Tensegrity Structures

RELATED APPLICATION(S)

[0001]This application claims the benefit of U.S. Provisional Application No. 63/742,144, filed on Jan. 6, 2025. T1he entire teachings of the above application are incorporated herein by reference.

BACKGROUND

[0002]Tensile integrity (tensegrity) structures are known in the art as being composed of compression and tension elements, as described, for example, in U.S. Pat. No. 3,063,521, issued from U.S. app. Ser. No. 837,073, the entirety of which is hereby incorporated by reference. Examples of tensegrity structures include functional structures, such as chairs and tables, and purely artistic structures, such as sculptures. These include so-called “floating” structures, such as floating tables and chairs; these structures lack continuous rigid members extending from the bottom to the top of the structure, creating the illusion that the top of the structure is floating above the base.

[0003]Some tensegrity structures, including many artistic structures, are designed to accommodate only static loads, such as those arising from the weight of the structure itself. Other structures, such as floating tables and chairs, must accommodate applied external loads, which can be variable.

SUMMARY

[0004]So-called “floating” tables and chairs, and other tensegrity structures, are comprised of rigid elements (e.g., struts) that connect to flexible tension elements (e.g., cables). The inventor recognized that the cables in such structures can induce bending moments in the struts, resulting in stress and material fatigue.

[0005]According to aspects of the present invention, these bending moments are reduced or eliminated by suitably routing the cables around pulleys or other “rotational couplings” that have been attached to the struts. For example, when a cable approaches and leaves a pulley at the same angle (i.e., at substantially equal and opposite angles, such as +30°/−30° or +90°/−90°) relative to the longitudinal axis of a strut, the theoretical net bending moment in the strut is zero. This happens because the pulley does not change the magnitude of the tension in the cable, but only its direction; and the geometry dictates that the resultant force exerted by the cable on the strut is entirely compressive, i.e., directed along the strut's longitudinal axis. This reduces stress and material fatigue in the strut.

[0006]As further appreciated by the inventor, the theoretical net bending moment remains zero regardless of the tension in the cable. This is a significant advantage, particularly where the cable carries the weight of a loadable portion of the structure, such as the top surface of a floating table or the seat of a floating chair. Regardless of loading (or changes in loading), bending moments in the struts are reduced or eliminated. In addition to reducing fatigue, this allows the strut to have a higher slenderness ratio than could otherwise be achieved, with attendant material and cost savings.

[0007]In some embodiments, the structure comprises a bottom member and a top member. The bottom member is configured to support the structure relative to a support surface and has an ascending section connected to a first rotational coupling at a first coupling region. The top member lacks rigid connectivity with the bottom member and has a descending section connected to a second rotational coupling at a second coupling region. The second coupling region has a vertical position lower than that of the first coupling region, where the “vertical position” represents an elevation along the gravitational axis when the structure is in a quiescent state. A support cable engages in turn with the bottom member, the first rotational coupling, the second rotational coupling, and the top member. A plurality of stabilizing cables, each connected to both the top member and the bottom member, stabilize the structure.

[0008]The structure can be a tensegrity structure, and it can be, for example, a table or chair. The rotational couplings can be frictionless, and can be, for example, pulleys. The cable can be braided wire. In some structures, the theoretical bending moment is zero along at least a portion of the ascending or descending and/or substantially all forces in the sections are compressive forces.

[0009]In some embodiments, at least one of the ascending section and the descending section comprises a first rigid elongate member, which is optionally substantially aligned with the gravitational axis. A second rigid elongate member can be attached at the end of the first rigid elongate member, and it can be oriented substantially perpendicularly to it.

[0010]In some embodiments, the cable supports the combined weight of the top member and any external load.

[0011]In some embodiments, the cable has a substantially vertical central section from the first rotational coupling to the second rotational coupling, and its theoretical tension varies substantially linearly with the supported load. If desired, the center of mass of the top member can be substantially colinear with the central cable section.

[0012]In some embodiments, the plurality of stabilizing cables collectively exerts a net downward force on the top member, which is carried by the central cable.

[0013]In some embodiments, the structure further comprises a first auxiliary rotational coupling connected to the bottom member at a first auxiliary coupling region; and a second auxiliary rotational coupling connected to the top member at a second auxiliary coupling region. The first auxiliary coupling region is farther from the central axis (the axis that runs vertically through the structure's center of mass) than is the first coupling region, and the second auxiliary coupling region is farther from the central axis than is the second coupling region. The support cable engages in turn with the bottom member, the first auxiliary rotational coupling, the first rotational coupling, the second rotational coupling, the second auxiliary rotational coupling, and the top portion.

[0014]In some embodiments, the descending section has a substantially vertical descending column having an upper end and a lower end; and a transverse lower beam having a distal end and a central end, the central end being nearer to the central axis of the structure. The lower end of the descending column connects to the transverse lower beam at a lower connection region located in a midsection of the lower beam between the distal end and the central end. The transverse lower beam has a proximal lower section extending from the central end to the lower connection region and a distal lower section extending from the lower connection region to the distal end. At least a portion of the proximal lower section and the distal lower section are cantilevered under the descending column. The first rotational coupling and the first auxiliary rotational coupling are connected to the proximal lower section and the distal lower section, respectively, of the transverse lower beam.

[0015]In addition, in some embodiments, the ascending arm likewise comprises a substantially vertical ascending column having a lower end and an upper end; and a transverse upper beam having a central end and a distal end. The upper end of the ascending column is connected to the upper beam at an upper connection region located in a midsection of the transverse upper beam between the central end and the distal end. The transverse upper beam has a central upper section extending from the central end to the upper connection region and a distal upper section extending from the upper connection region to the distal end. At least a portion of the central upper section and the distal upper section are cantilevered over the ascending column. The second rotational coupling and the second auxiliary rotational coupling are connected to the central upper section and the distal upper section, respectively, of the transverse upper beam.

[0016]The first rotational coupling, the first auxiliary coupling, the second rotational coupling, and the second auxiliary rotational coupling can be pulleys; the theoretical bending moments in the descending column or the ascending column can be zero; and/or substantially all forces in the descending column or the ascending column can be compressive forces.

BRIEF DESCRIPTION OF THE DRAWINGS

[0017]FIG. 1 is a front view of one example of a prior art tensegrity or “floating” table, illustrating characteristics and shortcomings as appreciated by the inventor.

[0018]FIG. 2A is a front view of an embodiment of the present invention, depicting the use of rotational couplings to engage the cables with the struts. As shown, the cable approaches and departs from each coupling at roughly the same angle with respect to the strut's longitudinal axis. This largely eliminates bending stress in the struts.

[0019]FIG. 2B is a front view of another embodiment of the present invention. Unlike in FIG. 2A, the cable does not approach and depart from each coupling at the same angle with respect to the strut's longitudinal axis. This reduces, but does not eliminate, bending stress in the struts.

[0020]FIG. 2C is a schematic of a rotational coupling (pulley) connected to a strut, and showing a cable approaching and leaving the coupling.

[0021]FIG. 3A is a front view of another embodiment of an improved tensegrity structure according to aspects of the present invention. The support cable is routed through a system of pulleys in a manner that neutralizes bending moments in the upright strut portion of the rigid support arms. These rigid arms also have a beam that is oriented transverse to the struts and is doubly-cantilevered over the ends.

[0022]FIG. 3B is a perspective view of the table shown in FIG. 3B.

[0023]FIG. 3C is a schematic depicting how the arrangement of the cable and pulleys in FIGS. 3A and 3B eliminates the bending moments in the upright strut segments of the rigid support arms.

[0024]FIG. 4A is a front view of another embodiment of an improved tensegrity structure. The support arms are inclined struts. Only one pulley is attached to each strut. The support cable is routed such that the theoretical net bending stress along the entire length of the inclined struts is zero, and the only stresses experienced by the struts are compressive.

[0025]FIG. 4B is a front view of another embodiment of an improved tensegrity structure, wherein a pulley is used to reduce the bending moment at only one of the two support arms.

[0026]FIG. 5 is a front view of yet another embodiment of the present invention, which has three vertical support struts, each engaged with the support cable by a single pulley.

[0027]The foregoing will be apparent from the following more particular description of example embodiments, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating embodiments.

DETAILED DESCRIPTION

[0028]A description of example embodiments follows.

[0029]As recognized by the inventor, in some tensegrity structures such as floating tables and chairs, rigid elements can experience not only compressive, axial forces, but bending moments as well. Consider, for example, a prior art floating table structure depicted in FIG. 1. The table has a top member 20, a rigid descending strut 25, a bottom member 10, and a rigid ascending strut 15. Flexible support cable 50 is connected to the upper end of ascending strut 15 and to the lower end of descending strut 25. Accordingly, both strut 25 and top member 20 are suspended from strut 15. Because the top member is above the point of suspension and because no rigid elements connect the top member to the bottom, this creates the illusion that the top member is “floating.” Four side cables 80 (two are shown in profile view) provide stability.

[0030]As recognized by the inventor, in such a design, struts 15 and 25 experience bending moments caused by the action of cable 50. In addition, as load 99 is varied, these bending moments change, creating dynamic fatigue. This can require the use of different, wider, or heavier materials of construction, which can reduce the achievable slenderness ratio in the struts, and increase cost and weight.

[0031]As recognized by the inventor, alternative designs of such structures can not only neutralize the bending moments in rigid support elements, but they can also do so across a range of applied loads.

[0032]According to aspects of the present invention, pulleys and other rotational couplings are attached to the struts of tensegrity structures and are used to route cables around the struts in geometries that neutralize the bending moments caused by the action of the cables on the struts. Because idealized pulleys are frictionless (and real pulleys are substantially frictionless over working load ranges), cable tension is essentially the same in the cable segments entering and leaving a pulley.

[0033]Accordingly, if the cable enters and exits a pulley at the same angle relative to the longitudinal axis of the strut, the theoretical bending moment in the strut is zero. In some embodiments, all the forces in the struts are compressive forces. Even though cable tension can scale with applied load, the cable tension in the strands entering and exiting the pulley remains equal. Although compressive forces in the struts will similarly scale with the applied load, the theoretical bending moment is still zero. Because many materials can resist compression better than bending, struts can achieve much higher slenderness ratios than would otherwise be possible.

[0034]As used herein, “rotational couplings” refers to couplings that engage with and change the direction (i.e., “rotate” the direction) of a tension element. They provide limited or no resistance to motion of the tension element along a travel path. The rotational coupling can have a connector that attaches it to another element, such as a strut, thereby “coupling” the tension element and the rigid element. One example is a pulley. Other examples include wheels, polished rods, or axes designed to reduce friction. Some rotational couplings permit movement of the cable primarily tangentially, rather than normally, to the point of attachment. Ball bearings are used in some rotational couplings.

[0035]As used herein, “cable” and “tension element” are used synonymously to refer to flexible structures that transmit tension. Examples include ropes, wires, filaments, aircraft cable, braided wire, wire rope, and braided cables. They can be made of any suitable material, including natural fibers, synthetic fibers (including polymers), and metals, including steel, stainless steel, copper, aluminum, or galvanized materials. They can be formed as a single strand or filament or as a plurality of strands braided together.

[0036]As used herein, “compression element” refers to materials designed to resist longitudinal compression under load to an extent necessary for structural stability and to prevent failure. Compression elements can include struts and beams, and they can be formed, without limitation, of wooden, metal, plastic, glass or carbon fiber, ceramic, composite, or other materials.

[0037]As used herein, “theoretical” or “ideal” means that ideal characteristics of the components are assumed, including that rotational couplings are frictionless; that rigid components do not deflect, deform, or compress; and that tension elements are infinitely flexible and massless. It should be appreciated that real devices are not ideal frictionless. For example, due to friction, the tension in a cable will not be the same on either side of a real rotational coupling, but rather will be different by some amount, such as 5%, 10%, 20%, or 30%.

[0038]FIG. 2A shows a front view of an embodiment of the present invention. Tensegrity structure 200A comprises a top member 220A and a bottom member 210A, which rests on support surface 213A. Top member 220A lacks rigid connectivity with bottom member 210A.

[0039]Bottom member 210A has an ascending section 234A, which connects to a first rotational coupling, 260A. Top member 220A has a descending section 233A, which is connected to a second rotational coupling 266A. Top surface 221A has a vertical position higher (relative to the support surface) than that of the first rotational coupling 260A. Second rotational coupling 266A has a lower vertical position than first rotational coupling 260A.

[0040]A flexible tension element, cable 250A, is fixedly attached to the bottom portion at connection point 251A. The cable proceeds as follows: vertically upwards via cable segment 253A to the first rotational coupling 260A, downwards via central support segment 255A to the second rotational coupling 266A, and then upwards again via segment 257A to a point of fixed connection 259A with the top member. Four stabilizing cables 280A (two are visible in profile view) provide stability to the top portion.

[0041]The tension in cable 250A exerts a force on section 234A via its connection to rotational coupling 260A. This force is represented by vector 245A. Vector 245A is aligned with the longitudinal axis of ascending section 234A, and it exerts mainly compressive forces on the section. At the thinnest section of section 234A, where failure is most likely to occur, the force is entirely compressive.

[0042]FIG. 2B differs from FIG. 2A in that the cable, 250B, is routed and oriented such that the net force 245B exerted on the ascending section 234B of the bottom portion 201B lies along a vector 245B that is not colinear with the longitudinal axis of the ascending section. As a result, the bending moment along arm 234B is reduced, but not entirely eliminated.

[0043]FIG. 2C shows details of an example of a rotational coupling that can be used in connection with some embodiments of the present invention. Pulley 70 is connected to strut 8 by connector 76 (an axle extending into the plane of the page). Cable 5 enters pulley 70 in the direction of vector 6A, takes path 72 over a cable contact surface, and leaves in the direction of vector 6B. Tension in the cable is exerted on the coupling normal to the cable contact surface, as depicted by the family of force vectors 7a. The cable is compressively engaged with wheel 75, which spins freely with respect to axle 76 because of ball bearing ring 74 (the interior bearings are not shown). The sum of the force vectors 7a is vector 77, which acts on strut 8 via axle 76. Pulley 70 is a rotational coupling; accordingly, it rotates the cable vector from direction 6A to direction 6B, across an angle 61. The supplement 65 of angle 61 is given by θ, and is known as the “bending angle” provided by the rotational coupling. A variety of bending angles can be used in accordance with the present invention. Typically, these bending angles are between 0° and 180°, including, without limitation about 10°, 20°, 30°, 45°, 50°, 60°, 90°, 120°, 150° or in a range between the foregoing.

[0044]An ideal rotational coupling will exert zero resistance to the travel of the tension element along the travel path, while real (non-ideal) couplings will, depending on configuration, provide a resistance force along the travel path of less than about 40%, 30%, 20%, 10%, or 5% of the tension in the tension element. Other rotational couplings can be used, including polished axles, where the cable slides without substantial (and ideally zero) friction along the contact surface.

[0045]FIGS. 3A and 3B, show a front and a perspective view, respectively, of another embodiment of the present invention. Floating table 300 has a bottom member 310 and a top member 320.

[0046]Bottom member 310 is comprised of a left lateral segment 310L, a right lateral segment 310R, a central segment 310C, and an ascending section 315. Ascending section 315 has a slender vertical ascending column 315U and a transverse upper beam 315T. Upper beam 315T has two pulleys: a first pulley 366 (located closer to the central axis 395), and a first auxiliary pulley 368, located farther from the central axis. (The central axis 395 of the structure runs vertically through its center of mass 397, as determined absent any external load.)

[0047]Top member 320 has a top surface 322, and is comprised of a descending section 325, which in turn comprises a slender vertical descending column 325D and a transverse lower beam 325T. Lower beam 325T has two pulleys: a second pulley 360 (located closer to the central axis) and a second auxiliary pulley 362 located farther from the center. Both the ascending and descending sections consist of vertical columns 315U, 325D, with beams, 315T, 325T, cantilevered over both ends of each column, i.e., they are doubly cantilevered beams.

[0048]The first pulley 366 and first auxiliary pulley 368 are located higher (along the vertical axis) than are the second pulleys 360 and second auxiliary pulley 362. For stability, four flexible stabilizing cables 380 connect the top member and the bottom, to which they are fixedly attached at connection points 370. An external load 399 is optionally applied to the top member.

[0049]Flexible support cable 350 is fixed to the top member at attachment point 359 and to the bottom member 310 at attachment point 351. Cable 350 passes from the bottom attachment point 351 as follows: up via first countermoment segment 353, over first auxiliary pulley 368, over first pulley 366, down via central support segment 355, under second pulley 360, under second auxiliary pulley 362, and up via second countermoment segment 357 to top connection point 359.

[0050]Because the tension in cable 350 is (ideally) uniform along its length: (1) the bending stress along the lengths of the ascending column 315U and the descending column 325D are greatly reduced (and reduced to zero for frictionless pulleys), thereby minimizing structural fatigue; and (2) the bending stress in the columns remains low (and is ideally zero) notwithstanding changes in the tension of cable 350 as the weight of the load 399 varies. It is precisely these changes in load that are simply and elegantly transmitted to opposite sides of the arms to provide continuous load-independent counterbalance.

[0051]FIG. 3C illustrates this concept with arrows depicting forces acting on the ascending and descending sections as a result of the tension in rope 350. In ideal conditions (for frictionless pulleys), the forces on the transverse beam of the ascending section are as follows: downward force 343D as applied by the first countermoment segment and an equal downward force 345D as applied by the central support segment. The pulleys are spaced on the beam evenly on either side of the column. As a result, there is no net bending moment, and no bending stress along the length of the ascending column.

[0052]The situation is similar for the descending section. Upward force 347U as applied by the second countermoment segment is balanced by an equal upward force 345U as applied by the central support segment. Again, the pulleys are spaced on the beam evenly on either side of the column. As a result, there is no net bending stress along the length of the descending column.

[0053]For this arrangement, and assuming frictionless pulleys, ideal counterbalance of forces across the transverse (cantilevered) segments of the rigid arms is achieved by an equal spacing of the pulleys about the columns and in which the countermoment rope segments are vertical, i.e. parallel with the columns. Other arrangements are possible to achieve zero net torque taking into account the force vectors applied by the respective rope segments (accounting for both magnitude and direction) and the displacement vectors from the point of application of the force to a point about which torque is to be determined.

[0054]It should be noted that in this design, which uses “T”-shaped assemblies for the ascending and descending sections, portions of the assembly (namely the transverse beams) will still experience bending moments. Columns 315U and 325D, however, do not experience any theoretical net bending moment. This permits at least that portion of the rigid assembly to be made more slender than would otherwise be possible using the same materials, design tolerances and specifications, with attendant material savings, cost savings, and flexibility for aesthetic and functional purposes.

[0055]In various embodiments, and depending on the shape of the support segment (e.g., column or other support arm), this bending stress-free section can extend for a substantial portion along one or more axes of the support sections (e.g., along the length of the thinnest dimension), such as for example, at least 20%, 40%, 60%, 80%, or 100% of the length. In some embodiments the bending stress in the material is zero across the entirety of the relevant material cross section. If desired, the bending stress-free segment can be chosen to coincide with an anticipated failure region of the segment. For example, the anticipated failure region in a support arm made of a uniform material could be the region subject to the highest bending stress relative to its cross-sectional area. In other embodiments, the bending stress-free region of an elongate support section is selected to correspond to at least the region having the smallest cross section.

[0056]Referring back to FIG. 3A, the central segment 355 of the cable is colinear with the center of mass 397 under unloaded conditions. This reduces to zero the torque caused by the action of the upward force from central cable section 355 on the top member. Other configurations are possible. Where the center of mass of the structure (or of the structure plus load) is not in line with the upward force applied by the support segment of the cable acting on the descending arm, then the torque on the top member can be balanced by differential tension in the stabilizing cables 380. The tension in each of the stabilizing cables can be different, according to principles of static equilibrium, and as a whole they can be pretensioned.

[0057]FIG. 4A is a drawing of another embodiment of a floating table according to principles of the present invention. Bottom member 410 is comprised of a left lateral segment 410L, a right lateral segment 410R, a central segment 410C, and an ascending section 415. Ascending section 415 is an inclined ascending column 415 to which a first pulley 466 is attached.

[0058]Top member 420 has a top surface 422, and is further comprised of a descending section 425. Descending section 425 has a second pulley 460. The first pulley 466 is located higher (along the vertical axis) than is the second pulley 460. For stability, four flexible stabilizing cables 480 connect the top member and the bottom member. An external load 499 is optionally applied to the top member.

[0059]Flexible support cable 450 is fixed to the top member at attachment point 459 and to the bottom member 410 at attachment point 451. Cable 450 passes from the bottom attachment point 451 as follows: up via first countermoment segment 453, over first pulley 466, down via central support segment 455, under second pulley 460, and up via second countermoment segment 457 to top connection point 459.

[0060]Because the tension in cable 450 is (ideally) uniform along its length: (1) the bending stress along the entire lengths of the ascending section and the descending section are greatly reduced (and reduced to zero for under ideal conditions), thereby minimizing structural fatigue; and (2) the bending stresses in the columns remains low (and is ideally zero) notwithstanding changes in the tension of cable 450 as the weight of the load 499 varies.

[0061]Consider, for example, ascending arm 415. Because the theoretical tension in the rope is constant across its length, central segment 455 and countermoment segment 453 both exert the same magnitude of generally downward force on ascending section 415 via interaction with pulley 466. The geometry is such that the angle formed by countermoment segment 453 with the longitudinal axis of arm 415 is equal to the angle formed between the longitudinal axis and the central cable segment 455. As a result, the net bending moment on arm 415 is zero, and the arm experiences only compressive forces along its longitudinal axis.

[0062]Here, the bending angle of cable 450 around pulley 466 is an acute angle. A variety of bending angles can be used.

[0063]FIG. 4B shows another embodiment according to aspects of the present invention. Here, the ascending section 415 is connected to a pulley, but the descending section is not, with cable 450 instead being fixedly attached to the lower end of the descending arm. Accordingly, the theoretical bending moment is eliminated in only the ascending section 415.

[0064]FIG. 5 is a drawing of another embodiment. Bottom member 510 is comprised of a left lateral segment 510L, a right lateral segment 510R, a central segment 510C, and two ascending sections: left ascending section 515L and right ascending section 515R, to which are attached a left pulley 566L and a right pulley 566R, respectively.

[0065]Top member 520 has a top surface 522 and is further comprised of a descending section 525. Descending section 525 has a central pulley 660. The left pulley 566L and the right pulley 566R are both located higher (along the vertical axis) than is the central pulley 560. For stability, four flexible stabilizing cables 580 connect the top member and the bottom. An external load 599 is optionally applied to the top member.

[0066]Flexible support cable 550 is fixed to the bottom member at attachment points 551L and 551R. Cable 550 passes from the bottom attachment point 551L as follows: up via left countermoment segment 553L, over left pulley 566L, down via left central support segment 555L, under central pulley 560, and up via right central support segment 555R, over right pulley566R and down again via right countermoment segment 553R to bottom attachment point 551R.

[0067]All three columns, 515L, 525, and 515R, experience zero theoretical bending moment, because the resultants of the pairs of torque vectors arising from the action of the rope segments on the arms are zero—with a net compressive force being directed axially along the length of each rigid column.

[0068]The teachings of all patents, published applications, and references cited herein are incorporated by reference in their entirety.

[0069]While example embodiments have been particularly shown and described, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the scope of the embodiments encompassed by the appended claims.