Nonlinear Programming Approach to Form-finding and Folding Analysis of Tensegrity Structures using Fictitious Material Properties

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1 Submtted to Int. J. Solds and Structures Nonlnear Programmng Approach to Form-fndng and Foldng Analyss of Tensegrty Structures usng Fcttous Materal Propertes *M. Ohsak¹ and J.Y. Zhang 2 1 Department of Archtecture, Hroshma Unversty, Kagamyama 1-4-1, Hgash-Hroshma , Japan. Currently, Department of Archtecture and Archtectural Engneerng, Kyoto Unversty, Kyoto-dagaku Katsura, Nshkyo, Kyoto , Japan 2 Department of Archtecture and Urban Desgn, Nagoya Cty Unversty, Japan *Correspondng author: ohsak@arch.kyoto-u.ac.jp Abstract An optmzaton approach s presented for form-fndng of tensegrty structures. It s shown that varous equlbrum shapes can be easly found by solvng a forced-deformaton analyss problem formulated as a mnmzaton problem consderng the nodal coordnates as desgn varables. The objectve functon s defned n terms of the member lengths, and t can be regarded as the total stran energy correspondng to fcttous elastc materal propertes. The self-equlbrum forces can be found from the optmalty condtons of the nonlnear programmng problem. Stablty of the self-equlbrum shape s nvestgated based on the local convexty of the objectve functon. Smlarty between form-fndng problem of a structure wth zero-unstressed-length cables and the problem of mnmum square-length network s also dscussed. Furthermore, foldng of a structure wth small unstressed-length cables s approxmately smulated usng affne transformaton of equlbrum shape. Keywords: Tensegrty, Form-fndng, Optmzaton, Stablty, Affne transformaton, Foldng analyss 1. Introducton Tensegrty structures consst of cables and struts that carry tensle and compressve forces, respectvely. Although the hstory and defnton of tensegrty structures are subject to 1

2 controversy (Hanaor, 212), the structure consdered here has dscontnuous struts and no support. In a broader defnton of tensegrty, the struts may contact wth each other; however, prestress s always needed for ensurng enough stffness aganst external loads. The weght of a tensegrty structure can be neglected f the prestress s large enough compared wth the selfweght; accordngly, the member forces due to prestress consttute a self-equlbrum state wthout external load. It s dffcult to obtan a desred shape of a tensegrty structure, because the structure s generally knematcally ndetermnate;.e., t s unstable n absence of prestress, and the shape of the structure defned by nodal coordnates at a self-equlbrum state depends on the dstrbuton of prestress. To overcome these dffcultes, varous analytcal and numercal approaches have been developed for form-fndng of tensegrty structures (Motro, 23; Zhang and Ohsak, 26; Skelton and de Olvera, 29). Form-fndng algorthms can be classfed nto knematcal method and statcal method (Tbert and Pellegrno, 23). The knematcal method mnmzes the total cable length or the sum of squares of cable lengths wth fxed total length of struts; alternatvely, t maxmzes the total strut length or the sum of squares of strut lengths wth fxed total length of cables. The statcal method drectly solves the equlbrum equatons. In the knematcal method, optmzaton approaches are effectvely used for mnmzng/maxmzng the functons of cable/strut lengths. Pagtz and Tur (29) presented a two-stage algorthm based on mnmzaton of the total potental energy by adjustng the cable lengths. Masc et al. (25) presented a method for mnmzng the quadratc error norm of the nodal locatons from the target locatons. They also formulated the problem of mnmzng the error of the equlbrum equatons. Mk and Kawaguch (21) proposed an approach to form-fndng of cable networks and tensegrty structures by solvng optmzaton problems wth varous objectve functons and constrants. Gasparn et al. (211) carred out formfndng usng nonlnear programmng (NP) approach. They also dscussed form-fndng wth constant stress elements as well as smlarty between the penalty functon approach and the constraned optmzaton approach. Chen et al. (212) used an ant-colony optmzaton method. et al. (21) solved a mnmzaton problem of the total potental energy usng a Mote Carlo method. Ohsak et al. (25) presented a method by mnmzng the error of member forces from the target values. Burkhardt (26) mnmzed the dfference between squared sums of lengths of cables and struts. Zhang and Ohsak (27b) proposed a multobjectve programmng approach consderng the lowest postve egenvalue of the tangent stffness matrx and the complance aganst specfed loads, where convexty 2

3 propertes of the objectve functons are fully utlzed to valdate enumeraton of the vertces of feasble regon. The method has been extended to a hybrd optmzaton-antoptmzaton problem ncorporatng the errors of the member forces (Ohsak et al., 212). Ehara and Kanno (21) developed a mxed-nteger programmng approach to topology desgn of a tensegrty structure consstng of dscontnuous struts. In the process of form-fndng of a tensegrty structure that s free-standng, rgd-body motons should be approprately constraned. Zhang et al. (214) proposed an approach based on random selecton of the sx dsplacement components to be constraned. Zhang and Ohsak (213) proposed a method utlzng sngular value decomposton or a generalzed nverse of the tangent stffness matrx. By contrast, for a cable net that has fxed supports, the equlbrum shape can be found by smply mnmzng the total stran energy, whch s a convex functon of the nodal coordnates, under convex constrants (Kanno et al. 22). It s well known that fcttous materal propertes wth varous stress-stran relatons can be used for form-fndng for specfed forces (Pagtz and Tur, 29; Mk et al., 213). Fcttous dampng s also wdely used for the dynamc relaxaton method for form-fndng. Schenk et al. (27) nvestgated the stffness of a tensegrty structure wth zero-unstressedlength cables, and showed that such structure has zero-stffness f the drectonal vectors of struts le on a projectve conc, whch means that the strut lengths do not change due to deformaton; accordngly, no external force s needed n the process of an affne transformaton or a smlarty transformaton (Masc et al., 25, Zhang and Ohsak, 27a). For foldng and deployment of a tensegrty structure, t s preferable to utlze a flexble/complant mechansm wth self-equlbrum forces (Ohsak and Nshwak, 25; Ohsak et al., 213), because such a structure s stable wthout addtonal restrant (Smal and Motro, 25, 27). Deployment of folded tensegrty structures s very mportant for applcaton to space structures. Infntesmal mechansm can also be utlzed for reducng energy requred for foldng (Sultan, 214). Arsenault and Gosseln (25, 26) analytcally nvestgated knematc propertes of tensegrty mechansms. The foldng process can be smulated by trackng the path of equlbrum state called equlbrum manfold (Mchelett and Wllams, 27; Sultan and Skelton, 23). Deployment of rng modules s studed by Rhode-Barbargos et al. (212). In vew of practcal applcaton of form-fndng of a tensegrty structure, t s very mportant to ensure stablty of the equlbrum state, because an unstable soluton cannot be actually manufactured. Guest (26) nvestgated stffness of prestressed framework, and presented a smple approach to dervaton of tangent stffness matrx. Ohsak and Zhang 3

4 (26) and Zhang and Ohsak (27a) nvestgated the condtons for stablty, prestressstablty, and super-stablty of tensegrty structures. Deng and Kwan (25) presented an energy-based approach to classfcaton of stablty of pn-jonted frameworks. In ths paper, we present a method for form-fndng of tensegrty structures usng a nonlnear programmng (NP) approach. Varous equlbrum shapes are found by solvng a forced-deformaton analyss problem formulated as a mnmzaton problem of a functon of the nodal coordnates n terms of member lengths. The problem s regarded as a mnmzaton problem of the total stran energy wth fcttous elastc materal propertes. The selfequlbrum forces can be found from the optmalty condtons of the NP problem. We apply the method to tensegrty towers wth blnear elastc stress-stran relaton for some selected cables. Stablty s guaranteed, except for the fcttous materal wth softenng property, from the local convexty of the objectve functon; hence, stffness matrces are not needed for nvestgaton of stablty. Smlarty between form-fndng problem of zerounstressed-length cables and the problem of mnmum square-length network s also dscussed. It s shown that a tensegrty structure can be folded wth small external forces, f the unstressed lengths of cables are small. However, the convergence property of the optmzaton problem s deterorated f the unstressed lengths of cables are very small. To resolve ths dffculty, we present a smple form for nvarance property of the equlbrum equatons wth respect to affne transformaton, and approxmately smulate the foldng of a structure wth small unstressed-length cables usng the affne transformaton of equlbrum shape. The proposed method s appled to foldng analyss of a tensegrty tower, and t s confrmed that the force densty matrx remans unchanged and the requred force s very small f cables wth small unstressed lengths are used. 2. Basc propertes and equatons Consder a tensegrty structure consstng of cables and struts. Although the defnton of a tensegrty structure s controversal, the structure consdered n ths paper s clearly defned as follows: 1. Cables and struts transmt only tensle and compressve forces, respectvely. 2. The structure has no support;.e., t s free-standng wth three translatonal and three rotatonal rgd-body motons. 3. Cables and struts are pn-jonted; therefore, the structure can be modeled usng truss elements, and no frcton or sldng occurs at the jonts. 4

5 4. The level of prestress s large enough compared wth the self-weght of structure; therefore, the self-weght s neglected, and there s no external nodal load at the selfequlbrum state. We defne the basc vectors and matrces below, for completeness of the paper. See, e.g., Zhang and Ohsak (26, 215), for detals. et n and m denote the numbers of nodes and members, respectvely. The topology of a tensegrty structure s defned usng the connectvty (ncdence) matrx m n C. If nodes and j ( j) are connected by member k, then the th and j th elements n the k th row of C are equal to 1 and 1, respectvely, whle all other elements n the k th row are (Skelton and de Olvera, 29; Zhang and Ohsak, 215). The coordnates of the th node at a self-equlbrum state s denoted by ( x, y, z ). The n vectors of x -, y -, z-coordnates are gven as x, y, z ( ), and the vector of all nodal coordnates s defned as 3 X ( x T, y T, z T ) T n. The length and axal force of the th member at the self-equlbrum state s denoted by and N, respectvely. et q N / denote the force densty of member, and defne the force densty vector as q ( q1,, q ) T m. Then, the force densty matrx n n Q s gven as T Q C dag( q) C (1) Alternatvely, the (, j) -component Q, j of Q s explctly gven as qk, f j ki Q, j qk, f nodes and jare connected by member k, otherwse where I s the set of members connected to node. Q as The self-equlbrum equaton wth respect to the nodal coordnates can be wrtten usng The coordnate dfference matrces Qx, Qy, Qz (3) x U, respectvely, of members are defned as y U, and U z m ( m ) (2) n x -, y -, and z-drectons, U x dag( Cx ), U y dag( Cy ), U z dag( Cz ) (4) The th dagonal components of U x, y U, and U z are denoted by x U, U, and U z, y 5

6 respectvely. The drectonal cosnes x, and z-axes, respectvely, are obtaned as, and x of member wth respect to x -, y -, x x x U, y y U, z z U (5) and ther matrx forms x Θ, y Θ, and are gven as z m Θ ( m ) Θ x dag( x 1,, x ), y y y 1 m Then, the equlbrum matrx D s wrtten as Θ dag(,, m ), Θ z dag( z 1,, z m ) (6) D C Θ T x C Θ T y T z C Θ Snce we neglect the self-weght, the self-equlbrum equaton wth respect to the member force vector N ( N1,, N ) T m s formulated as (7) DN (8) Suppose the unstressed lengths ( 1,, m) of members are gven. The members are connected to satsfy the specfed compatblty (connectvty) condtons at nodes. The member lengths ( X ) ( 1,, m) satsfyng the compatblty condtons are found by solvng a geometrcally nonlnear analyss problem subjected to forced deformaton; hence, ( X ) s regarded as a functon of X. The axal forces N( X ) ( 1,, m) are also found by solvng the analyss problem. Ths way, the form-fndng problem s formulated as a forced-deformaton analyss problem wth varable vector X. The square of member length s expressed n terms of coordnate dfferences as ( X) U ( X) U ( X) U ( X ) (9) 2 x 2 y 2 z 2 Dfferentaton of Eq. (9) wth respect to X leads to x y z U ( X) x U ( X) y U ( X) z ( X) U ( X) U ( X) U ( X) ( X) ( X) ( X) whch can be wrtten usng the drectonal cosnes as (1) x x y y z z ( X) ( X) U ( X) ( X) U ( X) ( X) U ( X ) (11) From Eqs. (4) and (11), the followng relaton holds: T x T x C Θ ( XNX ) ( ) CΘ ( X) ( X) ( X) C Θ ( XNX ) ( ) CΘ ( X) N( X) ( ) ( ) ( ) C Θ XNX CΘ X m N T y T y 1 T z T z (12) 6

7 whch s equvalent to Eq.(8). Hence, the self-equlbrum equaton can be wrtten as m N( X) ( X) (13) 1 3. Nonlnear programmng approach to form fndng usng fcttous materals 3.1 Optmzaton problem and equlbrum condtons We formulate the form-fndng problem as a mnmzaton problem of a functon wth respect to the nodal coordnates. Although the propertes of a tensegrty structure are defned based on the materals of the cables and struts, we can use a fcttous materal n the process of formfndng for generatng nodal coordnates and axal forces of varous self-equlbrum shapes. The structure s actually realzed usng the true materal, and the unstressed lengths are found so that the nodal coordnates and axal forces do not change after assgnng the true materal propertes. Foldng of a tensegrty structure wth small unstressed-length cables can also be smulated usng a fcttous cable materal that has zero-unstressed length. A contnuously dfferentable functon of X n terms of the member lengths ( X ) s defned as An unconstraned optmzaton problem s formulated as m F( X) F( ( X )) (14) Mnmze 1 m F( X) F( ( X )) (15) Note that X should be n the feasble regon satsfyng the sgn requrements of axal forces n cables and struts. The statonary condton of F( X ) s wrtten as 1 m F( ( X)) F( X) ( X) ( X) 1 (16) If F( X ) s a locally convex functon n the neghborhood of the soluton X s X satsfyng Eq. (16), then s X s a local optmal soluton, and the hessan 2 F( ) X of F( X ) s postve defnte at s X X. We consder a fcttous materal satsfyng F( ( X)) N( X) ( X) Then, Eq.(16) s equvalent to the equlbrum equaton (13). Ths way, a self-equlbrum 7 (17)

8 shape can be found by mnmzng a functon of member lengths. 3.2 Stablty of the self-equlbrum state Although mnmzaton of a functon of member lengths s not new for form-fndng of tensegrty structures, stablty of the obtaned self-equlbrum state has not been fully nvestgated. We present an approach to stablty nvestgaton wthout resort to stffness matrces. There are several defntons of stablty used n the feld of structural mechancs. The most common defnton s the postve defnteness of the tangent stffness matrx, whch s derved from the defnton of asymptotc stablty based on apunov s drect method (a Salle and efscetz, 1961) assumng exstence of approprate dampng. If we can regard F( X ) as a apunov functon, then the equlbrum state obtaned by solvng Problem (15) s stable, f F( X ) s a strctly quas-convex functon n the neghborhood of the equlbrum state and F( X ) corresponds to the restorng force of the structure (Ohsak, 23). On the other hand, the contnuous functon F( X ) s strctly quasconvex n the neghborhood of a local mnmum, f t s found by an NP algorthm. Therefore, the equlbrum state found by mnmzng F( X ) usng the fcttous materal s always stable. By contrast, f the objectve functon s not convex, then stablty of the equlbrum state obtaned by optmzaton s not guaranteed. Further dfferentaton of F( X ) n Eq. (16) wth respect to X leads to the Hessan of F( X ) as F( ( X)) F( ( X)) ( X) ( X) ( X) ( X) m 2 m 2 T 2 F 2 1 ( ) X 1 ( X) (18) 2 The Hessan F( X ) s postve defnte, when the optmzaton algorthm reaches an solated local mnmum of F( X ). 3.3 Mnmzaton of stran energy usng fcttous materal The total stran energy may be a natural choce for the objectve functon. For the gven unstressed member length, the stran energy of member s regarded as a functon of ( X ), whch s denoted by S ( ( X )). 8

9 et A, E, and denote the cross-sectonal area, Young s modulus and the unstressed length of member. The stran energy of member s defned as EA S ( ( X)) ( ( X ) ) (19) 2 2 Then, the total stran energy S( X ) s obtaned as m S( X) S( ( X )) (2) 1 The self-equlbrum shape s found by solvng an optmzaton problem of mnmzng S( X ). When no constrant s gven, the statonary condton of S( X ) s wrtten as m S( ( X)) S( X) ( X) ( X) 1 (21) At the optmal soluton satsfyng Eq. (21), the equlbrum equaton (13) s satsfed, because S ( ( X))/ ( X ) s the axal force N ( X ) of member as S ( ( X)) E A N ( X) ( ( ) ) X ( X) When the objectve functon s the total stran energy, the optmzaton problem to be solved for form-fndng s a standard analyss problem wth forced deformaton to satsfy the compatblty at nodes for specfed unstressed member lengths. Therefore, the prncple of mnmum total potental energy ensures stablty of the equlbrum shape obtaned by mnmzng the stran energy of the structure wthout external loads. Although the stffness matrces are not necessary n the process of fndng a stable selfequlbrum state, the tangent stffness matrx s used n the numercal examples for confrmaton of stablty of the equlbrum state. By ncorporatng S( X ) nto F( X ) n Eq. (18), the frst and second terms n the rght-hand sde turns out to be the lnear stffness matrx 3n3n K and geometrcal stffness matrx wthout dervaton and omttng the varable X as where T KE DKD, KG (22) 3n3n K G, respectvely, whch are gven Q Q Q (23) 9

10 K dag( E A /,, E A / ) (24) m m m See, e.g., Zhang and Ohsak (27a) for detals. Note that the member lengths at selfequlbrum state s usually used n Eq. (24). Then, the tangent stffness matrx 3n3n K T s gven as K K K (25) T E G 4. Fcttous materals of cables Fcttous materals are used for cables n the process of form fndng. The true materal and a large cross-sectonal area are used for struts so that they have suffcently larger stffness than cables. 4.1 Blnear materal A blnear elastc stress-stran relaton s gven for a cable as shown n Fg. 1. The materals n Cases 1 and 2 are lnear elastc and stffenng blnear, respectvely. Case 3 has a softenng blnear property wth a postve stffness n the second part to ensure local mnmum of S( X ) at the equlbrum state. Stress Case 2 Case 3 Case 1 ˆε Stran Fgure 1. near and blnear elastc stress-stran relatons for cables. The value of stran at the transton pont of the blnear elastc property s denoted by ˆ ;.e., the length U at the transton pont s U (1 ˆ). The elastc modulus for ( X ) of cable s denoted by E ˆ. Then, the stran energy for U formulated as U 2 U U U 2 j j j 2 2 ( X ) of cable s EA EA EA ˆ S ( ( X)) ( ) ( ( X) ) ( ( X ) ) (26) U 1

11 Dfferentaton of S ( ( X )) wth respect to X leads to S ( ( X)) EA EA ˆ ( X) ( ( X) ) ( X) U U X From Eqs. (13) and (27), the axal force at equlbrum s obtaned as follows at the soluton s X that mnmzes S( X ): et A and s U s U (27) EA EA ˆ N ( X ) ( ( X ) ) (28) E denote the cross-sectonal area and Young s modulus of the true lnear elastc materal. The unstressed length of the cable wth the true materal s found to satsfy the followng relaton so that ts axal force s equal to N X : s ( ) EA EA EA ˆ ( ( X ) ) ( ( X ) ) s U s U When an equlbrum state s found usng an optmzaton algorthm, t corresponds to a local mnmum of S( X ) usng the fcttous materal, whch means S( X ) s locally convex at X s also convex, f the followng relaton s satsfed: EA ˆ EA s X and the equlbrum state s stable. The stran energy functon usng the true materal In ths case, the structure wth true materal s also stable. Note that the condton n Eq. (3) s a suffcent but not a necessary condton for stablty. Member numbers are assgned, for smplcty, such that members (29) (3) 1,, mc are cables and members mc 1,, m are struts. For Case 2, we can formulate a constraned optmzaton problem wth upper bound U for cable as Mnmze S( X) S ( ( X)) m 1 U Subject to ( X), ( 1,, mc ) (31) et denote the agrange multpler for the th constrant n Eq. (31). The optmalty condtons (KKT condtons) for the constraned mnmzaton problem (31) s wrtten as S( ( X)) ( X) ( X) m mc 1 ( X) 1 11

12 , X, U ( ( ) ) X, ( 1,, mc ) (32) U ( ) Hence, the equlbrum equaton (13) s satsfed by regardng N ( X) S ( ( X))/ ( X ) for cables and N ( X) S ( ( X))/ ( X ) for struts. A smlar formulaton s derved by Mk and Kawaguch (21); however, ther purpose s to fnd an equlbrum shape by mnmzng a functon of cable lengths under constrants on the strut lengths. Note that the suffcent condton (3) for stablty of the structure wth true materal s not ensured for Case 2. Ths s observed from the KKT condton (32) that addtonal tensle force correspondng to the agrange multpler ( ) should be appled to cable to restrct the deformaton of the cable. 4.2 Zero-unstressed-length cable Schenk et al. (27) nvestgated the propertes of tensegrty structures, where each cable has zero unstressed length, and demonstrated that such cables can be manufactured usng conventonal sprngs attached alongsde bars. A cable wth small unstressed length can be modeled usng a flexble sprng or rubber. By assumng energy of member s formulated as Note that the coeffcent * 2 2 ( X) n Eq. (19), the stran EA S ( ( X)) ( ( X )) (33) EA dverges to nfnty n the lmt /(2 ). Ths formulaton s closely related to form-fndng analyss based on mnmum squarelength network that has the smallest total square length of cables for the gven strut lengths. Snce EA/ s the extensonal stffness of the cable, we can rewrte Eq. (33) as follows usng k E A / : * k 2 S ( ( X)) ( ( X )) (34) 2 From Eqs. (22) and (34), we obtan S * ( ( X)) N( X) k ( X) ( X) (35) Therefore, the force densty q of a cable at the equlbrum state the specfed stffness as X s X should be equal to 12

13 N X q s ( ) s ( X ) Note that the matrx Q n Eq. (3), whch s a lnear functon of the force densty vector q (,, ) T as n Eq. (1), should have four zero egenvalues n order to realze an q1 q m equlbrum shape n 3-dmensonal space (Zhang and Ohsak, 26). It s easly observed from the defnton n Eq. (2) that the vector (1,, 1) T s one of the egenvector correspondng to a zero egenvalue rrespectve of the values of q. Therefore, q should satsfy three nonlnear equatons to ensure a 3-dmensonal tensegrty structure, whch means that force denstes of cables generally cannot have specfed values as Eq. (36), and mnmzaton of the sum of k (36) S * ( ( X )) leads to a shape n a smaller dmenson;.e., plane, lne, or pont. Ths fact s consstent wth the observaton by Mk and Kawaguch (21) notng that mnmzng the sum of S * ( ( X )) does not easly lead to a stable 3-dmensonal confguraton. By contrast, f the unstressed length s moderately large, and S ( ( X )) n Eq. (19) s used, a wde range of force denstes can be covered, and there exsts more possblty of generatng a shape n 3-dmensonal space. Fgure 2 shows varaton of q k ( )/ wth respect to (.5 1) for k 1 and varous values of. As seen from the fgure, q has the constant value 1 for the case of zero unstressed length wth. If has a moderate value, then q can vary n a wde range; e.g.,.2 q.6 for.4. Force densty = =.2 =.4 =.6 = ength Fg 2. Varaton of force densty wth respect to member length. 13

14 4.3 Summary of optmzaton process The process of form-fndng usng optmzaton approach s summarzed as follows: Step 1. Assgn materal propertes for cables and struts ncludng fcttous materal propertes for some cables. Step 2. Defne a functon of member lengths as an objectve functon to be mnmzed. Step 3. Solve the optmzaton problem usng any optmzaton algorthm. Step 4. Assgn the true materal propertes, and compute unstressed lengths so that the axal forces do not change from those determned n Step 3. Step 5. Check stablty of the self-equlbrum state usng the true materal propertes. In the numercal examples, optmzaton s carred out usng SNOPT Ver.7 (Gll et al., 22) that utlzes sequental quadratc programmng (SQP). The senstvty coeffcents are computed analytcally, e.g., as Eqs. (22) and (27) for F( X) S( X ). et ( k) 3n 3n H denote the approxmate Hessan of F( X ) at the kth step of SQP. The ncrement of X s denoted as ( k) 3n dxx, whch s the varable vector of the QP sub-problem at the kth step. Then, the sub-problem for the unconstraned problem s formulated as ( k1) ( k) ( k) T 1 T ( k) Mnmze F ( d) F( X ) F( X ) d d H d (37) 2 When the approxmate Hessan s sngular at a step of SQP, a small dagonal term, whch ( k ) leads to a penalty term of the quadratc norm of the ncrement of varables, s added to H. Therefore, for the analyss problem of a free-standng tensegrty structure, the rgd-body motons are automatcally suppressed, and the nearest soluton from the ntal soluton s obtaned. 5. Constraned optmzaton and affne transformaton for foldng analyss 5.1 Constraned optmzaton for foldng analyss We consder a process of foldng a tensegrty tower that conssts of struts, vertcal cables, saddle cables, dagonal cables, and horzontal cables (Zhang and Ohsak, 28; Mchelett and Wllams, 27; Sultan, 214). The foldng process s slow enough so that dynamc effect need not be consdered, assumng exstence of approprate dampng; therefore, we consder a quasstatc process of foldng. 14

15 An example of three-layer tower s shown n Fg. 3. Foldng analyss s carred out by assgnng constrants on the dfference between z-coordnate rng and z-coordnate z k of the k th node n the bottom rng as U z j of the j th node n the top z z h, U j k R R ( j 1,, n ; k 1,, n ) (38) where R n s the number of nodes n each rng, and h s the specfed total heght. Horzontal Saddle Dagonal Vertcal Horzontal Fgure 3. A 3-layer tensegrty tower model. Foldng s carred out by suppressng the top rng to the bottom. The parametrc optmzaton problem for obtanng the equlbrum states n the foldng process s formulated as Mnmze S( X) S ( ( X)) m 1 U R R Subject to zj ( X) zk ( X) h, ( j 1,, n ; k 1,, n ) (39) where h s conceved as a parameter to be decreased from the ntal value to. et jk denote the agrange multpler correspondng to the constrant reacton forces U R j and rng, respectvely, are computed from z z h. The R k for the j th node n the upper rng and the k th node n the lower U j k R U j R n, k1 jk R k n R (4) j1 jk In the process of foldng, cables should reman n tensle state. Therefore, t s desrable that the unstressed length of each cable s small enough compared wth the length at equlbrum. However, as dscussed n Sec. 2, t s very dffcult to obtan an equlbrum 15

16 shape f unstressed cable lengths are too small. In ths case, we can utlze affne transformaton as shown below. 5.2 Foldng of a tensegrty structure wth zero-unstressed-length cables usng affne transformaton Masc et al. (25) proved that self-equlbrum equaton of a tensegrty structure s nvarant wth respect to an affne transformaton. Ths property can be easly observed usng the equlbrum equatons wth respect to the nodal coordnates. The self-equlbrum equaton can be wrtten usng the force densty matrx Q and the nodal coordnate vectors x, y, and z as Eq. (3). Obvously, these equatons are satsfed wth fxed Q and the coordnates after affne transformaton as xa xa y a zbi y a x a y a zb I za xa y a zbi (41) n where I s the vector that has 1 n all components. Usng the affne transformaton, varous equlbrum shapes can be found after fndng a sngle equlbrum shape. Suppose we found an equlbrum shape usng the zero-unstressed-length cables, and a forced deformaton s appled to the structure. For a zero-unstressed-length cable, the axal force N( X ) s proportonal to the length at equlbrum ( X ) as N( X) k ( X ), where k s the axal stffness;.e., k turns out to be the force densty, as shown n Eq. (36), whch does not depend on ( X ) n the process of forced deformaton. Therefore, t s easly seen that the force densty matrx does not change, and the structure can be deformed wthout external force, f there exsts an affne transformaton that does not change the strut lengths so that the force denstes of all members are unchanged. Ths property has been dscussed based on the theory of structural rgdty by Schenk et al. (27). Consder, for example, a tensegrty structure that s axsymmetrc wth respect to z-axs, and suppose there s only one class of struts;.e., a strut can be moved to any other strut through a rotaton around z-axs. In ths case, there always exsts an affne transformaton xdx, ydy, zcz (42) that preserves the strut lengths by approprately assgnng the coeffcents c and d. Ths way, foldng process s smulated wthout carryng out form-fndng analyss;.e, t can be smulated by successvely decreasng c to and fndng d n Eq. (42) that does not change strut lengths. 16

17 6. Examples of tensegrty tower 6.1 Form-fndng of 2-layer tower usng blnear fcttous materal To demonstrate the effectveness of usng fcttous blnear elastc materals, the optmzaton approach s appled to form-fndng of a 2-layer tensegrty tower as shown n Fg. 4(a). The unts are omtted, n the followng, for smple presentaton of the results. The tower has three struts n each layer, and the radus and heght of each layer are 2. and 22.5, respectvely, at the ntal state for solvng optmzaton problem for mnmzaton of the total stran energy. et e denote the rato of unstressed length to the ntal length. The values of e for cables and struts are assumed to be.8 and 1., respectvely;.e., the unstressed length of a cable s 8% of the length n the ntal shape n Fg. 4(a), whle the unstressed length of a strut s the same as the length n the ntal shape. The value of AE for struts s 1. As we demonstrated n Fg. 2, the values of e for cables should be suffcently small to obtan varous stable equlbrum shapes wthout slackenng of cables. (a) (b) (c) (d) Fgure 4. Intal and self-equlbrum shapes of a 2-layer tensegrty tower; (a) ntal shape, (b) equlbrum shape usng a lnear elastc materal (Case 1), (c) equlbrum shape usng a stffenng materal (Case 2), (b) equlbrum shape usng a softenng materal (Case 3). 17

18 Case 1: We fnd the equlbrum shape usng cables wth lnear elastc materal denoted by Case 1 n Fg. 1, where AE 1, whch s denoted by AE, for all cables. In ths case, the stffness of the true materal s the same as that used n the form-fndng process. The equlbrum shape obtaned by solvng the unconstraned optmzaton problem s shown n Fg. 4(b). The maxmum axal force among all cables s Egenvalue analyss s carred out for the tangent stffness matrx K T to fnd the 6th and 7th smallest egenvalues as lsted n the frst row of Table 1. Snce the 7th egenvalue s suffcently larger than the 6th egenvalue that s approxmately equal to, the equlbrum state s stable wth sx zero egenvalues correspondng to rgd-body motons. Table 1. Egenvalues of tangent stffness matrx usng true materal. Case 6th 7th Case 2: We next consder a fcttous materal wth blnear stress-stran relaton. The 6 vertcal cables are classfed nto sx groups connectng the nodes wth the same xy-coordnates n the horzontal plane of the ntal shape n Fg. 4(a). Ten cables n one of the sx groups are selected to have the stffenng blnear stress-stran relaton as ndcated as Case 2 n Fg. 1. The stran ˆ at the stffness transton pont s.1, and the value of AE n the frst and second parts are AE ( 1) and 1 ( 1) AE, respectvely. The equlbrum shape obtaned by optmzaton s shown n Fg. 4(c). The mnmum and maxmum values of strans among the members wth blnear stress-stran relaton are.128 and.13, whch are close to ˆ (.1). Ths way, a curved shape can be generated by assgnng large stffness for the cables that are vertcally algned at the ntal shape. We compute the unstressed member lengths, and carry out egenvalue analyss of the tangent stffness matrx usng the true materal wth constant value AE 1 for all cables. The 6th and 7th egenvalues are lsted n the second row of Table 1, whch shows that the 18

19 structure s stable, although the true materal has smaller stffness than the fcttous materal and the suffcent condton (3) for stablty s not satsfed. If we set the maxmum member length 1.1 n Eq. (31), whch s consstent wth ˆ.1, and solve the constraned U optmzaton problem, almost the same equlbrum shape as shown n Fg. 4(c) s obtaned. The axal forces and the agrange multplers of the constraned members n layers 1, 3, and 5 are lsted n Table 2. The axal forces obtaned usng the blnear materal are also lsted n the frst column of Table 2. It can be confrmed from these results that the cable forces at equlbrum can be obtaned approxmately as the sum of the dfferental coeffcent S / and the agrange multpler. Table 2. Cable forces at equlbrum of constraned members n layers 1, 3, and 5. Constraned optmzaton ayer Blnear (A) Dfferental (B) agrange (A) + (B) model coeffcent of stran multpler energy Case 3: Fcttous materal property s gven n the same vertcal cables as Case 2, where ˆ.1 also for ths case. We decrease the value of AE n the second part of the stress-stran relaton of the ten vertcal cables to AE /1 ( 1) as ndcated by Case 3 n Fg. 1. The equlbrum shape obtaned by solvng the unconstraned optmzaton problem s shown n Fg. 4(d). As seen from Fgs. 4(c) and (d), the tower can be bent to opposte drectons by ncreasng and decreasng the value of AE of the vertcal cables n the specfed group. The axal forces of the vertcal cables wth blnear stress-stran relaton are between 13. and 14., whch are close to the specfed value.1ae 1. We compute the unstressed member lengths, and carry out egenvalue analyss of tangent stffness matrx. The 6th and 7th lowest egenvalues are lsted n the thrd row n Table 1, whch confrms the stablty of structure. Snce the stffness of the fcttous materal s smaller than that of the true materal, the equlbrum shape wth the true materal s always stable, f the shape wth fcttous materal s stable. 19

20 6.2 Foldng analyss of 2-layer tower Foldng analyss s carred out for a 2-layer tensegrty tower wth three struts n each layer. Optmzaton problem s successvely solved by varyng the heght h to be constraned n Eq. (38). The value of AE for the struts s 1, whch s suffcently large, and a lnear elastc materal s used for cables. The unstressed lengths of cables should be suffcently small to prevent slackenng, whle those of struts are the same as those n the ntal shape. We frst consder the case e.2 ;.e., the unstressed length of each cable s 2% of the ntal length. The value of AE for cables s 1. The heght of each layer of the ntal shape s 4.,.e., the total heght h s 8, whereas ts radus r s taken as a parameter to solve the mnmzaton problem of the total stran energy and obtan varous self-equlbrum shapes. If r s small, the equlbrum shape approaches a straght lne as shown n Fg. 5(a). If r s large, a planar shape s obtaned as shown n Fg. 5(c). (a) (b) (c) Fgure 5. Equlbrum shapes for varous ntal radus for e.2 ; (a) r 1, (b) r 3, (c) r 8. We assgn r 3 for foldng analyss for varous values of e. The equlbrum confguraton wthout constrans s shown n Fg. 5(b), where the total heght s To prevent too large axal force at equlbrum, the value of AE s adjusted dependng on the unstressed length of cable as AE 2e (43) 2

21 whch leads to AE 1 for e.5. It should be noted that the numercal results below depends on the value of AE. The total reacton force at the top nodes durng the foldng analyss s plotted n Fg. 6 for each case of e wth respect to the heght h that s decreased to. For the range e.18, the maxmum reacton force ncreases as the unstressed length becomes smaller, and accordngly, the ntal stran becomes larger. No reacton force exsts at h 5 for e.16, because the heght of equlbrum shape s less than 5 wthout constrant (38) on the heght. The reacton forces are small for e =.12 and.1, and the equlbrum shape degenerates nto a plane as e approaches, whch agrees wth the fact that a tensegrty structure wth zero-unstressedlength cables cannot be obtaned by energy mnmzaton. 8 Reacton force Heght 8 e =.2 e =.3 e =.4 e =.5 Reacton force e =.12 e = Heght e =.18 e =.16 e =.14 (a) (b) Fgure 6. Varaton of reacton forces wth respect to the specfed heght; (a) e =.5,.4,.3,.2, (b) e =.18,.16,.14,.12,.1. The force denstes of struts are plotted wth respect to the heght n Fg. 7. The force denstes for the cases e.18 are normalzed by the values at h 5, whereas those for the cases are normalzed by the value at the maxmum heght wth non-zero reacton force. We 21

22 can see from Fg. 7(a) that the normalzed force denstes are close to 1, and are between 1 and 1.22 even for e.5 ; therefore, the varaton force densty n the foldng process s very small. Snce a shape varaton wth constant force densty matrx corresponds to an affne transformaton, t s expected that the foldng process can be approxmately smulated usng affne transformaton wthout sucessvely solvng the constraned optmzaton problem. Normalzed force densty Heght e =.5 e =.4 e =.3 e =.2 Normalzed force densty e =.12 e = e = Heght e =.18 e =.16 Fgure 7. Varaton of normalzed force densty of struts; (a) e =.5,.4,.3,.2, (b) e =.18,.16,.14,.12,.1. Although the value of e should be close to to smulate the behavor of zero-unstressedcable, we use e.1 to prevent numercal dffculty n the optmzaton process. Furthermore, the ntal radus r should be very large to prevent convergence to a thn shape as Fg. 7(a) when e has a small value; thus we set r 1. The total heght of the ntal equlbrum shape wthout constrant s h 133.9, whch s larger than the ntal heght 8, because the heght becomes larger due to shrnkage of cables. et c denote the rato of the total heght h to h. The rad of the crcles, where the top/bottom nodes and mddle nodes are located, at the equlbrum state are denoted by r 1 and r 2, respectvely. Affne transformaton n Eq. (42) s carred out to fnd the horzontal scale d so that the lengths of struts do not 22

23 change. The values of R 1, R 2, and the horzontal scale d are lsted wth respect to the heght rato c n Table 3. Table 3. Varatons of horzontal scale d, rad r 1 and r 2, and 7th and 8th egenvalues of tangent stffness matrx wth respect to heght rato c of a tensegrty structure wth zerounstressed-length cables. c d r r * r 1 * r We can see from Table 3 that the horzontal scale and rad ncrease as the heght s decreased. The values of rad obtaned by solvng constraned optmzaton problem (39) wth e.1 are lsted as 2 * R 1 and * R 2 n Table 3. Note that these values are slghtly dfferent from R 1 and R, because e.1 ndcates that the unstressed cable lengths do not vansh completely. It has been confrmed that each confguraton obtaned usng affne transformaton has the self-equlbrum forces that are almost the same as the forces at equlbrum obtaned by optmzaton wth e.1. Accordng to Eq. (34), the axal stffness k of zero-unstressedlength cable s equal to the force densty. Snce the equlbrum shape does not change when the stffnesses of all members are scaled proportonally, we multply to the force densty of each cable to obtan ts stffness. A large value s multpled to the absolute value of the force denstes of struts so that they can be assumed to be rgd compared wth cables. The 7th and 8th egenvalues 7 and 8 of the tangent stffness matrx are lsted n Table 3. It has been confrmed that the sx lowest egenvalues can be regarded as. Therfore, ths result ndcates that the structure n the foldng process s stable; however, a 23

24 flexble deformaton exsts n the drecton of the 7th egenmode. Stablty can also be confrmed for the equlbrum shape obtaned by optmzaton wth e.1, because the equlbrum shape s found by energy mnmzaton, and the reacton forces at the top and bottom have small postve values (compressve forces); e.g., for c.5, the reacton force s 836.7, whle the forces of struts are Conclusons An optmzaton approach has been presented for form-fndng and foldng analyss of tensegrty structures usng fcttous materal propertes. The followng conclusons have been obtaned from ths study: 1. Stablty of the equlbrum shape obtaned by optmzaton can be ensured wthout resort to tangent stffness matrces. The stablty s always guaranteed from local convexty of the objectve functon, f the gradent of objectve functon corresponds to the restorng force of a structure wth fcttous materal propertes. Furthermore, the self-equlbrum state usng the true materal property s stable f the fcttous materal used for formfndng has smaller stffness than the true materal. 2. Varous equlbrum shapes can be obtaned usng fcttous materal propertes wth blnear elastc stress-stran relatons. A curved tensegrty tower can be generated by assgnng stffenng/softenng materals for a group of vertcally algned vertcal cables. A softenng materal can be used for fndng a self-equlbrum state wth the specfed cable forces. It has been confrmed that the optmzaton problem wth stffenng blnear stressstran relaton s equvalent to a constraned optmzaton problem wth upper bound for the member lengths. 3. When the unstressed cable lengths are reduced to zero, the axal force s assumed to be proportonal to the length at equlbrum, and the axal stffness should be proportonal to the force densty. In ths case, there s generally no set of force denstes that satsfes the rank deffcency condton of a tensegrty structure n 3-dmensonal space; therefore, the equlbrum shape degenerates to a space of lower dmenson;.e., plane, lne, or pont. Ths property can be easly explaned through smlarty between form-fndng problem of zero-unstressed-length cables and the problem of mnmum square-length network. 4. Quas-statc foldng of a tensegrty structure can be smulated by solvng constraned optmzaton problems successvely reducng the total heght of the structure to be 24

25 specfed. The reacton forces, whch are computed from the agrange multplers for the heght constrants, frst ncreases and then decreas as the heght s reduced to. The force denstes of members are almost constant for cables wth small unstressed length compared wth the length at equlbrum; however, the convergence property of optmzaton deterorates when the unstressed cable lengths are very small. 5. Invarance of the equlbrum equatons wth respect to affne transformaton can be smply explaned usng the nvarance of force densty matrx. Foldng process of a structure wth small unstressed-length cables can be approxmately smulated usng affne transformaton. 6. The rgd-body motons need not be constraned when solvng the optmzaton problem usng an SQP method, because the quadratc programmng sub-problem s automatcally stablzed by assgnng small postve values n the dagonals of the approxmate Hessan of the arangan. Acknowledgements The authors apprecste prelmnary numercal nvestgaton by Mr. Naoto Fujta (former graduate student of Hroshma Unversty) and Mr. Tetsuo Taguch (graduate student of Hroshma Unversty). References Al, N.B.H, Rhode-Barbargos,., Smth, I.F.C., 211. Analyss of clustered tensegrty structures usng a modfed dynamc relaxaton algorthm. Int. J. Solds Struct. 48, Arsenault, M., Gosseln C., 25. Knematc, statc, and dynamc analyss of a planar onedegree-of-freedom tensegrty mechansm. J. Mech. Desgn, ASME 127, Arsenault, M., Gosseln C., 25. Knematc, statc, and dynamc analyss of a spatal threedegree-of-freedom tensegrty mechansm. J. Mech. Desgn, ASME 128, Burkhardt, R., 26. The applcaton of nonlnear programmng to the desgn and valdaton of tensegrty structures wth specal attenton to skew prsms. J. Int. Assoc. Shell and Spatal Struct. 47(1), Chen, Y., Feng, J., Wu, Y., 212. Novel form-fndng of tensegrty structures usng ant colony systems. J. Mech. Robotcs, ASME 4, No Deng, H., Kwan, A.S.K., 25. Unfed classfcaton of stablty of pn-jonted bar assembles. Int. J. Solds Struct. 42,

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