Transformable Structures and their Architectural Application. Sam Bouten. Supervisor: Prof. dr. ir.-arch. Jan Belis Counsellor: Ir.

Transcription

1 Transformable Structures and their Architectural Application Sam Bouten Supervisor: Prof. dr. ir.-arch. Jan Belis Counsellor: Ir. Jonas Dispersyn Master's dissertation submitted in order to obtain the academic degree of Master of Science in de ingenieurswetenschappen: architectuur Department of Structural Engineering Chairman: Prof. dr. ir. Luc Taerwe Faculty of Engineering and Architecture Academic year

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3 Transformable Structures and their Architectural Application

4 Acknowledgements Motivated by Prof. Mónica García Martínez, I attended a 2013 lecture on the work of Spanish architect Emilio Pérez Piñero, a pioneer in the field of deployable structures. My interest grew by taking part in a transformable design competition and congress dedicated in his honor. I decided to further deepen this - by then passionate - interest by writing this master s thesis on the subject. I m grateful to my supervisor, Prof. Jan Belis: he had both a highly motivating outlook and critical but ever constructive feedback that made me work more driven and precise. His open approach to the research allowed me to discover widely without losing focus on the important aspects. My gratitude also goes to Michiel Van Der Elst and Jonas Van Den Bulcke, fellow students whose shared interest and curiosity have resonated with mine, and often made me see the topic in new ways. Jonas knowledge of digital fabrication was instrumental in making some of the test models used throughout the thesis. Thanks also to Prof. Niels De Temmerman for his expertise-based tips and encouraging words. Finally and mostly, I d like to thank Silvia for her continuous support and kind listening. Permission for use of content The author gives permission to make this master dissertation available for consultation and to copy parts of this master thesis for personal use. In the case of any other use, the limitations of the copyright have to be respected, in particular with regard to the obligation to state expressly the source when quoting results from this work. Sam Bouten, May iii

5 Transformable Structures and their Architectural Application By Sam Bouten Master s dissertation submitted in order to obtain the academic degree of Master of Science in de ingenieurswetenschappen: architectuur Supervisor: Prof. Dr. Ir.-Arch. Jan Belis Counsellor: Ir. Jonas Dispersyn Department of Structural Engineering Chairman: Prof. Dr. Ir. Luc Taerwe Faculty of Engineering and Architecture Academic year Summary The field of transformable structures is remarkably varied since it transcends the borders of conventional disciplines and inscribes itself into the modern notion of adaptivity. The main aim of this thesis is to provide insight in the design of transformable structures on an architectural scale. In the first part of the thesis, an extensive literature research is done to show the possibilities that lie in the hands of the designer. Geometrical variations are complemented by examples of real-life use of each of the addressed categories: scissor-like structures, rigid-foldable origami and Jitterbug-like mechanisms. Many of them are identified as being variations on overconstrained linkages. The second part addresses the kinematic aspects such as the analysis of degrees of freedom and trajectories. A numerical model for a generalized deployable 4-bar structure is given. Materialization challenges and pitfalls in the scaling of transformable structures are further discussed, specifically joint design and actuation. The third part focuses more deeply on the Sarrus linkage and the different arrays that can be formed from it. A novel way of introducing a polar angle in the Sarrus linkage by means of a joint offset is given. Furthermore, a novel array, dubbed the overlap array, is analyzed and its geometrical aspects discussed. A parametric tool for the design of flat and polar Sarrus arrays is given. The trade-off between deployability and structural performance of the arrays is discussed and two case studies finally are used to structurally analyze the different arrays. Keywords Deployable structure, scissor-like structure, rigid-foldable origami, Jitterbug-like mechanism, overconstrained linkage, Sarrus linkage, Sarrus array iv

6 Contents Acknowledgements... iii Abstract...iv List of symbols... vii 1. Introduction Categorization Basic mechanical concepts... 3 Part I. Review of Literature 2. Scissor-Like Elements Geometrical possibilities Translational units Polar units Angulated units Architectural application Rigid-Foldable Origami Patterns and tessellations Miura-ori pattern Yoshimura pattern Waterbomb pattern Resch patterns Flat-foldability Architectural application Jitterbug-Like Linkages Geometrical possibilities Odd-valent vertices Planar variations Architectural application Overconstrained Linkages Bennett Linkages Goldberg and Myard linkages Bricard linkages Parallel manipulators Modified Wren platforms Sarrus linkages v

7 Part II. Design Tools 6. Kinematic studies Determining degrees of freedom Trajectories and envelopes Generalized trajectory of 4-bar deployable structures Materialization Challenges Joint design Thickness in rigid-foldable origami Actuators Locking systems Design criteria Part III. Uneven Sarrus Chains 8. Uneven Sarrus Chains Basic module Joint-to-joint arrays Polar module with joint offset Mobility of joint-to-joint arrays Overlap arrays Polar module with ellipse method Overlap factor Parametric tool for regular array design Secondary structural systems Case studies Case study 1: Pedestrian bridge Case study 2: Barrel vault Conclusions References Appendix A. Formal Studies Appendix B. Transformable Designs vi

8 List of Symbols Greek symbols Operating angle [rad] Operating angle between xy-plane and original bars in Sarrus modules [rad] Operating angle between xy-plane and original bars in Sarrus modules [rad] Deformation angle in xy-plane [rad] Polar angle [rad] Polar angle for joint offset method if only 2 different bar lengths are used [rad] Maximum polar angle [rad] Maximum polar angle for joint offset method [rad] Maximum polar angle for ellipse method [rad] Kink angle in angulated scissor-like elements [rad] Apex angle in Yoshimura based rigid-foldable origami [rad] η Amount of plate elements in curved direction for Yoshimura based origami Fold angle, operating angle for rigid-foldable origami [rad] Twist angle at joint of overconstrained mechanisms (I) [rad] Twist angle at joint of overconstrained mechanisms (II) [rad] Bar proportion between and bars in uneven Sarrus modules Form factor for snow load ξ Spatial deformation angle for rigid-foldable origami [rad] Maximum spatial deformation angle for rigid-foldable origami [rad] Air density [kg/m²] vii

9 Lower-case Latin symbols Damping coefficient of hyperbolic paraboloid surfaces Probability factor for wind load Overlap factor for Sarrus modules in overlap arrays Roughness factor for wind load Vertical height of Sarrus module in fully deployed state [m] Relative structural height of Sarrus module in fully deployed state Joint offset used to introduce polar angle in uneven Sarrus chains [m] Joint offset if only 2 different bar lengths are used [m] Joint offset for maximum polar angle [m] Bar length, by default assumed greater than bar length [m] Adapted bar length (from ) for joint offset method [m] Adapted bar length (from ) for maximum polar angle [m] Adapted bar length (from ) for ellipse method [mm] Adapted bar length (from ) for maximum polar angle [m] Terrain factor for wind load Bar length, by default assumed lesser than bar length [m] Projected bar length of [mm] Adapted bar length (from ) for ellipse method [mm] Adapted bar length (from ) for maximum polar angle [m] Length of translation vector in generalized trajectory of deployables (I) [m] Length of translation vector in generalized trajectory of deployables (II) [m] Distributed imposed load [KN/m²] Peak velocity pressure for wind load [KN/m²] viii

10 Snow load [KN/m²] Basic wind speed [m/s] Basic wind speed [m/s] Total wind pressure [N/m²] Internal wind pressure [N/m²] External wind pressure [N/m²] Wind load in global x direction [N/m²] Wind load in global y direction [N/m²] Reference height of structure for wind load [m] Upper-case Latin symbols Exposure factor for wind and snow load External wind pressure coefficient Internal wind pressure coefficient Temperature coefficient for snow load DaP Dihedral angle preserving (joint) DoF Degrees of freedom of a mechanism Total number of links in a mechanism Total load from combinations [N] Number of grounded links in a mechanism Turbulence intensity for wind load ISA Instantaneous Screw Axis Number of joints of order i in a mechanism ix

11 Foldline in rigid-foldable origami Mobility of a system, equals to degrees of freedom for rigid link mechanisms Internal moment around local y axis [Nm] Internal moment around local y axis [Nm] Internal axial force [N] Concentrated imposed load [N] Reaction force in global x direction [N] Reaction force in global x direction [N] Reaction force in global z direction [N] Number of mountain folds around vertex in rigid-foldable origami Deformation in global x direction [m] Number of valley folds around vertex in rigid-foldable origami Self-weight [kg] x

12 1. Introduction 1. Introduction Are you really sure that a floor can t also be a ceiling? - Escher M. C. The term itself, transformable structure, is an oxymoron: structure is what gives static shape to systems, while transformable is a word more at home in the world of the shifting and the unstable. It s between these two worlds that transformable structures strike a balance, looking for a trade-off between the mechanical and static qualities. In the transformation phase, controlled movements must be potentiated, but once the mechanism is locked in place, the resulting structure must be rigid and secure in its use. The terms structure and mechanism (or linkage) will hence be used freely in the thesis, sometimes referring to the very same geometries, depending on the state they are in. The fact that this field of study is located at an intersection point of many other domains makes it very diverse, and many points of view need to be reconciled in any transformable design. Not only structural and kinematical aspects, but also three-dimensional geometric patterning, transport and actuation play a part. In short, thinking about transformable structures inherently includes the fourth dimension of time,a factor often minimalized in the building industry, where static and unchanging constructions rule. Although transformable structures have been used throughout history - mostly on the fringes of architectural culture - their more recent popularization inscribes itself into a wider paradigm shift, where a dynamic lifestyle and durability are two key notions. Transformable structures can offer dynamic answers to modern problems, such as deployment for creating temporary spaces, responsiveness to climatic influences, and change of use. The design of transformable structures then is the design of change. Fig 1.1 Metamorphosis I (Escher M. C. 1937) 1

13 1. Introduction 1.1 Categorization Classification into different groups is useful to gain insight into shared underlying principles. The categorization used in this thesis, shown in Fig 1.2, is based on the work of Hanaor A. and Levy R. (2001) who discern two main axes that divide transformable structures. The primary axis, kinematics, describes the important difference in how transformability is achieved: mechanisms can be made up of rigid links that are connected at joints that offer controlled local motion. Structures made from deformable links can change shape due to the elastic properties of materials. Because of the large discrepancy between these two groups and the advantage of better control in the first group, only rigid-link mechanisms are discussed in this dissertation. The secondary axis, morphology, is more arbitrary and describes the basic shapes that make up the transformables, whether they are bar elements (lattices) or surface elements. A third axis added here, the mobility, denotes the freedom with which a transformable structure moves. A transformable with high mobility offers more options but is less easily controlled. For rigid link mechanisms, the mobility translates directly to degrees of freedom. Fig 1.2 Categorization of transformable structures 2

14 1. Introduction 1.2 Basic mechanical concepts Since transformable structures are inherently mechanisms, some basic notions about their mechanical systems and connections are necessary to understand them. Here, the concepts of degrees of freedom and joint classes are shortly explained, and a more expended analysis of the geometric-kinematic characteristics of the structures is given in chapter 6. Perhaps the most important thing to keep in mind when dealing with the mechanical aspects of transformable structures is that they behave as a closed system, and not just separate mechanisms joined together. However, since many of the transformables are made up of basic modules, analysis of each of the categories will for simplicity often start with these smallest building blocks Degrees of freedom Any possible motion a mechanical system can undergo is bound by its Degrees of Freedom (DoF), or Mobility (M). The degree of freedom of a mechanism is equal to the number of independent parameters (measurements) that are needed to uniquely define its position in space at any instant of time. Or in other words: [ ] the number of inputs that need to be provided in order to create a predictable output (Norton R. L. 1991). A host of mechanisms, including the ones found inside the domain of architecture, are of a single degree of freedom (1DoF), since they are easily driven and need only a single control parameter to function. As such, most mechanisms described in this dissertation are of 1DoF as well. However, with the rise of data-driven adaptability, the application of mechanisms with a higher DoF in architecture becomes more thinkable and even desirable. Particularly of interest are mechanisms in which each DoF is directly linked to a design parameter and can easily be locked without affecting the other DoF (and such, design criteria). An example would be the design for a singular façade structure that could independently regulate shade, heating, and ventilation, directly by changes made within each DoF Joint types Joints can defined as he motion-permitting connection of two or more links. There are several ways of categorizing joints, the main and most basic distinction being made in the literature being the one between lower pairs and higher pairs. As Reuleaux F. (1963) defines it, lower pairs are joints with surface contact (one element encompasses the other, such as in a spherical joint). Higher pairs are joints with point or line contact. Norton R.L. (1991) notes that, due to unavoidable practical imperfections, every joint is in fact made possible thanks to discrete contact points. Joints are further defined by the degrees of freedom they give between their connecting elements. The most-used and important joints in transformable structures are listed on the next page. 3

15 1. Introduction R-Joint P-joint Half-joint S-joint C-joint Fig 1.3 Joint types Revolute joint (R-): 1DoF. Often called Pin joint. Fixes the joined links on a mutual axis around which they have one rotational freedom. Most structures discussed in this thesis use solely R- joints. Prismatic joint (P-): 1DoF. The joined links have one relative translational freedom. Telescopic systems are made out of these joints. Half-joint: 2DoF. Links have one translational- and one coplanar rotational freedom. They are referred to as half because they limit half the DoF as the typical R- and P-joints. Spherical joint (S-): 3DoF. Links have three independent rotational freedoms, but all translations are bound. Cylindrical joint (C-): 2DoF. Links have one translational- and one perpendicular rotational freedom. 4

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17 Part I. Literature Review

18 2. Scissor-Like Elements 2. Scissor-Like Elements Scissor-Like Elements (SLEs), sometimes denominated as scissor units, pantographs or Nuremberg mechanisms are the most widely used mechanism type in larger-scale structures, thanks to their reliable synchronous movement, their compactness and their economic use of material. A basic SLE is formed by bars that are interconnected along their length by one or more revolute joints the Intermediate hinges - allowing one free revolution in their (common) plane. By linking SLEs together through articulated joints at their end nodes, planar and spatial grids can be formed that all possess a single DoF, being able to deploy easily from compact bundles to space-encompassing frameworks. The intermediate hinges that allow this synchronous movement of the bars are at the same time an encumbrance on the static-structural level. The continuity of the bars makes for a bending moment at the location of the central hinge, exactly where the material is at its least because of the need for a physical joint axis. Hence, the largest deformations will happen at the location of these weak spots. The deformability of SLE structures and the fact that they need to statically comply not only in deployed, but also in the intermediate states, make the design process iterative and often long-winded. As Gantes C. J. (2004) put it: From a structural point of view, deployable [SLE] structures have to be designed for two completely different loading conditions, under service loads in the deployed configuration, and during deployment. The structural design process is very complicated and requires successive iterations to achieve some balance between desired flexibility during deployment and desired stiffness in the deployed configuration. In this chapter firstly a geometric categorization and the different variations of SLEs are given, based mainly upon the theoretical work done by Escrig F., Langbecker T. and De Temmerman N. and the designs and patents of Hoberman C. Afterwards, a brief history of the architectural applications is given, showing the way SLE structures have been used as space-encompassing systems. 7

19 2. Scissor-Like Elements 2.1 Geometrical possibilities Based on variations in the basic SLE the shape of the bars and placement of the intermediate hinges three general subgroups can be identified: translational-, polar-, and angulated elements Translational units Defining the unit lines (dotted lines in Fig 2.1) as the imaginary lines that connect the articulated end nodes of the bars, the translational SLEs are characterized by the fact that these lines always stay parallel to one another. The curves connecting the R-joints are straight, but there are still many ways of varying within this group. The most basic scissor structure that is the repeated linkage of a symmetrical translation element, forming a straight framework, is called the lazy-tong and is shown in Fig 2.1. Fig 2.1 Translational unit Fig 2.2 Lazy-tong scissor mechanism 8

20 2. Scissor-Like Elements Translational flat variations To form spatial frameworks of SLEs, a multitude of options is open, starting by defining the shape of the formed array. Fig 2.3 shows the results of choosing a square grid array formed by connecting the edge nodes to each other at straight angles. This results in an unstable structure in the projected-plane, needing additional bracing for the square shapes to make it function optimally in static state. The hexagonal grid in Fig 2.4 is made up of equilateral triangles and offers in plane stability and greater strength, at the cost of compactability. Fig 2.3a Square grid array Fig 2.3b (Escrig F. 1991b) Fig 2.4a Hexa-triangular grid array Fig 2.4b (Escrig F. 1991b) 9

21 2. Scissor-Like Elements More complex framework shapes are derived when the planes of each of the lazy-tongs no longer cut each other perpendicularly, but obliquely. By doing so, the intermediate hinges of 2 or more SLEs intersect, and fewer bars meet at the end nodes. In Fig 2.5 the example of a triangular oblique grid is given. The intermediate hinge (marked in orange) has to be specifically designed for the three bars crossing each other. Fig 2.5a Oblique triangular grid array Fig 2.5b (Escrig F. 1991b) As Escrig F. (1986) observed, the perpendicular type of grids can be formed as a collection of prismatic elements (Fig 2.6), and the oblique grids from the collection of anti-prismatic elements (the bars cross through the center of the circumscribing prism) (Fig 2.7). For a more complete set of prismatic variations, the 1986 article of Escrig F. is highly recommended. Fig 2.6 Prismatic modules make up make up straight grid array Fig 2.7 Anti-prismatic modules make up oblique grid array 10

22 2. Scissor-Like Elements Translational curved variations Even though the unit lines of translational elements by definition stay parallel, a curved grid can be formed by varying the point on the bars that the intermediate hinge is connect. Fig 2.9 shows such a translational curved variation, in which the array consists of two mirrored halves so that a central peak is formed. It should be noted that the compacted bundle retains the original height of the completely deployed mechanism, making it less than ideal for the use in transportable structures. Fig 2.8 Translational unit (curved) Fig 2.9 Curved translational scissor mechanism Translational multi layered variations By raising the amount of intermediate hinges for each bar, multi-layered systems can be made from SLEs. As the amount of fixed points and the structural height per SLE are increased, the whole will be subject to a smaller maximal bending moment. That is, however, at the cost of more material and lesser compactness, and a multi-layered structural system is seldom used for deployable structures where a compact bundle is key. However, the rhombic shapes as in Fig 2.10 can form the basis of elaborate spatial structures, such as ruled surfaces, a category of SLE structures of which the applications are discussed further on in this dissertation. 11

23 2. Scissor-Like Elements Fig 2.10 Multi-layered translational scissor mechanism All of the variations mentioned above for translational elements can also be applied to the undermentioned polar- and angulated units, touching off even more possibilities of different shapes of regular frameworks. 12

24 2. Scissor-Like Elements Polar units By moving the intermediate hinge of the SLE away from the center by a certain eccentricity, the unit lines will move from being parallel to having a polar angle between them, which changes from being 0 at the completely (theoretical) folded state to being at its maximum at the fully-deployed state. This maximum angle is proportional to the eccentricity. By linking these polar SLEs together, a nearly planar compact bundle can deploy into a structure with constant curvature, as shown in Fig Fig 2.11 Polar unit Fig 2.12 Polar scissor mechanism Polar free-form variations More randomly curved shapes can be made by varying the size of the bars within each of the SLEs. If the condition of complete foldability (and thus, maximum compactness) is to be achieved, the sum of the partial-bar lengths of two adjoining elements must be equal. This is defined in an equation first given by Escrig F. (1988) that is from here on referred to as the compactability equation (Fig 2.13): [2.1] 13

25 2. Scissor-Like Elements Fig 2.13 Random bar length scissor mechanism Equation [2.1] was extended by geometrical description to three-dimensional SLE structures by Langbecker T. (1991, 1999). The constraint can also be unfulfilled for one of the adjoining SLEs, while the connecting one can completely collapse into a linear state, hence they will be partially foldable (De Temmerman N.). Finally, the compactability constraint forms the basis of the design of deployable structures, and applies much more widely than for SLEs. Polar singly curved variations By repeating the arches formed by polar SLEs in a linear fashion and connecting them by translational SLEs, cylindrical structures can be made. Barrel vaults have been researched by Escrig F. (1986, 1996) and geometrically and structurally investigated by Langbecker T. (2000). Fig 2.15 shows an alternative, braced cylindrical vault that was used as a calculation model of Langbecker. Fig 2.14 Barrel Vault (Escrig F. 1986) Fig 2.15 Barrel vault with cable substructure and X-bracing (Langbecker T. 2000) 14

26 2. Scissor-Like Elements Polar doubly - curved variations By using polar units in multiple directions, doubly curved spatial structures such as domes can be made. Escrig F. (1988) demonstrated different ways to form dome-shapes from polar units. The domes in Fig 2.16 and Fig 2.17 can be made using respectively square and triangular grid. The advantage of these regular grids is the modularity of the polar units. Domes can be generated from oblique grids as well, as is discussed in the section 2.2, architectural applications of SLEs. Fig 2.16 Dome from square modules (Escrig F. 1988) Fig 2.17 Dome from triangular modules (Escrig F. 1988) The domes made from these regular grids deviate from a pure spherical form. To approximate a more constant curvature, the edges of the polygons from any geodesic dome can be replaced by modular polar elements. The completely folded and completely deployed state don t show any geometric problems, but in certain intermediate phases there can be geometric incompatibilities, which have to be resolved by artifices that locally open more DoF or by material deformation (see bi-stable structures in chapter 6). An example is the geodesic dome in Fig

27 2. Scissor-Like Elements Fig 2.18 Geodesic dome from triangular modules (Escrig F. 1988) A last category mentioned by Escrig F. are the domes generated from a rhombic pattern; socalled lamella domes, which don t show any incompatibilities, and are easily designed (Fig 2.19). Fig 2.19 Lamella dome (Escrig F. 1988) Anticlastic surfaces such as hyperbolic paraboloids can also be made by flipping the side of the eccentricity of the intermediate hinge, or keeping a central hinge and using side-by-side compatible translational elements as demonstrated by Langbecker T. (2000). Fig 2.20 Anticlastic surface from translational units (Langbecker T. 2000) 16

28 2. Scissor-Like Elements Angulated units Fig 2.21 Angulated units Fig 2.22 Angulated unit variations (Hoberman C. 1990a) Angulated elements, popularized by Hoberman C. (1990a), possess a central kink of angle ε which causes a constant angle γ between the unit lines throughout the whole transformation process. As Hoberman also demonstrated, it is actually the relative location of the intermediate hinge that is of importance, and differently shaped figures can be used (Fig 2.22). The angle δ between the unit line and the adjoining semi-bar is hereby bound by the equation: [2.2] This opens the possibility to make ring structures as in Fig 2.24 that open and close around a central point O. This radial deployment has a lesser compactness than a linear one, since the minimum phase is constrained by the continuity of the chain of elements. Fig 2.23 Multi-angulated unit Fig 2.24 Radial scissor mechanism from angulated units 17

29 2. Scissor-Like Elements By introducing more kinks, a multi-angulated element can be formed. Equation [2.2] still holds and, as such, a radial kinematic system can be made that is denser and consequently will be able to carry more loads and span greater widths. An example of triply angulated elements is given in Fig Hoberman C. (1990, 2001) changed the size of the adjoining elements always complying the compactability equation [2.1] to make radial, non circle shaped elements. Likewise, You Z. and Pellegrino S. (1997) developed a general angulated unit for non-circular closed radial geometries. Fig 2.25 Radial scissor mechanism from multi-angulated units Doubly - curved spatial variations Approximately spherical surfaces can be created by using these closed-ring structures as the circumferences of a sphere, and making them intersect in a triangular grid by using a geodesic subdivision into hexagons and pentagons (Fig 2.26). Platonic solids such as the Icosahedron in Fig 2.27 can be made combining translational and angulated SLEs. These figures were also made famous by Hoberman C. These mechanisms all have a mobility of 1DoF. Fig 2.26 Expansion of geodesic dome from angulated units (Hoberman C. 1990a) 18

30 2. Scissor-Like Elements Fig 2.27 Expansion of icosahedron from angulated units (Hoberman C. 1990a) Gómez Lizcano D. E. (2013) has shown how a pendentive dome can be generated by grouping different angulated modules based on the intersections of a sphere and pyramidal shapes of different width. A myriad of different anticlastic shapes can be created using angulated SLEs by changing both their total length and kink angle, as Hoberman has demonstrated. In the work of Roovers K. et al. (2013 the conversion of any arbitrary continuous surface to a scissor mechanism is described geometrically. His approach makes it possible to optimize any form for maximum compactness. One should note, however, that freeform design almost always go at the loss of modularity of the composing units. Fig 2.28 Expansion of anticlastic geometry from angulated units (Roovers K. et al. 2013) 19

31 2. Scissor-Like Elements 2.2 Architectural application The usage of the SLE mechanisms in their different variations has led to some architectural applications in which the scale of the projects together with the kinematic behavior are nothing less than awe-inspiring. However, realizations are few because of the mentioned difficulties in structural calculation of each phase of the deployment. The fact that furthermore there is no single regulatory body for deployable structures makes them unknown and unused by the mainstream of designers. The examples used in this section consist only of real-life structures or models that were used to this goal: intents to materialize the discussed geometries. The father of scissor-like structures, and the original person responsible for their proliferation and the widespread research in the academic world, is undoubtedly the Spanish architect Emilio Pérez Piñero. The reason his work has not been mentioned until this point in the dissertation is that his methods were focused on generating real structures: he was not at all an academic figure, but analyzed the geometrical and structural principles behind his work just in order to obtain patents and to further their realization. Fig 2.29 Emilio Pérez Piñero and his design for a deployable theatre, oblique grid from polar units (adapted from 20

32 2. Scissor-Like Elements In Fig 2.29, a domical design with an oblique triangular grid is shown. It is deployable through the manipulation of 6 central joints (Escrig F. 1991a). It was in its time a novel structural concept and consequently won the London-based competition for a traveling theatre. Because of the press coverage and the consecutive travels of Piñero in which he promoted his designs, SLE structures were made known to many of his contemporary architects and engineers. Fig 2.30 Compacted bundle for transport (adapted from In Fig 2.30, the compact bundle for an oblique-grid dome structure can be seen. One of the most important inventions that made the realization his structures possible was the design of the central joints that connect the spatially intersecting SLEs, demonstrated in small scale metal models in Fig 2.31 for both quadrangular and triangular grids. Pérez Piñero went on to design both planar and curved scissor-like structures until his early death in Fig 2.31 Joint details in deployable test models done by Pérez Piñero (Cruz J. P. S. 2013) Having been inspired after seeing the work of Pérez Piñero, Spanish architect and engineer Escrig F. carried on his legacy to the academic world. Together with Sánchez J. and Valcárcel J. he not only geometrically and structurally analyzed SLE systems, but realized many new pantographic typologies. In Fig 2.32, their design for the deployable cover of the San Pablo swimming pool in Seville, based on a polar quadrangular grid, is shown. Notice the diagonal bracing elements used to stabilize the grid. 21

33 2. Scissor-Like Elements Fig 2.32 Deployable swimming pool from rectangular modules (Escrig F. 2012) As seen in Fig 2.33, Escrig and Sánchez also made use of a curved grid. This allows for large spans to be covered with efficient material use, but goes at the cost of the compactness of the elements and thus requires large-scale transportation. Fig 2.33 Deployable swimming pool from multi-layered curved bars (Escrig F, 2012) Aforementioned Hoberman C., a multidisciplinary designer, used his patented angulated units to build several pantographic structures at sculptural-architectural scale. All of them are radially deployable, either compacting to a central position as in the geodesic dome in Fig 2.34, or along their circumference as in the Iris dome in Fig

34 2. Scissor-Like Elements Fig 2.34 Expansion of triangulated geodesic dome from angulated units (Hoberman C. portfolio) Fig 2.35 Edge-to-center deployment of Iris dome from angulated units (Hoberman C. portfolio) As mentioned before, Gantes C. J. (1997) meticulously investigated stress-related effects of SLEs, in particular to make self-locking structures (see bi-stable structures in chapter 6). As mathematical analysis combined with finite-element modeling is of great importance before even attempting to construct these structures full-scale, few of them have been built. Raskin I. (1998) proposed and analyzed more simple systems such as deployable slabs and columns (Fig 2.36) for regular use in construction. He describes in detail how it is possible to go from the mechanical spatial grid to a stable structure by adding boundary conditions such as a top layer of rigid plate elements. Fig 2.36 Deployable column and slabs from translational units (Raskin I. 1998) 23

35 2. Scissor-Like Elements A system based on the angulated ring mechanism of Fig 2.25 was proposed by Kassabian P. E. (1997) to form a retractable roof by adding plates to the bar structure. Jensen F. V. and Buhl T. (2004) later improved upon this system by using only the plates as rigid elements, abolishing the need for any secondary bar system. A stadium designed by Lake Associates used this very system (Fig 2.37). Fig 2.37 Retractable stadium roof based on angulated units (Lake Associates) Another way of covering pantographic systems consists of using separate foldable plate elements that join in completely deployed state. This method was first used by Pérez Piñero E. in the design of glazing panels for the Dalí museum in 1970 and later improved upon by Valcárcel V. P. (Escrig F. 2012). Fig 2.38 displays the fish fold designed by the latter to cover a deployable dome. Naturally, by introducing two-dimensional elements into the pre-existing grid of quasi one-dimensional bars, the mechanism becomes less compact, as seen in the right-side image. Fig 2.38 Use of additional plate elements on translational grids (Escrig F. 2012) 24

36 2. Scissor-Like Elements De Temmerman N. (2007) combined the know-how of pantographic systems and tensile substructures to design quickly deployable structures in which the fabric works actively on a structural level. Fig 2.39 and Fig 2.40 demonstrates some new tent typologies designed by De Temmerman in this fashion. Fig 2.39 Textile substructure in barrel vault Fig 2.40 Textile structure attached to from variable polar units central mast from angulated units (De Temmerman N. 2007) Alegria Mira L. (2010) bridged the gap between translational-, polar- and angulated units by designing a Universal Scissor Component (USC). By giving the possibility to vary the central hinge, the eventual angles between the SLEs can be changed, and many singly- and doubly curved shapes can be made. The price paid for this is the maximum compatibility, especially of the translational and polar arrangements. Nevertheless, a higher structural strength of the units is gained in the zones where the largest bending moment is introduced, making it a versatile and effective design. In Fig 2.41, the basic USC is shown, together with an icosahedral variation. Fig 2.41 Universal Scissor Component (USC) and icosahedral variation (Alegria Mira L. 2010) 25

37 2. Scissor-Like Elements A last group that can be shaped by the use of SLEs is that of the ruled surfaces. A one-sheet hyperboloid type structure has been proposed by Escrig F. and Sánchez J. (2012) (Fig 2.42). The compactability of these hyperboloids is too low to be considered a potent alternative to single-layered pantographic structures, but their transformability can be useful e.g. for climatic adaptation. Fig 2.42 Deployable hyperboloid from multi-layered grid (Escrig F. 2012) Hyperbolic paraboloids - as being investigated by Maden F. and Teuffel P. (2013) - can also be formed by using multi-layered scissor elements. However, as the joints of the bars undergo a relative translation, these mechanisms cannot be formed merely by revolute joints, and a combination of P- and half-joints need to be specifically designed to allow for their mobility. For this reason, their design is still troublesome. Fig 2.43 demonstrated the possible use of 6 independently mobile saddle-surfaces used for shading in a public space. Fig 2.43 Adaptable hypar surfaces from multi-layered grid (Maden F. and Teuffel P. 2013) 26

38 3. Rigid-foldableOrigami 3. Rigid-Foldable Origami Rigid-foldable origami elements, Foldable Plate Elements (FPE s), or hinged plate elements form the basis of kinetic surfaces that are traditionally continuous, but may just as well be discontinuously connected. The rigid plate elements are connected by R-joints along their outer edges. The amount and the relative in-plane angle of these edges will determine the boundary conditions of the resulting kinetic surface. As such, by simply changing the relative inclination of the edges, systems can sometimes gravely shift their kinematic behavior. Because of the fact that the basic element is a surface unlike with the quasi-linear SLE s and because of the fact that these surface elements oftentimes need to be continuously connected, the maximum compactness achieved in foldable plate structures is much lower, and so the value of these structures is sometimes sought in the different qualities of their analogue phases: the advantages of adaptive structures versus those of typically deployable ones. On the flip side of the coin, they typically allow for an efficient distribution of forces because of their continuous and corrugated surfaces. The continuity of the hinges is of importance here, and special care has to be taken not to introduce local concentrated loads that unnecessarily change the static behavior of the structure for the worse. Additionally, the waterproofing of these hinges is an interesting theme to be addressed in the materialization process. Almost all foldable plate structures are based upon the ancient Japanese art of origami. Designing them often starts with a single sheet that is folded - without any cutting - into a threedimensional figure. When it is possible to generate an FP surface from this process, it is said to be developable (Solomon et al. 2012). Sometimes developability and continuity of the surface are unnecessary constraints when thinking of architectural applications. Nevertheless, the work done on paper origami models when designing FP structures proves to be invaluably for fully understanding the transformation processes. In this chapter, firstly the general geometric possibilities and variations are described. This part is supported by the works of major figures in the field of origami such as Miura K., Resch R., Tachi T. and Hull T. Subsequently, the limited body of noteworthy architectural applications is given, focusing only on the buildings or components that are truly transformable: static buildings inspired by origami structures are left aside. 27

39 3. Rigid-foldableOrigami 3.1 Patterns and tessellations Both modularity and homogenous mechanical behavior are of value to FP surfaces. An efficient way to design them is therefore to start with the basic repeated patterns made up of copied or similar elements. These are named tiling patterns or tessellations. All of the discussed patterns are based on foldable, developable surfaces. The following collection of origami tessellations is not meant to be exhaustive, but to offer a view on the most interesting patterns for application in the fields of engineering and architecture. Some have proven their use in large-scale application, while other still wait to be used. All of the patterns are made up of Mountain (M-) and Valley (V-) folds, folds around which the FPE s rotate clock- and counterclockwise, relative to the orientation of the surface. For each of the flatfolded patterns in this chapter, as in Fig 3.1, mountain folds are shown in orange lines and valley folds in blue dotted lines Miura-ori pattern Fig 3.1 Miura-ori fold pattern The name Miura-ori is a contraction of the Japanese ori fold and the name of the inventor of the pattern, Japanese astrophysicist Miura K. He devised the compactly foldable surface to use it as a deployable solar sail for a space unit (Fig 3.3). Another big advantage of the mechanism is the fact that it has only 1 DoF, making it easy to actuate. For these reasons, the Miura-ori pattern is one of the most widely studied patterns in contemporary engineering. Fig 3.2 Closed and semi-deployed Fig 3.3 Solar sail application of Miura-ori fold Miura-ori mechanism (Miura K. and Natori M. 1985) 28

40 3. Rigid-foldableOrigami In Fig 3.2, the expanding mechanism is shown. It consists of interconnected quadrilateral plate elements. The regular pattern is made up of identical parallelograms and unfolds into a (corrugated) planar geometry. However, variations are possible that disturb the planar pattern into a singly or doubly curved one. For any variation around a single internal vertex of the pattern, Tachi T. (2009) quotes the relationship between the fold angles from and Hull T. C. (2006) as: and [3.1] The minus value in the first equation is due to the fact that the fold line of is a mountain fold, while the other ones are valley folds. The relationship between these two pairs of angles is then given by: ( ) ( ) ( ) [3.2] In which ξ is the angle between and, in any configuration except for the folded one given by: ( ) [3.3] which reaches its maximum when the maximum operating angle is reached. thus equals only for flat-folded patterns. Using these equations parametric models of connected (repeating) vertices can be set up. Fig 3.3 Variation of Miura-ori fold around single internal vertex (adapted from Tachi T. 2009) Tachi continues his 2009 paper by giving the necessary condition for rigid foldability of any quadrilateral pattern in function of the lateral ( and ) and longitudinal fold angles (( and ), which in their turn can be defined in terms of the angles from equations [3.2] and [3.3]. Using this method, any rigid-foldable freeform Miura-ori pattern can be created. An example is given in Fig 3.4.Tachi furthermore developed software, called Freeform Origami, for the easy manipulation of (amongst others) Miura-ori patterns without affecting their foldability. 29

41 3. Rigid-foldableOrigami Fig 3.4 Expanding Miura-ori variation (Tachi T. 2009) Another operation on Miura-ori patterns is the removal of facets that are unnecessary for mobility, i.e. the single DoF of the pattern is left unaffected by this operation. Beatini V. and Korkmaz K. (2013) give the boundary conditions for this operation. An example of their work is seen in Fig 3.5. Fig 3.5 Flat Miura-ori variation with removed facets (Beatini V. and Korkmaz K. 2013) A final note on the geometrical characteristics of the Miura-ori pattern is the existence of a barshaped counterpart of the basic 4-facets tile in the form of a spherical linkage, as demonstrated in Fig 3.6 (Abdul-Sater K. et al. 2014). Analogously, any origami pattern can be replaced by an array of spherical linkages by replacing the foldlines by the rotational axes of compact R-joints. This method is not only interesting for the analysis of origami mechanisms, but also gives the possibility to develop more compact, simple, mechanisms based on plate elements. Fig 3.6 Corresponding bar linkage of single Miura-ori vertex (Abdul-Sater K. et al. 2014) 30

42 3. Rigid-foldableOrigami Yoshimura pattern Fig 3.7 Yoshimura fold pattern The Yoshimura pattern is made up of triangular facets, and typically folds into a singly curved corrugated surface, although also doubly-curved surfaces can be reached when the pattern is not flat-foldable. It has multiple DoF, but can be stabilized fairly well and shows a high rigidity when done so, which makes it suitable for engineering and architectural purposes. Fig 3.8 Barrel vault from Yoshimura pattern (De Temmerman N. 2007) The basic shape of interest that can be developed is the barrel vault, as shown in Fig 3.8 (De Temmerman N.). It results from a regular pattern as in Fig 3.7. For the pattern to be rigidfoldable, the apex angle of each triangular facet has to lie in between and. Furthermore, for each amount of facets in the curved direction, there exists a certain apex angle for which the mechanism can be fully folded into its most compact form, i.e. the form in which the semi-facets touch each other, as in the second diagram in Fig 3.8. De Temmerman gives the clear formula: ( ) ( ) [3.4] In which: : apex angle η: amount of plate elements in curved direction 31

43 3. Rigid-foldableOrigami E. g. for the pattern in Fig 3.8, where η = 7, an apex angle of gives the compact form. For a comprehensible design method based on the parameters of height, span, structural height of barrel vaults based on semi-regular Yoshimura patterns, the doctoral thesis of De Temmerman N. (2007) is highly recommended. Irregular tessellations can also be generated easily from the Yoshimura pattern, their only necessary condition for rigid-foldability being that the sum of 2 adjoining deformation angles stays constant (Fig 3.9). Following this rule, interesting shapes such as the ones in Fig 3.10 can be designed. Fig 3.9 Yoshimura variations retain rigid-foldability when the sum of two adjacent deformation angles is constant (Tonon O.L. 1993, as adapted by De Temmerman N. 2007) Fig 3.10 Rigid-foldable Yoshimura variations (Tonon O. L. 1993) By taking away the semi-facets at the top of the pattern and connecting the remaining full facets with each other, polar, doubly curved geometries can be made, as in Fig Different combinations of regular singly- and doubly-curved patterns are shown by De Temmerman for a maximum of 5 facets in the curved direction. It has to be noted that these geometries can only exist in the erected state shown, i.e. they can t be form a fully closed loop and at the same time stay rigid-foldable. 32

44 3. Rigid-foldableOrigami Fig 3.11 Doubly-curved Yoshimura variation in compacted and fully deployed state Fig 3.12 Combination of single- and doubly curved mechanism into static structures (De Temmerman N. 2007) Another possible geometry that applies the Yoshimura pattern is the fully closed cylinder that folds upon itself, as investigated by Guest S. D. and Pellegrino S. (1994) (Fig 3.13). Later research to use these cylinders structurally as inflatables has been done by Barker R. J. P. and Guest S. D. (2000).These cylindrical mechanisms are also referred to as origami booms in the literature. 33

45 3. Rigid-foldableOrigami Fig 3.13 Inflatable booms based on Yoshimura pattern (Guest S. D. and Pellegrino S. 1994) Waterbomb pattern Fig 3.14 Waterbomb fold pattern Another multi-dof pattern is the waterbomb pattern, being made up triangular facets with straight-angled apexes. The pattern typically introduces a double curvature in its surface, as seen in its folded state in Fig Its high mobility makes it less suitable for large-scale applications, but may prove its use in smaller design applications. Fig 3.15 Folded waterbomb pattern (Tachi T.) 34

46 3. Rigid-foldableOrigami Resch patterns Fig 3.16 Resch fold pattern The Resch patterns were developed by the geometrist and artist Ronald D. Resch during the 1960 s. Many variations on the basic pattern in Fig 3.15 have been proposed both by Resch and others, but the all share some characteristics: they typically have 2DoF per module, one twisting and one folding one. In the patterns two facet layers can be detected. Firstly, there is the front layer of which the facets barely undergo any out-of-plane rotation, but which rotate around normal axes through their centroids. Secondly, there is the back layer of which the facets undergo a complete relative rotation of between folded and unfolded state. This way, the back facets are tucked in between the front layers (Tachi T. 2013). The back layer is invariably made up of triangular facets, while the facets of the front layer can take on different complementing shapes. The partially folded pattern is shown in Fig In Fig 3.17 Resch and his arts- and architecture students are shown next to a large-scale test of the same regular triangular Resch-pattern, still from the documentary Paper and Stick Film. Fig 3.17 Hexagonal Resch pattern Fig 3.18 Large-scale folded Resch pattern (Resch R. D. and Armstrong E.) 35

47 3. Rigid-foldableOrigami Fig 3.17 Variations on Resch pattern (Piker D. 2009a) Some regular variations of the Resch pattern are seen in Fig 2.17, respectively with a square, triangular and hexagonal facets in the top layer. Because of the high mobility, the Resch patterns are able to be made irregular more easily than - for example - the Miura-ori and the Yoshimura patterns without affecting the foldability. Tachi T. (2013) makes use of this characteristic to generate freeform rigid-foldable origami of nearly any shape, as exemplified in Fig Note that the compact foldability and the transformative motion are not of interest here, but the final static shape is. Fig 3.18 Random foldable shapes with Resch pattern (Tachi T. 36

48 3. Rigid-foldableOrigami 3.2 Flat-foldability The compactability of many origami mechanisms will often be directly related to their flatfoldability, their ability to reach the compact folded state in which the plate surfaces are all parallel to each other. Take a single vertex with a surrounding crease pattern made up of mountain folds m (orange) and valley folds v (blue). The creases are named,,, with. Let the angle between any two creases and be. Three criteria for their flat-foldability will then be (Bern M. and Hayes B. 1996): - The sum of alternate angles around the vertex equals π (Kawasaki s theorem) - (Maekawa s theorem) In which: U: number of mountain folds V: number of valley folds - if, then and must have be opposite folds (U, V) Fig 3.19 Flat-foldability around a single vertex Kawasaki s theorem alone is however sufficient to predict flat-foldability of a single vertex, as the other theorems will follow directly out of this. Proof of this, and further explanation of the other theorems is given in the 1996 article of Bern M. and Hayes B. and the 2002 article by Belcastro S. and Hull T. (2012). Expanding to a global flat-foldability, i.e. of a multi-vertex plate structure, Kawasaki s theorem is a necessary but not sufficient condition. As of now, no algorithm is known to solve the global problem, since it has been proven to be a problem of NP complexity. 37

49 3. Rigid-foldableOrigami 3.3 Architectural application Application of rigid-foldable origami mechanisms in architecture is limited, due the lack of knowhow and tradition in materialization. When the problems of hinges, plate thickness, waterproofing and compactability are researched on a larger scale, many interesting designs will become possible. In what follows, some examples from the limited body of work are given, starting with the large scale theoretical projects and moving towards local design elements. Some more theoretical origami structures have been proposed by Tachi T. Fig shows a structure based on the waterbomb pattern. It consequently has multiple DoF, making it deformable to the user s needs or follies. The high mobility however also makes it structurally highly unstable, and a strong connection to the ground plane would be needed here to stabilize it. Another project by Tachi is the Miura-ori variation used to make a temporary and deployable connection between two buildings in a museum complex (Fig 3.21). Fig 3.20 Shape-shifting pavilion from Yoshimura pattern (Schenk M. 2012, project by Tachi T.) Fig 3.21 Deployable passageway between buildings, from Miura-ori pattern (Tachi T. De Temmerman N. (2007) developed a deployable shelter based on the Yoshimura fold (Fig 3.22). The plate elements here are substituted by bars lying on the perimeter of the facets, after which half of the bars are removed in locations where they were doubled. The joints used here are based directly on the vertices of origami folds. To give structural height to the resulting barrel vault a fabric screen is added as a tensile layer. 38

50 3. Rigid-foldableOrigami Fig 3.22a Corresponding bar structure Fig 3.22b Joint detail of Miura-ori pattern (De Temmerman N. 2007) A large-scale project that was really materialized is the retractable roof system developed by the Venezuelan architect Hernandez C. H. (2013) It was used first in the expo of 1992 in Sevilla and later in projects such as a pool cover in Venezuela (Fig 3.23a). It is a regular Miura-ori pattern with trapezoidal facets. Materialization was done in thin metal sheets, and the joints were designed especially for stabilizing and waterproofing the cover (Fig 3.23b). Note that the roof is not self-supporting, but carried by light-weight trusses. Fig 3.23a Deployable roof from Miura-ori pattern Fig 3.23b Joints detail (Hernandez C. H. 2013) A mobile bamboo pavilion was proposed by architect Tang M. The first steps in the opening process are shown in Fig The basic pattern is fairly simple, but it needed to be triangulated in order to be even mobile. The elegance of circular origami-fold mechanisms is that they can be locked and made static by simply fixing the opposing ends to each other. 39

51 3. Rigid-foldableOrigami Fig 3.24 Radial shelter from variable pattern (Schenk M. 2012, project by Ming Tang) Rigid-foldable origami has been applied successfully to kinematic facades. Most often the materialization here is less cumbersome, since the active folding angles are not big and designs taking into account the plate thickness are more easily achieved. A first and simple example thereof is the shading device designed by Ernst Giselbrecht + Partner (Fig 3.25), where single-fold and sliding mechanisms can each be actuated separately to ensure a pleasant inside climate. A second example is the iconic façade designed by Aedas Architects, where 1DoF modules of 6 elements can closely regulate the solar gains (Fig 3.26). Fig 3.25 Hinged facade (Ernst Giselbrecht + Partner) Fig 3.26 Triangulated foldable façade (Aedas Architects) 40

52 3. Rigid-foldableOrigami On a smaller scale, the acoustic panels designed by RVTR make for a very interesting project: a simple Resch pattern has been applied where the front layer of bamboo facets function as reflectors, while the tucked in facets work as absorbers (Fig 3.27a). By increasing and decreasing the operating angle, different acoustic atmospheres can be created. A central electronic panel with sensors can adapt the different actuators in real-time. The actuators themselves are simple P-joints between the facets, three per module (Fig 3.27b) Fig 3.27a Acoustic panels from Resch pattern Fig 3.27b Actuation system (RVTR) On a smaller scale yet, the kinematic characteristics of origami can be applied to create more strong and rigid meta-materials, in particular sandwich panels. Schenk M. and Guest S. D. s 2010 paper makes for a good introduction on the Miura-ori fold from the perspective of structural engineering. Engineers such as Miura K. (1972) introduced the sandwich panel (Fig 3.28) and modern engineering firms such as Tessellated Group (Fig 3.29) are intending to commercialize their origami sandwich products. The main advantage that these panels offer is their controlled deformability which makes them suitable for impact resistance. Fig 3.28 Miura-ori sandwich panel (Miura K. 1972) Fig 3.29 Corrugated sandwich panel (Tessellated Group) 41

53 3. Rigid-foldableOrigami To conclude, rigid-foldable origami can also be found applied more trivially in design elements such as the wood fabric created by Elisa Strozyk (Fig 3.30). The wooden facets are glued onto an underlying textile. It is an interesting concept to apply on a larger scale, where discrete hinges could be replaced by a continuous surface material connecting rigid facets. Fig 3.30 Wooden facets on fabric (Elisa Strozyk) What is interesting about smaller-scale projects is simply the fact that they are materialized, and during the design process they have likely gone through some difficult iterations that bare the problems involved in modeling rigid-foldable origami for real-world use. They can serve as stepping-stones to popularize know-how about its employment. Together with the academic studies of the origami thickness problem and the structural behavior in static state, they can form a matrix for new and better designs. 42

54 4. Jitterbug-Like Linkages 4. Jitterbug-Like Linkages The first Jitterbug-like linkages were nothing more than theoretical models professed firstly by Buckminster Fuller. It was in fact a single transformative octahedral model that sparked Fuller s interest and that he dubbed the Jitterbug, which was the name of a popular ballroom dance of the 1940 s that the movement of the mechanism reminded him of. A later-made physical model of this first Jitterbug throughout its transformation is shown in Fig 4.1. The mechanism in itself is a closed spatial loop that has a single DoF. For Fuller, the discovery of a mechanism that moved through different polyhedrons was paramount: In the mechanism the elementary geometric forms that have stood together since Plato s time as a set of regular solids are shown now to be a phase transition in a single process of metamorphosis. (Krausse J. and Lichtenstein C. 1999) Fig 4.1 Transformation of Jitterbug mechanism ( Fullers interest in the Jitterbug-like mechanisms was mainly intellectual, but he wrote extensively on [ ] how it could help understand the abstracted sciences of chemistry and physics by allowing us to see movements that are normally occurring invisibly all around us. (Krausse J. and Lichtenstein C. 1999). This would later prove to be true at least for the field of virology: certain viruses have been discovered using the expansive movement of Jitterbugs to negotiate their environments (Shim J. et al. 2012). The discovery and further research of Jitterbuglike mechanisms by Fuller has caused them often to be named Fulleroid-like linkages in the modern literature by prominent researchers in this field such as Röschel O. (2012). Because of the specific movements of the rigid elements relative to each other, the joints are a study of careful design, as will be discussed later in this chapter. This difficulty in synthesizing the joints led to the early physical models made by Fuller and his collaborators to be mechanically very unstable structures, requiring a supporting armature to keep them from collapsing. (Schwabe C. 2010) One of the original built models is displayed in Fig

55 4. Jitterbug-Like Linkages Fig 4.2 Model array of jitterbug-like mechanisms stabilized by cables ( 4.1 Geometrical possibilities The basic octahedral Jitterbug is taken here as an example for later generalization of the Jitterbug movement. The transformation each triangular element of the octahedral mechanism undergoes is a helical screw movement along an Instantaneous Screw Axis (ISA) that is normal to the plane of the element and goes through the circumcenter of the triangle. Since for the octahedron there exists an inscribed sphere that is tangent to the triangular facets at their circumcenters, the ISAs of all of the facets pass through the center of the inscribed sphere. Fig 4.3a Closed Jitterbug mechanism with Instantaneous Screw Axes (ISAs) Fig 4.3b Dilation of the base polyhedron during transformation 44

56 4. Jitterbug-Like Linkages The ISAs themselves are fixed in space, and hence the resulting movement of the whole octahedron is a dilation. This becomes more obvious when the underlying octahedrons are seen for each configuration of the Jitterbug : since all of the facets retain their original relative angles and they dilate at the same rate, planes parallel with each element intersect along the edges of a dilating octahedron, the base polyhedron. Fig 4.3a shows the ISAs of each facet of the octahedron, and an inscribed circle tangent to some of the facets at the intersection with their ISAs. Fig 4.3b shows the same geometry going through the Jitterbug transformation, in which the base octahedron dilates around its center, and in each facet of this base polyhedron the real triangular facets that define its edges undergo a rotation. The dilation ratio of the base polyhedron is ( ). Fig 4.4 Dilation of base polygons during transformation Fig 4.5 Maximum deployment with base polyhedron When the maximal configuration is reached, all the vertices of the triangular facets intersect the edges of the base octahedron in the centers of these edges, as in the hexahedron in Fig 4.5. For this case, the maximal value for is /3, and so the maximal deformation dilation is ( ( )). For the octahedral Jitterbug, the dilation is homogeneous, in other words it is a homothetic transformation. There exist Jitterbug mechanisms for which the transformation is not homogeneous, such as the cuboctahedron Jitterbug in Fig 4.6. Fig 4.6 Dilation of cuboctahedral Jitterbug-like mechanism (Kiper G. 2010, adapted from Röschel O.) 45

57 4. Jitterbug-Like Linkages Kiper G. shows in his 2010 doctoral thesis that a mobile homothetic Jitterbug can be obtained from any polyhedron of which the homothety centers of adjacent facets are in symmetrical positions relative to their common edge, i.e. when the ISAs (which intersect the homothety centers of their respective facets) of neighboring facets also intersect each other. The movement between two adjoining facets along the Jitterbug transformation is made physically possible thanks to the joints that link together those facets (Fig 4.7). For the purpose of visibility they are exaggerated here. Dreher D. reportedly developed the first prototypes of said joints when working as a student under Fuller (Schwabe C. 2010). The joints allow each facet to rotate inside its plane, while maintaining the dihedral angle between the facets. For this reason they are often referred to as Dihedral Angle Preserving (DaP) joints. In the literature, they are also sometimes referred to as gussets, double rotary joints (Wohlhart K. 1995), spherical double hinges (Röschel O. 2012) or because of their shape simply as V-joints (Kiper G. 2010). These joints obviously have two rotational DoF, and as such it is necessary to link the facets of the Jitterbug together into a closed loop in order to get a 1DoF system. Fig 4.7a Dihedral Angle Preserving (DaP) joint Fig 4.7b Joint detail (Verheyen H. 1989) This means that the normals of two adjacent faces have a constant angle between them, even though the plane which they define may undergo a rotation during the Jitterbug transformation. (When their plane on the other hand undergoes a pure translation, this implies that the Jitterbug transformation is a homothety). The Jitterbug transformation in this way is applicable to different polyhedron groups, the most obvious ones belonging to the Platonic- and Archimedean solids. As an example, in Fig 4.8 an icosidodecahedral geometry is shown transforming to a rhombicosidodecahedral geometry. 46

58 4. Jitterbug-Like Linkages Fig 4.8 Dilation of icosidodecahedral Jitterbug-like mechanism (Verheyen H. 1989) Odd-valent vertices It is important to note that the helicoidal movement of each facet of a Jitterbug is of a rotation opposite to that of its adjacent facets. Because of this, it was once presumed impossible to construct a mobile Jitterbug from a base polyhedron that has an uneven amount of facets coming together at any of its vertices. This would imply that a single facet has to be able to undergo both a clockwise- and a counterclockwise rotation. There are however some loopholes tricks that allow for a mobile Jitterbugs to be made out of base polyhedrons that have an odd number of facets intersecting at their vertices. These will be discussed in the following section of the chapter. Double facets dipolygonids A first and most important technique in making odd-valent vertex Jitterbugs mobile is simply doubling all of the facets and connecting them in an alternate manner. This doubles the valence of each of the vertices, thus making any odd-even vertex polyhedron possibly mobile. The facet doubles are attached to their original facets by means of a single R-joint whose axis is the ISA of the original facet. Using this method thus implies making a type of spatial Scissor-like elements, and any loop of four facets (two original ones and their doubles) form a single DoF mechanism. In other words, there is no need for a closed group of elements to gain the single DoF, and more stable Jitterbug-like mechanisms are the result. The idea was first proposed by Clinton J., a former student of Fuller. It was picked up and researched later in depth by Verheyen H. who named these mechanisms dipolygonids. Some examples from Verheyen s 1989 paper are shown in Fig 4.9a-c, showing respectively a cube-, dodecahedron- and icosahedron dipolygonids, and Fig 4.9d showing the movement of the icosahedron variation. 47

59 4. Jitterbug-Like Linkages Fig 4.9a Hexahedral dipolygonid Fig 4.9b Dodecahedral dipolygonid Fig 4.9c Icosahedral dipolygonid Fig 4.9d Dilation of Icosahedral dipolygonid (Verheyen H. 1989) 48

60 4. Jitterbug-Like Linkages Multiple elements per facet A more efficient way of making odd-valent vertex Jitterbugs mobile is to subdivide the facets of the base polyhedron in an even amount of smaller subfacets, so that at each vertex there are an even amount of them. This changes up the geometric transformation of the whole, and will almost never result in a homothetic transformation. In Fig 4.10 examples of this method for the tetrahedron and cube are given by Kiper G. (2010) (who adapted the figures to make them more readable from the work of Wohlhart K. 2001). Fig 4.10a Multi-facetted icosahedron Fig 4.10b Multi-facetted cube (Kiper G. 2010) In his thesis, Kiper G. also applies this technique to gain ring-like structures from dipyramidal base polyhedrons. An example is given here for a subdivided geometry of an octagonal dipyramid in Fig These shapes have great interest because their facets behave very much like SLEs, and hence have greatly varying dimensions in their open and closed states. Fig 5.11 Multi-facetted dipyramid (Kiper G. 2010) 49

61 4. Jitterbug-Like Linkages Offset elements A third and last known technique for obtaining mobile Jitterbugs is using offset elements between the vertices of connected polyhedron facets. This way, the total amount of elements around each vertex gets doubled, and a clockwise-counterclockwise movement becomes possible. An example is given by Kovács F. et al. The model they used in their research for the behavior of micro-organisms is an expendable dodecahedron with offset elements. Fig 4.12 Offset elements in dodecahedron (Kovács et al.) Also in the work of Wohlhart (2001) these offset Jitterbug-like mechanisms are described into detail. He uses the type mechanism to synthesize both regular and irregular dilating shapes. Examples are the expanding icosahedron in Fig 4.13 and the cylinder in Fig Fig 4.13 Offset elements Fig 4.14 Offset elements in cylinder (Wohlhart, 2001) in icosahedral geometry (Wohlhart, 2001) 50

62 4. Jitterbug-Like Linkages Planar variations When the ISAs of adjacent facets are taken to intersect at, they come to lie in the same plane. The Jitterbug movement now is degenerated into a two-dimensional one. An example of hexagonally connected triangles is seen in Fig Fig 4.15 Hexagonal dipolygonid pattern Edges of these planar mechanisms can be connected to create larger polyhedral mechanisms. In fact, this is exactly what is being done when a facet is subdivided to obtain even-valent vertices. When turning the edges, the movement of the facets on each side needs to be compatible with its subdivided neighbour. Making each edge congruent is a simple way of achieving this, as in the pyramidal dipolygonid by Verheyen H. (1989) in Fig Fig 4.16 Pyramidal dipoligonid with facetted faces (Verheyen H. 1989) 51

63 4. Jitterbug-Like Linkages 4.2 Architectural application The relative scarceness of designer familiarity with the Jitterbugs and the difficulty in synthesizing the DaP joints has led them to be underused at bigger-than-human scale in general, and the field of architecture in particular. Notwithstanding, they show potential for larger-scale structures, especially in the arts. Some of the few materializations of the Jitterbug mechanism are typically gadgets and furniture, such as the coffee table fashioned by Verheyen H. in Fig Fig 4.17 Coffee table from cube Jitterbug-like mechanism (Verheyen H. 1989) Further they have been used in the dancing arts by Tomoko Sato in her Synergetics performance (Schwabe C. and Ishiguro A. 2006), and other performers as seen in Fig 4.18 and Fig Free body expression with Jitterbug geometry (Schwabe C. and Ishiguro A. 2006) 4.19 Free body expression with Jitterbug geometry (Schwabe C. 2010) 52

64 4. Jitterbug-Like Linkages A first attempt to apply the Jitterbug mechanism on large scale was the Heureka project (Fig 4.20) that was the symbol of the Swiss national research exhibition in Zürich, It was a mobile sculpture with the original octahedron Jitterbug geometry and 8m side lengths of the triangular facets (Michaelis A. R. 1991). It was through the initial lobbying and design done by Schwabe C. that this sculpture would be realized. When it opened, its height doubled and the volume five-folded. After three months in operation it collapsed into the tetrahedral position, because the steel hinges and the connections to the triangles made of composite polyester were not well enough engineered. (Schwabe C. 2010) Fig 4.20 (Schwabe C. 2010) A last example of the use of dipolygonid Jitterbugs in architectural applications is by Gómez Lizcano D. E. (2013), who used a triangular grid applied to the polygons of an icosahedron Fig 4.2. The polygons are connected to their sides by means of P-joints. The joint connections between the facets here are more complex and require special maintenance. Although a prototype of one facet with sliding sides has been made, the whole project remains unbuilt. 53

65 4. Jitterbug-Like Linkages Fig 4.21a Triangular dipolygonid facet with sliding edges Fig 4.21b Icosahedral pavilion from dipolygonid facets Fig 4.21c Expandable dipolygonid pavilion in use (Gómez Lizcano D. E. 2013) Many more possibilities remain undiscovered, not in the least façade application for the opening and closing of planar variations, and public sculptures for the polyhedral Jitterbugs. Research in the development of sturdy DaP (Dihedral Angle Preserving) hinges is a definite requirement before designing these mechanisms on a large-scale can be made possible. 54

66 5.Overconstrained Linkages 5. Overconstrained Linkages In this chapter, different 1 DoF spatial mechanisms with a minimum of elements are discussed. The term overconstrained refers to the fact that they have more DoF than the analytically determined constraints predict. This will be discussed in more detail in chapter 5, Kinematic studies. Most of these mechanisms are well-studied in the fields of kinematics, but more recently have found their way into the field of deployable structures. As of yet, ways of chaining of these mechanisms together to form compactable wholes are unsatisfactory, but advances have been made in the last decade. Firstly three similar groups of mechanisms, namely the Bennett-, Myard-/Goldberg- and Bricard linkages, are analyzed as single- and multi-loop systems. They are respectively 4R, 5R and 6R mechanisms. The geometrical conditions for their mobility are given, clarifying their conceptual connection to one another. The three groups of linkages are strongly inter-related, so that it is convenient, if not necessary, to treat them all at once (Baker E. J. 1979). Secondly parallel manipulators, a group of mechanisms popular in contemporary literature, are discussed. The subgroups of Wren platforms and Sarrus linkages are analyzed further because of their interest in the field of deployable architecture. Especially the second group will show to be the basis for many known deployables that have been discussed in previous chapters. 5.1 Bennett linkages Forming a closed loop of four elements, each connected to one other by a total of four R-joints, there were formally only two known ways to construct a mechanism from the four parts. The first one was by making the axes of the R-joints intersect at, gaining a planar rhombus mechanism. The second way was making each of the four R-axes intersect at a common center point, which results in a spherical mechanism. In the early 1900 s, Bennett introduced a new single DoF mechanism of four looped elements that doesn t adhere to the last two classes of 4R mechanisms: a new spatial mechanism was born. (Bennett G. T. 1903) A Bennett loop is shown in Fig 5.1. Fig 5.1 Deployment of Bennett linkage (Gan W. W. Pellegrino S. 2003) 55

67 5.Overconstrained Linkages For the 4-bar loop to be mobile, certain geometrical conditions between the revolute joints and their axes must be satisfied. Here they are quoted from Bennett G. T. (1914) and Chen Y. (2003). Fig 5.2 Bennett linkage (adapted from Chen Y. 2003) - Two opposing elements have the same length: - Two opposing joints have the same twist: -The relationship between twists and lengths is fixed by: [5.1] - By these relationships, the operating angles become mutually dependent parameters, as they are bound by three equations, and so a single DoF mechanism is the result. 56

68 5.Overconstrained Linkages [5.2] [5.3] ( ) ( ) [5.4a] - In the special case of and, the resulting mechanism will be equilateral and equation [5.4a] then becomes [5.4b]. Many of the studied Bennett linkages in the field of deployable structures take on this form for the symmetry and simplicity of reproducing the bar elements. [5.4b] -Furthermore, if and, all the elements are congruent and the motion becomes discontinuous, since is no longer uniquely defined when. - In the degenerate case of and, the mechanism becomes a 2D rhombus. Chen Y. (2003) goes on to prove that a completely flat-folded state of Bennett linkages exists. Furthermore, she makes both mathematical and physical models of chained Bennett linkages, resulting in compactable Bennett chains of both planar (Fig 5.3) and cylindrical variations (Fig 5.4), as shown by You Z. and Chen Y (2011). Fig 5.3 Flat array of Bennett linkages (Chen Y. 2003) 57

69 5.Overconstrained Linkages Fig 5.4 Singly-curved array of Bennett linkages (You Z. and Chen Y. 2011) Melin N. O. is involved in the same field of study, attempting to materialize a long-span structure made of foldable Bennett units, as shown in Fig 5.5. These and other studies in the connecting of Bennett units reveal a difficulty in compacting the mechanism when multi-layer structures are desired in deployed state. More research on the connecting joints between units appears to be necessary to make these deployables competitive with existing SLE structures. Their small structural height further makes them less than ideal for large-scale applications. g 5.5a Deployment of curved array of overlapping Bennett linkages Fig 5.5b Diagram of deployed curved array (Melin N. O. 2004) 58

70 5.Overconstrained Linkages 5.2 Goldberg and Myard linkages The 5R Goldberg linkage is a single DoF five-bar mechanism that can be generated by joining two Bennett linkages together through a mutual element, as in Fig 5.6 (Huang Z. et al. 2013). If the first of the joined Bennett linkages has a pair of elements with length and twist, and another pair of elements with length and twist, then the second Bennett linkage will have a pair of links which share the link length and twist, and a last pair with length and twist. Adding these linkages together will then give a single DoF mechanism with 6 bars, of which the shared one is redundant. The relationships between the lengths and twists for this mechanism then become: [5.5] Fig 5.6 Synthesis of 5R Goldberg mechanism from 2 Bennett modules (adapted from Huang Z. et al. 2013) What was formerly seen as a separately discovered mechanism by Myard in 1931, was actually a plane-symmetric variation of the 5R Goldberg linkages. Two Bennett linkages are mirror images of each other, the mirror being coincident with the plane of symmetry of the resultant linkage (Baker E. J. 1979). So for this mechanism the twists and the remaining twist. A schematic Myard linkage is seen in Fig 5.7. Fig 5.7 5R Myard linkage (adapted from Chen Y. 2003) 59

71 5.Overconstrained Linkages These Myard linkages can be combined into circular arrays in which two bars and one revolute joint of any linkage is shared with each of its neighbors. The left mechanism shown in Fig 5.8 is such a combination of two linkages, the system on the right shows a circular array of six linkages. Since the movements between all the composing linkages are congruent, the resultant system also has 1DoF. Fig 5.8 Array central connection of Myard linkages (adapted from Huang H. et al. 2012) Another way of chaining together Myard linkages is by connecting them on the peripheral points of the loops. This is done in Fig 5.9: three linkages are connected together to form a 1 DoF mechanism that forms a planar triangle in its deployed state. Fig 5.9a shows this system for three connected Myard linkages, while Fig 5.9b shows a combination of both the central and peripheral connections to make a planar hexagon when deployed. Fig 5.9a Array from peripheral connection Fig 5.9b Array from both central and peripheral of Myard linkages connection of Myard linkages (Qi X. et al. 2013) The multiple-loop systems here are new advances in space engineering (antennae are the mainly proposed structures), but a good application of them may well lie in deployable architecture. If singly and doubly curved surfaces are tessellated in a way that allows these mechanism patterns to overlay them, efficient deployables might be distilled. 60

72 5.Overconstrained Linkages 5.3 Bricard linkages Between 1897 and 1927, Bricard R. established 6 new classes of 6R mechanisms that have 1DoF. The classes are divided according to the orientation of the members throughout the motion, but share basic geometric relationships. Here, a planar-symmetric variation (Chen Y. et al. 2005) is discussed, since it shows the most applicability to deployable structures. Link lengths in this variation are all equal, and twist angles of adjacent joints are supplementary: and The relationship between the operating angles is fixed: and Fig 5.10 Bricard linkage (Chen Y. et al. 2005) When these hybrid Bricard linkages fold flat to triangular shape, they can be chained together using two common elements with a central joint, as with SLEs. (Fig 5.11). This way, the 1DoF property is maintained. However, since the connections made into a single-layer mechanism need alternating orientations to function, it is not so trivial to fashion curved surfaces out of them. Fig 5.11 Flat array of Bricard linkages (Chen Y. 2003) 61

73 5.Overconstrained Linkages 5.4 Parallel manipulators In kinematics, a parallel manipulator typically consists of a moving platform that is connected to a fixed base by several limbs or legs in parallel. (Li Y. and Xu Q. 2007) An advantage of parallel manipulators is that geometric errors in one bar are normally compensated for by the others Modified Wren platforms Kiper G. (2010) discusses the known Wren platform mechanism that uses spherical joints to connect bars to the platforms (Fig 5.12). In the case where the legs are skew relative to each other, the Dof of the system is 1. If the mechanism moves to a state where the legs are all parallel, a 2DoF mechanism is the result. Fig 5.12 Wren platform in its two mobilities (Kiper G. 2010) To lift the possibility of the mechanism moving into a 2DoF state, Kiper G. (2010) proposed using the DaP connections of Jitterbug mechanisms to connect the legs and platforms. These modified Wren platforms indeed have 1DoF, but are difficult to chain together in deployable mechanisms, since they have multiple DoF when using the platforms as common elements (Fig 5.13a and Fig 5.13b). Fig 5.13a Modified Wren Fig 5.13b Array of modified Wren platform with DaP joints platforms (Kiper G. 2010) 62

74 5.Overconstrained Linkages Sarrus linkages The first overconstrained linkage ever to be studied was the 6R linkage published by Sarrus P. T. in It is a 1DoF parallel manipulator mechanism in which the legs consist of revolute pairs, connected to the bases again by revolute joints. The result is that the connected platforms undergo a straight line translation, which makes this mechanism interesting for a myriad of applications. For the mechanism to work, the minimum amount of leg pairs is 2. In the examples in Fig 5.14, the legs are perpendicular, but the necessary condition is simply that the axes connecting them to the base are not parallel. Fig 5.14a Sarrus linkage Fig 5.14b Diagram of Sarrus linkage (Chen Y. 2003) ( The Sarrus mechanism is so elegant in its simplicity that it has often unknowingly found its use in many of the transformable structures known today. A first group of transformables that is actually a chain of interconnected Sarrus linkages are the SLE deployables: often, they are described as being interconnected two-dimensional mechanisms. But another point of view shows that in three dimensions, their single DoF is due to the basic module of the Sarrus linkage contained in them. Fig 5.16a shows a rectangular, planar SLE grid, made up of Sarrus units as in Fig 5.15b. Fig 5.15a Rectangular SLE grid Fig 5.15b Applied Sarrus linkage 63

75 5.Overconstrained Linkages The double-faced Jitterbug-like mechanisms (dipolygonids) are another mobile group build out of Sarrus linkages. This can be seen when the 2DoF DaP joints are seen as kinked physical elements which are connected by 2 revolute joints. So, another definition of the platforms in Sarrus linkages is dihedral angle preserving links. The connected facets in Fig 5.16a are replaced by bar structures that are contained within them in Fig 5.16b, in order to show the identity of the two mechanism groups. Fig 5.16a Dipolygonid connection with DaP joints Fig 5.16b Applied Sarrus linkage Another place in transformable structures where the Sarrus linkages have found their use is in a certain type of cupola structures described by Wohlhart K. In his 2007 paper he describes the need to have a 1DoF building block in order to make 1DoF cupola mechanisms that aren t closed loops. One of his solutions is doubling the links in order to have each of the facets move like Sarrus linkages (Fig 5.17). This method implies physical interference between the links and hence low compactability. Fig 5.17 Application of Sarrus linkages in cupula mechanisms (Wohlhart K. 2007) 64

76 5.Overconstrained Linkages In his 1981 doctoral thesis, Calatrava Valls S. studied some overconstrained mechanisms, amongst which some were based on the Sarrus linkage. Fig 5.18 shows multiple Sarrus units joined together. Fig 5.18 Hexagonal array of Sarrus linkages (Calatrava Valls S. 1981) A last example of Sarrus chains is by Huang H. et al. (2012). In their paper, one of the proposed mechanisms is a group of perpendicularly connected Sarrus mechanisms. Notice in Fig 5.18 that the axes connecting the smaller (black) leg pairs are not normal to the surface of their connected larger (red) bars, since this would give a planar 3DoF mechanism. This slight twist in joints makes the model a 1DoF Sarrus chain. Fig 5.18 Alternative rectangular array of Sarrus linkage (Huang H. et al. 2012) The prevalence of the Sarrus linkages in many of the deployable structures is the proof of its practical usefulness. Studying them when faced with existing transformables can help better understand their behavior. New transformable structures can and will very likely be distilled from the overconstrained mechanisms mentioned here. The question of how to successfully link them together into different shapes that allow their movement is one the frontiers of the design of deployable structures. 65

77 Part II. Design Tools

78 6. Kinematic Studies 6. Kinematic Studies In this chapter fundamental concepts and methods for describing the geometric-kinematic characteristics of transformable structures are explained, using concise examples and offering references for further study. It is the understanding of these very basic methods that make it feasible to design and later analyze the kinematic behavior of more complex mechanisms. The concepts are generally applicable to any structural system that falls under the category of rigid links, such as SLEs, rigid-foldable origami, Jitterbug-like mechanisms, etc. Understanding these underlying concepts, the arbitrariness of the categorization and the subdivision between these groups also becomes clearer, and crossovers between the different transformable structure groups become thinkable. 6.1 Determining degrees of freedom A simple formula, known as Grüblers equation (Grübler, 1917), can be used to define the DoF of a planar mechanism. The formula stems from a few basic observations: Firstly, that the amount of independent displacements of any planar link is threefold: one rotation and two mutually perpendicular translations. Any free planar link thus has a DoF of 3. Restricting these three displacements makes for a grounded link, reducing its DoF to 0. Connecting two independent links by means of a joint takes away the independence of their respective movements by a degree that depends upon the type of joint. The most used of all joints, a planar R-joint, binds the two translational freedoms of the connecting links, thus reducing the DoF of the whole by 2. When a joint connects more than two links, it is said to be of a higher joint order or valence (Norton R. L. 1991), and accordingly reduces the mobility of the connected links. The order is equal to the amount of connected links minus one. For example, an R-joint connecting three links is of order 2, and reduces the mobility of the system by 4. An R-joint connecting four links is of order 3 and reduces the mobility by 6. Pouring these observations into a formula gives: [6.1] In which: M: mobility of the mechanical system E: number of links, including the grounded link G: number of grounded links number of joints of order i i: order of corresponding joint 67

79 6. Kinematic Studies As in the analysis of mechanisms there will always be a chosen frame in which the movement occurs, normally one single link is held to be grounded, thus simplifying the equation to: ( ) [6.2] Since joints of 2DoF, such as half joints which possess a translational and rotational freedom, also exist in planar mechanisms, Kutzbach modified Grübler s equation to account for these: ( ) [6.3] In which: : joints with 1DoF and order i : joints with 2DoF and order i For the structure in Fig 6.1, equation [6. 1] gives a mobility of 0, the structure thus is static and there is no movement possible. Fig 6.1 Static structure (2 x 1 st order, 3 x 2 nd order, 1 x 3 rd order) ( ) ( ) 68

80 6. Kinematic Studies Fig 6.3a 1DoF system with 4 dependent rotations Fig 6.3b Dependent deformations J = 6 (3 x 1 st order, 2 x 2 nd order, 1 x 3 th order) ( ) ( ) In Fig 6.3 one of the bars is removed, resulting in equation [6. 2] giving a single DoF: there are 4 joints in which a rotational movement is liberated. However, the rotation around one of the joints completely determines the rotation around the remaining three, as fixed by. The figure of the rhombus thus makes the movement of the structure in Fig 6.3b possible, and since the left square is static due to the triangulating diagonal, the mechanism can be reduced to its pure form. This rhombus system is one of the simplest (1DoF) mechanisms in existence. It is also commonly referred to as a planar 4-bar mechanism and is used extensively, for example forming the basis of the 1DoF scissor-like structures. Fig 6.3c Planar 4-bar mechanism 69

81 6. Kinematic Studies To turn the system in Fig 6.3 into a structure again, it is necessary to add one more bar to make its mobility M equal 0 (a static system), or a negative integer (a hyperstatic system). However, adding an extra link in just any place might give the desired result to Grüblers equation; it will not necessarily make a static structure out of the mechanism. The added bar in Fig 6.4 was not inserted in the mobile part of the mechanism (the rhombus). This is a trivial exception to Grüblers equation, since it can be taken as a rule that the DoF need to be spread evenly over the analyzed mechanism and no superfluous links should be counted. Fig 6.4 Superfluous links in a mechanism However, inherent exceptions to Grüblers equations validity do exist, and the mechanical systems that don t follow the rules set by the equation are called kinematic paradoxes. One of the simplest examples of these paradoxes is the triangle bar system with 3 (sliding) P-joints, as shown in Fig 6.5. Fig 6.5a Kinematic paradox (3P loop) Fig 6.5b Movement of 3P loop E = 3 J = 3 (3 x 1 st order) ( ) 70

82 6. Kinematic Studies Grüblers equation here gives what would be a static system, when it is clearly a mechanism. It is important to be aware of the limitations that are still found in this (and any non-geometric method) used to calculate DoF, since it cannot account for dimensional exceptions. Objectively speaking, there is a sharp contradiction between the theoretical formula [Grübler] and practical applications. There is an urgent need to resolve this contradiction in order to ensure the continuous invention and application promotion of new functional mechanisms. (Zhao J. et al. 2014). Different methods have been proposed for calculating the actual DoF of systems, among which the most successful are those based on screw theory. Hitherto no consensus has been reached for adopting any of these methods and equations. This stresses the need for designers to always test their ideas on both real prototypes and virtual models to get feedback on the kinematics. The method used to get to Grüblers equation can be applied to determine the DoF of spatial structures, adapting the formula to take into account that each joint has a total of 6 independent possible movements: one translation along each of the 3 independent axes, and one rotation around each of these axes. Spatially, any 1DoF joint will bind 5 of the formerly independent displacements of the connected links. The equations [6.2] and [6.3] become: [6.4] ( ) [6.5] In which: : number of joints with xdof and order i It s useful to understand that the positive terms in equations [6. 1] to [6. 5] are more general than joints the amount of independent movements that any part of a structure can undergo, while the negative terms are more generally than joints the restricting equations, or constraints, to these movements. As an example of the spatial formula, analyzing an open 3-bar mechanism with 2 R-joints with equation [6. 4] gives 2DoF, each being of course the independent rotations of the bars round the grounded bar (Fig 6.6). 71

83 6. Kinematic Studies Fig 6.6 Spatial mechanism L = 3 J = 2 (2 x 1 st order) ( ) There are also paradoxes to the equation for three-dimensional DoF. Fig 6.7 shows a 1DoF Bennett mechanism (see chapter 5). Applying equation [6.4] here gives a DoF of -2. Again, the general equation can t account for dimensional exceptions such as this one. As seen in chapter 5, these kinds of systems are referred to as overconstrained mechanisms, as alternative name for kinematic paradoxes, since the Grüblers equation predicts more constrains than are actually working on them. Many times, overconstrained mechanisms can be perceived as made up from sub-mechanisms with 1DoF each, joined together in a compatible way as to still produce a 1DoF system together. The Bennett mechanism in this case can be dissected into two 2-bar mechanisms with 1DoF each (Fig 6.). Points (A,B) and (A,B ) have the same trajectories: trajectory curves c and c are identical but displaced. When joining A with A and B with B, the axes going through them can be chosen as to always make c and c coincide, resulting in a 1DoF mechanism. These complimentary sub-mechanisms that share the same trajectory curves were dubbed cognate links by Hartenberg R. S. and Denavit J. (1959). 72

84 6. Kinematic Studies Fig 6.7a sub-mechanisms forming Bennett linkage Fig 6.7b Bennett linkage E = 4 J = 4 (4 x 1 st order) ( ) Overconstrained mechanisms give a kinematic advantage over regular mechanisms in the sense that their mobility is only given within specific geometric bounds. This means that unexpected degrees of freedom due to slight material deformations are less common in them, making them more reliable particularly for 1DoF mechanisms. This is also the reason why the most successful deployable structures are based on overconstrained mechanisms, giving them more controllability on the kinematic level. There is a flipside to this however, since for certain groups of overconstrained linkages, such as the Myard 5-bar mechanism, small production imperfections may cause the geometric conditions for mobility to not be met. This would cause the mechanisms to become practically static or unable to be assembled (Huang H. et al.). For this reason, additional joint clearance can be added to the mechanisms, for example giving additional rotational freedoms perpendicular to existing R-joints. 73

85 6. Kinematic Studies 6.2 Trajectories and envelopes The continuous sequence of movements of a structure its trajectory or displacement path can be displayed in different ways. A common way of doing so is by plotting out the complete range of motion of its members. A two- or three- dimensional surface, called the envelope, is marked by the linear trajectory described by the vertices. This envelope can then be used to predict the total space the structure takes in throughout its transformation process. Choosing the origin of the coordinate center is of importance for the eventual perception and vector equations of the trajectory (Fig 6.8). By choosing the origin on the vertex O results in the envelope being a regular ellipse, while the trajectory for the origin on the vertex O will give a lopsided ellipse. In Fig 6.9, the spatial trajectories of a Sarrus-based mechanism are shown for the origin on the side vertices of each of the 4 modules. Fig 6.8 Trajectories for: origin in O origin in O Fig 6.9 Trajectory for Sarrus-based mechanism (Calatrava Valls S. 1981) 74

86 6. Kinematic Studies 6.3 Generalized trajectory of 4-bar deployable structures Many of the known deployable structures based on overconstrained mechanisms (Bennett-, Myard-, Sarrus- and thus all SLEs) have comparable trajectories: when looking at each of the moving bar elements in these deployables, it can be observed that during the transformation the bars coincide with the changing rulings of a hyperbolic paraboloid. Furthermore, for 4-bar deployables, the bars will together form an inscribed hyperbolic paraboloid. For 1DoF deployables, the inscribed hypar surface will vary its shape by one variable parameter. Fig 6.10 shows this surface for a rectangular SLE module. Fig 6.10 Inscribed hypar surface of deployables The perpendicular sets of rulings of this hypar surface are given by fixing either x or y in the formula: [6.6] Depending on the damping coefficient c, a hypar surface will change shape between its two degenerate cases of a plane ( ) and a line ( ), all rulings are parallel. This indeed makes the hyperbolic paraboloid appear like the perfect base shape for deployables, which require high compactability. This shape can then be combined in different ways to form different modular arrays. 75

87 6. Kinematic Studies As stated, the trajectories of each of the bars are also hypar surfaces. For an operating angle between each bar and the xy-plane, the bars move along the different rulings of a set. Fig 6.11a shows the trajectory of one bar relative to the center, the bar itself is drawn perpendicular to the x-axis. Fig 6.11b shows the combination of the 4 bars in planar state, i.e. for The equations of these rulings as the bars rotate and move towards or away from the center of a rectangular hypar surface in point { 0 ; 0 ; 0 } are given as in [6.7] as parametric vectors. Fig 6.11a Trajectory of single bar in 4-bar deployable Fig 6.11b Trajectory of all 4 bars in deployable : [6.7] [ ] [ ] : [ ] [ ] In which: : operating angle : (joint to joint) bar length As the opening angle moves through the interval [0, ], the hypar surface formed by all bars shifts from open (planar) to closed position. The intersection points of the rulings (A, B, C, D) are the joint nodes of the deployable and are given by: 76

88 6. Kinematic Studies A: : C: : [6.8] [ ] [ ] B: : D: : [ ] [ ] As a check-up, these coordinates show that points A and B are indeed the symmetric images of points C and D respectively. Note that these equations don t account for physical joint size, since the bars would have to meet in a single point. Simple joint offsets can be inserted in the equations. Since the end points of the bars no longer intersect after the offset, all points A, B, C, D are split into two, each point of the pairs being the end point of one of the connected bars (see Fig 6.12). For joints as in Fig 6.12 these offsets give: : : [6.9] [ ] [ ] : : [ ] [ ] 77

89 6. Kinematic Studies : : [ ] [ ] : : [ ] [ ] In which: p: joint offset from center in tangent direction q: joint offset from center in perpendicular direction Fig 6.12 shows diagrammatic top views of the 4-bar deployable, Fig 6.12a for a joint offset p in the tangent direction of the bars, Fig 6.12b for a perpendicular offset. Fig 6.12a Tangent joint offsets Fig 6.12b Perpendicular joint offsets In physical joints p and q can t both be zero. Mostly, there will be opted for giving the joint a material thickness in either of the two directions: tangential while keeping the perpendicular offset 0, or vice versa. However, an offset in both directions is also possible with this method. More information on joint offsets is given in the cantilever and straddle joint mounts section under the joint design section in chapter 7. 78

90 6. Kinematic Studies Fig 6.13 Projected joint nodes Fig 6.14 In-plane deformation of deployable 4-bar For non-rectangular deployables, the projections of the original intersection points A, B, C, D on the xy-plane - A, B, C, D respectively - will move along the and bisectors as the in-plane deformation angle changes. The translated points along these bisectors,,,,, can then be given in function of, in relationship to the projections of the original intersection points in the rectangular configuration, for which. For the projected points A and C the magnitude of the translation vectors is, for point B and D the length of the translation vectors is : ( ) [6.10] ( ) In which: : projected bar length in xy-plane,, Taking the parameter as variable, the maximum compactability of a deployable will be reached when its corresponding hypar surface has its maximum area surface in planar state. This is achieved when the cross product, in other words when or. In a strict sense, the most efficient 4-bar closed loop deployables that can be made are rectangular. That is of course, neglecting the structural problems brought on by the lack of in-plane stability of rectangular grids. 79

91 6. Kinematic Studies Since the translations due to a changed deformation angle only affect the original intersection points A,B,C,D in the xy-plane, they can be applied easily to them as they were for their xyprojections. The new intersection points,,, and, are then calculated. The difference in signs before the addends is to compensate for the direction of the translational vectors m and n in reference to the x- and y-axes. [ ] ( ) ( ) [6.11] [ ] [ ] [ ] and likewise: ( ) ( ) [ ] [ ] ( ) ( ) [ ] [ ] ( ) ( ) [ ] [ ] 80

92 6. Kinematic Studies Here, the new point coordinates are written in function of and. The joint offset parameters are used and the trigonometric formulae are substituted by applying the half-angle formulae. Expanding these vector representations gives: ( ) ( ) [6.12] [ ] [ ] and likewise: [ ] Fig 6.15a Joint A [ ] [ ] Fig 6.15b Joint B 81

93 6. Kinematic Studies [ ] [ ] Fig 6.15c Joint C [ ] [ ] Fig 6.15d Joint D The new bars between and, between and, between and and between and will each have changed lengths from the original bar length by an amount of, owing to the joint offset in their tangent direction. The center points of the joints are found when both p and q assume a value of 0. [6.13] gives the final vector equations of the bars. Using expressions [6.12] and [6.13], the trajectories of closed-loop deployables can be described for variables (deformation angle) and and (joint offsets). Notice that the deformation angle should stay fixed to give a singular trajectory, in other words a 1DoF mechanism. The kinematic design of some deployables may come down to controlling this parameter for all of its closed loops. 82

94 6. Kinematic Studies * ( )+ = * ( )+ [6.13] [ ] = * ( )+ * ( )+ [ ] * ( )+ = * ( )+ [ ] * ( ) + = * ( )+ [ ] In which: : operating angle : deformation angle : distance between adjacent joint center actual bar length = -2p : joint offset from center in tangent direction : joint offset from center in perpendicular direction 83

95 6. Kinematic Studies 6.4 Auxetic Geometries An interesting geometric way of looking at transformable structures, is by defining them as a collection of auxetic structures, a term that is normally reserved for material analysis on the atomic scale. The term auxetic signifies a negative Poisson ratio. In other words: when an auxetic material is enlarged in one direction, it increases its size in all perpendicular directions. Likewise, when compressed, it decreases in all directions (Álvarez Elipe M. D.; Anaya Díaz J. 2012, 2013). The atomic and mathematical models used for auxetic structures coincide with many of the hitherto developed 1DoF architectural structures. By being aware of the overlap between the distinct disciplines on micro scale (material science and nanotechnology) and macro scale (architecture), independently discovered models could be exchanged, leading to new kinematic structures that are sized to human dimensions. Fig 6.16a Auxetic square re-entrant pattern Fig 6.16b Rigid-foldable waterbomb pattern (Tachi T. Fig 6.17a Auxetic triangular arrowhead pattern Fig 6.17b Spatial grid of pyramidal module ( 84

96 6. Kinematic Studies Both Fig 6.16a and Fig 6.17a have a framework of bars that rely on the same basic principle: the shifted repetition of a geometric figure is made foldable by inserting R-joints at the centers of the common bars. In the waterbomb pattern in Fig 6.18b the bars are replaced by the fold lines of the origami pattern, making the surfaces shift outside of their plane to allow for the rotation. In 6.19b the triangular pattern is extrapolated in 3D, devising a foldable spatial truss with loadbearing qualities. Fig 6.18a Auxetic rotational square pattern Fig 6.18b Square Resch pattern (Tachi T. Fig 6.19a Auxetic rotational triangular pattern Fig 6.19b Dipolygon pyramid (Verheyen H. 1989) The main merit of the auxetic models is that they simplify the sometimes complex kinematic structures into their most basic form. Oftentimes these auxetic patterns can be found when the transformables geometry is projected unto one of its symmetry planes. 85

97

98 7. Materilization Challenges 7. Materialization Challenges With the geometry of the transformable structure chosen, the materialization is the part of the design process where many problems may arise. Conceptual lines are now translated into bars, surfaces into panels and joints go from being simple meeting points of axes to complicated design exercises. As such, the main reason why transformables aren t more widely applied in architecture is the lack of know-how about translating geometric models into real structures: different parameters are at play than for designing static buildings and few people in the conservative construction sector appear to possess the experience to work with them. Furthermore, there is no industrial standardization as of yet in fields such as scissor structures and rigid-foldable origami, which have nonetheless been studied academically. If transformable structures are to be made more accessible to the mainstream, standardized and even catalogued joints designed for each category will be important. Concerning the choice of materials, transformable and deployable structures often rely on lightness, both for ease of transportation and to be able to manipulate the mechanisms with as little energy as necessary. Materials that can boast a relatively high structural resistance versus a low volumetric weight are especially suited. In this category one will find diverse materials such as aluminum, wood and its derivatives, cardboard, composite sandwich panels, textiles, etc. depending on the proposes design. For larger scale structures which have a longer life-span and less transformation cycles, the lightness of course becomes increasingly less important, and sometimes even steel elements are used in these cases. Other important parameters to consider are the ease of reproduction and the modularity of the elements: can they be constructed readily from industrially available parts, or even more easily from (recycled) by-products? In this chapter the design parameters and difficulties in materialization are addressed, in particular the design of joints, actuators and blocking mechanisms. A small part is dedicated to the different tactics available for solving the problem of thickness in rigid-foldable origami. 87

99 7. Materilization Challenges 7.1 Joint design As mentioned, the joint is where the design of a transformable structure can be rated to its quality. To start with, different joint types might be able to give comparable mechanical results, implying a first choice here. Afterwards, the many parameters of mechanical resistance, maintenance, production, ease of montage, waterproofing, etc. come into play. This part of the chapter offers some information about each of these. Joint types When one can choose between different types of joints, there are some practical considerations concerning friction and maintenance. The prevalent and simple revolute pin joints are in a clear advantage: they are easy to design and require little to no maintenance. They owe this last characteristic to the fact that they can keep a lubricant film trapped in the cavity between pin and whole by capillary working. This separation of the parts is called hydrodynamic lubrication (Fig 7.1). If necessary a seal can easily be provided around them to keep the lubricant surplus in and dirt parts out. Radial holes can allow replacement lubricant to be placed without disassembly (Norton R. L. 1991). Fig 7.1 Capillary influence on lubricant film in R-joints (Norton R. L. 1991) When choosing revolute joints in a design that is projected to have a long life span or considerably forces acting on the joints, the use of bearings can be provident. Different kinds of bearings offer different advantages. Ball bearings (Fig 7.2) are actually higher pair joints, connecting the pin only through point contact. They are easily lubricated, require little maintenance, but cannot withstand much load and are not impact-resistant. Cylindrical (needle) bearings (Fig 7.3) are generally more expensive and require more maintenance, but they can carry both radial and moderate axial loads while allowing for an almost frictionless gyration, making them ideally suited for highly stressed joints. When circumstances are not so demanding, the revolute joints can be fabricated with simply drilled holes and bolts, preferable free of screw-thread at the contact surface. 88

100 7. Materilization Challenges Fig 7.2 Ball bearings Fig 7.3 Cylindrical (needle) bearings ( The sliding or prismatic joints have some notable disadvantages: they generally give much more friction, are less forgiving in their imperfections and need more maintenance: the lubrication is not geometrically fixed in place and needs to be resupplied by manual regreasing or running the joint in an oil bath (Norton R. L. 1991). The rails they use to slide on need to be rigid and clean, but open rails tend to accumulate dirt particles, building up the friction up to points that it can grind the mechanism. The same applies to half-joints, where the sliding and rotational movements are combined. Sliding ball bearings exist for both prismatic and cylindrical joints, but require a near perfect surface to run on (Fig 7.4 and Fig 7.5). Simple wheel-rail systems are often more pragmatic and able to take on higher loads, with more friction (Fig 7.6). Fig 7.4 Sliding P-joints on rails ( Fig.5 Cylindrical joint ball bearings ( Fig 7.6 Variations of sliding P-joints (De Marco Werner C. 2013) 89

101 7. Materilization Challenges Cantilever and straddle mounted joints Fig 7.7 Cantilever and straddle mounted joints (Norton R. L. 1991) Joint elements can be supported one- or two-sidedly, denominated cantilever and straddle mounted respectively (Fig 7.7). The straddle mounting can avoid (excessive) bending moment being taken up by the link by keeping the forces on the same plane (Norton L. R. 1991). The double section of a straddle mount moreover means that both sides can take up shear force. Since this makes the cantilever joint inherently weaker, it seems obvious to discard it. It can boast its own advantages however: a more easy materialization and, especially for high valence joints, higher compactness. In deployable structures, the net sum of space saved by using cantilever joints can be considerable for categories such as the SLEs where there are many R-joints. The axonometric drawings and top plans of cantilever (left) and straddle mounted (right) joints are shown in Fig 7.8. The joints are hatched grey in the top view. In both cases the joint offsets are the same distance, but the cantilever offers a much higher compactness (it is not yet in its most compact state in the top view). When using cantilevered joints like this, the torsion in the joints themselves needs to be taken into account. Fig 7.8 Comparison of compactness between cantilever and straddle mounted joints 90

102 7. Materilization Challenges Ease of production Naturally, a reliable production process and the use of existing industrial parts is a main parameter to consider when designing the joints. For the examples in Fig 7.8, the cross-shaped joints on the right would have to be especially developed for a multi-step production process, while the square joints on the left could be made easily by slicing a metal profile of standardized cross-section. It is always interesting to check if any of the commercially available joints from sectors outside of construction might offer a valid alternative. E.g. Friction hinges or constant torque hinges are industrialized revolute joints that apply a fixed amount of torque between two adjoining elements, being able to (slightly) fix the parts in any position and thus making them especially suited for transformables that have a wide range of desirable in-between positions. When choosing aluminum as a material to work with, a complex two-dimensional drawing can easily be extruded. This could serve to design versatile joints that fit in multiple connection positions, joints that allow for the attachment of secondary systems such as cables or textiles, allow for locally adding more links that form part of an actuator or blocking mechanism, or give waterproofing to the connection. Because of local stress concentrations and their size determining the overall compactness of the system, it s often decided to fabricate the joints in a stronger material, e.g. steel joints used in aluminum systems or wooden block joints in cardboard structures. Ease of montage and replacement A quick and easy joint design will improve the efficiency in the montage of the whole structure. Any standardized joint should be either symmetric or should denote differences in angles very clearly. If (shoulder) bolts are used as revolute pin connections, they should be readily insertable without overlapping (a common problem in compact revolute joints of higher valence). Elements should be easily removable to change their orientation in the case of montage errors. Depending on the projected life-span of the structure, the ease of replacement needs to be taken into account. The material in the joints will typically suffer the most from extended normal (non failure-related) use. This holds true especially in larger-scale projects where there is a multitude of the same recurring joints, projects which are expected to last through many transformation cycles. In systems like these, the joints where the local friction and other residual stresses are higher will fail first and need replacement for the system to live out its time. 91

103 7. Materilization Challenges Waterproofing and sealants Waterproofing is oftentimes a bothersome issue in transformable structures, since there needs to be a continuous surface circumscribing an interior that might become entangled during the deployment process. Special care should go to the connection to any textile with the joints. An example of good detailing is the swimming pool designed by Escrig F. and Sánchez J. (Fig 2.34). Here, the textile is connected to the interior of an SLE structure, protecting the metal bars and joints from the erosive chlorine gasses. On another note, hydro-induced deformations may occur in the elements and joints themselves if they aren t waterproofed. The summation of these deformations may cause the mechanism to change its geometry to a point where it becomes unmovable. This goes especially for woodbased elements. The case of sealants is also a complicated one, since they need enough flexibility to not compromise the movement during deployment, but enough rigidity as to ensure sealing and even help the locking of parts (De Marco Werner C. 2013). Preformed joint sealants in the form of strips, layers or prisms are commonly available in rubber, plastics and foams and the specific characteristics such as mechanical resistance should be considered when choosing them. Also, thin-layered coverings that can deform elastically are available commercially. Unlike sealant strips, they are continuously connected to both of the linked pieces, making them highly compactable, but sealed during all stages of transformation (Fig 7.9). Fig 7.9 Different sealant techniques (De Marco Werner C. 2013) In some cases, sealants can be used efficiently as material joints. An example could be the use of flexible neoprene or rubber in the fold lines of rigid-origami structures, ensuring mobility and weatherproofing. Such a strategy could potentially solve the vertex difficulties that are typical in materializing rigid-foldable origami. 92

104 7. Materilization Challenges Joint examples Some example joints that try to incorporate different parameters are discussed below. While the first one is generally applicable for structures of the same category, the others are developed as highly individual for their projects and therefore are quite complex. The light-weight and waterproofing of the aluminum R-joints in Fig 7.10, typically used for doors in industrial spaces, make them very suitable and a tested solution in foldable architecture. The added top profile gives the joint a higher mechanical strength and could be used for waterproofing origami projects. The global compactness in some larger fold patterns might be compromised. Fig 7.10 Rigid-foldable origami joint ( An attempt to link multiple Bennett linkages was undertaken by Piker D. (2009b), who designed joints in which the connecting joints each have additional interdependent rotational DoF to allow for the compatible movement of paired Bennett modules. The author acknowledged that unwanted DoF still exist in the joint. Fig 7.11a Tent design from array of Bennett linkages Fig 7.11b Joint detail (Piker D. 2009b) 93

105 7. Materilization Challenges Fig 7.12 shows an S-joint with the possibility of reaching a very high DOF is currently being developed by Yokosuka Y. and Matsuzawa T. (2013) with the particular aim of using it in transformable architecture. The connected elements are being held together by a system that exerts a multidirectional push towards a sphere that sits inside the center of the joint a type of ball-bearing. All of the elements have 3 completely free, independent rotations. The generic name of the prototyped connection is the Multilink Spherical Joint, and allows for larger variable structures, since there is a lesser accumulation of joint tensions due to physical inaccuracies. Fig 7.12 Multi-DoF joint (Yokosuka Y. and Matsuzawa T. 2013) A more conceptual joint is given by De Temmerman N. (2007) (Fig 7.13) as a possible solution to the geometric incompatibilities in doubly-curved SLE grids. The fins that accept the bars are joined around a central cylinder, so that they can rotate small amounts around a vertical axis. Note that the primary function of the hollow cylinder is to conduct a cable that forms part of a complementary structure system that can co-determine the rigidity by the introduced tension. This joint tries to tackle different issues all in one design, but it would not be easy to materialize it without compromising its stress resistance. Fig 7.13 Joint with additional rotational freedom (De Temmerman N. 2007) 94

106 7. Materilization Challenges 7.2 Thickness in rigid-foldable origami Before making the leap to rigid-foldable origami applications on a larger scale, a necessary stepping stone is modeling the plate elements with real thickness. This especially applies to designs where the compactability is an issue and the maximum fold angle is to be reached. There exist several strategies for dealing with thick elements, each of which depending on the specific design has its advantages and disadvantages. Axis shift Hoberman C. (1988) proposed a method that alternates the axes of rotation on a facet and introduces a removal of half of the material at the places where the different facet would intersect (Fig 2.20). A complete folding motion of can be reached this was. The plate elements here have 2 levels of thickness, but compared to the other proposed methods, it is one of the easiest to geometries to materialize. Another important downside is that the application of this technique is limited to regular fold patterns, i.e. it only works for symmetrical and flat-foldable vertices (Tachi T. 2011). Fig 7.14 Axis shift in thick origami panels (Tachi T. 2011) Offset facets Hoberman C. (1990b) introduced a second strategy for coping with thickness: the introduction of increasingly bigger offset facets in triangular and trapezoidal shape (Fig 7.15 and Fig 7.16). It is in particular applicable to high-frequency Miura-ori and the Yoshimura fold patterns, where the addition of overlapping facets forms a real problem to the foldability. It is a method very suitable for materials in which foldlines are easily introduced and as such no separate hinges need to be added. All offset facets in a single row are by definition different, which forms a problem for easy industrial production. 95

107 7. Materilization Challenges Fig 7.15 Axis shift in thick origami panels (Hoberman C. 1990b) Fig 7.16 Expansion of model based on axis shift method for thick origami panels (Hoberman C. 1990b) Slidable hinges A third possible method is giving each of the revolute joints and extra translational degree of freedom, letting them slide accros each other to reach a quasi-foldable solution (Fig 7.17). It was proposed by Trautz M. and Künstler A. (2009) in order to materialize structures based on the Miura-ori fold. This method is not generally applicable because of the global behavior, since the summation of the translations can cause major geometric imperfections. It greatly complicates the joint design. Fig 7.17 Slidable hinges for thick origami panels (Trautz M. Künstler A. 2009) 96

108 7. Materilization Challenges Tapered Panels Tachi T. (2011) presented a fourth method: starting with the ideal (zero thickness) origami model, the facets are thickened by the same amount on both sides. Then, folding the model until quasifolded state, the volume of each of the panels is trimmed by the volume of its direct neighboring panels (Fig 7.18). Truncated pyramid shaped panels are the result. Depending on the thickness, a better approximation of the completely folded state of the ideal facet model can be achieved. Since this method is not based on any particular geometry, it is both locally and globally applicable. The downside however, is that the tapering of the panels is a three-dimensional process: an easy geometric design is traded for a more difficult panel production process. Fig 7.18 Tapered edges for thick origami panels (Tachi T. 2011) Beveled Panels A comparable method as the tapered panels has been proposed by Buffart H. C. and Traut M (2013). Here the volumetric coincident of adjoining plates is trimmed away in closes state, giving even more complex panels (Fig 7.9). The structural performance of the fine vertices is dubious. Fig 7.19 Beveled edges for thick origami panels (Buffart H. C. and Traut M. 2013) 97

109 7. Materilization Challenges 7.3 Actuators For a transformable system, any change of shape requires a change of energy inside its system. The ways energy is added to the system will depend on the scale and weight of the structure, as well as the required precision of the intermediate stages. The possible actuators for transformable systems are commented on below. Gravity and manpower The easiest solution to the actuation problem is using the energy sources that are readily available. For 1DoF mechanisms up to a certain scale, gravity may be a reliable source for at least the deployment phase. For singly or doubly curved bar mechanisms such as those based on SLEs, the uppermost point of the geometry should be lifted in its desired position, leaving the connected elements to unfold under their own self-weight. This process was actually used in the first experimental works of Emilio Perez Piñero, who proposed using a telescopic truss on the back of a truck to lift the transported SLE structure into the air, opening under its own weight. Blocking systems would then be inserted, after which the structure becomes self-supporting and the supporting truss and truck can be taken away (Fig 7.20). Fig 7.20 Gravitational actuation At a smaller scale, gravitational forces might not even be needed to unfold a structure. Many small deployables, such as the commercially available SLE tents used by market vendors and for small events, are easy to set up alone or with a few people. (Fig 7.21). Fig 7.21 Manual actuation ( 98

110 7. Materilization Challenges Using actuators in cases as these would be a waste of material and time. In the cases where manpower can just barely serve to actuate the transformable, mechanical aids such as pulleys, gear trains and leverages can of course be used to transfer the muscle energy more efficiently. Considering leverages, the system needs a substructure to ground itself in order for any momentum to be applied. The biggest leverage should be determined in completely closed position, where the system is hardest to actuate since there is no momentum. Cable and pulley systems Either actuated by human muscle or on a motorized coil, cable systems are among the most efficient for actuating deployable systems. In almost any transformable geometry there exist points that come to lie closer to each other when the system changes shape. Fig 7.22 shows this for a polar SLE structure. The cables between opposing joints are tensioned until the fully open form is reached. Fig 7.22 active cable system actuation Fig 7.17 shows a transformable project done by Laboratoria de Arquitectura, in which a rigid container is hinged around a floor plate. A counterweight is added and the manual coil is used together with a pulley in order to slowly lift and lower in between its closed and ( ) open state. Fig 7.23 Cable coil actuation system ( photo Pedro Kok) 99

111 7. Materilization Challenges Cable systems for mechanism control can be given a secondary structural role, forming a complementary tension network in a deployed structure. Intermediate tensors can be changed in compact state as to give more or less rigidity to the structure in open state. The lower cables in Fig 7.22 would serve exactly this purpose. Special joints could be designed which integrate a guiding track for cables (as the hollow cylinder in Fig 7.13), and at the same time being actuated locally so that by defining their position on the cable they open or close the deployable. The dome-shaped SLE structures by Emilio Perez Piñero were not only supported by a structural cable network on the interior convex side, but also on the outer side, as to offer resistance in the case of wind load reversing the bending moment of the whole. Some of the larger art pieces done by Hoberman Associates have been operated by the use of cable systems wrapped around a motorized axis, giving reliable results for pieces in near constant motion. An example can be seen in Fig 7.24 for a hypar surface made up of angulated SLEs. Fig 7.24 Model suspended from cables for actuation (Hoberman C. portfolio) Motorization For heavier structures, motors may be needed to actuate the systems. The rotational energy gained from the motors can be directly used on an R-joint of a deployable structure, turning one of the connected joints while keeping another fixed. Another option is to create a type of rockercrank subsystem or a cable system that is being operated by the motor. A motor is especially useful in case of constantly adapting structures, as opposed to deployables which only have 2 states of interest. 100

112 7. Materilization Challenges Motor types should be decided on based on their torque-speed curves, determining their speed sensitivity to load. Most of the time, slight variations on the movement speed in these type of mechanisms will not be important, but for certain applications, closed-loop servomotors might be a necessity for smooth movement. For more in-depth information and analysis of motors, the referenced work of Norton R. L. (1991) is advised. A notion to consider for motorized, mobile transformables is the availability of energy: will there always be an energy output at the intended locations? If not, would a generator be a valid energy provider, or do more efficient ways actuating the system exist? In most cases, motorization is only a valid design choice if the transformable structure is of considerable scale or fixed in location as a movable building part. Hydraulics and pneumatics Likewise useful for transformable structures that are fixed in location are hydraulic and pneumatic systems to actuate them. Hydraulic systems need a supporting infrastructure of pumps, tubes and pressurized containers. They are however efficient in both energy and space use, since a small hydraulic cylinder can exert a relatively large force, and can serve at the same time as a blocking mechanism. Examples of the use of motorized hydraulic cylinders can be found in the transformable building parts designed by Calatrava Valls S. The synchronous working of all of the cylinders is necessary for a good functioning, so that a feedback loop has to be inserted in some way. Fig 7.25a detail of hydraulic jack Fig 7.25b Hydraulic jacks in Hemisférico in operation ( 101

113 7. Materilization Challenges Another example of the use of hydraulic cylinders is on the roof of the Merck Serono building in Geneva, designed by Murphy/ Jahn Architects et al. for an international competition. The roof located above an atrium can open up to 5 meters to allow for natural ventilation to occur. Fig 7.26a Detail of hydraulic jacks Fig 7.26b Opened roof of Meck Serono building ( Pneumatic jacks can be used for smaller scale projects and are more energy-efficient. The initial energy needed can be inserted manually and they can take up to medium loads, which make them more suitable for mobile and low-budget projects. Heat deformation Used in smaller scale projects, heat deformations in the materials themselves might cause spectacular effects when the small differences are scaled up by summation or repetition. These are interesting nearly exclusively for light-weight projects, such as the Smartwindow project by Doris Sung, in which thermal bimetals are used to curl open or closed along a thermal curve, so that shading is only applied at moments when solar gains are highest (Fig 7.27). The bimetal surfaces are made up of continuously connected metal layers with different expansion coefficients, making them bend out of plane when heated. Fig 7.27 Smartwindow project actuated by heat deformation ( dosu-arch.com) 102

114 7. Materilization Challenges 7.4 Locking systems After the transformable structure has reached its desired phase, the geometry has to be locked in place. There are different methods to reach this goal, all of which spring from the basic concept of introducing a constraining relationship between two mobile links. They are listed here in order of commonness. Constraining elements The most obvious method of materializing the constraining relationships is by turning them straight into material components, forming fixed triangles with the existing geometry that reduce the local and global DoF, and in doing so stabilize the structure. For example, the changeable distance between two mobile bar link could be fixed by introducing a third bar link that has its end points on the two original links. Retaking the bar geometry of Fig 7.22, this constraining element would be the central orange bar, as in Fig This central bar prohibits the scissor system to close and to open further. However, if the mechanism here formed part of a larger polar network, the central bar could be replaced by a cable with the same effects. Whenever a deployable structure s movement is delimited to a certain phase by its own geometry, using tensile geometries can offer the advantage of simplicity and a locking mechanism less prone to fail suddenly under unexpected loading due to buckling of constraining bar elements. Furthermore, tensile constraining elements can be included in a secondary structural system. Fig 7.28 Constraining bar element There are other ways of dealing with unexpected loading (e.g. wind gusts can have a strong effect on typically light-weight deployables). A tested method for preventing plastic deformation and sudden failure is using spring-parts as constraining elements. An example from the industry of small deployables is seen in Fig 7.29, where gable springs are used to stabilize a square-grid SLE tent structure in a non-maximum position. When loading gets too high, the springs yield and reduce pressure on the connected bars, avoiding any damage to the structure while also guaranteeing enough rigidity of the system. 103

115 7. Materilization Challenges Fig 7.29a Constraining gable springs Fig 7.29b Joint detail ( Another option when using constraining elements is using the same hydraulic or pneumatic cylinders used for actuation, locking them in place to turn the mechanisms into structures, making them very efficient parts. An example of this can be seen in one of Calatrava s mobile designs, an entrance gate to the Valencian metro where a hinged hydraulic cylinder is fixed to keep them open or closed (Fig 7.30). Fig 7.30 Hydraulic jack for opening ( A final important aspect to constraining elements is their frequency or spread: how many and at what exact locations are they inserted into the mechanism. In 1DoF systems, the constraining elements will have to resist the forces in all of the parts that would normally cause the system to transform. Therefore high tensions will tend to form in and around the constraining elements, prompting the designer to distribute them evenly over the structure. This way, the accumulation of small local deformations will also be held in check. In designs for space-encompassing structures, an array of constraining elements can be installed along the connection to the ground, at the same anchoring the whole. 104

116 7. Materilization Challenges Toggle position The term toggle is used in mechanics to describe a stationary configuration of a mechanism, in which two elements connected by a revolute joint are collinear. For typical 4-bar linkages, the toggle position will mean that a triangular shape is formed, with one of its side composed of the two elements (Fig 7.31). Since the connected elements are collinear, no coplanar force can mobilize the system, making it pseudo-static. However, force eccentricities can make the system buckle, and therefore the toggle position should not be used as the only locking system in transformable structures that are intended for bearing load. It can serve as a place-holder for keeping the structure stable while other constraining elements are being locked, or for designs where loads are small and coplanar to the toggled elements. Norton R.L. (1991) writes: In other circumstance the toggle is very useful. It can provide a self-locking feature when a linkage is moved slightly beyond the toggle position and against a fixed stop. It must be manually pulled over center, out of toggle before the linkage will move. You have encountered many examples of this application, as in card table or ironing board leg linkages and also pickup truck or station wagon tailgate linkages. So a way of using toggle to its maximum advantage is by introducing constraining elements that keep the mechanism in its toggle position, as a safe-guard for when eccentricities cause the mechanism to buckle. This way, lesser stress will be taken up by the constraining elements. Fig 7.31 Mechanism in toggle position (Norton R. L. 1991) 105

117 7. Materilization Challenges Snap-through effect and bi-stable structures A third method for stabilizing transformables is using an inherent geometric incompatibility, which happens when a mechanism is impeded in its foreseen motions because they are at odds with the trajectory of a partnering mechanism. Most incompatibilities are found in complex systems, in which the trajectories of multiple mechanisms intertwine spatially. The transformable will undergo additional stresses when it is in phases of geometric incompatibility. If these stresses occur only at intermediate phases of the deployment or transformation, it is said that there is a snap-through effect: an additional amount of force is needed to deform the system and cause it to snap from one phase to the next. This phenomenon can even be of an added advantage, as Zeigler H. (1976) first used these stresses to attain a self-locking mechanism: the structure is fixed easily into its deployed state. Fig 7.32 Deformation of elements in bi-stable structure (De Temmerman N. 2007) If the elements are in a stress-free state before and after deployment, but go through an intermediate stage with deployment-induced stresses, they are called bi-stable structures. The SLE structure in Fig 7.32 is such a bi-stable structure: the marked-red bars are most stressed in an intermediate phase, locking the structure into a semi-stable state when fully deployed. To successfully design and calculate the structures for a snap-through analytical and computer models are needed to make sure the elements deform only elastically, not reaching yield point and destroying the structure in the process. 106

118 7. Materilization Challenges Joining kinematically incompatible mechanisms A last method of locking systems is joining generally incompatible mechanisms together in one exact phase in which they are compatible with one another. Doing so makes them into a stable structure in which the stresses due to constrains are not local, but spread globally. An advantage over bi-stable structures is that with this method no deformations should occur during the transformation process and hence its simples to calculate and materialize. Examples of this method can be found in the rigid-origami Yoshimura pattern based structures, in which a straight and curved mechanisms are joined together in erect state. This is the only state in which the edges of both mechanisms are coincident, hence connecting them in this way effectively makes for a stable structure (Fig 7.33). The method is however not exclusively useful for rigid origami structures, and can find its application in most kind of deployable systems. Fig 7.33 Joining of kinematically incompatible Yoshimura based origami mechanisms (De Temmerman N. 2007) 107

119 7. Materilization Challenges 7.5 Design criteria With the knowledge of the kinematic characteristics and the materialization difficulties applied to each of the transformable structure categories, a designer can make a well-founded evaluation of any kinematic design. Choosing one certain category and detailing method over another is sometimes very straightforward, while at other times a lot of iteration is needed. An example: for building a spaceencompassing structure, the design choice of using rigid-foldable plate elements over bar elements can be justified by their properties of continuity, offering a weatherproof surface and a self-supporting structure all at once. Then, it must be considered if these advantages weigh up against the higher compactability of bar systems. Finally, ease of maintenance and durability of the joints are important themes in repeated deployment. These and many more parameters come into play when designing and choosing a suitable transformable structure. To be able to methodologically evaluate any transformable structure design over another, Hanaor A. and Levy R. (2001) proposed a system with 9 criteria to evaluate the effectiveness and efficiency of any design. The criteria are geared towards self-supporting deployable structures (low weight, high compactability and transportability), but they touch on some important aspects for all kinematic structures. Fig 7.35 shows the criteria in a chart. Fig 7.34 Evaluation chart for deployable structures (Hanaor A. and Levy R. 2001) 108

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121 Part III. Uneven Sarrus Chains

122 8. Uneven Sarrus Chains 8. Uneven Sarrus Chains In chapter 5 it has been shown that the Sarrus mechanism forms the basis of many of the existing deployable structures, and new ways of linking Sarrus mechanisms together have been described both by Wohlhart K. (2007) and Calatrava Valls S. (1981) - although the latter wrongly analyzed their kinematic properties. In this chapter firstly a geometric and kinematic description is given of the basic Sarrus module. Secondly, different ways of chaining these basic modules into arrays are presented, their geometric possibilities (both flat and polar) shown and kinematic characteristics discussed. The geometric analysis of the arrays and their composing modules is then used to create a parametric design tool. Thirdly, two case studies using different array types are analyzed to their structural properties. 8.1 Basic module module Fig 8.1 Deployment of uneven Sarrus The degree of freedom of a minimal Sarrus mechanism, in Fig 8.2 formed by bars,,, and physical joint elements in and, is given by [6.4]: ( ) Which shows that indeed the Sarrus module is an overconstrained mechanism since it actually moves with one degree of freedom. This holds true for any deformation angle. 111

123 8. Uneven Sarrus Chains Fig 8.2 Uneven Sarrus module As the title implies, the basic Sarrus mechanism examined here is asymmetric. The bars connected to the top joint are shorter than the bars connected at the bottom joint. The proportion between the two bar lengths is from here on defined as: [8.1] The mechanism will open unevenly, since the uneven rhombus moves between a closed collinear position and an open position where the shorter bars are coplanar, while the longer bars are oriented diagonally. The nonlinear relationship between the operating angles and (Fig 8.3) is given by: = ( ) [8.2] The graphed version of this equation in Fig 8.3 shows the increasing divergence between the two angles as approaches. On a kinematic level this means that small imperfections may cause relatively large deviations of in the phase close to complete deployment, where the influence of is small. Care should be taken in the design process to avoid potential movements, for example by locking the shorter bars in their deployed position. 112

124 αk 8. Uneven Sarrus Chains α l = cos 1 ( l ) α k cos 1 ( l ) 1,57 1,256 0,942 0,628 0,314 0 λ = 0.9 λ = 1 0 0,314 0,628 0,942 1,256 1,57 rad αl Fig 8.3 Relation between opening angles In order to be able to fold closed completely, and thus reach maximum compactness, the sum of lengths of a shorter and longer bar of a pair needs to be constant throughout a module (and any connected modules). As shown in [2.1], this is called the compactability equation: [8.3] When completely deployed, the coplanarity of the bars makes for a toggle state in the mechanism. As was discussed in chapter 7, the toggle mechanism can serve as a locking system aid, as the bars are in a stationary configuration where no in-planar force can cause movement in the mechanism. From this toggle position, an inverted configuration (Fig 8.4) can be reached where. Note that the concave quadrilaterals formed here are equal ( and ). The inverted configuration can be desirable from the point of view that it offers greater compactness. Joints will however need to accommodate for bar thickness, resulting in differently designed joints in and. 113

125 8. Uneven Sarrus Chains Fig 8.4 Inverted configuration The lack of control to which configuration the bars will move after toggle position, in combination with the greater deviation of to close to this deployed position, call for adapted design solutions for the joint. Fig 8.5 shows an example of this where the joint is extended (here shortened slightly for visibility) to form a central connection bar to lock the joint into place in the unfolded position. Fig 8.5 Extended bottom joint The uneven Sarrus module forms the building block for different arrays that can eventually serve to span distances and carry loads. Fig 8.6 shows three different ways of chaining the Sarrus module together, for each of them the method(s) for introducing a polar angle is shown. 114

126 8. Uneven Sarrus Chains Scissor array Joint-to-joint array Overlap array Polar angles are introduced by changing the bar proportion. The max structural height is thus bound to the polar angle. Polar angles are introduced by keeping top bar lengths equal and using a joint offset with new bar lengths. Polar angles are introduced by using a joint offset and or using new bar lengths and derived by ellipse method. Fig 8.6 Array types and their methods for introducing polar angles The first array method is the widely used scissor connection, in which the bars are continuous. The second array joint-to-joint has no continuous bars, but introduces an intermediate module. In the third array the bars of the modules overlap to form intermediate rhombi. In what follows, the joint-to-joint array and overlap array are shown more extensively, the methods for introducing a polar angle in each of them are given, and their peculiarities are discussed. 115

127 8. Uneven Sarrus Chains 8.2 Joint-to-joint arrays As the name implies, joint-to-joint arrays are made by chaining together modules at their joint nodes so that a repetitive pattern of non-continuous bars is formed. They are one of the deployable structures that Calatrava Valls S. (1981) mentions in his dissertation. The physical model in Fig 8.7 and the render in Fig 8.8 both show a flat rectangular array. Note that there are two joint types: those that are on the inside of the Sarrus modules, and those that are on its edges, which receive the double amount of bars. Fig 8.7 Deployment of flat rectangular joint-to-joint array Fig 8.8 Flat rectangular joint-to-joint array 116

128 8. Uneven Sarrus Chains Fig 8.9 Deployment of flat hexagonal joint-to-joint array Fig 8.10 Flat hexa-triangular joint-to-joint array By changing the angles at which the bars meet each other in the joint nodes, different patterns such as the hexagonal-triangular one in Fig 8.9 and Fig 8.10 can be formed. The method of jointto-joint arrays has an important limitation in that only patterns of polygons with an even amount of sides can be formed. Triangulation in the xy-plane of the structure will thus need to be added by other means. 117

129 8. Uneven Sarrus Chains Polar module with joint offset Fig 8.11 Polar module with joint offset Fig 8.12 Introduction of polar angle by joint offset By partitioning the physical joint in into one part at its original location and a second part offset a vertical distance j upwards, two rhombi with different proportions are formed: and. The resulting module is still a 1DoF overconstrained mechanism. The compactability equation [8.3] is now expanded to: [8.4a] In which: j: joint offset is constant for both rhombi types While the rhombus reaches its fully opened state, the rhombus is partly opened. I.e. while the bars of the first rhombus are parallel, the bars of the second rhombus are at an angle, creating an asymmetrical polar Sarrus module suitable to array along singly curved surfaces. To find a certain j and corresponding to any chosen polar angle, [8.4a] is expressed in function of : ( ) ( ) [8.4b] In a right-handed coordinate system with origin in, the coordinates of A,, and become: [ ] [ ] [ ] [8.5] 118

130 8. Uneven Sarrus Chains Substituting [8.5] in [8.4b] and expanding within the square roots: ( ) ( ) [8.6] For which has one root: ( ) ( ( )) [8.7] In which: h =, the vertical distance between and when deployed Finally, j and can be expressed for : ( ) ( ) [8.8] ( ) ( ) [8.9] Fig 8.13 Maximum polar angle for joint offset method The maximum polar angle that can be achieved for any chosen bar lengths and is reached when the new bars can unfold to become collinear. The values for, and are then: ( ( ) ) [8.10] [8.11] [8.12] 119

131 8. Uneven Sarrus Chains Mostly, this maximum polar angle has little practical application, since it would mean a relatively large joint offset, which could easily potentiate a bending moment in the joint that is too big to compensate for. Fig 8.14 Joint offset giving equal bar lengths For reasons of simplicity and easier production, there can be opted for making the length of the new bars, so that only 2 different bar lengths are used. The corresponding and are fixed to: ( ) [8.13] [8.14] When placed in a joint-to-joint array, the polar module cannot go from toggle position to its inverted configuration, since the toggle position is not reached for both rhombi and (except for when is used). This characteristic can be exploited to effectively lock the mechanism in its open position by tensioning a cable between the central joints and. By using this method for generating polar modules, the minimal structural height h can be determined from the values chosen for bar lengths and. Independently, the curvature can be decided upon: using the polar angle, the joint offset j is calculated relative to the chosen bar lengths. If the joint offset is perceived as too large, iterations can be made. Fig 8.15 Polar modules with extended bottom joint 120

132 8. Uneven Sarrus Chains Taking a repetitive array of a rectangular polar module, cylindrical geometries can be made of which the curvature can be determined independently of the structural height. An example of a semi-arch consisting of modules with constant polar angle can be seen in Fig These arrays of polar modules are completely flat-foldable since the compactability equation [8.4a] is fulfilled. This also means that the changing polar angle converges to zero as the mechanism closes. Fig 8.16 Semi-arch joint-to-joint array from polar modules with joint offset A polar module can also be introduced locally in an array of compatible flat and/or polar modules. An example of this is given in Fig 8.17, where a double-sloped structure is created by connecting two flat arrays by a linear chain of polar modules with polar angle =. Fig 8.17 Locally introduced polar modules connecting flat rectangular joint-to-joint arrays 121

133 8. Uneven Sarrus Chains Mobility of joint-to-joint arrays The overconstrained joint-to-joint arrays typically have 1DoF, but some physical models show an additional, dependent, DoF that can occur due to material imperfections. Firstly, the inherent 1DoF movement is analyzed, here described for a minimal array of two flat Sarrus modules as in Fig Fig 8.18 Minimal joint-to-joint array of two flat 4 bar modules In the regular DoF, the bars trajectory lies on the generalized hypar surface as described in chapter 6. Accordingly, the location of joint nodes can be given for a variable operating angle between each of the bars and the xy-plane, as the mechanism opens and closes: A: C: [8.15] [ ] [ ] : : [ ] [ ] In which: 122

134 8. Uneven Sarrus Chains A secondary hypar trajectory is described by the bars, sharing the coordinates of the A and C nodes. In the same coordinate system, the locations of joint nodes and can then be derived. The x- and y- values will remain the same, while the z-value compensates takes into account the uneven bar proportion. : : [ ] [ ] Using [8.2] allows for writing these solely in function of : : : [8.16] [ ] [ ] It can be checked that indeed for a completely deployed state where thus, the z-value is ; while for the completely folded state where, the z-value becomes ( ). Additional mobility and solutions Physical models show an additional mobility that is notably more lenient close to the folded state, and lessens greatly to disappear as the mechanism approaches the deployed state. This shows that this dependent mobility is not inherent to the geometry but rather originates from small deformations that allow the two modules ( and ) to rotate slightly around the diagonal axis passing through nodes A and C. Meanwhile, the same joints in nodes A and C rotate around the perpendicular diagonals, turning outside of their original plane. Fig 8.19 shows the same physical model as in Fig 8.7; here the additional mobility is demonstrated. A small joint clearance allows for a small angular distortion near the opening phase. Fig 8.20 shows an equivalent fold-line model with the normal mobility of the Sarrus chain to the right and the additional mobility shown (exaggerated) in the bottom. The additional mobility will completely disappear in models with absence of joint clearance. 123

135 8. Uneven Sarrus Chains Fig 8.19 Additional mobility of joint-to-joint array in physical model Fig 8.20 Additional mobility for minimal joint-to-joint array 124

136 8. Uneven Sarrus Chains Since the additional mobility appears only during the deployment phase, the effect on the deployed structure can be made minimal if it is fixed in multiple locations, especially when extended joint-bars and secondary structural systems are used. However, in some cases the deployment process will need to be strictly controlled, and measures should be taken to make sure the mechanism doesn t deviate too much from its 1DoF trajectory. Some example solutions are given below. A first solution method is to locally introduce rhombic elements which have a shared bar connecting them, i.e. local pantographs. Fig 8.21a shows some possibilities for how to achieve this in the xz-plane of two chained Sarrus modules. Only two connected rhombi need to be formed; Fig 8.21b and Fig 8.21c show some examples of implementing this method with fewer bars. As can be seen in Fig 8.21c, the bars can be scaled - making for a good location to install an actuation mechanism. Fig 8.21a Possible locations for introduction of local pantographs to inhibit additional mobility Fig 8.21b Introduction of local pantograph at top bar layer Fig 8.21c Introduction of local pantograph at bottom bar layer 125

137 8. Uneven Sarrus Chains Since the additional mobility makes the opposing joints rotate out of their plane, another method is simply linking any 2 opposing joints by a 3R-mechanism that does not lie in a plane perpendicular to the xy-plane. Fig 8.22 shows this method with the 3R-mechanism lying in the xyplane itself. Fig 8.22 Introduction of transversal bars to inhibit additional mobility In deployed state, the diagonal bars can also serve for the triangulation of a rectangular array. The disadvantage of this method is that in compacted state, the bars that form part of the 3Rmechanism won t be collinear with the original bars, lowering the compactness of the whole. A last method for removing the additional mobility in joint-to-joint arrays is to locally introduce overlapping bar elements, also applying these added bars for the xy-triangulation of the structure. This is demonstrated further in Fig

138 8. Uneven Sarrus Chains 8.3 Overlap arrays The overlap arrays are made by letting the bar pairs of the Sarrus modules cross with corresponding bar pairs of other modules. This way, intermediate pantographic rhombi are formed that fold-closed to a line and deploy to a triangle (due to the uneven bar proportion ). The overconstrained overlap arrays have 1DoF and don t display the additional mobility noted in the joint-to-joint arrays. Fig 8.23 shows a simple overlap array of 4 flat Sarrus modules that is in the midst of its deployment. In Fig 8.24 a completely deployed array of the same modules can be seen. Fig 8.23 Semi-deployed rectangular overlap array Fig 8.24 Flat rectangular overlap array 127

139 8. Uneven Sarrus Chains As the hexa-triangular array in Fig 8.25 exemplifies, array patterns of odd joint count can be used, making for an easier xy-triangulation and giving more options to both planar and curved geometries. Fig 8.25 Flat hexa-triangular overlap array The joint-to-joint array and overlap array can be combined with locally overlapping bar pairs that both remove the additional mobility and give xy-triangulation to a typically even-numbered jointto-joint array. The original bar lengths of the modules in the joint-to-joint array should be used as to comply with the compactability equation [8.3]. After reaching the toggle position, this method will also prevent the mechanism from going into its inverse configuration (Fig 8.4). Fig 8.26 shows how two identical Sarrus modules with 3 equal pairs of legs each are fit together to get an overlap connection in the diagonal sense. Fig 8.26 Minimal joint-to-joint array with diagonal overlap bars 128

140 8. Uneven Sarrus Chains Polar module with ellipse method Fig 8.27 Points for which the compactability equation holds, lie on an ellipse Fig 8.28 Rotation of new top bar length around top joint node by polar angle In the overlap system, a polar angle can also be introduced by using the joint offset method, as was done for the joint-to-joint system and parametrized by equations [8.4] to [8.14]. However, a simpler method for the overlap array specifically consists of changing a pair of bar lengths, while keeping their sum constant. For the compactability equation [8.4] to be fulfilled, the new bar lengths and need to comply with: [8.17] Since both the original bars and, as the new bars and share the same end points and, this relationship can be visualized by plotting an ellipse with its center point on and its two foci on said joint nodes and. Fig 8.27 shows some of the different bar pairs that comply with [8.17], including the ones giving an inverted configuration. The length of the semi-major axis of the ellipse then equals ( ) and the lengths of the semi-minor axis ( ). Making the origin of the coordinate system coincide with the center of the ellipse then gives its general equation: ( ) + ( ) = [8.18] To find the values of and corresponding to any chosen polar angle, a point at horizontal distance from is introduced. Its coordinates as in Fig 8.28 are ( ). 129

141 8. Uneven Sarrus Chains The rotation of this point around with a certain angle will then place it on the ellipse. Substituting the x- and y- values of this rotated point into the ellipse equation [8.18] gives: ( ( )) ( ) + ( ( ) ) ( ) = [8.19] for which: solving for gives: = ( ) ( ( ) ( )) [8.20] and from [8.17] and [8.20]: = ( ) ( ( ) ( )) [8.21] Fig 8.29 Maximum polar angle for ellipse method The maximum polar angle that can be achieved for any chosen bar values and is reached when the new bars can unfold to become collinear. The values for, and are: ( ) [8.22] [8.23] Overlapping rectangular modules which have a polar angle in of their directions, cylindrical geometries can be generated. A physical test model as part of a singly curved overlap array like this is shown in Fig Doubly curved configurations which fold flat such as the lamella dome in Fig 8.31 can also be made. Their geometric description is outside the scope of this dissertation. 130

142 8. Uneven Sarrus Chains Fig 8.31 Singly-curved overlap array Fig 8.31 Lamella dome overlap array Overlap factor The modules analyzed above are then chained together by overlapping them by a certain amount, the overlap factor, and connecting the bars and bars with each other at the center of the overlap. The overlap factor is taken at zero when there is no overlap and at 1 if the two modules completely overlap. In a larger flat array, the maximum overlap factor for practical materialization is 0.5, since larger factors would make for a double overlap between 3 adjacent modules. In a rectangular grid array, there are two independent overlap factors that can be chosen. Fig 8.32 Overlap factor 131

143 8. Uneven Sarrus Chains 8.4 Parametric tool for regular array design Using the equations established in this chapter for making flat and polar arrays, a software tool was developed for parametrically designing scalable arrays of both joint-to-joint and overlap type. The tool was implemented in the Grasshopper plug-in for Rhinoceros 3D, and allows for parametrizing regular arrays in function of the parameters shown in the screen capture in Fig Fig 8.33 Parameters Fig 8.34 Array-specific parameters and output A simple Boolean toggle further allows the choice between said two array types, which each of them having their output parameters numerically plotted as well, allowing comparison between the maximum polar angle that can be used, bar lengths for a certain polar angle, and the bounding dimensions of the arrays (Fig 8.34). Since the paramatrized arrays are regular, the polar arrays result to cylindrical geometries, being suited for generating the geometries of barrel vaults, such as the one analyzed in later mentioned case study 2. The headroom output parameter in Fig 8.34 thus refers to the maximum interior height of the barrel vaults. Variations of regular arrays can be made by generating different but compatible regular arrays and joining them together, for example a repeated polar array and flat arrays of the same basic module. This technique was used for joining two flat arrays to a central polar array in later mentioned case study

144 8. Uneven Sarrus Chains A simplified step diagram of the tool is shown in Fig In reality the main parameters refer to almost all steps, and the geometries generated at intermediate steps also create output material. Fig 8.35 Step diagram of parametric tool The first step after entering the parameters is generating the Sarrus modules, which happens separately for each of the array types, since the methods for generating polar modules are different between them. Mostly, intersecting circles are used here in combination with the equations for polar modules ([8.4] to [8.14] and [8.17] to [8.23]). This ensures that bar lengths stay constant during deployment. The geometry of the basic modules is halved and copied around to form the smallest chain that is later copied into larger arrays. This intermediate step is mostly use to make certain that the borders of the arrays stop on continuous lines of bars, not having the edge and corner bar pairs sticking out. The overlap factors, parametrized separately for the flat and polar direction in the array, is inserted in this step as well. The smallest chains are then used to generate flat arrays, which in turn are copied into polar arrays. Attention has been given here to avoiding doubled geometries of the copied elements. Finally, the resulting arrays are translated and rotated to stand upon the xy plane. Numerical output from the mostly geometrically generated model is then given in data panels. To return this numerical data for both of the array types in order to compare the two, the input of the Boolean toggle that chooses between them is actually inserted at the end of the process, hiding one of both of the arrays in the graphical representation. 133

145 8. Uneven Sarrus Chains Changing the operating angle, the parametrized mechanism opens and closes in real time. The other parameters can be changed independently; e.g. for the joint-to-joint array in Fig 8.36 the changing bar proportion is shown, altering the structural height and the expandability. Flat arrays can be made by choosing a zero polar angle, e.g. for the overlap array in Fig Fig 8.36 Parametric tool applied to create singly-curved joint-to-joint array Fig 8.37 Parametric tool applied to create flat overlap array 134

146 8. Uneven Sarrus Chains 8.5 Secondary structural systems When a joint-to-joint or overlap array is locked in its fully deployed state, it can be said that the bars form a compressive layer, while the bars function as diagonals. In most cases an additional (tensile) layer may then be necessary to give structural height to the array. Firstly the relationship between the structural height and the bar proportions is described and, secondly some secondary structural systems are discussed here. The relative structural height is defined as the vertical distance between the and nodes in deployed phase, divided by the bar length: = = [8.24] Plotting to naturally gives the first quadrant of a circle (Fig 8.38) The graph can simply be read as the trade-off between the relative structural height and the compactability, since l k is directly proportional to how far the Sarrus module opens in the perpendicular direction. Trade-off: Structural height (h r ) vs Compactability ( ) 1 0,8 h r 0,6 0,4 0, ,2 0,4 0,6 0,8 1 Fig 8.38 Trade-off between relative structural height and compactability This trivial relationship makes it easy for designers to decide the proportion, which mostly won t go lower than the mark, since lower values give increasingly smaller returns to. To visualize this point, the optimization of the sum of relative structural height and the bar proportion is shown in Fig values between 0.6 and 0.8 give the best trade-off. 135

147 8. Uneven Sarrus Chains h r Optimization of (h r ) with as main parameter 1,5 1,4 1,3 1,2 1, ,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Fig 8.39 Trade-off with compactability as primary goal A first method for adding a secondary structural system to gain structural height is doubling the bars and connecting this new layer to the bottom joints. Fig 8.40a shows a semi-deployed jointto-joint array where the top layer of bars is to be added to the lower joint nodes. Fig 8.40b shows the same array where the lower layer of bars has been joined, here colored orange for visibility. Fig 8.40a Adding of lower bar layer Fig 8.40b Joint-to-joint array with lower bar layer in semi-deployed state A way of looking at the new mechanism created by adding the bottom bar layer is as two arrays of Sarrus modules interlocking upside-down and sharing their bars (diagonals). In Fig 8.41 the same array as in Fig 8.40 can be seen in its fully deployed state. 136

148 8. Uneven Sarrus Chains Fig 8.41 Joint-to-joint array with lower bar layer in fully deployed state Fig 8.42 shows how extended joints (here in blue) can be added to form vertical connecting struts when the mechanism closes, effectively forming a square-on-square truss grid in this case. They can also be introduced locally as constraining elements to lock the structure in deployed state. Fig 8.42 Joint-to-joint array with lower bar layer and extended bottom joints to form square-on-square grid truss A downside of this secondary bar system is of course a less compact bundle when the mechanism is closed. A design decision can then be to implement this secondary bar layer with the inverted configuration (such as in Fig 8.4) to save space. When adding a secondary bar layer to a polar array, the curvature will naturally make the bars in the inner curve smaller, but the same strategy for adding them can be applied. The advantage of the secondary bar layer method is that changing moments are accounted for, since both the top and bottom layer can function in compression. When potentially changing moments aren t a concern, the introduction of a tensile layer is the lighter and more compact solution. The tensile layer can exist of either cables or textile, or a combination of both. 137

149 8. Uneven Sarrus Chains Cables can be added either parallel to the upper layer of bars, or diagonally as to give structural stability to the xy-plane. This second method is shown in Fig 8.43 for a joint-to-joint array and in Fig 8.44 for an overlap array, where X-braces are formed. Fig 8.43 Joint-to-joint array with triangulating cable substructure Fig 8.44 Overlap array with X-brace cable substructure 138

150 8. Uneven Sarrus Chains In Fig 8.45 a polar joint-to-joint array model can be seen where a fabric is introduced to take up tensile forces. This method naturally offers triangulation to a quadrilateral grid, which is an issue for joint-to-joint arrays. Fig 8.45 Singly-curves joint-to-joint array with textile substructure A last secondary structure system mentioned here are added plate elements. They can be placed in between and over the compression bar layer to add rigidity and lessen deformations to the rest of the structure, serving also as constraining elements. Since they have to be placed on the structure after the transformation, they add time to the deployment process. Although added plate elements give stability in the xy-plane and increase the overall rigidity, no structural height is reached with them, and a cable or bar layer can be combined with them to this purpose. In Fig 8.46 a flat joint-to-joint grid is given a tensile layer and plate elements. Fig 8.46 Joint-to-joint array with added plate elements 139

151 8. Uneven Sarrus Chains 8.6 Case studies In this section, some designs are used to show some of the real-world applications and possibilities of the analyzed Sarrus arrays. A short geometric description of the designs is given and they are structurally analyzed both to global and local stability and section strength, using SCIA Engineer 14 software. Eurocode standards are used wherever applicable, and the unity checks are done both for ULS (Ultimate Limit State) and SLS (Serviceability Limit State) Case study 1: Pedestrian bridge The first case study is the design for an easily deployable bride to span a small river of 10 meters wide. It should be possible to quickly recover and compact the bridge during any floods, such as in the monsoon season. A joint-to-joint array materialized in aluminum is chosen as the primary structure, stainless steel cables give it structural height, and plywood plates form the walkway. Fig 8.47 Deployable pedestrian bridge from joint-to-joint array (all dimensions in meters) 140

152 8. Uneven Sarrus Chains In order for local boats to pass underneath and to introduce less of a bending moment, the bridge should be arched. However, the maximum inclination at any point of the bridge is chosen to be, so that it is accessible when materialized either with flat non-skid plates or stepped plates. These conflicting interests are solved by linking together two inclined flat arrays and a central polar array into one mechanism (Fig 8.48). Fig 8.48 Joining of two flat arrays to central polar array As can be seen in Fig 8.49, a central bar is fixed to the lower joint, while the upper joint slides over it. This bar does not affect the mechanism, but serves both for receiving the handrail and for triangulation in its lower part. A welded ledge on this central bar should stop the top joint just before toggle position so that the mechanism folds closed more easily and does not enter the inverted configuration. Fig 8.49 Central bar with P-joint serves for triangulation and receiving the handrail 141

153 8. Uneven Sarrus Chains Fig 8.50 Deployment stages of pedestrian bridge 142

154 8. Uneven Sarrus Chains The deployment stages are shown in Fig The compact package is delivered by either crane or boat. The deployment can happen using the self-weight of the aluminum bars, and is controlled by active cables running along the top joints (marked orange in the central drawing). Later, these light cables can be used to compact the mechanism again. After opening, the last two rows of joints on either side are pinned to premade foundations, making the structure static. Lastly, the plywood plates are added from the sides inwards. The plates further rigidize the structure once it s in use, helping to remove any residual mobility in the system. This works well with the joint-to-joint arrays, where an additional mobility was found to appear with larger joint clearance. In this sense, the combination of joint-to-joint grids and plate elements can also be suitable for deployable stage designs. The two different kinds of modules used in the design are shown in Tab 8.1. The relative structural height does not take into account the small offset that is necessary for attaching the cables to the bottom joints. The values of the joint offset and the new bar length of the polar module are calculated using [8.8] and [8.9]. Flat module = 900 mm Polar module = 1150 mm = = 720 mm = = mm = 1023 mm Tab 8.1 Due to the eccentricity that is inherent in the use of a joint offset, a bending moment will be introduced in the bottom joints of the polar modules. Care is needed when designing these joints, so that they can offer a great enough resisting moment. In this case, a hollow joint section is needed to connect the central bars anyway, and its relatively large dimensions are adequate for resisting said bending moment. Tab 8.2 shows the material properties used for each of the element types, along with their largest and smallest dimensions occurring in the design. Short elements and a relatively high bar proportion, together with the use of a medium-high strength aluminum alloy, allow the use of small sections. 143

155 8. Uneven Sarrus Chains Bar elements Cables Plate elements Material Aluminum EN AW 6082 T5 Stainless steel wire EN Elastic modulus [Gpa] Density [kg/m³] Dimensions [mm] 70 2,720 50x50x3 x 1150 x x60x4 x 200 x ,850 8 x 1700 x 1250 Plywood EN x 1800x900 Tab 8.2 Tab 8.3 shows the different loads used in the calculation with SCIA Engineer software. The primary structure excluding the plate elements has a total weight of kg, which makes it easy for transportation, and quite lightweight at 13.8 kg/m². The stainless steel cables make up 18.4 kg of this total weight, and they could be replaced with ropes if further reduction of weight were required. Type Load Self-weight W Permanent Primary structure: kg total Plywood plates: 20 kg/piece Distributed load k Variable 4kN/m² Concentrated load k Variable Local Wind load in x Variable direction (transverse) Wind load in y Variable direction (longitudinal) 4kN 1.02kN/m² 0.501kN/m² Tab 8.3 The variable load of 4kN/m² is an overestimate, since the bridge would be used only for pedestrians and light vehicles such as bikes and scooters. Snow load is omitted since the bridge here is proposed in equatorial climates where snow tends not to occur. Wind loads are calculated referring to EN , particularly the section about bridge design. The most important values and coefficients used for wind load calculation are shown in Tab

156 8. Uneven Sarrus Chains Basic wind speed 25 m/s Roughness factor Mean wind speed 19 m/s Turbulence intensity 0.25 Structure reference height 2.7m Exposure factor 1.59 Air density 1.25 kg/m³ Force coefficient Terrain factor (II) 0.19 Force coefficient 0.9 Tab 8.4 The model done in SCIA Engineer describes the physical joints as short (120 and 200mm) hollow section 60x60x5 profiles, receiving 4 or 8 bars orthogonally with one rotational freedom. The joints receive the bars head-on, i.e. not on the sides in cantilever, in order to minimize axis eccentricities. This goes at the cost of compactness, as shown in Fig 7.8. Fig 8.51 Hinged supports Hinged supports are modeled under both the bottom and side joints of the last row of flat modules, as they would be fixed to a base plate after deployment (shown in Fig 8.50). Four complete ULS load combinations exist, since the longitudinal and transverse wind loads are mutually exclusive. EN 1990 gives the coefficients for the governing load combinations, which simplify to: ULS1: [8.25] ULS2: ULS3: ULS4: in which: : self-weight : distributed load : concentrated load : transverse wind load : longitudinal wind load 145

157 8. Uneven Sarrus Chains max [kn] ULS1 max [kn] 7.38 ULS1 max [kn] ULS1 Tab 8.5 l bars k bars central bars cables max unity check ULS1 ULS1 ULS1 ULS2 max [kn] ULS2 ULS1 ULS2 ULS2 max [knm] ULS1 ULS1 ULS2 max [knm] ULS1 ULS1 ULS2 Tab 8.6 The deciding bar section has a maximum unity check of 0.674, which means that the structure could be further optimized. However, iterating over the next-in-range sections available industrially gives a failed unity check (>1), hence it s decided to use the 50x50x3 mm profile section. The deformations are checked for the SLS, the governing load combinations and their coefficients as given by EN 1990 become: SLS1: [8.26] SLS1: SLS3: SLS4: max [mm] 3.0 SLS1 max [mm] 3.3 SLS1 max [mm] -8.9 SLS1 Tab

158 8. Uneven Sarrus Chains The deformations being lower than, the structure is more than rigid enough for its intended use. The rigidity is owed partly to the use of the plate elements and the network of cables, them being necessary additions to the basic joint-to-joint array. The largest local deformations are to be found at about ¼ of the length and half the width of the structure, near the top of the deck, as shown in Fig This is where the flat Sarrus modules are not triangulated since there are no central bars used here to connect the top and bottom joints. Fig 8.52 Largest deformations occur at unsupported top joints (shown for ULS for clarity) Lesser rigidity around the top joint (connecting the bars) is inherent in joint-to-joint arrays, where the angle of top bars becomes more independent near fully deployed state (Fig 8.3). A possible solution would be to introduce an extended joint (as in Fig 8.4) that adds triangulation here as well. The design of this small bridge is meant as a first study of the possibility of using the joint-to-joint array consisting of both flat and polar modules. The rigidization by the plates that serve as a walkway and the central bars in the module that give triangulation are important design elements. Similar bridges of larger scale (and most likely, built in steel) could be based on the same design strategy. 147

159 8. Uneven Sarrus Chains Case study 2: Barrel vault The second case study analyzed here is a barrel vault made from a polar overlap array. A fabric is used as a secondary tensile system on the inside of the polar array, but is simplified in the SCIA Engineer model by applying cables in the polar direction and X-braced cables in the edge bays. Fig 8.53 Deployable barrel vault from overlap array (all dimensions in meters) 148

160 8. Uneven Sarrus Chains The analysis done here is a direct reference to the work done by Alegria Mira L. (2010), who structurally analyzed and optimized the Universal Scissor Unit she designed in the application of a barrel vault. Since a comparative structural study is interesting, Alegria s methodology was largely followed for this case study. The main geometric characteristics were reproduced: the amount of modules in the array in both polar and flat directions is the same. Due to the repeated overlaps, the resulting vault is smaller. The folded bundle of the overlap array is relatively compact because only one-dimensional bar elements are used. As this design focuses more on compactness and transportability, instead of using steel (S235), the lighter aluminum (EN AW 6082 T5) is chosen for the bar elements. Fig 8.54 Cantilever joints allowing high compactness The use of square-sectioned bars was decided upon early, since these offer the most compact configuration. In combination with a straddle-mounted joint, they form a very space-efficient system of which the top view can be seen in Fig For this case, said method gives a compact bundle of 4.9m³ that expands to 142 times its volume during deployment. The joints are modeled in SCIA Engineer as placeholders with the same offset as in reality (Fig 8.55) and they are not further analyzed in this case study. The circles in Fig 8.55 are SCIA Engineer s notation of rotational degrees of freedom, where the bars meet the joint. Fig 8.55 Modeled joint thickness 149

161 8. Uneven Sarrus Chains Fig 8.56 Deployment stages of barrel vault 150

162 8. Uneven Sarrus Chains From the parametric Grasshopper file the deployment of this barrel vault was analyzed, as shown in Fig 8.56 for an opening angle of up to complete deployment. Active cables as in Fig 7.22 would be used to pull the top and bottom joint of each of the modules together to prevent force concentrations. No active cables were included in the calculational model: their effects are beneficial to the rigidity of the locked structure, but were chosen to be ignored here. The structure is primarily locked by placing constraining bar elements at the edges of the barrel vault. These edge rows are also where the X-brace cables are located in the model, and hence where SCIA engineer predicts the largest risk of member buckling. Thus the additional constraining elements prevent overdimensioning these edge bars. After the constraining elements are in place, the structure is locked further by pinned or weighted supports along all of its bottom joints, which ensures a more even force distribution. Only one polar module (Tab 8.8) was used in the design, simplifying the production process. Calculating the polar bar lengths and was done in function of the desired polar angle, using equations [8.20] and [8.21] for overlap arrays. The overlap factor is taken constant for both the flat and polar directions of the vault. Polar module = 828 mm = = 1150 mm = = 862 mm = 1116 mm = 798 mm = 0.3 Tab 8.8 Tab 8.9 shows the material properties used for each of the element types, along with their largest and smallest dimensions occurring in the barrel vault. As noted, the membrane is simplified in the SCIA Engineer model to a set of steel cables in the polar direction. The elastic properties of the cables are used, but the weight of the fabric is used in calculating the total self-weight. 151

163 8. Uneven Sarrus Chains Material Elastic modulus Density Dimensions [Gpa] [mm] Bar Aluminum 70 2,720 60x60x4 x 1150 elements EN AW 6082 T5 kg/m³ x 828 Fabric PVC coated Polyester Cables Stainless steel wire EN kg/m² 190 7,850 kg/m³ (between supports) 8 x 1050 x 1159 Tab 8.9 In Tab 8.10 the different loads used in the calculational model are shown. The total structure, including the joints and the membrane, has a weight of kg, making it quite lightweight at 23.5 kg/m². Combined with the compactness the total package fits with the 2 x 2 x 1.3 m container the deployable barrel vault is suited for transportation and quick deployment. Type Load Self-weight W Permanent Primary structure: kg total Membrane: kg total Wind load in x Variable max kn/m² direction (transverse) Wind load in y Variable max kn/m² direction (longitudinal) Snow load s Variable 1 kn/m² Tab 8.10 Wind loads are here again calculated referring to EN The coefficients used can be seen in Tab For the calculation of the peak velocity pressure, the seasonal factor is taken into account to lower the statistically determined percentile rank, seeing that one use-cycle of the structure would likely be less than three months. Basic wind speed 25 m/s Roughness factor Mean wind speed 19 m/s Turbulence intensity Structure reference height 3.08 m Exposure factor Air density 1.25 kg/m³ Probability factor 0.85 Terrain factor (II) 0.19 Peak velocity pressure kn/m² Tab

164 8. Uneven Sarrus Chains EN prescribes the different zones that need to be defined for obtaining the internal and external pressure coefficients in the structure. The zonification of the cylindrical structures analyzed by Alegria Mira L. (2010) and De Temmerman N. (2007) serve as references here. Fig 8.57 shows the zonification and Tab 8.12 Tab 8.13 show the resulting wind loads in each of the zones. Fig 8.57 Zonification and wind load diagrams (adapted from Alegria Mira L. 2010) External pressure External pressure Internal pressure Internal pressure Total pressure w [kn/m²] coefficient [kn/m²] coefficient [kn/m²] A B C Tab 8.12 External pressure External pressure Internal pressure Internal pressure Total pressure w [kn/m²] coefficient [kn/m²] coefficient [kn/m²] D E F Tab

165 8. Uneven Sarrus Chains The snow load is shortly calculated from EN , taking zero load for the zones of the barrel vault that have an inclination greater than. The coefficients applied here are shown in Tab Form factor 2 Exposure coefficient 1 Temperature coefficient 1 Characteristic snow load on ground 0.5 kn/m² Tab 8.13 All loads except for the self-weight are introduced at the lower joints (through SCIA Engineer s load panels that redirect the forces to its vertices). Realistically, the supported membrane would indeed be suspended from the lower joints of the modules. Four complete ULS load combinations exist, since the longitudinal and transverse wind loads are mutually exclusive. EN 1990 gives the coefficients for the governing load combinations, which simplify to: ULS1: [8.27] ULS2: ULS3: ULS4: 1 1 s in which: : self-weight : transverse wind load : longitudinal wind load : snow load max [kn] ULS4 max [kn] 0.81 ULS3 max [kn] 4.41 ULS4 Tab 8.14 l bars (polar) k bars (polar) l bars (flat) k bars (flat) Cables max unity check ULS4 ULS4 ULS4 ULS4 ULS4 max [kn] ULS4 ULS4 ULS4 ULS4 ULS4 max [knm] ULS4 ULS4 ULS4 ULS4 max [knm] ULS4 ULS4 ULS4 ULS4 Tab 8.15 Results of SCIA Engineer s calculational model are shown in Tab The unity check for said loads is barely complied with for the lowest bars at the edge bays, due to the accumulated forces 154

166 8. Uneven Sarrus Chains almost exceeding their buckling resistance. However, the unity checks for the other bars on the lowest lines also result close to 1, meaning the structure is nearly optimized. The deformations are checked for the SLS, the governing load combinations and their coefficients as given by EN 1990 become: SLS1: [8.28] SLS2: SLS3: SLS4: s max [mm] 6.3 SLS4 max [mm] 0.6 SLS4 max [mm] SLS4 Tab 8.16 The vertical deformation occurring at the center of the barrel vault is relatively small, at less than 1/700 of the projected span. To further investigate how precise the SCIA model can describe the deformations (and further, rigidity) of the overlap array, smaller physical sample models should be analyzed under load and their deformations compared to the ones in the calculational tool. Effects of higher and lower joint clearance could then be checked and the eccentricity of the bars to each other and to the (straddle mounted) joints researched. The basic rigidity the overlap gives to the structure ensures that a changing moment, such as with upwards wind load during a storm, the structure can still offer a resisting moment without deforming excessively. To check this in the SCIA model, the transverse wind giving the biggest deformation is applied as a sole load on the model, ignoring the stabilizing self-weight and snow load. The results of the joint displacements are shown in Tab 8.17, and are well in acceptable range. max [mm] 6.4 transverse wind load max [mm] 0.4 transverse wind load max [mm] +6.3 transverse wind load Tab 8.17 Fig 8.58 Deformation by transverse wind 155

167 8. Uneven Sarrus Chains Fig 8.59a Large maximum bending moment at central hinge of scissor arrays Fig 8.59b Spread of bending moment over shorter elements in overlap arrays One of the structural shortcomings of regular scissor arrays is that the central connection introduces a bending moment in the connected bars at the location where the section is at its weakest (shown diagrammatically in Fig 8.59a). With the overlap array method, the bars are doubled in the central zone where more structural height and material is needed, lowering the maximum moment (Fig 8.59b). This way, the overlap method can ensure increased rigidity and strength both during and after deployment. The bars in an overlap array are shorter, making them more resistant to buckling. Furthermore, in an overlap array, the relative structural height in completely deployed state is independent of the polar angle, allowing this important factor for structural behavior to be determined freely; while in a scissor array the structural height and polar angle in deployed state are dependent, sometimes prompting the array to not fully deploy in order to gain structural height. The downside of using an overlap array over a regular scissor array is a lowered compactness, directly proportional to the overlap factor. Good detailing can guarantee that the top ( ) bars and the bottom ( ) bars lie in the same plane, preventing the compact bundle of becoming more bulky. Fig 8.60 Hinge detail example for compact overlap arrays 156

168 8. Uneven Sarrus Chains 8.7 Conclusions In this chapter the variations of the Sarrus mechanism and ways of chaining them together were researched. By primarily focusing on the basic mechanism as a module, important characteristics were derived. These include the relationship between the bar proportion, the different opening angles and the structural height. Three different array methods were discerned: the well-known scissor array, the joint-to-joint array and the overlap array. The latter two were investigated on further in this work. The joint-to-joint type has been used in the work of Wohlhart K. and Calatrava Valls S. (1981), but a more thorough analysis of the geometric possibilities and structural use of the arrays had been lacking. The overlap array, in turn, is novel and shows promising structural advantages. A novel way of introducing polar angles in a Sarrus module has been discovered. It uses a vertical joint offset and can be applied both to the joint-to-joint array and overlap arrays. A second method that uses changed bar lengths and the ellipse locus for introducing a polar angle in overlap arrays was analyzed. Both methods started from the assumption that the compactability constraint needs to be complied with, i.e. stating that the joined elements should be able to fold flat into a linear state as to offer compact deployables. The numerical relationships between the elements lengths and polar angle of a Sarrus module have been derived and, using the resulting equations, a parametric tool was developed in Grasshopper software. This tool can be used to generate both flat and polar arrays. An additional dependent mobility was noticed in physical models of joint-to-joint arrays and first attempts at describing this mobility have been made. Furthermore, various solutions for removing the dependent mobility in order to have a more controlled deployment have been offered. Information was given about the trade-off between the structural and kinematic aspects, and secondary structural systems intended for giving structural height to the arrays were addressed. The introduced double-bar system could be used to form deployable square-on-square truss grids. 157

169 8. Uneven Sarrus Chains Further work Some geometric possibilities of the joint-to-joint and overlap array were already given, but many more remain undiscussed. In particular the doubly-curved variations of the overlap array should be researched. Predictably, many of the doubly-curved geometries of the simple scissor arrays will be translatable directly to the overlap array. Variable overlap factors might also allow for new compactable geometries to be used. A more precise description of the additional mobility in joint-to-joint arrays is needed. Physical testing should reveal the influence of the joint clearance on this mobility. Test models should be made to test the effectiveness of the proposed methods for inhibiting it. Physical test models should also be made for both array types to check whether the structural responses coincide with those that calculational software describes, specifically with regards to the overall rigidity. A comparative study between overlap arrays and regular scissor arrays would be interesting, using compactness and structural strength and rigidity as main parameters, and plotting the exchange between them. Further study of the materialization of the joints is needed to make the real-world use of the structures possible. Tests should be done on the influence of the joints eccentricities, including the vertical joint offset, on the structural response. 158

170 References References Abdul-Sater K., Irlinger F., Lueth T. C. (2014): Four-Position Synthesis of Origami Evolved, Spherically Constrained Planar RR Chains Mechanisms and Machine Science, Vol Alegria Mira L. (2010): Design and Analysis of a Universal Scissor Component for Mobile architectural Applications Master s thesis, Free University of Brussels, Faculty of Engineering Álvarez Elipe J. C., Díaz Lantada A. (2012): Comparative studiesof auxetic geometries by means of computer-aided design and engineering Smart Materials and Structures 21.10: Álvarez Elipe M. D., Anaya Díaz J. (2013): Development of transformable structures and geometries New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P. Escuela Técnica Superior de Arquitectura, Sevilla Baker J. E. (1979): The Bennett, Goldberg and Myard linkages in perspective Mechanism and Machine Theory, Vol Barker R. J. P., Guest S. D. (2000): Inflatable triangulated cylinders IUTAM-IASS Symposium on Deployable Structures: Theory and Applications, Cambridge, UK Beatini V., Korkmaz K. (2013): Shapes of Miura mesh mechanism with mobility one International Journal of Space Structures, Vol. 28, No Belcastro, S.-M., Hull, T. C. (2002): A mathematical model for non-flat origami Origami^{3} Bennett G. T. (1903): A new mechanism Engineering, Vol Bennett G. T. (1914): The skew isogram mechanism Proceeding of London Mathematics Society 2, Vol Bern M., Hayes B. (1996): The complexity of flat origami Annual ACM-SIAM Symposium on Discrete Algorithms, Atlanta, Georgia

171 References Bricard R. (1927): Leçons de cinématique Tome II Cinématique Appliquée, Gauthier-Villars, Paris 7-12 Buffart H. C., Trautz M. (2013): Construction approach for deployable folded plate structures without transversal joint displacements New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P., Escuela Técnica Superior de Arquitectura, Sevilla Buhl T., Jensen F., Pellegrino S. (2004): Shape optimization of cover plate for retractable roof structures Computers & Structures, Vol. 82, No Calatrava Valls S. (1981): Zur Faltbarkeit von Fachwerken Doctoral thesis, Eidgenoessischen Technischen Hochschule Zürich Chen Y. (2003): Design of Structural Mechanisms Doctoral thesis, St Hugh s college, University of Oxford Chen Y., You Z., Tarnai T. (2005): Threefold-symmetric Bricard linkages for deployable structures International Journal of Solids and Structures, Vol. 42, No Cruz P. J. S. (2013): Structures and Architecture: New concepts, applications and challenges CRC Press, Florida, US De Marco Werner C. (2013): Transformable and transportable architecture Master s thesis, Escuela Técnica Superior de Arquitectura de Barcelona, Universidad Politécnica de Cataluña De Temmerman N. (2007): Design and analysis of deployable bar structures for mobile architectural applications Doctoral thesis, Free University of Brussels, Faculty of Engineering EN 1990: Basis of structural design, European Standards Eurocode EN 1991: Actions on structures, European Standards Eurocode EN 1993: Design of steel structures, European Standards Eurocode EN 1999: Design of aluminium structures, European Standards Eurocode 160

172 References Escrig Palláres F. (1985): Expandable space structures Space Structures Journal, Vol. 1, No Escrig Palláres F. (1986): Introducción a la geometría de las estructuras espaciales desplegables de barras Boletín Académico Escuela Técnica Superior de Arquitectura, Coruña, Vol Escrig Palláres F. (1991a) Las estructuras de Emilio Peréz Piñero Arquitectura Transformable, Escuela Técnica Superior de Arquitectura, Sevilla 9-32 Escrig Palláres F. (1991b) Geometría de las estructuras desplegables de aspas Arquitectura Transformable, Escuela Técnica Superior de Arquitectura, Sevilla Escrig Palláres F., Valcárcel V. P. (1988): Expandable curved space bar structures IETCC, Escuela Técnica Superior de Arquitectura, Sevilla Escrig Palláres F., Sánchez J. Valcárcel V. P. (1996): Two way deployable spherical grids International Journal of Space Structures, Vol. 11, No Escrig Palláres F., Valcarcel V. P. (2012): Modular, ligero, transformable: un paseo por la arquitectura ligera móvil Escuela Técnica Superior de Arquitectura, Sevilla Fuller R. B., Applewhite E.J. (1975): Synergetics, Explorations in the geometry of thinking Macmillan Publishing Co. Inc. Gan W. W., Pellegrino S. (2003): Closed loop deployable structures Structural Dynamics, and Materials Conference, Norfolk, Virginia American Institute of Aeronautics and Astronautics Gantes C.J. (1997): An improved analytical model for the prediction of the nonlinear behavior of flat and curved deployable space frames Journal of Constructional Steel Research, Vol Gantes C. J., Konitopoulou E. (2004) Geometric design of arbitrarily curved bi-stable deployable arches with discrete joint size International Journal of Solids and Structures, Vol. 41, No

173 References Gómez Lizcano D. E. (2013): An expandable pendentive dome and polyhedral mechanisms using angulated scissors with sliding elements New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P. Escuela Técnica Superior de Arquitectura, Sevilla Grübler M. F. (1917): Getriebelehre Springer Verlag, Berlin Guest S. D., Pellegrino S. (1994): The Folding of Triangulated Cylinders, Part I: Geometric Considerations Journal of Applied Mechanics, Vol. 61, No Hanaor A., Levy, R. (2001): Evaluations of deployable structures for space enclosures International Journal of Space Structures, Vol.16, No Hartenberg R. S., Denavit J. (1959): Cognate links Machine Design, 16/04/1959 Hernandez C. H. (2013): Experiences in deployable structures New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P. Escuela Técnica Superior de Arquitectura, Sevilla Hoberman C. (1988): Four sided central zone and pair of triangular flap hinges US patent A Hoberman C. (1990a): Reversibly expandable doubly-curved truss structure US Patent, number Hoberman C. (1990b): Reversibly expandable structures US patent Hoberman C. (2001): Connections to make foldable structures EU Patent, number A1 Hoberman C. : portfolio Accessed on 11/12/2013 Huang H., Deng Q., Li B. (2012): Mobile assemblies of large deployable structures Journal of Space Structures, Vol. 5, No

174 References Huang Z., Li Q., Ding H. (2013): Theory of parallel mechanisms Springer, Mechanisms and Machine Science series, Vol. 6 Huffman D. (1976): Curvature and creases: a primer on paper IEEE Transactions on Computers, Vol. 25, No Hull, T. C. (2006): Project Origami A. K. Peters Press Jensen F.V. (2004): Concepts for retractable roof structures Doctoral thesis, University of Cambridge, Department of Engineering Kassabian P.E. (1997): Investigation into a type of deployable roof structure Final year project, University of Cambridge, Department of Engineering Kiper G. (2011): Design methods for planar and spatial deployable structures Doctoral thesis, Middle East Technical University Krausse J., Lichtenstein C. (1999): Your Private Sky R.Buckminster Fuller Lars Müller Publishers Li Y., Xu Q. (2007): Kinematic analysis of a 3-PRS parallel manipulator Robotics and Computer-Integrated Manufacturing, Vol. 23(4) Langbecker T. (1991): Kinematic analysis of deployable scissor structures International Journal of Space Structures, Vol Langbecker T. (2000): Kinematic and non-linear analysis of foldable barrel vaults Engineering Structures, Vol Maden F., Teuffel P. (2013): Development of transformable anticlastic structures for temporary architecture New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P. Escuela Técnica Superior de Arquitectura, Sevilla Melin N. O. (2004): Application of Bennett mechanisms to long-span shelters Doctoral thesis, Magdalen College, University of Oxford 163

175 References Michaelis A. R. (1991): Heureka the Swiss national research exhibition 1991, Zürich Interdisciplinary Science Reviews, Vol. 16, No Miura K. (1972): Zeta-core sandwich its concept and realization ISAS Report, 37(6), University of Tokyo, No Miura K. (1980): Method of packaging and deployment of large membranes in space 31st IAF Congress, No. IAF-80-A31 Miura K., Natori M. (1980): 2-D array experiment on board a space flyer unit Space Solar Power Review, Vol Myard E. F. (1931): Contribution à la géométrie des systèmes articulés Societé mathématique de France, Vol Norton R. L. (1991): Design of Machinery McGraw Hill, University of Worcester Piker D. (2009a): Origami Electromagnetism Accessed on 24/01/2014 Piker D. (2009b): Networked Bennett linkages in Grasshopper Accessed on 24/01/2014 Qi X., Deng Z., Li B., Liu R., Gua H. (2013): Design and optimization of large deployable mechanism constructed by Myard linkages CAES Space Journal, Vol. 5, No Raskin, I. (1998): Stiffness and stability of deployable pantographic columns Doctoral thesis, University of Waterloo, Civil Engineering Department Resch R. D. (1968): Self-supporting structural unit having a series of repetitious geometrical modules US Patent A 164

176 References Resch R. D., Armstrong E. (1971): Paper and Stick Film Reuleaux F. (1963): The Kinematics of Machinery Dover Publications, New York Translation by A. B. W. Kenedy Roovers K., Alegria Mira L., De Temmerman N. (2013): From surface to scissor structure New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P. Escuela Técnica Superior de Arquitectura, Sevilla Röschel O. (2012): A Fulleroid - like mechanism based on the cube Journal for Geometry and Graphics, Vol. 16, No Sarrus P. T. (1853): Note sur la transformation des mouvements rectilignes alternatifs, en mouvements circulaires et réciproquement Académie des Sciences, Schenk M. (2012): Origami in engineering and architecture Presentation, University of Cambridge, Advanced Structures Group Schenk M., Guest S. D. (2010): Origami folding: a structural engineering approach Origami 5: Fifth International Meeting of Origami Science, Mathematics, and Education Schwabe C., Ishiguro A. (2006): Geometric Art Kousakusha publishers, Tokyo Schwabe C. (2010): Eureka and Serendipity: The Rudolf von Laban Icosahedron and Buckminster Fuller s Jitterbug Kurashiki University of Science and the Arts, Okayama Shim J., Perdigou C., Chen E. R., Bertoldi K., Reis P. M. (2012) : Buckling-induced encapsulation of structured elastic shells under pressure Proceedings of the National Academic of Science of the United States of America, Solomon J., Vouga E., Wardetzky M., Grinspun E. (2012): Flexible developable surfaces Eurographics Symposium on Geometry Processing 2012, Vol. 31, No Tachi T. : Freeform Origami Accessed on 25/01/

177 References Tachi T. (2009): Generalization of rigid foldable quadrilateral mesh origami Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures, Universidad Politécnica de Valencia Tachi T. (2011): Rigid-Foldable Thick Origami Fifth international Meeting of Origami Science, Mathematics, and Education Tachi T. (2013): Freeform Origami Tessellations by Generalizing Resch s Patterns ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference Tonon O. L. (1993): Geometry of Spatial Folded Forms 4 th International Conference on Space Structures, University of Surrey, Guildford, UK Trautz M., Künstler A. (2009): Deployable folded plate structures - folding patterns based on 4-fold-mechanism Evolution and Trends in Design, Analysis and Construction of Shell and Spatial Structures, Universidad Politécnica de Valencia Verheyen, H. F. (1989): The complete set of Jitterbug transformers and the analysis of their motion The International Journal of Computers and Mathematics With Applications, Vol Wohlhart K. (1995): New overconstrained spheroidal linkages 9th world congress on the theory of the machines and mechanisms, Milan Wohlhart, K. (2001): Regular polyhedral linkages 2nd workshop on computational kinematics, Seoul Wohlhart K. (2007): Cupola Linkages 12 th World Congress in Mechanisms and Machine Science, Besançon, France, Vol Yokosuka Y. and Matsuzawa T. (2013): Shape-finding for variable geometry structures formed by a Multilink Spherical Joint New Proposals for Transformable Architecture, Engineering and Design in the Honor of E. P. P., Escuela Técnica Superior de Arquitectura, Sevilla

178 References You Z., Pellegrino S. (1997): Foldable bar structures International Journal of Solids and Structures, Vol. 34, No You Z., Chen Y. (2011): Motion structures : deployable structural assemblies of mechanisms Taylor and Francis, Hoboken, NY Zhao J., Feng Z., Chu F., Ma N. (2014): Advanced theory of constraint and motion analysis for robot mechanisms Elsevier Science & Technology Books, 2013 Zeigler H. (1976): Collapsible self-supporting structures US Patent No

179 References Websites Accessed on 02/01/ Accessed on 12/02/ Accessed on 20/02/ Accessed on 22/02/ Accessed on 16/12/ Accessed on 22/02/ Accessed on 17/11/ Accessed on 24/11/ Accessed on 12/02/ Accessed on 23/02/ Accessed on 15 /11/ Accessed on 10/05/ Accessed on 18/02/ Accessed on 02/01/ Accessed on 22/02/

180 References Accessed on 21/02/ Accessed on 12/02/ Accessed on 18/02/ Accessed on 23/02/ Accessed on 18/10/ Accessed on 24/03/ Accessed on 11/01/ Accessed on 15/02/015 Software programs used for modeling Dassault Systèmes (2014): Solidworks 2014 McNeel R. (2010): Grasshopper Generative Modeling with Rhino Robert McNeel and associates, Seattle, WA, USA McNeel R. (2014): Rhinoceros 3D, version 5.0 Robert McNeel and associates, Seattle, WA, USA Nemetschek (2014): SCIA Engineer

181 Appendix A. Formal Studies

182 Appendix I. Formal Studies 171

183 Appendix I. Formal Studies 172

184 Appendix I. Formal Studies 173

185 Appendix I. Formal Studies 174

186 Appendix I. Formal Studies 175

187 Appendix B. Transformable Designs

188 Appendix II. Transformable Designs Acoustical shell for street musicians Categorization Rigid-foldable origami, Yoshimura pattern A deployable, lightweight shell for performers in the public space. Made out of a curved Yoshimura pattern, the identical triangles not only give rigidity to the whole, but in the unfolded state also act as acustical diffusers for the mid- to-high tones, creating a more balenced sonic field. Materialization challenges The primary difficulty is, as in almost all of compactable origami designs, how the detailing allows the folding of the facets, which should take in account the cumulative thickness. A second main design challenge is how to lock the structure into static shape. Another aspect here is the reflective properties on the interior of the facets, which will determine the usability of the piece. 177

189 Appendix II. Transformable Designs Façade plug-in Categorization Rigid-foldable origami This foldable facade plug-in is the result of a research on what advantages a dynamic metal sheet-facade can bring. The used parameters are all linked to interior comfort. In fully closed state the plug-in offers of solar shading without affecting the interior view. Indirect reflection ensures an evenly spread out light inside. At the same time the enclosed air in between the interior and exterior glass panes makes for a greenhouse effect, which lowers the wind chill. In opened state the plug-in allows for ventilation and offers shading, easily adapting the angle to the height of the sun. Boundary conditions The structure moves with a single degree of freedom and thus can be made functional by one actuator, possibly connected to temperature and humidity sensors. Materialization challenges Detailing is done with thin sandwich aluminum panels. The folds are made possible by fitting an elastic material as neoprene between the ends of the panels. 178 Design and detailing of this work where done in collaboration with Michiel Van Der Elst and Charlotte De Vreese

190 Appendix II. Transformable Designs Mobile padel court Categorization SLEs A mobile court designed for the increasingly popular racket game padel. The main mechanism used is a simple SLE grid that has a fixed maximum opening angle to lift the court some 50 cm of the ground. This increases the visibility of the players and, more importantly, allows for the court to be placed on uneven terrain, using the adaptable footings. The side columns upon which the characteristic panels are hung are unfolded by a 4R planar lever system. Like this, the workers can easily hang the upper panels and hoist them up, without lifting overhead or needing heavy machinery. Boundary conditions The end bars of this conventional SLE system slide down the side columns when opening, until the end of the rail where the desired maximum aperture is reached. To fix the mechanism into place, stage panels are clicked into the joints of the SLE. This way, the panels are supported each meter to prevent sag. Materialization challenges All of the structural bars would be made from aluminum. The difficulty here is to optimize the weight-stability relationship, to make it easily transportable and at the same time limit any deformations caused by the dynamic load of the athletes. 179

191 Appendix II. Transformable Designs Scissor gates Categorization SLEs A design for the acces gates to a deconstructionalist museum building, the client wanted these 5m gates to be dematerialized into 2 segments. In the final design, the lower part of the gate is prolonged to close off a recessed part in the building. Boundary conditions The gate is made up out of 2 parts supported by a central axis, which is in turn supported by a (half-joint) wheel in a rail. To ensure that the mechanism moves only in the desired way (1DOF), the 2 parts are each fixed to the wall at their extremities by a vertical (half-joint) wheel and a horizontal (R) wheel. 180

192 Appendix II. Transformable Designs Solar panel array, primary axis rotation Categorization 6R Planar Linkages Design for a solar panel array in which there is a 1DOF rotation around the primary axis. The main design concept is to make the panels rotate and at the same time undergo a relative translation, which ensures a greater distance between them as the sun changes angle. This way, casting shadows on one another is avoided. For the same reason of avoiding shadows, any structural pieces of the mechanism are placed below the panels. Furthermore, this movement has to be abled in both directions, to allow the rotation to respond to all of the suns cycle. The whole could be driven by a single actuator, powered directly by the energy collected by the panels. Boundary conditions The system is made up of a row of 6R linkages that each actuate a column of panels. The outer edges of the panels themselves form part of the system as rigid links. The bottom edges of the panels are centrally supported by half-joints that slide along a rail. Materialization challenges The multitude of joints makes the system susceptible to more friction. A simpler way of supporting the system can be sought, in which the supporting half-joints is replaced by a simpler solution. 181

193 Appendix II. Transformable Designs Solar panel array, secondary axis rotation Categorization Planar Jitterbug-like mechanism, SLEs A 1DOF set-up of solar panels which can rotate around their secondary axes. Any shadows on the solar cells themselves are avoided by the translation taking place when the translating the panels further apart with a greater aperture, and by placing all the mechanism links underneath the surfaces. Boundary conditions The mechanism is made up of triangular links of which the panel border forms one edge. R-joints connects bars to the center of the diagonal edge of the triangular link. In a scissorlike array these elements are finally connected to a rail by half-joints at their lower verteces. Looking at the triangular link as half a rectangle and the connecting bar as the diagonal of another rectangle, the system can be described as a type of planar Jitterbug mechanism. Materialization challenges The geometry is pretty straightforward and easy to materialize. Rows of panels could be connected to each other, making the rail in which the bars slide only necessary at both ends 182

194 Appendix II. Transformable Designs Foldable bar unit Categorization Rigid-foldable origami An easily compactable and deployable bar unit for temporary events. The main geometry is the repetition of a Miura-ori fold, while the edges are a variation that unfold perpendicularly. The zigzag floor plan and the edges give a minimum of transversal stability to the piece. Boundary conditions The single degree of freedom makes for easy deployment. The particular geometry of the fold pattern doesn t allow the plates to fold any further than rad. In its fully deployed position constraining elements are added underneath the top plates in order to lock the whole into place and add strength locally. Materialization challenges To account for the thickness of the plates, offset elements have been introduced at the location of the hinges. Controlling these offset distances, the top plates can be made to fold over the vertical plates. This way, both a compact bundle and a deployed state with fine edges can be secured. 183