Improved Symmetry Method for the Mobility of Regular Structures Using Graph Products
Publication: Journal of Structural Engineering
Volume 142, Issue 9
Abstract
Mobility analysis plays a key role in form finding and design of novel kinematically indeterminate structures. For large-scale or complex structures, it demands considerable computations and analyses, and, thus, efficient method is of great interest. Because many structures could be viewed as the product of two or three subgraphs, such structures are called regular structures and usually hold certain symmetries. Combining graph theory with group representation theory, this paper proposes an improved symmetry method for the mobility of kinematically indeterminate pin-jointed structures. The concepts of graph products are described and utilized, to simplify the conventional symmetry-extended mobility rule. Based on the definitions of the Cartesian product, the direct product, and the strong Cartesian product, the authors establish the representations of nodes and members for the graph products, respectively. The proposed method focuses on the simple subgraphs, which generate the entire structure, and computes the matrix representations of the nodes and members under each symmetry operation. Therefore, symmetry analysis of the entire structure is transformed into independent evaluations on the subgraphs. Mobility of symmetric structures with a large number of nodes and members is studied, and the static and kinematic indeterminacy of the structures is evaluated using the proposed method.
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Acknowledgments
This work has been supported by the National Natural Science Foundation of China (Grant Nos. 51278116 and 51508089), the Priority Academic Program Development of Jiangsu Higher Education Institutions of China, and Scientific Research Foundation of Graduate School of Southeast University (Grant No. YBPY1201).
References
Altmann, S. L., and Herzig, P. (1994). Point-group theory tables, Clarendon Press, Oxford, U.K.
Calladine, C. R., and Pellegrino, S. (1991). “First-order infinitesimal mechanisms.” Int. J. Solids Struct., 27(4), 505–515.
Ceulemans, A., and Fowler, P. W. (1991). “Extension of Euler’s theorem to symmetry properties of polyhedra.” Nature, 353, 52–54.
Chen, Y., and Feng, J. (2012). “Generalized eigenvalue analysis of symmetric prestressed structures using group theory.” J. Comput. Civ. Eng., 488–497.
Chen, Y., and Feng, J. (2014). “Efficient method for Moore-Penrose inverse problems involving symmetric structures based on group theory.” J. Comput. Civ. Eng., 182–190.
Chen, Y., and Feng, J. (2015). “Group-theoretic method for efficient buckling analysis of prestressed space structures.” Acta Mech., 226(3), 957–973.
Chen, Y., Feng, J., and Zhang, Y. T. (2014). “A necessary condition for stability of kinematically indeterminate pin-jointed structures with symmetry.” Mech. Res. Commun., 60, 64–73.
Chen, Y., Guest, S. D., Fowler, P. W., and Feng, J. (2012). “Two-orbit switch-pitch structures.” J. Int. Assoc. Shell Spatial Struct., 53(3), 157–162.
Connelly, R., Fowler, P. W., Guest, S. D., Schulze, B., and Whiteley, W. J. (2009). “When is a symmetric pin-jointed framework isostatic?” Int. J. Solids Struct., 46(3), 762–773.
Connelly, R., and Whiteley, W. (1996). “Second-order rigidity and prestress stability for tensegrity frameworks.” SIAM J. Discrete Math., 9(3), 453–491.
Dai, J. S., Li, D., Zhang, Q., and Jin, G. (2004). “Mobility analysis of a complex structured ball based on mechanism decomposition and equivalent screw system analysis.” Mech. Mach. Theory, 39(4), 445–458.
Ding, X. L., Yang, Y., and Dai, J. S. (2011). “Topology and kinematic analysis of color-changing ball.” Mech. Mach. Theory, 46(1), 67–81.
Fowler, P. W., and Guest, S. D. (2000). “A symmetry extension of Maxwell’s rule for rigidity of frames.” Int. J. Solids Struct., 37(12), 1793–1804.
Guest, S. D. (2006). “The stiffness of prestressed frameworks: A unifying approach.” Int. J. Solids Struct., 43(3–4), 842–854.
Guest, S. D., and Fowler, P. W. (2005). “A symmetry-extended mobility rule.” Mech. Mach. Theory, 40(9), 1002–1014.
Guest, S. D., and Fowler, P. W. (2007). “Symmetry conditions and finite mechanisms.” J. Mech. Mater. Struct., 2(2), 293–301.
Guest, S. D., Schulze, B., and Whiteley, W. J. (2010). “When is a symmetric body-bar structure isostatic?” Int. J. Solids Struct., 47(20), 2745–2754.
Kaveh, A., and Alinejad, B. (2009). “A general theorem for adjacency matrices of graph products and application in graph partitioning for parallel computing.” Finite Elements Anal. Des., 45(3), 149–155.
Kaveh, A., and Koohestani, K. (2008). “Graph products for configuration processing of space structures.” Comput. Struct., 86(11), 1219–1231.
Kaveh, A., and Nikbakht, M. (2008). “Stability analysis of hyper symmetric skeletal structures using group theory.” Acta Mech., 200(3–4), 177–197.
Kaveh, A., and Nikbakht, M. (2010). “Improved group-theoretical method for eigenvalue problems of special symmetric structures, using graph theory.” Adv. Eng. Software, 41(1), 22–31.
Kaveh, A., and Rahami, H. (2004). “An efficient method for decomposition of regular structures using graph products.” Int. J. Numer. Methods Eng., 61(11), 1797–1808.
Kaveh, A., and Rahami, H. (2005). “A unified method for eigendecomposition of graph products.” Commun. Numer. Methods Eng., 21(7), 377–388.
Kettle, S. F. (2008). Symmetry and structure: Readable group theory for chemists, Wiley, Chichester, U.K.
Koohestani, K. (2012). “Exploitation of symmetry in graphs with applications to finite and boundary elements analysis.” Int. J. Numer. Methods Eng., 90(2), 152–176.
Kovács, F. (2004). “Mobility and stress analysis of highly symmetric generalized bar-and-joint structures.” J. Comput. Appl. Mech., 5(1), 65–78.
Kuznetsov, E. N. (1991). “Systems with infinitesimal mobility: Part I—Matrix analysis and first-order infinitesimal mobility.” J. Appl. Mech., 58(2), 513–519.
Liu, R., Serré, P., and Rameau, J. (2013). “A tool to check mobility under parameter variations in over-constrained mechanisms.” Mech. Mach. Theory, 69, 44–61.
Maxwell, J. C. (1864). “On the calculation of the equilibrium and stiffness of frames.” Philos. Mag., 27, 294–299.
Ohsaki, M., and Zhang, J. Y. (2006). “Stability conditions of prestressed pin-jointed structures.” Int. J. Non-Linear Mech., 41(10), 1109–1117.
Pellegrino, S., and Calladine, C. R. (1986). “Matrix analysis of statically and kinematically indeterminate frameworks.” Int. J. Solids Struct., 22(4), 409–428.
Sabidussi, G. (1959). “Graph multiplication.” Math. Z., 72(1), 446–457.
Sultan, C. (2013). “Stiffness formulations and necessary and sufficient conditions for exponential stability of prestressable structures.” Int. J. Solids Struct., 50(14–15), 2180–2195.
Tarnai, T., and Szabó, J. (2000). “On the exact equation of inextensional, kinematically indeterminate assemblies.” Comput. Struct., 75(2), 145–155.
Vassart, N., Laporte, R., and Motro, R. (2000). “Determination of mechanism’s order for kinematically and statically indetermined systems.” Int. J. Solids Struct., 37(28), 3807–3839.
Wei, G. W., Chen, Y., and Dai, J. S. (2014). “Synthesis, mobility and multifurcation of highly overconstrained deployable polyhedral mechanisms.” J. Mech. Des., 136(9), 091003.
Whitehead, A. N., and Russell, B. (1927). Principia mathematica, Cambridge University Press, Cambridge, U.K.
Zhang, J. Y., Guest, S. D., and Ohsaki, M. (2009). “Symmetric prismatic tensegrity structures. Part II: Symmetry-adapted formulations.” Int. J. Solids Struct., 46(1), 15–30.
Zhang, P., Kawaguchi, K., and Feng, J. (2014). “Prismatic tensegrity structures with additional cables: Integral symmetric states of self-stress and cable-controlled reconfiguration procedure.” Int. J. Solids Struct., 51(25–26), 4294–4306.
Zhao, J. S., Chu, F. L., and Feng, Z. J. (2009). “The mechanism theory and application of deployable structures based on SLE.” Mech. Mach. Theory, 44(2), 324–335.
Zingoni, A. (2009). “Group-theoretic exploitations of symmetry in computational solid and structural mechanics.” Int. J. Numer. Methods Eng., 79(3), 253–289.
Zingoni, A. (2014). “Group-theoretic insights on the vibration of symmetric structures in engineering.” Phil. Trans. R. Soc. A: Math. Phys. Eng. Sci., 372(2008), 20120037.
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Published In
Journal of Structural Engineering
Volume 142 • Issue 9 • September 2016
Copyright
© 2016 American Society of Civil Engineers.
History
Received: Feb 25, 2015
Accepted: Jan 11, 2016
Published online: Mar 28, 2016
Discussion open until: Aug 28, 2016
Published in print: Sep 1, 2016
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