USING SYMMETRY FOR TENSEGRITY FORM-FINDING

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1 JOURNAL OF HE INERNAIONAL AOCIAION FOR HELL AND PAIAL RUCURE: IA UING YMMERY FOR ENEGRIY FORM-FINDING R. PANDIA RAJ n.d. GUE Deprmen of Engineering, Unieriy of Cmbrige, rumpingon ree, Cmbrige CB PZ, UK Grue uen, Reer, Eior Noe: e fir uor of i pper i one of e four winner of e Hngi Prize, wre for ouning pper re ubmie for preenion n publicion e nnul IA ympoium by younger member of e ociion (uner yer ol). I i re-publie ere wi permiion of e eior of e proceeing of e IA-APC ympoium: New Olympic New ell n pil rucure, el in Ocober in Beijing, Cin. UMMARY ymmery cn implify e form-fining proce for enegriy rucure; n i pper will ecribe one implificion ecnique. Our meo i be on e commonly ue force eniy meo, bu e clculion re one uing ymmery-pe coorine yem. e nr force-eniy meo ume known conneciiy for e rucure. A enion coefficien (enion iie by leng) mu en be foun for eery member o n equilibrium oluion i poible. Fining e nol coorine i rigforwr once uible e of enion coefficien i foun; bu fining uible enion coefficien my be non-riil. In i pper we implify e correc coice of enion coefficien by e ue of ymmery in iion o e conneciiy of e rucure, we ume e rucure cerin ymmery properie, grely reucing e ifficuly of fining poible configurion. e pper will ow imple emple of e meo were imple nlyicl oluion gie ll poible ymmeric enegriie wi gien conneciiy. Keywor: enegriy rucure, form-fining, re mri, group repreenion eory, ymmery-pe coorine yem.. INRODUCION enegriy rucure re rigiize by elf-re. e key ep in eign of ee rucure i formfining, e eerminion of elf-ree equilibrium configurion. Here, imple ecnique for enegriy form-fining i ecribe be on e commonly ue force eniy meo, bu e clculion re one uing ymmery-pe coorine yem. e nr force-eniy meo, preene in [], ume known conneciiy for e rucure. A enion coefficien (enion iie by leng) mu en be foun for eery member o n equilibrium oluion i poible. Fining e nol coorine i rigforwr once uible e of enion coefficien i foun; bu fining uible enion coefficien my be non-riil. We will ow uing ymmery cn elp.. EQUILIBRIUM AND ABILIY OF ENEGRIY RUCURE In e force-eniy meo, e equilibrium of e rucure i wrien uing re mri, (ofen known e force eniy mri), wic i efine follow. Conier wo noe i n j, poibly connece by member ij wic crrie enion coefficien ij. e coefficien of e re mri re ij ij ik if i j, if i j: ummion oer ll noe k connece o noe i if i n j re no connece () If e unknown nol coorine re wrien ree ecor,, y n z, n pplie nol force in e, y n z irecion re wrien p, p y, n p z, en unloe equilibrium configurion re oluion of e equion:

2 VOL. 7 () No. December n. c b Figure. Emple enegriy rucure wi D ymmery. () Iomeric iew, owing e noe n elemen numbering ceme ue. enion member re own by in line, compreion member by ick line. (b) Pln iew, owing e op ringle by e line, boom ringle by oli line, n e locion of ree C roion e,, b, c, ec of wic lie lf eig in e rucure. p y p z p y z () In orer for ree-imenionl rucure o ei, e equilibrium equion (Eq. ) mu e ree inepenen oluion, wic emele mu be inepenen of e uniform ecor [,,... ], wic will lwy be in e nullpce of ny properly efine re mri. u e enion coefficien in e member mu be coen uc nulliy of, i.e. i i rnk-eficien by. In iion, o gurnee e rucure i ble, we require o be poiie emi-efinie. In i pper we implify e correc coice of enion coefficien by e ue of ymmery. In iion o e conneciiy of e rucure, we ume e rucure cerin ymmery properie. We en wrie e re mri, uing ymmery-pe coorine yem i efine by e irreucible repreenion of e ymmery group o wic e rucure belong: e reuln re mri i imilr o, bu block-igonl form. e nulliy of e wole mri i now imply e um of nulliy of ec e ub-block of, n ence fining e require nulliy of i implifie.. IMPLEX ENEGRIY Conier e emple enegriy own in Fig. : we will ume i olly ymmeric e of elf-re were e enion coefficien re gien by for e igonl ru (member,, n ), for e orizonl cble (member ), n for e ericl cble (member 7, 8, n 9). e re mri (efine in Eq. ) i en gien by Eq..

3 JOURNAL OF HE INERNAIONAL AOCIAION FOR HELL AND PAIAL RUCURE: IA () Noe e re mri oe no epen on e cul configurion of e rucure. In e coenflie noion [] e rucure D ymmery i i rnforme ino n equilen configurion by i ymmery operion: e ieniy, E, roion by (C ) or roion by ( C ) bou e ericl i; wofol roion bou e ree e ( C ), b ( b C ), n c ( c C ). e irreucible repreenion of ymmery group [] own in ble, proie e men o fin ymmery pe coorine yem, ecribe in []. Applying meo ere gie n orogonl rnformion mri V, o, p V p, V were / V () V efine four ymmery ubpce. A i were lo, or coorine, re olly ymmeric uncnge by eery ymmery operion; for A, lo n coorine re preere by E, C n C, bu reere by C, b C n c C. E i wo-imenionl repreenion, woe ymmery ubpce ger nying no in A or A : i pli ino E(), quniie preere by C, n E(), quniie reere by C. e ymmery pe were y p y, p n, p z z cn be wrien V V () wic gie, e block form own in Eq. (). ble : Irreucible repreenion of ymmery group D D E C C C b C c C A A E A A E E

4 VOL. 7 () No. December n. ( ) ( ) ( ) ( ) () Eq. ow e block-igonlize re mri, coni of four inepenen ub-mri block, e A, A, E(), n E() block noe E() n E() re ienicl, wic i conequence of e ymmery. We will now conier ec of ee block eprely. A block A [ ] (7) A e lue of fir ( ) mri, mu be zero (nulliy ) for properly configure re mri, becue e um of ny row (or column) i zero, by efiniion. A A block [ ] (8) Wen (9) e econ block gie nulliy. E block E E ( ) ( ) () In orer o e nulliy of greer n zero for ec E() n E() block, e eerminn of E E n oul be zero. ( ) ( ) () (b) Figure. Emple enegriy rucure wi: () n (b). ^ ^ ^ ^

5 JOURNAL OF HE INERNAIONAL AOCIAION FOR HELL AND PAIAL RUCURE: IA Figure. enegriy rucure wi ymmery. e ouer ne of member own by in line (blue n yellow) re cble in enion n e inner br own by ick line (green) re ru in compreion. ( ) ( ) () o obin ol nulliy of, we nee o ify equion (9) n (), wi non-zero enion coefficien. e oluion re ± () ee oluion reul in ing nulliy of, n lo being poiie efinie. ere i coice in e lue of ; oweer, if we iionlly require bo orizonl cble n ericl cble o be in enion (Fig. ), wi n poiie, en we mu cooe. ( i lo li rucurl configurion, bu wi e role of igonl ru n ericl cble reere (Fig. b).) n eermine e re mri for e rucure (Eq. ) pr from ingle prmeer repreening e oerll mgniue of e e of elf-re. i prmeer oe no ffec e nullpce of, n ence i i poible o fin ble equilibrium configurion of e yem from e poible oluion of Eq... GROUP ENEGRIY RUCURE A econ emple, e rucure own in Fig. will be nlye. I i rucure wi poin group ymmery, e ymmerie of roion, bu no reflecion, of ereron. Le e enion coefficien ue o e prere be enoe by, n, for e cble in e ringle, e cble connecing e ringle, n e ru, repeciely. e re mri, cn be e up in erm of only ee ree enion coefficien. () were,

6 VOL. 7 () No. December n.,, n e ymmery group ree irreucible repreenion, e one-imenionl A, e woimenionl E, n e ree-imenionl. A wi e fir emple, i i poible o fin ymmery rnformion mri o gie e block-igonlize re mri, wic e rucure () () () E() E() A ()

7 JOURNAL OF HE INERNAIONAL AOCIAION FOR HELL AND PAIAL RUCURE: IA () () inefinie < () () poiie efinie > () inefinie ^ ^ / () > X X X X X X () () negie efinie < ^ ^ / Figure. oluion of Eq. 7,. e oluion lie on ree line, wic pli e plo ino region. Requiring, n ence o be poiie emi-efinie implie we re ineree in e oluion beween region () n (). Aiionlly requiring poiie, n negie, gie oluion mrke by croe. i ( ) mri; E A E i ( ); n re ( );,, n re ( ). A i gurnee o e nulliy of for ny properly conruce re mri.,, n re gurnee o be imilr o one noer, n ence if e nulliy of, en e ol nulliy of will be le e require. u, we require () For conenience, () cn be wrien A B C () n A, B, n C re foun o be. A... B C , n e oerll mgniue of re i no imporn, o we cn wrie Eq. A ^ ^ ^ ^ ( / )B ( / ) C (7) 7

8 VOL. 7 () No. December n. () (b) Figure. Poible configurion of enegriy rucure wi ymmery. e oluion o Eq. 7 re ploe in Fig.. Requiring o be poiie emi-efinie, wi cble crrying enion n ru compreion gie ingle line of poible oluion. Auming uible lue on i line, y. n.79, gie poiie emi-efinie re mri wi e require nulliy. e nullpce of i re mri ice poible coorine for i rucure, of wic one e gie e enegriy rucure own in Fig. (). i i no e only poible configurion, bu ny oer configurion mu be rece n roe erion of e rucure own. An lernie eign for. n.8, gie e enegriy rucure own in Fig. (b).. CONCLUION In i pper, i i own uing e re mri for ree-imenionl rucure o ei, e re mri mu e nulliy of four. e nullpce of e re mri en gie e configurion of e enegriy. Uing ymmery pe coorine yem mke i ey o ceck e rig nulliy. Furermore, i lo elp o fin e of enion coefficien ciee equilibrium configurion of enegriy rucure wic re prere ble n ence grely reuce e ifficuly of fining poible configurion.. ACKNOWLEDGEMEN R. Pni Rj grefully cknowlege e uppor of e Ge Cmbrige ru. REFERENCE [] iber, G., Pellegrino,., Reiew of formfining meo for enegriy rucure. In. J. pce ruc,, Vol. 8, No., 9. [] Biop, D. M., Group eory n cemiry, Doer Publicion, Inc. New York, 97. [] Almnn,. L., Herzig, P., Poin-group eory ble, Ofor: Clrenon, 99. [] Kngwi, R.D., Gue,.D., Pellegrino,., An inroucion o e nlyi of ymmeric rucure. Compuer n rucure, 999, Vol. 7,