Vibration Suppression in a Simple Tension-AlignedArray Structure

Tingli Cai,∗ Ranjan Mukherjee,† and Alejandro R. Diaz‡

Michigan State University, East Lansing, Michigan 48824-1226

DOI: 10.2514/1.J052127

Tension-aligned structures have been proposed for space-based antenna applications that require a high degree of

accuracy. The structure uses a compressionmember to impart tension on the antenna, thus helping tomaintain shape

and facilitate disturbance rejection. These structures can be very large and therefore sensitive to low-frequency

excitation. A simple control strategy for tension-aligned structures is proposed, based on the concept of stiffness

variation by sequential application and removal of constraints. The process funnels vibration energy from low-

frequency to high-frequencymodes of the structure, where it is dissipated naturally due to internal damping.A simple

model for the arrangement is used to demonstrate the effectiveness of the control strategy. It consists of a curvedbeam

modeled as a nonlinear elastica arch (the support structure), connected to an array of hinged panels (the antenna).

Two methods for stiffness switching are investigated: variable stiffness hinges in the panels and variable stiffness

elastic bars connecting the panels to the structure. It is shown in simulations that sequential application and removal

of the constraints is an effectivemechanism to remove energy from the system.Variable stiffness hinges are effective in

low-tension applications, while combining hinges with elastic bars is required in the presence of high-tension loads.

Nomenclature

A = cross-section area of the rod (elastica arch), m2

b = width of panel array, mD = nondimensional value of dd = vertical eccentricity of end load on the rod (elastica

arch), mE = Young’s modulus, Pae = axial strain of the rodF = nondimensional value of ff = horizontal end load on the rod (elastica arch), Nh = thickness of panel array, mI = area moment of inertia of the rod (elastica arch), m4

�I = nondimensional value of IK = nondimensional value of kK = stiffness matrixKA = stiffness matrix of the archKP = stiffness matrix of the hinged panel arrayKr = change in stiffnessmatrix due to the addition of springsK0 = constant stiffness matrixk = curvature of the rod (elastica arch), m−1

kr = change in stiffness due to the addition of aspring, N∕m

ks = static value of curvature of the rod (elastica arch),m−1

k1,k2, k3

= stiffness of springs in two-degree-of-freedomsystem, N∕m

L = length of the rod (elastica arch), mLP = length of each panel of panel array, mM = mass matrixMA = mass matrix of the archMP = mass matrix of the hinged panel arraym1, m2 = masses in the two-degree-of-freedom system, kgP = nondimensional value of p

p = axial force in the rod (elastica arch), Nps = static value of axial force in the rod (elastica arch), Nq1, q2 = modal coordinates of two-degree-of-freedom systemr = radius of the rod (elastica arch), mS = nondimensional value of ss = arc length along the centerline of the static

equilibrium shape of the rod, mT = nondimensional value of tt = time, stn = time at which stiffness is varied, s

Un = nondimensional value of unUt = nondimensional value of utu = displacement vector of a point on the rod (elastica

arch), mun = displacement of a point on the rod (elastica arch)

along the normal direction of the static equilibriumshape, m

ut = displacement of a point on the rod (elastica arch)along the tangential direction of the static equilibriumshape, m

Vn, Vt = nondimensional mode shapes

Wn,Wt = nondimensional shape functions

Wnc = work done by external forces, J

X = vector of generalized coordinatesXA = vector of generalized coordinates of the archXP = vector of generalized coordinates of the hinged

panel arrayx1, x2 = displacements of masses m1 and m2, respectively, mY, Z = vectors of nodal values of Vn and VtΓ = modal disparity matrixΓA = modal disparity matrix for method AΓAB = modal disparity matrix for methods A and B

combinedΔK = time-varying stiffness matrixϵn = unit vector along the normal direction of the static

equilibrium shape of the rod (elastica arch)ϵt = unit vector along the tangential direction of the static

equilibrium shape of the rod (elastica arch)ζ = damping ratioθ0 = angle of inclination of the rod (elastica arch) at

s � 0, radλ = modal disparity indexλA = modal disparity index for method AλAB = modal disparity index for methods A and B

combined

Received 20 June 2012; revision received 3 October 2013; accepted forpublication 3 October 2013; published online 31 January 2014. Copyright ©2013 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1533-385X/14 and $10.00 in correspondence with the CCC.

*Graduate Student, Department of Mechanical Engineering, 2555Engineering Building.

†Professor, Department of Mechanical Engineering, 2555 EngineeringBuilding; mukherji@egr.msu.edu (Corresponding Author).

‡Professor, Department of Mechanical Engineering, 2555 EngineeringBuilding.

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μ, ν = modal coordinates in the unconstrained andconstrained states

ΠT = kinetic energy, J�ΠT = nondimensional kinetic energyΠV = strain energy, J�ΠV = nondimensional strain energyρ = mass per unit length, kg∕m�ρ = material density, kg∕m3

Φ, Ψ = modal matrices in the unconstrained and constrainedstates

Ω1, Ω2 = natural frequencies of the two-degree-of-freedomsystem, rad∕s

ωj, �ωj = jth natural frequency of the unconstrained andconstrained systems, rad∕s

I. Introduction

L ARGE space structures are contemplated for use as space-basedradars for imaging and moving-object identification and

tracking. These radars will consist of a large support structure andphased array antennas attached to this structure. The complete systemhas to be designed so that it can be folded into a compact volume forease of transport, and the phased array antennas can maintain a highdegree of accuracy after deployment. A high degree of accuracy isdifficult to achieve because these structures are large and sensitive todisturbances that result in vibration. The structure can be modeledand deformations of the structure can be measured and correctedusing sensors and actuators in real time, but such systems areextremely challenging to engineer. This is because these structuresare designed with numerous and complex joints and mechanisms forfolding that introduce nonlinearities such as slipping, backlash, anddeadband. Model-based control also requires development andidentification of a high-dimensional mathematical model thataccounts for the nonlinearities and integration of significant controlsystem hardware to the structure. Vibration suppression in largespace structures is a challenging problem, and a practical solutionneeds passive or active control using few sensors and actuators.Tomeet the precision requirements of space-based radars, tension-

aligned structures have been proposed byMikulas et al. [1], and Joneset al. [2]. Similar to a bowwith a string, in a tension-aligned structure,the array antennas are attached to the support structure via tensionersat each end (see Fig. 1); the support structure is used as a compressionmember to impart tension to the array antennas pinned at each end.The tension in the antenna array helps maintain flatness, but moreimportantly, it increases the stiffness of the array, which is necessaryfor disturbance rejection [3]. Other benefits of tension-alignedstructures include elimination of the high-dimensional accuracyrequirements of the support structure; greater flexibility in design,because the support structure and the array antennas can be separatelypackaged and deployed; reduced effect of nonlinearities such asdeadband and backlash [2]; compensation for creep andmanufacturing tolerance build-up [4]; and increase in structuraldamping [5] which facilitates vibration suppression. The tension-

aligned architecture is equally well suited for radar designs where theantennas are an array of panels or are attached to a flexible membrane[6–8].To investigate the feasibility of tension-aligned architectures,

Jones et al. [2,9] studied the effect of tension on the stiffness of a largeaperture antenna. Using the DARPA ISAT (from “Innovative Space-Based Radar Antenna Technology”) as the representative platform[10], nonlinear finite-element methods were used to compute thesystem frequencies with sensor surfaces ranging from gossamers topaneled radars. For a free–free support structure, it was shown that itis not possible to find a tension/mass ratio combination that yields ahigher or even equivalent peak frequency to that of the structure. Thisimplies that addition of the sensor surface will only reduce thefundamental frequency of the overall system. This problem can bealleviated by introducing an offset such that the tension adds bendingmoments on the support structure. However, an offest increases thedeformation of the support structure, and the tension required toachieve the same fundamental frequency of the ISAT is close to thebuckling load. To eliminate problems related to buckling and largedeformation, Jones et al. [2] proposed to introduce connectorsbetween the support structure and the sensor surface as well as guywires to provide counter-tension; see Fig. 2.The tension-aligned architecture proposed by Jones et al. [2] can be

viewed as a passive method for vibration suppression where thelocation of the intermediate connectors and internal stress in thestructure are optimized to attain the same level of structural stiffnessas the ISAT platform. This may not be sufficient for meeting the highaccuracy requirements of space-based radars because spacestructures such as the ISAT platform are large and prone to low-frequency excitation. The tension required to achieve the desiredlevel of stiffness is also high and may not be suitable for long-termoperation. To address these challenges, a simple control method fortension-aligned structures is proposed in this paper. Based on theconcept of stiffness variation, this method applies and removesconstraints cyclically such that vibration energy is funneled into thehigh-frequency modes of the structure, where it can be dissipatedquickly and naturally due to high rates of internal damping.Compared to traditional active control, which relies on accuratemeasurement by sensors and careful compensation by actuators, oursimple control strategy requires fewer sensors and actuators andeliminates the need for extensive computations based on amathematical model of the structure.Several researchers [11–15] have explored stiffness variation as a

method for vibration suppression. In all of these works, variable-stiffness elements are placed in a state of high stiffness and energy isstored in them. Once the stored energy reaches a maximum value, thestiffness of the element is switched to a lowvalue todissipate energy. Inour earlier work [16–18], stiffness variation was achieved throughapplication and removal of constraints, and energy dissipation wasaccomplished through a targeted and purposeful energy redistributionfrom low-frequency modes to high-frequency modes. This paper is anextension of our earlier work to a simple tension-aligned structure.This paper is organized as follows. In Sec. II, we first illustrate the

simple control strategy with the help of a simple two-degree-of-freedom (2-DOF) mass–spring–damper system. We then extend thecontrol strategy for application to multi-degree-of-freedom linearsystems and define “modal disparity index”, a metric that can be usedto determine the efficacy of our control strategy. In Sec. III, weconsider a simple tension-aligned structure consisting of a planarelastica arch and an array of hinged panels and derive its lineardynamic model. We also discuss two methods for variation ofstiffness. In Sec. IV, we present numerical simulation results. Two

Support Structure in Compression

Sensor Surface in Tension

Compression Support Truss

Tensioned Radar Array Panels

Fig. 1 Tension-aligned structure comprised of a support structure andasensor surface (taken from Jones et al. [2]).

Fig. 2 Connectors and guy-wire for stiffening a tension-alignedstructure [2].

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methods of stiffness variation are explored with the hinged panelarray subjected to low tension and high tension. Concluding remarksare provided in Sec. V.

II. Vibration Suppression Through Stiffness Switching

A. Two-Degree-of-Freedom Illustrative Example

Consider the 2-DOF mass–spring–damper system in Fig. 3. Thetwo massesm1 andm2 are connected to fixed supports by springs ofstiffness k1 and k2 and to each other by the spring of time-varyingstiffness k3�t�. The displacements of the two masses are denoted byx1 and x2 and the springs are undeformedwhen themasses are in theirequilibrium configuration (i.e., x1 � x2 � 0). We assume�k1∕m1� ≠ �k2∕m2� such that the two masses have different naturalfrequencies when k3�t� � 0. The equation of motion of the twodegree-of-freedom system is given next:�

m1 0

0 m2

���x1�x2

���k1 � k3�t� −k3�t�−k3�t� k2 � k3�t�

��x1x2

���0

0

�

Now, consider the three cases where the stiffness k3�t� is chosendifferently

Unconstrained: k3�t� � 0 (1a)

Constrained: k3�t� � kr (1b)

Switched: k3�t� ��0 if t ∈ �ti; ti�1�kr if t ∈ �ti�1; ti�2�

; i � 0; 2; 4; · · ·

(1c)

It is assumed that kr is large compared to k1 and k2, and tn,n � 0; 1; 2; · · · , are chosen such that x2�tn� − x1�tn� � 0. Thisensures that no energy is removed from the systemwhen the stiffnessis switched from kr to 0 or added to the system when the stiffness isswitched from 0 to kr.

§ Assuming modal damping with uniformdamping ratio ζ, the equation of motion of the system can be writtenin modal coordinates as follows:

��q1

�q2

�� 2ζ

�Ω1�t� 0

0 Ω2�t�

��_q1

_q2

���Ω2

1�t� 0

0 Ω22�t�

��q1

q2

�

��0

0

�(2)

where q1 and q2 are the modal coordinates, and Ωj�t�, j � 1, 2, arethe natural frequencies of the system. For the three different cases, thenatural frequencies are denoted as follows.

Unconstrained: Ωj�t� � ωj (3a)

Constrained: Ωj�t� � �ωj (3b)

Switched: Ωj�t� ��ωj if t ∈ �ti; ti�1��ωj if t ∈ �ti�1; ti�2�

; i � 0; 2; 4; · · ·

(3c)

where ωj ��������������kj∕mj

p, j � 1, 2. The expressions for �ωj are

complicated and are not provided here.

Simulations were performed for the three cases discussedpreviously, using parameters in Table 1 and the same set of initialconditions. The natural frequencies of the unconstrained andconstrained systems were found to be

�ω1;ω2� � �1.4142; 1.0000�;� �ω1; �ω2� � �1.1547; 1.7321 × 102�

(4)

where the units are radians per second. One of the frequencies for theconstrained system is high relative to the other natural frequencies.This frequency is associated with the relative motion of the twomasses when they are connected by the stiff spring kr.The simulation results are shown in Figs. 4 and 5. The total energy

and the displacements of the unconstrained and constrained systemsare shown in Fig. 4. For the unconstrained system, the total energy ofthe system decays very slowly; only 10.5% is dissipated in 42.7 s.This is because of low internal damping associated with low naturalfrequencies of the system. For the constrained system, the totalenergy decays rapidly initially, but slowly thereafter; 35.9% isdissipated in 42.7 s. One natural frequency of the constrained systemis high, and rapid decay of the energy associated with this modecontributes to the initial rapid decay of the total energy. Thedisplacements of the two masses of the constrained system appear tobe identical. This is because of small relative motion of the masses, aconsequence of high stiffness of the spring connecting them.For both the unconstrained and the constrained systems in Fig. 4, a

small fraction of the energy is dissipated. In contrast, the energyof theswitched system (see Fig. 5) decays significantly faster; 83.4% isdissipated in 42.7 s. For the switched system, the two masses areinitially unconstrained. They are connected (constrained) by thespring at t1 � 11.24 s, released (unconstrained) at t2 � 21.25 s, andagain connected at t3 � 32.70 s. As mentioned earlier, t1, t2, and t3are chosen such that no energy is added to or subtracted from thesystem during the process of application or removal of the constraint(stiffness switching). At t � t1 and t � t3, application of theconstraint creates a high-frequencymode and funnels energy into thismode, where it is dissipated quickly; this can be verified from theenergy plots of the low- and high-frequency modes. As in the case ofthe constrained system in Fig. 4, the displacement plots of the twomasses for the switched system appear to be identical when they areconstrained by the spring. The plots of the modal coordinates showdiscontinuities at the times of constraint application and removal.This is because the modal coordinates have different functionaldescriptions in the constrained and unconstrained states. In theconstrained state, the high-frequencymode has a small amplitude (q2in Fig. 5), but its energy content is significant. This energy decaysrapidly each time after the system is switched from the unconstrainedstate to the constrained state. This can be seen from the magnifiedviews of q2 in the time interval �t1; t2� and again in the interval�t3; 42.7�. A comparison of the magnified views of q2 in the intervals�t1; t2� and �t3; 42.7� also confirms that switching results in funnelingof energy into the high-frequency mode.The previous example illustrates that energy dissipation is faster in

systems with switched stiffness than in systems with constantstiffness in the presence of modal damping. The faster rate ofdissipation is due not to direct removal of energy by the action ofswitching but to the funneling of energy into the high-frequency

Fig. 3 Two-degree-of-freedom mass–spring–damper system.

Table 1 Parameters used in simulations

m1 m2 k1 k2 kr ζ

1.00 kg 2.00 kg 2.00 N∕m 2.00 N∕m 20; 000 N∕m 0.001

§In this particular example, the stiffness is switched (low to high and high tolow) only when the relative displacement is zero. This is purposely done toillustrate that the change (decrease) in energy of the system is not due to thedirect action of stiffness switching.

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modes of the system. The ease with which energy can be funneledfrom the low-frequency modes to the high-frequency modes isdiscussed in the next section for a general multi-degree-of-freedomlinear system. In the numerical simulations presented later in Sec. IV,wewill ensure that stiffness increase occurs at the time of zero relativedisplacement; this will ensure that no energy is added to the system.However, when the stiffness is decreased, there is no need to ensurethat the relative displacement is zero because stiffness reduction withnonzero relative displacement will result in direct loss of energy. Thisdirect loss of energywill contribute to the total energy dissipation; wewill account for it in our simulations and show that energy funnelinginto the high-frequency modes is nevertheless the primarymechanism of energy dissipation.

B. Multi-Degree-of-Freedom Systems

Consider the following N-DOF linear system:

M �X�K�t�X � 0 (5)

where X � �X1; X2; · · · ; XN�T denotes the vector of generalizedcoordinates, M denotes the mass matrix, and K�t� denotes thestiffness matrix. The stiffness matrix K�t� consists of a constantstiffness matrix K0 and a time-varying stiffness matrix ΔK�t� asfollows:

K�t� � K0 � ΔK�t�; ΔK�t� ��0 if t ∈ �ti; ti�1�Kr if t ∈ �ti�1; ti�2�

;

i � 0; 2; 4; · · · (6)

where tj, j � 0; 1; 2; · · · , are chosen such that the change in stiffnessdoes not increase or decrease the total energy of the system. In theprevious equation, Kr is the change in the stiffness matrix due to theaddition of springs connecting pairs of generalized coordinates. Inthe simplest casewhere a single spring is used to connect generalizedcoordinates Xm and Xn, the entries of Kr ∈ RN×N can be obtainedfrom the Hessian of the additional strain energy �1∕2�kr�Xm − Xn�2:

0 10 20 30 40 0 10 20 30 40

0 10 20 30 40 0 10 20 30 400.0

0.2

0.4

0.0

0.2

0.4

0.0

-0.2

-0.4

0.2

0.4

0.0

-0.4

0.4

a)

c)

b)

d)Fig. 4 Total energy and displacements of the unconstrained system (a and b) and of the constrained system (c and d).

a)

b)

d)

e)

0 10 20 30 40 0 10 20 30 400.0

0.2

0.4

0.0

0.5

-0.5

0 10 20 30 40 0 10 20 30 400.0

0.2

0.4

0.0

0.5

-0.5

uc uc uc ucc cc c

c) f)

0 10 20 30 40 0 10 20 30 400.0

0.2

0.4

0.0

0.5

-0.5

Fig. 5 Switched system: a) total energy, b) energy in low-frequency modes, c) energy in high-frequency mode, d) displacements, and e, f) modaldisplacements. “uc” and “c” denote the unconstrained and constrained states.

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Kr�i; j� �(kr if �i; j� � �m;m� or �n; n�−kr if �i; j� � �m; n� or �n;m�0 otherwise

m ≠ n

(7)

where kr is the stiffness of the spring, which is large compared to themagnitude of the entries of K0.Let ϕi and μi, i � 1; 2; : : : ; N, denote the linearly independent

orthogonal mode shapes and the corresponding modal coordinates inthe unconstrained state. Similarly, let ψ i and νi, i � 1; 2; : : : ; N,denote the linearly independent orthogonal mode shapes and thecorrespondingmodal coordinates in the constrained state. At the timeof application of the constraint (ΔK changes from 0 to Kr), thegeneralized coordinates and their velocities can be expressed asfollows:

X�ti�1� ��PN

j�1 μj�ti�1�ϕj � Φμ�ti�1�PNj�1 vj�ti�1�ψ j � Ψv�ti�1�

i � 0; 2; 4; · · ·

(8)

_X�ti�1� ��PN

j�1 _μj�ti�1�ϕj � Φ_μ�ti�1�PNj�1 _vj�ti�1�ψ j � Ψ _v�ti�1�

i � 0; 2; 4; · · ·

(9)

where Φ � �ϕ1;ϕ2; : : : ;ϕN � and Ψ � �ψ1;ψ2; : : : ;ψN � are modalmatrices in the unconstrained and constrained states, respectively.Using Eqs. (8) and (9), the transition of the system from theunconstrained state to the constrained state can be described by thefollowing relations:

ν�ti�1� � Γμ�ti�1�; _ν�ti�1� � Γ_μ�ti�1�; i � 0; 2; 4; · · ·

(10)

where Γ is the modal disparity matrix [19,20] and is given by thefollowing relation:

Γ ≜ ΨTMΦ (11)

The transition of the system from the constrained state to theunconstrained state can be similarly described by the followingrelations:

μ�ti� � ΓTν�ti�; _μ�ti� � ΓT _ν�ti�; i � 0; 2; 4; · · · (12)

The transformation matrix Γ is the identity matrix whenKr � 0 (i.e.,when no stiffness variation is introduced). WhenKr ≠ 0, Γ�i; j� ≠ 0for some values of i and j, i ≠ j. This implies that energy will betransferred from the jth mode of the unconstrained state to the ithmode of the constrained state, and vice versa. If the frequency of theith mode of the constrained state is much higher than that of the jthmode of the unconstrained state, the energy transferred from the low-frequency mode to the high-frequency mode will be quicklydissipated. This follows from our discussion in the last section, wherewe assumed modal damping with uniform damping ratio. For theprocess to be repeated, the system has to be switched back from theconstrained state to the unconstrained state. To avoid energy flowfrom the high-frequency modes in one state to the low-frequencymodes in the other state, the system should be held in each statesufficiently long time such that energy in the high-frequencymodes isdissipated. This strategy for vibration suppression is explained withthe help of Fig. 6.The success of vibration suppression using stiffness switchingwill

depend on modal disparity created by the change in stiffness. Toquantify modal disparity, we define the metric

λ ≜XNi�j�1

XN−1j�1�i − j�jΓ�i; j�j (13)

where Γ�i; j� is the �i; j�th entry of themodal disparity matrix Γ. Thismetric is a weighted sum of the projections of the low-frequencymodes in the unconstrained state onto high-frequency modes in theconstrained state, and the weights are the difference of the indices ofthe modes in the two states. This metric will be used to determinebetter location of constraints in a simple tension-aligned structure,modeled and simulated in the following sections.

III. Model of a Simple Tension-Aligned Structure

In this section we present a finite-element model of a tension-aligned structure. The tension-aligned structure, shown in Fig. 7,consists of a planar elastica arch support structure in compression anda hinged panel array in tension. The planar elastica arch is initially astraight slender rod; it is bent into its curved shape by eccentric endloads that maintain equilibrium with the tension forces in the panels.

A. Support Structure: Planar Elastica Arch

1. Nonlinear Dynamic Model

The dynamic model of the elastica arch is reproduced from thework by Perkins [21]. The elastica arch, shown in Fig. 8, is assumedto be a slender rod of length L, held in static equilibrium under thehorizontal end load f and moment fd, where d denotes the verticaleccentricity of the end load f. In a disturbed state, a point on the rodhas a displacement ofu�s; t�, where s denotes the arc length along thecenterline of the static equilibrium shape, and t denotes time. u�s; t�can be decomposed into its tangential component and normalcomponents as follows:

u�s; t� � ut�s; t�ϵt � un�s; t�ϵn

ConstrainedState

UnconstrainedState

UnconstrainedState

constraintapplication

constraintremoval

MFHMFHMFH

MFLMFLMFL

one cycle of constraint application and removal

natural dissipationdue to internal damping

Fig. 6 Vibration suppression through energy funneling from low-frequency modes (LFM) into high-frequency modes (HFM).

support structure

hinged panel array

assembled tension-sligned structure

planar elastica arch

Fig. 7 Tension-aligned structure formed by connecting a support

structure (in compression) to an array of hinged panels (in tension).

st

nθ0

s = 0s = L

Fig. 8 Planar elastica arch.

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where ϵt and ϵn are unit vectors along the tangential and normaldirections of the static equilibrium shape, shown in Fig. 8.We follow Kirchhoff’s assumptions for rod deformation [22],

which are as follows.1) The rod is linearly elastic.2) The strain is small (although rotations may be large), and the

cross-sectional dimensions of the rod are small compared to itslength.3) The cross sections remain plane, undistorted, and normal to the

axis of the rod.4) The transverse stress and rotary inertia can be neglected.

Under these assumptions, the kinetic energy and the strain energy ofthe rod can be expressed as follows:

ΠT ≜1

2

ZL

0

ρ

��∂ut∂t

�2

��∂un∂t

�2�ds (14)

ΠV ≜1

2

ZL

0

�EIk2 � EAe2� ds (15)

where ρ,E,A, and I are constants and denote themass per unit length,Young’s modulus, cross-sectional area, and area moment of inertia ofthe rod, respectively. In Eq. (15), k � k�s; t� and e � e�s; t� are thecurvature and the axial strain. The expression for k�s; t� is obtainedfrom Love [23], and that of e�s; t� is obtained from Perkins andMote[24]:

k � ks �∂∂s

�∂un∂s� ksut

�(16)

e � p

EA

� psEA� ∂ut

∂s− ksun �

1

2

��∂ut∂s

− ksun�2

��∂un∂s� ksut

�2�(17)

wherep � p�s; t� is the axial force, andps and ks are the static valuesof p and k, respectively, in the static equilibrium configuration.The work done by external forces can be defined as

Wnc ≜ f�ut cos θ0 � un sin θ0�js�0 � fd�∂un∂s� ksut

�����s�Ls�0(18)

where θ0 is the angle of inclination of the rod at s � 0, which will bedetermined later. Substituting Eqs. (16) and (17) into Eq. (15),

neglecting terms that have degree three and higher of variables ut andun and their spatial derivatives, and using Hamilton’s principle,

δ

Zt2

t1

�ΠT − ΠV �Wnc� dt � 0 (19)

we get the nondimensional equations of motion in the normal andtangential directions [21]

−∂3

∂S3

�∂Un∂S� KUt

�� ∂

∂S

�P

�∂Un∂S� KUt

��

� K�P� 1

�I

��∂Ut∂S

− KUn�−∂2K∂S2� PK � ∂2Un

∂T2(20)

K

�∂2

∂S2

�∂Un∂S�KUt

��� ∂

∂S

��P� 1

�I

��∂Ut∂S

− KUn��

− KP�∂Un∂S� KUt

�� K ∂K

∂S� ∂P

∂S� ∂2Ut

∂T2(21)

In the previous equations, the nondimensional variables are definedas follows:

S ≜s

L; D ≜

d

LUt ≜

utL; Un ≜

unL;

K ≜ ksL P ≜psL

2

EI; F ≜

fL2

EI; �I ≜

I

AL2;

T ≜t

�ρL4∕EI�1∕2(22)

Together with the equations of motion, the following boundaryconditions are obtained from Hamilton’s principle:

��∂∂S

�∂Un∂S� KUt

�� K − FD

�δ

�∂Un∂S

��S�0���

−∂∂S

�∂∂S

�∂Un∂S� KUt

�� K

�� P

�∂Un∂S� KUt

�� F sin θ0

�δUn

�S�0

���K

�∂∂S

�∂Un∂S� KUt

��� K2 �

�P� 1

�I

��∂Ut∂S

− KUn�� P − FDK � F cos θ0

�δUt

�S�0

���

−∂∂S

�∂Un∂S� KUt

�− K � FD

�δ

�∂Un∂S

��S�1���

∂∂S

�∂∂S

�∂Un∂S� KUt

��K

�− P

�∂Un∂S� KUt

��δUn

�S�1

���

−K�∂∂S

�∂Un∂S� KUt

��− K2 −

�P� 1

�I

��∂Ut∂S

− KUn�� P� FDK

�δUt

�S�1� 0 (23)

2. Static Equilibrium Configuration

The static equilibrium configuration of the elastica arch dependson the values of f and d or, alternatively, on the nondimensionalvariablesF andD. For a tension-aligned structure,F andD are designvariables; the value of F will depend on the tension desired in thehinged panel array, and the value ofDwill depend on the stiffness ofthe slender rod (elastica arch) and the difference in lengths of thehinged panel array and the slender rod. Assuming that the valuesof F and D are provided, we determine the static equilibriumconfiguration by first substituting Ut � Un � 0 in Eqs. (20) and(21). This yields the following equations

−K 0 0 � KP � 0 (24)

P 0 � KK 0 � 0 (25)

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where 0 denotes the derivative with respect to S. Substituting Ut �Un � 0 in Eq. (23), and using the geometric boundary conditions,

δUn�S � 1� � 0; δUt�S � 1� � 0;

δUn�S � 0� � tan θ0 · δUt�S � 0�

we obtain the following natural boundary conditions:

K � FD at S � 0; 1

F� P cos θ0 − K 0 sin θ0 � 0 at S � 0 (26)

A closed-form solution to Eqs. (24–26) involves elliptic integrals ofthe first kind and can be found in [21]. The solutionsK�S�,P�S�, andθ0 determine the equilibrium configuration and the prestress in thisconfiguration.

3. Linear Dynamic Model

We use the Raleigh–Ritz method [25] to obtain the linear dynamicmodel of the elastica arch about its static equilibrium configuration.To this end, we substitute the equilibrium values of P � P�S� andK � K�S� obtained from the solutions of Eqs. (24–26) into thenondimensional version of Eqs. (16) and (17) and substitute theresults in the nondimensional form of the kinetic and strain energiesin Eqs. (14) and (15). Neglecting terms that have degree three andhigher of variables Un and Ut and their spatial derivatives, we havethe following expressions for the nondimensional strain and kineticand energies:

�ΠV �1

2

Z1

0

�K2�P2 �Iz}|{terms1and2

� 2K∂∂S

�∂Un∂S�KUt

�� 2P

�∂Ut∂S

−KUn�z}|{terms3and4

��∂∂S

�∂Un∂S�KUt

��2

��P� 1

�I

��∂Ut∂S

−KUn�2

�P�∂Un∂S�KUt

�2�dS (27)

�ΠT �1

2

Z1

0

��∂2Un∂T2

�2

��∂2Ut∂T2

�2�dS (28)

Note that �ΠV and �ΠT are related to ΠV and ΠT , respectively, by thefollowing relations:

�ΠV �L

EIΠT; �ΠT �

L

EIΠT

In Eq. (27), terms 1 and 2 of the integrand are functions of S alone andnot a function of time. The same is true for terms 3 and 4 because avariation of the integral of these terms can be shown to be zero. Thefirst four terms of Eq. (27) therefore result in constant strain energy,which does not contribute to the vibration of the system. We assumeUn and Ut to be of the form

Un�S; T� � Vn�S�eiωT; Ut�S; T� � Vt�S�eiωT (29)

where Vn�S� and Vt�S� are the mode shapes. The mode shapes arediscretized as follows:

Vn�S� �Xi

Wn;i�S�Yi � Wn�S� · Y (30)

Vt�S� �Xi

Wt;i�S�Zi � Wt�S� · Z (31)

whereWn�S� andWt�S� are vectors of known shape functions. Theyare constructed using piecewise polynomials (cubic and linear,

respectively), standard in finite-element discretizations, withdiscontinuities at nodes. Y and Z are vectors of nodal degrees offreedom [see Eq. (35) next] associated with Vn and Vt. SubstitutingEqs. (29–31) into Eqs. (27) and (28), we rewrite the nondimensionalstrain and kinetic energies as follows:

�ΠV �ei2ωT

2�YT ZT �KA

�YZ

�� C (32)

�ΠT �ei2ωT

2�−ω2��YT ZT �MA

�YZ

�(33)

where

C ≜1

2

Z1

0

�K2 � P2 �I � 2K

∂∂S

�∂Un∂S� KUt

�

� 2P

�∂Ut∂S

− KUn��

dS (34)

is the constant strain energy associated with the static equilibriumconfiguration, discussed before. The mass and stiffness matricesMA

andKA of the elastica arch (support structure) are associated with thegeneralized coordinate XA:

XA � eiωThYT ..

.ZTiT

�huln; θ

lA; · · · ; u

in; θ

iA; · · · ; u

rn; θ

rA...ult ; · · · ; u

it; · · · ; u

rt

iT (35)

where un and ut are the translational DOFs; θA is the rotational DOFof each node; and l, r, and i denote the left end node, right end node,and ith node, respectively, of the elastica arch.Note that elements ofYand Z need to be consistent with the following geometric boundaryconditions:

Vn�S � 1� � 0; Vt�S � 1� � 0;

Vn�S � 0� � tan θ0 · Vt�S � 0�

The previous boundary conditions will be changed when the elasticaarch is assembled with the hinged panel array.

B. Hinged Panel Array

The array of hinged panels is shown in Fig. 9. It wasmodeled usinga standard finite-element method. We used two-dimensional two-node frame elements with three DOFs at each node: two translationaland one rotational DOF. A geometric stiffness matrix¶ was added tothe standard frame stiffness matrix to model the effect of tension f. Ahinge between two panels is treated as a node in the finite-elementmodel. The left and right elements of the hinge node haveindependent rotations but have common translations. The DOFs ofthe hinged panel array are denoted by

XP��xl;yl;θlP; · · · ; xk;yk;θkP|{z}

nodekon panel array

; xk�1;yk�1;θk�1P|{z}node�k�1�on panel array

· · · ;xr;yr;θrP

�T

(36)

where x and y are the translational DOFs; θP is the rotational DOF ofeach node; and l, r, and k denote the left end node, right end node,and kth node, respectively, of the hinged panel array. For the

Fig. 9 Array of hinged panels.

¶This matrix is also referred to as the “initial stress matrix” [26].

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generalized coordinates XP, the mass and stiffness matrices areassembled as MP and KP.

C. Assembly of Elastica Arch with Hinged Panel Array

The elastica arch is assembled with the hinged panel array byconnecting their ends together using pin joints. After assembly, theend nodes of the two substructures share translations in the plane butmaintain independent rotationalDOFs. TheDOFs of the end nodes ofthe elastica arch and hinged panel array, shown in Fig. 10, satisfy therelations �

xl

yl

���cos θ0 sin θ0sin θ0 − cos θ0

��ultuln

�;�

xr

yr

���

cos θ0 − sin θ0− sin θ0 − cos θ0

��urturn

� (37)

where θ0 is the angle of inclination of the elastica arch at its leftboundary** and is obtained from the solution of the equilibriumconfiguration. The mass and stiffness matrices of the elastica arch(MA,KA) and the hinged panel array (MP,KP) are assembled into thetotal mass matrix and the total stiffness matrix; we denote them asMandK0 to be consistent with our discussion in Sec. II.B. The numberof DOFs of the assembled structure is four less than the sum of theDOFs of the two substructures. To simulate a free-floating spacestructure, boundary conditions are not applied to the assembledstructure. This gives rise to three (two translational and onerotational) rigid-body modes of the system; these rigid body modesare associated with the three zero eigenvalues of K0.

D. Methods of Stiffness Variation

Stiffness variation described by Eq. (6) is realized in the assembledtension-aligned structure by two methods. These two methods aredepicted in Fig. 11 and are described next.Method A: The rotations of two adjacent panels at their common

hinge, θkP and θk�1P , are connected by a rotational spring of time-varying stiffness. The stiffness of the rotational spring is changedfrom its low value to its high value at times tn such thatθk�1P �tn� � θkP�tn�.Method B: Node i on the elastica arch and node j on the panel array

are connected by a translational spring of time-varying stiffness. Thestiffness of the translational spring is changed from its low value to itshigh value at times tn such that uit�tn� � uin�tn� � xj�tn� � yj�tn�.For both cases A and B, the stiffness is changed from its high value

to its low value at times tn�1, where �tn�1 − tn� � c. Here, c is aconstant that will be chosen a priori such that a large fraction of theenergy transferred to the high-frequency modes is dissipated.Method A can be implemented by placing an electromagnetic

brake at the hinge of the adjacent panels. Turning on the brake will

prevent relative rotation between the adjacent panels and will beequivalent to constraining the degrees of freedom θkP and θk�1P by arotational spring of very high stiffness. Turning off the brake willrelease the degrees of freedom and will be equivalent to setting thespring stiffness to zero.MethodB can be implemented by connectingand disconnecting an elastic bar between a point on the arch and apoint on the panel. These two points will be chosen to coincide withnodes of the finite-element model for the purpose of simulation.The stiffness of the tension-aligned structure can be varied using

multiple springs of the type described in method A and/or method B.Because each of these springs can be in one of two states, the tension-aligned structure will have multiple stiffness states. In the nextsection, where we present simulation results, the stiffness of thestructurewill be switched cyclically between the lowest stiffness stateand the highest stiffness state via intermediate stiffness states. Thelowest and highest stiffness states are defined as the states with thelowest and the highest fundamental frequency.

IV. Numerical Simulation

A. Description of Tension-Aligned Structure

The material and geometric properties of the tension-alignedstructure are provided in Table 2. The structure is made of aluminumand the damping ratio of all modes is assumed to be ζ � 0.001. Thepanel array is comprised of eight panels of dimensions Lp × b × h;these dimensions are shown in Fig. 12. Each panel is modeled using10 beam elements. The support structure (elastica arch) is initially astraight rod of radius 0.04m and length≈8.00 m. It is modeled using80 elements. The eccentricity of the load applied to the supportstructure is 0.008 m. In the next two subsections, we simulate thebehavior of the structure with and without control, for low and hightension; the low and high tension values are provided in Table 2.

B. Vibration Suppression: Low-Tension Case

Here the tension in the hinged panel array was assumed to be1000 N. This is less than 5% of the buckling load of the straight rodwith free–free boundary conditions, which is≈21.4 kN.We simulatethe behavior of the structure for an initial condition where the secondjoint of the hinged panel array (see Fig. 12) was displaced verticallyby 0.01 m and released. The first 25 modes of the structure weresimulated; these do not include the rigid-body modes. The energy ofthe tension-aligned structure is shown in Fig. 13 for three differentcases, as described next.1) Unconstrained (uncontrolled) structure undergoes free

vibration.2) Stiffness of the structure is switched using method A. The high-

stiffness rotational springs in joints J1, J3, J4, and J6 are activatedsequentially when their adjacent panels are aligned and their stiffnessare then set to zero simultaneously. This is illustrated with the help ofFig. 14. The process is repeated 92 times in the simulation periodof 180 s.3) Stiffness of the structure is switched using methods A and B.

The high-stiffness rotational springs in joints J1, J3, and J4 areactivated sequentially when their adjacent panels are aligned. This isfollowed by connecting an elastic bar (high-stiffness translationalspring) between a point on the elastica arch and a point on the panelarray (see Fig. 12) in the manner described in Sec. III.D. Thestiffnesses of all four springs are then set to zero simultaneously. Theprocess is repeated 105 times within the simulation period of 180 s.It is clear from Fig. 13 that the energy of the uncontrolled structure

decays slowly compared to the structure with switched stiffness.After 180 s, the uncontrolled structure has ≈22.5% of its initialenergy left; in contrast, the structure with switched stiffness usingmethod A has ≈0.4% of its initial energy left. For vibrationsuppression to 0.4% energy level, methods A and B combinedrequires 76 s as compared to 180 s required bymethod A. For the twocases with switched stiffness, the joints are released simultaneously,not sequentially, to reduce the time required for each cycle ofconstraint application and removal. In simulations, where high-stiffness springs are used to constrain the joints, simultaneous releaseof the joints causes residual energy stored in the springs to vanish. A

hinged panel array

elastica arch

Fig. 10 Relationship betweenDOFs of the end nodes of the elastica archand hinged panel array.

**Because of symmetry, the angle of inclination of the elastica arch at itsright boundary is −θ0.

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bookkeeping of this energy over all cycles indicates that it does notexceed 0.1% of the total energy dissipated for the two cases withswitched stiffness. This small fraction can be attributed to the choiceof large magnitudes of the spring stiffnesses and is further explainedin the Appendix using the example of Sec. II.A. In practicalimplementation [20], where electromagnetic brakes may be used toconstrain the joints, release of the brakes will not result in direct lossof energy (because brakes can be viewed as rotational springs of

infinite stiffness) but facilitate energy transfer to the high-frequencymodes where they will be dissipated quickly.Although methods A and B combined shows better performance

than method A alone, it will involve higher cost and complexity. Thechoice between method A and methods A and B combined willtherefore depend on cost, complexity, and performance consid-erations. There may be cases where deployment considerations andaccuracy requirements will necessitate a higher level of tension. Ahigher level of tension in the hinged panel array will increase thenatural frequencies of the low-frequency modes of the structure; thiswill increase internal damping and reduce the effectiveness ofmethod A. In the next section, we simulate the behavior of thestructure with higher tension and demonstrate that combiningmethods A and B is necessary for the effectiveness of vibrationsuppression.

C. Vibration Suppression: High-Tension Case

Here, the tension in the hinged panel array was assumed to be10,000N,which is slightly lower than 50%of the buckling load of thestraight rod with free–free boundary conditions. Using 25modes, thebehavior of the structure was simulated for the same initial conditionthat was used in Sec. IV.B. The energy of the tension-alignedstructure is shown in Fig. 15 for the following three cases.1) Unconstrained (uncontrolled) structure undergoes free

vibration.2) Stiffness of the structure is switched using method A. The high-

stiffness rotational springs in joints J1, J3, J4 and J6 are activatedsequentially when their adjacent panels are aligned and their stiffnessare then set to zero simultaneously. The process is repeated 27 timeswithin the simulation period of 50 s.3) Stiffness of the structure is switched using methods A and B.

The high-stiffness rotational springs in joints J1, J3, and J4 areactivated sequentially when their adjacent panels are aligned. This isfollowed by connecting an elastic bar (high-stiffness translational

rotational springat hinge

node k

hinged panel array

elastica arch

node i

node j

a) b)

Fig. 11 Stiffness variation in the tension-aligned structure is realized using methods A and B; these are described in Sec. III.D.

Table 2 Properties of simulated tension-aligned structure

Material Aluminum

Young’s modulus E 69 × 109 PaDensity �ρ 2700 kg∕m3

Damping ratio ζ 0.001Panel number 8Panel length Lp 1.000 mPanel area b × h 0.500 × 0.015 mRadius of support rod r 0.040 mLength of support rod L 8.000 m (approximate)Eccentricity of connection d 0.008 mLow-tension f � flow 1000 NHigh-tension f � fhigh 10,000 N

side view

Fig. 12 Eight-panel tension-aligned structure used in simulations, bothfor low tension and high tension.

0 20 60 100 140 1800.000

0.020

0.010

Fig. 13 Plot of energy for the three cases discussed in Sec. IV.B.

Fig. 14 Sequence for activating and releasing the joints of the structure in Fig. 12; the dark blocks represent the activated joints.

0 10 20 30 40 500.0

0.1

0.3

0.2

Fig. 15 Plot of energy for the three cases discussed in Sec. IV.C.

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spring) between a point on the elastica arch and a point on the panelarray (see Fig. 12) in a manner such that no energy is added to thestructure. The stiffnesses of all four springs are then set to zerosimultaneously. The process is repeated 27 times within thesimulation period of 50 s.It is clear from Fig. 15 that the structure with switched stiffness

using method A has marginally better damping characteristics thanthe freely vibrating unconstrained structure. After 50 s, the structurewith switched stiffness has ≈21% of its energy left; in contrast, theuncontrolled freely vibrating structure has ≈28% of its energy left.The effectiveness of stiffness variation using method A is notsignificant due to the high tension in the hinged panel array, whichenhances internal damping by increasing the natural frequencies ofthe low-frequency modes. This can be verified from Table 3, whichshows the first six natural frequencies of the unconstrained structurefor the low-tension and high-tension cases.The effectiveness of vibration suppression using stiffness variation

can be regained by combining methods A and B. This can be verifiedfrom the energy plot in Fig. 15, which shows a residual energy of≈0.3% after 50 s, and Fig. 16, which shows the transversedisplacements of three points on the hinged panel array. Thisimprovement in effectiveness can be understood by examining themodal disparitymatrices for the two cases and comparing their modaldisparity indices. The modal disparity matrices for method A andmethods A and B combined, denoted as ΓA and ΓAB, are given next:

ΓA �

266666666666664

0.9995 −0.0226 0.0070 −0.0081 −0.0113 0.0047 · · ·

−0.0179 −0.9559 −0.2629 0.0859 0.0889 0.0059 · · ·

−0.0107 −0.2475 0.9559 0.1295 0.0629 −0.0277 · · ·

0.0044 0.0576 −0.0653 0.8549 −0.4146 0.2648 · · ·

0.0183 0.1399 −0.0854 0.4520 0.8437 −0.2084 · · ·

−0.0067 −0.0020 0.0174 −0.0370 0.0762 0.4616 · · ·

..

. ... ..

. ... ..

. ... . .

.

377777777777775

ΓAB �

266666666666664

0.4455 0.6268 −0.0810 −0.0543 −0.0209 −0.0215 · · ·

0.0258 −0.1794 −0.9812 −0.0065 −0.0204 0.0045 · · ·

−0.0060 −0.0758 0.0152 −0.9225 0.3340 −0.1463 · · ·

−0.1321 0.3701 −0.0631 −0.3370 −0.7316 0.2783 · · ·

−0.0418 0.0957 −0.0032 −0.0341 −0.1170 −0.4564 · · ·

−0.0175 0.0562 0.0042 −0.0212 0.1783 0.5359 · · ·

..

. ... ..

. ... ..

. ... . .

.

377777777777775

It can be seen that ΓA is predominantly diagonal, but ΓAB hasrelatively large off-diagonal entries. The modal disparity indices forthese two cases were computed as

λA � 22.21; λAB � 40.88 (38)

Because λAB is greater than λA, stiffness variation combiningmethodsA and B is more effective than method A in transferring energy fromthe low-frequency modes to the high-frequency modes.

Finally, consider the two constrained systems where 1) method Ais used to keep the rotational springs in joints J1, J3, J4, and J6 in theirhigh-stiffness state at all times, and 2)methodsA andB are combinedto keep the rotational springs in joints J1, J3, and J4 and thetranslational spring (elastic bar) in their high-stiffness state atall times.The first six natural frequencies of these two constrained systems

are shown in Table 4. It is clear from the entries in this table that thenatural frequencies of the two systems are comparable. This impliesthat the success of the control strategy combiningmethodsA and B isdue not to elevated natural frequencies of the system but to its ability

to transfer energy from the low-frequency modes to the high-frequency modes as measured by the modal disparity index.

V. Conclusions

Using linearized models, we demonstrate that it is possible totransfer energy from low- to high-frequency modes of a structurethrough stiffness variation. Furthermore, simulations show that thisenergy transfer together with internal damping is the mechanismresponsible for efficient vibration suppression. The results are

Table 3 Natural frequencies of the unconstrained tension-alignedstructure in radians per second

ω1 ω2 ω3 ω4 ω5 ω6

Low-tension case 3.820 6.999 9.129 13.144 15.859 20.213High-tension case 10.727 20.533 21.888 35.327 40.381 54.301

−0.005

0.000

0.005

0 10 20 30 40 50

−0.005

0.000

0.005

−0.005

0.000

0.005

Fig. 16 Plots of y1, y4, and y7 (see Fig. 12): the transverse displacementsof three points on the hinged panel array.

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especially promising for large tension-aligned structures, whoseperformance is susceptible to low-frequency disturbances becausethe low-frequencymodes of these structures are very lightly damped.Themodel derived in thiswork, which consists of an elastica arch anda hinged panel array attached to it, supports these findings. Themodelis simple, yet it incorporates essential features of tension-alignedstructures such as change in curvature of the support structure andvariation of natural frequencies of the combined structure due tochange of tension. Two methods for stiffness variation areinvestigated using our models: through locking and unlocking ofhinges in the panels, and through connecting and disconnecting anelastic bar between the arch and the panels. These methods can berealized in actual tension-aligned structures using relatively simpleimplementations. The overall results show that sequential switchingof system stiffness can be an operationally useful mechanism basedonwhich an effective strategy for vibration control of tension-alignedstructures can be designed for high-accuracy space applications. Ourfuture work will focus on optimal locations for placement of thestiffness variation mechanisms (hinges and connecting bars) as wellas optimal sequences and timing for varying the stiffness. The modaldisparity index introduced in this paper is likely to be useful insolving these optimization and optimal control problems.

Appendix: Energy Loss Directly Due to StiffnessSwitching

For the switched system in Sec. II.A, kr was chosen to be2 × 104 N∕m. To illustrate the role of the magnitude of kr in the rateof energy decay, we plot the total energy of the system and the energystored in the spring k3 for kr � f2 × 103; 2 × 104; 2 × 105g N∕mover the time interval t � �t1; t1 � 10� s. These plots, shown inFig. A1, indicate that higher values of kr lead to faster dissipation of

the total energy, as expected. More importantly, higher values of krresult in faster decay of the energy stored in the spring k3 itself. Thisimplies that switching k3 from kr to 0will not be accompanied by anyappreciable energy loss directly due to the action of switching if krhas a large magnitude and if the system spends some reasonable timein the constrained state. In the context of the stiffness variationmethod discussed in Sec. III.D, this implies that the value of c can besmall when the value of kr is large and yet the energy loss directly dueto stiffness reduction will be negligible.

Acknowledgment

The authors gratefully acknowledge the support provided by theU.S. Air Force Office of Scientific Research, grant FA9550-10-1-0500, for this work.

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Precision Deployable Space Structures Applied to an X-Band Phased

0.0

0.2

0.4

0.0

0.2

0.4

d)

e)

a)

b)

c) f)

0.0

0.2

0.4

11 15 19 11 15 190.0

0.2

0.4

0.0

0.2

0.4

0.0

0.2

0.4

11 15 19 11 15 19

11 15 19 11 15 19

Fig. A1 Plot of the total energy and the energy stored in the spring k3 of the switched system in Sec. II.A. Figures A1a, A1d correspond tokr � 2 × 103 N∕m; Figs. A1b, A1e correspond to kr � 2 × 104 N∕m; and Figs. A1c, A1f correspond to kr � 2 × 105 N∕m.

Table 4 Natural frequencies of the constrained tension-alignedstructure in radians per second

�ω1 �ω2 �ω3 �ω4 �ω5 �ω6

Method A 10.829 22.535 23.085 39.568 51.604 75.956Methods A and B 15.352 22.274 37.528 42.196 64.514 75.827

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Control, Vol. 131, No. 2, 2009, Paper 021011.doi:10.1115/1.3023126

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Multidiscilinary Optimization, Vol. 31, No. 1, 2006, pp. 1–7.doi:10.1007/s00158-005-0578-1

[20] Issa, J., Mukherjee, R., Diaz, A. R., and Shaw, S. W., “Modal Disparityand Its Experimental Verification,” Journal of Sound and Vibration,Vol. 311, Nos. 3–5, 2008, pp. 1465–1475.doi:10.1016/j.jsv.2007.10.038

[21] Perkins, N. C., “Planar Vibration of an Elastica Arch: Theory andExperiment,” ASME Journal of Vibration and Acoustics, Vol. 112,No. 3, 1990, pp. 374–379.doi:10.1115/1.2930518

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N. WereleyAssociate Editor

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