Static Analysis of Tensegrity Structures Part 1. Equilibrium Equations

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1 Sttc lss esegrt Structures Prt. Equlrum Equts Jul rre Jseph u rl. re III Reserch ssstt Grdute Reserch Pressr Pressr eter r Itellget ches d Rtcs, eprtmet echcl Egeerg, Uverst lrd, Gesvlle, 36 cmr@cmr.me.ul.edu strct I ths pper the mthemtcl mdel t perrm the sttc lss tprsm tesegrt structure suected t wde vret eterl lds s ddressed. he vrtul wr pprch s used t deduce the equlrum equts d methd sed the Newt s hrd w t ver the umercl results s preseted. I the secd prt the pper severl umercl emples re gve.. Itrduct esegrt structures re sptl structures rmed cmt rgd elemets (the struts) d elstc elemets (the tes). N pr struts tuch d the ed ech strut s cected t three cplr tes []. he struts re lws cmpress d the tes tes. he etre cgurt stds tsel d mts ts rm slel ecuse the terl rrgemet the tes d the struts []. esegrt s revt tes d tegrt. he develpmet tesegrt structures s reltvel ew d the wrs relted hve l ested r the 5 ers. Keer [3] estlshed the relt etwee the rtt the tp d ttm tes. e [] preseted prcedures r the geert tesle structures phscl d grphcl mes. Y [] ted Keer s results usg eerg csderts d ud the equlrum pst r the ulded tesegrt prsms. Ster [4] develped geerc desg equts t d the legths the struts d elstc tes eeded t crete desred gemetr. Sce eterl rces re csdered hs results re reerred t the ulded pst the structure. Kght [5] ddressed the prlem stlt tesegrt structures r the desg deplle te. I ths pper the prlem the determt the equlrum pst tesegrt structure whe eterl rces d eterl mmets ct the structure s ddressed.. Nmeclture gure shws tesegrt structure rmed struts ech e legth S. I ever structure t s pssle t det the tp tes, the ttm tes d the lterl r cectg tes whch re deted s, d respectvel. gure shws the sme structure. he ttm eds ech strut s leled csecutvel s E, E,, E,, E detes the rst strut d stds r the lst strut. Smlrl the tp eds the struts re leled s,,,,, rtrr ut ce t s chse t shuld t e chged. 3. Geerled rdtes d rsrmts trces. he select the rst strut s gure shws rtrr pt P lcted strut legth s. I reerece sstem whse s s lg the s the strut d wth ts rg lcted t the lwer ed the strut, the crdtes P re smpl (0,0,l). Hwever requetl s mre cveet r purpses lss t epress the lct P the gll reerece sstem.

2 p te ectg te Strut E E ttm te E () gure. Nmeclture r tesegrt structures. ) mpets; ) Strut eds. () E I the lwer ed e strut s cstred t mve the hrtl ple d ls the rtt ut ts lgtudl s s cstred, the strut c e mdeled uversl t. I ths w the t prvdes the 4 degrees reedm sscted wth the strut. he ttl sstem hs 4* degrees reedm whch mes there re 4* geerled crdtes. r ech strut the geerled crdtes re the hrtl dsplcemets, the lwer ed the strut tgether wth tw rtts ut the es the uversl t, ε d β. ε crrespds t the rtt the strut ut s d β crrespds t the rtt ut s, see gure. he lgmet the s the ed sstem wth the s the rd c e ccmplshed usg the llwg three csecutve trsrmts, [7]: trslt, t(,, 0), rtt ε ut the curret s s ( ) d rtt β ut the curret s ( ). he crdtes P epressed the gll reerece sstem re P P (),,0 ε β P l S β, t ε, () () gure. egrees reedm sscted wth e the struts tesegrt structure. ) rtrr pt the strut; ) Strut mdeled s uversl t.

3 ,,0 () cs ε s ε 0 ε (3) 0 s ε cs ε cs β 0 s β (4) s β 0 cs β β P ] [0 0 l (5) Susttutg the ve three epresss t () elds P l s β l s ε cs β l csε cs β (6) I ddt t the cstrt mpsed tht the lwer eds re t rem the hrtl ple d r ech strut t vd the rtt ut ts lgtudl s the llwg ssumpts re mde wthut lss geerlt: he eterl mmets re ppled lg the es the uversl ts. he struts re mssless. ll the struts hve the sme legth. Ol e eterl rce s ppled per strut. here re dssptve rces ctg the sstem. ll the tes re tes t the equlrum pst,.e., the tl legths the tes re greter th ther respectve ree legths. he ree legths the tp tes re equl. he ree legths the ttm tes re equl. he ree legths the cectg tes re equl. he stess ll the tp tes s the sme. he stess ll the ttm tes s the sme. he stess ll the cectg tes s the sme. 4. rdtes he Eds the Struts he rtes crdtes the lwer eds E, epressed the gll reerece sstem, re ted terms the geerled crdtes susttutg l (6) 0, d replcg d, E (7) 0 Smlrl the crdtes the upper ed the struts re evluted replcg l the legth the struts S (6)

4 4 s s β s s ε cs β (8) s csε cs β Equts (7) d (8) permt e t t epresss r the legths the tp, ttm d lterl tes terms the geerled crdtes s llws / (,, ) (,, ) (,, ) / ( E, E, ) ( E, E, ) ( E, E, ) / ( E ) ( E ) ( E ) ( (,,,,,, ) (9) ) (0) ) () the 5. he Prcple Vrtul Wr r esegrt Structures he vrtul wr r sstems le t stre ptetl eerg c e stted rm [6] W W c W c () W s the ttl vrtul wr, Wc s the vrtul wr perrmed r -cservtve rces d mmets d Wc s the vrtul wr perrmed cservtve rces. Wc c e represeted s W W W (3) c W s the ttl vrtul wr perrmed -cservtve rces d W s the ttl vrtul wr perrmed -cservtve mmets. he vrtul wr perrmed the cservtve rce, Wc s Wc V V s the ptetl eerg sscted wth the cservtve rce, therere the ttl ctrut the cservtves rces Wc s W c V (4) V s the summt ver ll the V preset the structure. Susttutg (3) d (4) t () elds W W W V (5) I equlrum the vrtul wr descred (5) must e er [6], the the equlrum cdts c e deduced rm W W V 0 (6) 6. he Vrtul Wr ue t the Eterl rces s t s ssumed tht there s l e eterl rce ctg ech strut, the vrtul wr perrmed ll the eterl rces s gve W r s the eterl rce ctg the strut d r s the vrtul dsplcemet r, ths s the vectr tht ges rm the rg the gll reerece sstem t the pt pplct the eterl rce. I the dstce etwee the pt pplct the rce d the lwer ed the strut s, see gure 3, the epress r r the gll sstem c e ted rm (6) replcg l d ts rectgulr crdtes re r r r r s β s ε cs β csε cs β W (7) (8)

5 5 r gure 3. ct the eterl rce ctg the strut. he vrtul dsplcemets c e deduced rm (8) the dervtves re te wth respect t the geerled crdtes ε, β, d r r r r cs β β csε cs β ε s ε s β β s ε cs β ε csε s β β (9) Susttutg (9) t (7), the geerl epress r the vrtul wr perrmed eterl rces s gve W ( [ csε cs β s ε cs β ] ε [ cs s s cs s ] β ε β ε β β ) (0) 7. he Vrtul Wr ue t the Eterl mets W he vrtul wr perrmed the eterl mmets s gve ε ε β β Prvded tht ths mdel the tesegrt structure the eterl mmets c e eerted l lg the s the uversl t, ε s cller wth ε d β s cller wth β, see gure 4, therere () s smpled t W ε ε 8. he Ptetl Eerg β β Sce the struts re csdered mssless the term relted t the ptetl eerg the prcple vrtul wr s the resultt the elstc ptetl eerg ctruts gve the tes. he ptetl elstc eerg r geerl te s gve, [6] () ()

6 6 β β ε ε, gure 4. Eterl mmets ppled t e the struts tesegrt structure. 0 ) ( w w V (3) V s the elstc ptetl eerg r te, the te stess, w the curret legth the te d w the ree legth the te. herere the deretl the ptetl eerg r te s w w w V ) ( 0 (4) he deretl the ptetl eerg r ll the tesegrt structure, V, s the resultt the ctruts the tp tes, the ttm tes d the lterl tes d c e epressed s ( ) ( ) ( ) V (5) re the stess the tp, ttm d cectg tes respectvel, d re the ree legths the tp, ttm d cectg tes respectvel d d re gve (9), (0) d () d re ucts sme the geerled crdtes.,, 0 0, 0, 9. he Geerl Equts Nw tht ech e the terms ctrutg t the vrtul wr hs ee evluted, the equlrum cdt r the geerl tesegrt structure c e estlshed. Susttutg (0), () d (5) t (6) d re-grupg elds β β β ε ε ε (6) ( ) ( ) ( ) (7) ( ) ( ) ( ) (8) [ ] ( ) ε β ε β ε ε cs s cs cs ( ) ( ) ε ε (9)

7 7 3 [ cs β s ε s β csε β ] β s,,..., ( ) ( ) ( ) β β β (30) Equt (6) must e stsed r ll the vlues the vrtul dsplcemets whch geerl re deret rm er, the s gve equts (7) t (30). Equts (3) represet strgl cupled sstem 4* equts depedg l the 4* geerled crdtes. he equlrum pst r geerl tesegrt structure s ted slvg umercll the set (3) r,, ε, β,...,,, ε, β. ter tht equts (7) d (8) eld eplctl epresss r the crdtes the eds the struts the gll crdte sstem. 0. Itl dts e le t slve (3) t s ecessr t d prper set vlues r the geerled crdtes the ulded pst. hs s ccmplshed usg Y s methd [], whch s preseted here wthut pr. γ R ( R R ) s 0 (3) γ R ( R R ) s 0 (33) [ cs( α γ ) csα ] 0 R R (34) s R 0 d R 0 (35) γ γ s s d the gles γ d α re gve π π π γ d α (36) s the umer struts. he three uws R, R d the legth the cectg tes re slved usg equts (3) thrugh (34). hese vlues re used t slve the llwg set geerled crdtes r the ulded pst. ( ( ) ),,,... ( ( ) ),,,..., 0 R cs γ, (37), 0 R s γ, (38) tε,0,0 R s( ( ) γ α) (39) H R cs( ( ) γ α),0 t β,0 (40),0 R s( ( ) γ α) s ε,0 (3)

8 8 d H s R R RR s (4) the γ. Verct the Numercl Results ecuse the cmplet the equlrum equts t s essetl t ver the swers ted. depedet vldt the results c e ccmplshed usg Newt s hrd w. I there re eterl mmets ctg slted strut, t s sucet t perrm the summt mmets wth respect t the lwer ed the strut. I there re eterl mmets the verct prcess vlves mre steps d eeds sme ddtl ccepts. he uted Plücer crdtes le g tw te pts (,, ) d (,, ), s s the cse the rces the tes, c e wrtte the gll reerece sstem s, [8] $ˆ [ N P Q R] (4) N,, d N (43) P, Q, R (44) he sude (43) d (44) detes the ed the te ttched t the curret strut d the sude s r the remg ed the te. urther the crdtes the eds the tes c e evluted usg (7) d (8). I, d N re smulteusl equl t er (4) must e mded t $ˆ [ P Q R] (45) P Q R Whe eterl rce d ts pt pplct re w, the Plücer crdtes re ted $ (46) r crrespds t the eterl rce epressed the gll reerece sstem d r s gve (8). he Plücer crdtes c e epressed ew sstem tht s trslted d rtted wth respect t the gll reerece sstem. I the ew sstem s the sstem, ths s the sstem deed r the es the uversl t, the epress tht reltes the Plücer crdtes the sstem d the sstem s, [8] $ e $ (47) R O3 e (48) R 3 R (49) 0

9 9 0 0 R 0 cs ε s ε (50) 0 s ε cs ε d 03 s 3 3 eres mtr. gure 5 shws the ree d dgrm r rtrr strut mdeled wth uversl t. I ddt t the rces the tes there s eterl rce whch s w, rect rce R pssg thrugh the lwer ed d rect mmet R t the lwer ed. Newt s hrd w epressed Plücer crdtes the sstem s $ˆ $ˆ E $ˆ E E E $ˆ E E E E $ˆ E E E $ˆ E $ ε $ˆ ε β $ˆ β $ $ 0 (5) he cecets,, represet the mgtudes the rces the tes, d re gve *(w-w) s the stess, w the curret legth d w the ree legth the te. he curret legths re gve (9) d () r the tp tes d cectg tes respectvel. It shuld e ted tht the mgtude des t deped the reerece sstem whch s used. he uted Plücer crdtes $ˆ r ech e the tes c e clculted the sstem usg (4) thrugh (44) d the cverted t the sstem usg (47) thrugh (50). he Plücer crdtes the eterl rce ctg the curret strut $, c e evluted the sstem usg (46) d the cverted t the sstem wth the d (47) thrugh (50). ε d β re the mgtudes the eterl mmets d ther uted Plücer crdtes the sstem re gve $ˆ ε [ ] d $ˆ β [ ]. $ R Sce the rect rce $ R epressed the sstem s pure rce d the rect mmet epressed the sstem s pure mmet the hve the rm [ R R R ] d [ R R R ] respectvel. urther the re the l uws (5). ter epdg (5), rws ur d ve represet the cmpets the d drects the summt mmets ut the lwer ed the strut. uversl t ct eert mmet lg ts w es. I ter umercl evlut, R d R re th er, the the equlrum mmets s mted slel due t the rces the tes d t the eterl lds ( ) d therere the curret pst s equlrum pst. R R E E E EE β E ε R R E EE gure 5. ree d dgrm r rtrr strut mdeled wth uversl t.

10 0. cluss he mdel develped here llws e t le geerl t-prsm tesegrt structure suected t wde vret eterl lds. he mdel s develped usg the vrtul wr pprch d ll the results re checed usg the Newt s hrd w. hs verct ssures e tht the swers prduced the umercl methd ccurtel crrespd t equlrum psts. themtcl mdels r vrts the sc cgurt tesegrt structures such s the rerced tesegrt prsms mght e develped llwg the sme prcedure preseted ths reserch. cwledgemets he uthrs wuld le t grteull cwledge the supprt the r rce Oce Scetc Reserch (Grt Numer ) d the U.S. eprtmet Eerg (Grt Numer E- G04-86NE37967). Reereces. Y, J.P., lss r the esg Sel-eplle esegrt d Rerced esegrt Prsms wth Elstc es, Reprt r the eter r Itellget ches d Rtcs, Uverst lrd, Gesvlle, e, R.S., Reprt Iqur t he Estece, rmt d Represett esle Structures, ster Idustrl esg hess, Prtt Isttute, New Yr, Keer, H., Gedesc th d Hw t Use It, Uverst lr Press, erele d s geles, lr, Ster, I.P. evelpmet esg Equts r Sel-eplle N-Strut esegrt Sstems, ster Scece hess, Uverst lrd, Gesvlle, Kght,.. eplle te Kemtcs usg esegrt Structure esg, ctr Phlsph ssertt, Uverst lrd, Gesvlle, ught, S., echcs ches, Jh Wle & Ss, Ic., New Yr, re,., u, J., Kemtc lss Rt pultrs, mrdge Uverst Press, New Yr, Kut, K.H., Kemtc Gemetr echsms, Ord, New Yr, 978.