Double Layer Tensegrity Grids

Transcription

1 Acta Poltechnca Hungarca Vol. 9, No. 5, 0 Double Laer Tensegrt Grds Tatana Olejnkova Department of Appled Mathematcs, Cvl Engneerng Facult Techncal Unverst of Košce Vsokoškolská, 0 00 Košce, Slovaka e-mal: tatana.olejnkova@tuke.sk Abstract: Ths paper descrbes the geometr of a double laer tensegrt grds assembled of three or four strut prsmatc cells. The elementar cells are self-equlbrated and so s ther assembl. The paper shows the creaton of a planar grds composed of elementar equlbrum and grds wth sngle or double curvatures composed of modfed equlbrum shapes. Kewords: tensegrt sstem; compresson; tenson; equlbrum; prsmatc cell; grd structures Introducton A tensegrt structure s a class of tenson structures consstng of tensle members and compressve members. In cvl engneerng, the lght-weght characterstc of tensegrt s recognzed as a sgnfcant advantage for space structures over conventonal structural sstems. Accordng to the defnton gven b Fuller (975), a tensegrt structure s a jonted structure consstng of contnuous tensle members (cables) and dscontnuous compressve members (struts). A tensegrt sstem s establshed when a set of dscontnuous compresson components nteracts wth a set of contnuous tensle components to defne a stable volume n space. However, t needs to be slghtl modfed, takng nto account the followng factors: the components n compresson are ncluded nsde the set of components n tenson, and the stablt of the sstem s self-equlbrum stablt. There are three tpes of tensegrt sstems: tensegrt grds, tensegrt frameworks and tensegrt domes. Ths artcle descrbes the geometrc relatons of tensegrt grds assembled of prsmatc cells. There are three orbts of members: horzontal cables, vertcal cables and struts. Each node s connected b two horzontal cables lng n a horzontal plane, one vertcal cable and one strut. The 95

2 T. Olejníková Double Laer Tensegrt Grds vertcal cables and struts connect nodes n dfferent horzontal planes. The members n each orbt are of equal length. The necessarl even number of nodes characterses the case of tensegrt sstems. If n s ths number, and f attenton s pad to the spatal case, the mnmal value of n s 6. Geometr of the Prsmatc Three-Strut Tensegrt Grds The geometr of a spatal retculate sstem s completel defned b ts relatonal structure and b the knowledge of the coordnates (k), (k), z(k), k =,,n, for ts n nodes n reference to a chosen as sstem. It s necessar for all nternal elements (struts) to have the same length s and for the eternal elements (cables) to have the same length c. If the length of the struts s nsuffcent, the set of envelope elements wll not have a defnte shape (the sstem s then knematcall ndetermnate). A frst geometr s defned as a trangular prsm (Fg. ). Ths sstem s unstable, and another shape can be defned (Fg. ) b relatve rotaton of the two trangles of cables n the parallel planes h 6 r 5 Fgure Fgure Trangular prsm Equlbrum, aonometrc and top vew For a gven value of the strut length, the totalt of the cable net takes a sngularl defnte shape, whch wll be referred to as null self-stress equlbrum geometr. The geometrc dstance between the nodes corresponds strctl to the length of manufactured elements. Two geometrcal ranges can be dentfed. If the relatonshp between the lengths of the struts and cables s rato s c, 67 and the relatve rotaton between trangles s 6. When the parameter r s specfed, then the length of the struts s s r, the length of the all cables s c r and the heght of the equlbrum s h r. 96

3 Acta Poltechnca Hungarca Vol. 9, No. 5, 0. B-Dmensonal Assembles Ths secton descrbes eamples of b-dmensonal assembles. Several juncton modes can be used: node on node, node on cable, cable on cable. In tpe a) n Fgure, the juncton s operated wth a sngle vertcal (bracng) cable: each of two ends of the strut les n dfferent horzontal planes, and t wll be used n a plane confguraton leadng to a double laer grd, n tpes b) and c), the juncton s operated wth horzontal cables. a) b) c) Juncton: node on vertcal cable Juncton: node on horzontal cable Fgure Juncton modes node on cable A planar double laer tensegrt grd (Fg. 5) s created b usng three-struts cells va a node on cable juncton of tpe a) (Fg. ). Fgure Three cells juncton n mode node on vertcal cable of tpe a) The node coordnates k, k, zk of one cell n the planar grd are n (): k k r cos k, k k r sn k, zk for k,, for k,5,6 0 k k - 0, h 0 k k -, h h r h () 97

4 T. Olejníková Double Laer Tensegrt Grds where the parameter k, r s a radus of a trangle created b the three 0 horzontal cables n the cell, h s a heght of the cell (Fg. ). Fgure 5 Planar double laer tensegrt grd b three-struts cells The node coordnates j,k,, j,k,, j,k, z of all cells n the planar grd dsplaed n Fgure 5 are j,k k d j,, j,k k d j, z, j,k k, z () j j j j d (j) v () cos( ) - v() sn( ), d(j) v () sn( ) v() cos( ) () where,...,6,j,...,6,k,...,6 (), and r v, v v v, coordnates k, k, zk 5 9 v 0, v v, v v, v v, v v, v v, v v, v 6 v, v 6 v, v 7 v, v 7 8v. r, v are epressed n v, v, B mantanng the prncple of elementar self-stressed cells, t s possble to modf the equlbrum shape so as to generate a double curvature sstem and sphercal surface wth a radus R. The elementar cell must be modfed b the () 98

5 Acta Poltechnca Hungarca Vol. 9, No. 5, 0 parameter k 0 n the equatons () of the node coordnates of the upper trangles, for k,5,6,,..., 6, j,..., 6. The transformaton equatons of the node coordnates of the lower laer of the planar grd to the sphercal surface wth a radus R and centre S(0,0,-R) for k,,,,...,6,j,...,6 and parameter k n equatons () are: z 0 z P z, j,k,, j,k,, j,k, j,k,, j,k,, j,k, j,k R cos u, j,k cos v, j,k, j,k R cos u, j,k snv, j,k, j,k R snu, j,k R P where d u, j,k v, j,k sgn, j,k, j,k, d R, j,k, j,k, j,k arccos d coordnates j, k,, j, k, j,k, j,k, are epressed n equatons (). The transformaton equatons of the node coordnates of the upper laer of the planar grd to the sphercal surface wth a radus R+h and a centre S(0,0,-R) for k,5, 6 and R h parameter k0 n equatons () are R z, j,k ( R h) cos u, j,k cos v, j,k, j,k ( R h) cos u, j,k sn v, j,k, j,k ( R h) snu, j,k R, (5) (6) (7) where d v, j,k, j,k, j,k, u, j,k, j,k sgn, j,k arccos d, j,k, j,k d, j,k R h, (8) Fgure 6 shows the grd composed of three-strut cells transformed to the sphercal surface. Fgure 7 llustrates the transformaton of the pont P k to the pont P, j,k, both located n the tangent plane of the sphercal surface and ts P, j,k located on the sphercal surface. Fgure 8 transformaton to the pont dsplas a determnaton of the parameter d, j, k and angles, j, k v, j, k, whch are used n equatons (8). u and 99

6 T. Olejníková Double Laer Tensegrt Grds, j,k P, j,k Fgure 6 Double curvature tensegrt grd sstem P, P k u(,j,k) P P, j,k j,k z 0 d(,j,k) v(,j,k) S Fgure 7 P, j,k Transform.: Pk P, j,k Fgure 8 Determnaton of d(,j,k),u(,j,k),v(,j,k) In Fgure 9 are dsplaed the nodes and cables of the trangles of the bottom laer on the sphercal surface. Fgure 0 dsplas the trangles of the bottom laer of the tensegrt grd located on the tangent plane of the sphercal surface, and n Fgure are these trangles placed on the sphercal surface together wth a sphercal surface. Fgure 9 Nodes and cables of the bottom laer of the sphercal grd 00

7 Acta Poltechnca Hungarca Vol. 9, No. 5, 0 Fgure 0 Trangles on the tangent plane Fgure Trangles on the sphere Geometr of the Prsmatc Four-Strut Tensegrt Grds Geometr of the four-strut cell s dependent upon onl one parameter a (Fg. ). When the topolog s defned, then the geometr s qualfed b the whole set of coordnates, whch s closel related to the self-stress equlbrum. When the parameter a s specfed, then the length of the struts s s a 0, the length of the cables of the bottom laer s c a s c a and the heght of the equlbrum s h a 5., length of the cables of the upper laer h 5 a Fgure Aonometrc, top vew and frontal vew of the -strut cell Planar double laer tensegrt grd s created b usng of four-struts cells b node P k k, k, z k for k,,, on node juncton. The node coordnates and k 5,6,7,8 of one cell n the planar grd are epressed n equatons (9) 0

8 T. Olejníková Double Laer Tensegrt Grds for k,,, and P k k -, z k k k, k, zk a cos k a, a sn k a, zk for k 5,6,7,8 P and k k -, z k k k, k, zk a cos k a, a sn k a, zk a 0, (9) where parameters a, h are llustrated n Fgure. A planar double laer tensegrt grd s created b usng four-struts cells b node on node juncton. The coordnates of the nodes n the planar grd for,...,, j,...,,k,..., 8 are n equatons (0) z, j,k k a,, j,k k a j,, j,k z k, where k, k, z k (0) are coordnates of the nodes of one cell epressed n (9). Fgure shows the planar double laer grd composed of 9 four-strut cells (top vew and frontal vew). Fgure Top and frontal vew of the planar double laer tensegrt grd of cells 0

9 Acta Poltechnca Hungarca Vol. 9, No. 5, 0 B mantanng the prncple of elementar self-stressed cells t s possble to modf the equlbrum shape so as to generate a sngle curvature tensegrt grd sstem (on the clndrcal surface). Then the node coordnates are epressed n transformaton equatons (), where the nodes located n the plane are transformated to the nodes located on the clndrcal surfaces wth rad R and P k P, j,k,,...,7, j,...,7 R+h, z, j,k kcos z k R sn,, j,k k a j,, j,k k sn z k R cos R, where parameter a for nodes k, for k 6,8 b a h R llustrated n Fgure, parameter 6 arctan a R () P n (0) s modfed on, R s radus of the curvature of the sngle curvature sstem., where Fgure Sngle curvature double laer tensegrt grd of 77 four-strut cells In Fgure s dsplaed a modfed four-strut cell and n Fgure 5 a modfed one. Fgure dsplas a sngle curvature double laer tensegrt grd contanng 77 modfed four-strut cells. The node coordnates of the grd wth double curvature (created surface wth two curvatures R and R ) are z, j,k kcos zk R sn,, j,k k,, j,k k sn z k R cos R, () 0

10 T. Olejníková Double Laer Tensegrt Grds 7 7 b 8 b 6 8 b 6 z Fgure modfed four-strut cell, j,k, j,k, j,k, j,kcos z, j,k R sn, j j, j,k, j,ksn z, j,k R cos R, a 5 j j Fgure 5 modfed four-strut cell where parameter a for nodes k, f k 6,8 b a h R, and for nodes P k, f k 5, 7 on b a h R arctan a r, arctan a r, 6, 6 j () P n (0) s modfed on,, where r sgn R and r sgn R are the rad of the curvatures of the double curvature sstem wth ts orentaton determned b the parameter sgn,. In Fgure 6 s dsplaed a double curvature double laer tensegrt grd contanng 77 modfed four-strut cells, where the rad of curvature are the same sze and the same orentaton r r. Ths grd has the form of a sphercal surface. 5 Fgure 6 Double curvature double laer tensegrt grd 77, r r 0

11 Acta Poltechnca Hungarca Vol. 9, No. 5, 0 In Fgure 7 the tensegrt grd wth the same rad but wth opposte orentaton r has the form of the translatonal surface created b the translaton of the r crcle wth radus R along the crcle wth radus R. In Fgure 8 the tensegrt grd wth dfferent rad and opposte orentaton r r has the form of the translatonal surface created b the translaton of the crcle wth radus R along the crcle wth radus R. Fgure 7 Double curvature double laer tensegrt grd wth rad r r Fgure 8 Double curvature double laer tensegrt grd wth rad r r 05

12 T. Olejníková Double Laer Tensegrt Grds Conclusons Ths paper amed to provde some eamples of planar or non-planar prsmatc tensegrt grd sstems. Some are onl geometrcal studes wthout equlbrum consderatons. It s not eas to defne tensegrt sstems. It could be clamed that everthng n the unverse s tensegrt wth propertes related to the contnuum of tensoned components. Tensegrt structures are the most recent addton to the arra of sstems avalable to desgners. The concept tself s about eght ears old and t came not from wthn the constructon ndustr but from the world of arts. Although ts basc buldng blocks are ver smple a compresson element and a tenson element the manner n whch the are assembled n a complete, stable sstem s b no means obvous. The dea was adopted nto archtecture n 960 when Macej Gntowt and Macej Krasńsk, archtects of Spodek, a venue n Katowce n Poland, desgned t as one of the frst major structures to emplo the prncple of tensegrt. The roof uses an nclned surface held n check b a sstem of cables holdng up ts crcumference. Another eample of a practcal mplementaton of the tensegrt sstem s Seoul Olmpc Gmnastcs Arena desgned b Davd Geger n 980 for the 988 Summer Olmpcs. The Georga Dome, whch was bult for the 996 Summer Olmpcs, s a large tensegrt structure of smlar desgn to the aforementoned Gmnastcs Hall. Acknowledgement Ths work was supported b the Slovak Grant Agenc of Mnstr of Educaton of the Slovak Republc wthn the project VEGA No. /0/ Theoretcal and epermental analss of adaptve cable and tensegrt sstems under statc and dnamc stress consderng the effect of wnd and sesmc. References [] Olejíková T.: Geometr of Tensegrt Sstems, Proceedngs of Scentfc Works Innovatve Approach to Modelng of Intellgent Constructon Components n Buldng 00, Košce, 00, pp [] Motro R.: Tensegrt. Structural Sstems for the Future. Kogan Page Lmted, London and Sterlng, 00 zob0sc&o=fnd&pg=pr7&dq=tensegrt:+structural+sstems+for+the+ Future&ots=85AsVSIo_l&sg=amBKpKIADZVPlpc0vVQ0g0OUI#v=o nepage&q&f=false [] Burkhardt R. W.: A Practcal Gude to Tensegrt. Cambrdge, USA, 008, MA [] Kmeť S., Platko P., Mojds M.: Desgn and Analss of Tensegrt Sstems, The Internatonal Journal Transport & Logstcs, Vol. 8, 00, pp