Abstract: In this paper, the dynamic behaviour of the tensegrity domes is explored. The consideration includes all cable structures called Geiger domes, i.e., two cases of configurations (with a closed and open upper section) and two variants of the nature of a dome (regular and modified) are taken into account. Particularly, the impact of the number of girders on the natural frequencies is analysed. A geometrically quasi-linear model is used, implemented in an original program written in the Mathematica environment. The results confirm that the number of girders affects the number of infinitesimal mechanisms. However, the dynamic behaviour does not depend on the number of mechanisms. The most important is the nature of a dome and the type of load-bearing girder. Especially, the behaviour of Geiger domes with a closed upper section is specific. In this case, not only the frequencies corresponding to the infinitesimal mechanisms depend on the prestress. There are additional frequencies that depend on prestress. The number of them, and the sensitivity on the initial prestress changes, depends on the number of girders.
Generally, for the same number of girders, the natural frequencies of regular domes are higher than for the modified ones.
B I B L I O G R A F I A[1] D.H. Geiger, “Roof structure”, U.S. Patent 4 736 553, 1988.
[2] D. Geiger, A. Stefaniuk, and D. Chen, “The design and construction of two cable domes for the Korean Olympics”, in Shells, Membranes and Space Frames, Proceedings IASS Symposium, 5–19 September 1986, Osaka. Elsevier, 1986, pp. 265–272.
[3] M. Ding, et al., “Experimental study and comparative analysis of a Geiger-Type ridge-beam cable dome structure”, International Journal of Civil Engineering, vol. 16, pp. 1739–1755, 2018, doi: 10.1007/s40999-018-0331y.
[4] Z. Gao, et al., “Four-step tensioning construction method and experimental study for rigid bracing dome”, International Journal of Steel Structures, vol. 18, no. 8, pp. 281–291, 2018, doi: 10.1007/s13296-017-1002-6.
[5] A. Zhang, C. Sun, and Z. Jiang, “Experimental study on the construction shape-forming process and static behaviour of double strut cable dome”, Journal of Zhejiang University – Science A (Applied Physics & Engineering), vol. 19, pp. 225–239, 2018, doi: 10.1631/jzus.A1700071.
[6] H. Deng, Q.F. Jiang, and A.S.K. Kwan, “Shape finding of incomplete cable-strut assemblies containing slack and prestress elements”, Computers and Structures, vol. 83, no. 21-22, pp. 1767–1779, 2005, doi:
10.1016/j.compstruc.2005.02.022.
[7] Q. Ma, et al., “Step-by-step unbalanced force iteration method for cable-strut structure with irregular shape”, Engineering Structures, vol. 177, pp. 331–344, 2018, doi: 10.1016/j.engstruct.2018.09.081.
[8] M. Saitoh, et al., “Design and Construction of Tension Strut Dome”, IEEE Journal of Solid-state Circuits, vol.1, pp. 67–78, 1994.
[9] F. Fu, “Non-linear static analysis and design of tensegrity domes”, Steel and Composite Structures, 2006, vol. 6, no. 5, pp. 417–433, 2006, doi: 10.12989/scs.2006.6.5.417.
[10] A. Albertin, et al., “Genetic algorithms in the optimization of cable systems”, Newsletter EnginSoft, vol. 9, no. 1, pp. 30–33, 2012.
[11] X. Wu, et al., “Measuring full static displacements of cable domes based only on limited tested locations”, Applied Mathematical Modeling, vol. 77, pp. 1054–1064, 2020, doi: 10.1016/j.apm.2019.08.018.
[12] L. Chen, et al., “Identification and adjustment of the pretension deviation in cable-strut tensile structures”, KSCE Journal of Civil Engineering, vol. 24, no. 1, pp. 143–152, 2020, doi: 10.1007/S12205-020-1473-4.
[13] Z. Jiang, et al., “Catenary equation-based approach for force finding of cable domes”, International Journal of Steel Research, vol. 19, no. 1, pp. 283–292, 2019, doi: 10.1007/s13296-018-0117-8.
[14] P. Zhang and J. Feng, “Initial prestress design and optimization of tensegrity systems based on symmetry and stiffness”, International Journal of Solids and Structures, vol. 106-107, pp. 68–90, 2017, doi: 10.1016/ j.ijsolstr.2016.11.030.
[15] Z. Wang, X. Yuan, and S. Dong, “Simple approach for force finding analysis of circular Geiger domes with consideration of self-weight”, Journal of Construction Steel Research, vol. 66, no. 2, pp. 317–322, 2010, doi: 10.1016/j.jcsr.2009.09.010.
[16] X. Yan, et al., “Mechanical properties of a hybrid cable dome under non-uniform snow distribution”, Journal of Constructional Steel Research, vol. 153, pp. 519–532, 2019, doi: 10.1016/j.jcsr.2018.10.022.
[17] K.Y. Volokh, O. Vilnay, and I. Averbuh, “Dynamics of cable structures”, Journal of Engineering Mechanics, vol. 129, no. 2, pp. 175–180, 2003, doi: 10.1061/(ASCE)0733-9399(2003)129:2(175).610 P. OBARA, M. SOLOVEI
[18] K. Seung-Dong and S. In-A, “A comparative analysis of dynamic instability characteristic of Geiger-Typed cable dome structures by load condition”, Journal of the Korean Association for Spatial Structures, vol. 14, no. 1, pp. 85–91, 2014, doi: 10.9712/KASS.2014.14.1.085.
[19] X. Wu, H. Deng, and D. Zhu, “Mode shape expansions for the dynamic testing of cable domes considering random pretension deviations”, Journal of Sound and Vibration, vol. 394, pp. 155–170, 2017, doi: 10.1016/j.jsv.2017.01.036.
[20] P. Obara, Dynamic and dynamic stability of tensegrity structures. Kielce, Poland: Wydawnictwo Politechniki Świętokrzyskiej, 2019 (in Polish).
[21] H. Murakami, “Static and dynamic analyses of tensegrity structures. Part 1. Non-linear equations of motion”, International Journal of Solids and Structures, vol. 38, no. 20, pp. 3599–3613, 2001, doi: 10.1016/S0020-
7683(00)00232-8.
[22] H. Murakami, “Static and dynamic analyses of tensegrity structures. Part II. Quasi-static analysis”, International Journal of Solids and Structures, vol. 38, no. 20, pp. 3615–3629, 2001, doi: 10.1016/S0020-7683(00)00233-X.
[23] P. Obara and J. Tomasik, “Parametric analysis of tensegrity plate-like structures: Part 2 – Quantitative analysis”, Applied Sciences, vol. 11, no. 2, art. no. 602, 2021, doi: 10.3390/app11020602.
[24] W. Gilewski, et al., “Application of singular value decomposition for qualitative analysis of truss and tensegrity structures”, Acta Scientiarum Polonorum Hortorum Cultus, vol. 14, no. 3, pp. 3–20, 2015.
[25] P. Obara, J. Kłosowska, and W. Gilewski, “Truth and myths about 2D tensegrity trusses”, Applied Sciences, vol. 9, no. 1, art. no. 179, pp. 1–19, 2019, doi: 10.3390/app9010179.
[26] P. Obara and J. Tomasik, “Parametric analysis of tensegrity plate-like structures: Part 1 – Qualitative analysis”, Applied Sciences, vol. 10, no. 20, art. no. 7042, 2020, doi: 10.3390/app10207042.
[27] S.-D. Kim and I.-A. Sin, “A comparative analysis of dynamic instability characteristic of Geiger-typed cable dome structures by load condition”, Journal of the Korean Association for Spatial Structures, vol. 14, no. 1, pp. 85–91, 2014, doi: 10.9712/kass.2014.14.1.085.
[28] EN 1993-1-11: 2006 Eurocode 3: Design of steel structures – Part 1–11: Design of structures with tension components. 
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