]> A Practical Guide to Tensegrity Design: 6.2.6 Dome Step 6: Make adjustments to fix problems

6.2.6 Dome Step 6: Make adjustments to fix problems

The same clearance goals that are used for the 4ν t-octahedron spherical truss in Section 8.2.3 seem appropriate for this structure. With these thresholds, eight member pairs are singled out as having poor clearances. The poor clearances are mostly between pairs of struts. Table 6.25 enumerates the member pairs involved and the corresponding clearances. In addition, the solution exhibits a substantial range in member forces in the tendons, from a minimum of 0.7076 (#143) to a maximum of 5.5859 (#99).

 Member 
Pair
 Preliminary 
 Clearance 
Final
 Clearance 
7-200.1373 0.1994
13-170.1596 0.1855
13-240.1587 0.1811
17-240.1591 0.1894
18-200.1403 0.1927
21-230.1741 0.1901
18-440.1101 0.1519
20-310.1106 0.1616
 
Table 6.25: 6ν T-Octahedron Dome: Preliminary and Final Values for Problem Clearances

The interference problem is the most fundamental one. A range of tendon forces can be dealt with at construction time by using different materials depending on the relative force for the tendons, though in some situations it might be worthwhile to see what can be done to moderate the range of forces at design time. The forces are greatest in the convergence and interlayer tendons and smallest in the binding tendons and guys.

The interference problem can mostly be attributed to the low frequency of the model. At lower frequencies, the inward-pointing tripods whose peaks are the inner convergence tendon triangles tend to be shallow. This means the non-adjacent component members approach each other too closely in the vicinity of the convergence triangle.

The interference problem can be fixed by decreasing the lengths of the outer binding tendons which constrain the extent of the base of the tripod. Since the binding tendons are all weighted members of the objective function in this model, this means increasing the weight corresponding to the outer binding tendon in question. The outer binding tendon to select is the one which most parallels the strut with the clearance problem. Increasing the weight on this tendon gives the strut a steeper trajectory on its path from the outer to the inner layer and thus keeps it from approaching nearby tendons and struts at the convergence too closely. Table 6.26 lists the outer-binding tendon corresponding to each strut with an interference problem and the new value selected for the tendon's weight.

 Strut 
#
 Outer Binding 
Tendon #
 Revised 
Weight
18138 0.4038
20140 0.4597
13133 0.4038
24144 0.3371
21141 0.3678
17137 0.3371
 
Table 6.26: 6ν T-Octahedron Dome: Member Weight Adjustments

The revised model is brought to convergence using three iterations with the exact method in conjunction with Fletcher-Reeves. The derivatives of the objective function with respect to the independent coordinate values are all less than 10-5 and all clearances are above their respective thresholds.

Tables 6.27 to 6.31 show the values for the final lengths and relative forces. As before, excluded members are marked with ‡. Tables 6.32 and 6.33 show the final values for the coordinates of the basic points. Figures 6.11 and 6.12 show how the final structure appears as viewed from the side and base of the structure respectively. For clarity, interlayer tendons have been excluded8 and members in the background have been eliminated by truncation. For reference, selected points are labeled.

 Member 
#
 Relative 
Force
 Member 
#
 Relative 
Force
1 -10.0652  13-10.1760 
2-10.0700  14-9.9202 
3-9.8692  15-10.1476 
4-11.3735  16-6.9104 
5-9.7762  17-9.8347 
6-9.9708  18-10.6229 
7-10.4024  19-6.8948 
8-10.0965  20-9.9960 
9-9.8578  21-10.0265 
10-10.0713  22-6.9341 
11-9.9785  23-9.5009 
12-11.2464  24-10.1033 
 
Table 6.27: 6ν T-Octahedron Dome: Final Strut Forces


 Member 
#
 Length   Relative 
Force
 Member 
#
 Length   Relative 
Force
25 2.21528   4.43056  49 2.03547  2.84966 
262.215634.43127 502.045562.86379
272.231604.46319 512.052612.87366
282.346554.69309 522.115272.96138
292.386184.77235 532.083142.91640
302.490134.98025 542.145633.00388
312.277454.55489 552.049002.86860
322.293314.58661 562.036442.85101
332.323544.64708 571.666522.33313
342.381954.76390 582.066012.89242
352.499564.99913 592.150963.01134
362.372084.74416 602.111992.95678
372.264194.52838 612.034122.84777
382.295684.59136 621.647812.30693
392.269714.53941 632.036002.85040
402.258784.51755 64‡N/AN/A
412.249214.49841 652.032542.84556
422.235454.47090 662.047162.86602
432.258874.51773 67‡N/AN/A
442.240994.48198 682.042862.86000
452.251454.50290 692.034392.84814
462.220344.44068 70‡N/AN/A
472.234554.46910 712.042092.85893
482.219444.43888 722.032212.84510
 
Table 6.28: 6ν T-Octahedron Dome:
Final Primary and Secondary Interlayer Tendon Lengths and Forces


 Member 
#
 Relative 
Force
 Member 
#
 Relative 
Force
734.94358 974.80745
745.26772 985.13742
755.16773 995.60404
764.99886 1004.16775
774.65185 1014.90834
783.74185 1024.01986
795.18694 1035.00085
805.03322 1044.84606
814.93701 1055.26904
825.14960 1064.86884
833.71641 1073.88380
844.878121084.23798
855.039401095.26776
865.078071105.32991
875.166351114.92827
88‡N/A112 4.59887
895.014781135.07911
905.467411145.78452
91‡N/A115 5.09214
924.288411164.80643
934.592981175.27596
94‡N/A118 4.72593
954.692911195.04795
964.928451205.43235
 
Table 6.29: 6ν T-Octahedron Dome:
Final Inner and Outer Convergence Tendon Forces


 Member 
#
 Length   Relative 
Force
 Member 
#
 Length   Relative 
Force
121 2.71468   0.83201  145 1.89573   1.39443 
1222.633230.80705 1461.835881.35041
1232.660600.71632 1472.010471.29907
1241.839310.91966 1481.219691.46362
1252.469340.75682 1491.286341.29525
1262.302500.96601 1501.286031.29493
1272.549560.78140 1511.287780.94725
1282.615120.70407 1521.655901.06997
1292.594381.08847 1531.283930.94442
1302.569520.78752 1541.254251.26293
1312.185080.91675 1551.271671.28048
1321.851500.92575 1561.224551.46946
1332.634191.06381 1571.750261.13094
1342.609381.09477 1581.326950.97606
1352.734340.83804 1591.396981.02757
136‡N/AN/A 1602.904321.09132
1372.696100.90895 1612.541330.95493
1382.728531.10191 1621.849111.19481
139‡N/AN/A 1632.947040.79343
1402.541471.16838 1642.526540.77435
1412.693280.99054 1651.795271.32054
142‡N/AN/A 1662.872120.99419
1432.662420.71680 1672.674650.81974
1442.682310.90430 1681.898721.39663
 
Table 6.30: 6ν T-Octahedron Dome:
Final Outer and Inner Binding Tendon Lengths and Forces


 Member 
#
 Length   Relative 
Force
175 2.13957   0.85583 
1762.037740.81510
1772.203110.88124
1782.064260.82570
1792.235460.89419
1802.002170.80087
 
Table 6.31: 6ν T-Octahedron Dome: Final Guy Lengths and Forces


 Point 
Coordinates
x y z
P1
P2
P3
P4
P5
P6
 1.80813   0.75948   3.17349 
 1.37576   1.63043   2.94000 
 -0.38534   2.33141   2.97035 
 0.93791   1.01456   3.59498 
 1.08789   -0.26100   3.66069 
 0.00032   0.20887   3.95059 
P7
P8
P9
P10
P11
P12
 -1.37989   2.26180   2.89276 
 -1.03831   1.21623   3.55502 
 -0.98267   0.24115   3.76980 
 0.13756   -0.21195   3.96804 
 1.10801   -1.24300   3.47290 
 2.34501   -0.88144   2.93375 
P13
P14
P15
P16
P17
P18
 -2.16855   0.81829   3.08325 
 -0.80566   -1.01805   3.78455 
 1.36294   -2.31666   2.73597 
 -2.65858   1.30563   2.36051 
 -2.80252   0.32432   2.48821 
 -1.69815   -0.70421   3.46056 
P19
P20
P21
P22
P23
P24
 -1.33769   -1.58608   3.15663 
 0.38282   -2.35663   2.93031 
 0.77488   -2.93589   2.21564 
 -3.73094   -0.08603   0.04784 
 -3.10232   -1.76219   1.09539 
 -1.60996   -3.20168   1.04252 
 
Table 6.32: 6ν T-Octahedron Dome: Final Inner Coordinate Values


 Point 
Coordinates
x y z
P1 '
P2 '
P3 '
P4 '
P5 '
P6 '
 3.14866   0.86815   3.60831 
 1.46480   2.92782   3.57404 
 0.49843   2.97345   3.82708 
 1.12403   1.16198   5.26616 
 1.52593   0.25847   5.41497 
 0.54407   0.39653   5.54496 
P7 '
P8 '
P9 '
P10 '
P11 '
P12 '
 -2.11381   2.32487   4.18070 
 -1.49949   1.78849   4.75941 
 -1.19400   -0.20205   5.60735 
 -0.20625   -0.35308   5.64651 
 1.60916   -1.87445   4.55170 
 2.49145   -1.52781   4.23324 
P13 '
P14 '
P15 '
P16 '
P17 '
P18 '
 -2.36532   1.39524   4.45003 
 -0.82018   -1.11302   5.43305 
 2.16560   -2.42225   3.92697 
 -3.97029   1.88345   2.29814 
 -3.70848   -0.43912   3.29623 
 -3.04238   -0.81407   3.94099 
P19 '
P20 '
P21 '
P22 '
P23 '
P24 '
 -1.41738   -2.90865   3.69479 
 -0.43534   -3.06546   3.79964 
 1.37815   -4.22827   2.17849 
 -4.47657   1.26415   1.69803 
 -3.50421   -1.41803   3.29141 
 -0.95692   -3.59216   3.12842 
P30 '
P31 '
P32 '
 -4.79594   -0.71196   1.73878 
 -3.05833   -3.26445   2.55366 
 -0.47757   -4.87734   1.58579 
 
Table 6.33: 6ν T-Octahedron Dome: Final Outer Coordinate Values


side view of dowel and fishing line drawing of 6v double-layer t-octahedron dome with point labels
Figure 6.11: 6ν T-Octahedron Dome: Side View


base view of dowel and fishing line drawing of 6v double-layer t-octahedron dome with point labels
Figure 6.12: 6ν T-Octahedron Dome: Base View

8 In Figure 6.11 the interlayer tendons at the base are included. In Figure 6.12 guys are also excluded.