Experimental and numerical studies on wrinkling control of an inflated beam using SMA wires

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Smart Mater. Struct. 19 (2010) 105019 (9pp) doi:10.1088/0964-1726/19/10/105019

Experimental and numerical studies onwrinkling control of an inflated beamusing SMA wiresC G Wang and H F Tan

Center for Composite Materials, Harbin Institute of Technology, Harbin 150080,People’s Republic of China

E-mail: wangcg@hit.edu.cn

Received 12 June 2010, in final form 21 July 2010Published 18 August 2010Online at stacks.iop.org/SMS/19/105019

AbstractA novel control strategy is proposed to control the wrinkling deflection and improve thebending performance of an inflated beam using shape memory alloy (SMA) wires. In thisstrategy, seven SMA wires are subtly attached to the stretched side which is opposite to thewrinkled/compressed region of the inflated beam. These electrically driven SMA wires generatethe recovery force to remove the wrinkles and improve the bending performance of the inflatedbeam. An equivalent nodal force (ENF) method incorporating direct perturb-force (DP)technology is proposed to simulate the wrinkling control of the inflated beam. Aphotogrammetric method is used to measure the wrinkling and control results, which agree wellwith the numerical predictions. The results obtained in this paper are good references to thewrinkling control and the applications of smart materials and structures in space membranestructures.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Membrane inflated booms have been widely used in variousinflatable space structures, such as inflatable antenna reflectors,truss structures, solar sails, and inflatable wings etc (Jenkins2001, Norris and Pulliam 2009). In these applications, themembrane inflated booms need to meet the requirements ofhigh axial-direction precision, load-carrying, and remainingwrinkle-free (Wang et al 2010). However, membrane inflatedbooms are very easily wrinkled and can even be collapsed bythe wrinkles. Inflated booms subject to bending will developshort wavelength periodic ripples on the compressed side, andthe inflated booms buckle locally and collapse soon after theoccurrence of the wrinkles (Wang et al 2009a). Therefore,controlling the wrinkling of the inflated booms is one ofthe most important issues concerning high-precision inflatablestructures.

Predicting the wrinkling of the inflated beams duringbending may be divided into problems in which the wallmaterial is regarded as either a true membrane or a thin-shell,

which results in two models. One of these two models maybe called the ‘membrane model’ (Comer and Levy 1963, Mainet al 1994, Wang et al 2008). The other is named the ‘thin-shellmodel’ (Wielgosz and Thomas 2002, Thomas and Wielgosz2004, Veldman et al 2006). The distinction between these twomodels is whether the bending and compression stiffnesses ofthe wall material are considered or not. For the ‘membranemodel’, a membrane with zero bending stiffness cannot resistany compression loads or bending moments. For the ‘thin-shellmodel’, the wall material is considered as a thin-shell with asmall but non-zero bending stiffness.

At present, a lot of film and wire type smart materials, suchas macrofiber composites, polyvinylidene fluoride (PVDF) andshape memory alloy (SMA) films or wires, are used to controlthe shape and vibration of space membrane structures andimprove the performance of inflatable structures. Fang et al(2007) used PVDF devices to control the shape of a largemembrane reflector, and DeSmidt et al (2006) explored thefeasibility of utilizing an active gore/seam cable based controlsystem to reduce global RMS (root mean square) figure errors

0964-1726/10/105019+09$30.00 © 2010 IOP Publishing Ltd Printed in the UK & the USA1

Smart Mater. Struct. 19 (2010) 105019 C G Wang and H F Tan

due to thermal loading and inflation effects in large gossamerinflatable membrane reflectors. Lee et al (2006) introduced theconcept of the configuration control of an inflated cylindricalbeam by using SMA wire actuators with appropriate heatingand cooling cycles. Peng et al (2006) developed a geneticalgorithm based control system for active shape control of aninflatable structure using a SMA wire actuator attached to theedge of the membrane surface structure.

However, only a few researchers have paid attention tothe wrinkling controls of the inflated beams. A typical casedone by Yoo et al (2007) is the study on the wrinklingcontrol of inflated booms using SMA wires. They developeda methodology to control wrinkling growth and the deformedconfiguration of the inflated boom structure with a SMA wireactuator. The SMA wires are attached to the compressedside of the inflated boom and generate a recovery force toremove wrinkling and restore the deformation of the boom.Through regulating the control force of the SMA wires, thewrinkling region can be reduced, and the wrinkling generationcan also be delayed. The wrinkling control is numericallymodeled using ABAQUS finite element programs with awrinkling algorithm developed based on the membrane theory.A bending–wrinkling test is performed to verify the numericalresults.

In order to accurately predict and control the wrinklingbehaviors of the inflated beams during bending, the detailedwrinkling characteristics, including wrinkling wavelength,wrinkling number and wrinkling region, need to be predictedfirst. A reasonable and effective control methodologyneeds to be developed to control the wrinkling deformations.Differently from the work of Yoo et al (2007), a new strategyin this paper is proposed to remove and control the wrinklesusing SMA wires attached to the edge of the stretched side ofthe inflated beam.

To control wrinkling of the inflated beam during bending,the key is to search for a smart material which can generatea high driving force and a small deflection. In addition, forits application in membrane structures, a lightweight smartmaterial is also necessary. The SMA wire is electrically drivento generate high recovery forces; its deflection is small. It isalso easy to attach the SMA wires to the membrane surfaceby using adhesive tape. Several SMA wires can generate veryhigh driving forces, thus there is no need for high densitydistributions. Therefore they can control the wrinkles by usinga few weights. The SMA wires are good electric materialswhich may be easily driven by introducing current. In addition,SMA wires have achieved good effects on the shape andwrinkling controls of membrane structures (Lee et al 2006,Peng et al 2006, Yoo et al 2007). Based on the aboveconsiderations, the SMA wires are used to control the wrinklesof the inflated beam during bending.

In addition, the detailed wrinkling deformations arepredicted by using direct perturb-force (DP) simulationtechnology (Wang et al 2009b) incorporating the ANSYSnonlinear thin-shell post-buckling program. A wrinklingcontrol test of the inflated beam during bending using SMAwires is used to verify the numerical results.

2. Wrinkling prediction and control simulations

The wrinkling is simulated by using a direct perturb-force(DP) technology which is based on the modified displacementcomponent (MDC) method (Wang et al 2009b). Thismethod is better than previously used methods. Comparedwith membrane models, the MDC method is based on thebifurcation theory, which may predict the detailed wrinklinginformation, such as wrinkling wavelength and number.Compared with thin-shell models, the MDC method has twocharacteristics. One is that the initial imperfections areintroduced by using the quantitive disturbing forces. Theother is that the initial imperfections are removed after thefirst wrinkle occurs. In previous methods, the imperfectionswere introduced by using disturbing displacements. Theseimperfections were not removed, which influenced the post-wrinkling behaviors. In some other cases, although theimperfections are introduced by using random disturbingforces, they are removed only in the last load step, which alsohas effects on the post-wrinkling behaviors (Wang et al 2009b).

In the MDC method, the accurate introduction and thetimely removal of the initial imperfections are two key stepswhich are introduced into the ANSYS nonlinear thin-shellpost-buckling program based on DP technology. The basis ofthe DP technology is to apply some small out-of-plane forceson the membrane surface to induce the imperfections, andfurther to induce the wrinkle. After that, these imperfectionsare removed from the model by deleting those out-of-planeforces. Two key problems should be taken into consideration.Firstly, the imperfections generated by the out-of-plane forcesare quantitative, but not randomly distributed. Secondly, theequal numbers of positive and negative out-of-plane forcesshould be applied on the membrane surface so that the net out-of-plane forces remain equal to zero, which meets the forceequilibrium condition and mainly initiates the analysis intothe post-wrinkling phase. The locations of these out-of-planeforces are based on the pattern of the critical wrinkling mode.The DP method is thus restricted to wrinkling modes with aneven number of wrinkle waves, so that they meet the momentequilibrium as well.

In addition, several effective strategies are used toimprove the convergence, including the stress stiffening effect,the displacement control technique, the stabilized Newton–Raphson iteration, and the dichotomy method. The flowchartof the wrinkling simulation is shown in figure 1.

Three thousand shell elements (ANSYS SHELL181) areused to model the inflated beam. The SHELL181 elementis suitable for thin to moderately thick shell structures; it isalso well-suited for nonlinear, large rotation, and/or large strainnonlinear applications. The thin-shell element can simulatemembrane and bending effects simultaneously after inputtingthe real constant of material thickness. The FE (finite element)model of the inflated beam is shown in figure 2.

In order to consider the recovery force generated bythe SMA wires that could remove wrinkling of the inflatedbeam, the equivalent nodal force (ENF) method is used inthe numerical analysis of the wrinkling control effect. Thedriving forces of SMA wires are simulated by introducing

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Smart Mater. Struct. 19 (2010) 105019 C G Wang and H F Tan

Figure 1. Flowchart of wrinkling simulation using DP technology.

Figure 2. FE model of inflated beam.

some equivalent nodal forces to the nodes of the thin-shellelements, as shown in figure 3. The SMA wires will generatethe same deflection when they are electrically driven. Thus thedriving force per unit length of SMA wire is equal. Accordingto this effect, the SMA wire is considered as many springs inseries. The total driving force is divided into many nodal forceswhich are seeded equivalently on the specified nodes of thethin-shell elements.

In the simulation, seven SMA wires are attached to theedge of the stretched side which is opposite to the wrinklingregion of the inflated beam (figure 2). When a single SMA wire

is pre-stretched by 4% in the electrically driven performanceexperiment, the total generated recovery force is 30 N. Therecovery force is then divided into thirty parts, and is loadedon thirty nodes of the thin-shell element continuously inthe opposite area of wrinkling region (figure 2). Thus theequivalent nodal force is 1 N. The flowchart of the wrinklingcontrol simulation using the ENF method is shown in figure 4.

The inflated beam is made of a cylindrical membrane; itslength is 0.5 m, its radius is 0.05 m and its wall thickness is0.03 mm. The material modulus and Poisson’s ratio are 3 GPaand 0.34, respectively. The beam is inflated to 10 kPa internal

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Smart Mater. Struct. 19 (2010) 105019 C G Wang and H F Tan

k k k k k k

F

F = i-kΔR

i-1

Figure 3. Sketch of the equivalent nodal forces.

Figure 4. Flowchart of wrinkling control simulation using ENF method.

pressure; it is also subject to a 7 N tip load. Based on theENF method, the results before and after wrinkling control areshown and compared in figure 5.

It can be observed from figures 5(a) and (b), that there are37 wrinkles before wrinkling control and that they are reducedto 23 after wrinkling control. Here the obvious yellow tip inthe figures is measured as one wrinkle. Both the axial andhoop wrinkling evolutions are effectively controlled using theSMA wires. After wrinkling control (as shown in figure 5(b)),the value of the minimum principal stress is reduced by 50%,which reveals the decreased wrinkling degree. In addition,the wrinkling region is greatly reduced from an initially large‘triangle’ to a final small ‘corn’ shape, which is interpreted infigure 6.

For an inflated beam during bending, the maximummoment occurs at the fixed edge and the minimum momentoccurs at the free edge. The wrinkles will be formed when thestress in the compressed side reaches the buckling stress of theinflated beam, which corresponds to a negative compressivestress, just as shown in the stress distribution of cross-sectionI–I. For the fixed edge, the maximum moment is obtainedwhich corresponds to the maximum compressive stress, that isto say, the maximum wrinkling region may be observed alongthe hoop direction of the cross-section of the inflated beam(cross-section A–A). When the compressive stress reaches zeroin the compressed side, the wrinkles will disappear (cross-section B–B). From cross-section A–A to cross-section B–B,both the compressive stress and the wrinkling region along the

hoop direction of the inflated beam decrease as the momentdecreases. From the view of X O Z , the wrinkling regionis a ‘triangular’ shape. While, from the view of Y O Z , thewrinkling region is a ‘corn’ shape.

In addition, as observed in figure 5, the reduction inwrinkling is not uniform; the wrinkling shape is also differentbefore and after wrinkling control. This relates to the controlmechanism and the deformed characteristics of the flexiblemembrane structures. On the stretched side, the SMA wiresare separately distributed and electrically driven to generaterecovery forces which improve the bending performance ofthe inflated beam. These recovery forces are transferredthrough the membrane surface to the compressed side to reducethe wrinkles indirectly. Here there is a nonlinear deformedcharacteristic in the membrane structures, which leads to auniform reduction in wrinkling and a difference in shape. Inaddition, the SMA wires are attached to the membrane surfaceby using adhesive tape which is simulated using the ENFmethod in the simulation. This difference is also a possiblereason for the different shape and the uniform wrinklingreduction.

The edge wrinkling patterns before and after wrinklingcontrol are also obtained and shown in figure 7. Figure 7indicates that the wrinkling pattern before control is afluctuant/unsmoothed curve with many wrinkling waves. Thewrinkling region reaches a length of 0.14 m along the axialdirection of the inflated beam. After wrinkling control, thewrinkles almost disappear except in the range of the inner

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Smart Mater. Struct. 19 (2010) 105019 C G Wang and H F Tan

Figure 5. Results (a) before and (b) after wrinkling control.

0.02 m of the length near the fixed edge of the inflated beam. Inthis region (less than 0.02 m), five large wrinkles are reducedto two small wrinkles by using wrinkling control. In addition,the structural performance is greatly improved after wrinklingcontrol. In the range of 0.16 m of axial length, the tip deflectionof the inflated beam is 2.8 mm which is greatly reduced to0.9 mm by using wrinkling control. The total tip deflectionof the inflated beam is shown in figure 8.

It can be seen in figure 8 that the tip deflection increaseslinearly as the tip load increases before wrinkling control.When the tip load is 7 N, the SMA wires are electricallydriven to generate recovery forces. These recovery forcesamount to an improvement in the bending performance of theinflated beam. When the tip load increases beyond 7 N, the tipdeflection of the inflated beam will decrease dramatically. Therecovery of the tip deflection after wrinkling control reaches40.7% of that before control, indicating the validity of thewrinkling control.

The detailed and quantitive comparisons of the resultsbefore and after wrinkling control are listed in table 1.

In table 1, α = |P−−P+|P−

× 100% is the wrinkling controlparameter which reflects the wrinkling control ability usingSMA wires, in which P− and P+ denote before and afterwrinkling/deformed parameters, respectively. The wrinklingnumber is defined as the number of the wrinkling peaks. Thewrinkling wavelength is defined as the distance between two

Table 1. Comparisons of results before and after wrinkling control.

ItemsWrinklingnumber

Wrinklingwavelength(mm)

Wrinklingarea (m2)

Tipdeflection(m)

Beforecontrol, P−

37 4.297 0.012 48 0.0113

Aftercontrol, P+

23 4.714 0.006 85 0.0067

α (%) 37.8 9.7 45.1 40.7

adjacent wrinkling peaks. Based on the results in table 1,the control parameters of the wrinkling number and area are37.8% and 45.1%, which indicate the wrinkling number andarea can be effectively controlled using SMA wires. Forthese two wrinkling parameters, the control ability using SMAwires reaches almost 50%. For the control of the wrinklingwavelength, the control using SMA wires is restricted to smallwrinkles occurring in the inflated beam with limited structuralsize and loading conditions. Thus, the control effect of thewrinkling wavelength is not obvious which corresponds to α =9.7%. For the whole structure, the control parameter of the tipdeflection of the inflated beam reaches 40.7%, which indicatesthat control using the SMA wires is also a very effective wayto improve the structural bending performance.

3. Wrinkling control experiment

A wrinkling experiment based on photogrammetric measure-ment is performed to verify the above numerical results. Pho-togrammetry is a three-dimensional measurement techniquebased on the analysis of two-dimensional photographs and theprinciples of triangulation. It is a fully non-contact methodand can be used to obtain detailed data on the wrinkling shape(Wang et al 2007).

There are five major steps in photogrammetric measure-ment: (i) camera calibration; (ii) determination of target;(iii) camera arrangement; (iv) taking photos and image pro-cessing; and (v) post-processing. The camera calibration isused to recompute the physical parameters in camera. Toobtain accurate test results for small-amplitude wrinkles in asmall-scale structure, a printed-dot target (Wang et al 2007) isused to capture the wrinkles. Such targets have merits and canbe printed onto the membrane (as shown in figure 9). Accord-ing to the requirements of the test, 7500 target dots are uni-formly arranged in the 0.16 m ×0.2 m wrinkled region and thedistance between two adjacent cross dots is 2 mm. The diame-ter of each target dot is 0.5 mm. In order to capture the imageof the detailed wrinkling configuration, a camera is arrangedto take pictures form six different shooting angles (60◦ angu-lar interval). These pictures are then analyzed using image-processing software. The triangulation relationship, using col-inearity between different three-dimensional coordinates in ob-ject space and the two-dimensional image coordinates, is fun-damental to the photogrammetry image-processing technique.

The devices for the wrinkling test are shown in figure 9,and the wrinkling results are shown in figure 10.

Some small wrinkles can obviously be observed infigure 10(a), which corresponds to the regions with unmarked

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Smart Mater. Struct. 19 (2010) 105019 C G Wang and H F Tan

Figure 6. Moment and stress distributions of inflated beam in bending.

Figure 7. Edge wrinkling pattern of the inflated beam.

white wrinkled regions of figure 10(b). For the wrinkles nearthe fixed edge of the inflated beam, they are compressed andin contact with each other which shields the target dots markedon these wrinkles. The wrinkles in figure 10(a) are thus relatedto the white unmarked regions in figure 10(b). In addition,the sinuate dotted curves and surface in figure 10(b) are also

Figure 8. Tip load versus tip deflection of the inflated beam.

evidence of the wrinkling ripples. The wrinkling number, thewrinkling wavelength, the wrinkling area and the structural tipdeflection can be obtained based on these test results.

In the wrinkling control test, the inflated beam has thesame material and structural size as those described in thenumerical simulation. Seven NiTi (49.5% Ni) SMA wires areequidistantly attached to the stretched side which is oppositeto the wrinkling region of the inflated beam. The length ofa single SMA wire is 0.3 m, and the diameter of each SMAwire is 0.5 mm. The distance between two adjacent SMAwires is 0.01 m. Seven SMA wires are arranged as an ‘S’shape, shown in figure 11. This ‘S’ shape connection is aconnection in series, which will generate a uniform current,

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Smart Mater. Struct. 19 (2010) 105019 C G Wang and H F Tan

Figure 9. Wrinkling test devices.

and further generate a uniform recovery force. This ‘S’ shapeis an optimized connection which may be electrically drivento generate a uniform recovery force. During the test, theSMA wires in the ‘S’ shape series connection are observed togenerate a simultaneous and uniform contraction.

The driving force of a single SMA wire is obtained bya pre-tensile test and an electrically driven test based onthe shape memory effect of shape memory materials. Thepre-tensile tests are completed on an INSTRON5500 tensiontesting machine. The named ‘pre-tensile’ is a tension test ofthe SMA wires before they are electrically driven. SMA wiresgenerate recovery forces only if they are pre-tensile. SMAwires with different pre-stretches generate different recoveryforces. Four different pre-stretches 1%, 2%, 3% and 4% aremade separately to the SMA wires and the calibrated lengthis 0.165 m. The largest driving forces of SMA wires underdifferent pre-stretches are obtained and shown in table 2. Forall cases, the largest driving force is 30 N which correspondsto the 4% pre-stretch. This driving force value is then selectedin the wrinkling control test (it is same as the value of therecovery force in the simulation), and it was driven by a 2 Adirect current plus a 2 A pulse.

Based on the wrinkling photogrammetric measurement,the wrinkling and tip deflection of the inflated beam before and

Table 2. The driving force of SMA wires.

Drivingcurrent (A)

1% pre-stretch (N)

2% pre-stretch (N)

3% pre-stretch (N)

4% pre-stretch (N)

1.5 1.75 9.75 17.25 221.75 2.5 12 18.2 26.252 3.75 16.25 20 30

after control are obtained and compared with numerical results(table 1) in table 3.

In table 3, is shown the difference between the tests andthe simulations δ = |Rs−Re|

Rs× 100%, where Rs and Re denote

the results of the simulation and the experiment, respectively.That is to say, the differences δ in table 3 are obtained bycomparing the experimental results obtained from table 3 andthe simulation results in table 1. Take an example, for thewrinkling number: it is 37 in the simulation, and it is also37 in the test before control. Here the difference in wrinklingnumber δ = |37−37|

37 × 100% is zero before control. Thewrinkling number is 23 in the simulation, and it is 26 in thetest after control. Thus the difference in wrinkling numberδ = |23−26|

23 × 100% is 13% after control.According to these comparisons, there is no difference

(δ = 0) between the experiment and the simulation for thecase of the wrinkling number before control. The numericalpredictions agree very well with the experimental results;almost all differences are smaller than 5% except the wrinklingnumber after control and the tip deflection before control. Thediscrepancy between the numerical and experimental results isbelieved to be caused by following factors.

(a) There is an approximation between the ideal numericalmodel and the actual solid model, including the boundarycondition and the material model.

(b) In the test, the SMA wires are continuously glued/attachedto the membrane surface. However, the recovery forceusing SMA wires is simulated by the ENF method whichis separately loaded on the nodes.

(c) There is no consideration of the variations in the volumeand pressure of the inflated beam during the simulation.

(d) There is an error in the photogrammetric measurementsespecially for small wrinkles.

(a) (b)

Figure 10. (a) Wrinkling photo and (b) configuration test results.

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Smart Mater. Struct. 19 (2010) 105019 C G Wang and H F Tan

Figure 11. Connection and distribution of SMA wires.

Table 3. Wrinkling control tests and their comparisons with simulations.

ItemsWrinklingnumber

Wrinklingwavelength (mm)

Wrinkling area(m2) Tip deflection (m)

Experiment Before control 37 4.189 0.012 16 0.0097After control 26 4.531 0.007 17 0.0064

δ (%) Before/after 0/13 2.5/3.9 2.6/4.7 14.1/4.5

4. Conclusions

A novel control strategy is presented to control and remove thewrinkles in an inflated beam using SMA wires. The SMA wiresare subtly attached to the stretched side of the inflated beam,which is a big difference compared to the prior methods. Thestretched side is opposite to the wrinkled/compressed regionof the inflated beam. The wrinkling simulation is performedby using the DP method, and the wrinkling control simulationusing SMA wires is realized by introducing the ENF method.The wrinkling control experiments using SMA wires arecarried out based on the photogrammetric measurements.A good agreement is obtained between the numerical andexperimental results.

(a) SMA wires have been verified to be effective for thecontrol and removal of the wrinkles in the inflated beam.Based on the numerical and experimental results, thewrinkling area and number can be greatly decreased byusing SMA wires. For these two cases, the control abilityusing SMA wires reaches almost 50%. A similar controlability can also be obtained for the case of tip deflectioncontrol of the inflated beam. The slight control ability ofthe wrinkling wavelength is due to the limited structuralsize and loading conditions.

(b) The ENF method gives a fairly good prediction ofthe wrinkling control, which agrees well with theexperimental results. Almost all differences between thenumerical and experimental results are smaller than 5%except the wrinkling number after control and the tipdeflection before control.

(c) The largest driving force and the distribution of the SMAwires are two keys to control the wrinkles of the inflatedbeam effectively. A deep study is needed to understandhow they can be suitably arranged to obtain the bestwrinkling control effects.

Acknowledgments

This project was supported by the National Natural ScienceFoundation of China, 10902027; the Specialized ResearchFund for the Doctoral Program of Higher Education ofChina, 200802131046; and the China Postdoctoral ScienceFoundation funded major project, 200801290.

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