Pre-stressed sinusoidal member in assembly and applications

Abstract

Sinusoidal shaped member units and support member units are parts that form pre-stressed assemblies having flexural properties. Sinusoidal shaped members are relaxed material members that have been elastically deformed. Support members maintain the elastically deformed state of the sinusoidal shaped members. The sinusoidal shaped members and support members are organized into pre-stressed curvilinear assemblies containing stored elastic potential energy that is equal to the work done by the forces that deformed their pre-stressed structure. The assemblies' sinusoidal shaped members and support members are adapted to use materials having exceptional mechanical properties and flexural strength. This includes nano-composites. The assemblies' pre-stressed state enhances its mechanical, electrical and structural performance. The size, number, density and possible geometric configurations of the sinusoidal shaped member units and support member units within an assembly/structure is vast. Products of this sinusoidal building system have mechanical and structural applications and can be manufactured using an automated process.

Claims

1. A flexure for structural and mechanical applications having an assembly axis comprising at least two prestressed sinusoidal members with residual tension and compression along a length of each of said at least two prestressed sinusoidal members and formed of an elastic material that allows for reversible deformation without compromising a strength of said at least two prestressed sinusoidal members, each of said at least two prestressed sinusoidal members supported by a first support member at a distal antinode and connected together longitudinally at a proximal antinode by a locking mechanism on a second support member, holding the at least two prestressed sinusoidal members in tension and compression.

2. The flexure of claim 1 wherein the elastic material is a composite material.

3. The flexure of claim 2 wherein the composite material is a carbon fiber reinforced polymer or a nano-composite material.

4. The flexure of claim 1 wherein each of said at least two sinusoidal members is formed from a discrete, straight, relaxed linear member.

5. The flexure of claim 4 wherein the discrete, straight, relaxed linear member has a prismatic or non-prismatic geometry.

6. The flexure of claim 5 wherein the prismatic or non-prismatic geometry of the discrete, straight, relaxed linear member or an elastic property varies along the length, a width or a depth of the discrete, straight, relaxed linear member.

7. The flexure of claim 1 wherein the first support member is under a compressive force or the second support member is under a tensile force.

8. The flexure of claim 7 wherein the first support member under the compressive force is a ring, n-gon, linear strut or inflatable member.

9. The flexure of claim 7 wherein the second support member under the tensile force is an axial member under a tension.

10. The flexure of claim 9 wherein each of the at least two prestressed sinusoidal members are fixed to the axial member at ends of the at least two prestressed sinusoidal members such that the tensile force is evenly distributed along a length of the at least two prestressed sinusoidal members.

11. The flexure of claim 9 wherein the axial member is periodically attached to the at the least two prestressed sinusoidal members at an antinode.

12. The flexure of claim 1 wherein the first support member is under a compressive force and a plurality of first support members are spaced apart at a spatial frequency that corresponds to a wavelength of said at least two prestressed sinusoidal members.

13. A flexure for structural and mechanical applications having an assembly axis comprising at least two sinusoidal prestressed members with residual tension and compression along a length of each of said at least two prestressed sinusoidal members and formed of an hyperelastic material that allows for substantial reversible deformation without dissipation of energy, being disassemblable and reassemblable into said flexure, each of said at least two prestressed sinusoidal members supported by a first support member at a distal antinode and connected together longitudinally at a proximal antinode by a flexural locking mechanism on a second support member.

14. The flexure of claim 13 wherein each of said at least two prestressed sinusoidal members is formed from a discrete, straight, relaxed linear member.

15. The flexure of claim 13 wherein the at least two sinusoidal prestressed members have a same wavelength and phase.

16. The flexure of claim 15 wherein the at least two prestressed sinusoidal members have a same amplitude.

17. The flexure of claim 13 wherein the at least two prestressed sinusoidal members have a different wavelength, amplitudes or phases.

18. The flexure of claim 13 wherein an internal force is evenly distributed and corresponds to a center of mass/centroid axis of the flexure such that the flexure is an un-deflected structure.

19. The flexure of claim 13 wherein an internal forces is unevenly distributed and falls outside a center of mass/centroid axis of the flexure such that the flexure is deflected.

20. A planar, radial or polygonal flexural assembly formed from two or more of the flexures of claim 13, connected at antinodes of the at least two prestressed sinusoidal members.

Description

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

(1) In the accompanying drawings, in which several of various possible embodiments of the invention are illustrated, corresponding reference characters refer to corresponding parts throughout the several views of the drawings in which:

(2) FIG. 1A is an orthogonal view showing an exemplary conception of a relaxed member used to form a sinusoidal shaped member.

(3) FIG. 1B is an orthogonal view showing an exemplary conception of a sinusoidal shaped member formed from a relaxed member that is linear, slender and made of a high performance composite material.

(4) FIG. 2A is a perspective view depicting a relaxed prismatic beam having rectangular cross section.

(5) FIG. 2B is a perspective view depicting a relaxed prismatic beam having triangular cross section.

(6) FIG. 2C is a perspective view depicting a relaxed prismatic beam having pentagonal cross section.

(7) FIG. 2D is a perspective view depicting a relaxed prismatic beam having square cross section.

(8) FIG. 2E is a perspective view depicting a relaxed non-prismatic beam having hexagonal cross section.

(9) FIG. 2F is a perspective view depicting a relaxed non-prismatic beam having rectangular cross section.

(10) FIG. 3A is a perspective view of an exemplary relaxed prismatic member depicting its cross-section before being formed into a sinusoid.

(11) FIG. 3B is a perspective view of an exemplary relaxed prismatic member depicting its cross-section after being formed into a sinusoid.

(12) FIG. 4A is an orthogonal view depicting the physical wave parameter of wavelength.

(13) FIG. 4B is an orthogonal view depicting the physical wave parameter of amplitude.

(14) FIG. 4C is an orthogonal view depicting physical wave phase relationships.

(15) FIG. 5A is an orthogonal view of a relaxed slender tapered beam.

(16) FIG. 5B is an orthogonal view of a slender tapered beam after becoming a pre-stressed damped sinusoidal shaped member.

(17) FIG. 6A is an orthogonal view of a sinusoidal shaped member having regular period.

(18) FIG. 6B is an orthogonal view of FIG. 6A's spatial frequency and is oriented to show their shared phase geometry.

(19) FIG. 6C is an orthogonal view of a damped sinusoidal shaped member.

(20) FIG. 6D is an orthogonal view of FIG. 6B's spatial frequency and is oriented to show their shared phase geometry.

(21) FIG. 6E is a perspective view of the members axis and spatial frequency.

(22) FIG. 6F is an orthogonal view showing bending moments as force arrows shaping a relaxed member into elastic sinusoidal shaped member.

(23) FIG. 6G is a perspective view showing the relationship between the sinusoidal shaped member, its spatial frequency and support member location.

(24) FIG. 7A is a perspective view showing a sinusoidal assembly labeling the sinusoidal shaped members, support members, assembly's axis and the axis of an individual sinusoidal member that is shaded.

(25) FIG. 7B is a perspective view showing the assembly's sinusoidal shaped members and identifying the location of the nodes and antinodes that punctuate the assembly's length.

(26) FIG. 7C is an orthogonal view depicting bending moments as force arrows along the sinusoidal shaped members' lengths.

(27) FIG. 7D is an orthogonal view depicting the sinusoidal shaped members and the unit support members that punctuate the assembly's length.

(28) FIG. 7E is an orthogonal view of exemplary sinusoidal structures formed from various sinusoidal shaped members and various support members types.

(29) FIG. 7F is a front view of an exemplary sinusoidal structure that exhibits branching.

(30) FIG. 7G is a side view of an exemplary sinusoidal structure that exhibits branching.

(31) FIG. 7H is a top view of an exemplary sinusoidal structure that exhibits branching.

(32) FIG. 7I is a perspective view of an exemplary sinusoidal structure that exhibits branching.

(33) FIG. 8A is a perspective view of an exemplary sinusoidal planar assembly.

(34) FIG. 8B is a lengthwise top view of the planar orientation and organization of the sinusoidal shaped members in assembly.

(35) FIG. 8C is a perspective view of an exemplary radial assembly.

(36) FIG. 8D is a lengthwise top view of the radial organization and planar orientation of the sinusoidal shaped members in assembly.

(37) FIG. 8E is a perspective view of an exemplary closed form polygon assembly.

(38) FIG. 8F is a lengthwise top view the 3-gon organization of the sinusoidal shaped members in assembly

(39) FIG. 9A is a perspective view or an exemplary 4-gon sinusoidal assembly.

(40) FIG. 9B depicts the sinusoidal shaped members 4-gon configuration in lengthwise top view.

(41) FIG. 9C depicts the sinusoidal shaped members and support member 4-gon configuration in lengthwise top view.

(42) FIG. 9D is a perspective view or an exemplary 6-gon sinusoidal assembly.

(43) FIG. 9E depicts the sinusoidal shaped members 6-gon configuration in lengthwise top view.

(44) FIG. 9F depicts the sinusoidal shaped members and support member 6-gon configuration in lengthwise top view.

(45) FIG. 9G depicts another closed form polygon assembly in perspective view using a different style of support member.

(46) FIG. 9H shows line drawings of the 6-gon test specimen depicted in FIG. 9G.

(47) FIG. 9I is a perspective view or an exemplary 8-gon sinusoidal assembly.

(48) FIG. 9J depicts the sinusoidal shaped members' 8-gon configuration in lengthwise top view.

(49) FIG. 9K depicts the sinusoidal shaped members' and support members' 8-gon configuration in lengthwise top view.

(50) FIG. 10A is a perspective view of a flexural lattice assembly.

(51) FIG. 10B is a perspective view showing one of the flexural lattice's twenty-one sub-assemblies.

(52) FIG. 10C is a lengthwise top view of the lattice assembly.

(53) FIG. 11A shows perspective views of two flexural lattice assemblies built from 4-gon sub-assemblies.

(54) FIG. 11B shows lengthwise top views of these lattice assemblies; this includes the sinusoidal shaped members and support members.

(55) FIG. 12A is a perspective view of a tubular/n-gon assembly formed from lengthwise oriented sinusoidal shaped members and ring type support members.

(56) FIG. 12B is a perspective view of an impossible (unsupportable) tubular/n-gon assembly formed from lengthwise oriented sinusoidal shaped members and strut type support members.

(57) FIG. 12C is a perspective view of a tubular assembly formed from circumferentially oriented sinusoidal shaped members and strut type support members.

(58) FIG. 13A is a perspective view of an assembly having radially oriented sinusoidal shaped member and free antinodes because it uses an axial support member.

(59) FIG. 13B is a perspective view of an assembly having radially oriented sinusoidal shaped member with antinodes attached to ring type support members.

(60) FIG. 14A is a orthogonal view of an un-deflected assembly.

(61) FIG. 14B is a orthogonal view of a deflected assembly.

(62) FIG. 14C is a perspective view of a deflected assembly imposing forces on a solid body.

(63) FIG. 15A is a perspective view of a complete assembly that incorporates equilateral triangle support members.

(64) FIG. 15B depicts the assembly length wise, with and without support members.

(65) FIG. 15C is a perspective view of a complete assembly that incorporates a type of flexible support member.

(66) FIG. 15D depicts the assembly length wise, with and without support members.

(67) FIG. 15E is a perspective view of a complete assembly that incorporates a type of flexible support member.

(68) FIG. 15F depicts the assembly length wise, with and without support members.

(69) FIG. 15G is a perspective view of a complete assembly that has free antinodes and truss type support members.

(70) FIG. 15H is a perspective view of a complete assembly that has free antinodes and incorporates a type of flexible support member.

(71) FIG. 15I depicts the FIG. 15H assembly length wise, with and without support members.

(72) FIG. 16A shows the beam assembly un-deflected.

(73) FIG. 16B shows the assembly bending due to external forces shown as force arrows.

(74) FIG. 16C shows further compression at the interior of the assembly's bend and the subsequent deformation and the twisting of the slender sinusoidal shaped members.

(75) FIG. 16D depicts the assembly's further deformation and the snap buckling (looping) of sinusoidal members at the interior of the bend.

(76) FIG. 17 is a perspective view of an assembly that illustrates the use of various support member designs to maintain its sinusoidal shaped members' pre-stressed state.

(77) FIG. 18A is perspective views depicting support members that are not oriented perpendicularly to an assembly's axis.

(78) FIG. 18B is a detail of FIG. 18A

(79) FIG. 18C is an orthogonal view of a sinusoidal structure that is supported by an axially oriented inflatable support member.

(80) FIG. 18D is a orthogonal view of an assembly formed from relaxed members, strut support members and an inflatable support member (104) that has not placed force on the relaxed members (100) to deflect them.

(81) FIG. 19A shows two perspective views of a flexural beam assembly composed of sinusoidal shaped members having two different amplitudes.

(82) FIG. 19B shows orthogonal views of the sinusoidal members of different amplitudes and also lengthwise top views of the sinusoidal members in assembly.

(83) FIG. 20A show orthogonal views of the sinusoidal member of different wavelength and also the complete assembly they are part of.

(84) FIG. 20B shows lengthwise top views of the sinusoidal members in assembly.

(85) FIG. 21A is an orthogonal view of an assembly containing sinusoidal members that have an anti-phase relationship.

(86) FIG. 21B is an orthogonal view of an assembly containing sinusoidal members that are out of phase.

(87) FIG. 22A is an orthogonal view of a sinusoidal plane organized to form a sinusoidal structure with members that are out of phase.

(88) FIG. 22B is an orthogonal view of the sinusoidal plane being deformed to become an n-gon tube.

(89) FIG. 22C is an orthogonal view of the sinusoidal plane formed into a tube composed of skewed sinusoidal members.

(90) FIG. 22D is an orthogonal view of a sinusoidal tube depicting its helical secondary structure.

(91) FIG. 23A is a perspective view of a prismatic beam assembly formed from damped sinusoidal shaped members and support members.

(92) FIG. 23B is an orthogonal view of one of the assembly's damped sinusoidal shaped members.

(93) FIG. 23C is a lengthwise top view of the assembly's sinusoidal shaped members.

(94) FIG. 23D is a lengthwise top view of the assembly's support members.

(95) FIG. 23E is a lengthwise top view of the sinusoidal shaped members and support members.

(96) FIG. 23F is an orthogonal view of a damped and tapered sinusoidal shaped member.

(97) FIG. 23G is an orthogonal view of an organized assembly of six constituent damped and tapered sinusoidal shaped members.

(98) FIG. 23H is an orthogonal view of the assembly's support member organization along its length.

(99) FIG. 23I is a perspective view of a non-prismatic beam assembly formed from damped and tapered sinusoidal shaped members and ring support members.

(100) FIG. 23J is a perspective view of damped sinusoidal shaped members forming an assembly having a fan shape and planar geometry.

(101) FIG. 23K is a line drawing of a fan shape planar geometry test specimen formed from damped sinusoidal shaped members and axial support members

(102) FIG. 24A is a perspective view of an assembly built from sinusoidal shaped members having depth (a rectangular cross section), oriented to form a plane.

(103) FIG. 24B is a perspective view of an assembly having sinusoidal shaped members having a rectangular cross section oriented radially.

(104) FIG. 24C is a lengthwise view of the assembly depicted in FIG. 24B, Note: the slender flexible ring support members.

(105) FIG. 24D is a perspective view of a lattice assembly formed from 4-gon oriented sub-assemblies.

(106) FIG. 24E is a lengthwise view of the sinusoidal shaped members depicted in FIG. 24D.

(107) FIG. 24F is a perspective view of the support members depicted in FIG. 24D.

(108) FIG. 25A is a perspective view of a planar assembly formed from sinusoidal shaped members having depth and a rectangular cross section.

(109) FIG. 25B is a perspective view of planes formed from sinusoidal shaped members having axes intersecting in assembly forming geometric lattices.

(110) FIG. 26A is a perspective view of a lattice structure having a large number of multi axis flexural regions giving the assembly high performance flexural elastic properties.

(111) FIG. 26B is perspective view cross-section detail.

(112) FIG. 27A is a perspective view of an assembly having specific degrees of freedom as un-deflected.

(113) FIG. 27B is a orthogonal view of an assembly having specific degrees of freedom as deflected.

(114) FIG. 28A is a perspective and side view of a flexure system in assembly built from a number of pre-stressed sinusoidal shaped members.

(115) FIG. 28B is a perspective and side view of a flexure system being articulated.

(116) FIG. 29A is a perspective view of pre-stressed flexures formed from four sinusoidal shaped members having a rectangular cross-section held in assembly by axial and strut support members.

(117) FIG. 29B is a perspective view of pre-stressed flexures formed from four sinusoidal shaped members having a rectangular cross-section held in assembly by strut support members.

(118) FIG. 30A is a perspective view depicting a sinusoidal structure as a flexure coupler that is being articulated.

(119) FIG. 30B is a perspective view depicting a sinusoidal structure as a flexure coupler in its relaxed state.

(120) FIG. 31A depicts perspective views of exemplary pre-stressed lattice structures that may be used to reinforce a solid material.

(121) FIG. 31B depicts lengthwise views of the exemplary lattice assemblies depicted in FIG. 31A.

(122) FIG. 32A is a perspective view of an assembly having unevenly distributed pre-stressing forces.

(123) FIG. 32B is a section view that depicts an assembly with unevenly distributed pre-stressing forces having its deflection normalized by placing it under tension.

(124) FIG. 33A is a section view of an assembly shown encased in a solid material.

(125) FIG. 33B is a perspective view of a curved sinusoidal structure composed of sinusoidal shaped members having differing amount of stress applied to them during assembly.

(126) FIG. 33C is a perspective view of assemblies formed to have a predetermined shape.

(127) FIG. 34A is a perspective view of a sinusoidal assembly following a helical path.

(128) FIG. 34B is a perspective view of a sinusoidal assembly following a helical path with its helical spirals joined to form a tube.

(129) FIG. 34C is a perspective view of a sinusoidal assembly's helical spirals being joining (“zippered”) one to another.

(130) FIG. 35A is a perspective view of a tapered hexagonal sub-assembly.

(131) FIG. 35B is a perspective view of interconnected tapered hexagonal sub-assemblies.

(132) FIG. 35C is a perspective view of interconnected tapered hexagonal sub-assemblies forming a curved surface.

(133) FIG. 36A is an orthogonal view of an exemplary support member.

(134) FIG. 36B is an orthogonal view of an exemplary relaxed member.

(135) FIG. 36C is an orthogonal view of an exemplary sinusoidal shaped member.

(136) FIG. 36D is a perspective view of a sinusoidal shaped member being supported by support members.

(137) FIG. 36E is a perspective view of an exemplary sinusoidal assembly.

DETAILED DESCRIPTION OF AT LEAST ONE PREFERRED EMBODIMENT OF THE INVENTION

(138) The invention's method of pre-stressing is inspired by wave behavior in nature and how waves efficiently move, store and transfer energy within a system. This “natural” method/system of energy storage and energy transfer guides the building and pre-stressing of assemblies built from physical wave shaped members formed from high performance composite materials.

(139) Waves in nature absorb and transfer energy within the context of other waves they interact with. Similarly, wave shaped members (sinusoidal shaped members) held in assembly with other pre-stressed sinusoidal shaped members act to transfer potential energy among the other members in assembly. The dynamic and complex character of wave phenomena in nature when applied to the building of manmade physical structures may transcend conventional methods of assembly in technology.

(140) This system uses pre-stressed material members and material support members to form sparse structures whose materials are deformed during assembly. That deformation is elastic, the stress-strain relationship is called Hooke's law and the elastic strain energy is stored in the structure's material members.

(141) During the assembly process bending moments result in the elastic deformation of the material members. This deformed material will return to its original shape if the forces applied during assembly are released. The forces applied to the materials during assembly result in material bending, stress and strain at the molecular level. The physical and behavioral characteristics of this stress and strain are defined by the material's elastic modulus and/or flexural modulus.

(142) The strain energy reservoir held within the assembled system's structure results from the deformation of its elastic material. This energy reservoir is intended to enhance the structure's performance.

(143) These assemblies are energetic composite material structures that are elastically deformed through the bending of their component parts during assembly. These structures contain strain energy as an assembly that is not present in the individual straight relaxed linear members and support member parts. The assembly process flexes and places stress on all the pre-assembled straight linear material members. The assembly process bends and naturally curves the pre-assembled straight structural members into sinusoidal shaped members. The post-assembled sinusoidal shaped members and support members become a system of flexure designs working as a unit. The structure formed is pre-stressed, can be stable and its internal stresses are in equilibrium. The support member parts may also experience elastic deformation and “carry” the stress during and after assembly.

(144) FIG. 1A depicts a discrete, straight, relaxed linear member (100) unit having length and width. FIG. 1B depicts the same relaxed linear member elastically deformed into a discrete sinusoidal shaped member (102) unit having elastic potential energy. The relaxed linear member and the elastically deformed sinusoidal shaped member are topologically equivalent.

(145) As depicted in FIGS. 2A-F, relaxed members (100) with length may have various transverse cross-sectional geometries. FIGS. 2A-D depict exemplary relaxed members as prismatic beams having different cross-sectional shapes; rectangular, triangular, pentagonal and square respectively. Members having non-prismatic geometry (geometry that changes longitudinally) may also function as slender elastic material members.

(146) Examples of non-prismatic beams are depicted in FIGS. 2E-F. Elastic material strips, rods and other slender members are fundamental physical structures that because of their simplicity can exist as viable structures at many different dimensions and scales within the context of this invention. FIG. 2E depicts a tapered shaft having hexagonal cross section. FIG. 2F depicts a tapered beam having a rectangular cross section.

(147) FIGS. 3A-B depict a relaxed linear member's transverse cross-section and the sinusoidal shaped member's transverse cross-section as being topologically equivalent. The deformation/pre-stressing of the relaxed member (100) depicted in FIG. 3A into the exemplary sinusoidal shaped member (102) depicted in FIG. 3B may not alter its cross-sectional shape. The cross-sectional shape is shaded.

(148) A sinusoidal shaped member is an articulated beam having the physical wave parameters of spatial frequency, amplitude and wavelength. FIGS. 4A-C depict examples of the physical wave parameters that may be exhibited by physical sinusoidal shaped members in assembly. FIG. 4A depicts several discrete sinusoidal shaped member units (102) having the property of amplitude. FIG. 4B depicts several discrete sinusoidal shaped member units having the property of wavelength/spatial frequency. FIG. 4C depicts several discrete sinusoidal shaped member units having the property of phase offset.

(149) The physical “wave parameters” of the pre-stressed sinusoidal shaped member can vary over the length, width and depth of that member. The physical dimensions and material properties of the relaxed linear member that forms the sinusoidal shaped member can also vary along its length, width and depth. An example of this could be a composite material damped sinusoidal shaped member formed from a tapered relaxed linear member. FIG. 5A depicts a tapered/non-prismatic relaxed linear member (100). FIG. 5B depicts the same relaxed linear member elastically deformed into a discrete sinusoidal shaped member (102) unit that is damped and tapered along its length. Being tapered reflects a gradual change in amplitude (attenuation) along its length. Both the sinusoidal shaped member and the relaxed member forming it are tapered and both are non-prismatic beams. The wave parameters manifested by a sinusoidal shaped member can vary along its length, width and depth.

(150) FIGS. 6A-D depict a sinusoidal shaped member's structure and spatial frequency. The spatial frequency is a characteristic of any structure that is periodic across its position in space. This spatial frequency corresponds to the mathematical curve that describes the sinusoidal shaped member's smooth repetitive oscillation. Frequency based spatial damping may also apply to the sinusoidal shaped member's periodicity. FIG. 6A depicts a sinusoidal shaped member and FIG. 6B depicts its regular spatial frequency. λ/2 regions of a sinusoidal shaped member are represented by its spatial frequency. These λ/2 flexural regions punctuate the length of a sinusoidal shaped member to from a sequence of contiguous flexures. FIG. 6C depicts a damped sinusoidal shaped member and FIG. 6D depicts its spatial frequency attenuation characteristics. The damped sinusoidal shaped member is formed from a sequence of contiguous flexures that reflect the sinusoidal shaped member's gradually changing wave properties.

(151) FIGS. 6E-G depict the properties of an exemplary sinusoidal shaped member. FIG. 6E depicts the spatial frequency's (a) relationship to the sinusoidal shaped member's (102) wave structure and sinusoidal axis (b). FIG. 6F depicts a single sinusoidal shaped member (102) with periodic out of phase bending moments being applied along its length. These oscillating deformative forces and/or alternating positive and negative bending moments are represented as arrows. The force arrow's orientation and size correspond to the direction and strength of the forces being exerted on the member. FIG. 6G depicts a single sinusoidal shaped member (102) with support members (104) positioned along its length that correspond to its spatial frequency (a). The sinusoidal shaped member can be a slender sinusoidal member having spatial period and flexural strength. A sinusoidal shaped member cannot exist as an independent unit; it must connect to other sinusoidal shaped members through support members to maintain its shape.

(152) FIGS. 7A-E depict aspects of sinusoidal shaped members and support members in an exemplary assembly. FIG. 7A depicts exemplary pre-stressed sinusoidal shaped members (102) and support members (104) in assembly to form a sinusoidal structure. One sinusoidal shaped member is shaded. The assembly's axis (c) and the sinusoidal member's axis (b) are represented longitudinal to the length of the assembly. This assembly specifically contains two pre-stressed sinusoidal shaped members and thirteen support members. Each sinusoidal shaped member contains six complete wave cycles, fourteen nodes and thirteen anti-nodes. The support members' position mirrors the sinusoidal shaped members' physical wave parameters. A minimum of two sinusoidal shaped members is required to form an assembly. The sinusoidal shaped members have an anti-phase relationship within the spatially periodic assembly.

(153) FIGS. 7B-D depict aspects of this exemplary pre-stressed assembly's sinusoidal structure. In FIG. 7B anti-nodes (d and d*) are shown as points along a sinusoidal member's length. Anti-nodes represent the location where the sinusoid undergoes the maximum displacement. Nodes (e) are shown as points at zero displacement and exist along each sinusoidal member's axis (b). The anti-nodes location along a sinusoidal member's length corresponds to its spatial frequency. Anti-nodes when in assembly are of two categories: distal anti-nodes (d) and proximal anti-nodes (d*). This is based on their relative position to the assembly's axis (c). The distal anti-nodes are the greatest distance from the assembly's axis, the proximal the least. Distal and proximal anti-nodes identify regions of the sinusoidal shaped member's geometry where connections may take place. There are instances where the anti-nodes within an assembly do not form connections. An assembly containing free (not connected to another sinusoidal shaped member) anti-nodes is termed unsaturated. A saturated sinusoidal assembly has all of its anti-nodes connected or interconnected to other sinusoidal shaped members. Two sinusoidal shaped members are said to connect; if more than two join, they are interconnected.

(154) In FIG. 7C the orientation and magnitude of bending moments along the assembly's length are shown as force arrows. The distribution of first and/or second order bending moments along a sinusoidal shaped member's length defines its shape and flexural properties. First order bending (pre-stressing) shapes the member's wave; second order bending refers to the possible bending/deformation of the complete assembly. In FIG. 7C there is no deformation present. Second-order bending or lateral deflection of the sinusoidal members within an assembly is depicted in FIG. 33B-C.

(155) In FIG. 7D two sinusoidal shaped members (102) are held in a pre-stressed state by support members (104); in this example, the support members are specifically designed to support two sinusoidal shaped members having identical amplitude, wavelength and transverse section. The support members' placement along the sinusoidal shaped member's length corresponds to its spatial frequency. In this example, the forces within the assembly are balanced and the sinusoidal shaped members are identical opposites. Their interaction can be thought of as destructive interference. The forces/moments within each sinusoidal member are of equal magnitude and act in opposite directions in assembly.

(156) The two sinusoidal members depicted in FIGS. 7A and 7D are oriented anti-phase and connected anti-node to anti-node by support members. In this example, proximal (inner) anti-nodes (d*) and the distal (outer) anti-nodes (d) connect through the support members (104) to balance the sinusoidal shaped members (102) pre-stressing forces. Because the pre-stressed sinusoids in assembly contain identical internal forces/elastic potential energy, they are in static equilibrium, and because those forces are evenly distributed along the assembly's axis (c), there is no second order bending.

(157) FIG. 7E depicts various exemplary pre-stressed sinusoidal structures formed from different types and orientations of support members and sinusoidal shaped members.

(158) More complex sinusoidal assemblies are also possible. FIGS. 7F-I depict an exemplary sinusoidal assembly that exhibits dichotomous branching. This exemplary structure is composed of 12 sinusoidal shaped members held in a pre-stressed state and oriented using 6-gon support member that follow a trajectory that branches to form sister structures each composed of 6 sinusoidal shaped members held in position by 3-gon support members. FIGS. 7F-I depict the assembly's sinusoidal shaped members (102), each shaped from a single piece of elastic material that extends the length of the assembly. As these sinusoidal shaped members (104) traverse the length of the assembly, they separate into groups to form a branching structure. This separation of the members into groups is facilitated by the support member type and/or design variant used. FIG. 7I depicts the support member (102) types used before and after branching occurs. The sinusoidal shaped member's wave properties may altered as they follow their branching trajectory, specifically at the branching regions where the members separate into groups.

(159) Assembly Orientation and Organization

(160) The presented method of pre-stressing and building includes, but is not limited to, the relative placement (organization) and orientation of sinusoidal shaped members within an assembly in three basic types, planar, radial and n-gon. This method of pre-stressing is possible through the sinusoidal shaped members being oriented, connected and/or interconnected. These orientations are depicted herein. They are specifically presented in FIGS. 8A-F and FIGS. 9A-K in various views, including top views along the assembly's length and/or cross-sectional.

(161) FIGS. 8A-F depict three methods of positioning and orienting sinusoidal shaped members in assembly, linearly, radially and as a closed form polygon respectively. The three exemplary assembly configurations depicted contain pre-stressed sinusoidal shaped members (102) and their associated support members (104). In these examples, the discrete support member unit's geometry is specifically designed to support and hold the sinusoidal shaped members in assembly in a pre-stressed state. The support member's design is specific to the geometry of the given assembly. These exemplary assembly configurations may serve as geometric templates for assemblies containing a plurality and variety of pre-stressed sinusoidal shaped members and support members.

(162) Planar

(163) FIGS. 8A-B depict six sinusoidal shaped members (102) having planar orientation (parallel to the reference plane) connected longitudinally at their anti-nodes through support members (104). FIG. 8A depicts the sinusoidal plane having a single sinusoidal shaped member shaded. FIG. 8B depicts the planar orientation of the assembly's six sinusoidal shaped members from above lengthwise. Note: the structure's distal anti-nodes (d) may remain unconnected to other sinusoidal members.

(164) Radial

(165) FIGS. 8C-D depict three sinusoidal shaped members (102) having radially symmetric orientation to the assembly's axis (c). The sinusoidal shaped members are connected longitudinally at their anti-nodes (d) through support members (104). Note: the assembly uses two different support member designs. FIG. 8C depicts the radial symmetric assembly having a single sinusoidal shaped member shaded. FIG. 8D depicts the radially symmetric configuration of the assembly's three sinusoidal shaped members from above lengthwise. Note: the structure's peripheral anti-nodes may remain unconnected to other sinusoidal shaped members.

(166) Polygonal

(167) FIGS. 8E-F depict three sinusoidal shaped members (102) having an equiangular n-gon orientation connected longitudinally at their anti-nodes (d) through support members (104). FIG. 8E depicts the 3-gon assembly having a single sinusoidal shaped member shaded. FIG. 8F depicts the equiangular regular 3-gon configuration of the assembly's sinusoidal shaped members from above lengthwise. From top view, the sides of the polygon are the profile of the sinusoidal shaped members' orientation and represent their amplitude. FIGS. 8E-F 3-gon assembly is one of many possible closed form polygon geometries. Note: polygonal co-planar sinusoidal organization that is open, closed, regular and irregular are possible.

(168) FIGS. 9A-K depict 4-gon, 6-gon and 8-gon sinusoidal shaped member configurations. Each of these assemblies is composed of identical sinusoidal shaped members in an equiangular/regular polygon configuration with equal/even numbers of sinusoidal shaped members in an anti-phase orientation. The pre-stressing forces are symmetrical to the assemblies' axes (c). Because the assemblies have evenly distributed internal forces (relative to the centroid axis), there is no distortion to their structure. There is no second order bending. The assemblies depicted in FIGS. 9A-K contain sinusoidal shaped members oriented to form tubular polygons. FIG. 9H are line drawings of a 6-gon test specimen and FIG. 9G is a representative drawing of that specimen. The 6-gon structures depicted in FIGS. 9D and 9G-H use different support member (104) variants to maintain their sinusoidal shaped members' pre-stressed state. The support members depicted in FIGS. 9G-H and FIGS. 15E-F experience similar forces.

(169) The secondary structure is the shape of the assembly; that shape can be deflected or un-deflected. The secondary structure/shape is formed by the assembly's internal pre-stressing forces. A deflected assembly's structure could be a curve, irregular sphere, helix or other geometric shape. The term secondary structure is generally used to describe the shape of biomolecules. The pre-stressing forces within an assembly that can cause a secondary structure to form are similar to the intra-molecular forces (pre-stressing) within a molecule (assembly) that cause a protein to bend and impact its ability to fold and/or buckle.

(170) The exemplary assemblies depicted in FIGS. 8A-F and FIGS. 9A-K may function as sub-assemblies and/or serve as geometric templates used to a form complex lattice and/or adaptable structures containing large numbers of discrete parts assembled according to a prescribed geometry.

(171) Structures labeled as sub-assemblies can exist as discrete assemblies and sub-assemblies. They may also share parts with adjacent “sub-assemblies” when part of a greater lattice assembly, making them indiscrete structures when in the greater assembly. Sub-assemblies may serve as the actual building blocks for a lattice or be utilized as engineering design project tools in the conception of a lattice.

(172) FIGS. 10A-C depict aspects of a pre-stressed flexural lattice structure. The lattice assembly depicted in FIG. 10A is formed from a number of sub-assemblies having radially symmetric geometry. A single component sub-assembly that forms the lattice is depicted in FIG. 10B. The sub-assemblies join one to another at their distal anti-nodes (d) through support members to form the lattice assembly. FIG. 10C depicts the lengthwise top view of this lattice.

(173) FIGS. 11A-B depict pre-stressed flexural lattice structures of regular 4-gon geometry. The lattice geometry is defined by the sinusoidal members' parallel orientation forming a regular four sided polygon pattern. A single sinusoidal support member in the assemblies is shaded. FIG. 11B depicts the lengthwise top views of the lattice structures.

(174) Within the network of internal forces that may exist in a complex pre-stressed assembly, torques may exist that force an assembly to take on a secondary shape. Force couples can result in a torque and can also be put into equilibrium by a moment or another couple of equal magnitude and opposite direction anywhere in the same plane or in a parallel plane within the assembly.

(175) Forces within Assembly

(176) Sinusoidal shaped members in assembly have flexural properties. As depicted in FIGS. 12A-B, flexural regions (f) can exist between the support members (104) within an assembly. The assemblies are punctuated by flexural regions (f). The flexural regions are defined by an assembly's spatial frequency. The support members within the assembly have spatial frequency that corresponds to the sinusoidal shaped member's mathematical curve describing its smooth repetitive oscillation.

(177) The portion of a sinusoidal shaped member that exists between support members composes its flexural regions. The lengths of the sinusoidal shaped members that exist between support members in assembly form that assembly's flexural regions. A flexural region can form a length of sinusoidal assembly encompassed by a number of support members. A flexural region's length can be defined by the distance between the terminal support members that punctuate it. A flexural region requires at least two support members.

(178) The tubular (12-gon) assembly depicted in FIG. 12A has sinusoidal shaped members that are oriented along its length and ring type support members having period placement. In FIG. 12B the sinusoidal shaped members are also oriented along the length of a tubular (12-gon) assembly, but the support members used are struts/ties and as shown unable to maintain the pre-stressed shape depicted without additional support members. An example of a similar system being supported using an additional (inflatable) member is depicted in FIG. 18C.

(179) FIG. 12C depicts circumferentially oriented sinusoidal shaped members (one is shaded) in a cylindrical/tubular assembly. The orientation of the sinusoidal shaped member in relation to an assembly's axis changes the orientation of the spatial frequency and flexural regions. The orientation of the flexural regions will affect an assembly's structural property. FIGS. 12A-B show one sinusoidal shaped member as shaded.

(180) As depicted in FIGS. 13A-B, an assembly may or may not contain an axial support member. FIG. 13A depicts an assembly having an axial support member (104) that extends the length of the structure. This central axial support member is under tension and functions similarly to the periodically placed ring shaped support members (104) depicted in FIG. 13B. These ring-shaped support members are under compression. The pre-pressing forces imposed on the support members by the sinusoidal shaped members are represented as force arrows. If the sinusoidal shaped members have identical dimensions and mechanical properties, the forces experienced by the support members depicted in FIG. 13A and the forces experienced by the support members depicted in FIG. 13B are equivalent.

(181) The tensile stress experienced by the axial support member depicted in FIG. 13A can be uniformly distributed (normal stress) or periodically distributed (localized stress) along its length. If an axial support member is only fixed to the sinusoidal shaped member's ends and allowed to float within the rest of the assembly's length, the tensile force is evenly distributed along the assembly's length. If the axial support member is periodically fixed to the sinusoidal shaped member (102) at its anti-nodes (d), the axial support member's (104) tensile stress will be periodic, and in effect the axial support member becomes a series of members experiencing localized tensile stress. If the assembly experiences a transverse load or loads the axial member being fixed or unfixed to the sinusoidal shaped members' anti-nodes will affect the assembly's flexural strength. The assembly depicted in FIG. 13A has free anti-nodes and is considered an unsaturated assembly. Its linear support member's position conforms to the assembly's neutral axis.

(182) Support members (104) in an assembly can vary in type and design, but all have a similar function. Support members function to hold the sinusoidal shaped members in a pre-stressed state and function to transfer forces among the elastically deformed sinusoidal shaped members within assembly. If an assembly's internal forces are evenly distributed and correspond to the assembly's center of mass/centroid axis, the assembly may be an un-deflected structure. If an assembly's internal forces are unevenly distributed and do not correspond with the centroid axis, the assembly will be a deflected structure. FIGS. 14A-B depict an assembly having pre-stressing forces (internal forces) that are unevenly distributed and do not correspond to its centroid axis. The uneven distribution of forces in relation to the assembly's centroid axis results from the sinusoidal shaped members (102) not transferring their pre-stressing force equally to the anti-phase oriented sinusoidal shaped members in the assembly. The transfer of those internal forces from one sinusoidal shaped member to another occurs through the support members (104). FIG. 14A depicts an assembly having pre-stressed sinusoidal shaped members that are oriented to cause the assembly's neutral axis and centroid axis not to converge. This causes the assembly to deflect as depicted in FIG. 14B. A deflected assembly could function as a shaped and pre-stressed structure designed to have specific degrees of freedom and/or function under specific load applications. For example, FIG. 14C depicts a deflected assembly being used to pre-stress a solid material beam. The pre-stressed and pre-deflected assembly is held and encased/cast within a solid material (like concrete) shape that is designed to accept a directional load. The axial support member (104) extends the length of the pre-deflected assembly and is placed under tension prior to it being encased. The axial support member is later released transferring its internal forces to the surrounding material.

(183) Support Members in Assembly

(184) Support members can control the organization of the sinusoidal shaped members in assembly. Support members are the assemblies' scaffold. By analogy, support members (104) and scaffolding molecules serve similar functions and experience similar forces in assembly. An analog may be the role histone and non-histone scaffolding proteins play in the formation of the chromosomal structures. As in the scaffold loop model of chromosomes, where the scaffolding proteins determine the shape of the native meta-phase chromosomes and function as a basis for organizing the chromatin in specific domain loops. It is proposed that during chromosome condensation, domain loops organize under stress into a regular sinusoidal structure. Like sinusoidal shaped members and support members in assembly, the structure of a chromosome contains mechanical stress and that stress may be organized having a sinusoidal form.

(185) FIGS. 15A-I presents examples of pre-stressed assemblies having a variety of support member designs. Support members may be flexural members subject to tension, compression and bending within their structure. A support member design functions to efficiently support, hold and shape sinusoidal members. Support members may experience compressive, tensile and/or shear forces. They may also have flexural strength. Support members (104) may take on a variety of forms based on their support function.

(186) Support members may connect to and be fixed to sinusoidal shaped members or connect to and allow the sinusoidal members to articulate within their body, i.e. the sinusoidal shaped member may rotate within the support member or be fixed to it. Support member types may be but are not limited to linear struts, tension rings, compression rings, inflatable members and n-gon forms. Note: polygonal shaped support members that are open, closed, regular and irregular are possible.

(187) Support member design and construction requires the consideration of multiple factors. The design and fabrication process will generally include considerations: 1) design requirements, 2) likely failure modes, 3) stress analysis, 4) material selection and behavior, 5) fabrication, and 6) testing.

(188) FIGS. 15A-I present the variety of support member types and designs used to hold and support sinusoidal shaped members in assembly. FIGS. 15A, 15D, 15E, 15H and 15I depict complete assemblies having various engineered support member designs (15H-I depict unsaturated assemblies). FIGS. 15C, 15B, 15F and 15G show the orientation (all have the same orientation) of sinusoidal shaped members in assembly and also top views of the complete assembly including the support members.

(189) FIGS. 15A-B depicts an assembly made up of nine sinusoidal shaped members and thirteen support members. These support members may be described as being a combination of regular n-gon struts. The slender sinusoidal shaped members in this assembly all have the same wave parameters and their pre-stressing forces are evenly distributed within the assembly. There is no deflection.

(190) FIGS. 15C-D depicts an assembly with support members having the flexural-torsional properties of a ring or circular arch. An assembly's support members' versatility is seen in the context of the overall engineering design project goals. For example, in order to achieve a reduction in weight, increase in flexural strength, reduction in cost, specific load capacity, range of flexural behavior and flexural strength, the engineering design, material composition, and methods of fabrication must all be considered.

(191) FIGS. 15E-F depicts an assembly made up of nine sinusoidal shaped members and thirteen support members (104). These support members can be described as having the tensile and compressive strength of a ring and/or strut.

(192) FIG. 15G-I depicts assemblies formed from support members that do not require a connection be made to all the available anti-nodes (d) within the structure. In these assemblies, the sinusoidal shaped members maintain their pre-stressed shape while having free anti-nodes (d). An assembly having free anti-nodes is considered unsaturated and may have physical and behavioral properties that a saturated assembly does not. Assemblies having free anti-nodes may be well adapted to form flexural mechanisms.

(193) FIG. 16A-D illustrates the articulation and flexural buckling of an unsaturated beam assembly in a progression sequence. Along with being more flexible than an assembly having saturated anti-nodes, an assembly with unsaturated/free anti-nodes may function as a device/mechanism that buckles and/or folds. The slender sinusoidal shaped members in assembly make this possible. As shown in FIGS. 16A-D, when a free anti-node region is increasingly compressed, the slender sinusoidal shaped member increasingly bends, twists and buckles to form a slender loop. This torsion flexural buckling behavior may function within an assembly as a singular flexure mechanism or function in groups as a method of morphing a lattice. A structure capable of buckling can have two stable states, unbuckled and buckled as depicted in FIGS. 16A and 16D. This state of being snap buckled is an equilibrium state that can be maintained in the assembly a length of time and may be reversible. The assembly may embody parts in different states of equilibrium including stable equilibrium, unstable (or labile) equilibrium, and metastable equilibrium. The state of stable equilibrium (the stable state) can be characterized by being brought out of the state of stable equilibrium by some external disturbance (force). When this disturbance is removed, the system will return to the initial state of equilibrium by itself. If a system is in a state of unstable equilibrium (the labile state), the system will not return to the initial state and will pass into a new state, a state of stable equilibrium. Even the slightest external disturbance (force) can be sufficient to move a system out of its state of labile (unstable) equilibrium. Finally, the state of relatively stable equilibrium (the metastable state) is defined as the state in which a system remains for a long period of time, and any slight disturbance causing the system to deviate from the metastable state does not result in the system passing into another state; as soon as the external disturbance (force) is removed, the system will return into the initial metastable state. On the other hand, a sufficiently strong disturbance will put the system out of the metastable state, and the system will pass into a new state of stable equilibrium. Thus, the metastable state is intermediate in relation to the stable and unstable states.

(194) FIG. 17 depicts an assembly containing various support member (104) types and designs engineered to maintain its sinusoidal shaped members' pre-stressed state. By combining various types of support members, including but not limited to, n-gons, rings, struts, inflatables and axial support members in the same assembly, diverse structural and mechanical applications can be served. Those applications can range from complex morphing lattice structures to simple compliant mechanisms.

(195) Unlike the previous examples, FIGS. 18A-B depicts an assembly having support members that are oriented parallel and/or not perpendicular to the longitudinal axis of the assembly. These support members do not transect the axis of the assembly and attach to the sinusoidal shaped members at their anti-nodes. FIG. 18A shows a single sinusoidal shaped member shaded within the assembly. FIG. 18B is a close up depicting the orientation of the support members in assembly. The ring type support members depicted in FIGS. 18A-B may have flexural strength, exhibit flexural behavior and store elastic potential energy when in assembly. A support member's orientation within an assembly is not limited to being orthogonal, perpendicular or parallel to the assembly's axis.

(196) Support members having the ability to expand and contract may also function to maintain the sinusoidal shaped members in assembly. FIG. 18C depicts an assembly having an inflatable support member (104) that is axially oriented and applies the force required to maintain the shape of the pre-stressed sinusoidal structure. FIG. 18D depicts the assembly having relaxed members (104) and strut type support members (104) prior to the inflation of its inflatable support member and the application of force to the rest of the assembly. As force is applied to the structure, the strut and/or the support members experience tensile stress. The Inflatable tubular support member's diameter changes in sync with the wavelength and amplitude of the sinusoidal shaped members in assembly.

(197) An inflatable support member design may take on a verity of shapes and the inflation dynamics of an inflatable structure may be enhanced when used to support a sinusoidal structure. The inflatable support member and the other components in assembly may interconnect during assembly to guide one another's formation.

(198) Sinusoidal Shaped Members in Assembly

(199) Amplitude

(200) Sinusoidal shaped members having the same spatial frequency may interconnect at their anti-nodes through support members independent of their amplitude to form an assembly. For example, FIGS. 19A-B depict four pre-stressed sinusoidal shaped members in assembly, all having the same wavelength and two different peak to peak amplitudes. In this example, strut support members are used to connect and hold the members in a pre-stressed state. The sinusoidal shaped members (102) in assembly depicted in FIG. 19A are pre-stressed into two different shapes and specific amplitudes. There are two sinusoidal shaped members of the specific amplitudes represented in the assembly. Their amplitudes may reflect the amount of stored potential energy their elastically deformed material contains if the members all have the same material properties. FIG. 19B depicts the two different amplitudes represented by the sinusoid members in assembly and top views of the assembly. The sinusoidal shaped members of the same amplitude values are oriented parallel to one another to form a prismatic beam assembly having rectangular cross-section. In this polygon orientation, the pre-stressing forces are symmetric in relation to the centroid axis and neutral axis of the assembly. The orientation of even/balanced number of sinusoidal members having identical amounts of stored elastic energy along the assembly's axis allows the structure to be pre-stressed, balanced and un-deflected. Combinations of sinusoidal shaped members with different amplitudes cannot form equilateral or regular polygon cross-sections.

(201) Wavelength/Spatial Frequency

(202) Sinusoidal shaped members having the same amplitude may interconnect at their anti-nodes independent of their wavelength and/or spatial frequency to form an assembly. For example, FIGS. 20A-B depict six pre-stressed sinusoidal shaped members in assembly, all having the same amplitude and two different wavelengths. In this example, strut support members connect, support and hold the sinusoidal members in a pre-stressed state. The sinusoidal shaped members (102) in assembly depicted in FIG. 20A are pre-stressed to form shapes having two different wavelengths. There are four members of one wavelength and two of another. FIG. 20B depicts the top view of the assembly. The sinusoidal shaped members having different spatial frequency are oriented parallel to one another to form a prismatic beam assembly having 6-gon cross-section. In this regular polygon orientation, the pre-stressing forces are symmetric to the axis of the assembly. The orientation of an even number of sinusoidal members having identical amounts of stored elastic energy along the assembly's axis allows the structure to be pre-stressed, balanced and un-deflected. Combinations of sinusoidal shaped members having different amplitudes cannot form equilateral or regular polygon cross-sections.

(203) Sinusoidal shaped members having different spatial frequencies and different amplitudes may also connect to form an assembly. The interconnection of sinusoidal shaped members having a range of physical wave parameters may result in the deformation of these members' sinusoidal shape, for example, the skewing of the interconnected wave forms in assembly. The distortion or skewing of the sinusoidal members results from the transfer and equalization of forces from one member to another within the assembly.

(204) Phase Relationship

(205) Sinusoidal shaped members having predetermined wavelengths and/or amplitudes along with the required support member designs could be interconnected independent of phase relationship to form an assembly. The possible combinations and orientations of sinusoidal shaped members and support members is vast.

(206) Sinusoidal shaped members of the same wavelength may connect at their anti-nodes while having different phase relationships. For example, FIGS. 21A-B depict radially symmetric assemblies containing four identical sinusoidal shaped members. In FIG. 21A the sinusoidal shaped members in assembly all have an anti-phase relationship. In FIG. 21B the sinusoidal shaped members in assembly form wave pairs having an anti-phase relationship and these pairs are also 180 degrees out of phase. In both these examples (FIGS. 21A-B) the assembly's proximal anti-nodes (central) (d*) connect to an axial support member (104) and the distal anti-nodes (peripheral) (d) are free and not connected to a support member (104). FIG. 21A depicts the assembly's sinusoidal shaped members being attached at their anti-nodes in anti-phase and held in a pre-stressed state by support members. FIG. 21B depicts the assembly's two pairs of pre-stressed sinusoidal shaped members 180 degree out of phase and connected at their anti-nodes through an axial support member (104) that is under tension. The sinusoidal shaped member forming pairs are connected anti-phase. The two wave pairs are oriented 90 degrees in relation to one another along the assembly's axis. These sinusoidal shaped members are 180 degrees out of phase and not skewed or distorted in the assembly. The pre-stressing forces in FIGS. 21A-B are symmetric to the assembly's axis. The axial support members depicted in FIGS. 21A-B are shown shaded.

(207) Another method of connecting sinusoidal shaped members out of phase is by distorting them. An example would be the formation of a closed (helical) polygonal assembly from a planar structure. FIG.22A-D illustrates the sequential articulation of a plane formed from sinusoidal shaped members that are out of phase into a closed n-gon (tube). The out of phase sinusoidal shaped members orient to form a closed n-gon shape that is helical and pre-stressed. Unlike FIG. 21B this method of connecting sinusoidal shaped members out of phase distorts their wave form in assembly. Points identify reference anti-nodes (d) that can be followed to show the trajectory of the plane's distortion and the sinusoidal shaped members out of phase relationship in assembly. FIG. 22A depicts the assembly as an undistorted plane prior to it becoming closed, out of phase and distorted. The six identical pre-stressed sinusoidal shaped members are connected at their anti-nodes. FIG. 22B depicts the sinusoidal shaped members planar assembly distorted out of phase a total of 360 degrees. FIG. 22C depicts the plane as a closed form 360 degrees out of phase. The sinusoidal waveforms must distort to accommodate the out of phase relationship. The sinusoidal members are skewed in the assembly and form a sawtooth waveform. FIG. 22D superimposes a helical shape over the assembly to emphasize the geometry created by the sinusoidal shaped members' out of phase relationship. The assembly's helical geometry may give it useful flexural properties (flexural strength and flexural modulus) not found in assemblies without a secondary structure.

(208) Period

(209) Sinusoidal shaped members having spatial frequency/wavelength that is not of a regular period along their length may connect to form an assembly. A damped oscillation is not a periodic function, and therefore a damped sinusoidal shaped member's spatial frequency is not uniform along its length. FIGS. 23A-E depict damped sinusoidal shaped members used to form a prismatic beam. FIG. 23B depicts a constituent damped sinusoidal shaped member unit. The beam is formed from three radially symmetric sub-assemblies. Each sub-assembly is composed of three damped sinusoidal members. FIG. 23C depicts the nine sinusoidal shaped members' orientation in assembly from above lengthwise. FIG. 23D depicts the support members in assembly from above lengthwise. FIGS. 23A and 23E depict the assembly in total. One of its twenty-two flexural regions/areas (f) is labeled.

(210) A non-prismatic beam assembly composed of damped sinusoidal shaped members is depicted in FIGS. 23F-I. Each damped sinusoidal members in the assembly is formed from a tapered relaxed member. A constituent tapered and damped sinusoidal shaped member is shown in FIG. 23F. The sinusoidal shaped members in assembly are specifically depicted in FIG. 23G. The structure's sinusoidal members are polygonally organized but are not co-planar. The support members are specifically depicted in FIG. 23H. The design and location of its support members results in the assembly's damped period and non-prismatic shape. FIG. 23I depicts the tapered assembly in total and show its flexural regions (f) corresponding to a damped function.

(211) Damped sinusoidal shaped members forming exemplary pre-stressed planar assemblies are depicted in FIGS. 23J-K. In FIG. 23J a fan shaped structure's component sinusoids attach at their antinodes. As the shape expands/spreads the sinusoids amplitude increases and wavelength decreases. A variation on this pre-stressed fan shaped member orientation is depicted in a line drawing of a test specimen. The test specimen depicted in FIG. 23K has sinusoidal shaped members connected harmonically. Note: the harmonic sinusoids form connections at their nodes and anti-nodes.

(212) Sinusoid Dimensions and Orientations

(213) Sinusoidal shaped members having differences in length-width-depth dimensions can be used to form assemblies. FIGS. 24A-E depict sinusoidal shaped member of various dimensions and orientations in assembly. FIG. 24A depicts sinusoidal shaped members (102) having rectangular cross-sections in a planar orientation. The sinusoidal shaped members are connected longitudinally at their anti-nodes (d) through support members (104). FIGS. 24B-C depicts sinusoidal shaped members (102) having rectangular cross-sections in a radially symmetric orientation connected longitudinally at their anti-nodes through support members (104). FIGS. 24D-E depicts sinusoidal shaped members (102) having rectangular cross-sections in an n-gon configuration connected longitudinally at their anti-nodes through support members (104). This assembly is made up of congruent regular polygon shapes and may form a 4-gon tessellation. The sinusoidal shaped members in this columnar lattice assembly maintain their pre-stressed state through ring-shaped support members (104). The 4-gon sub-assemblies that can be seen as composing the lattice are congruent and share sinusoidal members.

(214) Sinusoidal shaped members and support members can be oriented across multiple axes within a pre-stressed assembly. FIGS. 25A-B depict pre-stressed sinusoidal shaped members (102) having a rectangular cross-section being connected through support members (104). FIGS. 25A and 24A depict a sinusoidal assembly in the form of a plane. Flexural regions (f) punctuate the assemblies. FIG. 25B depicts two planes intersecting at 90 degrees. The planes' axes intersect as a “woven” assembly. A plurality of interconnected planar assemblies may form a lattice with flexural properties.

(215) FIGS. 26A-B depict a woven lattice as an exemplary flexural structural beam that is illustrative of the principles of the invention. The beam's flexural lattice embodies the invention's anisotropic structural properties. The large number of flexural regions within the beam would give it extraordinary flexural strength. For example, pre-stressed composite material sinusoidal shaped members in a lattice assembly may function as “tissue” tendons. Tendons stretch and then recoil during locomotion, storing and returning elastic strain energy; dense sinusoidal lattice structures may serve a similar energy saving function when applied as a locomotive mechanism that loses and regains kinetic energy.

(216) Applications

(217) The present invention demonstrates the applicability of pre-stressed sinusoidal shaped members to form engineering design projects for structural and mechanical applications. Exemplary assemblies for functional applications are presented herein. The engineering design process is used to produce an assembly that meets the specific application's requirements.

(218) This method of pre-stressing material may be used to form assemblies having flexural properties and specific degrees of freedom. FIGS. 27A-B depict a sinusoidal beam assembly having compliant directions that permit specified motions and prevent motion in all other directions. FIG. 27B depicts the assembly in side view from above being deflected by a force. The beam's deflection is an accumulative deflection manifested through the plurality of flexural regions (f). The greater the number of flexural regions per unit length, the smoother the deflection curve.

(219) A flexure is a flexible element or combination of elements engineered to be compliant in specific degrees of freedom. A flexure can be a part of a sinusoidal shaped member or the assembly it composes.

(220) Compliant mechanisms are flexible structures that utilize elastic deformation to achieve their desired motions. A compliant mechanism's flexural behavior can come from the deflection of its constituent flexural members and in turn the flexure elements that compose them. Compliant mechanisms built using this invention can be formed from sinusoidal flexure members composed of contiguous flexure elements. These parts go to form a flexure system. The flexure system's members and the elements that compose them have flexural properties.

(221) FIGS. 28A-B depict composite material cross-axis flexural pivots built from pre-stressed sinusoidal shaped member flexures. Several of these flexure joints could constitute a mechanism. The pre-stressed members' dimensionality and sinusoidal geometry may help to improve a flexure pivot's support stiffness which is generally greatly reduced during large deflections. Flexure pivots may be used efficiently as precision machine elements due to their excellent resolution characteristics, low-cost characteristics, and ease with which they may be fabricated. Because of the strain energy associated with the bending of the flexible members, compliant mechanisms often have higher actuation effort compared to traditional mechanisms. Static balancing is one way of reducing this actuation energy. The pre-stressing/static balancing of flexures through the use of the system's sinusoidal shaped members may reduce a compliant mechanism's actuation energy.

(222) The flexure depicted in FIG. 29A uses axial support members (104) and strut support members (104) to maintain its sinusoidal shaped members' pre-stressed state. The flexure depicted in FIG. 29B uses strut support members to maintain its sinusoidal shaped members' (102) pre-stressed state. These assemblies can be seen as series flexures. A flexure composed of a series of flexural regions (f) may have advantages over other types of flexure designs. Factors that determine a mechanism's flexibility (and thus, its range of motion) are: the type of support members used, material type (its elastic properties and strength), boundary conditions (how it is attached to whatever needs movement), and its geometry (thickness and length for rectangular cross sections). The sinusoidal shaped member's cross-sectional geometry and the sinusoids wave parameters are both considered the flexure's geometry.

(223) Flexure systems are important to precision motion and positioning applications used within precision motion stages. A flexure system is a flexural system composed of many flexures that transmit loads by way of a material's basic stress states of compression, shear and tensile stress. FIGS. 30A-B depict a lightweight flexure coupler. This system of pre-stressing composite material assemblies may be used to form exemplary flexure mechanism components. Their pre-stressed state and compliance characteristics may provide exceptionally smooth precision motion characteristics.

(224) Pre-stressed lattice structures may function to reinforce and/or pre-stress concrete beams or other solid material shapes. FIGS. 31A-B depict specifically engineered sinusoidal lattice assemblies having pre-determined structural and behavioral properties tailored to reinforcing and/or pre-stressing a concrete beam. A pre-stressed sinusoidal assembly encased in a solid material beam/shape may affect that beam's oscillatory characteristics and natural frequency.

(225) A pre-stressed assembly may be engineered to deflect in a predetermined way. This is achieved by determining the required pre-stressing force distribution characteristics within the assembly. A sinusoidal assembly's deflection may be used to enhance the effect of the pre-stressing forces imposed on the concrete member it is encased in. As depicted in FIGS. 32A-B a number of axial support members (104) may be placed under tension to normalize the assembly's deflection prior to be being encased in concrete. The tension placed on the axial support members is then released. This allows the residual deflection of the pre-stressed concrete member to respond to external loads.

(226) Pre-stressed lattice structures encased in a solid may not exhibit deflection or be directionally compliant. FIG. 33A depicts a pre-stressed un-deflected planar structure formed from interconnected 12-gon sub-assemblies engineered to reinforce a material slab on grade.

(227) The assembly's pre-stressing forces are evenly distributed and the assembly's centroid axis and neutral axes (c) are congruent and the structure is not deflected.

(228) Pre-stressed lattice structures having flexural properties may also function as stand-alone structural components used to build larger structural systems for applications on earth and in space. Such applications could include terrestrial bridges, towers and buildings to expansive deployable structures in water and in space.

(229) Sinusoidal shaped members in assembly may form structures having a variety of three-dimensional shapes. These shaped structures may be connected to form more complex assemblies having tertiary and quaternary structure.

(230) One method of forming assemblies having a predetermined three dimensional shape is through modulating the distribution of the structure's potential energy/pre-stressing energy during the assembly process. If that potential energy is not evenly distributed across the assembly's cross-sectional shape, the centroid axis and the neutral axis are incongruent and the assembly will take on a deflected secondary shape. FIGS. 33B-C depict assemblies having a secondary three dimensional structure that mirrors their stress profile. Their curved shape results from their sinusoidal shaped members being given different amounts of stored potential energy during assembly. In FIG. 33B the sinusoidal members having the greatest potential energy are shaded.

(231) The increased potential energy and longer wavelength of the shaded member in assembly causes the structure to curve. By designing a sinusoidal structure composed of interconnected sinusoidal shaped members having specific wave properties, the structures could be designed to follow specific irregular three dimensional trajectory paths, as depicted in FIG. 33C.

(232) An assembly could also follow a predefined geometrical path as depicted in FIGS. 34A-C. FIGS. 34A-C and FIGS. 8C-D both depict assemblies composed of three sinusoidal shaped members having radially symmetric organization; they differ in that FIG. 34A-C is formed to follow a path. FIG. 34A depicts a sinusoidal assembly that follows a curved helical path.

(233) In FIG. 34B the spirals of the helical path are attached to form a tube. The attachment of the helix's adjacent spirals is depicted in FIG. 34C where turns of the helix are “zippering” together to form a hollow tube. The attachment of the sinusoidal assembly's helical spirals may use a locking mechanism/method that allows for their reversible attachment. Such locking methods may also be used to attach sub-assemblies to one another and also be used to attach sinusoidal shaped members to one another through the support members. The locking mechanism(s) may involve a part of the sinusoidal shaped member and/or part of the support member and may require complimentary parts like clips, pins or adhesives.

(234) Assemblies having a secondary structure may be seen as sub-assemblies (a shape) that can interconnect to form tertiary and quaternary structures.

(235) Sinusoidal shaped members displaying amplitude characteristics that vary along their length may form sub-assemblies that can interconnect to form curved three-dimensional surfaces and shapes. FIGS. 35A-C depicts non-prismatic sub-assemblies (tapered building blocks) interconnecting to form a curved surface. This surface could function to form a pre-stressed dome structure or a variety of predetermined modeled surfaces. FIG. 35A depicts a tapered non-prismatic sinusoidal sub-assembly. FIG. 35B shows the fundamental tapered 6-gon shape of said sub-assemblies. FIG. 35C depicts the tapered shapes assembled.

(236) By analogy, observed wave behavior as found in nature can be applied to the invention and its use of three-dimensional physical waves. For example, when a particle is moved from its original position (e.g. by an excitation) a restoring force then acts on it. This force can be calculated by the Hooke's law and acts in the direction of the original position. This leads to a propagation of energy through the medium. Because the particles in matter are linked to each other, the displacement of one particle leads to a displacement of its surrounding particles. Transverse and longitudinal waves are types of elastic waves that need a propagation medium. Similarly, a three-dimensional physical wave shaped from linked partials of composite material matter having elastic properties can become a vehicle of energy storage and transfer. The principles of superposition, destructive and constructive interference can all be applied to the interaction/interference that takes place among pre-stressed energy rich sinusoidal shaped members in an assembly. The support members serve as the conduit that allows the members' internal forces to entangle and reach a static equilibrium within the assembly. The more even the distribution of internal forces within the assembly, the less deflection present in the greater structure.

(237) As an overview, the parts and identifying characteristics of the system are presented in FIGS. 36A-E. FIGS. 36A-C depict an exemplary support member, relaxed member and sinusoidal shaped member respectively. FIG. 36D depicts a single sinusoidal shaped member with its associated support members. FIG. 36E depicts the assembly in total. The parts and characteristics listed below are labeled in the FIGS.

(238) Material Parts

(239) (100) Relaxed member: Relaxed member variants have dimensions and are composed of elastic material.

(240) (102) Sinusoidal shaped member: Sinusoidal shaped member variants are flexural members having different wave parameters and dimensions.

(241) (104) Support member: Support member variants may be but are not limited to axial support members, ring support members, n-gon support members, inflatable support members and strut support members.

(242) Definitions of Sinusoid Characteristics

(243) (a) Spatial frequency: Spatial frequency is the characteristic of any structure that is periodic across a position in space.

(244) (b) Sinusoidal axis: The Sinusoidal axis is the sinusoidal shaped member's neutral axis.

(245) (c) Assembly axis: The Assembly's axis is the assembly's neutral axis.

(246) (d) Distal Anti-node: Distal Anti-nodes are located on the assembly's periphery.

(247) (d*) Proximal Anti-node: Proximal Anti-nodes are located central to the assembly.

(248) (e) Node: Nodes are points on the sinusoidal shaped member that follow its neutral axis.

(249) (f) Flexural regions: Flexural regions are located between support members.

(250) While preferred embodiments of the invention are disclosed herein, many other implementations will occur to one of ordinary skill in the art and are all within the scope of the invention. Each of the various embodiments described above may be combined with other described embodiments in order to provide multiple features. Furthermore, while the foregoing describes a number of separate embodiments of the apparatus and method of the present invention, what has been described herein is merely illustrative of the application of the principles of the present invention. Other arrangements, methods, modifications, and substitutions by one of ordinary skill in the art are therefore also considered to be within the scope of the present invention, which is not to be limited except by the claims.