Construction of a Unit Cell Tensegrity Structure

Transcription

1 The 14th IFToMM World Congress, Taipei, Taiwan, October 25-3, 215 DOI Number: /IFToMM.14TH.WC.OS13.17 Construction of a Unit Cell Tensegrity Structure A. González 1 A.N. Luo 2 H.P. Liu 3 Harbin Engineering University Harbin Engineering University Harbin Engineering University Harbin, China Harbin, China Harbin, China Abstract: A tensegrity structure is a special kind of system composed of three or more bars and strings, where any external load is distributed exclusively as axial forces along its members. Each individual unit is made of compressive and tensile elements, and presents tensegrity properties such as With only two geometric parameters (radius r and height l ) and the number of bars p, the nodes matrix N can be generated and the model built. Following steps guide through the creation of the connectivity matrices C S and C B. With the aid of the software Matlab, the internal force density method is used to confirm the equilibrium of the unit cell and suggest modifications. Finally, a three-bar tensegrity structure is built to confirm the results. Keywords: Tensegrity structure, Unit cell, Internal force density Since his early work, Snelson [6] combined a series of -shaped unit structures, joined only by tensile elements, to make his sculptures. As shown in Fig.1, using this technique he was able to create various figures, from masts to animal-shaped. Although his work was focused on sculptures and based on observation, it has inspired further research in the fields of physics, mathematics and engineering, to give applications in science and technology. I. Introduction A tensegrity structure is a special kind of system composed of three or more bars and strings, where any external load is distributed exclusively as axial forces along its members. Guest [1] gives a similar definition, but also states that the compression members in the tensegrity must not touch each other. The applications for these structures vary from sculptures, communications towers and emergency relief shelters to satellite antennas. A second definition by Motro [2] establishes that a tensegrity is a system in a stable self-equilibrated state comprising a discontinuous set of compressed components inside a continuum of tensioned components. This self-equilibrium, or self-stress, state is the initial condition for the structure, before any load is applied; and the structure should have the ability to find a new equilibrium position after being loaded. For Gómez Jauregui [3], the self-equilibrium condition can be considered as sufficient for distinguishing between real and fake tensegrities. Inspired by the nature, where animals and plants are made of cells, many complex tensegrity structures such as telescopes and masts can be composed of simple unit cell structures. Some authors, like Pollard [4] and Liedl [5], from the biology discipline even make cases for these cells being a kind of tensegrity structure, with compressive-tensile cytoskeletons. 1 andres@hrbeu.edu.cn 2 luoani@hrbeu.edu.cn 3 liuheping@hrbeu.edu.cn Fig. 1. Dragon, sculpture made by Snelson using a series of unit cells combined [7] Another notable scientist, Skelton, has studied the concept of using numerous unit cell tensegrities for creating complex structures, specially focusing on the minimal mass design. His books and papers give an approach to the analysis of these unitary tensegrities; from the forces involved to the application of control methods for shape-controlling. This latter concept is another important advantage of tensegrity simplex cells. Skelton and his group [8] have proposed that small variations in one or more of the strings of the structure could lead to a change of the shape, while maintaining the tensegrity conditions of stability and Using this concept, they have been able to design complex structures such as parabolic tensegrity rigid structures like the one shown in Fig.2. This is composed of many prismatic units. Controlling the tension on the strings on one or more units, the system can compensate the effects of gravity adjusting the curvature of the telescope.

2 is the characteristic angle of regular polygons, the base nodes are given by (x, y, z) = (r cos θ, r sin θ, ) The top nodes are calculated the same way, but the rotation angle α must be included (x, y, z) = (r cos(θ + α), r sin(θ + α), l ) Fig. 2. Parabolic telescope composed of unit cell structures, designed by Skelton [8] II. Prismatic unit cells From a topographic point of view, prismatic tensegrity structures are basically two polygons with the same number of sides located on parallel planes and joined through their vertices by tensile and compressive elements. If the bottom polygon is considered the base of the cell, the top polygon is rotated counterclockwise an angle α. Depending on the desired application, these polygons could present a different circumradius. However, for this paper only regular polygons with the same radius will be considered. This way, the shape of the unit structure can be simplified as a cylinder, whose geometry is defined by a radius r and a height l. A third necessary parameter is the number of compressive elements p. A representation of this set of parameters is given in Fig. 3. l o n 3 n 4 r n 5 θ Z n 2 n 1 n 6 Fig. 3. Modelling of the unit cell and definition of the parameters Motro [2] and Connelly et al. [9] propose the idea that each node could be connected by just two strings; however, in this paper the condition of three strings m per node given by Skelton [1] is used. Each vertex of the polygon is considered as a node n i of the tensegrity. If α Y θ = 2π/p (1) It can be proved from a static analysis that the angle α must be in the range α π/2. However, Skelton [1] and Motro [2] define the optimal angle α = π/2 π/p (2) the conclusion comes from the work of Connelly [11]. The previous n i nodes are then arranged in a node matrix N, where each column of the matrix represents the coordinates of a node N 3 n = [n 1 n 2 n n ] 3 n The node coordinates are also arranged as a nodes vector n of size 3n n 1 n 2 n 3n 1 = [ n n ]3n 1 To represent the connectivity between the nodes the bars connectivity matrix CB and the strings connectivity matrix CS are defined. These matrices are size p n and m n respectively and are created as follows 1 if (i, j) is the initial node of member p i C S(i,j) = { 1 if (i, j) is the final node of member p i if (i, j)has no relation with member p i The CB matrix is generated in the same way, for every m i member. Then the connectivity matrix C becomes C = [ C B C S ] Finally, the bars matrix B and the strings matrix S are defined. These matrices are size 3 p and 3 m, and contain the direction vectors of the bar and strings members respectively B = NC B T (3) S = NC S T (4) The previous steps defined the topology of the unit cell; the next step is to establish if the desired configuration is stable according to the external load. Connelly [11] states that if the equilibrium stresses in a structure are positive on the external edge cables, and negative on the internal struts, this tensegrity structure is super stable. Among other methods found throughout the literature, the internal force density method can be considered as convenient for engineering purposes, since it is based on matrices and only requires the designer to know the geometry of the structure and the external forces involved. This method calculates the stresses on each member of the structure as a function of their respective length; these relations are called internal force densities. λ i corresponds

3 to the bar internal force density and γ i to the strings internal force density. An intermediate step is the creation of the external forces matrix W. In this matrix the columns are the vectors of the external forces wi acting on the node i. W 3 n = [w 1 w n ] 3 n The external force vectors wi are also arranged in an external forces vector w. w 1 w 3n 1 = [ w n ]3n 1 In their paper, Luo and Skelton [12] present the equilibrium equation where W = NK (5) K = C S T γ C S C B T λ C B (6) γ and λ represent the internal force density matrices for strings and bars, respectively. For the solution of this equation, the forces vector w and the nodes vector n are used and (5) is transformed into w 1 SC S1 w 2 SC [ = S2 w n ]3n 1 [ SC Sn BC B1 BC B2 γ [ λ ] (m+q) 1 BC Bn]3n (m+q) The algebra of this transformation can be studied in [12]. To simplify computations, the left part of the new matrix is named Aγ and the right part is named Aλ. C Si and C Bi are the diagonal square matrices whose entries are the i columns of CS and CB respectively. If A = [A γ A λ ] 3n (m+q) the force equilibrium function can now be written as the linear system (7) w 3n 1 = A [ γ λ ] (8) The force vector w is known, and the matrix A is calculated from the geometry of the structure. The solution of the equilibrium problem is obtained from the values of the internal force densities vector ifd ifd = [ γ λ ] the matrices are obtained. However, these results must be adjusted according to the rank of the matrix A; Luo and Skelton [12] and Jia [14] give a complete description of this process. Finally, the pseudoinverse A + of A is + A (m+q) 3n = V 1 Σ 1 T 1 U 1 (1) and the solution of (8) according to the ifd vector is obtained from [ γ λ ] = V 1Σ 1 1 U 1 T w (11) The values in this vector correspond to the internal force density for each of the members of the structure. Although the condition established by Connelly requires the bar stresses to be negative, in (7) a minus (-) is introduced so both force densities of bars and strings are expected to be positive. For the analysis of the results, Luo and Skelton [12] present three situations - If every γ i and λ i, then the structure is in - If any γ i <, the corresponding m i string must be replaced by a bar and then the structure will be in - If any λ i <, the corresponding p i bar is in tension and can be replaced by a string. III. Application to a three-bar tensegrity prism The method described above is used to create a three-bar model. A height l of 1 and a radius r of 5 are supposed. Using Matlab for the calculations, for these conditions, the nodal coordinates are as presented in Table I Node x y z TABLE I. Nodal coordinates Fig.4 shows a graph that represents the tensegrity system; the thick red lines represent the compressive members (bars) and the thin blue lines represent the tensile members (strings). This model is used as a visual confirmation of the model; this way, it can be confirmed that the nodes follow the desired geometry; also is confirmed that the bars don t collide. In order to solve it, the Singular Value Decomposition SVD method is used to obtain the pseudoinverse of the system. This method allows obtaining the inverse of singular or non-square matrices. Pellegrino [13] introduced the utility of this method in the analysis of frameworks. According to Jia [14], the SVD method decomposes the matrix A in coefficient matrices A = UΣV T (9) If the matrix A is input in the Matlab function [U, Σ, V]=svd(A)

4 Y : bar : string Z Y Fig. 4. Matlab generated prismatic tensegrity unit cell. Top and perspective view 6 5 for loop to calculate each side of A. For the left side, the code is for i=1:1: n a=s*diag(cs(:,i)); end where n represents the total number of nodes. The right side of the matrix A is calculated in a similar way. Finally, the solution of (11) for the three bar tensegrity structure is given by ifd = [.33 ] The values of the positions 7, 8 and 9 (corresponding to the vertical strings) have a negative value. According to the situations previously mentioned, these strings should be replaced by bars. After this, the unit cell tensegrity structure will be in Fig.5 represents a real full scale model of the tensegrity structure proposed in this paper. The compressive members are made of wood bars and the tensile members are cotton strings. This model confirms that the structure can stand by itself and can even bear loads applied on its top. Following the connectivity rules previously described, the CB is 1 1 C B = [ 1 1 ] 1 1 And the CS C S = [ 1 1 ] The matrices B and S are not shown here because they require only the multiplication of matrices (3) and (4) as previously shown. The final input required is the external force matrix W. If a force of 1 is assumed in the z axis, the matrix W is then W = [ ] The equation (6) can be programmed in Matlab creating a Fig. 5. Full-scale unit cell tensegrity model IV. Conclusions In this paper a simple design method is presented for unit cell tensegrity structures, based on the internal force density method, where the only required information are

5 the number of bars, the height and the radius of the unit cell. This procedure generates the nodes matrix for the structure and calculates the force density on every member. A computer generated example is given, and a real-scale model is presented, showing that the model generated by the method is stable and self-standing. One apparent deficiency of this procedure is that, although the force density of the vertical strings has a negative value, the structure is stable. The reason for this is that the method doesn t consider the level of pre stress, but if pre-tension is applied to the vertical strings, the structure is stable and can bear loads, as designed. As a further work, this method could be useful in the analysis of non-symmetric unit cell tensegrity structures. V. Acknowledgements This paper is funded by the International Exchange Program of Harbin Engineering University for Innovation-oriented Talents Cultivation, SKLRS (HIT) 214-ZD-5, 215-MS-1 and Natural Science Foundation of Heilongjiang Province E References [1] Guest S. The stiffness of tensegrity structures. IMA Journal of Applied Mathematics, 76(1):57-66, 211. [2] Motro R. Tensegrity. Structural systems for the future, London (UK): Kogan Page Science, 23. [3] Gómez Jauregui V. Estructuras tensegríticas: Ingeniería y arquitectura novedosas. Ingeniería Civil, 152:87-94, 28. [4] Pollard T.D. et al. Actin, a Central Player in Shape and Movement, New York (USA): Science 326, pp , 29. [5] Liedl T. Högberg B., Tytell J., Ingber D.E. and Shih W.M. Self-assembly of three-dimensional prestressed tensegrity structures from DNA, London (UK): Nature Nanotecnology 5, pp , 21. [6] Snelson K. The Art of Tensegrity. International Journal of Space Structures, 27(2&3):71-8, 212. [7] new.jpg, December 214 [8] Skelton R.E. Designing minimal-mass tensegrity telescopes of optimal complexity, (USA): SPIE Newsroom. 2 February 213. DOI: / [9] Zhang J.Y., Guest S.D., Connelly R., Ohsaki M. Dihedral Star Tensegrity Structures. International Journal of Solids and Structures, 47(1):1-9, 21. [1] Skelton R.E., de Oliveira M.C. Tensegrity Systems. Springer, 29. [11] Connelly R, Back A. Mathematics and Tensegrity. American Scientist, 86(2): , [12] Luo A.N., Skelton R. E. Connectivity Matrices, Internal Force Density, Mass of Tensegrity Structure System. Unpublished paper, 213. [13] Pellegrino S. Structural Computations with the Singular Value Decomposition of the Equilibrium matrix. International Journal of Solids and Structures, 3 (21): , 1993 [14] Jia Y.B. Singular Value Decomposition. University of Iowa, Unpublished paper, 214.