Copyright © 2004 by Bob Burkhardt

Hub ConnectivityThis shows how hub vectors (hv01.1 ...) are applied to the strut end points (p01 ...) to derive hub points (hp01.1 ...). Hub points are where tendons are connected to the strut. <VecPt> hp01.1 p01 + hv01.1 <VecPt> hp01.2 p01 + hv01.2 <VecPt> hp01.3 p01 + hv01.3 <VecPt> hp05.1 p05 + hv05.1 <VecPt> hp05.2 p05 + hv05.2 <VecPt> hp05.3 p05 + hv05.3 <VecPt> hp09.1 p09 + hv09.1 <VecPt> hp09.2 p09 + hv09.2 <VecPt> hp09.3 p09 + hv09.3 <VecPt> hp11.1 p11 + hv11.1 <VecPt> hp11.2 p11 + hv11.2 <VecPt> hp11.3 p11 + hv11.3TransformationsThis shows how transforms are defined and how they are applied to derive transformed objects from basic objects. # rotation matrices <XMat> x4 cos(2*pi/4) (-sin(2*pi/4)) 0 sin(2*pi/4) cos(2*pi/4) 0 0 0 1 <CompositeXform> x4^2 x4 x4 <CompositeXform> x4^3 x4 x4^2 # transform points and vectors <XformPt> p02 p01 x4 <XformPt> hp02.1 hp01.1 x4 <XformPt> hp02.2 hp01.2 x4 <XformPt> hp02.3 hp01.3 x4 <XformPt> p04 p05 x4^3 <XformPt> hp04.1 hp05.1 x4^3 <XformPt> hp04.2 hp05.2 x4^3 <XformPt> hp04.3 hp05.3 x4^3 <XformPt> p07 p09 x4 <XformPt> hp07.1 hp09.1 x4 <XformPt> hp07.2 hp09.2 x4 <XformPt> hp07.3 hp09.3 x4Member Descriptions[name, end point names, weight (if in objective function), second power of length (if a constraint), member category, Obj/Con/Exc (put in objective function, use as a constraint or exclude from computations), flags] For assembly purposes, only the name and end point names are of interest. The other information may be of interest after A Practical Guide to Tensegrity Design has been consulted. # struts <Member> strut1 p01 p09 0.00 sqr(317) 1 Con CalcClear * <Member> strut2 p05 p11 0.00 sqr(317) 1 Con CalcClear * # square tendon <Member> sqten hp01.1 hp02.2 0.00 sqr(129.2792) 2 Con * # triangle tendons <Member> triten1 hp04.1 hp09.1 0.00 sqr(129.2792) 2 Con * <Member> triten2 hp09.2 hp11.1 0.00 sqr(129.2792) 2 Con * <Member> triten3 hp11.2 hp04.2 0.00 sqr(129.2792) 2 Con * # zig-zag tendons <Member> zzten1 hp02.3 hp11.3 1.00 0.00 3 Obj * <Member> zzten2 hp04.3 hp07.3 1.00 sqr(129.2792) 3 Con *Hub ConstructsThese items are just vectors corresponding to each of the struts. At some point the software will be modified so these items aren't necessary since it should be possible to treat any member as a vector without explicit constructs. <DiffVec> strut1v p01 p09 <DiffVec> strut2v p05 p11Hub ConstraintsThese are constraints the hub vectors must meet. For these constraints to be specified, a set of center vectors (cv01, ...) is introduced. There is one center vector for each strut end point. The center vectors are constrained to be orthogonal to the strut. For each tendon connected to a strut end point, there is a hub vector. All the hub vectors at a strut end point must be orthogonal to the corresponding center vector and 4 mm long. 4 mm is the outer radius of the screw eye. The center vectors are constrained to be of length one just to make their values well defined. <VecDotVec> hubdot01 strut1v cv01 1.0 0.0 Con <VecLength> cvlen01 cv01 1.0 sqr(1.0) Con <VecDotVec> hubdot01.1 cv01 hv01.1 1.0 0.0 Con <VecDotVec> hubdot01.2 cv01 hv01.2 1.0 0.0 Con <VecDotVec> hubdot01.3 cv01 hv01.3 1.0 0.0 Con <VecLength> hvlen01.1 hv01.1 1.0 sqr(4.0) Con <VecLength> hvlen01.2 hv01.2 1.0 sqr(4.0) Con <VecLength> hvlen01.3 hv01.3 1.0 sqr(4.0) Con <VecDotVec> hubdot09 strut1v cv09 1.0 0.0 Con <VecLength> cvlen09 cv09 1.0 sqr(1.0) Con <VecDotVec> hubdot09.1 cv09 hv09.1 1.0 0.0 Con <VecDotVec> hubdot09.2 cv09 hv09.2 1.0 0.0 Con <VecDotVec> hubdot09.3 cv09 hv09.3 1.0 0.0 Con <VecLength> hvlen09.1 hv09.1 1.0 sqr(4.0) Con <VecLength> hvlen09.2 hv09.2 1.0 sqr(4.0) Con <VecLength> hvlen09.3 hv09.3 1.0 sqr(4.0) Con <VecDotVec> hubdot05 strut2v cv05 1.0 0.0 Con <VecLength> cvlen05 cv05 1.0 sqr(1.0) Con <VecDotVec> hubdot05.1 cv05 hv05.1 1.0 0.0 Con <VecDotVec> hubdot05.2 cv05 hv05.2 1.0 0.0 Con <VecDotVec> hubdot05.3 cv05 hv05.3 1.0 0.0 Con <VecLength> hvlen05.1 hv05.1 1.0 sqr(4.0) Con <VecLength> hvlen05.2 hv05.2 1.0 sqr(4.0) Con <VecLength> hvlen05.3 hv05.3 1.0 sqr(4.0) Con <VecDotVec> hubdot11 strut2v cv11 1.0 0.0 Con <VecLength> cvlen11 cv11 1.0 sqr(1.0) Con <VecDotVec> hubdot11.1 cv11 hv11.1 1.0 0.0 Con <VecDotVec> hubdot11.2 cv11 hv11.2 1.0 0.0 Con <VecDotVec> hubdot11.3 cv11 hv11.3 1.0 0.0 Con <VecLength> hvlen11.1 hv11.1 1.0 sqr(4.0) Con <VecLength> hvlen11.2 hv11.2 1.0 sqr(4.0) Con <VecLength> hvlen11.3 hv11.3 1.0 sqr(4.0) ConIn-Situ Member LengthsThese are the lengths of the members when they are in place and prestress is applied. The strut lengths are from screw-eye center to screw-eye center. The tendon lengths are from screw-eye rim to screw-eye rim. Since the model has been scaled appropriately, these values are in millimeters. strut1: 317 strut2: 317 sqten: 129.279 triten1: 129.279 triten2: 129.279 triten3: 129.279 zzten1: 129.279 zzten2: 129.279Relative Member Prestress Force MagnitudesThese values are useful for developing an assembly strategy for the structure. The tighter tendons are much easier to tie in place early on, while the looser tendons can be left to the last. This information is also used to adjust tendon lengths since the measured length of a tendon will be shorter for a highly-stressed tendon with the same in-situ length as a tendon which is not so stressed. strut1: -129.702 strut2: -133.842 sqten: 79.7366 triten1: 83.7261 triten2: 32.1339 triten3: 150.757 zzten1: 129.279 zzten2: 135.186Construction Lengths (in millimeters and halves)The construction length of a tendon is less than the in-situ length since when the tendon is measured off it isn't under any prestress force. The construction length for the strut represents the length of the wooden dowel. Prestress forces are assumed to affect tendon lengths and not strut lengths. Here no ad hoc adjustment is necessary for the tendons since the hubs are included explicitly in the mathematical programming problem. Elongation of Tendon of Unit Cross Section Under Force of Average Magnitude (fraction)> 0.02 Length Scale Factor> 1.0 (Things are scaled so model and construction units are the same.) Strut and Tendon Hub Adjustments - s;t> 6.5 0 (Hub connections were handled explicity in the mathematical programming problem, so no ad hoc adjustment is needed here to account for the tendon hub connections. The 6.5 mm adjustment for the strut represents the amount the screw-eye center extends from the dowel.) strut1: 304 0 strut2: 304 0 sqten: 127 1 triten1: 127 0 triten2: 128 1 triten3: 125 1 zzten1: 126 0 zzten2: 126 0Construction Lengths -- ad hoc adjustmentFor comparison, these are the construction lengths computed from the model where all members are assumed to meet at a single point and a simple ad hoc adjument is applied to account for hub geometry. If the the ad hoc adjustment is working well, the values here should be the same as the ones above. All the tendon values are 0.5 mm off, which is within assembly tolerances. This is good, because it means the ad hoc adjustment works. It is much easier to do the ad hoc adjustment than to model the hubs explicitly. Elongation of Tendon of Unit Cross Section Under Force of Average Magnitude (fraction)> .02 Length Scale Factor> 317/2.31755 Strut and Tendon Hub Adjustments - s;t> 6.5 4.0 (The 6.5 mm adjustment for the strut is the amount the screw-eye center extends from the dowel. The 4 mm adjustment for the tendon is half the outer diameter of the screw eye.) strut1: 304 0 strut2: 304 0 sqten: 127 0 triten1: 126 1 triten2: 128 0 triten3: 125 0 zzten1: 125 1 zzten2: 125 1

View of the Tensegrity Tulip

with Point Labels

structure files: flower/x4flower1.rc (ad hoc) flower/x4flower1c.rc (exact) variable files: flower/x4flower1.dat flower/x4flower1c.dat digit list: src/mm.dls |

CONTACT:

Bob Burkhardt

Tensegrity Solutions

Box 426164

Cambridge, MA 02142-0021

USA

e-mail: bobwb@juno.com