Advertisement

Group theoretic approach to large-deformation property of three-dimensional bar-hinge mechanism

  • Ryo Watada
  • Makoto Ohsaki
  • Yoshihiro Kanno
Original Paper Area 2
  • 15 Downloads

Abstract

A group-theoretic approach is presented for investigation of large-deformation property of bar-hinge mechanisms with dihedral symmetry in three-dimensional space. The number of the compatibility conditions at bar-ends is reduced by formulating them with respect to the null space of the linear compatibility matrix. It is shown that the system of reduced compatibility equations inherits the group equivariance from the original compatibility equations. This inheritance is used to develop a method to judge whether the frame has a finite mechanism mode. Sufficient conditions for large deformation mechanisms are derived based on the symmetry properties of infinitesimal mechanism modes and generalized self-equilibrium force modes. The detailed procedure of the method is shown through the numerical examples.

Keywords

Bar-joint mechanism Artibtrarily inclined hinge Group theory Dihedral group 

Mathematics Subject Classification

20-04 70B15 

References

  1. 1.
    Zingoni, A.: Group-theoretic exploitations of symmetry in computational solid and structural mechanics. Int. J. Numer. Meth. Eng. 79, 253–289 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ikeda, K., Murota, K.: Imperfect Bifurcation in Structures and Materials, 2nd edn. Applied Mathematical Sciences. Springer, New York (2010)CrossRefGoogle Scholar
  3. 3.
    Ikeda, K., Ohsaki, M., Kanno, Y.: Imperfection sensitivity of hilltop branching points of systems with dihedral group symmetry. Int. J. Non-Linear Mech. 40, 755–774 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Zhang, J.Y., Guest, S.D., Ohsaki, M.: Symmetric prismatic tensegrity structures: Part I. Configuration and stability. Int. J. Solids Struct. 45(1), 1–14 (2009)CrossRefGoogle Scholar
  5. 5.
    Kanno, Y., Ohsaki, M., Murota, K., Katoh, N.: Group symmetry in interior-point methods for semidefinite program. Optim. Eng. 2, 293–320 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, Y., Feng, J.: Generalized eigenvalue analysis of symmetric prestressed structures using group theory. J. Comput. Civil Eng. 26(4), 488–497 (2012)CrossRefGoogle Scholar
  7. 7.
    Ohsaki, M., Kanno, Y., Tsuda, S.: Linear programming approach to design of spatial link mechanism with partially rigid joints. Struct. Multidisc. Optim. 50, 945–956 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ohsaki, M., Tsuda, S., Miyazu, Y.: Design of linkage mechanisms of partially rigid frames using limit analysis with quadratic yields functions. Int. J. Solids Struct. 88–89, 68–78 (2016)CrossRefGoogle Scholar
  9. 9.
    Guest, S.D., Fowler, P.W.: Symmetry conditions and finite mechanisms. J. Mech. Mater. Struct. 2(2), 293–301 (2007)CrossRefGoogle Scholar
  10. 10.
    Schulze, B., Guest, S.D., Fowler, P.W.: When a symmetric body-hinge structure isostatic? Int. J. Solids Struct. 51, 2157–2166 (2014)CrossRefGoogle Scholar
  11. 11.
    Watada, R., Ohsaki, M.: Series expansion method for determination of order of 3-dimensional bar-joint mechanism with arbitrarily inclined hinges. Int. J. Solids Struct. (2018).  https://doi.org/10.1016/j.ijsolstr.2018.02.012 CrossRefGoogle Scholar
  12. 12.
    Ikeshita, R.: Bifurcation Analysis of Symmetric Mechanisms by using Group Theory (in Japanese), Graduation thesis, Department of Mathematical Engineering and Information Physics, School of Engineering, The University of Tokyo (2013)Google Scholar
  13. 13.
    Cheng, H., Gupta, K.C.: An historical note on finite rotations. J. Appl. Mech. 56(1), 139–145 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Haug, E.J.: Computer Aided Kinematics and Dynamics of Mechanical Systems: Basic Methods, vol. 1. Allyn and Bacon, Boston (1989)Google Scholar
  15. 15.
    Sattinger, D.H.: Group Theoretic Methods in Bifurcation Theory. Lecture Notes in Mathematics, vol. 762. Springer, Berlin (1979)Google Scholar
  16. 16.
    Thompson, J.M.T., Hunt, G.W.: A General Theory of Elastic Stability. Wiley, New York (1973)zbMATHGoogle Scholar
  17. 17.
    Marsden, J.E., Ratiu, S.R.: Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, vol. 17, 2nd edn. Springer-Verlag, New York (1999)CrossRefGoogle Scholar
  18. 18.
    Meyer, C.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia (2000)CrossRefGoogle Scholar
  19. 19.
    Bishop, D.M.: Group Theory and Chemistry. Dover Publications, Oxford (1973)zbMATHGoogle Scholar
  20. 20.
    Ginsberg, J.H.: Advanced Engineering Dynamics. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar

Copyright information

© The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Architecture and Architectural EngineeringKyoto UniversityKyotoJapan
  2. 2.Takenaka CorporationOsakaJapan
  3. 3.Mathematics and Informatics CenterThe University of TokyoTokyoJapan

Personalised recommendations