contents VOL. 47 (2006) No. 3 n. 152 December

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2 VOL. 47 (006) No. 3 n. 5 December conens Memorial Saemens Leer from he Presien J.F. Abel Memorial o Prof. Wolfgang Zerna W. B. Kräzig Memorial o Prof. Mircea Mihailescu I. Bucur-Horváh e al. IASS Repors IASS-APCS 006 Symposium Repor S.D. Xue IASS Tsuboi Awars 005 N. K. Srivasava IASS Hangai Prizes 006 M. Takayama IASS Technical Aciviies Repor, Y. B. Yang Inernaional Workshop On Compuaional Morphogenesis 006 H. Ohmori Inernaional Colloquium of Free Form Design 006 A. Borgar 5 h Inernaional Conference on Mealwork in Civil Engineering 006 O. Shimanobsky & O. Golonov Tsuboi Proceeings Awar Paper for 005 Experimenal Suies of Tower wih Hysereic Dampers Y. Ookouchi, T.Takeuchi, T.Uchiyama, K.Suzuki, T.Sugiyama, T.Ogawa & S. Kao Hangai Prize Papers for 006 Approximaing Complex Shapes wih Inelligen Embee Design Inelligence in Sysems for he Early Phases of Design J. Coeners Using Symmery for Tensegriy Form-Fining R. Pania Raj an S. D. Gues Developmen an Design of Glass Fole Plae S. Tromeer an M. Krupna Response Analysis of Naional Saium Uner Spaially Varying Earhquake Groun Moions W. H. Liu, Q. S. Yang an Y. J. Tian Technical Papers Felix Canela, Elegance an Enurance An Examinaion of he Xochimilco Shell N. Burger an D. P. Billingon Buckling of Concree Shells Numerical Simulaion M. Anres an R. Hare Buerfly Srucure for Spaial Enclosures T. C. Tran an J. Y. R. Liew Hierarchical Moular an heir Geomerical Configuraions Fuure Large Space Sysems N. Kishimoo an M. C. Naori Book Review Euaro Torroja Works an Projecs, by P. Chías Navarro & T. Aba Balboa Review by R. E. Shaeffer COVER Phoo of Euaro Torroja book IASS Secrearia CEDEX-Laboraorio Cenral e Esrucuras y Maeriales Alfonso XII, 3; 804 Mari, Spain Tel ; Fax ; iass@ceex.es; hp// SODEGRAF ISSN Depósio legal M

3 JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES IASS HIERARCHICAL MODULAR STRUCTURES AND THEIR GEOMETRICAL CONFIGURATIONS FOR FUTURE LARGE SPACE SYSTEMS NAOKO KISHIMOTO an M.C. NATORI Deparmen of Space an Maerials The Insiue of Space an Asronauical Science, Japan Aerospace Exploraion Agency Eior s Noe Manuscrip submie 3 January 006; revision receive 7 Sepember 006; accepe for publicaion 3 Ocober 006. This paper is open for wrien iscussion, which shoul be submie o he IASS Secrearia no laer han Augus 007. SUMMARY Presenly i akes oo much ime an cos o consruc large space sysems such as he Inernaional Space Saion. Someimes we mus moify or aler he mission an configuraion of such sysem as require o comply no only wih echnical bu also social or poliical affairs. Therefore basic conceps for fuure large space sysems nee o be able o ajus o such various changes. Also he life cycles of such space srucural sysems nee o ake accoun of such change. In his paper, hierarchical moular srucure sysems suiable for such fuure srucures are propose. These sysems inclue srucural shapes wih properies of fracals. The propose srucural sysems are compose of ienically shape moules which are hierarchically assemble using sysemaic rules. They can sysemaically form srucures of various shapes an sizes. The geomerical aspecs of moular space srucures an wo-imensional an hree-imensional examples of hierarchical moular srucural sysems base on symmery group are escribe. Keywors Large Space, Hierarchical Sysems, Moular.. INTRODUCTION Generally i akes oo much ime an cos o consruc large an complex space sysems seen in he case of he Inernaional Space Saion. During consrucion of such srucures, some changes of iniial esigns or schemes are ineviable. Reasons for he changes are no only echnical bu also social or poliical affairs. Therefore, basic esign principles for fuure large space srucures nee o be able o ajus o such changes. Aapaion o environmenal changes means emporal ransiion of srucural configuraions from one phase o anoher. Environmenal changes involve bounary coniions, applie forces, sizes, an shapes of srucures. Temporal ransiion is a basic aspec of a life cycle concep. A life cycle of an arificial srucure consiss of all phases an emporal ransiions from o be planne o o be reuse. Research on life cycle engineering of inusrial goos an builings on groun has evelope remarkably. Recenly research on isribuion services an proucion sysems has Design Re-Arrangeable Evacuaion Self-Assemble Cycle B Reconfigurable Resorable Cycle A Consrucion Funcion Deployable Convenional Acive (Shape an Vibraion Conrol) Figure. A Life Cycle for Fuure Space Srucure Sysems

4 VOL. 47 (006) No. 3 December n. 5 also been carrie ou []. Bu in his research, human aciviies insea of srucural funcions change he phases of heir life cycles, because i is easy o irecly access an suppor hem. On he oher han, space srucures are no easily accessible. Therefore, space srucures mus have funcions which can change he phases of heir cycles. Figure shows a life cycle moel of fuure space sysems wih such funcions. There have been few suies concerning life cycle approach o space srucures []. In his paper, we propose hierarchical moular srucures for fuure space sysems, which inclue he srucures wih fracal properies. The propose srucures consis of a number of moules wih ienical shapes which are hierarchically assemble. They can form various sizes an shapes wih he same-shape moules assemble by sysemaical rules. We also inrouce assembly rules base on symmeric group heory. Such rules easily generae hierarchically symmeric srucures. Twoimensional an hree-imensional examples are emonsrae in his paper.. CONCEPT AND BACKGROUND OF HIERARCHICAL MODULAR STRUCTURES Aapive srucures can ajus o environmenal changes which may inclue changes o shapes, sizes, bounary coniions an applie forces. The ajusabiliy o environmenal changes is one of ypical feaures of hings in naure. One of he reasons ha srucural sysems of hings in naure are able o efficienly aap o environmenal changes coul be heir hierarchical sysems, since hierarchical sysems are able o creae sufficien variey using limie resources an assembly rules. Typical examples of hierarchical srucures in naure are srucures wih fracal properies [3]. Some mechanical properies of represenaive russ srucures wih fracal properies are escribe in [4]. The resuls inicae some avanages for large or complex srucures from he viewpoin of ajusabiliy o environmenal changes. Figure shows a concep of hierarchical moular srucures propose in his paper. The concep of hierarchical moular srucures is a combinaion of hierarchical srucures in naure an curren moular srucures. They consis of a number of ienically shape moules, which are hierarchically assemble. Sysemaical assembly rules are expresse in he following equaions. The assembly rules give hierarchical moular srucures useful geomerical properies a) geomerical symmery, b) exenibiliy of shapes an sizes, an c) regular openings. Hierarchical Things in Naure Self- Similar <Aapabiliy o Environmens> Diversiy, Appropriae Degrees of Freeom Hierarchical Moular <Geomerical Characerisics> Symmery, Exenibiliy, Moular Regular Openings <Tasks of Srucure Engineering> Fuure Robus, Regularizaion of Complex Sysems wih Muli-Degrees of Freeom PasWeigh Reucion, Opimizaion Moular Srucure Srucure Arifacs Figure. Concep of Hierarchical Moular G = A (M, M,, M), G = A (G, G,, G ). G k = A k (G k-, G k-,, G k- ) = A k (A k- (G k-,..., G k- ),..., A k- (G k-,..., G k- )) =... Equaion k where M is an iniial member, A is kh assembly k rule o generae a nex-generaion srucure an G is a kh-generaion srucure. This expression shows k ha one G is compose of some previous k generaions G - s. In paricular, a uniform k A generae he srucures wih fracal properies. The hierarchical moular srucures have he same avanages as he curren moular srucures like he Japanese engineering es saellie ETS-VIII [5]. For example, i is easy o prouce or es each one moule of he srucure, o ranspor he moules in he limie space like a rocke cargo, an o analyze mechanical behavior using some sub-srucuring mehos. However, he curren moular srucures

5 JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES IASS have some isavanages uplicae members increase heir weigh, an heir sysems are oo complicae ue o heir muliple egrees of freeom. The geomerical properies of hierarchical moular srucures make i possible o eliminae hese isavanages. In he following some geomerical configuraions of hierarchical moular srucures are shown an iscusse. 3. TWO-DIMENTIONAL GEOMETRICAL CONFIGURATIONS OF HIERARCHICAL MODULAR STRUCTURES Hierarchical moular srucures consis of a number of sysemaically assemble of ienically shape moules. Le us specify -imensional assembly rules. Le assembly rules A k in equaion () be replace by close-loop configuraions erive from roaion mappings. Close-loop configuraions have some avanages over serial or branching configuraions from he viewpoin of srucural engineering, as hey can easily form srucures wih geomerical symmery an higher siffness. They can also reuce influence from local amage or failure of heir members. The naure of he configuraions is bes illusrae by means of he following examples. One basic moule of regular a n-gon shape is generae by roaion mapping hrough an angle π /n abou a fixe poin on he symmery axis of he one imensional member (Fig.3 (a) ; n = 3). A secon generaion is generae by roaion mapping hrough an angle p /m abou a fixe poin on he symmery axis of he firs-generaion (Fig.3 (b); m = 3 ). Consiering roaion mappings an axes, we efine mahemaical expressions of a firsgeneraion an secon generaion as n 0 an m ( n q 0 ), respecively. q inicaes he symmery axis (arrows in Fig.3). A hir generaion is generae by roaion mapping hrough an angle p/ l abou a fixe poin on he symmery axis of he secon generaion (Fig.3 (c); l = 3 ). We 3 express his hir-generaion as ( l q 3 mq ( n 0 ). In he same way, we can generae an express following generaions. In paricular, 3 3 ( ( p 3p 3 0 ) shown in Fig.3(c) is known as he hirgeneraion Sierpinski-gaske, which is one of he famous shapes wih fracal properies. This inicaes ha our hierarchical moular srucures inclue oher srucures wih fracal properies. (a) s generaion G Size (b) n generaion M G = A ( M, M, M) = A ( G, G, G ) (c) 3r generaion 3 3 π[3 π[3 0]] (,, ) ( (,, ), ) Figure 3. Examples of Composiion of Hierarchical Moular Figures 4 an 5 emonsrae various 4h-generaion moels, which consis of 54 hexagonal moules. These plane examples emonsrae ha he propose configuraions provie he hree geomerical properies menione before (geomerical symmery, exenibiliy of shapes an sizes an regular openings). Moreover his figure inicaes ha our meho can be applie saring from a finie number of moules or from an overall srucural shape. This propery is helpful when esigning arificial srucures. In conras, he convenional expressions like he Schlafli symbol represen only local relaionship beween polygon eges an faces. G 3 M = A G G G 3 3 = A A G G G L 6363A 6363B 6363C 6363D 6363E G 6363H Figure 4. Examples of Hierarchical Moular Compose of Hexagonal Moules (Type 6363)

6 VOL. 47 (006) No. 3 December n A 6633B 6633C 6633D of a basic eraheron wih is symmerical plane. These hree projecions are erive from hree irecions in Fig.8(b), respecively. Replacing regular polygon by hese projecions an replacing symmerical axis by he projecion of symmerical plane, plane configuraions menione in he previous secion can be applie o spaial configuraions. This configuraion can be applie o any oher polyheron moules. 6633E 6633F 6633G 6633H Figure 5. Examples of Hierarchical Moular Compose of Hexagonal Moules (Type 6633) Figure 6 (a) shows firs moe frequencies of frame moels (4h generaions from Fig.4). In his case, connecive members beween moules are consiere. Toal number of he connecive members (Fig.6 (b)) an oal weigh (Fig. 6 (c)) are also shown. The figure inicaes ha he propose configuraions, because of heir variey of shapes, can escribe various hierarchical moular srucures wih various mechanical properies. 4. GEOMETRICAL CONFIGURATIONS OF THREE-DIMENTIONAL HIERARCHICAL MODULAR STRUCTURES The generalizaion of our configuraion of hierarchical moular srucures ino hree imensions is invesigae in his secion. Roaion mapping mus be exene o he finie roaion group in Eucliean 3-imensional space [6]. Subsiuing assembly rules base on hreeimensional group ino he assembly rules A k in equaion gives us 3-imensional hierarchical moular srucures. Finie roaion group involves he hree subgroups he cyclic group, he iheral subgroup, an he polyheron group. Le us specify he assembly rules in 3D space. The naure of our meho is bes illusrae by means of examples base on each subgroup. Here we rea a regular polyheron as a basic moule of firs generaion. Figure 7 shows secon an hir generaions wih eraheral moules base on he cyclic group. This paern is he so-calle Waxman ype of ocaheral plane russ srucures. In his case, using projecion a spaial configuraion is simplifie o a plane configuraion. Figure 8(a) shows hree projecions frequency [HZ] number oal weigh[kg] Type 6363 Type 6633 (a) firs moe frequencies (b) number of connecive members (c) oal weigh Figure 6. Firs Moe Frequencies of Frame of Type 6363 an Type 6633 (a) n generaion 4 π/4[4 0] (b) 3r generaion 4 3 π/4[4 π/4[4 0]] Figure 7. n an 3r examples base on he cyclic groups

7 JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES IASS projecion A projecion projecion n generaion s generaion 3r generaion (a) projecions of a eraheron irecion A irecion B reflecion plane 4h generaion reflecion plane 5h generaion irecion C (b) irecions of Figure 8. Projecion for 3-imensional hierarchical moular srucures base on he cyclic groups Figure 9 shows a configuraion wih ocaheral moules base on he iheral group. Saring from a firs generaion, ierae reflecions can provie he following generaions. This paern also leas o anoher ype of ocaheral plane russ srucures. In he polyheron group symmery of he verexes is foun in he regular polyheron. An expression of a firs generaion shape eraheron is 4, which means ha 4 verexes form he eraheron symmery. By he same rule, expressions of cube, ocaheron, oecaheron, an icosaheron are 8 c, 6 o, 0, an i, respecively. Nex, for example, 4( 4 ) inicaes a secon generaion which consiss of 4 eraherons place on each verex of a larger eraheron. Figure 0 illusraes secon generaions prouce by mapping 4 [] base on he eraheral symmery group. Repeaing his operaion prouces he nex generaion. Two examples of hir generaions prouce by mapping 3 4 [] are shown in Fig.. These configuraions can be applie o he oher regular polyheral symmery groups. Table show secon generaions erive from various mappings an polyheral firs generaions. In paricular, secon generaions marke # in Table are he hree imensional hierarchical srucures wih fracal properies. Figure 9. 3-imensional hierarchical moular srucures base on he iheral groups 4 [4 ] 4 [8 c] 4 [6 o] 4 [0 ] 4 [ i] Figure 0. Secon generaion moels of 3-imensional srucure generae by mapping 4 []

8 VOL. 47 (006) No. 3 December n. 5 (a) 4 3 [4 [4 ]] (b) 4 3 [4 [6 o]] Figure. 3r-generaion moels of 3-imensional srucures Table. Secon Generaion of Three Dimensional Hierarchical Moular base on regular polyheron group eraheron cube ocaheron oecaheron icosaheron s generaion mapping for n generaion 4 eraheron 8 c cube 6 o ocaheron 0 oecaheron i icosaheron 4 [] eraheron 8 c[] cube 6 o[] ocaheron 0 [] oecaheron i [] icosaheron êë úû 4 8 êë cúû * 4 6 êë oúû * 4 0 êë úû * 4 4 [4 ] 4 [8 c] 4 [6 o] 4 [0 ] 4 [êë i] iúû * 4 4 *# 8 4 êë úû 8 c[4 ] 66 o[4 o 4 êë úû] 8 c 8 êë cúû # 8 c 6 êë oúû 8 c 8 c[8 c] 8 c[6 o] 8 êë úû 8 c c[0 ] 8 c[ êë ii] úû êë úû êë úû # êë úû 66 o[8o 8 cc] 6 6o[6 o 6 oo] 66 o[0o ] 6 6o[ o i i] 0 4 êë úû 0 8 êë cúû 0 6 êë oúû 0 0 êë úû # 0 [4 ] 0 0 o [8 c] 0 [6 o] 0 [0 ] 0 [ êë ii] úû i 4 êë úû i 8 êë cúû 6 êë oúû i i[4 ] i[8 c] i[6 o] i[0 êë ] úû i[ i êë i] iúû # * Mapping for n generaion is base on a eraheoran(see Fig. 0) # Relaion beween mapping an mappe polyheron is self-similar êë úû

9 JOURNAL OF THE INTERNATIONAL ASSOCIATION FOR SHELL AND SPATIAL STRUCTURES IASS ( ) 3 The moel expresse as 4 4() 4 (Fig.0(a)) is equal o a hir generaion of spaial Sierpinskigaske. We can generae a k-h generaion expresse as l k polygon(k)(m k- polygon(k-)( (n polygon()))), which provies a k-imensional able of k-h generaions. These spaial examples also prove ha he propose configuraions have he imporan geomerical properies menione before geomerical symmery, exenibiliy of shapes an sizes, an regular openings. In he hree imensional case, our configuraions can sar from a finie number of moules or from an overall srucural shape, since he shape expressions can explicily inicae he number an shape of each generaion. This propery is useful when esigning arificial srucures. In conras, he convenional expressions like Shlafli symbols an he Miller inices in crysallographic suy [7] represen only a local relaionship beween polyheral eges an faces. 5. CONCLUSION This paper shows a concep an geomerical configuraions of hierarchical moular srucures, which consis of sysemaically assemble ienically shape moules. The concep of hierarchical moular srucures is inspire by hierarchical srucures in naure which have he capabiliy o ajus o environmenal changes. The aapaion o environmenal changes will be require for fuure large space srucures from he viewpoin of heir auonomous life cycles. Also shown were geomerical configuraions of - imensional an 3-imensional hierarchical moular srucures. Geomerical configuraions of -imensional hierarchical moular srucures are hierarchical close-loops of polygonal moules. The assembly rule is base on a roaion mapping in plane. The srucures generae from hese configuraions have imporan an useful geomerical properies geomerical symmery, exenibiliy of sizes an shapes, an regular openings. These properies can eliminae some rawbacks of he curren moular srucures an provie some avanages for fuure large space srucures. The propose configuraions inclue shapes which have properies of fracals. Geomerical configuraions of 3-imensional hierarchical moular srucures are hierarchically assemble regular polyheral moules. The assembly rule is base on he finie roaion group in 3-imensional Eucliean space, which is equivalen o roaion mapping in -imenional plane. The finie roaion group inclues he cyclic subgroup, he iheral subgroup, an he polyheron subgroup. Some examples generae by mapping base on each subgroup are shown. They prove ha hese 3-imensional configuraions also have he same geomerical properies as hose of he -imensional case. The propose configuraions again also inclue shapes which have properies of fracals. REFERENCES [] Kimura, F., Life Cycle esign for inverse manufacuring, Proceeings of he EcoDesign Conference, 999, pp [] Naori, M.C., Huang, S., Umeani, Y., Higuchi, K., Nakano, S., an Kakua, H., Basic conceps on objec-oriene archiecure of inelligen aapive srucures, Fifh Inernaional Conference on Aapive, 994, pp [3] Manelbro, B.B., The Fracal Geomery of Naure, W.H.Freeman an CO., 98. [4] Kishimoo, N., an Naori, M.C., Basic consieraion of srucures wih fracal properies, Forma, 5, 000, pp.3-9. [5] Misugi, J., an Yasaka, T., A moular approach o buil a large space anenna, 4n Congress of he Inernaional Asronauical Feeraion, 99, IAF [6] Weyl, H., Symmery, Princeon Universiy Press, 95. [7] Ashcrof, N.W. an Mermin N.D., Soli Sae Physics, Harcour, 976.