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Eur. J. Mech. A/Solids 20 (2001) 1023–1031

2001 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS0997-7538(01)01181-0/FLA

Vibration of an elastic tensegrity structure

Irving J. Oppenheima, William O. Williamsb,∗

aDepartments of Civil and Environmental Engineering and Architecture, Carnegie Mellon University, 15213, Pittsburgh, PA, USAbDepartment of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

(Received 16 August 2000; revised and accepted 10 July 2001)

Abstract – The dynamic behavior of a simple elastic tensegrity structure is examined, in order to validate observations that the natural damping ofthe elastic elements in such a structure is poorly mobilized, due to the natural flexibility of the equilibrium position of the structure. It is confirmed,analytically and numerically, that the energy decay of such a system is slower than that of a linearly-damped system. 2001 Éditions scientifiques etmédicales Elsevier SAStensegrity / elastic structures / vibrations

1. Introduction

Following sculptures first created by Snelson in 1948, in 1961 Buckminster Fuller patented a class ofcable–bar structures which he called tensegrity structures (Fuller, 1976; Snelson, 1973). These consisted ofarrangements of bars and cables, with the bars not connected to one another, so that structural integrity wasmaintained by the tension in the cables. Hence “tension-integrity”, compressed to “tensegrity”. A notableexample is Snelson’s Needle Tower at the Hirshhorn Gallery in Washington, DC. Later work of both engineers,e.g., (Calladine, 1978, 1982, Pellegrino and Calladine, 1986; Calladine and Pellegrino, 1991; Kuznetsov, 1984),and mathematicians, e.g., (Connelly, 1980, 1982; Roth and Whiteley, 1981; Whiteley, 1987; Connelly andWhiteley, 1996) generalized the term “tensegrity structure” to include any pin-connected structural frameworkin which some of the elements are tension-only cables or compression-only struts. The simplest three-dimensional example of this type of structure is shown infigure 1. This example possesses the distinguishingcharacteristics of tensegrity structures: It is a form-finding structure, an under-constrained structural system,prestressable while displaying an infinitesimal flexure even when the constituent elements are undeformable. Itforms only at nodal geometries in which the statics matrix becomes rank deficient.

Since tensegrity structures have been proposed to be used in various constructions where weight is atpremium, it is of importance to consider their dynamical response in the vicinity of their equilibrium positions.Usually they are constructed with (essentially) rigid rods and (more) elastic cables, so presuming that the pin-connections at the nodes are efficient, the elastic response and intrinsic damping of the cables determine thevibrational behavior of such a structure. However, these structures, by the nature of their construction mustshow an infinitesimal flexibility in the equilibrium position, and it is easy to suspect that this will lead to a lessefficient mobilization of the cable’s damping than would be true in a more conventional structure. In previouswork (Oppenheim and Williams, 2001) we have noted this fact, and presented some numerical calculations

∗ Correspondence and reprints.E-mail address:[email protected] (W.O. Williams).

1024 I.J. Oppenheim, W.O. Williams

Figure 1. Simple tensegrity structure (not in equilibrium).

and a sketch, for a two-dimensional example structure, of the computations which verify it. This note gives theunderlying computations for the structure offigure 1, quantifying the claims both analytically and numerically.

2. The model

The problem will be examined for the tensegrity structure as pictured infigure 1, further simplified byassumptions of symmetry and uniformity. The structure is taken as a right regular prism, in which the end facesare equilateral triangles, perpendicular to the axis joining their centroids. The unique equilibrium position thenis known to require a 5π/6 relative rotation between the triangles.

We have examined the response to external loading of this structure previously (Oppenheim and Williams,2000), and we will follow the notations and calculations which we used there. The nodesA, B, andC arepinned to ground, the legsAa, Bb, andCc are inextensible bars, the elementsab, bc, andca are inextensiblecables, while the cross-cablesAc, Ba, andCb are taken to be linearly elastic, but with damping. Consideringonly the cross-cables as elastic simplifies the calculations, but captures the essence of the elastic response ofsuch systems.

We assume that:

• The trianglesABC andabc are congruent equilateral triangles.• The legs have the same length.• The cross-cables are identical in length and in elastic modulus and damping coefficient.• The mass of the system is localized in three equal masses at each of the nodesa, b, andc.

As a result of these simplifications the motion of the system can be described by a single parameter. We chooseθ , the angle of rotation of the upper triangle about its centroid measured from the position of equilibrium.

We can use the calculations from the earlier work (Oppenheim and Williams, 2000): of the base and uppertriangles isa and the length of the rigid legs isL, then we find that the length of the cross-cablesλ is given as:

λ(θ)2 = L2 − 2√

3a2 cosθ, (2.1)

Vibration of an elastic tensegrity structure 1025

and the heighth of the upper triangle as:

h(θ)2 = L2 − a2(2+ √3cosθ + sinθ). (2.2)

Note that from (2.1) it is implicitly assumed thatL>

√2√

3a.

If we assume that the cross-cables are linearly elastic, with a spring constant ofκ , then the elastic energystored in the three totals to:

�(θ)= 3

2κ(λ(θ)− λN

)2, (2.3)

whereλN is the natural length of the cable. The kinetic energy is given as:

K(θ, θ) = 3

2ma2(θ)2 + 3

2m(h)2, (2.4)

accounting for the complementary rotational and vertical parts.

The final element is the damping. We assume that the damping force is isolated in the cross-cables and thatit is linear in the rate of change of length of each cable. The total rate of dissipation of energy by these forces,then, is:

�(θ)= 3γ (λ)2. (2.5)

The equation of motion is obtained from the above as:

∂K

∂θθ + ∂K

∂θθ + d�

dθθ +�(θ)= 0. (2.6)

We may divide the common termθ from the equation and expand some terms to arrive at a non-linear second-order equation which governs the vibrations of the system:

ma2[1+ a2

h2sin2

(5π

6+ θ

)]θ

+ ma4

h2sin

(5π

6+ θ

)[cos

(5π

6+ θ

)+ a2

h2sin2

(5π

6+ θ

)]θ

+ 3γa4

λ2sin2 θθ + √

3κa2λ− λN

λ2sinθ = 0. (2.7)

The equation is rather complicated, but its essential behavior can be seen by looking at the leading terms in anexpansion:

θ + α(λ(0)− λN

)θ + βθ3 + τθ2θ = 0. (2.8)

The termα(λ(0) − λN) is the initial elastic modulus. If the cross-cables are slack in the equilibrium positionthen this term is zero and the elastic response is cubic, modulated byβ, which always is positive. For us, themost important point is that the damping, with coefficientτ , is modulated byθ2 (sin2 θ in the full equation(2.7)). This leads, as we make more precise in Section 4, to a strongly reduced damping effect.

By considering a quadrature for the solution of (2.8) withα andτ zero, one can show (Stoker, 1950) that theproduct of the period and the amplitude of that solution is a constant, so that the period is inversely proportionalto the amplitude. We shall see below that the solutions of (2.7) show corresponding period-lengthening with


decay. (Of course the exact inverse proportionality of the special case does not apply in the general case of(2.8).)

3. Numerical calculations

In order to visualize the behavior of the system (2.7), we set parameters to nominal but consistent values:

a = 1, L =√

6+ √3, m = 1, γ = 40, κ = 100, λN =

√6− √

3. (3.1)

The last condition ensures that the cross-cables are unstressed at equilibrium. A plot of a numerical solution ofan initial value problem for the equation with these values is shown infigure 2.

Figure 2. Theta versus time for no prestress.

Figure 3. Energy decay. Upper curve is for the given system, lower is equivalent linearly damped system. Inset: fit to data ofe(0)/(1+At)2.


Notice the two significant phenomena: the period increases as the amplitude decays, and the decay isrelatively slow. The decay is highlighted by plotting the change in energy of the system infigure 3. In thesame plot, we show the energy of the system, as altered by replacing the sin2 θ damping by a linear dampingbased on an average value of sin2 θ over the interval between zero and the starting value ofθ . Finally, in aninsert we show a fit of the energy function toe(0)/(1+At)2, which is the estimate arrived at in the next section.

To illustrate the effect of pre-stress in the cables, we leave all parameters as before, except thatλN is reducedto a value which corresponds to a 1% strain in the cross-cables in the equilibrium position. The oscillation due tothe same initial displacement is shown infigure 4. Of course the frequency increases due to the prestress. Similarperiod-lengthening and decay of amplitude occur, although both are less pronounced than in the previous case.

The reduced efficiency of damping is illustrated first infigure 5by contrasting the response for the givensystem and one with linear damping introduced as before. Second, infigure 6we make the same sort of energy-decay comparison as for the previous case, along with a fit of a curve of the forme(0)/(1 + At) per ourestimates in the next section.

Figure 4. Theta versus time with prestress.

Figure 5. Theta versus time for equivalent linearly damped system with prestress, contrasted with naturally damped system.


Figure 6. Energy decay with 1% prestrain. Upper curve is for the given system, lower is equivalent linearly damped system. Inset: fit to data ofe(0)/(1+At).

4. Dissipation calculations

To substantiate the relative slowness of the decay of the vibrations, we will estimate the energy of the systemas it evolves in time. The total energy of the system isK(θ(t), θ (t))+�(θ(t)), where these are given by (2.4),(2.3). The base stored energy is:

�(0) = 3

2κ(λ0 − λN), (4.1)

whereλ0 = λ(0) is the extension of the cables due to prestrain. We will consider the evolution of the transientenergy:

e =K(θ, θ)+�(θ)−�(0)

= 3

2ma2θ2 + 3

2mh2 + 3

2κ[(λ− λN)

2 − (λ0 − λN)2] (4.2)

through the energy equation (2.6):

e = −3γ(

dλ

dθ

)2

θ2. (4.3)

Our main estimates are:

PROPOSITION 4.1:For solutions of(4.3)within the feasible range(−π/6� θ(0) � π/6)

• if λ(0) > λN ( prestressed equilibrium) there is a constantµ such that

e(t) � e(0)

1+ µ

λ0−λNe(0)t

. (4.4)


• if λ(0) = λN (slack equilibrium) there is a constantν such that

e(t) � e(0)

(1+ ν√e(0)t)2

. (4.5)

Thus the decay is at best like1/t2 ast increases, in contrast to the exponential decay characteristic of linearlydamped systems.

To verify these estimates, note that from (4.2):

e � 3ma2

2θ2 (4.6)

and

e � 3κ

2

[(λ− λN)

2 − (λ0 − λN)2]. (4.7)

Equation (4.6) yields an estimate forθ2. We use (4.7) to obtain an estimate for(dλ/dθ)2.

In the feasible range:

λ� λ0 � λN, (4.8)

and we use (2.1) to see that: (dλ

dθ

)2

= 3a4 sin2 θ

λ2� 3a4

λ02 sin2 θ. (4.9)

To bound the last term, note that its derivative with respect toθ is:

6a4

λ02 cosθ sinθ � 6a4

λ02 sinθ, (4.10)

and similarly that, in the feasible range:

d

dθ(λ− λ0) =

√3a2 sinθ√

L2 − 2√

3a2 cosθ�

√3a2

√L2 − 3a2

sinθ. (4.11)

Hence

d

dθ(λ− λ0) � d

dθ

(1

A

3a4

λ02 sin2 θ

), (4.12)

whereA = 2√

3a2√L2 − 3a2/λ0

2.

Since each of the functions is zero whenθ = 0:

(λ− λ0) � 1

A

3a4

λ02 sin2 θ � 1

A

(dλ

dθ

)2

, (4.13)

which we rewrite as:

(λ− λN)− (λ0 − λN) � 1

A

(dλ

dθ

)2

. (4.14)


Henceforth, we consider two cases. First, ifλ0 >λN (prestressed case), we use (4.14) to deduce that:(dλ

dθ

)2

�A(λ− λN)

2 − (λ0 − λN)2

(λ− λN)+ (λ0 − λN)

� A

2(λ0 − λN)

[(λ− λN)

2 − (λ0 − λN)2]. (4.15)

Thus, from (4.7): (dλ

dθ

)2

� A

3κ(λ0 − λN)e. (4.16)

Substituting this and (4.6) into the energy equation (4.3) yields:

e � − 2Aγ

3κma2(λ0 − λN)e2 = − µ

λ0 − λN

e2. (4.17)

Integration of this equation yields the first bound for the transient energy.

In the second case, if there is no prestress,λ(0) = λN , and we find from (4.14):

(dλ

dθ

)2

� A(λ− λ0) (4.18)

and (4.7) leads immediately to:(

dλ

dθ

)2

� A

√2

3κ

√e. (4.19)

This can be substituted into the energy equation together with (4.6) to obtain:

e � −γA

√2

3κ

2

ma2e3/2, (4.20)

or

e � −2νe3/2, (4.21)

which we integrate to obtain the second estimate.

5. Angular damping

In order to obtain more efficient damping in the system, one can introduce a linear damping term byenhancing the natural damping in the pin connections of the frame. For example, we introduce a dampingwhich resists the rate-of-angulation of the legs to the upper cables as follows. The angleα which the legs maketo thexy-plane is given through the equation:

sin2α = h2

L2= 1− a2

L2(2+ √

3cosθ + sinθ)

= 1− 2ρ2(1− cosφ), (5.1)


whereφ is the total angle of rotation,θ + 5π/6, andρ = a/L. We suppose that the damping mechanism isto be applied between the leg and the nearest upper cable, and one can show that the angleτ between theseelements obeys:

cosτ = cosα cos(φ/2). (5.2)From (5.1) and (5.2) it follows that:

τ = − ρ cosφ√1− ρ2(1− cos2φ)

θ . (5.3)

If we assume that the damping is linear inτ , the energy dissipates at a rate 3%(τ )2 and so the additional termin (2.7) is:

3%ρ2 cos2φ

1− ρ2(1− cos2φ)θ . (5.4)

The linear approximation to this nearθ = 0 is:

3%[

3ρ2

4− ρ2+ 8

√3ρ2(1− ρ2)

(4− ρ2)2θ

]θ , (5.5)

which confirms the linear nature of the damping, and ensures exponential decay of solutions of the equation.

6. Conclusions

We have demonstrated that the natural geometric flexibility inherent to a tensegrity structure at equilibriumleads to inefficient mobilization of the natural damping in the elastic cables of the structure, leading to amuch slower rate of decay of amplitude of vibration than might be expected (order at best 1/t2 as opposedto exponential decay). This effect, readily apparent in models, could be a serious drawback in practical usages.

To control the effective damping, it seems that augmenting the natural damping would be inefficient.However a natural mode of damping, by ensuring damping of the angular motion between the structuralelements leads to linearly damped equations and hence exponential decay of free vibrations.

References

Calladine, C.R., 1978. Buckminster Fuller’s ‘tensegrity’ structures and Clerk Maxwell’s rules for the construction of stiff frames. Int. J. SolidsStruct. 14, 161–172.

Calladine, C.R., 1982. Modal stiffness of a pretensioned cable net. Int. J. Solids Struct. 18, 829–846.Calladine, C.R., Pellegrino, S., 1991. First-order infinitesimal mechanisms. Int. J. Solids Struct. 27, 505–515.Connelly, R., 1980. The rigidity of certain cabled networks and the second order rigidity of arbitrarily triangulated convex surfaces. Adv. in Math.37,

272–299.Connelly, R., 1982. Rigidity and energy. Invent. Math. 66, 11–33.Connelly, R., Whiteley, W., 1996. Second-order rigidity and prestress stability for tensegrity frameworks. SIAM J. Discrete Math. 9, 453–491.Fuller, R.B., 1976. Synergetics: Explorations in the Geometry of Thinking, Vol. 27. Macmillan, New York.Kuznetsov, E.N., 1984. Statics and geometry of underconstrained axisymmetric 3-nets. J. Appl. Mech. 51, 827.Oppenheim, I.J., Williams, W.O., 2000. Geometric effects in an elastic tensegrity structure. Journal of Elasticity 59, 51–69.Oppenheim, I.J., Williams, W.O., 2001. Vibration and damping in three-bar tensegrity structure. J. Aerospace Eng. 14, 85–91.Pellegrino, S., Calladine, C.R., 1986. Matrix analysis of statically and kinematically indeterminate frameworks. Int. J. Solids Struct. 22, 409–428.Roth, B., Whiteley, W., 1981. Tensegrity frameworks. Trans. Am. Math. Soc. 265, 419–446.Snelson, K., 1973. Tensegrity Masts. Shelter Publications, Bolinas, CA.Stoker, J.J., 1950. Nonlinear Vibrations. Interscience, New York.Whiteley, W., 1987. Rigidity of graphs. Pacific J. Math. 110, 233–255.