Tensegrity table

The maker's "floating table", rebuilt for real hardware and a maker's questions. A plywood hook on the base reaches over the centre, a plywood hook on the top reaches under it, and one short steel rope hangs the top from the base. Three to six outer cables pull the top back down and hold it level, so nothing seems to touch. Say how high and how wide, how many cables, and what it must carry: the Jig works out every tension, sizes the fittings from real catalogues, and refuses a load the hooks or the rope cannot take.

What it shows

• The form asks what the maker knows: table height and width, the number of cables, the load in kg, the sheet in the workshop, the router bit. The hook sizes, tensions and rope lengths come out.
• The form changes the structure. cable_count sets three to six cables and three to six sides of a polygon top, or a round top with that many anchors, all from one chained ring (sketch.pattern with chain and angle), and the statics follow it.
• Ranges that follow the design (formulas in variable.publish): the height range comes from the hooks, the rope and turnbuckle lengths and from what the posts can carry in buckling; the hook overlap from the eye nuts and rope eyes; the load limit from the fittings, the hooks and where the table rocks.
• A choice recolours the plates (feature.when on one chain per veneer, with material.preset patterns).
• Revolves about a slanted line (feature.revolve on a construction line in a datum.plane): rope, ferrules and turnbuckle bosses of one leaning cable are one profile. feature.torus makes the eyes and rope loops, feature.pattern turns all of it round the table.
• Measured against the statics. Assertions compare the model with the closed forms the load limit uses: volumes, the top's centre of mass, the hook's lever and the anchor circle (see Checks).

A static model is one table. This one is a family of them, and each knows what it may carry.

The real thing

How the table stands

The top hangs. The central rope pulls the top's hook up and the base's hook down. The outer cables pull the top down and the base up, so the whole table is in self-stress, the principle Snelson and Fuller named tensegrity (Snelson, US 3,169,611; Fuller, US 3,063,521). With the outer cables tight and the central rope inside their circle, the top cannot tip whichever way the load stands.

The old version of this Jig had vertical cables on any top, a cable through a loose hole in the hook arm, crimp sleeves on the floor under the base, and glued tabs nobody checked. This one differs in six ways:

• Hooks with a real elbow and a flared foot. The inside radius is 0.75 x the hook's depth and the foot flares to 1.7 x it, so the corner carries its curved-beam stress (the Winkler formula gives a stress factor of 1.37 at this radius, and the capacity uses it) and the tabs stand further apart.
• Cables that lean a little, and a base to match. Base anchors sit cable_lean further out than the top's, and the base grows with them, so the table is harder to tip. The lean barely stiffens the top against sway, because a push also tilts the top: 3.1 mm under 10 N at the defaults, 3.6 mm with vertical cables (the output, see Sway below).
• Real hardware with catalogue working loads, drawn at its catalogue size (see Hardware).
• A cross dowel in each hook arm. A vertical bolt through the hook's edge is unmakeable in 18 mm plywood; a steel M8 barrel nut (12 mm across, 18 mm long) through the arm's thickness takes the load in bearing instead, an M8 rod runs from it to an eye nut outside the arm, and the rope eye hangs on that. Bearing is checked net of the rod bore on the loaded side (see Statics).
• Flat faces stay flat. Each anchor bolt lies in a pocket on the plate's outer face (19 mm across, 7.6 mm deep, for a socket head and a DIN 9021 washer), flush with the surface. A soft rubber strip 12 mm wide round the base's rim stands the table on its rim, not on inner feet, so a load anywhere on the top lies inside what supports it. Anchors stay 25.5 mm inside the corners, so a pocket keeps 16 mm of plywood to the rim and the strip stays clear of it, on a triangle too.
• CNC reliefs where a square tab must fit a round-cornered slot: 270-degree dog-bones at the slot corners and at the tab roots, sized from the router bit, so the shoulder really seats.

Statics

The load is Q, anywhere on the top up to its furthest corner. W is the top's weight (off-centre by the hook, e), and n cables stand on a circle of radius r around the central rope. In the Jig's variables (document.inspect shows them):

alpha   = 2 R / (n r)                    share of a load at reach R on one cable (pile-group rule)
D_w     = 2 W e / (n r)                  the same for the top's own weight
T_v0    = 1.25 (alpha Q + D_w + Q/n) + 10 N     vertical part of the pretension of each cable
T_set   = T_v0 / cos(lean)               rope tension to tune to
T_max   = (T_v0 + alpha Q + D_w) / cos(lean)    worst outer tension
T_min   = (T_v0 - alpha Q - D_w - Q/n) / cos(lean)   least outer tension (stays > 0, 25 % of the swing spare)
T_c     = n T_v0 + W + Q                 central rope force, taken at its worst (outer cables keep their pretension)

The share alpha is the elastic method of a pile group, with a rigid top on a regular polygon of cables. Vertical load is taken at its worst for each check: as lost by the outer cables (for slack), as carried by the central rope (for its force). Both are safe-side bounds and cannot both happen.

Q is the heaviest load you ask for (the form's Heaviest load it will carry). The tension in the cables is set for it, so putting more on the table than that is not covered: at the defaults (tensioned for 5 kg) a cable goes slack above 7.5 kg at the edge (A cable goes slack above…). The form shows both that and the most the table can be set for. Set the load, then tension to the value the form gives for that load.

Hook capacity. The post is a beam-column. It takes T_c with lever p, the distance from the rope to the post's centre line, and with the Winkler stress factor at the elbow, against a design stress of f = 24.8 x 0.6 / 1.2 / 1.5 = 8.3 MPa: the characteristic compression strength of 13-ply birch plywood (24.8 MPa across the face grain, the lowest strength used) with EN 1995-1-1's kmod = 0.6 for a permanent load, gamma_M = 1.2, and a load factor of 1.5 on top (gamma_F, the EN 1990 value for a variable load, applied to all of it). Buckling of the post out of the hook's plane uses the Euler load of a cantilever (buckling length 2 x the post) with the creep-reduced modulus 7452 / 1.8 MPa (kdef = 0.8) and a factor 3; the two combine as 1/T = 1/T_bend + 1/T_buckle. The tab joint (rolling-shear of the plywood at the glue line, 2.34 x 0.6 / 1.2 / 1.5 = 0.78 MPa over the tab's four walls, lever between the tabs; a stand-in for the glue line, not a test of it), the plate's bending at the foot and the bearing of the cross dowel in the arm are limits too. The dowel's bearing uses the plywood's compression value (8.3 MPa, on the safe side of its embedment strength) over 12 mm by the 9.5 mm of plywood that remains beside the 8.5 mm rod bore, 942 N, a limit that governs the widest thick tops.

Rocking. The table stands on the rim strip, so its support reaches base_apothem - 6 mm. A load on the top beyond that line would rock it; the most any point of the top overhangs that line is 6 mm - lean x cos(pi/n). The load limit takes the smaller of the statics and that rocking limit, M g (support) / (overhang), which never governs here: at least 113 kg above the limit over 5906 sampled combinations. It assumes a floor flat enough for the strip to bear along the edge: a continuous rim can wobble on an uneven floor where three feet would not, which is why the strip must be soft (sponge rubber, not hard rubber or felt), and it may need shimming.

The load limit is the largest load for which every capacity holds.

Over the range the load limit runs from 2 kg (a tall table at its full hook overlap) to 17 kg (a wide top in 24 mm plywood), in 5906 random combinations of the formulas: median 9 kg. The hook post governs in 93 % of them, the dowel in 4 %, the hardware in 3 %. The form always shows it, and the load stops there.

Independent check. The closed forms above are not checked against themselves. A separate 6-degree-of-freedom stiffness solve (a rigid top on elastic cables with E = 57.3 kN/mm², the central rope in series with both hooks in bending, plus the geometric stiffness of the pretension; 72 load directions round the top) was run at 12 points across the range, each tensioned for and loaded with its own load limit. Its outer cable changes bear the closed forms out: the closed form over-estimates the largest outer tension by 8 to 43 %, under-estimates the least by 5 to 15 N (the least closed-form value stays at least 21 N, so no cable goes slack), and the central rope's real increase is 10 to 35 N where the closed form allows the full 69 to 137 N. At the default table at its 8 kg limit: largest outer tension 120 N solved against 135 N closed form, least 33.2 N against 26.0 N. This is an unpublished check: the script is not in this repository, and this is what it gave.

Sway. Push across the hooks (their weak direction) at the top surface. The form's estimate is a closed solution of the sideways shift and the tilt together, with the outer cables' axial and tension stiffness, the central rope's tension stiffness and both hooks bending sideways in series. It over-estimates: under the same assumptions, without the hooks' side-flex, the 6-degree solve gives 1.62 mm where the closed form gives 1.91 mm (default table at its limit; 2.29 mm with the hooks' flex, what the form shows then). It says how the top moves, not how far it can safely go.

What the load limit means. It is the largest static mass that may rest anywhere on the top, including right at its edge over a cable, with the cables tensioned for that load, and every rope, fitting and hook within its working load, and with the factors stated above. It is not for sitting, standing or climbing on, and not for children to hang from. The top hangs from one rope and has no second load path: if the central rope or a hook fails, the top falls. Keep it to books and plants. The form says this too (the Load group and the description). What it cannot prove: your batch of plywood, the glue, the ferrules, the floor. The eye nut is not derated for the lean (up to 8°; see below), and the cross dowel has no catalogue rating; before you trust it near people, have someone competent look at it. A table this size tips over when pushed: the form gives the push at the top that tips it, about 33 N at the defaults, a light shove. Do not stand it where it can be knocked over.

Hardware

Sized for stainless indoor use, with suppliers' declared values:

The ropes are loaded to at most a sixth of their minimum breaking load, the 6:1 factor of S3i's 7x7 rope data sheet for safe working load. An eye made with a pressed ferrule and thimble is weaker than the rope; the factor of 6 is meant to cover that, an assumption of the Jig's. The rope's stiffness uses E = 57.3 kN/mm² on the nominal area (S3i). The working limit in the form is the smallest of rope, turnbuckle and eye nut: for the Light set the 2 mm rope at 405 N. An eye nut's rating holds for an axial pull; cable_lean stops at 8 degrees, so the eye nuts are pulled within 8 degrees of their axis, and the WLL is used unreduced there, a judgement the Jig states and which a certified lifting design would not make. The cross dowel has no catalogue working load: Essentra recommends its M8 barrel nut for boards of 22 mm and up, so at 18 to 21 mm this Jig relies on its own bearing check, a judgement. It stops the sheet at 18 mm, the barrel's length.

Making it

Birch plywood, 18 to 24 mm: 18 mm gives a real table; WISA's and Metsä's declared values for 13-ply 18 mm are the characteristic strengths above (other thicknesses differ by a few per cent over that table, Metsä DoP). The veneer is a colour only: walnut, oak and birch are veneers on the same birch core, so all three keep the structural values.

1. Cut on a CNC router. Export the Cut sheet (below); 6 mm bit by default, set your own. Grain of the hooks along the post. Cut the pockets (19 mm, 7.6 mm deep) on each plate's outer face, or drill them with a Forstner bit, and drill the anchor holes 6.6 mm.
2. Drill the hooks. A 12 mm hole across the arm's thickness for the cross dowel, and an 8.5 mm hole down into the arm's edge, centred in the thickness, to reach the dowel: use a drill press and a fence. At 18 mm plywood that leaves 4.75 mm of wall each side.
3. Glue the hooks to the plates with a rigid glue (epoxy or PU, not PVA: it creeps under a permanent load). Glue the tabs into the slots square to the plate, shoulders tight.
4. Hardware. Bolt an eye nut at each anchor with an M6 socket screw and a DIN 9021 washer from the outer face. Put the cross dowels in, thread the M8 rod through the arm into the dowel, and turn the eye nut on the rod against the arm's edge.
5. Ropes. Press a ferrule and thimble on the ropes, ending the cable from each eye nut in a loop on the turnbuckle's eye. The outputs give each rope's length between its eyes; add the rope that forms the two eyes. Set the turnbuckles about a third open so there is thread left both ways.
6. Tension. Hang the top on the central rope. Bring every outer cable to just taut, then turn all the turnbuckles evenly, in small steps, by the form's Step 1 output: about 0.4 turn at the defaults. The form works this out from the cable and the hooks' stiffness (a stiff plywood value, so by this estimate the turns reach the target tension or fall short of it rather than over-tighten). A luggage scale cannot read a cable that is already installed. Pluck each cable: they should sound alike (equal pitch, equal tension); the pitch of a short stiff rope does not give its tension in newtons. Check again after a day: rope beds in and plywood creeps. Never turn past the form's maximum (0.68 turn at the defaults, against 0.40 to reach the tension). Over-tension cannot be seen or read, and one turn is about 150 N per cable there. The maximum is where the central path or a cable reaches its design limit with your load on, worked out with the stiff value above, so counted from a true just-taut the real tension at that many turns is no higher; turns taken up before the cables were really just taut are not counted, so judge just-taut carefully. At the load limit the two nearly meet (0.60 and 0.63 turn at 8 kg): if you cannot set a turnbuckle that finely, choose a lighter load.
7. Proof test. Keep hands and feet clear of the top and lower the load you tensioned for gently onto the table's edge, right over a cable: that cable must stay taut. If it goes slack the top tilts while the other cables still hold: lift the load off, add a little turn to every turnbuckle and repeat, never past the maximum. If it still goes slack at the maximum, the table is not good for that load: set a lighter one. Do not load beyond what the form says. The form gives these steps too.

Parameters

Outputs in the form: the load the table is tensioned for, the most it can be set for, where a cable goes slack at the edge, the tension of each outer cable, the turnbuckle turns to reach it and the turns never to pass, the proof-test load, largest outer tension and the working limit of a cable and its fittings, central rope force and what the central path carries, sway under a 10 N push, the push that tips the table, rope lengths between their eyes (lower and upper outer pieces, central), table mass.

Checks

The 20 assertions, each measured on the model and compared with a value the statics use:

• Height of the whole table as set; the hooks clear of each other by 10 mm; hooks and fittings clear of the cables.
• Closed-form volumes of the top plate (outline, holes, slots with reliefs, pockets, within 0.2 %) and of a hook (within 0.5 %).
• The top's centre of mass where the statics assume it (within 1 mm); the hook foot at the statics' lever position; the anchors at the statics' circle.
• The hook's height and the top's reach as the capacities take them (post_length_mm, load_reach_mm).
• Cable count from the top's holes; the cut sheet holds the two plates and two hooks; the plywood wall beside the rod bore at least 3 mm; the rim gasket's closed-form volume, and the gasket at least 5 mm clear of the anchor bolts.
• Each cable's lower eye ring on its base anchor, upper eye ring on its top anchor, and the cable's height the statics' span.
• No overlap of plates and hooks, or of bolts, dowels and rods with the wood they pass.

No assertion merely restates the load limit: that is a formula, and the published range is its check. What the assertions guard is that the model's geometry is the geometry the statics assume. The statics themselves are the closed forms above and the independent solve. The geometry assertions compare the model with the very variables the capacities read (lever_mm, anchor_r_mm, span_mm, post_length_mm, load_reach_mm). Mutation test on a scratch copy: adding 20 mm to lever_mm, 15 to anchor_r_mm, 30 to span_mm, 30 to post_length_mm, 20 to load_reach_mm or 3 to thickness_mm, reversing the cable's lean, moving the base anchors 20 mm or the top anchors 10 mm, and thinning the rim gasket each fail an assertion.

tools/jigs.py check runs check-range (the defaults, 64 corners of the box of ranges, every choice and 32 random points): all pass, none unproven. A further run of 306 points (300 random, seed 23) on this final model also passes. An earlier longer run found a boolean failure the standard points missed, a cutter end cap on the arm's face at some values, now fixed. The combinations tried by hand to break the Jig, and what the form does:

Checked by hand, not a limit in the load limit: the DIN 9021 washer on the plywood (220 mm², about 2 MPa at 405 N).

Not checked: the true strength of your plywood batch, a glue line in your shop, creep of the rope and ferrules over the years, a floor that is not flat, or any fall.

Export

• File › Export, Sheet of set: Cut sheet (SVG or DXF): the top plate, base plate and both hooks at true size, on one sheet, with holes and slots, ready for CAM. meta.json proves both formats, at the defaults and at a six-sided round top. Pockets are not on the sheet as cuts of their own: they are the 19 mm circles round the anchor holes in the description above, machined or drilled at the depth stated (the cut sheet shows the through holes). Pockets, hook drilling, and rope assembly follow the Making it list.

The README in the model's repository has the rest (Files, Rebuild, Difficulty, Platform notes, Licence): the whole README on GitHub.