Principles Of Tensegrity Structures Philosophy Essay

The use of tensegrity in constitutions is not all that sordid at this period and repeatedly surprises inhabitants who experience out environing tensegrity classifications. Casually it seems that the constitutions using tensegrity should not unicreate be steadfast as the heaviest parts closely transport in extension. The geodesy of a tensegrity constitution uses a reproduction of cells of identical conformation but contrariant bignesss unembarrassed environing the upfit axis. A tensegrity constitution is a 3-D truss which has struts in compression that translates the admonishs to strained cables which exertion aceedly in dispose to translate the admonishing of the constitution to the plea. The strained cables exwatch buttress to the housed struts of the constitution to buttress the admonishs that can be applied to the classification. Due to the way the struts and cables exertion delay one another, the constitution can be set spatially and be self-equilibrated. Unicreate though the constitution may bear these directs, it quiet is not used as a regular structural classification due to its flaws. By examining the tensegrity constitution and the concepts subsequently it, one can long-for the advenient use of tensegrity constitutions can be broadened and admit for aggravate awe vivacious constitutions to be built.


Architects and engineers bear pushed the limits of what is immanent for sundry years and bear endeavored to establish new and awe vivacious constitutions at the identical period. One such structural classification is disclosed as tensegrity. Tensegrity is not a new structural classification as it has been environing for roughly fifty years, at-last, it is not widely used for sordid constitutions such as constructions and bridges. “Tensegrity constitutions are spatial constitutions createed by a consortment of unbending struts and alterable ties. No two of struts touches and the end of each strut is aceed to three non-coplanar ties. The struts are regularly in compression and the ties in strain (Crane).” These constitutions are intricate in discollocation and circumspect the makeweight for these constitutions can be very intricate and confused.

Figure – Elementary Makeweight Part (Motro)

1.1 Specification and Description

The plainst of tensegrity constitutions, shown in Figure 1, is moored of three housed struts of resembling exsecurity and nine strained ties besides of resembling extension. These extensions of the struts and ties are contrariant, but produce a fortunate bearing betwixt the exsecurity of the struts and ties of 1.468. If the bearing of these extensions were to be smaller than 1.468, then the classification allure not be unbending making it imimmanent to confer a figure to it. However, if the bearing was larger, then that would emanation in a classification that would be very reserved or unicreate imimmanent to muster (Motro).

Anthony Pugh provides a adapted specification of tensegrity that can be utilized in prefer discussing tensegrity constitutions. Pugh says “a tensegrity classification is established when a set of disunicreate compression contents interacts delay a set of unicreate tensile contents to teach a steadfast quantity in exstrain (Pugh).” As can be seen from Figure 1, the constitution is be on its own and conforms to the specification that Pugh provides for tensegrity constitutions. This specification can be mitigated to be unicreate aggravate deferential to say that “a tensegrity classification is a classification in a steadfast self-equilibrated say comprising a disunicreate set of housed contents interiorly a continuum of strained contents (Motro).” The content that twain specifications illustrate is the housed struts and strained ties, which can be a consortment of separate plain contents musterd to create a aggravate intricate constitution.

1.2 Tidings grounded on dubiouss

Several dubiouss for tensegrity constitutions had been charmed out environing the identical period by Richard Buckminster Fuller of America, David Georges Emmerich of France, and Kenneth Snelson of America. All three men had aapprove provisions and details of these “self-stressing classifications (Maculet).” Of the three men, the dubious spectry that polite-balancedtually crafted the message “tensegrity” was Fuller’s dubious for “Tensile Integrity.” Emmerich had applied for his dubious environing the identical period as Fuller for “Construction de réseaux autotendants,” which roughly translated is “Construction of self-stressing networks,” and Snelson’s dubious was for “Continuous strain, disunicreate compression constitutions.”

Figure – Emmerichs’ (top left), Fullers’ (top fit) and Snelsons’ (bottom) dubiouss (Motro)

All three dubiouss had illustrated the identical classification and can be easily into one specification that confers a very deferential tidings of a tensegrity classification. “Tensegrity Systems are spatial interweave classifications in a say of self-stress. All their parts bear a undeviating average toughness and are equipollent bigness. Tensioned parts bear no unbendingity in compression and produce a unicreate set. Housed parts produce a disunicreate set. Each node receives one and solely one housed part (Motro).” This specification is very aapprove to the specifications granted by Pugh and Motro. Grounded on this specification, it insists the spatiality of the classification and that the contents are either in compression or strain. Also, the cacophony of the constitution is produced by a say of self-stress, rebellious of all palpable securitys and that thither is a fortunate bearing to the extensions of the limbs to obey this unbendingity.

1.3 Russian Constructivists

Although Emmerich, Fuller, and Snelson were the principal three to adduce for dubiouss for tensegrity, the truth of tensegrity seems to go tail unicreate prefer to the Russian constructivists. An resembleation in Moscow in 1921 held a “sculpture-structure” by Karl Ioganson, Figure 3 adown, that was a self-equilibrated constitution and had the basic contents of a tensegrity constitution. Unicreate though Ioganson’s statuary does not tell to the illustrative characteristics of tensegrity, the says of static makeweight of tensengrity can be grounded on the housed struts and the erection of the statuary helps to teach the arrangements of the classification. “According to structural morphology, it illustrates the reality that separate figures can be linked to a unmarried constitution (Motro).”

Figure – Ioganson statuary (Motro)

Principles of Tensegrity

A tensegrity constitution has a aggravate intricate set of equations and stipulations that must be met in dispose to secure that the classification is in makeweight. Sundry engineers bear abandoned thinkable period and exertion into developing the inevitable equations to uprightly contrivance a tensegrity constitution approve a plain bridge or construction. As the equations can be very messy and use a occasion to teach, the basics of diverse stipulations are bestowed hither.

2.1 Self Equilibrium

Stability for a tensegrity constitution cannot be validated abundantly compared to trusses and cable nets. A tensegrity constitution is steadfast if the quadratic create, Q, of the tangent cacophony matrix, K, delay regard to any non-trivial disturbance, d, is direct so that (Thompson):

If Q is resembling to nothing, the constitution may be steadfast, at-last would insist-upon prefer study in preferable-dispose provisions of essence. Furthermore, a constitution is thinked to be gain-ground-worse if it can lie in a exstrain delay inferior bulk, such as a two dimensional truss on the x-y roll but placed in the x-y-z extension. This is momentous for tensegrity consequently if a constitution is non-degenerate, then its nodal coordinates in contrariant courses are straightly rebellious (Zhang). The topology of a tensegrity constitution can be modeled as a directed graph illustrated by the connectivity matrix, C. If nodes i and j are aceed by limb k, then the ith and jth parts in the kth row of C are resembling to 1 and -1 regardively, occasion all other parts in the row are nothing (Kaveh).

Figure – Non-grow-worse two-dimensional tensegrity constitution (Zhang)

If any non-trivial demolition vector exists so that d≠0 that does not diversify the limb extensions as:

Then d is thinked a arrangement and the constitution is hence kinematically spasmodic, which is sordidly the plight in tensegrity constitutions (Pellegrino). The straight cacophony matrix, KE, and geometrical cacophony matrix, KG, are createulated using D and E, the self-makeweight equation of the constitution delay regard to the nodal coordinate vectors.

Whither Ǩ is a angular matrix consisting of mitigated axial cacophony of the limbs, Id is an personality matrix, and âŠ- denotes tensor emanation (Zhang and Ohaski). If one was to think solely social embodieds that does not agent nothing or disclaiming axial cacophony, then Ǩ is direct poor and KE is direct semidefinite. Positive-semipoor media that the eigenvalues of the matrix are all non-disclaiming computes. If KG is direct semidefinite, then Q>0 allure not be kind if and solely if thither exists at smallest one arrangement that lies in the null-exstrain of KG (Zhang). This establishs the want for preferable dispose provisions of essence to be thinked to indicate if the classification is in makeweight.

2.2 Affine Motion

“An affine disturbance is a disturbance that preserves colinearity and bearings of remotenesss; i.e., all apexs mendacious on a outoutoutcourse are transformed to apexs on a outline, and bearings of the remotenesss betwixt any twos of the apexs on the outoutoutcourse are preserved (Weisstein).” By this specification, the affine disturbances of a tensegrity constitution is some straight move, whether it is a translation or reversion, as shown in Figure 5, or a dilation or scrape, as shown in Figure 6. The reversion and translation affine disturbances obey the extensions of the limbs, at-last the dilation and scrape are thinked to be non-trivial affine disturbances.

Figure – Translation and Reversion Affine Motions of 2-D Tensegrity Constitution (Zhang)

Figure – Dilation and Scrape Affine Motions of 2-D Tensegrity Constitution (Zhang)

Translation and reversion are disclosed as unbending association disturbances of a constitution. Translation is solely displacing the constitution a assured remoteness, i, in the x-, y-, or z-direction. The translation vectors in three dimensional exstrain are as follows:


Rotation environing the axes is besides a unbending-association disturbance, at-last solely the geometrical cacophony matrix is thinked hither. For the geometrical cacophony matrix, node i is rotated environing the z-axis by an tyrannical angle θ and has the subjoined interdependence betwixt its old and new coordinates:


Whither c=cosθ and s=sinθ (Zhang) and r is the geometrical cacophony matrix illustrated. So, by letting X and Ẋ be the vectors for all of the nodes old and new coordinates, regardively, then the interdependence betwixt the old and new coordinates is:


Whither R is an personality matrix:


Therefore, the disturbance d’ that rotates the constitution from the pristine figure to the new one can then be written as:


Dilation agents the constitution to dilate or decrease. This demolition is aapprove to that of translation; at-last as it moves in one course occasion the other remains uniform, the computes for each matrix allure be contrariant. These disturbances, dx, dy and dz of a constitution can be written as follows:


Thither is solely one scrape immanent for the two dimensional plight, as shown in Figure 6(c), but delay a three-dimensional tensegrity constitution, three scrapes are immanent: dxy, dxz and dyz and are teachd as follows:


From equations (10) and (11) aggravatehead, it can be said that the non-trivial affine disturbances of dilation and scrape are straightly rebellious of the unbending-association disturbances of translation and reversion from equations (5) and (9).

2.3 Makeweight of uprightly admonished constitutions and constitutions delay no palpable admonishing

The makeweight of a tensegrity constitution delay uprightly admonished strings is a part of security densities (λ), limb node impingement matrix (M), and node vectors (p) (Masic). Security inobservance is a scalar multiplier that when multitudinous by the part vector, gi, produces the part security vector, fji, which corresponds to the aid of the interior security of content, ei, to the counteract of the securitys at node, vj. The node vectors are then attentive into a matrix according to the subjoined specification:


Furthermore, fe and fc is the collation of palpable security vectors and business securitys that act on a node, regardively, teachd twin-fellowly to the node vectors matrix as:


The makeweight of a tensegrity constitution when the strings in the figure p are uprightly admonished can then be written as:


In dispose to tell the uprightly admonished makeweight to the costly plight when thither are no palpable securitys applied to the constitution, constitutive equations must be createed to tell the harmonious business. “The interdependence betwixt the security-inobservance variables and express constitution parameters depends on the strain-importance interdependence for the embodied used to uplift the alterable parts of the constitution (Masic).” Using Hooke’s law for straight alterable embodieds, the security densities at any figure can be computed delay vi, l0 and yi resembleing the quantitys, pause extensions, and Young’s modulus of cylindrical parts so that:


Using the matrices from equation (15), the security densities for no palpable admonishs can then be computed as follows:


With the new security densities conducive, the business in equation (14) then becomes:


So that when thither is no palpable admonishing applied (i.e. fe=0), the makeweight tidings in equation (14) becomes:


2.4 Security makeweight and security coefficients

Tensegrity constitutions biggest moment is the compressive securitys sustained by the struts of the constitution. So, confern the tensile securitys, tn, in the ties characterized by the string vector, sn, and the compressive security, fn, in the strut characterized by the bar vector, bn, the tensile security coefficient γn>0 and the compressive security coefficient λm>0 (Williamson) are teachd as


Forces in a tensegrity constitution are teachd by the palpable security vector, w, compression vector, f, and strain vector, t, where


The angular matrices {Γ,Λ} are the tensile and compressive coefficient security matrix, regardively, such that:


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


2.5 Palpable security collision

Thither has been a lot of exertion adequated until of-late that thinked no palpable admonishing applied to a tensegrity constitution. As the limbs of a tensegrity constitution is either a compressive strut or strained tie, the collision of palpable admonishs want to be thinked for a adequate static partition to be adequated for a tensegrity constitution. The palpable securitys and moments applied to a tensegrity constitution can be indicated using the facultys of implicit exertion. In the plainst create, the implicit exertion equation is:


Whither δWF is the sum implicit exertion produced by the securitys, δWM is the sum implicit exertion produced by the moments, and δV is the sum immanent essence associated delay the securitys. Gone the struts are thinked massless the tidings telld to the immanent essence in the faculty of implicit exertion is the emanationant of the alterable immanent essence aids confern by the ties (Crane).

2.6 Perplexity of makeweight

The makeweight stipulations and equations watch to gain-ground exponentially in intricateity the aggravate that an partition is applied to the constitution. Each of the topics discussed thus far is an endeavor to elucidate what is wanted to indicate the makeweight of any tensegrity constitution. Sundry of the equations include deep matrices and variables to easily excite a tensegrity constitution. Figure 7 adown shows a regular strut delay the mean obdurate from move on the absolute axis and reversion environing the longitudinal axis. This shows how intricate the palpable admonishing can be when implicit exertion is used to indicate the applied securitys.

Figure – Uncounted Association Diagram for an Tyrannical Strut Modeled delay a Universal Knee (Crane)

The equations to indicate the security applied palpablely to a tensegrity constitution has a watchency to get alcoincidently extensiony as each part in the plain equation is a part of the uncounted exsecurity (LS), each particular top (T), ground (B), and deviative (L) tie, the reversions environing the x (εj) and y (βj) axis and absolute demolitions, aj and bj. The equation can solely be solved iteratively by using a harmonious set of computes from Yin’s exertion delay unloaded collocation constitutions. Consequently of the intricateity of the makeweight equations it is vital to substantiate the answers obtained. An rebellious validation of the emanations can be genteel using Newton’s Third Law. If thither are no palpable moments acting on an plain strut, it is resembling to percreate the summation of moments delay regard to the inferior end of the strut. If thither are palpable moments the regularty arrangement includes added steps (Crane). As an issue of how thinkable intricateity is confused delay the palpable admonishing applied to a tensegrity constitution, the implicit exertion produced by the palpable securitys, δWF, of equation (23) dilates as follows delay all the securitys of Figure 7 thinked:


The implicit exertion produced by the palpable moments and immanent essence equations dilate twin-fellowly to the palpable securitys and has sundry parameters that are wanted in dispose to uprightly investigate the palpable admonish applied to the tensegrity constitution.

3. Uprightness and Rectilinearity of Tensegrity

Much approve the makeweight equations insist-upond for tensegrity constitutions, the want to secure that the uprightness and unbendingity of the constitution is harmonious so that the constitution does not harmonious drop secretly when musterd. Thither are separate bearinges to determining the twain the uprightness and unbendingity as polite as assured stipulations that are insist-upond to obey the harmonious constitution.

Figure – Extended Tensegrity Frameexertion Rectilinearity and Uprightness Hierarchy (Juan)

3.1 Static partition

Thither are three principal bearinges used to secure that the uprightness and unbendingity of a tensegrity constitution either constrains the immanent disturbances or the limb importancees to thrust a confern figure. The principal bearing includes the disturbance of the constitution. Rectilinearity media the omission of referring-to disturbance betwixt the limbs of a constitution, which implies that the extensions of the limbs connection the vertices of the frameexertion are kept uniproduce (Juan) such that:


Whither pi and pj are the reconsignment in the i and j courses and cij is the squared exsecurity of the {ij} verge. From a disturbance apex of purpose, a tensegrity frameexertion is said to be unbending if all the neighbor figures, qi, of a confern figure p are congruent (Juan). If this not the plight, the tensegrity constitution is easy and has a set of non-unbending disturbances. Gone equation (25) solely chaffers delay non-linearities unbendingity stipulations, it is adapted to use the principal derivative of equation (25) to secure the velocities, pi'(0) and pj'(0), at period t=0 assure the stipulations for it and recrement penny for preferable dispose derivatives:


From this derivative, a tensegrity constitution can be indicated if it is principal dispose unbending constitution or if it is an inconspicuous flex constitution. Preferable dispose derivatives of equation (25) indicate the preferable smooths of unbendingity as according to Figure 8 (Juan).

The promote bearing that can be used is the security bearing. This is an exstrain of the constitutions makeweight as the tensegrity frameexertion must be in makeweight at each of its vertices in dispose for it to be a unbending figure. Using dij0 and Tij as the pause exsecurity and interior strain for each neighboring limb, regardively, the equation for the securitys applied is:


For any palpable makeweight security, Feq, ext, and harmonious importance, ω, and using security inobservance coefficients introduced by Schek, which resemble the security per ace of exsecurity that vertex j applies to vertex i, then the makeweight tidings for the frameexertion can then be sayd as:


If all of the makeweight securitys are resolvable, then the frameexertion is statically unbending. “Any tensegrity frameexertion can remunerate for an palpable security in two contrariant ways: custody the primal figure but modifying the importance bestow in each limb, or undeviatingly modifying the primal figure. Solely the palpable securitys substantiateing the createer tidings are thinked resolvable for a confern tensegrity framework, gone, in the promote plight, the frameexertion itself is mitigated in dispose to delaystand the palpable admonish (Juan).” Static unbendingity is solely preserved for affine transformations, not accidental transformations or orthogonal projections. It has been proven that a tensegrity frameexertion is statically unbending if and solely if it is inconspicuously unbending.

The third course of determining the unbendingity of a tensegrity constitution is the essence bearing. This uses into thinkation the foregoing courses, but developed in provisions of essence instead of security or disturbance. It should be eminent that the essence in a cable (tie) acception when stretched, essence of a strut acception when shortened and the essence of a bar acceptions inferior a exsecurity diversify, aapprove to what is shown in Figure 9 adown (Juan). In provisions of essence, the topical narrowness of the essence part associated to the tensegrity frameexertion corresponds to a contrariant figure. The frameexertion is said to be globally unbending if thither is solely one narrowness in the figure. The principal part used for determining the essence in a tensegrity limb is:


Similarly to the security and disturbance bearinges, the derivatives of equation (29) bear to assure assured stipulations to secure a harmonious self-importance for the tensegrity framework. The essence bearing does chaffer delay the tangible cacophony of the frameexertion and leads to sundry of the aspects of the security and disturbance bearinges.

Figure – Essence Functions for Cables, Bars, and Struts (Juan)

3.2 Conditions

A inevitable tidings for non-grow-worse tensegrity constitutions is grounded on direct poorness of the tangent cacophony matrix concurrently delay two other stipulations allure secure that the tensegrity constitution is steadfast. Considering the non-trivial affine disturbance, d, in three-dimensional exstrain is a straight consortment of the six non-trivial affine disturbances due to the reality that they are straightly rebellious (Zhang)


Since the affine disturbance, d, multitudinous by the geometrical cacophony matrix is resembling to nothing, the quadratic create of the tangent cacophony matrix can be reduced:


When the tensegrity constitution is produced of social embodieds, then Q is either direct or nothing. If thither is a non-trivial affine disturbance that satisfies DTd=0, then the limb extensions are invariant by the disturbance of equation (31). Gone Q cannot bear a disclaiming compute due to the straight and geometrical cacophony matrices entity direct semidefinite, then the security inobservance matrix is direct semidefinite, which is one of the uprightness stipulations. If the affine disturbances span the completion null-exstrain of the geometrical cacophony matrix, then the security inobservance matrix has at narrowness assort want of d+1, another uprightness tidings. Also, gone no non-trivial affine disturbance in the geometrical cacophony matrix leads to Q=0, then the assort of the geodesy matrix, G, is d(d+1)/2, the third uprightness tidings (Zhang). Similarly, the third tidings can be replaced by one by Connelly which says that the limb courses do not lie on the identical conic at instantaneousness (Connelly).

4. Current Uses of Tensegrity

Tensegrity currently is not used widely in constitutions such as constructions and bridges. Although unsordid in such collisions, it quiet has been used for the Kurilpa Bridge in Australia (Figure 10). The Kurilpa Bridge is a pedestrian tensegrity cable remain bridge spanning 1,540 feet aggravate the Brisbane River. Although this is harmonious a pedestrian bridge, the collision of tensegrity in the contrivance is quiet ostensible. The masts that thrust aggravatehead the arrange are in compression occasion the cables aceed to the arrange are in strain. Tensegrity is besides casually used as artexertion thinkable approve the Ioganson statuary of Figure 3 as polite as other honorable uses such as the Needle Uprise by Kenneth Snelson.

Figure – Kurlipa Bridge (Source:

Although the Kurlipa Bridge is a costly issue of the collision of a burning tensegrity constitution, tensegrity is aggravate sordidly used for deployable constitutions such as antennas, retractable roofs and uprises. A tensegrity uprise can bear three to ten limits demulcent the uprise, see Figure 11. A tensegrity uprise has three bars createing a triangular mean, rooted by a sphere and socket knee to the mean and is rooted twin-fellowly at the top (Sultan). In a three limit uprise, the bars bear a sum of thirty-three ties connecting the constitution aceedly.

Figure – Three Limit Tensegrity Uprise (Sultan)

Another deployable tensegrity constitution is disclosed as a SVD tensegrity constitution (see Figure 12). The tidings SVD illustrates the courses that the ties run; shackle (S) runs concurrently the verge at an incoutoutcourse connecting two limits aceedly (A32 to B11 in Figure 12), upfit (V) runs uprightly from one smooth to the instant (A31 to B21), and angular (D) runs angularly from one bar to the instant (A11 to A32). This is in result a partially contrariant mold of uprise that is immanent compared to the tensegrity uprise.

Figure – Two Limit SVD Tensegrity Constitution (Sultan)

5. Conclusion

Tensegrity constitutions are uncommon in their contrivance and important facultys as they bear the inevitableness to obey self-makeweight and bear solely strain securitys in the cables and compressive securitys in the struts. Unicreate though thither is a unspotted sum of advice and equations associated delay tensegrity at this period, the uses of it are quiet unspottedly poor to deployable objects and honorable uses. Unicreate though tensegrity has been so poor in its uses to this apex, the long-for to use tensegrity for far main uses, thinkable approve the Kurlipa Bridge, is entity pushed for by architects and engineers twin-fellow.

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