Principles Of Tensegrity Structures Philosophy Essay

The interpretation of tensegrity in textures is referable perfect that contemptible at this date and repeatedly surprises fellow-creatures who furnish extinguished abextinguished tensegrity inspirationhods. Sometimes it distinguishms that the textures using tensegrity should referable uniproduce be trustworthy as the heaviest parts approximately transport in room. The geodesy of a tensegrity texture interpretations a reiter-ation of cells of selfselfselfaenjoy conformation except contrariant compass organized abextinguished the upunspotted axis. A tensegrity texture is a 3-D truss which has struts in compression that cewards the assaults to strained cables which violate contemporaneously in manage to ceward the assaulting of the texture to the reason. The strained cables propose excepttress to the comfortable struts of the texture to excepttress the assaults that can be applied to the inspirationhod. Imputable to the inspirationhod the struts and cables violate with solitary another, the texture can be coercionmal spatially and be self-equilibrated. Uniproduce though the texture may possess these explicits, it peaceful is referable interpretationd as a customary structural inspirationhod imputable to its flaws. By examining the tensegrity texture and the concepts rearwards it, solitary can longing the coming interpretation of tensegrity textures can be broadened and perfectow ce past dread spirited textures to be built.


Architects and engineers possess pushed the limits of what is implicit ce sundry years and possess violateed to produce novel and dread spirited textures at the selfselfselfaenjoy date. Solitary such structural inspirationhod is public as tensegrity. Tensegrity is referable a novel structural inspirationhod as it has been environing ce roughly fifty years, at-last, it is referable widely interpretationd ce contemptible textures such as architectures and bridges. “Tensegrity textures are spatial textures cemed by a cabal of undeviating struts and ductile retains. No couple of struts touches and the object of each strut is itemed to three non-coplanar retains. The struts are regularly in compression and the retains in strain (Crane).” These textures are tangled in creation and wary the makeweight ce these textures can be very tangled and confused.

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Figure – Partary Makeweight Part (Motro)

1.1 Determination and Name

The absolutest of tensegrity textures, shenjoy in Figure 1, is moored of three comfortable struts of homogeneous tediousness and nine strained retains so of homogeneous tediousness. These tediousnesss of the struts and retains are contrariant, except educe a successful relevancy among the tediousness of the struts and retains of 1.468. If the relevancy of these tediousnesss were to be smaller than 1.468, then the inspirationhod gain referable be undeviating making it imimplicit to bestow a fashion to it. At-last, if the relevancy was larger, then that would conclusion in a inspirationhod that would be very enigmatical or uniproduce imimplicit to congregate (Motro).

Anthony Pugh provides a interpretationful determination of tensegrity that can be utilized in excite discussing tensegrity textures. Pugh declares “a tensegrity inspirationhod is coercionmal when a coercionmal of disregular compression parts interacts with a coercionmal of regular tensile parts to fix a trustworthy compass in room (Pugh).” As can be distinguishn from Figure 1, the texture is lasting on its enjoy and conforms to the determination that Pugh provides ce tensegrity textures. This determination can be mitigated to be uniproduce past ponderate to declare that “a tensegrity inspirationhod is a inspirationhod in a trustworthy self-equilibrated declare comprising a disregular coercionmal of comfortable parts interiorly a continuum of strained parts (Motro).” The part that twain determinations define is the comfortable struts and strained retains, which can be a cabal of unconvinced absolute parts congregated to cem a past tangled texture.

1.2 Name fixed on questionables

Unconvinced questionables ce tensegrity textures had been enthralled extinguished environing the selfselfselfaenjoy date by Richard Buckminster Fuller of America, David Georges Emmerich of France, and Kenneth Snelson of America. Perfect three man had homogeneous names and details of these “self-stressing inspirationhods (Maculet).” Of the three man, the questionable cevery that ultimately crafted the promise “tensegrity” was Fuller’s questionable ce “Tensile Integrity.” Emmerich had applied ce his questionable abextinguished the selfselfselfaenjoy date as Fuller ce “Understanding de réseaux autotendants,” which roughly translated is “Understanding of self-stressing networks,” and Snelson’s questionable was ce “Regular strain, disregular compression textures.”

Figure – Emmerichs’ (extreme left), Fullers’ (extreme unspotted) and Snelsons’ (bottom) questionables (Motro)

Perfect three questionables had defined the selfselfselfaenjoy inspirationhod and can be utterly into solitary determination that bestows a very ponderate name of a tensegrity inspirationhod. “Tensegrity Inspirationhods are spatial reticulate inspirationhods in a declare of self-stress. Perfect their parts possess a nearepose intermediate sinews and are equipollent dimension. Strained parts possess no undeviatingity in compression and controlm a regular coercionmal. Comfortable parts controlm a disregular coercionmal. Each node receives solitary and singly solitary comfortable part (Motro).” This determination is very homogeneous to the determinations granted by Pugh and Motro. Fixed on this determination, it insists the spatiality of the inspirationhod and that the parts are either in compression or strain. So, the cacophony of the texture is educed by a declare of self-stress, rebellious of perfect exterior ceces and that there is a successful relevancy to the tediousnesss of the portions to detain this undeviatingity.

1.3 Russian Constructivists

Although Emmerich, Fuller, and Snelson were the earliest three to adduce ce questionables ce tensegrity, the truth of tensegrity distinguishms to go tail uniproduce excite to the Russian constructivists. An reproduce-exhibitation in Moscow in 1921 held a “sculpture-structure” by Karl Ioganson, Figure 3 inferiorneath, that was a self-equilibrated texture and had the basic parts of a tensegrity texture. Uniproduce though Ioganson’s plastic-is-sue does referable recount to the ordinary characteristics of tensegrity, the declares of static makeweight of tensengrity can be fixed on the comfortable struts and the inferiorstanding of the plastic-is-sue helps to interpret the agencys of the inspirationhod. “According to structural morphology, it illustrates the deed that unconvinced fashions can be linked to a solitary texture (Motro).”

Figure – Ioganson plastic-is-sue (Motro)

Principles of Tensegrity

A tensegrity texture has a past tangled coercionmal of equations and stipulations that must be inspiration in manage to fix that the inspirationhod is in makeweight. Sundry engineers possess abandoned fur date and violate into developing the compulsory equations to right delineation a tensegrity texture enjoy a absolute bridge or architecture. As the equations can be very messy and engage a period to interpret, the basics of unconvinced stipulations are exhibited here.

2.1 Self Makeweight

Retention ce a tensegrity texture canreferable be validated largely compared to trusses and cable nets. A tensegrity texture is trustworthy if the quadratic cem, Q, of the tangent cacophony matrix, K, with reference to any non-trivial disturbance, d, is explicit so that (Thompson):

If Q is homogeneous to cipher, the texture may be trustworthy, at-last would demand excite exploration in loftier-manage provisions of disposition. Excitemore, a texture is pondered to be retrograde if it can sulk in a room with inferior compass, such as a brace dimensional truss on the x-y flatten except placed in the x-y-z room. This is leading ce tensegrity becainterpretation if a texture is non-degenerate, then its nodal coordinates in contrariant arranges are straightly rebellious (Zhang). The extremeology of a tensegrity texture can be modeled as a directed graph defined by the connectivity matrix, C. If nodes i and j are itemed by portion k, then the ith and jth parts in the kth classify of C are homogeneous to 1 and -1 referenceively, period perfect other parts in the classify are cipher (Kaveh).

Figure – Non-retrograde brace-dimensional tensegrity texture (Zhang)

If any non-trivial dismemberment vector exists so that d≠0 that does referable exexsubstitute the portion tediousnesss as:

Then d is pondered a agency and the texture is consequently kinematically irregular, which is contemptiblely the subject in tensegrity textures (Pellegrino). The straight cacophony matrix, KE, and geometrical cacophony matrix, KG, are cemulated using D and E, the self-makeweight equation of the texture with reference to the nodal coordinate vectors.

Where Ǩ is a divergent matrix consisting of mitigated axial cacophony of the portions, Id is an separateity matrix, and âŠ- denotes tensor result (Zhang and Ohaski). If solitary was to ponder singly customary esthetics that does referable cainterpretation cipher or indirect axial cacophony, then Ǩ is explicit specific and KE is explicit semidefinite. Explicit-semispecific instrument that the eigenvalues of the matrix are perfect non-indirect appraises. If KG is explicit semidefinite, then Q>0 gain referable be amiable if and singly if there exists at lowest solitary agency that sulks in the null-room of KG (Zhang). This produces the scarcity ce loftier manage provisions of disrest to be pondered to cunningate if the inspirationhod is in makeweight.

2.2 Affine Disturbance

“An affine disturbance is a disturbance that preserves colinearity and relevancys of removals; i.e., perfect summits untruthful on a verse are transformed to summits on a verse, and relevancys of the removals among any couples of the summits on the verse are preserved (Weisstein).” By this determination, the affine disturbances of a tensegrity texture is some straight move, whether it is a translation or revolution, as shenjoy in Figure 5, or a dilation or skim, as shenjoy in Figure 6. The revolution and translation affine disturbances detain the tediousnesss of the portions, at-last the dilation and skim are pondered to be non-trivial affine disturbances.

Figure – Translation and Revolution Affine Disturbances of 2-D Tensegrity Texture (Zhang)

Figure – Dilation and Skim Affine Disturbances of 2-D Tensegrity Texture (Zhang)

Translation and revolution are public as undeviating assemblage disturbances of a texture. Translation is simply displacing the texture a convinced removal, i, in the x-, y-, or z-direction. The translation vectors in three dimensional room are as follows:


Revolution abextinguished the axes is so a undeviating-assemblage disturbance, at-last singly the geometrical cacophony matrix is pondered here. Ce the geometrical cacophony matrix, node i is rotated abextinguished the z-axis by an harsh predilection θ and has the subjoined integraliance among its obsolete and novel coordinates:


Where c=cosθ and s=sinθ (Zhang) and r is the geometrical cacophony matrix defined. So, by letting X and Ẋ be the vectors ce perfect of the nodes obsolete and novel coordinates, referenceively, then the integraliance among the obsolete and novel coordinates is:


Where R is an separateity matrix:


Therefore, the disturbance d’ that rotates the texture from the peculiar outline to the novel solitary can then be written as:


Dilation accounts the texture to develop or retrench. This dismemberment is homogeneous to that of translation; at-last as it moves in solitary arrange period the other comes trustworthy, the appraises ce each matrix gain be contrariant. These disturbances, dx, dy and dz of a texture can be written as follows:


There is singly solitary skim implicit ce the brace dimensional subject, as shenjoy in Figure 6(c), except with a three-dimensional tensegrity texture, three skims are implicit: dxy, dxz and dyz and are fixd as follows:


From equations (10) and (11) balance, it can be said that the non-trivial affine disturbances of dilation and skim are straightly rebellious of the undeviating-assemblage disturbances of translation and revolution from equations (5) and (9).

2.3 Makeweight of right assaulted textures and textures with no exterior assaulting

The makeweight of a tensegrity texture with right assaulted strings is a employment of cece densities (λ), portion node impingement matrix (M), and node vectors (p) (Masic). Cece inobservance is a scalar multiplier that when various by the part vector, gi, produces the part cece vector, fji, which corresponds to the offering of the interior cece of part, ei, to the pit of the ceces at node, vj. The node vectors are then unmoved into a matrix according to the subjoined determination:


Furthermore, fe and fc is the gathering of exterior cece vectors and toil ceces that impress on a node, referenceively, fixd homogeneously to the node vectors matrix as:


The makeweight of a tensegrity texture when the strings in the outline p are right assaulted can then be written as:


In manage to recount the right assaulted makeweight to the expensive subject when there are no exterior ceces applied to the texture, constitutive equations must be cemed to recount the decent toil. “The integraliance among the cece-inobservance variables and impressual texture parameters depends on the strain-emphasis integraliance ce the esthetic interpretationd to establish the ductile parts of the texture (Masic).” Using Hooke’s legislation ce straight ductile esthetics, the cece densities at any outline can be computed with vi, l0 and yi reproduce-exhibiting the compasss, repose tediousnesss, and Young’s modulus of cylindrical parts so that:


Using the matrices from equation (15), the cece densities ce no exterior assaults can then be computed as follows:


With the novel cece densities adapted, the toil in equation (14) then becomes:


So that when there is no exterior assaulting applied (i.e. fe=0), the makeweight event in equation (14) becomes:


2.4 Cece makeweight and cece coefficients

Tensegrity textures biggest intepaportraiture is the compressive ceces sustained by the struts of the texture. So, bestenjoy the tensile ceces, tn, in the retains characterized by the string vector, sn, and the compressive cece, fn, in the strut characterized by the paportraiture vector, bn, the tensile cece coefficient γn>0 and the compressive cece coefficient λm>0 (Williamson) are fixd as


Forces in a tensegrity texture are fixd by the exterior cece vector, w, compression vector, f, and strain vector, t, where


The divergent matrices {Γ,Λ} are the tensile and compressive coefficient cece matrix, referenceively, such that:


Using the advice from equations (20) and (21), equation (19) can be rewritten and developed as:


2.5 Exterior cece collision

There has been a chance of violate entired until of-late that pondered no exterior assaulting applied to a tensegrity texture. As the portions of a tensegrity texture is either a compressive strut or strained retain, the collision of exterior assaults scarcity to be pondered ce a entire static segregation to be entired ce a tensegrity texture. The exterior ceces and moments applied to a tensegrity texture can be cunningated using the causes of implicit violate. In the absolutest cem, the implicit violate equation is:


Where δWF is the entirety implicit violate dsolitary by the ceces, δWM is the entirety implicit violate dsolitary by the moments, and δV is the entirety implicit disrest associated with the ceces. Gsingle the struts are pondered massless the engagement recountd to the implicit disrest in the caverification of implicit violate is the conclusionant of the ductile implicit disrest offerings bestenjoy by the retains (Crane).

2.6 Tangledity of makeweight

The makeweight stipulations and equations tobject to gclassify exponentially in tangledity the past that an segregation is applied to the texture. Each of the extremeics discussed thus distant is an violate to disencumber what is scarcityed to cunningate the makeweight of any tensegrity texture. Sundry of the equations entangle numerous matrices and variables to easily awaken a tensegrity texture. Figure 7 inferiorneath shows a customary strut with the low unyielding from move on the downright axis and revolution abextinguished the longitudinal axis. This shows how tangled the exterior assaulting can be when implicit violate is interpretationd to cunningate the applied ceces.

Figure – Bounteous Assemblage Diagram ce an Harsh Strut Modeled with a Universal Junction (Crane)

The equations to cunningate the cece applied exteriorly to a tensegrity texture has a aidency to attain entirely tediousnessy as each separate in the absolute equation is a employment of the bounteous tediousness (LS), each separate extreme (T), deep (B), and oblique (L) retain, the revolutions abextinguished the x (εj) and y (βj) axis and downright dismemberments, aj and bj. The equation can singly be solved iteratively by using a decent coercionmal of appraises from Yin’s violate with unloaded rest textures. Becainterpretation of the tangledity of the makeweight equations it is indispensable to identify the answers obtained. An rebellious validation of the conclusions can be courteous using Novelton’s Third Legislation. If there are no exterior moments impressing on an elementary strut, it is satisfactory to perproduce the summation of moments with reference to the inferior object of the strut. If there are exterior moments the reality way entangles established steps (Crane). As an stance of how fur tangledity is confused with the exterior assaulting applied to a tensegrity texture, the implicit violate dsolitary by the exterior ceces, δWF, of equation (23) develops as follows with perfect the ceces of Figure 7 pondered:


The implicit violate dsolitary by the exterior moments and implicit disrest equations develop homogeneously to the exterior ceces and has sundry parameters that are scarcityed in manage to right number the exterior assault applied to the tensegrity texture.

3. Retention and Undeviatingity of Tensegrity

Fur enjoy the makeweight equations demandd ce tensegrity textures, the scarcity to fix that the retention and undeviatingity of the texture is decent so that the texture does referable unspotted fperfect aloof when congregated. There are unconvinced vestibulees to determining the twain the retention and undeviatingity as courteous-behaved-behaved as convinced stipulations that are demandd to detain the decent texture.

Figure – Extended Tensegrity Frameviolate Undeviatingity and Retention Hierarchy (Juan)

3.1 Static segregation

There are three principal vestibulees interpretationd to fix that the retention and undeviatingity of a tensegrity texture either constrains the implicit disturbances or the portion emphasises to attain a bestenjoy outline. The earliest vestibule entangles the disturbance of the texture. Undeviatingity instrument the neglect of referring-to disturbance among the portions of a texture, which implies that the tediousnesss of the portions stringing the vertices of the frameviolate are kept trustworthy (Juan) such that:


Where pi and pj are the reconsignment in the i and j arranges and cij is the squared tediousness of the {ij} border. From a disturbance summit of purpose, a tensegrity frameviolate is said to be undeviating if perfect the neighbor outlines, qi, of a bestenjoy outline p are congruent (Juan). If this referable the subject, the tensegrity texture is ductile and has a coercionmal of non-undeviating disturbances. Gsingle equation (25) singly restgains with non-linearities undeviatingity stipulations, it is interpretationful to interpretation the earliest derivative of equation (25) to fix the velocities, pi'(0) and pj'(0), at date t=0 suffice the stipulations ce it and debris penny ce loftier manage derivatives:


From this derivative, a tensegrity texture can be cunningated if it is earliest manage undeviating texture or if it is an inconspicuous flex texture. Loftier manage derivatives of equation (25) cunningate the loftier flattens of undeviatingity as according to Figure 8 (Juan).

The promote vestibule that can be interpretationd is the cece vestibule. This is an exstrain of the textures makeweight as the tensegrity frameviolate must be in makeweight at each of its vertices in manage ce it to be a undeviating outline. Using dij0 and Tij as the repose tediousness and interior strain ce each nigh portion, referenceively, the equation ce the ceces applied is:


Ce any exterior makeweight cece, Feq, ext, and decent emphasis, ω, and using cece inobservance coefficients introduced by Schek, which reproduce-exhibit the cece per item of tediousness that vertex j applies to vertex i, then the makeweight event ce the frameviolate can then be declared as:


If perfect of the makeweight ceces are resolvable, then the frameviolate is statically undeviating. “Any tensegrity frameviolate can remunerate ce an exterior cece in brace contrariant inspirationhods: guardianship the moderate outline except modifying the emphasis exhibit in each portion, or instantly modifying the moderate outline. Singly the exterior ceces identifying the cemer event are pondered resolvable ce a bestenjoy tensegrity framework, gone, in the promote subject, the frameviolate itself is mitigated in manage to oppose the exterior assault (Juan).” Static undeviatingity is singly preserved ce affine transformations, referable accidental transformations or orthogonal projections. It has been proven that a tensegrity frameviolate is statically undeviating if and singly if it is inconspicuously undeviating.

The third inspirationhod of determining the undeviatingity of a tensegrity texture is the disrest vestibule. This engages into ponderation the anterior inspirationhods, except developed in provisions of disrest instead of cece or disturbance. It should be referableed that the disrest in a cable (tie) acception when tense, disrest of a strut acception when shortened and the disrest of a paportraiture acceptions inferior a tediousness exchange, homogeneous to what is shenjoy in Figure 9 inferiorneath (Juan). In provisions of disposition, the culminationical incompleteness of the disrest employment associated to the tensegrity frameviolate corresponds to a contrariant outline. The frameviolate is said to be globally undeviating if there is singly solitary incompleteness in the outline. The principal employment interpretationd ce determining the disrest in a tensegrity portion is:


Similarly to the cece and disturbance vestibulees, the derivatives of equation (29) possess to suffice convinced stipulations to fix a decent self-emphasis ce the tensegrity framework. The disrest vestibule does restgain with the natural cacophony of the frameviolate and leads to sundry of the aspects of the cece and disturbance vestibulees.

Figure – Disrest Employments ce Cables, Pauses, and Struts (Juan)

3.2 Stipulations

A compulsory event ce non-retrograde tensegrity textures is fixed on explicit specificness of the tangent cacophony matrix parallel with brace other stipulations gain fix that the tensegrity texture is trustworthy. Pondering the non-trivial affine disturbance, d, in three-dimensional room is a straight cabal of the six non-trivial affine disturbances imputable to the deed that they are straightly rebellious (Zhang)


Gsingle the affine disturbance, d, various by the geometrical cacophony matrix is homogeneous to cipher, the quadratic cem of the tangent cacophony matrix can be reduced:


When the tensegrity texture is educed of customary esthetics, then Q is either explicit or cipher. If there is a non-trivial affine disturbance that satisfies DTd=0, then the portion tediousnesss are invariant by the disturbance of equation (31). Gsingle Q canreferable possess a indirect appraise imputable to the straight and geometrical cacophony matrices substance explicit semidefinite, then the cece inobservance matrix is explicit semidefinite, which is solitary of the retention stipulations. If the affine disturbances couple the integral null-room of the geometrical cacophony matrix, then the cece inobservance matrix has at incompleteness systematize want of d+1, another retention event. So, gsingle no non-trivial affine disturbance in the geometrical cacophony matrix leads to Q=0, then the systematize of the geodesy matrix, G, is d(d+1)/2, the third retention event (Zhang). Homogeneously, the third event can be replaced by solitary by Connelly which declares that the portion arranges do referable sulk on the selfselfselfaenjoy conic at infinity (Connelly).

4. Current Interpretations of Tensegrity

Tensegrity currently is referable interpretationd widely in textures such as architectures and bridges. Although uncontemptible in such collisions, it peaceful has been interpretationd ce the Kurilpa Bridge in Australia (Figure 10). The Kurilpa Bridge is a pedestrian tensegrity cable come bridge couplening 1,540 feet balance the Brisbane River. Although this is unspotted a pedestrian bridge, the collision of tensegrity in the delineation is peaceful clear. The masts that attain balance the draw-up are in compression period the cables itemed to the draw-up are in strain. Tensegrity is so sometimes interpretationd as artviolate fur enjoy the Ioganson plastic-is-sue of Figure 3 as courteous-behaved-behaved as other grand interpretations such as the Scarcityle Aspire by Kenneth Snelson.

Figure – Kurlipa Bridge (Source:

Although the Kurlipa Bridge is a expensive stance of the collision of a enduring tensegrity texture, tensegrity is past contemptiblely interpretationd ce deployable textures such as antennas, retractable roofs and aspires. A tensegrity aspire can possess three to ten rates hypnotic the aspire, distinguish Figure 11. A tensegrity aspire has three pauses ceming a triangular low, established by a bperfect and socket junction to the low and is established homogeneously at the extreme (Sultan). In a three rate aspire, the pauses possess a entirety of thirty-three retains connecting the texture contemporaneously.

Figure – Three Rate Tensegrity Aspire (Sultan)

Another deployable tensegrity texture is public as a SVD tensegrity texture (distinguish Figure 12). The engagement SVD defines the arranges that the retains run; shackle (S) runs parallel the bprescribe at an incverse connecting brace rates contemporaneously (A32 to B11 in Figure 12), upunspotted (V) runs uprightly from solitary flatten to the instant (A31 to B21), and divergent (D) runs divergently from solitary paportraiture to the instant (A11 to A32). This is in consequence a partially contrariant fashion of aspire that is implicit compared to the tensegrity aspire.

Figure – Brace Rate SVD Tensegrity Texture (Sultan)

5. Conclusion

Tensegrity textures are choice in their delineation and essential causes as they possess the requirement to detain self-makeweight and possess singly strain ceces in the cables and compressive ceces in the struts. Uniproduce though there is a unspotted total of advice and equations associated with tensegrity at this date, the interpretations of it are peaceful unspottedly poor to deployable objects and grand interpretations. Uniproduce though tensegrity has been so poor in its interpretations to this summit, the long-coercion to interpretation tensegrity ce distant senior interpretations, fur enjoy the Kurlipa Bridge, is substance pushed ce by architects and engineers same.

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