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... He also provided detailed descriptions of various existing tensegrity structures and developed pattern-based classification. In contrast, Kenner explored tensegrity structures using an analytical approach based on geometry, mathematics, and engineering mechanics (Kenner 1976). Ken ner treated these structures as spatial diagrams of forces whose equilibrium defined stable structural configurations. ...
... Calladine's approach was used by many other researchers who applied matrix-based methods for studying statics of tensegrity structures, which included discovering and form-finding new tensegrity configurations (e.g. Kenner 1976;Tarnai 1980;Hanaor 1988), and studying first-and higher-order infinitesimal and finite mechanisms, equilibrium configurations, self-stress states and resolvable and non-resolvable forces (e.g. Pellegrino and Calladine 1986;Pellegrino 1989;Vilnay 1990;Calladine andPellegrino 1991, 1992). ...
... An important task in understanding the static behaviour of tensegrity structures is to find their shape. Kenner, Tarnai and Hanaor developed ana lytical solutions for form-finding of tensegrity structures with simple sym metrical geometries (Kenner 1976;Tarnai 1980;Hanaor 1988). Sultana and colleagues generalized the form-finding problem using innate characteristics of the tensegrity structures and attempted to obtain closed-form solutions of the associated prestressability problems using symbolic mathematical mod elling (Sultan et al. 2001;Sultan and Skelton 2003b). ...
... Spherical tensegrity is a type of tensegrity structure where all vertexes are located on a sphere. It has been shown [19] that all vertexes of any spherical tensegrity are coincident with the vertexes of a polyhedron. For example, the vertexes of a 6-strut spherical tensegrity are the vertexes of a regular icosahedron. ...
... Since these two structures are different sizes, CSK was surrounded by a sphere with the same radius as the cytoplasm, but the NSK was surrounded by a smaller sphere with a radius equal to the diameter of the nucleus. Any tensegrity structure may be constructed using several hypothetical planes passing through the center of the sphere [19]. The struts are located in these planes. ...
... the diameter of the nucleus. Any tensegrity structure may be constructed using several hypothetical planes passing through the center of the sphere [19]. The struts are located in these planes. ...
A tensegrity model can be used to describe the mechanical behavior of living cells. A finite element model (FEM) was used to assess the mechanical contribution of subcellular organelles. Continuum parts like the cytoplasm and membrane were modeled as continuous elements, while the tensegrity was chosen to model the cytoskeleton and nucleoskeleton. An atomic force microscope load was implemented to simulate the external load. The cell components were loaded separately to evaluate their mechanical contributions. The analysis started with a single cytoplasm and each of the cell components was added in consecutive steps. The results showed that the cytoskeleton carried the largest part of the reaction force. The cytoplasm was the second important component of the cell’s mechanical response. It was shown that the nucleoskeleton has a stiffer structure than the membrane and cytoplasm. The cytoskeleton supported approximately 90% of the reaction force, while the cytoplasm carried 9% and the shell parts and nucleoskeleton were responsible for about 1%.
... Spherical subdivision based on the ortho-polyhedron is a process that the surfaces of an ortho-polyhedron are subdivided into a number of ortho-polygons and then their vertices are mapped onto the sphere as the positions of the sub-eyes. Typified by the ortho-icosahedron, this subdivision mode has absolutely strict symmetry and reduce the distribution unevenness of sub-eyes to 138% [13]. It is a more preferable subdivision method, but still falls short of the ideal. ...
... In 2011, Son et al. proposed an optimization to the traditional subdivision based on orthoicosahedron [13], effectively diminishing the distribution unevenness of sub-eyes to 118.8% [12]. It obtained the optimum solution by introducing deformation coefficients and constraints. ...
To achieve fast location, precise tracking and accurate identification over a large field of view (FOV), we have proposed a heterogeneous compound eye camera (HeCECam), which consists of a heterogeneous compound eye array, an optical relay system and a CMOS detector. However, the current HeCECam can hardly acquire high-precision 3D information of the targets to realize these applications. To solve this challenge, we propose a scheme on optimizing the structure of the HeCECam to improving the detection performance, including the optimization of the distribution uniformity of the sub-eyes with the proposed “Three-direction center-of-gravity subdivision (TGS)” and the enhancement of the compatibility between heterogeneous compound eyes and the optical relay system with the proposed compensation method for tilt. The TGS significantly reduces the distribution unevenness of sub-eyes down to 117% from the previous 152%, and provides symmetry to the heterogeneous compound eye array. The tilt compensation effectively addresses previous imaging defects, such as distortion of sub-images, increased stray light, and support structures being imaged, and it improves the imaging clarity of the system, especially in external FOV. Based on two proposed methods, we re-design and fabricate the heterogeneous compound eye array to obtain a high-performance prototype. To verify the imaging capacities of the optimized HeCECam, a series of comparison experiments are performed, including blank scene imaging, FOV tests, resolution verification and real-world scene imaging. The results show that the previous imaging defects have been well eliminated, and the optimized prototype has stronger resolving power and wider FOV. This allow the HeCECam to perform better in subsequent practical applications, such as wide-area surveillance, forewarning, and navigation.
... The development of tensegrity structures is relatively new and the works related have only existed for approximately twenty five years. Kenner, 1976, established the relation between the rotation of the top and bottom ties. Tobie, 1976, presented procedures for the generation of tensile structures by physical and graphical means. ...
... where the terms l i , m i , and n i have been defined in (5), (6), and (7). Equation (12) may be rearranged as 3 1 1 2 2 ...
... a longitud de los tensores que delimitan los polígonos superior e inferior (Kenner, 1976), se obtiene con la ecuación (4) (Kenner, 1976). Para comenzar con la elaboración del modelo y poder averiguar tanto la cantidad y módulos simplex a utilizar, se realiza mediante un tipo de figura geométrica para este ejemplo el triángulo, de igual manera teniendo un radio deseado y la altura; se aplican las cuatro (4) fórmulas matemá ticas de la ecuación anterior permitiendo obtener la longitud de los elementos de tensión y de comprensión que interactúan entre sí en del sistema de tensegridad. ...
... a longitud de los tensores que delimitan los polígonos superior e inferior (Kenner, 1976), se obtiene con la ecuación (4) (Kenner, 1976). Para comenzar con la elaboración del modelo y poder averiguar tanto la cantidad y módulos simplex a utilizar, se realiza mediante un tipo de figura geométrica para este ejemplo el triángulo, de igual manera teniendo un radio deseado y la altura; se aplican las cuatro (4) fórmulas matemá ticas de la ecuación anterior permitiendo obtener la longitud de los elementos de tensión y de comprensión que interactúan entre sí en del sistema de tensegridad. ...
El presente trabajo se desarrolló el diseño de cubiertas ligeras específicamente en el área de la tensegridad, ya que este tipo de estructuras pueden llegar a cubrir claros importantes, se estudiara su modelo de diseño para realizar un prototipo experimental, el cual podrá determinar qué tipo de uniones pueden manufacturar en este sistema estructural, este simulado por el software Robot Analysis Structural y Solid Work, por lo que esta disertación se centró en el estudio de una metodología del diseño para generar modelos constructivos estructurales, adecuados para realizar cubiertas de tensegridad, por lo cual se ejecutó una metodología para construir dichos sistemas y fuesen más sencillos de diseñar, con ellos generamos ejemplos de aplicación y uso de este tipo de sistemas estructurales, por lo que se presentan algunos casos académicos que se desarrollaron en los talleres de diseño de la carrera de Arquitectura y el tipo de cubiertas modeladas que pueden generar un uso practico.
... The specified PMSSM rotor diameter (D r = 152.4 mm) and stator/rotor air gap (g = 2.0 mm) enabled direct calculation of each major diameter p i [23]. The stem diameter d, stem length l and plate thickness t were optimized to provide adequate volumes for the copper windings, while simultaneously restricting the outer diameter of the stator (D s = 213.5 mm). ...
... The specified PMSSM rotor diameter (D = 152.4 mm) and stator/rotor air gap (g = 2.0 mm) enabled direct calculation of each major diameter [23]. The stem diameter d, stem length and plate thickness were optimized to provide adequate volumes for the copper windings, while simultaneously restricting the outer diameter of the stator ( = 213.5 mm). ...
The development of direct-drive spherical motors offers a potential solution to the limitations of conventional multiple degree-of-freedom (DOF) actuators, which typically utilize single-DOF joints (rotational and/or prismatic), arranged in series or parallel and powered by multiple single-DOF actuators. These configurations can be accompanied by kinematic singularities, backlash, limited power density and efficiency, and computationally expensive inverse kinematics. This paper details the design, fabrication and experimental testing of permanent magnet synchronous spherical motors (PMSSM) for multi-DOF servo-actuation. Its stator-pole arrangement is based on a Goldberg polyhedron, with each pole comprised of hexagonal or pentagonal inner and outer plates. The stator geometry and winding configurations are optimized using electromagnetic finite element analysis. A custom-made controller board includes a microcontroller, servo drivers, a wireless serial interface, and a USB PC interface. Angular orientation is sensed using an inertial measurement unit in wireless communication with the microcontroller. A PID controller is implemented and demonstrated for time-varying reference trajectories.
... Gambar 1. Regular octahedron (Sumber: Kenner, 2003) Gambar sebelah kiri pada gambar 1. menunjukkan bentuk regular octahedron, yang memiliki 8 bentuk segitiga. Untuk gambar sebelah kanan pada gambar 1 menunjukkan pembagian segitiga menjadi 4 bagian lebih kecil dengan menyambungkan titik tengah dari pinggir segitiga. ...
... Gambar 2. Modified octahedron (Sumber: Kenner, 2003) Konsep dari frekuensi ditentukan dengan jumlah dari segmen dimana prinsip dari subdivisi. 2V berarti segitiga yang dibagi menjadi 2 segmen sama besar, untuk 3V berarti membagi 3 segmen sama besar dan seterusnya. ...
The geodesic dome consists of steel rod elements joined together to form a single structure. Generally, these geodesic domes are analyzed by assuming the joints of the gusset points are joints and only receive axial forces on the rods. However, in reality, it is not easy to apply gusset joints as pure joints in construction. This research will analyze the geodesic dome by modeling the joints of the gusset points as joints where there is only axial force arising on the rods and modeling the rigid gusset points where there will also be moments and shear on the rods. The analysis will only be carried out by comparing the value of the displacement at each gusset joint modeling and checking the cross-sectional dimensions of the internal forces that arise with the help of the MIDAS GEN program in modeling the geodesic dome structure which has a diameter of 20000 mm and a height of 10000 mm with the type of steel profile. used is a pipe profile using two types of geodesic dome, namely type 2V and 3V. For loads that are calculated, namely dead load, live load, and wind load. The results showed a relatively small difference in translational displacement and the axial force was relatively the same in the internal force analysis, so it is better if the analysis by modeling the gusset connection as rigid.ABSTRAKKubah geodesik terdiri dari elemen batang baja yang disambung menjadi satu kesatuan struktur. Umumnya kubah geodesik ini dianalisis dengan menganggap sambungan titik buhulnya berupa sendi dan hanya menerima gaya aksial saja pada batang - batangnya. Namun pada kenyataannya untuk mengaplikasikan sambungan titik buhul sebagai sendi murni pada konstruksi tidaklah mudah. Penelitian ini akan menganalisis kubah geodesik dengan memodelkan sambungan titik buhulnya sebagai sendi dimana hanya ada gaya aksial saja yang timbul pada batang - batangnya dan memodelkan titik buhulnya rigid dimana akan terjadi juga momen dan geser pada batang tersebut. Analisis hanya akan dilakukan dengan membandingkan nilai dari perpindahan pada setiap pemodelan sambungan titik buhul dan pengecekan dimensi penampang terhadap gaya – gaya dalam yang timbul dengan bantuan program MIDAS GEN dalam memodelkan struktur kubah geodesik yang mempunyai diameter 20000 mm dan tinggi 10000 mm dengan jenis profil baja yang digunakan adalah profil pipa dengan menggunakan dua tipe kubah geodesik yaitu tipe 2V dan 3V. Untuk beban yang diperhitungkan yaitu beban mati, beban hidup, dan beban angin. Hasil penelitian menunjukkan nilai perbedaan yang relatif kecil pada perpindahan translasi dan diperoleh gaya aksial yang relatif sama pada analisis gaya dalam yang timbul, sehingga sebaiknya analisis dengan pemodelan sambungan titik buhul sebagai rigid.
... Habitable structures for arctic, desert, space or underwater environments where cylinders form closed rings (donut shape) Semicircular Buildings which can be assembled flat on a slab then erected by pulling the 2 edges (Points of "D") together and become rigid when full height is reached. (Nissen or Quonset Hut Design) In these designs the earth acts as a tension member (Kenner 2003). A demonstration model has been made using rectangular pieces and is shown in fig. ...
... Combine these rigidity methods with tensegrity and geodesic techniques (Kenner, 2003). ...
Contact: cesnur@iimetro.com.au, Webpages: http://www.modularbikes.com.au/index.html Cover Picture: Flat Version of Model shown in Figure 15.
... Polega ona na postawieniu wieloparametrowego problemu minimalizacyjnego w postaci funkcji wraz z nałożonymi ograniczeniami [Pellegrino: 1986] Statyczne rozwiązanie analityczne polega na znalezieniu takiego układu elementów wstępnie założonej konfiguracji, aby w każdym z węzłów bez działania obciążenia zewnętrznego występowała równowaga sił [Kenner: 1976]. Zakłada się przy tym dodatnie siły normalne w cięgnach i ujemne w zastrzałach. ...
... Można udowodnić [Kenner: 1976], że wzajemny kąt obrotu podstaw modułów typu sim- ...
Idea tensegrity po raz pierwszy została opisana około 50 lat temu. Dotyczy specyficznych struktur, w których elementy ściskane oraz rozpierane przez nie elementy rozciągane wzajemnie stabilizują się, mimo że w strukturach występują mechanizmy. W rozprawie postawiono pytanie o to, czy i w jakim stopniu możliwe jest wykorzystanie idei tensegrity w konstrukcjach budowlanych, a w szczególności w obiektach mostowych. Aby na nie odpowiedzieć, podjęto rozważania zarówno na temat podstaw teoretycznych, jak i aspektów praktycznych projektowania tego typu konstrukcji. Bazując na mechanice konstrukcji sformułowano autorskie definicje „czystych tensegrity” i „konstrukcji o cechach tensegrity”. Określono najważniejsze cechy tensegrity, takie jak występowanie mechanizmów infinitezymalnych oraz stanów samonaprężenia, co pozwoliło na opracowanie opisu matematycznego tego typu konstrukcji oraz podanie własnej propozycji sposobu poszukiwania formy tensegrity. Wybrano metody obliczeniowe adekwatne do wspomnianych cech konstrukcji. Posługując się nimi, przeanalizowano właściwości statyczne i dynamiczne podstawowych modułów tensegrity. Podjęto rozważania na temat możliwości łączenia modułów pod kątem ich wykorzystania w konstrukcjach mostowych, a także wpływu sposobu łączenia na właściwości otrzymywanych w ten sposób struktur. Omówiono najważniejsze zagadnienia technologiczne związane z wykorzystaniem tensegrity w mostownictwie. Zbadano odpowiedź wybranych konstrukcji na obciążenia statyczne i dynamiczne w kontekście ich zastosowania jako ustroje nośne kładek dla pieszych. Dokonano oceny i klasyfikacji istniejących projektów oraz realizacji obiektów mostowych pretendujących do miana tensegrity. Przedstawiona analiza pozwoliła na sformułowanie wniosku mówiącego o możliwości zastosowania tensegrity w mostownictwie, głównie w zakresie ograniczonym do konstrukcji o cechach tensegrity. Zastosowanie „czystych tensegrity” może być uzasadnione w przypadku konstrukcji o specjalnym przeznaczeniu, takich jak mosty sterowane i rozwijane.
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The concept of tensegrity was first described about 50 years ago. It refers to a particular type of structure, in which the compression and tensile components stabilize each other, despite the existence of mechanisms in the structure. A question, whether and to what extent it is possible to use the idea of tensegrity in building constructions, especially in bridges, was raised in the dissertation. To answer it, the author elaborated on both theoretical basis and practical aspects of the design of this type of structures. Two own definitions, based on mechanics of structures are presented. These are “pure tensegrities” and “structures having tensegrity features” definitions. The main features of tensegrity, such as the existence of infinitesimal mechanisms and self-stress states were defined. This made it possible to develop a mathematical description of this type of structures and to propose a new form-finding method. Computational methods appropriate for aforementioned properties were chosen. Static and dynamic properties of the basic tensegrity modules were analyzed using the selected methods. The possibility of combining modules for the use in bridge structures was discussed, as well as the impact of connection type on the properties of thus obtained structures. The most important technological issues associated with the use of tensegrity in bridges were discussed. The response of selected structures on the static and dynamic loads was examined in the context of their use in construction of footbridges. Existing projects and realizations of bridges called tensegrity were classified and evaluated. The analysis allowed to conclude, that there is a possibility of using tensegrity in bridge structures, limited mainly to the structures having tensegrity features. The use of “pure tensegrities” may be justified in the case of special-purpose structures, such as adaptive and deployable bridges.
... When working on a rectangular (A, θ) grid, this can be achieved by using a weight factor w k = sin θ k , which is proportional to the area on the unit sphere covered by the cell around the k-th grid point. Alternatively, triangular vertex grids may be employed, which are also known as "icospheres" or "geodesic polyhedra" (Kenner 2003). Starting from an icosahedron, polyhedra with a higher number of vertices and decreasing size of the triangular faces are constructed by consecutive division of the edges and projection of the resulting points on the unit sphere. ...
Over the past decade, the Global Positioning System has released pre-flight calibrations for the transmit antennas of the Block IIR/IIR-M, Block IIF, and GPS III satellites that make up the current GPS constellation. Frequency-specific phase variations (PHVs) provided as part of these data sets are of key interest for an accurate and consistent modeling of GNSS carrier phase observations in precise point positioning applications as well as orbit and clock offset determination of the GPS satellites themselves. For proper utilization of the manufacturer calibrations, complementary information on the phase center offset (PCO) from the spacecraft center-of-mass is required. We describe necessary processing steps for converting the raw phase calibrations of Lockheed Martin and Boeing into a representation compatible with antenna models of the International GNSS Service (IGS), and provide a detailed discussion of inherent assumptions for combining PHVs and PCOs from different sources. Comparison with estimated antenna data from globally distributed monitoring stations shows good consistency of PHVs and suggests the use of manufacturer-calibrated, azimuth-dependent patterns in future releases of the IGS antenna model. In terms of PCOs, the new Block IIF calibrations exhibit a systematic bias of about 12 cm from PCOs estimates based on the IGS20 reference frame. This value closely matches the bias observed for manufacturer calibrations of GPS III and Galileo satellites, and suggests a careful review of the contribution that GNSS can make to the scale definition of the International Terrestrial Reference Frame (ITRF).
... As mentioned above, with the pioneer works of Emde, Refs. [3,4], Fuller, Stuart, Clinton, Ref. [5], Tarnai, Kenner Ref. [6] and Pavlov, Ref. [7] as background, a special geometric construction method was devised to obtain a minimum of hexagon types which were almost planar, that is, within "tolerable" deviations from the plane. Based on the practical experience of the manufacturer, Foiltec, these "non-planarities" were expected to be within the range of fabrication and installation tolerances and dead-weight deformations. ...
The Eden Project in Cornwall, UK, has probably the largest greenhouse complex in the world, comprehending eight interconnected spherical geodesic domes. The geometric design and optimisation of the double-layered spherical structures focussed primarily on the hexagonal grids of the external dome surfaces, where optimisation consisted of minimising the number of component types, being these components, faces, line lengths and angle combinations at grid nodes. The present article reviews the general geometric definition of the Eden Domes and elaborates, for the first time since the domes design around the year 2000, on their geometric optimisation. The results of this rationalisation are presented in colour images, which were produced with state-of-the-art (2021) software tools 20 years after the design development. The images reveal the geometric ‘economy’ – here, the extreme reduction of geometric component types – and the high symmetrical order of these structural configurations. The aim of the present paper is to present the thorough geometric specification of the Eden Domes in a single document that answers in detail the question of how the Eden Domes grids were geometrically constructed.
... The unique microphone spreading of geodesic sensor arrays allows good detection of complicated modal radiation patterns. Therefore, further investigation of other classes of geodesic geometry [27] could allow a better coupling of the TCS sensor array with the modal information radiated by the inlet with different flow and liner configurations. ...
p>An inverse technique for determining the mode amplitudes generated by turbofan inlets both for tonal and broadband noise is proposed using pressure measurements made in the near-field. The motivation of this research is to make use of the Turbulence Control Screen (TCS). This TCS offers a useful platform for locating microphones to implement a non-intrusive inverse technique since it is often fitted to aero-engines during ground testing to remove the integrated flow. The knowledge of such model content is very useful for characterizing source mechanisms of broadband noise and for determining the most appropriate mode distribution model for duct liner predictions and for sound power measurements of the radiated sound field. The near-field sound pressure radiated from a duct is modelled by directivity patterns of cut-on modes. The resulting system of equations is ill-posed and it is shown that the conditioning of the inverse problem, which depends greatly on the positions of the microphones, is important in assessing the sensitivity of the modal solution to measurement noise and thus the modal reconstruction accuracy. An optimal array geometry for robust inversion is investigated. It is then shown that the presence of modes with eigenvalues close to a cut-off frequency results in a poorly conditioned directivity matrix. A physical interpretation of the Singular Value Decomposition (SVD) of the directivity matrix throws light on the understanding of the issues of ill conditioning as well as the detection performance of the radiated sound field by a given sensor array. The detection of broadband modes generated by a laboratory-scaled fan inlet is performed using the optimal array geometry. This experiment provides a milestone for detecting modal content of broadband noise produced by real fan inlet engines.</p
... The struts are placed together in a triangle-like structure and the head or end of the strut are connected to the nearest adjacent vertices through four cables. The 12-strut tensegrity model is completed by adding 48 cables [21]. In the cytoskeleton, the end of each microtubule which is close to the membrane is connected to the integrin. ...
... Méthodes géométriques : Tibert et Pellegrino [TP03] mentionnent trois méthodes géométriques de recherche des configurations de tenségrité d'un système donné. La première, proposée par Kenner [Ken03] est purement analytique et s'appuie sur des propriétés de symétrie. Si les polygones formés par les extrémités supérieures et inférieures des barres présentent une symétrie de rotation autour d'un axe, on peut exprimer la rotation d'un polygone à un autre par un angle. ...
Une structure de tenségrité est un assemblage d'éléments en compression (barres) et d'éléments en traction (câbles, ressorts) maintenus ensemble en équilibre. La tenségrité est connue en architecture et en art depuis plus d'un siècle et est adaptée à la modélisation des organismes vivants. Les mécanismes de tenségrité ont été étudiés plus récemment pour leurs propriétés prometteuses en robotique telles que leur faible inertie, leur compliance naturelle et la capacité de déploiement. Un mécanisme de tenségrité est obtenu lorsqu'un ou plusieurséléments sont actionnés. Cette thèse s'inscrit dans le cadre du projet AVINECK, auquel participent des biologistes et des roboticiens dans un but double : avoir une meilleure compréhension de la nature et des cous d’oiseaux, et également de s’inspirer de la nature afin de concevoir des manipulateurs bioinspiré innovants. En premier lieu, une étude a été menée sur la cinématique des vertèbres d’oiseaux. Une modélisation des contacts entre surface articulaire a permis de reconstruire en 3D le mouvement d’une vertèbre d’oiseau quelconque par rapport à une autre, à partir de scans 3D de vraies vertèbres. Ensuite, plusieurs mécanismes simples ont été comparés afin de déterminer le meilleur candidat à la modélisation de la cinématique des cous d’oiseaux. En second lieu, deux mécanismes de tenségrité à 1 degré de liberté (DDL) potentiellement intéressants pour la modélisation de cous d’oiseaux et la réalisation de manipulateurs ont été modélisées. A partir d’une démarche de conception optimale, deux mécanismes optimaux selon un critère donné ont été dimensionnés, puis comparés afin de déterminer le mécanisme le plus intéressant d’un point de vue conception de manipulateurs. Le mécanisme à 1 DDL retenu a été un antiparallélogramme actionné de manière antagoniste, appelé mécanisme en X. Un prototype de manipulateur à 2 DDL a ensuite été conçu et réalisé. Une étude du modèle géométrique d’un tel manipulateur a mis en évidence des phénomènes intéressants comme le comportement cuspidal du robot sous certaines conditions. Une étude cinéto-statique du manipulateur a ensuite été effectué. Le comportement du manipulateur dépendant fortement de la stratégie d’actionnement, celles-ci ont été énumérées et certaines d’entre elles comparées. Les résultats obtenus ont été vérifiés et mis en évidence sur un prototype. Enfin, la modélisation de manipulateurs plans a été étendue aux manipulateurs à N DDL. Un modèle dynamique complet a été obtenu, et une démarche de conception a été menée afin d’obtenir les dimensions et spécifications d’un manipulateur plan bio-inspiré à 10 DDL.
... DGGS have precedent in Buckminster Fuller's development of geodesic structures, and inspiration in the scientific community's general enthusiasm for "buckyballs" (Fuller 1954;Kenner 1976; American Chemical Society 2010). Astronomy and remote sensing have devised various tessellations for planetary and celestial applications (Tegmark 1996;Fekete and Treinish 1990;Gorski and others 2005;Seong 2005). ...
In the same way that discrete global grid systems (DGGS) are used to index data on the spherical Earth, they can aggregate point data, with their spherical polygons serving as bins. DGGS are particularly useful at multiple map scales because they are spatially hierarchical and exist on the sphere or ellipsoid, allowing large or small scale binning without projection distortion. We use DGGS in a free and open-source pedagogical tool for teaching students about the modifiable areal unit problem (MAUP). Our software application uses Dutton’s quaternary triangular mesh (QTM) to bin global data points geodesically with counts or measures of any theme at multiple levels. Users can interactively select the level to which the data are binned by the QTM, as well as translate the whole tessellation east or west so that points fall into and out of different bins. These two functions illustrate the scaling and zoning aspects of the MAUP with dynamically-drawn choropleths on the surface of a virtual globe that the user can zoom and rotate, allowing visualization at virtually any cartographic scale. Users may also select various quantile classifications to further explore issues in visualizing aggregate data. In addition to presenting this new tool, we highlight the importance, especially at smaller scales, of using geodesic point-in-polygon intersection detection, rather than the projected 2D methods typically used by geographic information systems.
... In one hand, Pugh wrote "An Introduction to Tensegrity" [7] that is important because of the diversity of types that it explains and its strict typology and classification. In the other hand, Kenner wrote "Geodesic Math and How to Use It" [17] that explains the calculus "for any degree of accuracy", the relevant aspects of the geometry of the geodesic and tensegritic structures (angles and lengths of the bar pattern) and investigates 3 Casually, while Fuller was patenting his "geodesic domes" in 1954 (US 2,682,235), Emmerich patented the "stereometric domes" in 1967 (US 3,341,989). their potential. ...
"Push-and-pull" efficient structures have been inconceivable between XVIII centuries. It is because of the incapacity of obtain an efficient behaviour of tensioned material. Since XVIII centuries, architecture began to experiment new processes of adaptation defining new typologies in the engineering and architecture that had not been experienced until that moment. Tensegrities and tensioned structures were developed from the study of tension and geometry. Since then it has been emerged studies of this type by researchers who have continued or have been interested in these works.Tensegrities are bar and cable structures that work only in compression or tension efforts. Bars and cables are balanced, but in appearance the growth is disorderly. Most of deployable structures are based on tensegrity systems.This paper presents a synthesis of the art state of the set of technical solutions to the design of architectures based on tensegrities and tensioned structures that will be applied to the building construction. The research of adequate materials to tension efforts will be crucial in this study.This study provides a strong documental foundation for research on the significance of new structural and constructive systems in the production of the current architecture based on tensegrities and tensioned structures.
... My work is influenced by the scientific works of Hugh Kenner (Kenner, 1976), Joseph Clinton (Clinton, 1965(Clinton, , 1971 and Edward Popko (Popko, 2012) (among others) and is inspired by the counter-cultural movement of the sixties and seventies. I delve into the complexities of geometry to expand the frontiers of my knowledge, with curiosity as a driving force and with no pretensions other than exploration itself. ...
... Grünbaum and Shephard proposed a set of conjectures about the equilibrium of tensegrity frameworks that served as a starting point for tensegrity design [27]. Hugh Kenner presented one of the first mathematical investigations on the geometry of tensegrity systems which he incorporated in his book on geodesic mathematics [28]. Fuller adopted these ideas and used simple geometric shapes like regular prisms (tetrahedron, octahedron) to construct tensegrity geodesic domes [18]. ...
Tensegrity structures are reticulated frameworks composed of axially loaded elements in a stable self-equilibrated prestressed state. Their self-equilibrated prestressed state is a result of their topology and geometry which define their statical (self-stress) and kinematical (mechanisms) properties. They are materially and mechanically efficient form-found structures that can also integrate sensors and actuators. Therefore, they are good candidates for smart applications. Although much progress has been made in advancing research into tensegrity systems and their application in science and engineering, there are currently no universally accepted design guidelines for their dialectic form finding, tuning their behavior, and guiding their transformation into active systems. With the aim of controlling the shape and mechanically behavior of tensegrity structures, this thesis proposes a novel bio-inspired generative design method for tensegrity structures: cellular morphogenesis. In an analogy with biological cells that are combined through adhesion and fusion to form complex living organisms, tensegrity cells (infinitesimally rigid self-stressed units) are combined to form complex structures. It is shown that cells can be used to compose complex tensegrity structures with a desired shape and self-stress, as well as explain mechanism creation. The method gives also insights on sensor/actuator placement in tensegrity systems that can be used to inform the transformation process for smart applications. Consequently, cellular morphogenesis offers a new perspective for the design and analysis of tensegrity structures in smart structures, soft robotics, material design, and cellular modeling
... To make the arrangement of the sub-eyes on the curved compound eye denser and more even, the icosahedral subdivision method [32,33] is used to determine the exact positions of the sub-eyes on the curved surface. Figure 2 shows the procedure for calculating the positions of the sub-eyes using this method. ...
The compound eye of insects has many excellent characteristics. Directional navigation is one of the important features of compound eye, which is able to quickly and accurately determine the orientation of an objects. Therefore, bionic curved compound eye have great potential in detecting the orientation of the target. However, there is a serious non-linear relationship between the orientation of the target and the image obtained by the curved compound eye in wide field of view (FOV), and an effective model has not been established to detect the orientation of target. In this paper, a method for detecting the orientation of the target is proposed, which combines a virtual cylinder target with a neural network. To verify the feasibility of the method, a fiber-optic compound eye that is inspired by the structure of the bee’s compound eye and that fully utilizes the transmission characteristics and flexibility of optical fibers is developed. A verification experiment shows that the proposed method is able to realize quantitative detection of orientations using a prototype of the fiber-optic compound eye. The average errors between the ground truth and the predicted values of the horizontal and elevation angles of a target are 0.5951 ° and 0.6748°, respectively. This approach has great potential for target tracking, obstacle avoidance by unmanned aerial vehicles, and directional navigation control.
... The subdivision of the DGGS cells involves the choice of the shape and partitioning strategy. There are 4 types of shapes that are common for partitioning; triangles (Kenner, 1976) (Fig. 1b), diamonds (White, 2000) (Fig. 1c), hexagons (Kimerling et al., 1999) (Fig. 1d), and squares (Alborzi and Samet, 2000) (Fig. 1e). Hexagons are the most compact, they quantize the plane with the smallest average error (Conway and Sloane, 1998), and they provide the greatest angular resolution (Golay, 1969) among all the regular polygons that tile the plane. ...
Discrete global grid systems (DGGS) have been proposed as a data model for a digital earth framework. We introduce a new data model and analytics system called IDEAS – integrated discrete environmental analysis system to create an operational DGGS-based GIS which is suitable for large scale environmental modelling and analysis. Our analysis demonstrates that DGGS-based GIS is feasible within a relational database environment incorporating common data analytics tools. Common GIS operations implemented in our DGGS data model outperformed the same operations computed using traditional geospatial data types. A case study into wildfire modelling demonstrates the capability for data integration and supporting big data geospatial analytics. These results indicate that DGGS data models have significant capability to solve some of the key outstanding problems related to geospatial data analytics, providing a common representation upon which fast and scalable algorithms can be built.
... We know from Hugh Kenner's book Geodesic Math and How to Use It [5] that T-prism Tensegrity end n-gons are rotated relative to one another by the angle α, where the following applies, where n is the number of struts in the Tensegrity: ...
This paper presents the results of practice-based research into Tensegrity systems, formfinding, structural analysis, testing and use in a 'real-life' project: a demountable Tensegrity pavilion with a tensile fabric canopy-'Tension Pavilion'. Tensegrity has rarely been utilised in the built environment for a building structure. The term is often misunderstood and wrongly applied to structures that use different structural systems, or have been adapted to such an extent that they are no longer purely Tensegrity. This demountable pavilion was built with a chain of simplex Tensegrities that form an undulating ring, warped to follow a sine wave, creating three arches and three valleys. The Tensegrity ring and fabric canopy have been designed to resist 75mph wind loads, and tested for an 80kg load at the apex of an arch. In this paper we present preparatory practice-based research, parametric modelling, formfinding, structural analysis, physical testing, detailing, fabrication and construction. The benefits and challenges of using Tensegrity are discussed, and recommendations for future use and further study are made.
... The shape of the tensegrity is maintained by the balance between compression and tensile forces of the bars and strings [1]. Since no element experiences bending or shear forces, the tensegrity allows lightweight and strong designs compared to other approaches [2], [3]. ...
... All chord factors are found by exactly the same routine, and all are found independently, errors are isolated, and do not affect other computations. By using a good pocket calculator, which can quickly change angles into their sines and cosines, permits us to run through the chord-factor equation for each pair of points in less than a minute, even when coordinates are cumbersome [3]. ...
The paper reports an attempt at assessing the effect of geodesic dome mesh refinement on the structure critical load capacity. The initial surface of geodesic domes is the sphere divided into spherical triangles. Consecutive divisions of the spherical triangles decide the mesh refinement of the geodesic domes. Two computational models (3V and 4V) were examined. For 4V dome, two types of structure, namely truss and frame ones were considered. For 4V frame system, the values of the critical load capacity were determined for three buckling lengths of the bars μ: 0.5, 0.7 and 1.0.
... To do so, we use the geodesic sphere ray launching [49]. A geodesic sphere (Figure 3) is formed by tessellating faces of a regular polyhedron and extrapolating the intersection points on the surface of a sphere [24]. The geodesic vertices provide equivalent angular separation around the entire sphere [41]. ...
Judicious control of indoor wireless coverage is crucial in built environments. It enhances signal reception, reduces harmful interference, and raises the barrier for malicious attackers. Existing methods are either costly, vulnerable to attacks, or hard to configure. We present a low-cost, secure, and easy-to-configure approach that uses an easily-accessible, 3D-fabricated reflector to customize wireless coverage. With input on coarse-grained environment setting and preferred coverage (e.g., areas with signals to be strengthened or weakened), the system computes an optimized reflector shape tailored to the given environment. The user simply 3D prints the reflector and places it around a Wi-Fi access point to realize the target coverage. We conduct experiments to examine the efficacy and limits of optimized reflectors in different indoor settings. Results show that optimized reflectors coexist with a variety of Wi-Fi APs and correctly weaken or enhance signals in target areas by up to 10 or 6 dB, resulting to throughput changes by up to -63.3% or 55.1%.
... The name Tensegrity is the contraction between two words "Tensional" and "Integrity". Kenner (1976) and Clinton (1965) demonstrated the math to find the coordinates of the nodes that will generate the geodesic dome. They also classified the geodesic domes according to method is used to find the coordinates. ...
... Statistical comparisons require meshes with a common resolution where each point represents homologous locations across individuals. A geodesic sphere has many advantages for this purpose: ease of computation, edges of roughly similar sizes and, if the resolution is fine enough, edge lengths that are much smaller than the diameter of the sphere (Kenner 1976). We compared 4 different interpolation methods, described below, at each of 3 different mesh resolutions: IC3 (642 vertices and 1280 faces), IC5 (10 242 vertices and 20480 faces) and IC7 (163 842 vertices and 327 680 faces); for the comparison between VLBW and controls, the resolution used was IC7, with nearest neighbor interpolation. ...
Cortical surface area is an increasingly used brain morphology metric that is ontogenetically and phylogenetically distinct from cortical thickness and offers a separate index of neurodevelopment and disease. However, the various existing methods for assessment of cortical surface area from magnetic resonance images have never been systematically compared. We show that the surface area method implemented in FreeSurfer corresponds closely to the exact, but computationally more demanding, mass-conservative (pycnophylactic) method, provided that images are smoothed. Thus, the data produced by this method can be interpreted as estimates of cortical surface area, as opposed to areal expansion. In addition, focusing on the joint analysis of thickness and area, we compare an improved, analytic method for measuring cortical volume to a permutation-based nonparametric combination (NPC) method. We use the methods to analyze area, thickness and volume in young adults born preterm with very low birth weight, and show that NPC analysis is a more sensitive option for studying joint effects on area and thickness, giving equal weight to variation in both of these 2 morphological features.
Cell’s shape is dependent on the cytoskeleton mechanical properties. Hybrid models were developed that combine the discrete structure for the cytoskeleton and continuum parts for other cell organelles. Tensegrity-based structures that consist of tensile and compression elements are useful models to understand the cytoskeleton mechanical behavior. In this study, we are looking to examine the reaction of the cell to a variety of substrate stiffnesses and explain the relationship between cell behavior and substrate mechanical properties. However, which tensegrity structure is appropriate for modeling a living cell? Is the structure’s complexity play a major role? We used two spherical tensegrities with different complexities to assess the impact of the structure on the cell’s mechanical response versus substrate’s stiffness. Six- and twelve-strut tensegrities together with membrane, cytoplasm, nucleoskeleton, and nucleus envelope were assembled in Abaqus package to create a hybrid cell model. A compressive load was applied to the cell model and the reaction forces versus deflection curves were analyzed for number of substrate stiffness values. By analyzing the difference due to two different tensegrities it became clear that the lower density structure is a better choice for modeling stiffer cells. It was also found that the six-strut tensegrity is sensitive to higher range of substrate stiffness.
Tensegrity systems are structures in equilibrium under a self-stress state prior to the application of external loads. Such self-stress state is the result of the superposition of self-stress basis vectors. Tensegrities have complex nonlinear behavior when subjected to external actions (loads or imposed displacements). In this work, the robustness of this particular structural typology is evaluated by numerical simulations. Robustness is the ability of a locally damaged structural system not to behave in a manner disproportionate to the original failure, i.e., the insensitivity to local damage. This concept and its insights have been thoroughly analyzed for several structural systems, as in the case of frame structures. To date, limited studies have been performed on the robustness of tensegrity systems to unexpected conditions. Among the potential damages, one can consider the settlement of one of the supports, the reduction of prestress and the failure of one of the cables. The present study reports the results of a set of damage simulations on a simple T-3 prismatic tensegrity structure. Several risk factors are considered to study the response of the structure to unexpected damage. In particular, the tensegrity configuration is subjected to different stress loss in the constituent elements. The results are obtained through simulations performed in MSC.visualNastran 4D (vN4D), which run dynamic simulations by merging CAD, motion, FEA, and controls technologies into a single functional modeling system. The models were established to obtain different equilibrium configurations and conditions of loss of equilibrium, with the corresponding large displacements measured for the most significant points of the structure, assumed as markers for the resulting effects on the structure.
O processo de projeto em meio digital evolui desde as tecnologias de representação, documentação técnica, simulação, avaliações de desempenho, até a abordagem do desenho paramétrico, que tem influenciado e subsidiado a produção atual de arquitetura. Antoni Gaudí e Frei Otto são considerados os pioneiros desta abordagem em seus processos de projeto, ainda que em modelos analógicos. Os arquitetos desenvolveram modelos físicos aptos a modificações, a partir do uso de malhas suspensas e controladas por pesos variáveis. Para o momento atual dos processos digitais na arquitetura, as malhas expandiram seus usos junto à parametria, como soluções de representação, tesselação, análise e otimização de processos, soluções estruturais e também a possibilidade de simulação da atuação de forças físicas para gerar geometrias curvas e otimizadas. Porém, o currículo dos cursos de arquitetura no Brasil ainda não incorpora tais saberes de maneira sistemática e disciplinar, o que dificulta sua apropriação no contexto profissional. Entende-se que é necessária uma base geométrica que propicie o avanço até níveis maiores de complexidade exigidos para o domínio do desenho paramétrico. Neste processo, é importante tratar com aplicações da geometria na arquitetura para além das questões estéticas, compreendendo como uma forma não trivial pode apresentar atributos funcionais que evidenciem a qualidade de um projeto para o contexto atual. Busca-se, então, a partir de uma teoria didática, explicitar o saber envolvido relacionado com aplicações de malhas na arquitetura e compreender os conceitos geométricos que integram suas propriedades funcionais, para motivar a apropriação destes na prática de arquitetura por meio do projeto paramétrico. Tendo-se por base um estudo teórico de projetos referenciais, tenta-se compreender o processo de projeto desenvolvido para a definição formal e os conceitos e técnicas associadas, tendo-se por base a noção estruturada de um saber, de Yves Chevallard. A partir dessa teoria, define-se um método de estruturação do saber envolvido em cada um dos casos estudados, apresentando-se um dos projetos estudados. Como resultado da sistematização, tem-se uma abordagem integrada entre desempenho das superfícies, a estruturação formal e construtiva, de análise e requisitos essenciais ao processo de projeto contemporâneo, propiciada pela modelagem paramétrica e o saber da geometria.
The compound eye system has many unique advantages, enabling organisms to quickly and accurately obtain the three‐dimensional spatial information of the target. Therefore, its bionic applications have great potential in object localization, dynamic tracking, and 3D reconstruction. However, the current bionic compound eye systems still have great difficulties in target detection. Most of the bionic compound eye systems only verify the possibility of three‐dimensional (3D) detection, and there is no suitable calibration and detection scheme. In this article, a fiber‐optic stereo bionic compound eye is designed, and a simple and rapid calibration method suitable for this system is selected by comparing two effective optical information. Further, the system quantitatively analyzes the detection performance of the target to achieve precise positioning and dynamic trajectory tracking. In the static detection experiment, the average detection errors of the 3D position at the horizontal angle, the elevation angle and the depth are 0.33°, 0.32° and 1.84 mm, respectively. In the dynamic trajectory tracking experiment, the circle radius errors in the X‐Z plane and Y‐Z plane are 0.04 and 2.20 mm, respectively. The proposed system provides a new perspective to understand the nature of the compound eye.
This book presents the proceedings of 5th International and 20th National Conference on Machines and Mechanisms (iNaCoMM 2021) held at PDPM IIITDM Jabalpur during 9-11 December 2021. The conference was held in collaboration with the Association of Machines and Mechanisms (AMM) India and International Federation for the Promotion of Mechanism and Machine sciences (IFToMM). Various topics covered in this book include kinematics and dynamics of machines, compliant mechanisms; gear, cams and power transmission systems; mechanisms and machines for rural, agricultural and industrial applications; mechanisms for space applications; mechanisms for energy harvesting; robotics and automation; human-centric robotics; soft robotics; man-machine system, mechatronics and micro–mechanisms; CAD and CAGD; control of machines; vibration of machines & rotor dynamics; acoustic and noise; tribology; condition monitoring and failure analysis; fault diagnosis and health monitoring; biomedical engineering; and composites and advanced materials. Given the contents, the book will be useful for researchers and professionals working in the various domains of mechanical engineering.
Cell is a fundamental unit of life. Mechanical stimuli in cells are the driving factor in the many essential life processes of the cell such as growth, proliferation and metastasis. Many dynamic behaviours of cells like natural frequency, response and mode-shape are promising factors giving insight into the wellbeing of a cell. Dynamic behaviour helps in the treatment of many cells-related diseases. Cancer is one such disease leading to abnormal cell growth with the potential to spread to other parts of the body. Despite the rapid medical advancement in developing techniques to cure cancer, these cannot completely cure cancer at all stages and also comes along with side effects. In the present work, a six-strut tensegrity model is developed to mimic the cytoskeleton of MCF-7 cell, considering the effect of prestress. Dynamic analysis is performed, and the first five natural frequencies are in the range of 5–17 kHz. The mode shapes corresponding to the first four natural frequencies are reported.
This paper presents the design and dynamic modelling of a soft pneumatic actuator that can be used to mimic snake or worm-like locomotion. The bond graph technique is used to derive the dynamics of the actuator. To validate the accuracy of the derived dynamic model, we conduct numerical simulations using 20-sim software. Experimental results demonstrate that the soft actuator achieves bidirectional bending and linear displacement, which is essential for mimicking snake or worm-like locomotion.KeywordsBond graphControlDynamicsSnake robotsSoft robots
Sandwich constructions with a honeycomb core have recently become popular for high strength and dynamic load. The purpose of this study is to look at the shock wave resistance performance of two distinct types of honeycomb core sandwich structures in terms of face plate deflection and energy absorption when subjected to a blast load. This study employed square and hexagonal honeycomb core structures to determine the minimal face deflection under blast conditions. The honeycomb sandwich panel is composed of steel that is very ductile. In the sandwich construction, both the front and rear plates are solid, and the core structure is of the shell type. To administer the air-blast loads of 1, 2, and 3 kg TNT, a 10-cm stand-off distance is used from the front face. The dynamic response of the sandwich constructions is determined using the ABAQUS/explicit finite element method (FEM). For both square honeycomb and hexagonal honeycomb core sandwich panels, the front and rear face plate deflections were measured. Under comparable blast loading circumstances, the front and rear plates of the hexagonal sandwich panel showed less deformation than the square honeycomb core sandwich panel.
This dissertation aims to relate the aspects of Biomimetics to Buckminster Fuller's geodesics through an investigation of the fundamentals that guide these subjects in order to demonstrate the constructive advantages of applying these as a Biodesign strategy. The research uses qualitative methods through a more extensive
bibliographical survey on the concepts, methodological processes, techniques and applications related to the investigation of natural elements and structures and its applicability in design projects, as well as investigates the geodesics with their foundations, characteristics and applications in the contemporary context, to establish a link between the two subjects and justify the use of these emblematic configurations as coherent solutions to a scenario of complexity and sustainability. In the end, we will present a brief Biodesign experiment with the domes that will bring the strategy of building two new configurations of geodesics with reference in elements of nature to
evidence the potential of new studies of these structures.
This paper intends to describe an educational experiment accomplished in the Geometry and Architecture course of the first year in the Faculty of Architecture of the University of Porto in 2017. In this activity, students were introduced to digital three-dimensional modelling as an additional tool to develop their knowledge of geometry. The subject of solid tessellations was selected as leitmotif because of the structural and architectonic interest and creative potential that the situations in which polyhedra, other than the cuboid, fill space may have for aspiring architects. The time limitations of the academic year impaired the desired breadth for the task, so students had to focus their attention only in six uniform solid tessellations, out of the possible 28. Besides acquiring digital design modelling skills useful for their scholarly and professional practice, this experiment and collaborative assignment allowed students to improve knowledge of polyhedral theory and apply newfound IT skills in architectural design.KeywordsSolid TessellationsPolyhedraGeometryThree-Dimensional SoftwareArchitectural Design.
In order to adequately address the design and construction of geodesic structures, it is necessary to analyze the multiplicity of factors that so far have been considered as unequivocal. These start with the subdivision methods by which geodetic meshes can be generated, through the configuration of sets of specific tools to work on their definition and finally their adaptation to specific construction methods.
A given geodesic geometry cannot, in fact, be applied to different construction methods without adapting it to their characteristics and peculiarities; each of them configures its own specific geometry. For this reason, many of the data and concepts necessary for the definition of specific construction methods are not found in the existing bibliography.
The approach we propose in our research not only allows for a new reading of the existing methods of designing geodesic structures that addresses and tackles the source of their recurrent problems, but also allows the configuration of new construction methods that avoid these problems.
Geodesic domes are 3D reticulated structures used to cover large areas. As lightweight structures, they are sensible for dynamic effects. The structural behaviour of domes is affected by the kind of connections between their members. Semi -rigid joints produce loss of stiffness compared with rigid joints. Dynamic tests may capture the effect of semi-rigid joints. The objective of this work is to conduct both static and dynamic tests on a Geodesic Dome at the laboratory scale and compare the results with a linear elastic FE model. The research method involved designing, building and testing a prototype constructed of aluminum wit h overall dimensions of 2000 mm in diameter and 800 mm in height. A 3D finite element model was developed using beam and shell elements. Experimental and numerical results were compared in terms of displacements and frequencies. Load—displacement curves are presented for the static loading. The dynamic behaviour is evaluated by hitting an impact hammer on members and nodes to allow measurement of accelerations. An FFT technique is used to determine the frequency spectrum. The results show that the laboratory prototype presented about 20% less stiffness than predicted by the linear elastic numerical model. The difference is attributed to the transmissibility condition between nodal connectors and emphasizes the influence of semi-rigid joints. Despite being largely used to assess dome structures, standard FE models are shown to be insufficient in capturing local non-linearities.
During the last 10 years, the architecture firm Ctrl + Z has conducted continuous research in the field of geometric structures. Almost thirty models have been built with different scales and objectives. Their purpose was not to present proposals or results, but to be integrated into the research as tools. They were initially conceived as an external support for the mind, capable of retaining part of the information to carry out new operations on it. Their construction proved to be immediately functional for the understanding of the rules governing both the specific geometry and its materialization. Once realised, they constituted a record of the process and played a central role in the dissemination of the research, promoting contact with other experiences. They are also a suitable tool for teaching activities. Models prove to be a functional tool in different phases of the research, taking on a different leading role in each of them.
Bowl, saddle, tube, hemisphere or squished bowl? A geometric measure was introduced to reveal the geometric factors that determine the large, nanoscale molecular structures comprising phenine panels.
Abstract
A vector was introduced to quantitatively describe the pyramidalization of 1,3,5-trisubstituted benzene (phenine). The vector, named curved phenine normal vector (CPNV), defined the direction of pyramidalization of the trigonal phenine panel and quantified the degree of pyramidalization. The relative orientation of the two CPNVs further defined a dihedral angle, which quantified interphenine torsions. The CPNV analysis visually showed the geometric features of nanometer-sized molecules by revealing the important role of torsions. Biased electrostatic potentials were also correlated with CPNV pyramidalizations.
A synthetic strategy to construct large geodesic structures of phenine (1,3,5‐trisubstituted benzene) was devised. In this strategy, five pentagons were assembled on an omphalos pentagon, and bridging peripheral pentagons furnished five additional hexagons. Thirty phenine units were synthetically assembled to afford a large C220H180 molecule with a phenine framework isoreticular to a hemispherical, bisected segment of C60. Although a hemispherical structure of the phenine framework was suggested by solution‐phase NMR spectra, crystallographic analysis revealed an oval‐like deformation of the molecular shape. In‐depth structural analyses, including theoretical calculations, showed that structural fluctuations observed as variations in the biaryl torsion angles allowed structural deformations and, at the same time, that the dynamic fluctuations resulted in the spectroscopic observation of a hemisphere as a time‐averaged structure.
5+(5×5+5×6)=Hemisphäre: Ein Omphalos-Fünfeck wurde mit 5 Fünf- und 5 Sechsecken dekoriert, um ein hemisphärisches Molekül durch eine Polygonassemblierungsstrategie zu bilden. Dreißig Phenineinheiten wurden zu einem großen C220H180-Molekül mit einem Pheningerüst zusammengesetzt, das isoretikulär zu einem hemisphärisches Segment von C60 ist.
Abstract
A synthetic strategy to construct large geodesic structures of phenine (1,3,5-trisubstituted benzene) was devised. In this strategy, five pentagons were assembled on an omphalos pentagon, and bridging peripheral pentagons furnished five additional hexagons. Thirty phenine units were synthetically assembled to afford a large C220H180 molecule with a phenine framework isoreticular to a hemispherical, bisected segment of C60. Although a hemispherical structure of the phenine framework was suggested by solution-phase NMR spectra, crystallographic analysis revealed an oval-like deformation of the molecular shape. In-depth structural analyses, including theoretical calculations, showed that structural fluctuations observed as variations in the biaryl torsion angles allowed structural deformations and, at the same time, that the dynamic fluctuations resulted in the spectroscopic observation of a hemisphere as a time-averaged structure.
Viruses have evolved protein containers with a wide spectrum of icosahedral architectures to protect their genetic material. The geometric constraints defining these container designs, and their implications for viral evolution, are open problems in virology. The principle of quasi-equivalence is currently used to predict virus architecture, but improved imaging techniques have revealed increasing numbers of viral outliers. We show that this theory is a special case of an overarching design principle for icosahedral, as well as octahedral, architectures that can be formulated in terms of the Archimedean lattices and their duals. These surface structures encompass different blueprints for capsids with the same number of structural proteins, as well as for capsid architectures formed from a combination of minor and major capsid proteins, and are recurrent within viral lineages. They also apply to other icosahedral structures in nature, and offer alternative designs for man-made materials and nanocontainers in bionanotechnology.
In this chapter, we will discuss in detail some simple tensegrity structures. In the context of tensegrity structures, various mathematical tools are illustrated clearly which can be applied for tensegrity form-finding. Some of the tensegrity structures are simple enough so that the member lengths can be expressed in algebraic formulas. With the help of a computer, the numerical calculation codes are provided to experimentation the formulations. These native methods deal clearly with distances and angular measurements related to the algebra and trigonometry problems.
A saddle‐shaped macromolecule has been synthesized. The molecule was designed as a geodesic saddle using 1,3,5‐trisubstituted benzene (named phenine) as the fundamental unit. The phenines were woven into a polygonal framework that was composed of 168 sp2‐carbon atoms. The saddle‐shaped structure with unique symmetry showed atypical conformational changes. The biaryl linkages in this molecule had a small energy barrier for rotations, and these structural fluctuations resulted in an NMR observation of 7 1H resonances representing 84 aromatic hydrogen atoms. Nevertheless, the overall saddle shape of the molecule was persistent, and the "up" and "down" orientations of phenines circulated to give average 1H resonances. The structural characteristics of this molecule, including the anomalous entropy‐driven dimerization, may deepen our understanding of defect‐rich graphitic sheets.
A saddle‐shaped macromolecule has been synthesized. The molecule was designed as a geodesic saddle using 1,3,5‐trisubstituted benzene (named phenine) as the fundamental unit. The phenines were woven into a polygonal framework that was composed of 168 sp2‐carbon atoms. The saddle‐shaped structure with unique symmetry showed atypical conformational changes. The biaryl linkages in this molecule had a small energy barrier for rotations, and these structural fluctuations resulted in an NMR observation of 7 1H resonances representing 84 aromatic hydrogen atoms. Nevertheless, the overall saddle shape of the molecule was persistent, and the "up" and "down" orientations of phenines circulated to give average 1H resonances. The structural characteristics of this molecule, including the anomalous entropy‐driven dimerization, may deepen our understanding of defect‐rich graphitic sheets.
Discrete Global Grid Systems (DGGSs) are spatial references that use a hierarchical tessellation of cells to partition and address the entire globe. They provide an organizational structure that permits fast integration between multiple sources of large and variable geospatial data sufficient for visualization and analysis. Despite a significant body of research supporting hexagonal DGGSs as the superior choice, the application thereof has been hindered owing in part to the lack of a rational hierarchy with an efficient addressing system. This paper presents an algebraic model of encoding scheme for the Aperture 3 Hexagonal (A3H) DGGS. Firstly, the definition of a grid cell, which is composed of vertices, edges, and a center, is introduced to describe fundamental elements of grids. Secondly, by identifying the grid cell with its center, this paper proves that cell centers at different levels can be represented exactly using a mixed positional number system in the complex plane through the recursive geometric relationship between two successive levels, which reveals that grid cells are essentially special complex radix numbers. Thirdly, it is shown that through the recursive geometric relationship of successive odd or even levels, the mixed positional number system can also be applied to uniquely represent cell centers at different levels under specific constraint conditions, according to which the encoding scheme is designed. Finally, it is shown that by extending the scheme to 20 triangular faces of the regular icosahedron, multi-resolution grids on closed surfaces of the icosahedron are addressed perfectly. Contrast experiments show that the proposed encoding scheme has the advantages of theoretical rigor and high programming efficiency and that the efficiency of cross-face adjacent cell searching is 242.9 times that of a similar scheme. Moreover, the proposed complex radix number representation is an ideal formalized description tool for grid systems. The research ideas introduced herein can be used to create a universal theoretical framework for DGGSs.
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