Symbolc Stffness Optmzaton of Planar Tensegrty Structures BRAM DE JAGER, * AND ROBERT E. SKELTON Department of Mechancal Engneerng, Technsche Unverstet Endhoven, P.O. Box 5, 56 MB, Endhoven, The Netherlands Department of Mechancal and Aerospace Engneerng, Unversty of Calforna at San Dego, La Jolla, Calforna, 99-4, USA ABSTRACT: Durng the desgn of mechancal systems one normally explots numercal analyss and optmzaton tools. We make a plea for symbolc computaton and gve an example where structural dsplacements under load are computed symbolcally. Geometrcal desgn parameters enter n ths computaton. The set of equlbrum condtons, lnear n the dsplacements, but nonlnear n the desgn parameters, s solved symbolcally. The resultng expressons reveal the geometry whch yeld optmal propertes for stffness or stffnessto-mass. Ths technque s appled to a class of repettve mechancal systems, namely tensegrty structures. A large scale example wth 5 degrees-of-freedom s computed successfully. The results make t possble to optmze the structure wth respect to stffness propertes, not only by approprately selectng (contnuous) desgn parameters, nfluencng geometry, but also by selectng the number of stages used to buld up the structure (a dscrete desgn parameter), nfluencng topology. Key Words: structrual optmzaton, topology optmzaton, symbolc computaton, tensegrty INTRODUCTION THE desgn of mechancal systems explots numercal analyss tools, e.g., Fnte Element Method packages for statc or dynamc analyss of contnuous models, or Mult Body Dynamcs packages for dscrete models. In practce, parameter optmzaton s restrcted, because normally the optmzaton s performed numercally and often requres a large number of cases to be computed. The assocated long computng tmes are cumbersome n practce, and when dscrete desgn parameters are nvolved, the computatons are even more nvolved. Furthermore, from an educatonal pont of vew, the numercal results are not partcularly enlghtenng. In ths respect, analytcal expressons are stll preferred, but dffcult to obtan for larger problems, even when consderng lnear ones. We consder a powerful class of mechancal systems, namely tensegrty structures of class, that are relatvely easy to analyze, and so permt analytcal solutons. Ther man advantage s that an equlbrum s possble n dfferent confguratons or shapes. Tensegrty structures are web-lke mechancal structures that consst of two types of members: tensle ones (tendons) and compressve ones (bars). Ths class of systems has been studed for a long tme, see, e.g., Maxwell (89), whose termnology conssted of tes and struts nstead of tendons and bars. In a class *Author to whom correspondence should be addressed. E-mal: A.G.de.Jager@wfw.wtb.tue.nl tensegrty structure (Skelton et al.,, ) the bar endponts,.e., the nodal ponts, are only connected to tendons, not to other bars. Tendons are exclusvely loaded n tenson, otherwse they would buckle because they are very slender. Bars are normally loaded n compresson only and not n tenson. No bendng s assumed to occur n bars. By changng the length of tendons under load, a tensegrty structure can be made very stff for any statc load actng on the nodal ponts. It s not always feasble to change the length of all tendons, so the stffness propertes nherent to the structure are mportant. Snce those propertes depend on the geometry of the structure t s of nterest to study ths nfluence. Our man goal s to obtan gudelnes n the desgn of planar tensegrty structures of class. Structural aspects studed are changes n stffness and stffness-to-mass rato due to varatons n the geometry of a planar tensegrty structure. The procedure we employ s as follows. Two desgn parameters are ntroduced that determne the geometry of a cantlever-beam-lke structure bult up from basc tensegrty crosses, namely the overlap factor and the slenderness rato of the structure. The structure s repettve, where the basc pattern, the tensegrty cross, s repeated n tmes. Usng equlbrum condtons (Wllamson and Skelton, 998) and consderng small devatons from the equlbrum, a set of equatons lnear n the dsplacements can be formulated for each value of n. The desgn parameters enter nonlnearly n these equatons. JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 5 March 4 8 45-89X/4/ 8 $./ DOI:.77/4589X49 ß 4 Sage Publcatons
8 B. DE JAGER AND R. E. SKELTON These sets of equatons can be solved symbolcally because they are lnear n the unknowns, and the parameter dependency can be parameterzed polynomally when over-parameterzaton s used,.e., three nstead of two parameters. Gven the analytcal soluton for the dsplacements of the degrees-of-freedom (DOF), as functons of the desgn parameters, the superfluous desgn parameter can be elmnated, and t s straghtforward to obtan values for the two desgn parameters that mnmze the dsplacement of a specfc pont of the structure for a gven load at the nodal ponts, or to obtan values that optmze the stffness or stffness-tomass rato. Large scale examples wth up to 5 DOF are computed successfully. Results are obtaned for three dfferent consttutve equatons for the tendons, namely a stress stran relaton that gves a constant stffness k, or a stffness that depends on the (unstressed) length l of the tendon as EA=l or EV=l. The results depend on the type of materal behavor selected and on the number n of tensegrty crosses that are used to buld up the structure. Besdes structure optmzaton we also study possble geometres of a specfc tensegrty structure that stll allows for an equlbrum that satsfes the tensle force requrements for the tendons. Ths s necessary, because not every value of the desgn parameters wll result n a stable tensegrty structure. In the rest of ths paper we frst dscuss several aspects of modelng tensegrty systems of class wth an ncreasng number n of basc buldng blocks. Then the results are presented and dscussed, and gudelnes for the desgn of these structures are formulated. PLANAR TENSEGRITY STRUCTURES A tensegrty structure conssts of bars and tendons, arranged n such a way that the structure has ntegrty and s not a mechansm. Ths s acheved by prestressng the tendons by a tensle force. The term tensegrty stems from the words tensle and ntegrty, that appear n the defnton of a tensegrty structure. A planar tensegrty structure s one that only extends n the plane. A tensegrty structure can be of class, where bars are only connected by tendons, and do not connect drectly, or of class, where a connecton can connect up to two bars and a number of tendons. Ths can be generalzed to a class k defnton. Often a tensegrty structure s made up of nested tensegrty structures, gvng t a fractal character. Ths s benefcal for analyss and desgn, because only a lmted number of structures needs to be nvestgated. Those structures can then be used to buld up a more complex structure. Descrpton of Planar Tensegrty Structures An elementary stage, numbered, of a planar tensegrty structure of class s gven n Fgure. Ths stage can be repeated ndefntely, by replcatng t, shfted some dstance of the horzontal dmenson, to buld up a planar structure n x-drecton. It could also be replcated n y-drecton or both. Indcated are the numberng of the tendons that belong to stage, gven by t,wth. Also ndcated are tendons of stages and þ that are connected to the four endponts (nodes) n 4 of the two bars of stage. Note that the number of tendons s not mnmal. For nstance, all dagonal tendons t 4, 5, 8, 9 can be removed, whle the structure stll has ntegrty and does not become a mechansm. Dagonal tendons are ncluded because t avods nfntesmal movements of the stages relatve to each other wthout causng nfntesmal correctng forces. Wthout dagonal tendons the stffness s derved from second order effects (.e., t s zero n the lnear approxmaton, except for prestress). So, a better approach to get a mnmal number of tendons s to elmnate vertcal tendons and keep some dagonal tendons. Ths s done for the symbolc model. The left sde of the structure has to be modfed for the boundary condton, and s gven n Fgure. Besdes modfcatons for the dfferences n boundary geometry, n 4 t7 n t n 4 t7 n + t + n 4 t7 + n + n y x t! 9 t 8 t 6 t5 t4 t t n t n t9 t + t + 5 t t4 + t8 t + 9 t + 8 t6 n + t + n t6 + n + Fgure. Sngle stage of planar tensegrty structure. Bars:, tendons: ---.
Symbolc Stffness Optmzaton of Tensegrty Structures 8 n t n + t + n 4 t7 + n + t t5 t + t + 5 t t 4 t4 + t + 9 t + 8 n y x t n + t + n t6 + n + Fgure. Left stage of planar tensegrty structure, ¼ : n 4 t 7 n t t 9 n 4 t 7 n 4 t 9 y n x t 8 t 6 t5 t4 t t8 n t n t6 n t t Fgure. Rght stage of planar tensegrty structure, ¼ n: the left sde removes the three DOF of the rgd body, n effect, t restrcts movement of the upper left node n both x and y-coordnate drecton,.e., the node s translatonally fxed, and of the lower left node n the x-drecton. A result of the restrctons s that the vertcal left tendon t of the structure cannot rotate, although both bars of stage ¼ are stll free to rotate. Note that tendons t 6 9 no longer appear for ¼ and that some tendons connect to other nodes than n the prevous fgure. The rght sde s n Fgure. There are no restrctons specfed at ths boundary. Only dfferences n geometry are taken nto account, the connecton of some tendons s to dfferent nodes than n Fgure. The man goal of the models s to enable evaluaton of measures of performance. The symbolc lnear model presented later s useful n dervng analytcal expressons for statc stuatons, but not for evaluaton of dynamc performance. The nonlnear model can be used to evaluate the results wth smulatons, to check stablty, and to assess robustness ssues. TENSEGRITY STRUCTURE MODELS Two models are developed, a symbolc (lnear) model for small dsplacements and a numerc (nonlnear) model for arbtrary dsplacements and dynamc analyss. The symbolc model s needed to get analytcal expresson for stffness and stffness-to-mass rato as a functon of the geometrcal parameters. The numerc model s used to determne statc equlbrum forces, whch s done by solvng a lnear programmng problem that ncludes postvty constrants for the tendon forces. The basc assumptons n settng up the models are:. A bar s straght and of unform cross secton and densty. The central moment of nerta for rotaton of a bar around ts prncpal axs s zero. A bar s of fxed length, so nfntely stff axally 4. A tendon s massless 5. A tendon has no torsonal or bendng stffness, but has fnte axal stffness 6. A bar has two nodal endponts, whch are of zero dmenson 7. A tendon s connected to a bar at a nodal pont only 8. External loads are only appled at a nodal pont 9. External loads do not nclude bendng or torsonal loads. There are no potental felds (e.g., gravty) Owng to these assumptons, the bars are axally loaded only, except durng transents. Although members n a tensegrty structure are axally loaded only, the structure tself has a fnte stffness for bendng and torson.
84 B. DE JAGER AND R. E. SKELTON Nonlnear Dynamc Model The model of the complete structure s qute elementary, beng bult up of rgd bars that are connected by flexble tendons, and can best be developed by a classcal Newtonan formulaton, because we are also nterested n forces nternal to the structure. The model for a sngle bar, see Fgure 4, movng n the plane s m p ¼ F b J ¼ M b usng as bar coordnates the poston p of the center of mass and the orentaton angle around ths center. The mass m and central moment of nerta J are the physcal parameters that specfy the dynamcs of the bar. We can compute the forces F b and moment M b from the nodal force vectors f n and f n4, assumed gven n Cartesan components, by F b ¼ f n þ f n4 M b ¼ l b ½sn cos Š f n þ l b ½ sn cos Š f n 4 : The model for a tendon can be derved from classcal contnuum mechancs. A smple model, lnear elastc, for materal behavor s ¼ E" wth E the modulus of elastcty, and where ¼ F t =A, the stress, s the rato of tendon force and cross-sectonal area, and " ¼ l=l, the stran, s the elongaton l ¼ l l dvded by the unstressed length l. Ths gves F t ¼ EA ðl l Þ¼k t ðl l Þ l to compute the tendon force magntude F t gven l and l. To compute the unstressed length when both F t and l are known, use l ¼ l þðf t =EAÞ : ðþ Note that the length l can be computed as the Eucldean norm of a tendon vector t, l ¼ktk. A tendon vector t s computed as the dfference of the two nodal pont vectors that the tendon connects to, and taken to pont n up/rght drecton, where rght takes precedence, t ¼ p nj p nk. The Cartesan coordnates p n of the nodal ponts can be computed as " # p n ¼ p l cos b sn " # p n4 ¼ p þ l cos b : sn The stressed length l determnes the tendon force magntude F t. The drecton of the tendon force vector f t comes from the tendon vector t because those vectors are algned f t ¼ F t ktk t where the tendon vector needs to be scaled by ts Eucldean norm. Nodal forces f n are computed by summng tendon forces f t for those tendons connected to a partcular node, takng account of the sgn conventon, f n ¼ X f t þ w n, where w n s an external load actng on nodal ponts. The equatons for ndvdual bars can be composed n the usual way to form the followng set of dfferental equatons M q ¼ Tðq, wþ q T ¼½p,,..., p nb, nb Š where the generalzed coordnate q gathers the bar postons p and orentatons, the load w gathers the nodal loads w n, the generalzed force T follows from Fgure 4. Elementary bar ( p, ) n planar tensegrty structure.
Symbolc Stffness Optmzaton of Tensegrty Structures 85 the bar forces F b and moment M b, and M s the mass matrx, composed from the m and J terms for the bars. For a statc model q s equated to zero and the resultng algebrac equatons, Tðq, wþ ¼, represent the equlbrum condtons. For a planar tensegrty structure wth gven geometry and topology,.e., gven the postons of the nodal ponts and the connectons between those ponts, the requred tendon forces to make a certan confguraton q under load w an equlbrum can be computed by solvng a lnear programmng problem. The computed tendon forces requred for an equlbrum are not unque and can also be scaled. To see why a lnear programmng formulaton s possble one has only to realze that the tendon forces F t and load w appear affnely n the equlbrum condtons, so t holds that AðqÞF ¼ w represents the equlbrum condtons, wth AðqÞ a fat matrx of full rank and F a vector of tendon forces F t. The lnear programmng problem that needs to be approached for feasblty s mn, sub AðqÞF ¼ w, F > : The problem s feasble,.e., has a soluton for AðqÞF ¼ w wth postve tendon forces F, f can be made smaller than. As soon as that s obtaned the optmzaton can be cut short. See Fgure 5 for possble Planar tensegrty, standard confguraton y-poston [m].5.5 4 5 6 7 Planar tensegrty, possble confguratons for bar y-poston [m].5.5 4 5 6 7 Planar tensegrty, possble confguratons for bar y-poston [m].5.5 4 5 6 7 Planar tensegrty, possble confguratons for bar y-poston [m].5.5 4 5 6 7 x-poston [m] Fgure 5. Overvew of possble confguratons q for planar tensegrty.
86 B. DE JAGER AND R. E. SKELTON equlbrum confguratons q of a three-stage planar tensegrty structure. Ths result s obtaned by grddng for a sngle nodal coordnate, gvng several q s, and testng feasblty for all grd ponts. The shaded regon n pcture of Fgure 5 llustrates the set of possble locatons (equlbra wth postve tendon forces) of the nodal pont located at (,) n the top pcture. Lkewse, n pcture of Fgure 5, the shaded regon represents the set of possble equlbra for the nodal pont at (,) n the top pcture. The fourth pcture descrbes the possble locatons of the nodal pont at (,). The shaded regons extend n an obvous way outsde the boundares of the pctures. Note that only three possble nodal confguratons are presented. Other confguratons follow from symmetry relatons. For cases where two or more ponts are smultaneously perturbed the computatonal burden to establsh feasblty becomes prohbtve, due to the grddng, and a more drect approach would be benefcal. Lnear Symbolc Statc Model The symbolc model s derved for a planar tensegrty system, as seen n Fgure 6 for a three-stage structure, wth a mnmal number of tendons, so compared to Fgure 5 the nner vertcal tendons and the uneven pars of dagonal tendons are removed. Only the horzontal and the left and rght vertcal tendons and the dagonal tendons that cross the overlap are ncluded n the model. The equlbrum condtons for small perturbatons of the DOF are used to derve a set of equatons that s lnear n the perturbatons of the DOF. Loadng the structure and computng the deflecton wll then gve nsght n the stffness and stffness-to-mass propertes of the structure. For the stffness analyss t s assumed that the load s a vertcal force at the top/rght node of the structure and the relevant dsplacement s of ths node, although arbtrary load condtons can be specfed. The goal s to optmze the geometry, characterzed by the overlap between the stages of a multstage tensegrty structure and the angles of the bars. The optmum depends on the assumpton on the stffness of Fgure 6. Tensegrty wth mnmal number of tendons. Bars:, tendons: - - -. the tendons, on the number of stages, and on the slenderness of the structure. The number of stages, overlap, and slenderness together determne the bar angle, so not all factors are ndependent. To get an easy parameterzaton, the overlap and slenderness are used. STIFFNESS AND STIFFNESS-TO-MASS RATIO To characterze the geometry, two nondmensonal parameters are used, the slenderness rato l ¼ l x =l y, wth l x and l y the horzontal and vertcal dmensons of the structure, and the overlap factor s n the overlap h ¼ sl x between stages. The overlap h s the dstance between the rght nodes of stage and the left nodes of stage þ. The parameters s and l wll be vared n characterzng the solutons. The external force w and the stffness factor k of the tendons always appear n the combnaton w=k n the deflecton. In the stffness the force w drops out and k appears affnely. For the stffness-to-mass rato the mass s computed assumng a constant cross-sectonal area A b and the same specfc mass for all bars. The mass of the bars s then proportonal to ther lengths, whch can be expressed as a functon of l y, l, and s, where l y appears affnely. The horzontal projecton of a bar s l ¼ ll y ððn Þsþ Þ=n wth n the number of stages, the vertcal projecton of a bar l ¼ l y, the length of a bar s l b ¼ðl þ l Þ=. The lengths of the tendons are l, l h, h, l h, or l d ¼ðl þ h Þ =, respectvely, dependng on the tendon. The tendon lengths depend on s. Only for < s < = ðn þ Þ the topology s well-defned, except for n ¼ where < s < holds. Outsde ths range for s we wll have tendons that run parallel to each other, so an equlbrum wth postve tendon forces s not possble, and the topology has to be changed, see (De Jager and Skelton, ). To reduce the number of symbols and to speed up the symbolc computatons a unt load s assumed. Furthermore, l y and the constant factor n k are equated to, because ther nfluence on the results s easy to determne. Then, n the equatons three parameters are used, l, s, and l d, although the length of the dagonal tendons, l d, s a known functon of l and s. Ths s done to prevent a square root appearng n the equatons. Now only ratonal polynomals are nvolved. Ths overparameterzaton allows the computaton to proceed even for a large number of stages. The relaton for l d s substtuted n the soluton of the set of equatons to get the fnal results reported later. To compute the results the symbolc (analytcal) model of planar tensegrty structures of class s used. See Fgure 6 for an example of a three stage tensegrty
Symbolc Stffness Optmzaton of Tensegrty Structures 87 structure. For the stffness of the tendons three cases are explored:. k t ¼ k s constant and the same for all tendons. Here the length of a tendon does not nfluence ts stffness propertes.. k t ¼ EA=l t, wth l t the tendon length, E the modulus of elastcty, and A the cross-sectonal area of a tendon. It s assumed that the tendons are all of the same materal and have the same cross-sectonal area, so EA s the same for all tendons. Ths relaton can gve negatve stffness, e.g., when s <.. k t ¼ EV=lt. Ths relaton s relevant when t s assumed that n the prevous relaton for k t the cross-sectonal area A vares nversely proportonal to the length, due to a constant volume restrcton. Ths relaton always gves postve stffness. The model s vald for nfntesmally small varatons n the DOF of the structure around an equlbrum and only equates the equlbrum condtons n those perturbed states due to a certan load, t s not a dynamc model and t also does not account for prestress, ths havng a small effect n our structure due to the use of dagonal tendons. Wth ths assumpton we can get a model consstng of a square system of equatons lnear n the unknown perturbatons n the DOF, wth the loads as a forcng functon (rght-hand-sde). Ths lnear system of dmenson 6n wll be solved symbolcally. The left sde of the structure s fxed to earth, removng three DOF. So for n ¼ 56 a square system of sze 5 5 s solved symbolcally, a remarkable feat. The results gve the dsplacements of the DOF, and also all results that can be derved thereof, as a functon of the load condtons w, the tendon stffnesses k t, and the geometrcal parameters l y, l, and s. For optmzng the geometry we use a sngle load condton and two crtera to optmze. The load s a vertcal force w at the top/rght node. The frst crteron s the stffness of the structure regarded as a beam, w=y, the rato of force on and dsplacement of the top/rght node. The second crteron s the stffness-to-mass rato, w=y=m tot,wthm tot ¼ nm the total mass and m the mass of a sngle bar, equal to A b l b. The resultng relatons are proportonal to k, EA, or EV and nversely proportonal to varous powers of l y, dependng on the stffness model, and further have a polynomal denomnator (ncludng powers of square roots) n s and l. To obtan statonary ponts these relatons are dfferentated wth respect to s, and then equated to zero. The resultng algebrac relaton can be solved for s analytcally, but only when the degree of the polynomal s not too large, but ths s the case here, as wll become clear from the results. In general ths soluton s qute complex for polynomals of degree and larger, and a graphcal soluton wll gve addtonal nsght. RESULTS FOR STRUCTURAL ANALYSIS We now present and dscuss the results for several cases:. three tendon stffness models,. stffness and stffness-to-mass rato,. structures wth dfferent number of stages. Case : k t ¼ k For ths case the structural stffness K s of the form a K ¼ k b l s þ b l s þ b l : þ b Table gves the numerc values of the parameters n ths relaton. When the tendons have dfferent stffnesses k t, the parameters b j are composed from terms lke =k t. In ths case the optmal values of s are ndependent of l, l y, and k, and are also gven n the table. Inspecton shows that the optmum s equal to s ¼ðn Þ= ð5n Þ. In Fgure 7 ths relaton s depcted. Except for n ¼ all values are outsde the range < s < =ðn þ Þ for whch the assumed topology Table. Parameters for stffness for k t ¼ k and optmal value of s. n a b b b b s 4 7 4 / 4 8 9 4 5 6 7/9 8 6 9 7 55 64 5/ 6 95 65 56 /79 64 64 667 4 /5 64 8 6699 54 795 496 7/9 8 56 75 59 445 684 85/ Optmal overlap factor s.4.9.8.7.6.5.4. 4 6 8 4 Number of stages n Fgure 7. Optmal overlap factor s as functon of number of stages n.
88 B. DE JAGER AND R. E. SKELTON makes sense. However, for ths specfc tendon stffness relatonshp, other approprate topologes gve the same result for the structural stffness, and so also for the optmal geometry. In fact, the optmum tends to be lke a super tensegrty cross, where the overlap factor s so large that stages are only slghtly shfted wth respect to each other, see Fgure 8. For the stffness-to-mass rato, we have to dvde the prevous relaton for K by the mass M of all bars, whch s of the form pffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff M ¼ A b l y a l s þ a l s þ a l þ a The optmal s for the stffness-to-mass rato s a functon of l, but not of l or k. For n ¼ and n ¼ the optmal s s gven n Fgures and. The negatve K/M x.5 wth the parameters a ¼ n n þ, a ¼ ðn Þ, a ¼, and a ¼ n. See Fgures 9 and to get an dea how the stffness-to-mass rato looks as a functon of l and s. Here, the stffness-to-mass rato s scaled by A b l y. Note that the dependency on s s not that large, whch ndcates that the choce of overlap s s not crtcal. The dependency on l s much more mportant. Furthermore, the stffness decreases dramatcally f the number of stages ncreases. Ths could be expected, because ncreasng the number of stages ncreases the number of tendons and, because the tendon stffness s constant, therefore the stffness of the structure decreases. s l Fgure. Stffness-to-mass rato for n ¼ and k t ¼ k:.4.. Fgure 8. Super tensegrty cross. Bars:, tendons: - - -..4 4 6 8 Fgure. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ k..4 K/M... s l Fgure 9. Stffness-to-mass rato for n ¼ and k t ¼ k:...4 4 6 8 Fgure. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ k.
Symbolc Stffness Optmzaton of Tensegrty Structures 89 solutons for s are outsde the range of valdty of the model. The negatve overlap s reduces the length of the bars, but t leaves us wth a stuaton that some tendons are requred to transmt a compressve force, whch s not possble. A soluton would be to move members from the tendon class to the class of bars, causng nonsmoothness n the mass relaton. Probably ths wll cause s ¼, or a class tensegrty, to be the optmal soluton. For more nformaton on these topologes see (De Jager and Skelton, ). Case : k t ¼ EA=l t In ths case the stffness s gven by K ¼ EA a= b l s þ b =ð þ l s Þ = þ b l s l y þ b l s þ b l þ b : The stffness s stll nfluenced by the number of stages, but n general less than n the case of constant tendon stffness. One could expect the stffness to be ndependent of the number of stages, but ths s not true. For l ¼ and s ¼, where only a=ðb = þ b Þ s mportant, the stffness depends manly on the number of dagonal tendons, so decreases proportonal to =n, as for the frst stffness model. For larger l the nfluence of the dagonal tendons dmnshes, so the stffness s less nfluenced by n. Now for both stffness and stffness-to-mass rato the optmal values of s depend on l and are gven n Fgures 5 8. The optmal values for s do change, and smaller values of s are preferred n ths case. Ths s caused by the stffness relaton, where now for s <, or n other cases were the topology becomes napproprate, the computed stffness become negatve, whch s qute proftable for the stffness of the structure. Table gves the numerc values of the parameters n ths relaton. The stffness-to-mass rato s gven n Fgures and 4. Note that these fgures show a larger senstvty for s, so a proper choce of s s mportant. x Table. Parameters for stffness for k t ¼ EA/l t. K/M.5 n a b b = b b b b 8 4 5 4 4 48 9 6 8 8 448 85 64 6 5 5 84 65 45 4 56 48 589 744 87 9 65 4 64 89 54 5848 77 89 546 496 8 768 54 8768 56 8 845 684 s l Fgure 4. Stffness-to-mass rato for n ¼ and k t ¼ EA=l t :.4. Fgure. Stffness-to-mass rato for n ¼ and k t ¼ EA=l t. 4 6 8 Fgure 5. Optmal s for stffness, n ¼ and k t ¼ EA/l t.
9 B. DE JAGER AND R. E. SKELTON Case : k t ¼ EV=l t In ths case the stffness s gven by! K ¼ EV ly a= b 44l 4 s 4 þ b 4l 4 s þ b 4l 4 s þ b l s.4. þb 4 l 4 s þ b 4 l 4 þ b : 4 6 8 Fgure 6. Optmal s for stffness, n ¼ and k t ¼ EA/l t. Table gves the numerc values of the parameters n ths relaton. Two examples of the stffness-to-mass rato are n Fgures 9 and. Note the steep drop when n ¼ for values of s that are away from the (small) values of s that wll appear to be optmal. Also n ths case the stffness may decrease as the number of stages ncreases. The stffness for the case l ¼ and s ¼, proportonal to a=b decreases proportonal to =n, as for the other stffness models. A larger number of stages mples a shorter tendon length and hgher stffness, whch s offset by the ncrease n the number of tendons. Here, the number of tendons ncreases proportonal to n whle the stffness of tendons generally ncreases wth n. However, the length of the vertcal and dagonal tendons equals l y f s ¼, and appears to be the lmtng factor n the stffness. Ths also provdes an answer to the observaton that the dependency on l for s ¼ s very small, so the stffness does hardly change when the structure s extended n x-drecton, the stffness manly determned by the vertcal and dagonal tendons. Ths may ndcate that not only staggerng n x-drecton, but also n y-drecton may be needed to obtan optmal stffness propertes for ths stffness relaton, or that the dagonal tendons need a larger EV factor for ncreasng n..4.4.....4 4 6 8 Fgure 7. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ EA/l t..4 4 6 8 Fgure 8. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ EA/l t. Table. Parameters for stffness for k t ¼ EV=l t. n a b 44 b 4 b 4 b b 4 b 4 b 6 9 4 8 6 5 6 4 8 4 48 84 56 8 4 5 46 949 768 85 496 6 89 9585 776 64658 88 984 4 6556 6556 6949 65544 6949 66 88 65 48576 64 5488 798 99 4487 44 684 546 67776 8 4944 4445 49494856 7494 567668 55688 845 6845456
Symbolc Stffness Optmzaton of Tensegrty Structures 9.4 K/M... s l Fgure 9. Stffness-to-mass rato for n ¼ and k t ¼ EV/lt.. 4 6 8 Fgure. Optmal s for stffness, n ¼ and k t ¼ EV/lt. K/M x.5 s l Fgure. Stffness-to-mass rato for n ¼ and k t ¼ EV/lt..4. 4 6 8 Fgure. Optmal s for stffness, n ¼ and k t ¼ EV/lt. Agan the optmal values of s for both stffness and stffness-to-mass rato depend on l, and are gven n Fgures 4. Note that n ths case the optmal values of s are mostly wthn the valdty of the model. It appears that the optmum s reached n such a way that no extreme length of tendons occurs. It s clear that a long tendon would negatvely nfluence the stffness of the structure due to the stffness relaton..4.. Dscusson To llustrate the nfluence of the number of stages on the stffness and stffness-to-mass rato we dscuss two stuatons. Frst, when l ¼ ands ¼, so only a=b or a=ðb = þ b Þ s relevant for K, stffness decreases proportonal to =n, so n the same way, for all three stffness models. Ths result s expected, because the dagonal and vertcal tendons only nfluence the stffness, and ther length s the same and ther number.4 4 6 8 Fgure. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ EV/lt. ncreases wth n, so when l y ¼, stffness s also the same for the three models. In ths case the mass ncreases proportonal to n, so the stffness-to-mass rato s proportonal to =n.
9 B. DE JAGER AND R. E. SKELTON Second, when l! and s ¼, so only a=b j, j ¼,..., 4, s relevant, the stffness decreases for the frst two stffness models, fast for the frst and slow for the second, but ncreases for ncreasng n for the thrd model, see Fgure 5. Ths s also as expected. The ncrease n the number of stages decreases the length of the dagonal and the horzontal tendons, so they become stffer n the last two stffness models but also the number of tendons ncreases, so for the second model a constant stffness could be expected. Snce the change n length of the dagonal tendons s less than proportonal to =n, but ther number ncreases wth n, there s stll a.4...4 4 6 8 Fgure 4. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ EV/lt. slght decrease n stffness for the second model. For the thrd model the stffness ncrease domnates, so the stffness of the structure ncreases for ncreasng n. In ths case the mass s ndependent of n, as long as l s much larger than n, so the stffness-to-mass rato relaton s the same as the stffness relaton. As the second stffness model s probably the most relevant, one can conclude that t s best to choose the lowest number of stages possble, preferably equal to, gvng the hghest stffness. Other consderatons, e.g., falure modes of the structure or shape requrements, may lead to a larger number of stages. Because the stress n the horzontal tendons wll probably not vary very much f the number of stages vares, falure of the bars due to bucklng wll be the domnant falure mode for larger l that could gve rse to the use of a larger number of stages. Other consderatons could be geometry of the structure, e.g., f the outsde or closure of the structure has to have some specfed shape, whch may lead to a larger number of stages. Besdes an optmal choce of s, the results lead to a desgn guded by possble falure of the structure, strength, and bucklng consderatons, to obtan the mnmal number of crosses necessary. A comparson wth results obtaned for optmal stffness-to-mass rato for contnua, the Mchell truss (Mchell, 94), reveals that t may be better to not use tensegrty structures of class when dealng wth undrectonal statc loads, e.g., loads due to gravty, but to use tensegrty structures of class, where n a Stffness for l =, s = k t = k k t = EA/ l t k t = EV/ l t Stffness K Number of stages n Fgure 5. Stffness K as functon of number of stages n for l ¼,s¼.
Symbolc Stffness Optmzaton of Tensegrty Structures 9 nodal pont up to two bars can meet, at least for the planar case. In our setup ths s equvalent wth s ¼. For space applcatons, or other applcatons where a domnant drecton of forces s not prevalent, tensegrty structures of class have ther merts. REFERENCES De Jager, Bram and Skelton, Robert E.. Stffness of Planar Tensegrty Beam Topologes, In: Proc. Thrd World Conference on Structural Control, Wley, Chchester, pp. 45 4. Maxwell, James Clerk 89. The Scentfc Papers of James Clerk Maxwell, Dover, New York. Mchell, A.G.M. 94. The Lmts of Economy n Frame Structures, Phl. Mag., 8(47):589 597. Skelton, Robert E., Helton, J. Wllam, Adhkar, Rajesh, Pnaud, Jean-Paul and Chan, Waleung. An Introducton to the Mechancs of Tensegrty Structures, Chapter 7 In: The Mechancal Systems Desgn Handbook: Modelng, Measurement, and Control, CRC Press, Boca Raton, pp. 5 88. Skelton, Robert E., Pnaud, Jean-Paul and Mngor, D.L.. Dynamcs of the Shell Class of Tensegrty Structures, J. Frankln Inst., 8( ):55. Wllamson, Darrell and Skelton, Robert E. 998. A General Class of Tensegrty Systems: Equlbrum Analyss, In: Proc. th ASCE Engneerng Mechancs Conf., La Jolla, CA., pp. 76 79.