THE design of mechanical systems exploits numerical
|
|
- Liliana Owens
- 5 years ago
- Views:
Transcription
1 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 X/4/ 8 $./ DOI:.77/4589X49 ß 4 Sage Publcatons
2 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: ---.
3 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.
4 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.
5 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] Planar tensegrty, possble confguratons for bar y-poston [m] Planar tensegrty, possble confguratons for bar y-poston [m] Planar tensegrty, possble confguratons for bar y-poston [m] x-poston [m] Fgure 5. Overvew of possble confguratons q for planar tensegrty.
6 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
7 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 / / / / / / / Optmal overlap factor s Number of stages n Fgure 7. Optmal overlap factor s as functon of number of stages n.
8 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: 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: Fgure. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ k.
9 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 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 Fgure 5. Optmal s for stffness, n ¼ and k t ¼ EA/l t.
10 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 : 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 Fgure 7. Optmal s for stffness-to-mass rato, n ¼ and k t ¼ EA/l t 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
11 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 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 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 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.
12 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 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.
13 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 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): 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 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 A General Class of Tensegrty Systems: Equlbrum Analyss, In: Proc. th ASCE Engneerng Mechancs Conf., La Jolla, CA., pp
Analysis of Continuous Beams in General
Analyss of Contnuous Beams n General Contnuous beams consdered here are prsmatc, rgdly connected to each beam segment and supported at varous ponts along the beam. onts are selected at ponts of support,
More informationModule 6: FEM for Plates and Shells Lecture 6: Finite Element Analysis of Shell
Module 6: FEM for Plates and Shells Lecture 6: Fnte Element Analyss of Shell 3 6.6. Introducton A shell s a curved surface, whch by vrtue of ther shape can wthstand both membrane and bendng forces. A shell
More informationFinite Element Analysis of Rubber Sealing Ring Resilience Behavior Qu Jia 1,a, Chen Geng 1,b and Yang Yuwei 2,c
Advanced Materals Research Onlne: 03-06-3 ISSN: 66-8985, Vol. 705, pp 40-44 do:0.408/www.scentfc.net/amr.705.40 03 Trans Tech Publcatons, Swtzerland Fnte Element Analyss of Rubber Sealng Rng Reslence Behavor
More informationComplex Numbers. Now we also saw that if a and b were both positive then ab = a b. For a second let s forget that restriction and do the following.
Complex Numbers The last topc n ths secton s not really related to most of what we ve done n ths chapter, although t s somewhat related to the radcals secton as we wll see. We also won t need the materal
More information2x x l. Module 3: Element Properties Lecture 4: Lagrange and Serendipity Elements
Module 3: Element Propertes Lecture : Lagrange and Serendpty Elements 5 In last lecture note, the nterpolaton functons are derved on the bass of assumed polynomal from Pascal s trangle for the fled varable.
More informationS1 Note. Basis functions.
S1 Note. Bass functons. Contents Types of bass functons...1 The Fourer bass...2 B-splne bass...3 Power and type I error rates wth dfferent numbers of bass functons...4 Table S1. Smulaton results of type
More informationHermite Splines in Lie Groups as Products of Geodesics
Hermte Splnes n Le Groups as Products of Geodescs Ethan Eade Updated May 28, 2017 1 Introducton 1.1 Goal Ths document defnes a curve n the Le group G parametrzed by tme and by structural parameters n the
More informationElectrical analysis of light-weight, triangular weave reflector antennas
Electrcal analyss of lght-weght, trangular weave reflector antennas Knud Pontoppdan TICRA Laederstraede 34 DK-121 Copenhagen K Denmark Emal: kp@tcra.com INTRODUCTION The new lght-weght reflector antenna
More informationKinematics of pantograph masts
Abstract Spacecraft Mechansms Group, ISRO Satellte Centre, Arport Road, Bangalore 560 07, Emal:bpn@sac.ernet.n Flght Dynamcs Dvson, ISRO Satellte Centre, Arport Road, Bangalore 560 07 Emal:pandyan@sac.ernet.n
More informationA mathematical programming approach to the analysis, design and scheduling of offshore oilfields
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 A mathematcal programmng approach to the analyss, desgn and
More informationReview of approximation techniques
CHAPTER 2 Revew of appromaton technques 2. Introducton Optmzaton problems n engneerng desgn are characterzed by the followng assocated features: the objectve functon and constrants are mplct functons evaluated
More informationA MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS
Proceedngs of the Wnter Smulaton Conference M E Kuhl, N M Steger, F B Armstrong, and J A Jones, eds A MOVING MESH APPROACH FOR SIMULATION BUDGET ALLOCATION ON CONTINUOUS DOMAINS Mark W Brantley Chun-Hung
More informationMathematics 256 a course in differential equations for engineering students
Mathematcs 56 a course n dfferental equatons for engneerng students Chapter 5. More effcent methods of numercal soluton Euler s method s qute neffcent. Because the error s essentally proportonal to the
More informationCable optimization of a long span cable stayed bridge in La Coruña (Spain)
Computer Aded Optmum Desgn n Engneerng XI 107 Cable optmzaton of a long span cable stayed brdge n La Coruña (Span) A. Baldomr & S. Hernández School of Cvl Engneerng, Unversty of Coruña, La Coruña, Span
More information6.854 Advanced Algorithms Petar Maymounkov Problem Set 11 (November 23, 2005) With: Benjamin Rossman, Oren Weimann, and Pouya Kheradpour
6.854 Advanced Algorthms Petar Maymounkov Problem Set 11 (November 23, 2005) Wth: Benjamn Rossman, Oren Wemann, and Pouya Kheradpour Problem 1. We reduce vertex cover to MAX-SAT wth weghts, such that the
More informationR s s f. m y s. SPH3UW Unit 7.3 Spherical Concave Mirrors Page 1 of 12. Notes
SPH3UW Unt 7.3 Sphercal Concave Mrrors Page 1 of 1 Notes Physcs Tool box Concave Mrror If the reflectng surface takes place on the nner surface of the sphercal shape so that the centre of the mrror bulges
More informationProper Choice of Data Used for the Estimation of Datum Transformation Parameters
Proper Choce of Data Used for the Estmaton of Datum Transformaton Parameters Hakan S. KUTOGLU, Turkey Key words: Coordnate systems; transformaton; estmaton, relablty. SUMMARY Advances n technologes and
More informationModeling of Fillets in Thin-walled Beams Using Shell/Plate and Beam Finite. Elements
Modelng of Fllets n Thn-walled Beams Usng Shell/Plate and Beam Fnte Elements K. He Graduate Research Assstant W.D. Zhu Professor and Correspondng Author Emal: wzhu@umbc.edu; Tel: 410-455-3394 Department
More informationProblem Definitions and Evaluation Criteria for Computational Expensive Optimization
Problem efntons and Evaluaton Crtera for Computatonal Expensve Optmzaton B. Lu 1, Q. Chen and Q. Zhang 3, J. J. Lang 4, P. N. Suganthan, B. Y. Qu 6 1 epartment of Computng, Glyndwr Unversty, UK Faclty
More informationAccounting for the Use of Different Length Scale Factors in x, y and z Directions
1 Accountng for the Use of Dfferent Length Scale Factors n x, y and z Drectons Taha Soch (taha.soch@kcl.ac.uk) Imagng Scences & Bomedcal Engneerng, Kng s College London, The Rayne Insttute, St Thomas Hosptal,
More informationROBOT KINEMATICS. ME Robotics ME Robotics
ROBOT KINEMATICS Purpose: The purpose of ths chapter s to ntroduce you to robot knematcs, and the concepts related to both open and closed knematcs chans. Forward knematcs s dstngushed from nverse knematcs.
More informationHelsinki University Of Technology, Systems Analysis Laboratory Mat Independent research projects in applied mathematics (3 cr)
Helsnk Unversty Of Technology, Systems Analyss Laboratory Mat-2.08 Independent research projects n appled mathematcs (3 cr) "! #$&% Antt Laukkanen 506 R ajlaukka@cc.hut.f 2 Introducton...3 2 Multattrbute
More informationSupport Vector Machines
/9/207 MIST.6060 Busness Intellgence and Data Mnng What are Support Vector Machnes? Support Vector Machnes Support Vector Machnes (SVMs) are supervsed learnng technques that analyze data and recognze patterns.
More informationTopology Design using LS-TaSC Version 2 and LS-DYNA
Topology Desgn usng LS-TaSC Verson 2 and LS-DYNA Wllem Roux Lvermore Software Technology Corporaton, Lvermore, CA, USA Abstract Ths paper gves an overvew of LS-TaSC verson 2, a topology optmzaton tool
More informationAnalysis of 3D Cracks in an Arbitrary Geometry with Weld Residual Stress
Analyss of 3D Cracks n an Arbtrary Geometry wth Weld Resdual Stress Greg Thorwald, Ph.D. Ted L. Anderson, Ph.D. Structural Relablty Technology, Boulder, CO Abstract Materals contanng flaws lke nclusons
More informationActive Contours/Snakes
Actve Contours/Snakes Erkut Erdem Acknowledgement: The sldes are adapted from the sldes prepared by K. Grauman of Unversty of Texas at Austn Fttng: Edges vs. boundares Edges useful sgnal to ndcate occludng
More informationCompiler Design. Spring Register Allocation. Sample Exercises and Solutions. Prof. Pedro C. Diniz
Compler Desgn Sprng 2014 Regster Allocaton Sample Exercses and Solutons Prof. Pedro C. Dnz USC / Informaton Scences Insttute 4676 Admralty Way, Sute 1001 Marna del Rey, Calforna 90292 pedro@s.edu Regster
More informationEngineering Structures
Engneerng Structures 37 (2012) 106 124 Contents lsts avalable at ScVerse ScenceDrect Engneerng Structures journal homepage: www.elsever.com/locate/engstruct Topology optmzaton for braced frames: Combnng
More informationImprovement of Spatial Resolution Using BlockMatching Based Motion Estimation and Frame. Integration
Improvement of Spatal Resoluton Usng BlockMatchng Based Moton Estmaton and Frame Integraton Danya Suga and Takayuk Hamamoto Graduate School of Engneerng, Tokyo Unversty of Scence, 6-3-1, Nuku, Katsuska-ku,
More informationA Binarization Algorithm specialized on Document Images and Photos
A Bnarzaton Algorthm specalzed on Document mages and Photos Ergna Kavalleratou Dept. of nformaton and Communcaton Systems Engneerng Unversty of the Aegean kavalleratou@aegean.gr Abstract n ths paper, a
More informationFor instance, ; the five basic number-sets are increasingly more n A B & B A A = B (1)
Secton 1.2 Subsets and the Boolean operatons on sets If every element of the set A s an element of the set B, we say that A s a subset of B, or that A s contaned n B, or that B contans A, and we wrte A
More informationDesign of Structure Optimization with APDL
Desgn of Structure Optmzaton wth APDL Yanyun School of Cvl Engneerng and Archtecture, East Chna Jaotong Unversty Nanchang 330013 Chna Abstract In ths paper, the desgn process of structure optmzaton wth
More informationContent Based Image Retrieval Using 2-D Discrete Wavelet with Texture Feature with Different Classifiers
IOSR Journal of Electroncs and Communcaton Engneerng (IOSR-JECE) e-issn: 78-834,p- ISSN: 78-8735.Volume 9, Issue, Ver. IV (Mar - Apr. 04), PP 0-07 Content Based Image Retreval Usng -D Dscrete Wavelet wth
More informationPose, Posture, Formation and Contortion in Kinematic Systems
Pose, Posture, Formaton and Contorton n Knematc Systems J. Rooney and T. K. Tanev Department of Desgn and Innovaton, Faculty of Technology, The Open Unversty, Unted Kngdom Abstract. The concepts of pose,
More informationX- Chart Using ANOM Approach
ISSN 1684-8403 Journal of Statstcs Volume 17, 010, pp. 3-3 Abstract X- Chart Usng ANOM Approach Gullapall Chakravarth 1 and Chaluvad Venkateswara Rao Control lmts for ndvdual measurements (X) chart are
More informationSmoothing Spline ANOVA for variable screening
Smoothng Splne ANOVA for varable screenng a useful tool for metamodels tranng and mult-objectve optmzaton L. Rcco, E. Rgon, A. Turco Outlne RSM Introducton Possble couplng Test case MOO MOO wth Game Theory
More informationParallelism for Nested Loops with Non-uniform and Flow Dependences
Parallelsm for Nested Loops wth Non-unform and Flow Dependences Sam-Jn Jeong Dept. of Informaton & Communcaton Engneerng, Cheonan Unversty, 5, Anseo-dong, Cheonan, Chungnam, 330-80, Korea. seong@cheonan.ac.kr
More informationFeature Reduction and Selection
Feature Reducton and Selecton Dr. Shuang LIANG School of Software Engneerng TongJ Unversty Fall, 2012 Today s Topcs Introducton Problems of Dmensonalty Feature Reducton Statstc methods Prncpal Components
More informationLS-TaSC Version 2.1. Willem Roux Livermore Software Technology Corporation, Livermore, CA, USA. Abstract
12 th Internatonal LS-DYNA Users Conference Optmzaton(1) LS-TaSC Verson 2.1 Wllem Roux Lvermore Software Technology Corporaton, Lvermore, CA, USA Abstract Ths paper gves an overvew of LS-TaSC verson 2.1,
More informationOptimization Methods: Integer Programming Integer Linear Programming 1. Module 7 Lecture Notes 1. Integer Linear Programming
Optzaton Methods: Integer Prograng Integer Lnear Prograng Module Lecture Notes Integer Lnear Prograng Introducton In all the prevous lectures n lnear prograng dscussed so far, the desgn varables consdered
More informationA unified library of nonlinear solution schemes
A unfed lbrary of nonlnear soluton schemes Sofe E. Leon, Glauco H. Paulno, Anderson Perera, Ivan F. M. Menezes, Eduardo N. Lages 7/27/2011 Motvaton Nonlnear problems are prevalent n structural, flud, contnuum,
More informationNUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS
ARPN Journal of Engneerng and Appled Scences 006-017 Asan Research Publshng Network (ARPN). All rghts reserved. NUMERICAL SOLVING OPTIMAL CONTROL PROBLEMS BY THE METHOD OF VARIATIONS Igor Grgoryev, Svetlana
More informationON THE DESIGN OF LARGE SCALE REDUNDANT PARALLEL MANIPULATOR. Wu huapeng, Heikki handroos and Juha kilkki
ON THE DESIGN OF LARGE SCALE REDUNDANT PARALLEL MANIPULATOR Wu huapeng, Hekk handroos and Juha klkk Machne Automaton Lab, Lappeenranta Unversty of Technology LPR-5385 Fnland huapeng@lut.f, handroos@lut.f,
More informationThe Codesign Challenge
ECE 4530 Codesgn Challenge Fall 2007 Hardware/Software Codesgn The Codesgn Challenge Objectves In the codesgn challenge, your task s to accelerate a gven software reference mplementaton as fast as possble.
More informationFEATURE EXTRACTION. Dr. K.Vijayarekha. Associate Dean School of Electrical and Electronics Engineering SASTRA University, Thanjavur
FEATURE EXTRACTION Dr. K.Vjayarekha Assocate Dean School of Electrcal and Electroncs Engneerng SASTRA Unversty, Thanjavur613 41 Jont Intatve of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents
More information3D vector computer graphics
3D vector computer graphcs Paolo Varagnolo: freelance engneer Padova Aprl 2016 Prvate Practce ----------------------------------- 1. Introducton Vector 3D model representaton n computer graphcs requres
More informationS.P.H. : A SOLUTION TO AVOID USING EROSION CRITERION?
S.P.H. : A SOLUTION TO AVOID USING EROSION CRITERION? Célne GALLET ENSICA 1 place Emle Bloun 31056 TOULOUSE CEDEX e-mal :cgallet@ensca.fr Jean Luc LACOME DYNALIS Immeuble AEROPOLE - Bat 1 5, Avenue Albert
More informationParallel matrix-vector multiplication
Appendx A Parallel matrx-vector multplcaton The reduced transton matrx of the three-dmensonal cage model for gel electrophoress, descrbed n secton 3.2, becomes excessvely large for polymer lengths more
More informationLecture 5: Multilayer Perceptrons
Lecture 5: Multlayer Perceptrons Roger Grosse 1 Introducton So far, we ve only talked about lnear models: lnear regresson and lnear bnary classfers. We noted that there are functons that can t be represented
More informationVirtual Machine Migration based on Trust Measurement of Computer Node
Appled Mechancs and Materals Onlne: 2014-04-04 ISSN: 1662-7482, Vols. 536-537, pp 678-682 do:10.4028/www.scentfc.net/amm.536-537.678 2014 Trans Tech Publcatons, Swtzerland Vrtual Machne Mgraton based on
More informationUNIT 2 : INEQUALITIES AND CONVEX SETS
UNT 2 : NEQUALTES AND CONVEX SETS ' Structure 2. ntroducton Objectves, nequaltes and ther Graphs Convex Sets and ther Geometry Noton of Convex Sets Extreme Ponts of Convex Set Hyper Planes and Half Spaces
More informationOverview. Basic Setup [9] Motivation and Tasks. Modularization 2008/2/20 IMPROVED COVERAGE CONTROL USING ONLY LOCAL INFORMATION
Overvew 2 IMPROVED COVERAGE CONTROL USING ONLY LOCAL INFORMATION Introducton Mult- Smulator MASIM Theoretcal Work and Smulaton Results Concluson Jay Wagenpfel, Adran Trachte Motvaton and Tasks Basc Setup
More informationNAG Fortran Library Chapter Introduction. G10 Smoothing in Statistics
Introducton G10 NAG Fortran Lbrary Chapter Introducton G10 Smoothng n Statstcs Contents 1 Scope of the Chapter... 2 2 Background to the Problems... 2 2.1 Smoothng Methods... 2 2.2 Smoothng Splnes and Regresson
More informationTN348: Openlab Module - Colocalization
TN348: Openlab Module - Colocalzaton Topc The Colocalzaton module provdes the faclty to vsualze and quantfy colocalzaton between pars of mages. The Colocalzaton wndow contans a prevew of the two mages
More informationRange images. Range image registration. Examples of sampling patterns. Range images and range surfaces
Range mages For many structured lght scanners, the range data forms a hghly regular pattern known as a range mage. he samplng pattern s determned by the specfc scanner. Range mage regstraton 1 Examples
More informationDouble Layer Tensegrity Grids
Acta Poltechnca Hungarca Vol. 9, No. 5, 0 Double Laer Tensegrt Grds Tatana Olejnkova Department of Appled Mathematcs, Cvl Engneerng Facult Techncal Unverst of Košce Vsokoškolská, 0 00 Košce, Slovaka e-mal:
More informationThe Greedy Method. Outline and Reading. Change Money Problem. Greedy Algorithms. Applications of the Greedy Strategy. The Greedy Method Technique
//00 :0 AM Outlne and Readng The Greedy Method The Greedy Method Technque (secton.) Fractonal Knapsack Problem (secton..) Task Schedulng (secton..) Mnmum Spannng Trees (secton.) Change Money Problem Greedy
More informationDynamic wetting property investigation of AFM tips in micro/nanoscale
Dynamc wettng property nvestgaton of AFM tps n mcro/nanoscale The wettng propertes of AFM probe tps are of concern n AFM tp related force measurement, fabrcaton, and manpulaton technques, such as dp-pen
More informationAssignment # 2. Farrukh Jabeen Algorithms 510 Assignment #2 Due Date: June 15, 2009.
Farrukh Jabeen Algorthms 51 Assgnment #2 Due Date: June 15, 29. Assgnment # 2 Chapter 3 Dscrete Fourer Transforms Implement the FFT for the DFT. Descrbed n sectons 3.1 and 3.2. Delverables: 1. Concse descrpton
More informationMachine Learning: Algorithms and Applications
14/05/1 Machne Learnng: Algorthms and Applcatons Florano Zn Free Unversty of Bozen-Bolzano Faculty of Computer Scence Academc Year 011-01 Lecture 10: 14 May 01 Unsupervsed Learnng cont Sldes courtesy of
More informationAPPLICATION OF MULTIVARIATE LOSS FUNCTION FOR ASSESSMENT OF THE QUALITY OF TECHNOLOGICAL PROCESS MANAGEMENT
3. - 5. 5., Brno, Czech Republc, EU APPLICATION OF MULTIVARIATE LOSS FUNCTION FOR ASSESSMENT OF THE QUALITY OF TECHNOLOGICAL PROCESS MANAGEMENT Abstract Josef TOŠENOVSKÝ ) Lenka MONSPORTOVÁ ) Flp TOŠENOVSKÝ
More informationTPL-Aware Displacement-driven Detailed Placement Refinement with Coloring Constraints
TPL-ware Dsplacement-drven Detaled Placement Refnement wth Colorng Constrants Tao Ln Iowa State Unversty tln@astate.edu Chrs Chu Iowa State Unversty cnchu@astate.edu BSTRCT To mnmze the effect of process
More informationSolitary and Traveling Wave Solutions to a Model. of Long Range Diffusion Involving Flux with. Stability Analysis
Internatonal Mathematcal Forum, Vol. 6,, no. 7, 8 Soltary and Travelng Wave Solutons to a Model of Long Range ffuson Involvng Flux wth Stablty Analyss Manar A. Al-Qudah Math epartment, Rabgh Faculty of
More information5 The Primal-Dual Method
5 The Prmal-Dual Method Orgnally desgned as a method for solvng lnear programs, where t reduces weghted optmzaton problems to smpler combnatoral ones, the prmal-dual method (PDM) has receved much attenton
More informationSLAM Summer School 2006 Practical 2: SLAM using Monocular Vision
SLAM Summer School 2006 Practcal 2: SLAM usng Monocular Vson Javer Cvera, Unversty of Zaragoza Andrew J. Davson, Imperal College London J.M.M Montel, Unversty of Zaragoza. josemar@unzar.es, jcvera@unzar.es,
More informationChapter 6 Programmng the fnte element method Inow turn to the man subject of ths book: The mplementaton of the fnte element algorthm n computer programs. In order to make my dscusson as straghtforward
More informationAn Iterative Solution Approach to Process Plant Layout using Mixed Integer Optimisation
17 th European Symposum on Computer Aded Process Engneerng ESCAPE17 V. Plesu and P.S. Agach (Edtors) 2007 Elsever B.V. All rghts reserved. 1 An Iteratve Soluton Approach to Process Plant Layout usng Mxed
More informationKiran Joy, International Journal of Advanced Engineering Technology E-ISSN
Kran oy, nternatonal ournal of Advanced Engneerng Technology E-SS 0976-3945 nt Adv Engg Tech/Vol. V/ssue /Aprl-une,04/9-95 Research Paper DETERMATO O RADATVE VEW ACTOR WTOUT COSDERG TE SADOWG EECT Kran
More informationTopology optimization considering the requirements of deep-drawn sheet metals
th World Congress on Structural and Multdscplnary Optmsaton 7 th - th, June 5, Sydney Australa Topology optmzaton consderng the requrements of deep-drawn sheet metals Robert Denemann, Axel Schumacher,
More informationAVO Modeling of Monochromatic Spherical Waves: Comparison to Band-Limited Waves
AVO Modelng of Monochromatc Sphercal Waves: Comparson to Band-Lmted Waves Charles Ursenbach* Unversty of Calgary, Calgary, AB, Canada ursenbach@crewes.org and Arnm Haase Unversty of Calgary, Calgary, AB,
More informationMATHEMATICS FORM ONE SCHEME OF WORK 2004
MATHEMATICS FORM ONE SCHEME OF WORK 2004 WEEK TOPICS/SUBTOPICS LEARNING OBJECTIVES LEARNING OUTCOMES VALUES CREATIVE & CRITICAL THINKING 1 WHOLE NUMBER Students wll be able to: GENERICS 1 1.1 Concept of
More informationy and the total sum of
Lnear regresson Testng for non-lnearty In analytcal chemstry, lnear regresson s commonly used n the constructon of calbraton functons requred for analytcal technques such as gas chromatography, atomc absorpton
More informationVery simple computational domains can be discretized using boundary-fitted structured meshes (also called grids)
Structured meshes Very smple computatonal domans can be dscretzed usng boundary-ftted structured meshes (also called grds) The grd lnes of a Cartesan mesh are parallel to one another Structured meshes
More informationComputer Animation and Visualisation. Lecture 4. Rigging / Skinning
Computer Anmaton and Vsualsaton Lecture 4. Rggng / Sknnng Taku Komura Overvew Sknnng / Rggng Background knowledge Lnear Blendng How to decde weghts? Example-based Method Anatomcal models Sknnng Assume
More informationLOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit
LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE
More informationRepeater Insertion for Two-Terminal Nets in Three-Dimensional Integrated Circuits
Repeater Inserton for Two-Termnal Nets n Three-Dmensonal Integrated Crcuts Hu Xu, Vasls F. Pavlds, and Govann De Mchel LSI - EPFL, CH-5, Swtzerland, {hu.xu,vasleos.pavlds,govann.demchel}@epfl.ch Abstract.
More informationSimulation: Solving Dynamic Models ABE 5646 Week 11 Chapter 2, Spring 2010
Smulaton: Solvng Dynamc Models ABE 5646 Week Chapter 2, Sprng 200 Week Descrpton Readng Materal Mar 5- Mar 9 Evaluatng [Crop] Models Comparng a model wth data - Graphcal, errors - Measures of agreement
More informationInverse Kinematics (part 2) CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Spring 2016
Inverse Knematcs (part 2) CSE169: Computer Anmaton Instructor: Steve Rotenberg UCSD, Sprng 2016 Forward Knematcs We wll use the vector: Φ... 1 2 M to represent the array of M jont DOF values We wll also
More informationINVERSE DYNAMICS ANALYSIS AND SIMULATION OF A CLASS OF UNDER- CONSTRAINED CABLE-DRIVEN PARALLEL SYSTEM
U.P.B. Sc. Bull., Seres D, Vol. 78, Iss., 6 ISSN 454-58 INVERSE DYNAMICS ANALYSIS AND SIMULATION OF A CLASS OF UNDER- CONSTRAINED CABLE-DRIVEN PARALLEL SYSTEM We LI, Zhgang ZHAO, Guangtan SHI, Jnsong LI
More informationFitting: Deformable contours April 26 th, 2018
4/6/08 Fttng: Deformable contours Aprl 6 th, 08 Yong Jae Lee UC Davs Recap so far: Groupng and Fttng Goal: move from array of pxel values (or flter outputs) to a collecton of regons, objects, and shapes.
More informationInvestigations of Topology and Shape of Multi-material Optimum Design of Structures
Advanced Scence and Tecnology Letters Vol.141 (GST 2016), pp.241-245 ttp://dx.do.org/10.14257/astl.2016.141.52 Investgatons of Topology and Sape of Mult-materal Optmum Desgn of Structures Quoc Hoan Doan
More informationSENSITIVITY ANALYSIS IN LINEAR PROGRAMMING USING A CALCULATOR
SENSITIVITY ANALYSIS IN LINEAR PROGRAMMING USING A CALCULATOR Judth Aronow Rchard Jarvnen Independent Consultant Dept of Math/Stat 559 Frost Wnona State Unversty Beaumont, TX 7776 Wnona, MN 55987 aronowju@hal.lamar.edu
More informationSENSITIVITY ANALYSIS WITH UNSTRUCTURED FREE MESH GENERATORS IN 2-D AND 3-D SHAPE OPTIMIZATION.
SENSITIVITY ANALYSIS WITH UNSTRUCTURED FREE MESH GENERATORS IN 2-D AND 3-D SHAPE OPTIMIZATION. P. Duysnx, W.H. Zhang, C. Fleury. Aerospace Laboratory, LTAS, Unversty of Lège B-4000 LIEGE, BELGIUM. ABSTRACT.
More informationStructural Optimization Using OPTIMIZER Program
SprngerLnk - Book Chapter http://www.sprngerlnk.com/content/m28478j4372qh274/?prnt=true ق.ظ 1 of 2 2009/03/12 11:30 Book Chapter large verson Structural Optmzaton Usng OPTIMIZER Program Book III European
More informationSum of Linear and Fractional Multiobjective Programming Problem under Fuzzy Rules Constraints
Australan Journal of Basc and Appled Scences, 2(4): 1204-1208, 2008 ISSN 1991-8178 Sum of Lnear and Fractonal Multobjectve Programmng Problem under Fuzzy Rules Constrants 1 2 Sanjay Jan and Kalash Lachhwan
More informationDesign for Reliability: Case Studies in Manufacturing Process Synthesis
Desgn for Relablty: Case Studes n Manufacturng Process Synthess Y. Lawrence Yao*, and Chao Lu Department of Mechancal Engneerng, Columba Unversty, Mudd Bldg., MC 473, New York, NY 7, USA * Correspondng
More informationVibration Characteristic Analysis of Axial Fan Shell Based on ANSYS Workbench
Internatonal Conference on Logstcs Engneerng, Management and Computer Scence (LEMCS 2015) Vbraton Characterstc Analyss of Axal Fan Shell Based on ANSYS Workbench Lchun Gu College of Mechancal and Electrcal
More informationModule Management Tool in Software Development Organizations
Journal of Computer Scence (5): 8-, 7 ISSN 59-66 7 Scence Publcatons Management Tool n Software Development Organzatons Ahmad A. Al-Rababah and Mohammad A. Al-Rababah Faculty of IT, Al-Ahlyyah Amman Unversty,
More informationVISUAL SELECTION OF SURFACE FEATURES DURING THEIR GEOMETRIC SIMULATION WITH THE HELP OF COMPUTER TECHNOLOGIES
UbCC 2011, Volume 6, 5002981-x manuscrpts OPEN ACCES UbCC Journal ISSN 1992-8424 www.ubcc.org VISUAL SELECTION OF SURFACE FEATURES DURING THEIR GEOMETRIC SIMULATION WITH THE HELP OF COMPUTER TECHNOLOGIES
More informationSupport Vector Machines. CS534 - Machine Learning
Support Vector Machnes CS534 - Machne Learnng Perceptron Revsted: Lnear Separators Bnar classfcaton can be veed as the task of separatng classes n feature space: b > 0 b 0 b < 0 f() sgn( b) Lnear Separators
More informationMulti-posture kinematic calibration technique and parameter identification algorithm for articulated arm coordinate measuring machines
Mult-posture knematc calbraton technque and parameter dentfcaton algorthm for artculated arm coordnate measurng machnes Juan-José AGUILAR, Jorge SANTOLARIA, José-Antono YAGÜE, Ana-Crstna MAJARENA Department
More informationStiffness modeling for perfect and non-perfect parallel manipulators under internal and external loadings
Stffness modelng for perfect and non-perfect parallel manpulators under nternal and external loadngs Alexandr Klmchk, Damen hablat, Anatol Pashkevch o cte ths verson: Alexandr Klmchk, Damen hablat, Anatol
More informationAP PHYSICS B 2008 SCORING GUIDELINES
AP PHYSICS B 2008 SCORING GUIDELINES General Notes About 2008 AP Physcs Scorng Gudelnes 1. The solutons contan the most common method of solvng the free-response questons and the allocaton of ponts for
More informationOn Some Entertaining Applications of the Concept of Set in Computer Science Course
On Some Entertanng Applcatons of the Concept of Set n Computer Scence Course Krasmr Yordzhev *, Hrstna Kostadnova ** * Assocate Professor Krasmr Yordzhev, Ph.D., Faculty of Mathematcs and Natural Scences,
More informationBoundary Condition Simulation for Structural Local Refined Modeling Using Genetic Algorithm
2016 Internatonal Conference on Artfcal Intellgence: Technques and Applcatons (AITA 2016) ISBN: 978-1-60595-389-2 Boundary Condton Smulaton for Structural Local Refned Modelng Usng Genetc Algorthm Zhong
More informationGenetic Tuning of Fuzzy Logic Controller for a Flexible-Link Manipulator
Genetc Tunng of Fuzzy Logc Controller for a Flexble-Lnk Manpulator Lnda Zhxa Sh Mohamed B. Traba Department of Mechancal Unversty of Nevada, Las Vegas Department of Mechancal Engneerng Las Vegas, NV 89154-407
More informationCluster Analysis of Electrical Behavior
Journal of Computer and Communcatons, 205, 3, 88-93 Publshed Onlne May 205 n ScRes. http://www.scrp.org/ournal/cc http://dx.do.org/0.4236/cc.205.350 Cluster Analyss of Electrcal Behavor Ln Lu Ln Lu, School
More informationIntra-Parametric Analysis of a Fuzzy MOLP
Intra-Parametrc Analyss of a Fuzzy MOLP a MIAO-LING WANG a Department of Industral Engneerng and Management a Mnghsn Insttute of Technology and Hsnchu Tawan, ROC b HSIAO-FAN WANG b Insttute of Industral
More informationPolyhedral Compilation Foundations
Polyhedral Complaton Foundatons Lous-Noël Pouchet pouchet@cse.oho-state.edu Dept. of Computer Scence and Engneerng, the Oho State Unversty Feb 8, 200 888., Class # Introducton: Polyhedral Complaton Foundatons
More informationShape Optimization of Shear-type Hysteretic Steel Damper for Building Frames using FEM-Analysis and Heuristic Approach
The Seventh Chna-Japan-Korea Jont Symposum on Optmzaton of Structural and Mechancal Systems Huangshan, June, 18-21, 2012, Chna Shape Optmzaton of Shear-type Hysteretc Steel Damper for Buldng Frames usng
More information