Tensegrity structure

A tensegrity structure is a combination of compression members and tension cables. I have already discussed the advantages and disadvantages of this structure based on some reputed journals.

1. A SEMINAR ON TENSEGRITY STRUCTURE- RESPONSE TO DYNAMIC LOADING BY SHRIDHAN (170918023) Kurilpa Bridge, Brisbane, Australia

2. CONTENT INTRODUCTION ADVANTAGES DISADVANTAGES APPLICATION METHODOLOGY CONCLUSION REFERENCES

3. INTRODUCTION Tensegrity means Tension Integrity structure. It is one type of truss structure which consist of pin-jointed tension and compression members. A strict tensegrity is composed of a set of continuous cables in tension, and a set of discontinuous struts in compression . Tensegrity structure are spatial, reticulated and light weight structures. Stability is provided by the self stress state in tensioned and compressed members. The simple tensegrity structure is a 3-bar tensegrity system contains 3 struts and 12 pre-stress cables. [2, 4, 5, 7]

4. Tensegrity structures exist as a pre-stressed stable connection of bars and strings. [4] Fig-1: The Simplest 3-bar Tensegrity system, 1: Bar, 2: Declining string, 3: Upper level string, 4: Lower level string, [11]

5. ADVANTAGES It offering maximum amount of strength for a given amount of building material. Due to short length of components in compression, buckling is very rare. Very useful in terms of absorption of shocks and seismic vibration so desirable in areas where earthquake are a problem. For large construction, the process would be easy to carry out, since the structure is self-scaffolding. It provides minimal mass structures for a variety of loading conditions. [2, 10]

6. DISADVANTAGES Difficulties during manufacturing process. There is no specific standard design code. In addition to material cost, tensegrities require additional pre-tensioning labour and careful assembling on site. Higher labour cost. [7]

7. APPLICATION It is used for light weight structures. Solar facades and wind generators. Satellite solar panel. Light weight footbridge, roof truss, domes, stadium. This tensegrity concept is also used in architecture, biology, robotics and aerospace engineering. Elegant appearance with high strength to weight ratios. [7, 10]

8. 1 . Static and dynamic analysis of Tensegrity structures. Part-1. Nonlinear equation of motion by Murakami ( 2001) Objectives: A linearized equation of motion are used to conduct modal analysis of a three bar Tensegrity module and a Tensegrity beam. Tensegrity structure: Here, Bars and cables are assumed to be made of steel with Young’s modulus Y =200 GPa , and mass density ρ =7860 kg/m 3. Height of three bar module is 1 m. The radius of cables 0.001 m and the inner and outer radii of hollow circular cylindrical bars 0.018 m and 0.022 m. The element force of cables changed from -200 N to -1000 N. The corresponding compressive force in bar was between 0.4 and 1.99 MPa.

9. For beam six-stage Tensegrity system is considered. It built by stacking the three bar modules with alternating twisting directions and by adding additional strings. Methodology: The dynamic response can be investigated by conducting modal analysis. Linearized equations of motion is given by: M L Ÿ+K T Y-f L (t)=ç(0) Where , M L = mass matrix, K T = Tangent stiffness matrix, f L = External force vector Consider a harmonic motion of the form : u=ū exp ( i ω t), Where, ū=amplitude , ω =angular frequency. K t ū = ω 2 Mū Fig-2: Three bar Tensegrity module

10. Conclusion: Table-1: The natural frequency of the lowest four modes for various pre-stress. Fig-3: (a) The first mode and (b) The second mode of the Three-bar Tensegrity module .

11. In the first mode the axial Deformation accomplished by adjacent counter-twisting sets of three bars. Therefore the first mode is globally Longitudinal mode. The second mode appears as A flexural mode. The first mode represents an infinitesimal mechanism mode of beam so natural frequency is much smaller Fig-4: (a) The first mode and (b) the second mode of a six-stage than second mode. Tensegrity beam.

12. 2 . Design optimization and dynamic analysis of a tensegrity-based footbridge by Nizar et al. ( 2010) Objectives: This paper presents design optimization and dynamic analysis of a tensegrity-based civil structure. A pentagonal hollow rope is used as a structural system for a 21.6 m-span footbridge. The dynamic behaviour of the tensegrity-based footbridge is also analysed. Footbridge model: The used steel grade is S355 with modulus of elasticity of 210,000 MPa and yield stress of 355 MPa. Cables are made of stainless steel with a modulus of elasticity of 120,000 Mpa . Footbridge deck weight assumed as 100 kg/m2.

13. This footbridge is designed so as to have 2.0 width internal space for walking and a clearance of 2.5 m. A six pentagon module used which contains 15 struts and 30 cables. The length of pentagon is 360 cm, Length of diagonal and intermediate struts are 488 cm. Layer cable have a length of 330 cm and while x-cables are 250 cm length. As per Swiss code SIA 261: Consider UDL of 4kN/m2 and Point load 10kN . Considering uniform wind pressure of 1.3kN/m2 to 1.4kN/m2. The nodes of the structure at both extremities are supposed to be fixed in all three translation directions.

14. Fig-5: The tensegrity footbridge Fig-6: Illustration of pentagon module members

15. Methodology: The linearized equation of motion of a tensegrity structure is given as: [M] { ű } + [ K T ] {u} = {F} Where, [M] and [K T ] are mass and tangent stiffness matrix. {F} is a applied load vector, { ű } and {u} are vector of nodal acceleration and displacement. For a small harmonic motion {u} = {ū} sin( ω t) Where, ω = angular frequency, {ū }= Displacement amplitude vector. [K T ] {ū }= ω 2 [M]{ ū } The weight of footbridge deck is taken into account in the mass matrix. The self weight of the pedestrian is taken as 2pedestrian/m2 to 4pedestrian/m2.

16. Load combination: 1. For Ultimate Limit State (ULS): 1.35G+1.5Q+0.8P+0.6W and 1.35G+1.5Q+1.2P+0.6W 2. For Serviceability of Limit State (SLS): 1.0G+0.4Q+1.0P+0.6W Where, G= Dead load, Q= Live load, P= Loads due to self stress, W= Wind load

17. Conclusion: This study shows that tensegrity system with connected compression Table-2: Natural frequency of Tensegrity footbridge elements could compete with traditional structural system Eigen-frequency analysis results showed that the tensegrity based footbridge can withstand dynamic loads induced by a small number of pedestrians. Fig-7: Variation of natural frequency w.r.t. the no. of pedestrian on the footbridge

18. Fundamental frequency of tensegrity footbridge is not directly influenced by the self-stress level. Frequency influenced by other design parameters such as the c/s sizes of x-cables. Fig-8: Variation of natural frequency w.r.t. the Fig-9: Variation of natural frequency w.r.t. the c/s area of diagonal struts. c/s area of layer cable.

19. Fig-10: Variation of natural frequency w.r.t. the Fig-11: Variation of natural frequency w.r.t. the c/s area of intermediate struts. c/s area of x-cable.

20. 3. Dynamic behaviour and vibration control of a Tensegrity structures by Hadj Ali, and Smith ( 2010) Objectives: Dynamic behaviour of the Tensegrity structure is experimentally and analytically identified through testing under dynamic excitation. Laboratory testing is carried out for multiple self-stress levels and for different excitation frequencies . Tensegrity structure: There was 5 modules and rest on 3 supports. Surface area=15 m2, Height=1.2 m, Distributed DL=300 N/m2, There is composed of 30 struts and 120 tendons. Struts are fibre reinforced polymer tubes with modulus of elasticity of 28 GPa , and a specific mass of 2420 Kg/m2 with dia. Of 60 mm and c/s area of 703 mm2.

21. Tendons are stainless steel cables with modulus of elasticity of 115 GPa and c/s area of 13.85 mm2. Also equipped with 10 active actuated struts are used for strut length controlling by the way self-stress state in Tensegrity structure. Methodology: Analytical Method: 1. Equation of motion. 2. Stiffness matrix is a combination of Linear stiffness matrix (K) and Geometrical stiffness matrix (Kg). 3. Convert the mass and the stiffness Matrix to global co-ordinates. Fig-12: Experimental Tensegrity system (model)

22. 4. The generalised Eigen problem is then obtained considering a small harmonic motion of the form: u=ū sin( ω t) Where, ω =Angular frequency and ū=Amplitude vector. K t ū = ω 2 Mū Evaluate the first 5 natural frequencies is studied with respect to the self stress level. Then active strut length varied from 1295.5 mm to 1308.5 mm by the steps of 1 mm. This length changes the geometry of the system. Free vibration Test: The load is applied and suddenly removed and measured the displacement. The load value, Loaded node and Measured nodes shown in table. Table-3: Loading for free vibration testing

23. Forced vibration test: The shaker is used in this study was an electro-magnetic device composed of an electric motor to a linearly activated mass. For all test the shaker was connected to node-43 and displacement measured at the top surface nodes. Stress levels are varied around a reference stress (Ref) through increments of mm elongations and contractions. Fig-13: Active struts of tensegrity structure Fig-14: Basic module of Tensegrity structure.

24. Conclusion: For Analytical model: Table-4: Natural frequency for FEM method Due to active strut movement: Fig-15: Number of slack cables for different struts Fig-16: Tension in a reference cable for different active strut movements

25. Movements Table-5: Natural frequencies compared with Free vibration Fig-17: First natural frequency for different active strut movement.

26. Fig-18: Time history of vertical displacement of node 39 (600 N at Node-16)

27. Fig-19: Time history of vertical displacement of node 50 (600 N at Node-16)

28. Comparison with theoretical and experimental data Fig-20: Evaluations of first natural frequency with respect to self-stress level

29. 4 . Analysis of tensegrity structures subjected to dynamic loading using a Newmark approach by Faroughi and Lee (2015) Objective: The main objective is to highlight the Newmark approach, as a direct time integration scheme, can be adopted to investigate the behaviour of tensegrity structures subjected to dynamic loading. Assumptions: 1. The materials are assumed linear elastic and members are prismatic. 2. Struts are elements that carry compressive forces . 3. Strings are elements that carry only axial tensile forces. 4. The tensegrity systems are subjected to external load only at nodes.

30. Methodology: Three numerical examples are considered, Two examples are 3-D grid structure and the other is 2-D cantilever problem. 1. 6 x 6 quadruplex tensegrity model 2. 5 x 4 quadruplex tensegrity model 3. 3 module cantilever model Assume Boundary conditions: 1) The one lower corner node is completely fixed, 2) All nodes along the two edges are fixed in the Z and Y directions only .

31. From the equation of motion: Here Mg, Cg, and K T g are respectively mass, damping and tangent stiffness matrices in global direction and F is external dynamic loading. K T g = K E g + K G g K T g is composed of 2 parts. (1) K E g material stiffness matrix and (2) K G g geometrical stiffness matrix. Where, C l = β 1 m l + β 2 K T l Here β 1 and β 2 denotes Rayleigh damping coefficients. Initial neglect damping and external force vector. ω 2 = (K T g )(M g ) -1 From this equation calculate the eigenvalues for all three types of models.

32. Newmark’s time integration formula: Where, ∆t= time interval, α and β = Constant for estimation approach, This Newmark’s method is stable if β≥ 0.5 and α≥ 0.5 β Out of 3 models first discussing 5 x 4 quadruplex model . This structure composed of 79 nodes, 209 cables and 80 struts. The harmonic load applied in a form of: f(t)= f sin(2 πω 1 t) Where, f = 10 N, magnitude of harmonic input. ω 1 = 2.2023 Hz, First Eigen value of Tensegrity structure Then analyse the structure up to 10 s with time step 10e-4 s.

33. Fig-21: (5 X 4) quadruplex modules Table-6: Properties of module parameters

34. Results: Fig-22: Behaviour of the 5x4 tensegrity grid under external loads Table-7. Comparison the first six eigen values of tensegrity grid structure

35. Comparison between all three 6 x 6 quadruplex model, 5 x 4 quadruplex model and 3-module cantilever beam. Table-8: Initial properties for all three models

36. Hggghgh Fig-23: 2D 3-module cantilever model Fig-24: 3D 6 x 6 quadruplex module.

37. Conclusion The effect of damping and slackening condition on the dynamic analysis of tensegrity structure were addressed. Through comparison with classical method, Newmark method is efficient to solve the linearized tensegrity structure.

38. 5 . Dynamic behaviour and vibration mitigation of a spatial Tensegrity beam by Feng et al. (2018) Objectives: This paper focuses on employing an efficient control strategy to vibration mitigation for a spatial Tensegrity beam. To evaluate the differences between the nonlinear and linear models. Assumptions: 1. Cables and struts are connected by pin joins, 2. Strut members carry compressive forces, 3. Cable members only carry axial tensile forces, 4. The external loads only act on the nodes of whole system, 5. The local and global bucking of struts are neglected.

39. Methodology: Consider 5 single quadruplex modules consisting 28 nodes, 56 cables and 20 struts as shown in Fig. 1. The Boundary conditions are: Node 1 is fixed in all direction, Node-2 has motion along Z-axis, Node-26 is only fixed in Z-direction, Node-27 is free to move in Y-direction. Table-9: Geometric and mechanical Fig-25: Spatial quadruplex beam with 5 units Properties of beam

40. Dynamic load applied: F(t)=8sin(2t+50) in all direction. Time step: 0.02 s Unit-3 (U3) was selected for load and displacement calculation. In the U3 unit node-13 and node-16 are selected for results of displacement and for force calculation a cable (node 11-16) and a strut (node 11-15) was used. There are 5 different cases depends on the values of α and β which are the coefficient for weighting matrices. Based on the position of actuator there are 5 different cases .

41. Table-10: Control scenario for struts and cables Table-11: Initial forces for spatial Tensegrity beam

42. Conclusion: Fig-26: Response of displacement scenario A: (a) full record of displacement of node 13, (b) zoomed view 40–50 s of node 13, (c) full record of displacement of node 16, (d) zoomed view 40–50 s of node 16.

43. Fig-27: Response of displacement scenario A: (a) full record of displacement of node 13, (b) zoomed view 40–50 s of node 13, (c) full record of displacement of node 16, (d) zoomed view 40–50 s of node 16 .

44. The dynamic response of nonlinear Tensegrity system are depends on the actuator number, placement, and parameters of linear quadratic regulator. The placement of actuators in struts give better performance than in cable for both displacement and internal force. This is mainly due to weight of strut. For a large Tensegrity systems, the controls using continuous cables seems much more reasonable. Table-12: Response of displacement scenario A

45. Conclusion The stability and natural frequency of tensegrity structure is directly depends on the length of cables. Tensegrity structures promise to be highly efficient in the ratio of material to both performance and maintenance . It is the light weight structure so material time saving. It is acts as a scaffolding so no need to use a scaffolding. There was no any specific code related to tensegrity structure so design of structure is critical.

46. REFERENCES [1] H. Murakami, Static and dynamic analysis of Tensegrity structures. Part-1. Nonlinear equation of motion, International Journal of Solids and Structures, 38 (2001) 3599-3613. [2] R. E. Skelton et. al, An introduction to the mechanics of Tensegrity structures, IEEE conference, (2001) USA. [3] C. Sultan, M. Corless , E. Skelton, Linear dynamics of Tensegrity structures, Engineering Structures, 24 (2002) 671-685 . [4] D. Williamson, R. E. Skelton, J. Han, Equilibrium conditions of a tensegrity structure, International Journal of Solids and Structures, 40 (2003) 6347-6367 .

47. [5] J. Y. Zhang, M. Ohsaki , Y. Kanno , A direct approach to design of geometry and forces of tensegrity systems, International Journal of Solids and Structures, 43 (2006) 2260-2278 . [6] N . Bel Hadj Ali, I.F.C. Smith, Dynamic behavior and vibration control of a tensegrity structure, International Journal of Solids and Structures, 47 (2010) 1285-1296 . [ 7 ] N. Bel Hadj Ali, L. Rhode Barbarigos , Alberto A. Pascual Albi, Ian F. C. Smith, Design optimization and dynamic analysis of a tensegrity-based footbridge, Engineering Structures, 32 ( 2010) 3650-3659. [8] S. Faroughi , J. Lee, Analysis of tensegrity structures subject to dynamic loading using a Newmark approach, Journal of Building Engineering, 2 (2015) 1–8.

48. [9] F. Fabbrocino , G. Carpentieri , Three-dimensional modeling of the wave dynamics of tensegrity lattices, Composite Structures, 173 (2017) 9–16 . [10] M. C. Cimmino et. al, Composite solar facades and wind generators with Tensegrity architecture, Composites Part-B, 115 (2017) 275-281. [ 11] H. Liu, J. Zhang, M. Ohsaki , New 3-bar prismatic tensegrity units, Composite Structures, 184 (2018) 306-313 . [ 12] Xiaodong Feng , Mohammad S. Miah , Yaowen Ou , Dynamic behaviour and vibration mitigation of a spatial tensegrity beam, Engineering Structures , (2018 ).

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