Computational Modeling of Tensegrity Structures: Art, Nature, Mechanical and Biological Systems

© Springer Nature Switzerland AG 2020

Buntara Sthenly GanComputational Modeling of Tensegrity Structureshttps://doi.org/10.1007/978-3-030-17836-9_1

1. All About Tensegrity

Buntara Sthenly Gan¹ 

(1)

Department of Architecture, Nihon University, College of Engineering, Koriyama, Japan

Keywords

TensegrityArtArchitectureMechanicsBiologyComputationalMathematical modelingFinite element methodWorkshop

1.1 Definition

Tensegrity is an abbreviation between the words of "tensional and integrity." There are ambiguities regarding the definitions of tensegrity because they are based on the principles, geometral, or mechanical points of view to describe the same structure. Below are some definitions that we can find in the early invention of the tensegrity.

The word tensegrity (tensile-integrity) was first used by Fuller (1962) to describe a structure consisting of struts and cables, where the struts are imagined to be floating surrounded by cables in no gravity space. Figure 1.1 illustrates the imagination of the concept stated by Fuller (1962).

Fig. 1.1

The illustrative concept of compression struts surrounded by tension cables in space

In another occasion, Emmerich (1963) used the term self-stressing to describe the similar structures, where inside the tensegrity all the struts and cables are in the equilibrium state by tensioning and compressing each other.

Later, Snelson (1965) who is a contemporary sculptor and photographer patented the similar structure of tensegrity under a definition that the tensile members (cables) have to be continuous and the compressive members (struts) have to be separated from each other.

Denoting Fuller’s invention, Pugh (1976) stated a definition of tensegrity almost the same with the definition that Snelson (1965) stated with an addition that all the members have constant volume in space which implies that the form of the tensegrity has to be in a stable condition.

After three decades, Motro (2003) compiled the histories and definitions of tensegrity in his book comprehensively.

The sculptors, artists, and architects have long been fascinated by the magnificence of tensegrity structures ever since they were first exhibited. In the arts, the tensegrity structures are of high interest because of their rich aesthetic values (Motro 1992). Tensegrity structures have been famous for their appearances in unique geometrical arrangements of struts and cables that show the aesthetic monuments. In the last few decades, direct applications of tensegrity structures started to grow in the civil engineering and architecture fields.

1.2 Computational Tensegrity

In modern engineering mechanics, the role of computation is very prominence in advancing the understanding of the fundamental laws of nature. The study of modern engineering systems which are governed by the laws of mechanics, physics, and mathematics by using a computer has been started followed by the fast computer’s CPU (central processing unit) performance in few decades.

The performance of CPU is related to the number of transistors used for the calculations. The transistor which is a semiconductor device has been the driving force to the advancement and development of the semiconductor industry in decades.

In 1965, Moore who is the co-founder of Fairchild Semiconductor and Intel now predicted that the number of transistors in a sparsely integrated circuit doubles about every 2 years. Moore’s prediction is being proved to be correct for several decades and has been used in the semiconductor industry to lead long-term planning and to set objectives for research and development.

The computing speed performance of the computer has been enormously increased in the logarithmic pace of development. In measuring the computing speed of a computer, there are two units of measure, i.e., floating point operations per second (FLOPS) and million instructions per second (MIPS) . MIPS as a performance benchmark is adequate when a computer is used in database queries, word processing, or spreadsheets. FLOPS is handy in the fields of scientific computations that require floating point calculations of very large or very small real numbers that require a large dynamic range.

Figure 1.2 shows a tremendous drop of cost per MFLOPS (megaFLOPS) since the 1960s up to 2017 in the logarithmic scale. The costs are approximated by considering the inflation rate to have equal values of the dollar in the year 2017. The computing speed of the computer in MFLOPS which is worth 1 dollar can then be calculated and depicted as the climbing curve logarithmically as shown in the figure.

Fig. 1.2

Historical review of computational cost and speed

In lieu to the rapid advancement in computing speed performance of a computer, we can explore more complex and cumbersome calculations in design, analysis, and simulation of an engineering system much faster than ever done in the past centuries.

Computational tensegrity uses advanced computational mechanics and mathematics in the study of modern tensegrity systems governed by the laws of structural mechanics.

1.3 Where Can We Find Tensegrity?

We can find the concept of tensegrity in art and architecture, mechanical devices, and living human and creature.

1.3.1 Tensegrity in Nature and Biology

To control a movement , the bones and muscles of human being and animals are linked in such a way every mild and fine movements are following the command from the brain properly. The bones and muscles (Fig. 1.3) have been evolved to control the movement, where the bones will be subjected to compression act and the muscles carry out the stabilizing tension act which is necessary to move. Every single movement can then be triggered by the bones and muscles using combinations of relief contraction actuating mechanisms inside the body.

Fig. 1.3

The illustrative concept of relief contraction between bones and muscles

In a red blood cell membrane, the inner side of the membrane is called the lipid bilayer, where about more than 30,000 units of tensegrity-like structures attached. In each unit, a protofilament (junctional complex) acts as the strut, and the so-called spectrin, a cytoskeletal protein, acts as the cables which become the tensile connectors between lipid bilayer and the protofilament at the connecting point.

The cytoskeleton is a tensegrity system inside a cell which consists of the molecular microfilament, intermediate filament, and microtubule. The microtubule is a hollow cylinder which plays a role under compression and decompression in controlling the cellular response to mechanical stress on a cell. During the compression of the microtubule, the filaments are subjected by tension for balancing the tensegrity system in the cytoskeleton. An actin which is a multifunctional protein in the filaments plays an important role in the cell contraction and extension of cellular mechanisms .

1.3.2 Tensegrity in Art and Architecture

In 1951, at the Festival of Britain , a free, slender, vertical cigar-shaped tensegrity named the Skylon was built near the Thames between the Westminster Bridge and Hungerford Bridge in London. The Skylon is made of a steel lattice steel frame supported by cables through three steel pillars before being fixed to the ground. The bottom was nearly 15 m from the ground and the top of the lattice steel frame about 90 m high. The lattice steel frame was covered in aluminum louvers which lit from inside during the nighttime. A year after the installation, the Skylon was removed.

An inclined hanging roof of a modern supermarket in Warsaw, Poland, was built in 1962 at Lublin. The roof was designed by using the principle of tensegrity. In the 1980s, the Seoul Olympic Gymnastics Arena was prepared for the 1988 Summer Olympics. The roof was built by self-supporting cable dome which is the first of its kind of tensegrity structure. The Georgia Dome was built to prepare the 1996 Summer Olympics in Atlanta City which is a large tensegrity structure of gymnastic hall.

The first largest hybrid tensegrity bridge was built in 2009, the Kurilpa Bridge, in Queensland, Australia. The bridge which crosses the Brisbane River is a multiple mast, cable-stayed structure based on the principle of tensegrity producing a self-balance between tension and compression components of the light structure.

The design of National Library of Belarus where the famous collections of Leonardo da Vinci are exhibited, envisaged the construction of an original building in the shape of a rhombicuboctahedron, a complex polyhedron of 18 squares and 8 triangles resting on a supporting podium (stylobate).

As a contemporary sculpture in the century, Snelson’s works (Heartney 2009) have been built in public collections and public spaces throughout the United States and other countries.

1.3.3 Tensegrity in Mechanical Engineering

In mechanical engineering , the concept of tensegrity structures has been adopted in furniture manufacturing, robots (Shibata and Hirai 2009; Rovira and Tur 2009), electrical transducers (Carlson et al. 1999), underwater morphing wing applications (Moored and Bart-Smith 2007), and flight simulators (Sultan and Corless 2000).

Deployable tensegrity has been the research subject over the decades to reduce the volume of objects in the space. With the development of numerical models and the folding process, the application of tensegrity opened up significantly. Folding a tensegrity system requires the understanding of instability usually by changing the length of cables. Kwan and Pellegrino (1993) introduced the notion of active and passive cables. Active cables are cables that when relaxed can simultaneously lengthen special cables required to create movements. Pulling these cables causes the tensegrity to unfold. Passive cables enforce a limit on the unfolding of the tensegrity .

1.4 Mathematical Modeling of a Pin-Joint Structure

The classical analyses in Newtonian mechanics are pioneered by Euler and Lagrange in the eighteenth century and further developed by Hamilton, Jacobi, and Poincaré are useful formulations in mechanics. Along with the rapid acceleration of modern computers, the invasions of computational mechanics in research trends are massively progressing. The finite element method (FEM) becomes the most popular numerical method for the scientist, researcher, engineer, and practitioner.

In FEM, the actual physical system under study is idealized into a mathematical model from where the model is then discretized into typical elements by using the finite element concept (Fig. 1.4). The solutions of theories behind these elements are derived and interpolated at all the element’s nodes usually distributed along the boundary perimeter of an element.

Fig. 1.4

The idealization of the physical system into the finite element modeling

One good example to show a physical system to elucidate the concept of tensegrity is a harp as shown in Fig. 1.5. A harp is a musical instrument that has some individual strings mounted at an angle to its closed V-shape wood of frame to produce sound. The strings are plucked with the fingers. Harps have been recognized since antiquity in Asia, Africa, and Europe, dating back at least as early as 3500 BC.

Fig. 1.5

Modeling of a harp by using the finite element method

The instrument had great reputation in Europe during the Middle Ages and Renaissance, where it evolved into a wide range of variants with new technologies and was disseminated to Europe’s colonies, finding particular reputation in Latin America. Although some ancient members of the harp family deceased out in the Near East and South Asia, descendants of early harps are still played in Myanmar and parts of Africa, and other obsolete variants in Europe and Asia have been still utilized by musicians in the modern era.

Figure 1.6 shows a harp being modeled using a different type of elements (line, plane, block). Modeling a real physical system requiring an engineering judgment of the modeler to choose what type of element is preferred. Of course, the results of prediction and the errors of solutions are depending on the type of element being used in the model.

Fig. 1.6

In the finite element modeling different types of elements can be assembled

The elements shown in Fig. 1.7 can be assembled across the different types to build a model by connecting their nodes to their adjacent element(s). Under a defined rule of connection given to a node, the node can act as a connecting terminal between two or more elements which have similar rules.

Fig. 1.7

The concept of discretization in finite element modeling

Selection of element type for FEM modeling depends on the nature of the problems; most of the element types used in modeling a structure are usually line type, plate type, and block type. There are variations to these element types by adding a node or using curvatures at the boundaries. There are plenty of rules that governed the DOF depending on the theories assumed and the type of elements used in a model.

In FEM, this rule is called the assembling process, where each node in the element is facilitated with freedoms to move or rotate capabilities. These freedoms are called the degrees of freedom (DOFs) which are playing an important role in the assembling process. The appropriate freedoms of movements (displacements) or rotations of the nodes have to be jointed properly to connect the elements in a structure (Fig. 1.8). A line member (strut or cable) in a tensegrity structure has two nodes with 3-DOF attached at