Document Type : Research Paper

Authors

1 Aerospace Research Institute of Iran, Ministry of Science, Technology and Research, Tehran, Iran

2 M.Sc., Islamic Azad University, North Tehran Branch, Tehran, Iran

10.30699/jsst.2023.1349

Abstract

This paper deals with form-finding and free vibration analysis of a pre-stressed class-one triplex tensegrity structure. The form-finding is performed via a two-step procedure, the nodal coordinates connectivity matrix, and structural element force density determination. Accordingly, the possible states for the nodal coordinates and the structural force density of the triplex prism have been determined by trial and error (based on topology and member type knowledge) to satisfy the force density, and equilibrium matrices rank requirements. Based on different structural topologies, the equation of the motion in the frequency domain for free vibration analysis of the system is derived using the spectral element approach and dynamic shape functions. Simulations are provided for different system heights and the top-bottom aria ratios and compared with the FEM. The numerical simulations in the form of a comparative study of the natural frequencies of triplex tensegrity prism with different heights and cross-sections represent the system’s robustness with different topologies for single or multi-stage applications.

Keywords

Main Subjects

[1]     A. Tibert and S. Pellegrino, "Deployable tensegrity reflectors for small satellites," J. of Spac. and Rockets, vol. 39, No 5, pp 701-709, 2002.
[2]     C. Sultan and R. Skelton, "Deployment of tensegrity structures," Int. J. of Solids and Structures, vol. 40, No 18, pp. 4637-4657, 2003.
[3]     C. Russell and G. Tibert, "Deployment simulations of inflatable tensegrity structures," Int.J. of space struct., vol. 23, No2, pp. 63-77, 2008.
[4]     H. Furuya, "Concept of deployable tensegrity structures in space application," Int. J. of Space Struc., vol. 7, No 2, pp. 143-151, 1992.
[5]     K. Yildiz and G. A. Lesieutre, "Sizing and prestress optimization of Class-2 tensegrity structures for space boom applications," Eng. with Comp., Vol 38, No 2, pp. 1-14, 2020.
[6]     R. E. Skelton and M. C. De Oliveira, Tensegrity systems vol. 1, New York, Springer, 2009.
[7]     A. M. Popescu, R. Goyal, and M. Majji, "Design and Control of a Tensegrity Torus Spacecraft Composed of Reconfigurable Units," in AIAA Scitech 2021 Forum, p. 1387, 2021.
[8]     X. Li, W. Kong, and J. He, "A task-space form-finding algorithm for tensegrity robots," IEEE Access, vol. 8, pp. 100578-100585, 2020.
[9]     R. Goyal, M. Majji, and R. E. Skelton, "Integrating structure, information architecture and control design: Application to tensegrity systems," Mech. Sys. and Signal Proc., vol. 161, p. 107913, 2021.
[10]   L. Wen, F. Pan, and X. Ding, "Tensegrity metamaterials for soft robotics," Sci. Robot., vol. 5, No 45, p. eabd9158, 2020.
[11]   Y. Lu, X. Xu, and Y. Luo, "Path planning for rolling locomotion of polyhedral tensegrity robots based on dijkstra algorithm," J. of the Int. Association for Shell and Spatial Struc., vol. 60, No 4, pp. 273-286, 2019.
[12]   N. B. Kahla, M. H. E. Ouni, N. B. H. Ali, and R. A. Khan, "Nonlinear Dynamic Response and Stability Analysis of a Tensegrity Bridge to Selected Cable Rupture," Latin American J. of Solids and Struc., vol. 17, 2020.
[13]   D. E. Ingber, N. Wang, and D. Stamenović, "Tensegrity, cellular biophysics, and the mechanics of living systems," Reports on Progress in Physics, vol. 77, No4, p. 046603, 2014.
[14]   W. J. Lewis, Tension structures: form and behaviour: Thomas Telford, 2003.
[15]   Y. Wang and X. Xu, "Prestress design of tensegrity structures using semidefinite programming," Adv. in Civil Eng., vol. 2019, 2019.
[16]   X. Xu, Y. Wang, Y. Luo, and D. Hu, "Topology optimization of tensegrity structures considering buckling constraints," Journal of Structural Engineering, vol. 144, No 10, p. 04018173, 2018.
[17]   Y. Wang, X. Xu, and Y. Luo, "Topology design of general tensegrity with rigid bodies," Int. J. of Solids and Struc., vol. 202, pp. 278-298, 2020.
[18]   A. Pugh, An introduction to tensegrity: University of California Press, 1976.
[19]   A. Tibert and S. Pellegrino, "Review of form-finding methods for tensegrity structures," Int. J. of Space Struc., vol. 18, No 4, pp. 209-223, 2003.
[20]   S. H. Juan and J. M. M. Tur, "Tensegrity frameworks: Static analysis review," Mechanism and Machine Theory, vol. 43, No 7, pp. 859-881, 2008.
[21]   N. Vassart and R. Motro, "Multiparametered formfinding method: application to tensegrity systems," Int. J. of space struc., vol. 14, No 2, pp. 147-154, 1999.
[22]   K. Koohestani, "On the analytical form-finding of tensegrities," Composite Structures, vol. 166, pp. 114-119, 2017.
[23]   X. Xu, Y. Wang, and Y. Luo, "Finding member connectivities and nodal positions of tensegrity structures based on force density method and mixed integer nonlinear programming," Eng. Structures, vol. 166, pp. 240-250, 2018.
[24]   N. B. H. Ali, L. Rhode-Barbarigos, and I. F. Smith, "Analysis of clustered tensegrity structures using a modified dynamic relaxation algorithm," Int. J. of Solids and Stru., vol. 48, No 5, pp. 637-647, 2011.
[25]   Y. Chen, Q. Sun, and J. Feng, "Improved form-finding of tensegrity structures using blocks of symmetry-adapted force density matrix," J. of Struc. Eng.g, vol. 144, No 10, p. 04018174, 2018.
[26]   J. Cai, X. Wang, X. Deng, and J. Feng, "Form-finding method for multi-mode tensegrity structures using extended force density method by grouping elements," Composite Structures, vol. 187, pp. 1-9, 2018.
[27]   L.-Y. Zhang, S.-X. Zhu, S.-X. Li, and G.-K. Xu, "Analytical form-finding of tensegrities using determinant of force-density matrix," Composite Structures, vol. 189, pp. 87-98, 2018.
[28]   R. Burkhardt, "The application of nonlinear programming to the design and validation of tensegrity structures with special attention to skew prisms," J. of the Int. Association for Shell and Spatial Struct., vol. 47, No 1, pp. 3-15, 2006.
[29]   H. C. Tran and J. Lee, "Advanced form-finding of tensegrity structures," Computers & structures, vol. 88, No 3-4, pp. 237-246, 2010.
[30]   G. G. Estrada, H.-J. Bungartz, and C. Mohrdieck, "Numerical form-finding of tensegrity structures," Int. J. of Solids and Struct., vol. 43, No 22-23, pp. 6855-6868, 2006.
[31]   L. Zhang, Q. Gao, Y. Liu, and H. Zhang, "An efficient finite element formulation for nonlinear analysis of clustered tensegrity," Eng. Comp Vol 33, No 1, pp 252-273, 2016.
[32]   M. Pagitz and J. M. Tur, "Finite element based form-finding algorithm for tensegrity structures," Int. J. of Solids and Stru., vol. 46, No 17, pp. 3235-3240, 2009.
[33]   S. Yang and C. Sultan, "Free vibration and modal analysis of a tensegrity-membrane system," in ASME 2016 Int. Des. Eng. Tech. Conf. and Com. and Inf. in Eng. Conf., Volume 6: 12th International Conference on Multibody Systems, Nonlinear Dynamics, and Control. Charlotte, North Carolina, USA. August 21–24, 2016. V006T09A023. ASME. https://doi.org/ 10.1115/ DETC2016-59292
[34]   Z. Kan, H. Peng, B. Chen, and W. Zhong, "Nonlinear dynamic and deployment analysis of clustered tensegrity structures using a positional formulation FEM," Composite Structures, vol. 187, pp. 241-258, 2018.
[35]   X. Liu and J. Banerjee, "Free vibration analysis for plates with arbitrary boundary conditions using a novel spectral-dynamic stiffness method," Com. & Stru., vol. 164, pp. 108-126, 2016.
[36]   N. Ashwear and A. Eriksson, "Vibration health monitoring for tensegrity structures," Mech. Sys. and Signal Proc., vol. 85, pp. 625-637, 2017.
[37]   S. Zhang, R. Shen, T. Wang, G. De Roeck, and G. Lombaert, "A two-step FEM-SEM approach for wave propagation analysis in cable structures," Journal of Sound and Vibration, vol. 415, pp. 41-58, 2018.
[38]   H. Sharma, S. Mukherjee, and R. Ganguli, "Stochastic strain and stress computation of a higher-order sandwich beam using hybrid stochastic time domain spectral element method," Mech. of Adv. Mat. and Struc.,Vol 29, No 4, pp. 1-14, 2020.
[39]   J. M. Thompson, "A general theory for the equilibrium and stability of discrete conservative systems," Zeitschrift für angewandte Mathematik und Physik ZAMP, vol. 20, pp. 797-846, 1969.
[40]         H. Murakami and Y. Nishimura, "Initial shape finding and modal analyses of cyclic right-cylindrical tensegrity modules," Comp. & Struc., vol. 79, issue 9, pp. 891-917, 2001.