Folding to Curved Surfaces: A Generalized Design Method and Mechanics of Origami-based Cylindrical Structures

Transcription

1 Supplementay Infomaton fo Foldng to Cuved Sufaes: A Genealzed Desgn Method and Mehans of Ogam-based Cylndal Stutues Fe Wang, Haoan Gong, X Chen, Changqng Chen, Depatment of Engneeng Mehans, Cente fo Nano/Mo Mehans, AML Tsnghua Unvesty, Bejng 00084, Chna Columba Nanomehans Reseah Cente, Depatment of Eath and Envonmental Engneeng, Columba Unvesty New Yok, NY 007, USA Coespondng autho. Tel/Fax: ; Emal: henq@tsnghua.edu.n (C.Q. Chen)

2 Cylndal sufaes geneated usng the Method Fgue S Some ylndal sufaes by the n-plane method. (a) Type-: taget plane uves (a le and ellpse), 3D ogam onfguaton and D ease patten. (b) Type-: taget plane uves (a hypebol uve and staght lne), 3D ogam onfguaton and D ease patten. () Type-: taget plane uve (an equangula spal), 3D ogam onfguaton and D ease patten. (d) Type-: taget plane uve (an ellpse), 3D ogam onfguaton and D ease patten. (e) In 70 (4 adans), moe ponts pked make the appoxmaton moe auate. Whle ths also leads lage to the fundamental elatons of equaton (), smalle p pehosen. Aodng ae then obtaned, leadng the tapezods fomed flatte. Cteons of geomet ompatblty fo thk Mua-o The neessay and suffent ondtons fo lne segment AB to nteset wth DE, as an be seen n Fg. b, ae:

3 AD AD AB, DE (S) os os Fo an Type- Mua-o shown n Fg., wth onstant paametes,, l, and l, equaton (S) has the fom of: l os l, os l3 l, os os l ' l l '' l ''' (S) wth ' l lsn sn, '' ' l l los l os and ''' ' l l los l os beng sde lengths of the quadlateals (see Fg.Sa). Upon futhe smplfaton, we obtan the onstants fo, and l l, as shown n equaton (4). Fo an Type- Mua-o shown n Fg. d, All the quadlateals ae onguent. As shown n Fg. Sb, beause of symmety of the soseles tapezods, thee ae 3 ases afte foldng and uttng off mateals. The fst ase CO s O D, n whh the fnte-thkness plate mantan geomet ompleteness. Ths eques that os. The seond ase s CO OD n whh thee wll be a l3 l4 noth left. Ths eques os l l 4os aodng to equaton (S). Whle the 3 4 last ase s that pont O loates on extenson of DC, whh means BC wll be ut off and geomet nfomaton wll be lost(length of the bottom edge l 4 and heght of the tapezod h ). Note that all the above umstanes ae dsussed when 45. Coespondng examples ae shown n Fg. d. Fgue S types of fold pattens when foldng thk Mua-o. (a) Type- Mua-o, n whh,, l and l ae ndependent paametes. (b) Type- Mua-o. Note that quadlateals ABCD and

4 ADEF ae onguent soseles tapezods. The poston of nteseton pont O (o O on AF ) afte foldng detemnes ompleteness of ths type of thk Mua-o. Geomet elatonshps, ompatblty and Posson s ato of OCSs The 4-ease ogam patten studed n ths pape has sngle DOF. Wth zeo-thkness models, fou fold lnes of the OCSs meet at one vetex, atng as evolute jonts to make the ogam at as spheal lnkages (Fg. S3). Aodng to spheal tgonomety, Geomet elatonshps between the lne angles and dhedal angles an be desbed as: f f os f os os os sn sn 3 3 os f = os os os sn sn (S3) whee f -4 ae dhedal angles fomed plates of folded-state ogam, -4 oespondng lne angles, -4 ae ae auxlay dhedal angles between sufae AOC and fou plates DOA, AOB, BOC and COD, espetvely, and s the lne angle AOC. The fou auxlay angles an be wtten as: os os osos sn sn 4 os os os os sn sn 3 os os osos sn sn os os os os sn sn (S4) Fo Mua-o patten, lne angles ae ethe equal o supplementay to eah othe, that s: =, 3 4 (S5) Whee and ae the onstant paametes of OCSs. By substtutng equatons (S4-S5) nto S3, dhedal angles have the elatons: f f, f f (S6) 3 4 Note that dhedal angles ae defned n the onstant 0 f, so we have

5 f f f = 3 4 os os tan os f (S7) Beause of and, and hange altenately along the umfeental deton, whle the dhedal angles keeps onstant, as an be seen n Fg. 3a and 3b. So, the supplementay angle s adopted to haateze the foldng poess, takng nto aount that neases as OCSs fold. Substtutng wth,, espetvely, the ental angle and ad have the elatonshps:, and wth Rsn lsn Rsn lsn (S8) Expessng all the vaables n tems of, equatons (5-6) ae obtaned. Fgue S3 Geomet elatons between foldng paametes. (a) and (b) Cease patten of a spheal lnkage and oespondng ts folded state. Relatons between -4 and -4 f ae expessed n equatons S3 and S4. () -, unt ental angle and ad, R of an OCS. To mantan geomet ompatblty, the shotest length of all the quadlateal sdes n OCSs should be postve (see Fgs. and 3), whh eques: sn l l os los 0 sn (S9)

6 Togethe wth the onstant 0, Fg. S4 ae obtaned aodng to dffeent atos l l. Fgue S4 Geomet ompatblty ondtons of OCSs. Admssble egons of and assoated wth dffeent values of l l. Note that the small egons ae subset of lage ones. The nset shows an nompatble example when, and l l ae mpopely olloated. In OCSs, the ouplng of somet defomaton n axal and umfeental detons an be desbed by the vaable /, whh s an analogy of physal quantty Posson s vz Rato. The OCS s stan n the detons ae defned as follows: z dw z W R dr d R dr d R R (S0) Cumfeental stan s vefed by the expesson of stan n the ylndal pola oodnates : u u (S) when dr d R d dr = R R R (S) In Eq. (S), u, u ae adal and umfeental dsplaement, espetvely. Mehans of OCS: gd foldng/unfoldng

7 Analytal expessons of both the adal and axal balaned foe ae obtaned wth the method of mnmum potental enegy (equaton (0)). Dffeent load pattens and bounday ondtons (Fgs. S5a and S5b) detemne the oespondng mehanal esponses of foldng/unfoldng poess (As a ompason, llustaton of the lne foe and bounday ondtons of nhomogeneous elast defomaton s also pesent n Fg. S5). Both of the poesses ae mplemented usng the 9 5 OCS model. Aodng to equaton (0), extenal wok ndued by the somet foe s: d T F d F d d nt 0 (S3) Fo the axal foe, the postve deton s defned n the ompessve/foldng deton, so the dsplaement: nt nt a W W mlsn os os (S4) Fo the adal foe, the postve deton s also defned n the ompessve deton, but note that a ompessve load leads an unfoldng poess n ths ondton. Smple suppoted bounday ondton s appled on the endponts of R, so the adal dsplaement: nt H H (S5) whee H s the se of the shell stutue (Fg. 3b), whh an be desbed as: os n H R R (S6) Note Equatons (5) and (6): os R l tan tan s / tan,, tan sn tan sn tan sn tan sn os Fnally we obtan the non-dmensonal adal foe: F F k nt sn lmn mng mn G sn (S7)

8 Fa nt sn Fa n n G n G k m sn sn sn (S8) Whee nt os os os os os G = sn sn nt sn os sn os sn os nt (S9) Fgue S5d shows the pedted F vesus of a 9 5 OCS wth =45, 60, and nt =50, espetvely. The OCS an be ompletely unfolded wth F neasng fom 0 to about 90. Whle as t folds, F neases shaply at 6.4. Futhe analyss of equaton (7) shows that as, 0,whh means F ( ). Ths phenomenon, theefoe, s alled a lmtng foldng state n ths pape. Note that doesn t nfluene the value of. Also nluded ae the FEM smulated esults (shown as blue nt dotted lne n Fg. S5d). Exellent ageement between the analytal and FEM pedtons s obtaned. Fgue S5e shows the vaaton of as vaes, whh gves a geomet ntepetaton of the exstene of. Futhemoe, by ontollng paametes m 9 and n 5, we gve a zeo equpotental sufae of as and hange (, should satsfy the losue ondtons) to gude alteng elast popetes and desgn paths of OCSs. As shown n Fg. S5f, evey pont on ths sufae ndates a lmtng foldng poston and the oespondng (, ). We an see fom the phase paagaph that the phase dagam has banhes, whh ndates one sngle pa (, ) may oespond to dffeent f foldng/unfoldng stats fom bottom/top, espetvely.

9 Fgue S5 Mehanal esponses of OCSs Fst 3 fgues ae dffeent load pattens and bounday ondtons, wth ed lnes fo foe ponts, aows fo foe detons and blue lnes fo boundaes, espetvely. (a) Radal load and smply suppoted boundaes. (b) Axal load and smply suppoted boundaes. () Radal lne foe and lamped boundaes fo defomaton of elast OCS. Fgues n the seond ow ae OCSs esponses unde adal loads. (d) Analytal and FEM F uve of a 9 5 OCS unde adal loads ( =45, 60, nt =50 ). 3 numeal foldng states at 0, nt =50 and 60 ae llustated nsde, espetvely. (e) uve of an OCS wth n 5, 45 and 60. Monotonty of detemnes exstene of. (f) The equpotental sufae of when ontollng n 5. Any staght lne pependula to the bottom plane plane) epesents a foldng/unfoldng poess. In the last ow ae addtonal nfomaton (

10 mehanal esponses of axal loads. (g) Equpotental sufae expands as neases n nt spae. (h) Equpotental sufae shnks as nt neases n spae. FEM models of gd foldng and nhomogeneous defomaton Fo the gd foldng/unfoldng poess of an OCS, models onsstng of gd plates and lnea elast hnges ae adopted. The popetes of a eased sheet (Refe to Ref. 5 n the man text) s detemned by the elast onstant k of the hnges,.e., 3 ( ), whee k Eh v 0.5 s the toson ato, E =4 GPa s the Young s modulus, 0.38 s the Posson s ato and h 0.005m s the thkness of the sheet. In both analytal expessons and FEM smulatons, sde length of the OCS unts L s set to be unt length fo non-dmensonalzaton. It should be ponted out that, to model the plates n gd foldng/unfoldng, a lage modulus of 00 GPa s employed n FEM smulatons. Moeove, to onnet lage-modulus plates wth lnea elast hnges and auately quantfy the hnges elast popetes n FEM, pogammng language PYTHON s used as a seonday development tool fo the smulaton n ABAQUS. Some of the FEM modelng esults ae show n Fg. S6b. The assoated PYTHON odes used to geneate the models ae avalable on equest. To ealze dffeent foldng o defomaton modes of the OCS, pope load pattens and bounday ondtons need be aefully adopted, whh ae shown n Fg. S5. Shea defomable shell elements ae used n the FEM. The equvaleny s ensued by assumng that an OCS and ts EHCS have the same amount of mateal, n addton to the same adus R and ental angle,.e., mn 4l l sn h = W R h (S0) o e whee h o and h e ae the thknesses of of the OCS and EHCS, espetvely. Fnally the thkness ato of EHCS to OCS n FEM s: h h e o tan tan nt nt s tan os (S)

11 (a) (b) Fgue S6 (a) Foldng/unfoldng poess of a 9 5 gd OCS n FEM, olos n whh epesent the dsplaement along x deton. (b) Modellng esults of lnea elast hnges n FEM. Evey lttle tangle epesents a dsete hnge n the mesh. Refeenes. Sato, K., Tsukahaa, A. & Okabe, Y. New Deployable Stutues Based on an Elast Ogam Model. J. Meh. Des. 37, 040 (05).. Landau, L. D. & Lfshtz, E. M. Theoy of Elastty. (986).