported by the National Natural Science Foundation of China, and projects (Nos. 2013 M541358 and 2015 T80358) funded by China Postdoctoral Science Foundation, and Project (No. WH20150108) supported by Discipline Construction Guide Foundation in Harbin Institute of Technology at Weihai.


  1. 1.
    Silverberg, L.M., Park, S.: Interactions between rigid-body and flexible-body motions in maneuvering spacecraft. J. Guid. Control. Dyn. 13(1), 73–81 (1990)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ryan, R.: Simulation of actively controlled spacecraft with flexible appendages. J. Guid. Control. Dyn. 13(4), 691–702 (1990)CrossRefGoogle Scholar
  3. 3.
    Utku, S., Shoemaker, W.L., Salama, M.: Nonlinear equations of dynamics for spinning paraboloidal antennas. Comput. Struct. 16(1), 361–370 (1983)CrossRefGoogle Scholar
  4. 4.
    Shoemaker, W.L.: The nonlinear dynamics of spinning paraboloidal antennas. PhD. Dissertation, Duke University, Durham, North Carolina (1983)Google Scholar
  5. 5.
    Shoemaker, W.L., Utku, S.: On the free vibrations of spinning paraboloids. J. Sound Vib. 111(2), 279–296 (1986)CrossRefGoogle Scholar
  6. 6.
    Leissa, A.W., Kang, J.H.: Three-dimensional vibration analysis of paraboloidal shells. JSME Int. J. 45(1), 2–7 (2002)CrossRefGoogle Scholar
  7. 7.
    Kang, J.H., Leissa, A.W.: Free vibration analysis of complete paraboloidal shells of revolution with variable thickness and solid paraboloids from a three-dimensional theory. Comput. Struct. 83(31–32), 2594–2608 (2005)CrossRefGoogle Scholar
  8. 8.
    Al-Khatib, O.J.: Vibration of Paraboloidal Shells. PhD. Dissertation, Tennessee Technological University (2006)Google Scholar
  9. 9.
    Al-Khatib, O.J., Buchanan, G.R.: Free vibration of a paraboloidal shell of revolution including shear deformation and rotary inertia. Thin-Walled Struct. 48(3), 223–232 (2010)CrossRefGoogle Scholar
  10. 10.
    Boutaghou, Z.E., Erdman, A.G., Stolarski, H.K.: Dynamics of flexible beams and plates in large overall motions. Asme. T. 59(4), 991–999 (1991)CrossRefGoogle Scholar
  11. 11.
    Yoo, H.H., Chung, J.: Dynamics of rectangular plates undergoing prescribed overall motion. J. Sound Vib. 239(1), 123–137 (2001)CrossRefGoogle Scholar
  12. 12.
    Liu, J.Y., Hong, J.Z.: Dynamic modeling and modal truncation approach for a high-speed rotating elastic beam. Arch. Appl. Mech. 72(8), 554–563 (2002)CrossRefGoogle Scholar
  13. 13.
    Park, J.H., Kim, J.H.: Dynamic analysis of rotating curved beam with a tip mass. J. Sound Vib. 228(5), 1017–1034 (1999)CrossRefGoogle Scholar
  14. 14.
    Sugiyama, H., Koyama, H., Yamashita, H.: Gradient deficient curved beam element using the absolute nodal coordinate formulation. J. Comput. Nonlinear Dyn. 5(2), 1090–1097 (2010)Google Scholar
  15. 15.
    Pan, K.Q., Liu, J.Y.: Geometric nonlinear dynamic analysis of curved beams using curved beam element. Acta. Mech. Sin. 27(6), 1023–1033 (2011)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bauchau, O.A., Choi, J.Y., Bottasso, C.L.: On the modeling of shells in multibody dynamics. Multibody. Syst. Dyn. 8(4), 459–489 (2002)CrossRefGoogle Scholar
  17. 17.
    Lubowiecka, I., Chróścielewski, J.: On dynamics of flexible branched shell structures undergoing large overall motion using finite elements. Comput. Struct. 80(9), 891–898 (2002)CrossRefGoogle Scholar
  18. 18.
    Betsch, P., Sänger, N.: On the use of geometrically exact shells in a conserving framework for flexible multibody dynamics. Comput. Method. Appl. Mech. Eng. 198(17–20), 1609–1630 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Goyal, A., Dörfel, M.R., Simeon, B., Vuong, A.V.: Isogeometric shell discretizations for flexible multibody dynamics. Multibody. Syst. Dyn. 30(2), 139–151 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tzou, H.S., Ding, J.H.: Optimal control of precision paraboloidal shell structronic systems. J. Sound Vib. 276(1), 273–291 (2002)MathSciNetzbMATHGoogle Scholar
  21. 21.
    You, B.D.: Analysis and control of dynamic pointing accuracy for satellite antenna. PhD. Dissertation, School of Astronautics, Harbin Institute of Technology (2011)Google Scholar

Copyright information

© American Astronautical Society 2018

Authors and Affiliations

  • Bindi You
    • 1
    Email author
  • Yiming Sun
    • 1
  • Peibo Hao
    • 1
  • Dong Liang
    • 1
  • Zhihui Gao
    • 1
  1. 1.School of Naval Architecture and Ocean EngineeringHarbin Institute of TechnologyWeihaiChina

Personalised recommendations