Skip to main content
SpringerLink
Log in

A self-equilibrated load method to locate singular configurations of symmetric foldable structures

  • Original Paper
  • Published:
Acta Mechanica Aims and scope Submit manuscript

Abstract

Because of the geometric specialization, a foldable structure could be singular during folding along its ideal motion path. Singularity tends to result in undesired geometric configurations, and thus, it is necessary to locate singular configurations of foldable structures and to study their kinematic bifurcations. Using the steepest descent method and the nonlinear prediction–correction algorithm, we will present a self-equilibrated load method to locate singular configurations of foldable structures holding certain symmetries. The change of symmetry representation of internal mechanisms can reveal the singularity. The increment size for each iteration step depends on the product of the self-equilibrated load and the internal mechanisms. The singularity of both foldable pin-jointed structures and over-constrained structures is studied. Numerical results show that the method is effective and has a good convergence. Given certain load patterns, the proposed load method is capable of detecting the exact singular configurations in a small number of iteration steps. Independent mechanism modes associated with low-order symmetry representations will be induced, if a symmetric foldable structure reaches the singular configuration.

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.

Similar content being viewed by others

References

  1. Wei, G., Dai, J.S.: Mobility and geometric analysis of the Hoberman switch-pitch ball and its variant. J. Mech. Robot. 2, 031010 (2010)

    Article  Google Scholar 

  2. Ding, X.L., Yang, Y., Dai, J.S.: Topology and kinematic analysis of color-changing ball. Mech. Mach. Theory 46, 67–81 (2011)

    Article  MATH  Google Scholar 

  3. Chen, Y., Guest, S.D., Fowler, P.W., Feng, J.: Symmetry Adapted Analysis of the Hoberman Switch-Pitch Ball. In: IABSE-IASS-2011 Symposium (2011)

  4. Kawaguchi, K.I., Hangai, Y., Pellegrino, S., Furuya, H.: Shape and stress control analysis of prestressed truss structures. J. Reinf. Plast. Comp. 15, 1226–1236 (1996)

    Google Scholar 

  5. Tanaka, H., Hangai, Y.: Rigid body displacement and stabilization conditions of unstable truss structures. Shells, membranes and space frames. Proc. IASS Symp. 2, 55–62 (1986)

    Google Scholar 

  6. Guest, S.D., Fowler, P.W.: Symmetry conditions and finite mechanisms. J. Mech. Mater. Struct. 2, 293–301 (2007)

    Article  Google Scholar 

  7. Viquerat, A.D., Hutt, T., Guest, S.D.: A plane symmetric 6R foldable ring. Mech. Mach. Theory 63, 73–88 (2013)

    Article  Google Scholar 

  8. Gan, W.W., Pellegrino, S.: Numerical approach to the kinematic analysis of deployable structures forming a closed loop. P. I. Mech. Eng. C-J. Mec. 220, 1045–1056 (2006)

    Article  Google Scholar 

  9. Wei, G.W., Chen, Y., Dai, J.S.: Synthesis, mobility and multifurcation of highly overconstrained deployable polyhedral mechanisms. J. Mech. Des. Trans. ASME 136, 091003 (2014)

    Article  Google Scholar 

  10. Kumar, P., Pellegrino, S.: Computation of kinematic paths and bifurcation points. Int. J. Solids Struct. 37, 7003–7027 (2000)

    Article  MATH  Google Scholar 

  11. Lengyel, A., You, Z.: Bifurcations of SDOF mechanisms using catastrophe theory. Int. J. Solids Struct. 41, 559–568 (2004)

    Article  MATH  Google Scholar 

  12. Yuan, X., Zhou, L., Duan, Y.: Singularity and kinematic bifurcation analysis of pin-bar mechanisms using analogous stiffness method. Int. J. Solids Struct. 49, 1212–1226 (2012)

    Article  Google Scholar 

  13. Chen, Y., You, Z.: Two-fold symmetrical 6R foldable frame and its bifurcations. Int. J. Solids Struct. 46, 4504–4514 (2009)

    Article  MATH  Google Scholar 

  14. Altuzarra, O., Salgado, O., Petuya, V., Hernandez, A.: Computational kinematics for robotic manipulators: Jacobian problems. Eng. Comput. 25, 4–27 (2008)

    Article  MATH  Google Scholar 

  15. Bandyopadhyay, S., Ghosal, A.: Analysis of configuration space singularities of closed-loop mechanisms and parallel manipulators. Mech. Mach. Theory 39, 519–544 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Macho, E., Altuzarra, O., Amezua, E., Hernandez, A.: Obtaining configuration space and singularity maps for parallel manipulators. Mech. Mach. Theory 44, 2110–2125 (2009)

    Article  MATH  Google Scholar 

  17. Chen, Y., Guest, S.D., Fowler, P.W., Feng, J.: Two-orbit switch-pitch structures. J. Int. Assoc. Shell Spat. Struct. 53, 157–162 (2012)

    Google Scholar 

  18. Nagaraj, B.P., Pandiyan, R., Ghosal, A.: Kinematics of pantograph masts. Mech. Mach. Theory 44, 822–834 (2009)

    Article  MATH  Google Scholar 

  19. Zhang, J.Y., Ohsaki, M.: Self-equilibrium and stability of regular truncated tetrahedral tensegrity structures. J. Mech. Phys. Solids 60, 1757–1770 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, Y., Feng, J.: Generalized eigenvalue analysis of symmetric prestressed structures using group theory. ASCE J. Comput. Civil Eng. 26, 488–497 (2012)

    Article  Google Scholar 

  21. Kaveh, A., Rahami, H.: Block circulant matrices and applications in free vibration analysis of cyclically repetitive structures. Acta Mech. 217, 51–62 (2011)

    Article  MATH  Google Scholar 

  22. Zingoni, A.: Group-theoretic insights on the vibration of symmetric structures in engineering. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 372, 20120037 (2014)

    Article  MathSciNet  Google Scholar 

  23. Tran, H., Lee, J.: Form-finding of tensegrity structures with multiple states of self-stress. Acta Mech. 222, 131–147 (2011)

    Article  MATH  Google Scholar 

  24. Chen, Y., Feng, J.: Efficient method for Moore-Penrose inverse problems involving symmetric structures based on group theory. ASCE J. Comput. Civil Eng. 28, 182–190 (2014)

    Article  Google Scholar 

  25. Chen, Y., Feng, J., Ren, Z.: Numerical approach for detecting bifurcation points of the compatibility paths of symmetric deployable structures. Mech. Res. Commun. 71, 7–15 (2016)

    Article  Google Scholar 

  26. Dai, J.S., Li, D., Zhang, Q., Jin, G.: Mobility analysis of a complex structured ball based on mechanism decomposition and equivalent screw system analysis. Mech. Mach. Theory 39, 445–458 (2004)

    Article  MATH  Google Scholar 

  27. Hangai, Y., Kawaguchi, K.: Analysis of shape-finding process of unstable link structures. Bull. Int. Assoc. Shell Spat. Struct. 30, 116–127 (1989)

  28. Kaveh, A., Nikbakht, M.: Stability analysis of hyper symmetric skeletal structures using group theory. Acta Mech. 200, 177–197 (2008)

    Article  MATH  Google Scholar 

  29. Chen, Y., Feng, J., Zhang, Y.T.: A necessary condition for stability of kinematically indeterminate pin-jointed structures with symmetry. Mech. Res. Commun. 60, 64–73 (2014)

    Article  Google Scholar 

  30. Sultan, C.: Stiffness formulations and necessary and sufficient conditions for exponential stability of prestressable structures. Int. J. Solids Struct. 50, 2180–2195 (2013)

    Article  Google Scholar 

  31. Altmann, S.L., Herzig, P.: Point-Group Theory Tables. Clarendon Press, Oxford, UK (1994)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Prestress Engineering Research Center, Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education, Southeast University, Nanjing, China

    Yao Chen, Jian Feng & Zelun Qian

Authors
  1. Yao Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jian Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zelun Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jian Feng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, Y., Feng, J. & Qian, Z. A self-equilibrated load method to locate singular configurations of symmetric foldable structures. Acta Mech 227, 2749–2763 (2016). https://doi.org/10.1007/s00707-016-1652-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-016-1652-z

Keywords

Search

Navigation