Tensegrity Structures Prestressabiity Investigation Corne Sutan* and Robert Sketon** *Harvard University Boston MA 0115 USA **University of Caifornia La Joa CA 909-0411 USA (Received 10th October 001 revised version received 1th March 00) ABSTRACT: A methodoogy for the investigation of tensegrity structures prestressabiity is presented The methodoogy is aimed at identifying particuar prestressabe configurations for which the prestressabiity conditions can be expressed in a simpe form Exampes of the appication of the methodoogy to a cass of tensegrity structures caed tensegrity towers are presented INTRODUCTION Tensegrity structures represent a cass of space structures composed of a set of soft members and a set of hard ones The difference between the soft and hard members is that the soft ones must carry tension forces in order to avoid entangement (the best exampe being that of eastic tendons or membranes) These members cannot carry compressive forces (for exampe a tendon cannot be compressed) Because of this property we sha refer to these members as tensie members On the other hand the hard members are characterised by the fact that they can carry any type of force or moment The representative exampe is that of bars which can carry tension compression forces bending moments etc A structure composed of soft and hard eements as described above is a tensegrity structure if it has the property of prestressabiity This consists of the structure s abiity to maintain an equiibrium shape with a tensie members in tension and in the absence of externa forces or torques Tensegrity structures integrity is guaranteed by the tensie members in tension hence their denomination tensegrity an acronym of tension-integrity coined by R B Fuer A perspective view of a tensegrity tower composed of eastic tendons (soft members) 9 rigid bars a rigid base and a rigid top (hard members) is given in Fig 1 Figure 1Three stage tensegrity tower From the technoogica point of view tensegrity structures offer severa important advantages First as mentioned by Emmerich (1996) many tensegrity structures do not have bar to bar connections Thus compicated joints are eiminated Second tensegrity Internationa Journa of Space Structures Vo 18 No 1 00 15
Tensegritty Structures Prestressabiity Investigation structures offer exceent opportunities for physicay integrated structure and controer design since their members can serve simutaneousy as sensors actuators and oad carrying eements Having incorporated sensors and actuators tensegrity structures have considerabe promise as smart structures (see Sketon 1997 Sutan 1997 1998) Third most tensegrity structures are ightweight structures making them amenabe for various space appications such as depoyabe structures (see Sutan 1998) space teescopes (see Sutan 1999) robotic arms etc Fourth tensegrity structures are capabe of arge dispacement beonging to the cass of fexibe structures As mentioned before the fundamenta characteristic of tensegrity structures is their prestressabiity defined as the abiity to yied equiibrium configurations under no externa forces or torques and such that a tensie members are in tension The corresponding conditions are caed prestressabiity conditions and the equiibrium configurations prestressabe configurations A derivation of the prestressabiity conditions for certain types of tensegrity structures simiar to the one in Fig 1 under very genera modeing assumptions has been presented in Sutan (001) Numerica methods to investigate the prestressabiity probem have been proposed over the years Research of the prestressabe configurations has been deveoped simutaneousy with infinitesima mechanisms determination (see Peegrino 1986 1990 whose work is based on equiibrium matrix anaysis) Vassart (1999) deveoped a method aowing a mutiparametered form-finding The method expoits the force density method and its appication to tensegrity structures is shown to provide the designer with an efficient way to achieve interesting new prestressabe configurations The dynamic reaxation method has aso been appied successfuy to form-finding probems of tensegrity prisms (see Motro 1994) Hanaor (1988) deveoped an agorithm based on the fexibiity method of structura anaysis for the anaysis and optimum prestress design of prestressabe pin-jointed assembies incuding tensegrity structures Hanaor (199) used the stiffness method in obtaining prestressabe configurations of doube ayer tensegrity domes The previousy mentioned approaches to prestressabiity probem soution are numerica ones and as with a numerica methods they might suffer from computationa inneficiency On the anaytica soutions front Tarnai (1980) extending some pinjointed space trusses resuts presented geometries for which the equiibrium matrix is singuar (which in this case of square equiibrium matrix is a necessary but not sufficient condition for prestressabe configurations to occur) Kenner (1976) had previousy arrived at the same resuts for tensegrity prisms based on the assumption that in a prestressabe configuration cabe engths are a minimum Sutan (001) reported the discovery of casses of prestressabe configurations for severa tensegrity structures for which compete anaytica soutions of the prestressabiity probem are given In this paper we present a methodoogy for the investigation of the prestressabiity conditions The methodoogy which combines symboic and numerica computation is aimed at identifying particuar prestressabe configurations for which the prestressabiity conditions are expressed in a simper form This simper form aows for further anaytica investigations or efficient and accurate numerica soutions Hence the advantage of the method is computationa efficiency in configurations possessing high reguarity Exampes of the appication of the proposed methodoogy are incuded PRESTRESSABILITY 1 Modeing A ssumptions Consider a tensegrity structure composed of E eastic tendons (the tensie members) and R rigid bodies (the hard members) We assume that a the joints of the structure are affected at most by kinetic friction This means that the friction forces/torques acting at a joint are zero if the reative inear/anguar veocities between the eements in contact at the joint are zero Aso the tendons are affected at most by kinetic damping This means that the damping force introduced by a tendon is zero if its eongation rate is zero Hence the system is not affected by static (Couomb) friction We aso assume that a constraints on the system are hoonomic This impies that we can find a set of independent generaised coordinates q i i = 1 N to describe the mechanica motion of the system Additionay we assume that a constraints are sceronomic and biatera In other words they are not time dependent and they are not mathematicay expressed as inequaities Lasty we assume that the externa constraint forces are workess which means that they do no work in a virtua dispacement consistent with the geometric constraints We negect the forces 16 Internationa Journa of Space Structures Vo 18 No 1 00
C Sutan and R Sketon exerted on the structure by externa force fieds (eg gravitationa) Prestressabiity Conditions Sutan (001) proved that under these genera modeing assumptions the prestressabiity conditions can be expressed as a set of noninear equations and inequaities: A(q)T = 0 T i > 0 i = 1 E (1) where the eements of the matrix A(q) caed the equiibrium matrix are given by L A ij (q) = j i = 1N j = 1E () q i and T is the vector of tendon tensions Here L j is the ength of tendon j q is a vector of independent generaised coordinates used to describe the configuration of the system and N is the number of independent generaised coordinates METHODOLOGY In the foowing we present a methodoogy which aows for the identification of particuar prestressabe configurations and of the corresponding simper prestressabiity conditions The proposed methodoogy consists of severa steps as foows 1 Identify an appropriate set of independent generaised coordinates q i i = 1 N Symboicay compute the equiibrium matrix A(q) A ij (q) = L j q i = 1N j = 1E Define a cass of configurations of interest and parameterize its geometry using fewer independent coordinates p i i = 1 ˆN This resuts in q = ˆq(p) 4 Substitute q= ˆq(p) in A(q) resuting in Â(p) = A(ˆq(p)) 5 Numericay sove the prestressabiity conditions Â(p)T = 0 for one soution (p p T p ) 6 Based on the previous numerica soution identify the structure of T and symboicay substitute T into Â(p)T = 0 This is done as foows: T p has a certain structure (for exampe T p 1 = Tp = ) Then the vector T is assumed to have the corresponding structure T = [T a T b T a ] T and this expression of T is symboicay substituted into Â(p)T = 0 Next eiminate the redundant equations in Â(p)T = 0 The resuting set of equations and inequaities on the tensions (eg T a > 0 T b > 0 etc) wi be caed the essentia prestressabiity conditions Steps 4 and 6 can be carried out using symboic computationa software (Mape Mathematica etc) The structure of T is indicated just by one numerica soution The essentia prestressabiity conditions characte rize a the eements of the cass of prestressabe configurations of interest for which the structure of T is the same as the one indicated by the numerica soution If T p does not have any particuar structure then no simpification of the prestressabiity conditions is possibe In order for T to have a specia structure it is recommended that the prestressabe configurations of interest have some geometrica reguarity because it is expected that this reguarity woud resut in some symmetry in the vector of tendon tensions T The main advantage this methodoogy presents over the cassica numerica methods is that famiies of prestressabe configurations can be identified for which the prestressabiity conditions can be more easiy soved numericay or anayticay Hence not a singe soution of the prestressabiity conditions is generated but a whoe set parameterized by certain geometrica characteristics of the structure under investigation 4 TENSEGRITY TOWERS In the foowing we iustrate the above methodoogy through its appication to a cass of tensegrity structures caed tensegrity towers 41 Description A tensegrity tower is composed of n stages (where n ) each stage having bars The endpoints of a bar are abeed A ij and B ij i =1 j =1n where the A point denotes the bottom of the bar and the B point its top Bar A ij B ij wi be referred to as bar ij The indices of bar ij respect the foowing rue: the second index j denotes the number of the stage the bar beongs to whie the first index i indicates the number of the bar in a stage Stage number j is composed of bars ij i =1 The bars of the first stage are attached via Internationa Journa of Space Structures Vo 18 No 1 00 17
Tensegritty Structures Prestressabiity Investigation ba and socket joints to a base and the bars of the n-th stage are attached via ba and socket joints to a top We wi refer to the base by the points of the first stage bars which are attached to it that is A 1 A A Simiary the top wi be referred to as B n B n B n The end points of the bars are connected through a tota of 15n 1 tendons These are defined as foows Tendons associated with the first stage: A 11 A 1 A 11 B 1 A 1 A A 1 B 1 A 1 A A 1 B 11 Tendons associated with stage j j even j < n: A 1j B j A 1j A j+1 A j B 1j A j A 1j+1 A j B j A j A j+1 B 1j-1 B j B j-1 B 1j B j-1 B j Tendons associated with stage j j odd 1 < j < n: A j B 1j B 1j-1 B 1j A 1j B j B j-1 B j A j B j B j-1 B j A 1j A 1j+1 A j A j+1 A j A j+1 Tendons associated with the n-th stage n 4 even: B n-1 B 1n A n B 1n B n-1 B n A n B n B 1n-1 B n A 1n B n Tendons associated with the n-th stage n odd: B 1n-1 B 1n A n B 1n B n-1 B n A 1n B n B n-1 B n A n B n Tendons separating stages j and j + 1 (where 1 j n-1): B 1j A 1j+1 B j A 1j+1 B j A j+1 B j A j+1 B j A j+1 B 1j A j+1 The spatia poygon composed of the 6 tendons separating stages j and j + 1 wi be referred to as the j-th sadde We ascertain that there are three different types of tendons: the ones abeed A * B * and caed vertica tendons the ones abeed A * A * and B * B * caed diagona tendons and the ones abeed B * A * and caed sadde tendons We introduce the vector of tendon engths L bj associated with stage j; its components are the engths of the tendons associated with stage j j = 1 n in the order they are isted above We aso introduce the vector L sj of tendon engths associated with the j-th sadde j = 1 n-1 defined in the same manner as L bj The vector L of tendon engths is defined by assembing these vectors as foows: L = [L T b 1 L T s 1 L T b L T s n-1 L T b n ] T () Anaogousy we introduce the tension vector T bj associated with the j-th stage j = 1 n and the tension vector T sj associated with the j-th sadde j = 1 n-1 The vector of a tensions is: T = [T T b 1 T T s 1 T T b T T s n-1 T T b n ] T (4) For further mathematica modeing and anaysis we assume that the tendons are massess and inear eastic the base and the top are rigid the bars are rigid axiay symmetric and for each bar the rotationa degree of freedom around the ongitudina axis of symmetry is negected The base is assumed to be inertiay fixed We negect the forces exerted upon the structure by externa force fieds (eg gravitationa) The system is affected at most by kinetic friction/damping We introduce an inertia frame of reference ˆb 1 ˆb ˆb as a dextra (right-handed) set of unit vectors whose origin coincides with the geometric center of the triange A 11 A 1 A 1 Axis ˆb is orthogona to A 11 A 1 A 1 pointing upward whie ˆb 1 is parae to A 11 A 1 pointing towards A 1 We introduce another reference frame ˆt 1 ˆt ˆt caed the top reference frame which is fixed in the top rigid body Its origin coincides with the geometric center of the top triange B 1n B n B n ˆt is orthogona to B 1n B n B n and points upward whie ˆt 1 is parae to B 1n B n pointing towards B n In the foowing we sha reate the anaysis to the methodoogy previousy described 4 Step 1: Generaised Coordinates The independent generaised coordinates necessary to describe the configuration of this system are: d ij a ij the decination and the azimuth of bar ij i = 1 j = 1 n defined as foows: d ij Õ is the ange between A ij Bij and ˆb and a ij is the Õ ange between ˆb 1 and the projection of A ij Bij onto pane (ˆb 1 ˆb ) (see Fig ) Figure decination and azimuth of bar ij 18 Internationa Journa of Space Structures Vo 18 No 1 00
C Sutan and R Sketon x ij y ij z ij the inertia Cartesian coordinates of the mass center of bar ij i = 1 j = n-1 c f u the Euer anges for a -1- sequence to characterize the orientation of the top reference frame with respect to the inertia frame X Y Z the inertia Cartesian coordinates of the geometric center of B n B n B n We introduce the foowing vectors: ~ q 1 = [d 11 a 11 d 1 a 1 d 1 a 1 ]T where i = 1 j = n-1 Here x Aij y Aij z Aij i = 1 j = 1n-1 are the inertia coordinates of A ij x Bij y Bij z Bij i = 1 j = 1n-1 are the inertia coordinates of B ij and ij i = 1 j = 1 n-1 is the ength of bar ij The inertia Cartesian coordinates of the noda points of the n-th stage A in and B in i = 1 are expressed using the transformation matrix for a -1- sequence C 1 : [ x Ain ] [ X - in sin (d in ) cos (a in ) R Ain = y Ain = Y - in sin (d in ) sin (a in ) z Ain Z - in cos (d in ) ] ~ q j = [d 1j a 1j x 1j y 1j z 1j d j a j x j y j z j d j a j x j y j z j ]T ~ q n = [d 1n a 1n d n a n d n a n ]T j = n-1 (5) ~ q T = [c f u X Y Z] T Here q ~ j j = 1 n is the vector of independent generaised coordinates associated with the j-th stage and q ~ T is associated with the rigid top The vector of independent generaised coordinates is q = [ q ~ T ~ 1 q T q ~ T ~ n q T T ]T (6) The number of independent generaised coordinates is equa to the number of tendons: E = N = 15n-1 4 Step : Geomet ry and Equiib rium Matrix The inertia Cartesian coordinates of the noda points can be determined as foows x Bi1 = x Ai1 + i1 sin(d i1 ) cos(a i1 ) y Bi1 = y Ai1 + i1 sin(d i1 ) sin(a i1 ) z Bi1 = i1 cos(d i1 ) (7) ij x Aij = x ij - sin(d ij ) cos(a ij ) y Aij = y ij - sin(d ij ) sin(a ij ) z Aij = z - ij cos(d ij ) (8) ij ij x Bij = x ij + sin(d ij ) cos(a ij ) y Bij = y ij + sin(d ij ) ij sin(a ij ) z Bij = z + ij cos(d ij ) (9) ij ij [ X Bin ] + C 1 Y Bin (10) 0 [ x Bin ] R Bin = y Bin = z Bin [ ] [[ + C 1 X Bin ] Y Bin (11) 0 Here X Bin and Y Bin are the Cartesian coordinates of B in with respect to the top reference frame and C 1 = [ X Y Z c(c ) c(u ) - s(c ) s(f ) s(u ) s(c ) c(u ) + c(c ) s(f ) s(u ) - c(f ) s(u ) c(c ) s(u ) + s(c ) s(f ) c(u ) s(c ) s(u ) - c(c ) s(f ) c(u ) c(f ) c(u ) where c(f ) = cos(f ) s(f ) = sin(f ) ] (1) Once the noda points coordinates are expressed in terms of the independent generaised coordinates symboic computation can be used to determine the engths of the tendons L i (q) i = 1 N and the L eements of the equiibrium matrix A ji = i i = 1 q j N j = 1N - s(c ) c(f ) c(c ) c(f ) s(f ) 44 Step : Particuar Configurations We sha now define a set of particuar configurations caed the cass of cyindrica symmetrica configurations This set is defined as foows Internationa Journa of Space Structures Vo 18 No 1 00 19
Tensegritty Structures Prestressabiity Investigation Figure Top view of a tensegrity tower in a cyindrica symmetrica configuration Trianges A 11 A 1 A 1 and B 1n B n B n are congruent equiatera trianges of side ength b A bars have equa ength A bars have the same decination d Bars 11 and [i]i i = n are parae; bars 1 and [1 + i]i i = n are parae; bars 1 and [ + i]i i = n are parae In these formuas [x] is a periodic function such that [x] = [x-] if x 4 and [x] = x if x = 1 ; hence bar [4]4 is actuay bar 14 A noda points A ij B ij i = 1 j = 1 n ie on the surface of a rectanguar cyinder The projections onto the base of the j-th sadde points A j+1 B 1j B j A j+1 B j A 1j+1 make a reguar hexagon Panes A 1j A j A j and A 1j+1 A j+1 A j+1 j = 1 n -1 are parae The distance between A 1j+1 A j+1 A j+1 and B 1j B j B j is the same for a j = 1 n - 1 and it is caed the overap h being positive if B 1j B j B j is coser to A 11 A 1 A 1 than A 1j+1 A j+1 A j+1 A top view of such a configuration is given in Fig The cass of cyindrica symmetrica configurations can be parameterized using two independent parameters a and h The corresponding vaues of the independent generaised coordinates are: d ij z ij = = d where i = 1 j = 1n j - 1 cos(d ) - (j-1)h where i = 1 j = n-1 0 Internationa Journa of Space Structures Vo 18 No 1 00
p C Sutan and R Sketon a [i]i = a a [i+1]i = a + 4p a [i+]i = a + p where i = 1n b x [p-1]p-1 = - + sin(d ) cos(a ) y [p-1]p-1 Î = - b + sin(d ) sin(a ) 6 x [p]p-1 = sin(d ) cos(a 1 ) y [p]p-1 Î = b + sin(d ) sin(a 1 ) b x [p+1]p-1 = + sin(d ) cos(a 1 ) y [p+1]p-1 Î = - b + sin(d ) sin(a 6 1) where p = and p - 1 < n b x [p-1]p = - - sin(d ) cos(a 1 ) y [p-1]p Î = b - sin(d ) sin(a 6 1) b x [p]p = - sin(d ) cos(a ) y [p]p Î = b - sin(d ) sin(a 6 ) x [p+1]p = - sin(d ) cos(a ) y [p+1]p Î = - b - sin(d ) sin(a ) where p = 1 and p < n f = u = X = Y = 0 Z = n cos(d ) - (n-1)h a if n = + 6k if n = 4+ 6k if n = 8 + 6k where k is a natura number Here a and d are reated by the constraint that a noda points ie on the surface of a cyinder: b sin(a + p_ sin(d ) = ) (14) ÎW In this geometry a tendons of a given type (sadde vertica and diagona) have the same engths given by S = Î WWWW b +h V = Î WW - b WW WW WW cos(a ) - b WW WW sin (a ) (15) ÎW D = Î WW + h WW - WW b WW WW WW WW W cos (a ) - h cos(d ) respectivey 5p 45 Step 4: Prestressabiity Conditions for Particuar Configurations Cyindrica symmetrica configurations which are prestressabe configurations wi be caed symmetrica cyindrica prestressabe configurations (SCPC) In order to anayze them we substitute (1) into the prestressabiity equations A(q)T = 0 yieding Â(a h)t = 0 (16) Since Â(a h) is a square matrix a necessary condition for prestressabiity is det(â(a h)) = 0 (17) So far the first four steps of the methodoogy were appied In order to go to the next step invoving numerica computation of one soution of the prestressabiity conditions we need to fix the number of stages Consider first the three stage tensegrity tower (n = ) in Fig 1 for which E = N = c = p a - if n = 5 + 6k p if n = 6 + 6k p a + if n = 7 + 6k (1) 46 T hree Stage Towers; Step 5: Numerica Soution A numerica soution of Â(a h)t = 0 T i > 0 i = 1 is found as foows First we fix = 04 m b = 07 m a = 5 deg (18) Internationa Journa of Space Structures Vo 18 No 1 00 1
Tensegritty Structures Prestressabiity Investigation and numericay sove det(â(ha )) = 0 for h using a ine search combined with a Newton-Raphson method This resuts in h p = 00875 m We next determine the kerne of Â(h p a ) Every vector T p in this kerne can be expressed as: T p = P[1 074 1 074 1 074 17 111 17 111 17 111 096 064 096 064 096 064 064 064 064 17 111 17 111 17 111 1 074 (19) 1 074 1 074] T where P is an arbitrary scaar (the rank of Â(h p a ) is ) Ceary for P > 0 a eements of T p are positive 4 7 Step 6: Essent ia Prest ressabiit y Conditions Based on this soution we assume the foowing structure for T: T = [ T Ś1 T Ś1 T Ś1 T V T V T V (0) T Ś1 T Ś1 T Ś1 ] T investigations and can be very easiy numericay soved aowing for the foowing facts to be estabished The essentia prestressabiity equation (det(a e ) = 0) yieds at most a cubic equation in h For a = 0 this equation reduces to a inear one whose soution is h = Î WWW - b (5) Numerica experiments showed that for fixed a the essentia prestressabiity equation can yied severa rea soutions for the overap h but ony one might satisfy the inequaities on the tensions (T ri > 0 i = 1 6) 5 1 Numerica experiments showed that at a symmetrica cyindrica prestressabe configuration for three stage towers the rank of A e is aways 5 thus the genera soution for tensions is: T r = PT 0 = P[0 T Ś10 0 T V0 0 0 ] T (6) Substitution of (1) (14) and (0) into (1) eads to a system of equations and 6 inequaities Of the equations ony 6 are independent the ones associated with the rows of A whose numbers are 1 7 9 8 and (see Sutan 1999 for detais) They yied the essentia prestressabiity conditions: A e T r = 0 T ri > 0 i = 1 6 (1) where A e is a function of h and a whose nonzero eements are given in the Appendix and T r = [ T S1 T V ] T () The ast equation in A e T r = 0 is h h - cos(d ) ( + T S1 ) + ( + ) = 0 () S D which since a tensions must be stricty positive resuts in 0 < h < cos(d ) (4) Thus the overap must be positive and ess than the height of a stage The much simper essentia prestressabiity conditions are further amenabe to anaytic where P is an arbitrary positive scaar (the pretension coefficient) and T 0 is a base of Ker(A e ) Equiibrium equations of the noda points yied the forces in the bars A bars of the first and third stage experience the same forces equa to C r1 whie the forces in a bars of the second stage are equa to C r These forces can be expressed as C ri = PC 0i i= 1 We give in Figs 4-6 the soution of the essentia prestressabiity conditions potting the overap as a function of a the variation of the corresponding height of the structure the decination of the bars and the basis tensions (T 0 ) and compressive forces (C 0 ) for a three stage tensegrity tower with = 04 m and b = 07 m (in these Figures SCPC stands for symmetrica cyindrica prestressabe configurations) We note that T 0 has been normaized such that T s Eucidean norm is 1 Finay we note that numerica experiments with the prestressabiity conditions A(a h)t = 0 T i > 0 i = 1 did not identify other particuar structures for the vector of tensions T Internationa Journa of Space Structures Vo 18 No 1 00
C Sutan and R Sketon Figure 4 Prestressabe configuration parameters for a stage tensegrity tower 48 Four Stage Towers; Step 5: Numerica Soution As a second exampe we consider a four stage tensegrity tower In this case the number of tendons (and independent generaised coordinates) is 48 For the numerica investigation step we assign b and a the same vaues as in (18) and sove the prestressabiity conditions for h P and T P We identify the foowing soution: h P = 00808 m T P = P [1 069 1 069 1 069 141 107 141 107 141 107 097 088 097 088 097 088 05 05 05 145 145 145 145 145 145 097 088 097 088 097 088 05 05 05 107 141 107 141 107 141 1 069 1 069 1 069] T (7) where P > 0 is an arbitrary scaar Internationa Journa of Space Structures Vo 18 No 1 00
Tensegritty Structures Prestressabiity Investigation Figure 5 Basis tensions for a stage tensegrity tower Figure 6 Basis compressions for a stage tensegrity tower 4 Internationa Journa of Space Structures Vo 18 No 1 00
C Sutan and R Sketon 4 9 Step 6: Essentia Prest ressabiit y Conditions Based on the numerica soution the foowing structure of T is assumed: T = [ T S1 T S1 T S1 T V T V T V T D T D T D T S T S T S T S T S T S (8) T V T V T V T D T D T D T S1 T S1 T S1 ] T Proceeding in the same way as for the three stage tensegrity towers we obtain a system of 48 equations of which ony 8 are independent (the ones associated with rows number 1 7 8 9 10 11 48) and 8 inequaities on the tendon tensions (see Sutan 1999 for detais) These are the essentia prestressabiity conditions expressed as A e T r = 0 T ri > 0 for i = 18 (9) where the structure of A e is * * 0 * 0 0 0 * * * 0 * 0 0 0 * * * * 0 * * * * A e = * * * 0 * * * * * * * 0 * * * * * * * 0 * * * * * * * 0 0 * * * * * 0 0 0 * 0 * The interested reader is refered to Sutan (1999) for the nonzero eements of A e Further investigations ed to the foowing concusions The ast ine of A e yieds the condition that the overap must be positive and ess than the height of a stage: 0 < h < cos(d ) (1) The essentia prestressabiity equation (det(a e ) = 0) yieds at most a poynomia equation of degree 4 in h For a = 0 this equation reduces to a quadratic one: 1h 15h Î WW - b + 4( - b ) = 0 () Numerica experiments showed that for fixed a the essentia prestressabiity equation can yied severa rea soutions but ony one might satisfy the inequaities on the tensions (T ri > 0 i = 18) Numerica experiments showed that at a symmetrica cyindrica prestressabe configuration of four stage towers the rank of A e is aways 7 thus the genera soution for tensions is: T r = PT 0 () where P is an arbitrary positive scaar (the pretension coefficient) and T 0 is a base of Ker(A e ) The forces in a bars of the first and fourth stage are equa to C r1 whie the forces in the second and third stage bars are equa to C r and can be expressed as C ri = PC 0i i = 1 Figs 7-9 show the soution of the essentia prestressabiity conditions (the overap) the decination of bars height of the structure as we as the variation of the basis tensions T 0 and compressions C 0 with a As for the three stage tower T 0 has been normaized such that T s Eucidean norm is 1 Numerica experiments with the prestressabiity conditions A(a h)t = 0 T i > 0 i = 148 did not identify other structures for the vector of tensions T except for the one indicated by (8) 5 GENERALIZATION The methodoogy has been successfuy appied to tensegrity towers with to 10 stages The same cass of symmetrica cyindrica prestressabe configurations has been investigated Considerabe reduction in the size of the prestressabiity conditions was obtained Our numerica experiments aowed for some important concusions as foows For fixed b and a ony one soution for h of the genera prestressabiity conditions has been identified In genera the corresponding overap h decreases with the number of stages as shown in the next tabe The data in this tabe were obtained using = 04 m b = 07 m a = 5 deg (4) Tabe 1: Overap (h) vs Number of Stages (n) n 4 5 6 7 8 9 h (mm) 87 80 77 75 7 7 71 Internationa Journa of Space Structures Vo 18 No 1 00 5
Tensegritty Structures Prestressabiity Investigation Figure 7 Prestressabe configurations parameters for a 4 stage tensegrity tower At a soution of the prestressabiity conditions the rank of A is N - 1 thus T = PT 0 with P > 0 arbitrary (here T 0i > 0 i = 1N) The tension vector T has a structure which impies symmetry of tensions with respect to the midde stage(s) Mathematicay this is expressed as foows: 1 If n is odd n = p + 1 T s structure respects the foowing rues: (a) The first and the ast (n-th) stage tension vectors are T b1 = T bn = [ ] T (b) The midde stage m = p + 1 tension vector is T bm = [T Vm T Dm T Vm T Dm T Vm T Dm T Dm T Dm T Dm ] T (c) The tension vectors of the i-th stage i = n-1 i m is T bi = [T Vi T Vi T Vi ] T and the tension vector of stage number n + 1 - i is T bn + 1-i = [T Vi T Vi T Vi ] T (d) The tension vector of the i-th sadde i = 1 n-1 is equa to the tension vector of the n-i-th sadde: T si = T sn-i = [T Si T Si T Si T Si T Si T Si ] T If n is even n = p: (a) The first and ast stages tension vectors are T b1 = T bn = [ ] T (b) The tension vectors of a other stages have the foowing structure: 6 Internationa Journa of Space Structures Vo 18 No 1 00
C Sutan and R Sketon Figure 8 Basis tensions for a 4 stage tensegrity tower Figure 9 Basis compressions for a 4 stage tensegrity tower Internationa Journa of Space Structures Vo 18 No 1 00 7
Tensegritty Structures Prestressabiity Investigation T bi = [T Vi T Vi T Vi ] T i = n-1 (c) The tension vector of the midde sadde m = p is T sm = [T Sm T Sm T Sm T Sm T Sm T Sm ] T (d) The tension vector of the i-th sadde i = 1n-1 i m is T si = [T Si T Si T Si T Si T Si T Si ] T and the tension vector of the n-i-th sadde is: T si = [T Si T Si T Si T Si T Si T Si ] T After identification of the structure of the tension vector T and symboica substitution of the expressions for the generaised coordinates and for T in the genera prestressabiity conditions we ascertained the foowing facts: The corresponding essentia prestressabiity conditions resut in a poynomia equation in h and inequaities on the tendon tensions There are 5n- inequaities if n is odd and 5n-4 if n is even Numerica experience showed that for fixed b and a there is aways at most one soution of this poynomia equation satisfying the inequaities on the tendons thus ony one soution of the essentia prestressabiity conditions Numerica experience showed that at a prestressabe configuration the rank of the essentia prestressabiity conditions matrix (A e ) is aways equa to the number of coumns of A e minus one It can be easiy proved in the same manner as for the three and four stage tensegrity towers that at a cyindrica symmetrica prestressabe configuration the overap is ess than the height of a stage and positive: 0 < h < cos(d ) (5) An important issue is the stabiity of these prestressabe equiibria It is worth mentioning that our numerica experience indicated that these equiibria are stabe This has been ascertained by computing the stiffness matrix of the structures - the Hessian of the potentia energy - at these prestressabe configurations The stiffness matrices turned out to be positive definite indicating stabiity of these equiibria 6 CONCLUSIONS Symboic and numerica computation have been combined in a methodoogy aimed at simpifying the prestressabiity conditions for tensegrity structures The proposed methodoogy has been successfuy appied to some compicated tensegrity structures termed tensegrity towers for the investigation of certain prestressabe configurations These studies reveaed that through the appication of the proposed methodoogy the reduction in the size of the prestressabiity conditions is considerabe For exampe for three stage tensegrity towers the prestressabiity conditions consist of noninear equations and inequaities For a certain cass of prestressabe configurations this system reduces to 6 equations and 6 inequaities whose numerica soution is much simper 7 APPENDIX The structure of A e for three stage tensegrity tower is: A e = * * * 0 0 * * * * 0 0 * * * 0 * * * * * 0 * * * * * * 0 * * * * 0 0 * * 8 Internationa Journa of Space Structures Vo 18 No 1 00
p p p p p p C Sutan and R Sketon and its nonzero eements are given by: A e11 = (-Îãb cos(d ) sin(a ) + cos(d ) sin(d ) - 6h sin(d )) 6S A e1 = ( Îãb cos(d ) sin(a - ) + sin(d )(cos(d ) - h)) 6S A e1 = - b cos(d ) cos(a ) A e16 = (-h sin(d ) + sin(d ) + Îãb cos(d ) sin(a - )) V D A ÎW e1 =- (b sin(d ) cos(a ) - sin ÎW (d )) A e = (b sin(d ) sin(a + ) - sin (d ) 6S 6S 6 b A e = b sin(d ) sin(a ) A e6 = sin(d ) sin(a + ) V DÎW 6 bcos(d ) A e1 = Îãb cos(d ) sin(a ) - h sin(d )) A e = (-h sin(d ) + cos(a + )) S S ÎW 6 cos(d ) A e4 = (- sin(d ) - b cos(a + )) A e5 = (9 cos(d ) sin(d ) + Îãb cos(d ) sin(a ) - 6h sin(d )) 4V 6D A e6 = (-b cos(d ) cos(a ) + 6h sin(d ) + Îãb cos(d ) sin(a )) 6D 1 A e41 = - (Îãb sin(a ) + 6b sin (a ) + 6 sin(d ) cos(a ) - 6Îã sin(d ) cos(a )) 6S 1 A e4 = (- Îãb sin(a ) 6b cos (a ) - b + 1 sin(d ) cos(a )) 6S 1 A e44 = ( sin(d ) cos(a ) - b + Îã sin(d ) sin(a )) V 1 A e45 = (-6b sin (a ) + sin(d ) cos(a ) + Îã sin(d ) sin(a ) - Îãb sin(a )) 6D 1 A e46 = ( -ÎWb sin(a ) - 6b cos (a ) - b + 1 sin(d ) cos(a )) 6D b A e51 = - (-b sin(a ) - sin(d ) sin(a ) + ÎW sin(d ) cos(a ) - bîw cos(a )) S b A e5 = - (b sin(a ) + Îã sin(d ) cos(a ) - sin(d ) sin(a )) S Internationa Journa of Space Structures Vo 18 No 1 00 9
Tensegritty Structures Prestressabiity Investigation b sin(d ) sin(a ) A e5 = A e55 = (sin(a ) + ÎW cos(a )) V D b sin(a ) ( sin(d ) - b cos(a )) h A e56 = A e61 = A e6 = - A e65 = A e66 = ( cos(d ) - h) D S D b where b sin(a + d = arcsin( p ) ) ÎW REFERENCES 1 Emmerich D G 1996 Emmerich on sef-tensioning structures Internationa Journa of Space Structures 11 (1 and ) 9-6 Sketon R E and Sutan C 1997 Controabe tensegrity a new cass of smart structures Mathematics and Contro in Smart Structures Proc SPIE 4th Symposium on Smart Structures and Materias 09 166-177 Sutan C and Sketon R E 1997 Integrated design of controabe structures Adaptive Structures and Materia Systems Proc ASME Int Congress and Exposition 54 7-7 4 Sutan C and Sketon R E 1998 Force and torque smart tensegrity sensor Mathematics and Contro in Smart Structures Proc SPIE 5th Symposium on Smart Structures and Materias 57-68 5 Sutan C 1999 Modeing design and contro of tensegrity structures with appicatio ns PhD dissertation Purdue University Schoo of Aeronautics and Astronautics 00 pages 6 Sutan C Coress M and Sketon R E 001 The prestressabiity probem of tensegrity structures Some anaytica soutions Internationa Journa of Soids and Structures 8(0-1) 5-55 7 Peegrino S and Caadine C R 1986 Matrix anaysis of staticay and kinematicay indetermined frameworks Internationa Journa of Soids and Structures (4) 409-48 8 Peegrino S 1990 Anaysis of prestresse d mechanisms Internationa Journa of Soids and Structures 6(1) 19-150 9 Vassart N and Motro R 1999 Mutiparametered formfinding method: appication to tensegrity systems Internationa Journa of Spaee Structures 14() 147-154 10 Motro R 1994 Form-finding numerica methods for tensegrity systems Spatia Lattice and Tension Structures Proceedings of the IASS-ASCE Int Symposium 704 71 11 Hanaor A 1988 Prestressed pinjointed structures - fexibiity anaysis and prestress design Internationa Journa of Soids and Structures 8(6) 757-769 1 Hanaor A 199 Aspects of design of doube ayer tensegrity domes Internationa Journa of Spaee Structures 7() 101-11 1 Tarnai T 1980 Simutaneous static and kinematic indeterminacy of space trusses with cycic symmetry Internationa Journa of Soids and Structures 16(1) 47-59 14 Kenner H 1976 Geodesic math and how to use it University of Caifornia Press Berkeey 0 Internationa Journa of Space Structures Vo 18 No 1 00