SANDIA REPORTSAND82–0345 ● UC–60Unlimited ReleasePrinted October 1982

Finite Element Analysis and ModalTesting of a Rotating Wind Turbine

.

Thomas G. Carrie, Donald W. Lobitz,Arlo R. Nerd, Robert A. Watson

Prepared by

Sandia National LaboratoriesAlbuquerque, New Mexico 87185 and Livermore, California 94550for the United States Department of Energy

under Contract DE-AC04-76DPO078g

..

1“-.....,_. ....

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I

SAND82-0345

FINITE ELEMENT ANALYSIS AND MODAL TESTINGOF A ROTATING WIND TURBINE*

T. G. CarrieD. W. LobitzA. R. Nerd

R. A. WatsonSandia National Laboratories

Albuquerque, NM

Abstract——

A finite element procedure, which includes geometricstiffening, and centrifugal and Coriolis terms resulting fromthe use of a rotating coordinate system, has been developed tocompute the mode shapes and frequencies of rotatingstructures. Special application of this capability has beenmade to Darrieus, vertical axis wind turbines. In a paralleldevelopment effort, a technique for the modal testing of arotating vertical axis wind turbine has been established tomeasure modal parameters directly. Results from the predictiveand experimental techniques for the modal frequencies and modeshapes are compared over a wide range of rotational speeds.

* This work was performed at Sandia National Laboratories andwas supported by the U. S. Department of Energy undercontract number DE–AC04–76DPO0789.

1-2

I.

II.

III.

Iv.

v.

VI.

VII.

VIII.

Table of Contents

Page

Introduction. . . . . . . . . . . . . . . . . . . 5

Finite Element Theory for the VAWT . . . . . . . . 6

Finite Element Results for the 2-m VAWT . . . . . . 9

Experimental Procedure . . . . . . . . . . . . . . 12

Comparison and Discussion of Results . . . . . . . 16

Conclusions. . . . . . . . . . . . . . . . . . . 17

Acknowledgements . . . . . . . . . . . . . . . . . 17

References. . . . . . . . . . . . . . . . . . . . 17

3-4

Introduction

Over the past several years, windenergy has been considered as apotentially viable option in thesearch for renewable energy sources.The feasibility of tapping thissource depends on the ability todesign extraction devices which areboth safe and reliable, and, can beproduced at low cost. The extractiondevice of interest here is thevertical axis wind turbine (VAWT)shown in Figure 1. As dynamiceffects can be substantial due to theperiodic nature of the aerodynamicloading, the design process mustdepend on accurate predictive toolsand/or measurement techniques fordetermining the dynamic responsecharacteristics of the turbine.

Determination of modal charac-teristics is of paramount importancein that, with this information,resonant behavior can be avoided. Inrotating structures, the acquisitionof these characteristics is compli-cated by their dependence on thestructure’s rotational speed.Therefore, special methods have to beemployed for both the predictive andexperimental procedures.

In this paper predictive as wellas experimental techniques aredescribed for identifying the modalcharacteristics of a VAWT. Resultsfrom each method are obtained andcompared for a two–meter turbine.

Fig. 1 The Sandia 17-m Research Turbine

The predictive techniquedeveloped here is based on the finiteelement method in order to takeadvantaqe of its superiorversatility. As the turbine ismodeled in a frame which rotates atthe turbine operatinq speed (assumedconstant), rotating coordinate systemeffects must be taken into account.Within this frame, the motions of theturbine are assumed to be small.

The finite element method hasbeen used previously to predict themodal characteristics of rotatingstructures. Severalinvestigators, 1-4 have applied themethod to structures which arecomprised of rotatinq beamsreminiscent of fan or helicopterblades. However, in these cases someof the rotating coordinate systemeffects are neqlected. This may bejustified since, for these types ofstructures, motions normal to theaxis of rotation are relativelysmall. For the vertical axis windturbine, this is not the case andthese effects cannot be omitted. Amethod described in Ref. 5 usesfinite elements to obtain theeigenvalue solution to the suinninqskylab problem. This method, whichis similar to the one used here forthe analysis of the VAWT, includesall rotatinq coordinate systemeffects.

Other techniques which have beenused to obtain dynamiccharacteristics of rotatinqstructures include various Galerkinprocedures, the modal method, and thecomponent mode method in combinationwith various eigensolution schemes.In Ref. 6, the Garlerkin method isused in conjunction with anasymptotic expansion technique (themethod of multiple time scales) toexamine resonance in a spinning,nutating plate. A Galerkin procedurewas also used in Ref. 7 where thedynamic response of a horizontal axiswind turbine was studied with FloquetTheory. Refs. 8-10 employ thecom~onent mode method in combinationwith relatively standard eiqenvalueextraction Procedures to investigatethe dynamic characteristics of a VAWT.

A major experimental obstacle inobtaining the natural frequencies andmode shapes of a rotatinq structureis that mechanisms for exciting thestructure while it is rotatinq mustbe devised. Also slip-rinqs ortelemetry must be employed for datatransmission from the structure.These complications may account for

5

the minimal amount of reporting forthis type of modal testing in theopen literature. One reference,llhowever, was found where modal testdata were cited for a model propellerblade as a function of rpm. Theoriginal work, Ref. 12, may bedifficult to acquire.

The remainder of this paper isdivided into five sections. Thefirst describes the details of thefinite element method used forpredicting the modes and frequenciesof the rotating turbine. In thesecond, details of the finite elementmodel are presented. Theexperimental procedure for therotating modal test is described inthe third. The fourth providescomparisons of the predicted andexperimental data. The final sectioncontains the conclusions.

Finite Element Theory for the VAWT——

The effects which must heincluded for an accuraterepresentation of the dynamics of aVAWT relative to a coordinate systemrotating at a constant speed, includetension stiffening, and centrifugaland Coriolis terms. Manifestationsof the latter primarily appear in theinertia terms of the equations ofmotion. However, in the finiteelement context, they result inadditions to the damping andstiffness matrices, and the forcevector. Specifically, the Coriolisterms produce a skew-symmetriccontribution to the damping matrix,and the centrifugal forces result ina steady force plus a negativecontribution to the stiffness matrix,referred to here as centrifugalsoftening. The resulting finiteelement equations are represented by

Mti + CL - SU + KU = Fc + Fg,

where M is the classical mass matrix,C is the Coriolis matrix, S is thecentrifugal softening matrix, K isthe usual stiffness matrix, Fc is astatic load vector representing thesteady centrifugal force, and Fgrepresents the gravitational forceswhich are also steady due to theorientation of the axis of rotationof the turbine. To obtain the modesand frequencies of the turbine asobserved in the rotating system,equation (1) is reduced to thefollowing form:

Thus the solutions correspond tosmall motions about a prestressedstate.

Aeroel astic effects have not beenincluded in this anal,ysis. Theseeffects have, however, beenincorporated in stabilityinvestigations of the VAWT13 and,while stronqly influencing stability,seem to have little effect on thenatural frequencies and modes of theturbine. This, coupled with theqeneral experimental observation thatVAWTS are relatively stable machines,tends to justify this omission.

The coordinate system emoloyed inthis analysis rotates at the turbineoperatinq speed with the oriqin fixedin soace at the base of the tower.As shown in Fiqure 2, the anqularvelocity vector is directedvertically along the z axis.

z

x

MU+ Cd + (K +KG - S) u = O, (2)

where KG is the geometric stiffnessmatrix resulting from the steadycentrifugal and gravitational loads.

(1) Fig. 2 A Beam Element in the RotatingFrame

The finite element matrices aredeveloped on the basis of the motionsthat remain small within thisrotating frame. The matrices, M, K,and KG are presented in a number ofcurrently available finite elementtexts and will not be derived here.On the other hand, the matrices, Cand S, which are due to rotatingcoordinate system effects, are notcommonly encountered, and thereforewill be developed.

Due to the structural nature ofVAWTS, the finite element equationsneed only be developed for beamelements and concentrated masses, In

6

order to develop inertia matrices forthe beam elements, the displacements,velocities and accelerations areassumed to vary linearly along thelength of the element. The mass ofthe element is concentrated along itsaxis so that the rotational kineticenergy of the element about its axisis neglected. Using the followingequation for the total accelerationat a point in the rotating system,

*t=&+ 2Qx~+

lx[sx(~+~)l’(3)

where

%t ‘s the total acceleration vector,excluding gravitationalacceleration,

~, ~, ~ and ~ are the originalposition, the displacement,velocity, and accelerationvectors, respectively, asobserved in the rotatinqcoordinate system, and

J is the fixed angular veln”i+yvector of that system,

an expression for the inertia forcedue to the elemental mass at thatpoint can be developed. Adopting thelinear form for the motion, theinitial position, displacement,velocity, and acceleration relative tothe rotating frame can be cast interms of the nodal point values asfollows (see Figure 2):

where

[

1-s/! o 0 s/1. o 0p.f) 1-SIL o 0 s/L o

0 0 l-s/R o 0 1Slk ,

~T=u

[

u‘1 ‘Y1 Z1 lJx2 ‘Y2 UZ2 1, etc.,

s is the arc length variable, and

k? is the length of the element.

Note that the overbars denote nodalpoint values. Specializing ~ to \T= [0 O nz], the second and thirdterms of equation (3) become

.2Q x ~ = 2QzQi , (5)

where

Q.

o -1+s/!, o 0 -s/L oI-s/k o 0 SJL O 0

1 0 0 0 0 00

[

l-s/.L o 0 s/,L o 0R=O 1-s/! o 0 s/1 o1

1 0 000 0 OJ,

Using r!’Alembert forces alonq with theconcept of virtual work, the requiredmatrices are obtained as follows:

where .SUT is the variation in thenodal displacements and o is the massper unit length. The first inteqralon the riqht hand side of equation (7)corresponds to the mass matrix; thesecond, the Coriolis matrix; thethird, the centrifugal softeninqmatrix; and the fourth, the centrifqalforces . Of special interest here arethe second and third terms which arepresented in detail below:

Coriolis

Zplflz

o -1/3

1/3 D

o 0

0 -1/6

1/6 o

0 0

Centrifugal So fteninq

[1/3 o

o

0

0

0

0

0

0

0

0

0

1-1/6 O

1/6 00

0 00

0 -1/3 o

1/3 00

0 00

11/6 O 0

0 1/6 O

0 00

1/3 o 0

1/3 o

0

IIO‘1a

Y1

(1‘1 (8)

(1X2

oY2

‘*2

H

uxl

‘Yl

u‘1 (9)

u‘2

‘Yz

‘*2

7

Concentrated masses, such as bladejoints, flanges and support bearings,are handled in an entirely similarmanner, resultinq in the followinqexpressions:

Coriolis

[1

o -1 0

2mOz 1 0 0

000

These concentrated masses possess norotational kinetic energy. They must,however, be attached to a point whichis on the rotating structure.

The matrices, C and S, ofequations (1) and (2) are obtained byassembling the contributions from allthe beam elements and concentratedmasses of the finite element model.

A critical factor in thedevelopment of this method is thatvector transformations are notrequired between stationary(groundbased) and rotating coordinatesystems. For most VAWTS, the physicalconnections of the rotor to the groundoccur through the tiedown cables andthe tower base connection. Thestiffness of both of these connectionmechanisms is usually isotropic(independent of azimuthal position)and therefore can be represented bymassless linear springs which rotatewith the turbine and act to restore itto its original position. Theassociated mass can usually beadequately modeled by concentratedmasses placed at appropriate pointsthe rotating frame.

n

The eigensystem represented byequation (2) is Hermetian due to theskew–symmetry of the Coriolis matrixConsequently, it can be shown 14 thatthe eigenvectors are, in general,complex; but the eigenvalues, whichare the squares of the circularfrequencies, are real. Thus, withthis system, the natural frequenciesare purely real or purely imaginarydepending on the sign of the realeigenvalues. If structural damping isincluded, the system loses itsHermetian character and theeigenvalues as well as theeigenvectors are complex. For thepresent analyses, structural dampinghas been excluded due to the lightlydamped nature of most VAWTS.

Instead of developing a completelyindependent package for theeigensolution of equation (2), anexisting code was modified. With this

approach, duplication of such thingsas input, output, plotting, solutionprocedures, etc., is avoided. TheMacNeal-Schwendler version ofNASTRAN15 was selected here becausethe modification required was minimaland could be accomplished via DMAP~rogramming, a feature which allowsthe NASTRAN user to modify the codewithout actually dealing with theFORTRAN codinq. This version containscomplex eiqens,ystem solutionprocedures and also permits thestiffness, mass, and dampinq matricesto be modified throuqh an inputoption. Thus, the special matricesrequired in equation (2), specificallythe Coriolis (C) and softeninq (S)matrices, can be qenerated externallyand read into NASTRAN as input. Asthe NASTRAN code handles non-symmetricas well as symmetric matrices, nospecial problems occur due to theskew–symmetry of the Coriolis matrix.The mass (M) and stiffness (K)matrices are generated internally,complete with the effects of qeometricstiffening in the stiffness matrix.Additional details on this NASTRANprocedure are presented in the nextsection.

This analysis package hasundergone limited verification due tothe lack of available closed formsolutions for rotatinq structures.However, the exact solution for thecompressed, whirlinq shaft with pinnedends provides at least one test case.If the shaft is not spinning, itsnatural frequencies and corresoondinqmodal solutions are qiven by

n . (&y (#’2 [, - & (+)2]’2, (,1)

Hu 11

Cos Vnt

‘J1=sin~z o 9

11

u o=sin~z

\l3

‘2 Cos Vnt

(12)

where

P is the axial load,

u, v are the transversedisplacements, and

E, I, P, A, L are the usualphysical properties of the shaft.

On the otherspinning itsUn, relativeare

‘n =

where Vn is tand Q is theshaft. Thus,

hand, when the shaft isnatural frequencies,to the rotating frame

tfll,‘n (13)

iven by equation (11)angular velocity of thethe non-spinning natural

frequencies split into twofrequencies, one increasing withangular velocity at a rate of 1 Hz/Hz,and the other decreasing at 1 Hz/hz.Moreover, the modal solutions become

Hu [

Cos tint= sin ~ z

‘11Cos (Unt + Tr/2) ,

II

(14)u

. sin ~ ~1

Cos Unt

“2 L ICos (Unt + >) .

Whereas the u and v motions werepreviously uncoupled in the modes ofthe non-spinning shaft, now they arecoupled, with the motion in onedirection ninety degrees out of phasewith that in the other. This behavioras well as the natural frequencyvariation with angular velocity isprecisely predicted by the finiteelement analysis package.

Finite Element Results for the 2-m VAWT———..————.-—-- -----———-———__———__

In this section the details of thefinite element model used to predictthe modes and frequencies of theSandia 2-m research turbine as afunction of rpm are presented. Beforedc)ing this, however, a shortdescription of the NASTRANcalculational procedure is in order.

This procedure is based on version61 of MSC-NASTRAN15 and utilizes twoof the available rigid formatoptions. Using the first of tworequired data decks, a static analysiswith geometric stiffening effects(Rigid Format 64) is performed on themodel under the action of centrifugal,gravitational and boundary forces.The resulting modified stiffnessmatrix is then retained via a modestamount of DMAP programming for use inthe subsequent complex eigenvalueanalysis (Rigid Format 67) of thesecond data deck. Along with thenecessary structural data, this seconddeck also includes the “Direct MatrixInput at Grid Points” (DMIG) cards,through which the Coriolis and

Centrifugal Softeninq matricesdescribed in the previous section areinput. Because the actual number ofDMIG cards required tends to be larqe,a pre–processing proqram, FEVD, waswritten for automation purposes. FEVDreads the NASTRAN structural data deckand extracts information it needs fromGRID, CFiAR, PBAR, MAT1, CONM2, andRFORCE cards. With this informationit constructs the necessarv matricesand casts them in terms of DMIGcards. It then inserts these into thesecond NASTRAN data deck. In thismanner, the exercise of includinqrotating coordinate system effects inNASTRAN is relatively transparent tothe user.

NASTRAN provides the user with achoice of two complex eigensolvers.The one which employs the upperHessenberq method is very efficientbut limited to sixty degrees offreedom (the actual number may varydepending on computer installation).The other which utilizes the inverseDower method is limited in the numberof degrees of freedom only by economicconsiderations, but requires insiqhtinto the location of the desirednatural frequencies in the complexplane. For the analysis in thispaper, the upper Hessenberq method wasused in conjunction with Gu,vanreduction (also a NASTRAN option) toreduce the number of deqrees offreedom from that of the actual modelto sixtv.

The finite element model of the2-m VAWT is shown in Fiqure 3. Theactual hardware is displayed in aphotograph in Figure 4. The trusstower, which adds a

400 Ibf

1~

400 lbf

1~

r

1600 lbl/111- ,...,m, A ,!:%3>

ml”)bf/in

\ !..”, ! .“. ”,

“

(:01 ‘“M

TY(:0) “m

(1710’%* ~:~o~ol lbf/in (1 : 106) ~ (:%0:) “’’’”

2700 ~ 2700 @l#

(65,0001 rad (65,000)

Fig. 3 Finite Element Model of the 2-mVAWT

9

Fig. 4 The Sandia 2-m Turbine withInstrumentation

fair degree of complexity to themodel, is represented by 150 beamelements. The blades are modeledwith 20 beam elements each. The massof the hardware associated with theupper bearing and tie–down

ModeNumber

12345678

1:

connections is represented by aconcentrated mass placed at the top ofthe tower. The tie-down cables aremodeled with horizontal linear springsand a vertical downward force as shownin Figure 3. The mass of theinstrumentation installed on the rotorfor the test (which is not negligiblecompared to the rotor mass) isincluded through appropriateconcentrated masses. The base ismodeled as shown in Fiqure 3, with aconcentrated mass in combination withtorsional and linear surinqs. Asindicated the sprinqs are equal in thetwo orthogonal directions, thusDroducinq the desired isotropic effectmentioned in the orececling section.

Using Darked test data, the modelwas moderately tuned by adjusting theparameters associated with the basewhich, in general, must be estimated.The modified values are qiven inparentheses in Fiqure 3. Theresultinq oarked frequencies for thefirst ten modes are qiven in Table 1alonq with the experimental values.As computed from the table, theaverage error in the frequencies wasreduced from 12 to less than 1 percent, after tuning. Only the secondsymmetric flatwise mode deviated bymore than 1 per cent, which is mostProbably caused by inaccurate modelingof the blade–to–tower joint.

Mode identification terminology isgiven in the second column of Table 1for the first ten parked modes, andcorresponding three–views in Figure5. In Fiqure 6, for purposes ofcomparison, predicted and measuredthree–views of the 2nd rotorout-of-plane mode are qiven. Thisagreement is representative of thefirst ten modes.

Table 1 Parked Frequency Comparison

Mode Name

1st Antisymmetric Flatwise1st Symmetric Flatwise1st Rotor Out-of-Plane1st Rotor In-PlaneDumbbell2nd Rotor Out-of-Plane2nd Rotor In-Plane2nd Symmetric Flatwise2nd Antisymmetric Flatwise3rd Rotor Out-of-Plane

MeasuredModal

Frequency (Hz)

12.312.515.315.824.426.228.329.731.536.5

InitialAnalytical

Frequency (Hz)

12.512.617.117.222.630.530.630.939.742.3

ModifiedAnalytical

Frequency (Hz)

12.312.415.215.924.426.228.030.631.736.5

10

1 2 3 4

&

6 7

c’Fig. 6

8 9

@

Fig. 5 Predicted Mode Shapes for theParked 2-m VAl~T

/-<,----- --- 2..

. i2--- .’* ,

iA Comparison of the Predicted ;and Measured 2nd Rotor Out-of- —Plane Mode of the Parked Turbine ~2c

~Using the tuned parameters, the E

fan plot of Figure 7 was produced. s

The mode numbers on the variouscurves correspond to the modesidentified in Table 1. The lC

variations of the frequencies withrpm are quite complex. While mostmonotonically increase with rpm, somedecrease monotonically. Othersincrease and then decrease and vice aversa. Althouqh more complicated,

‘L’

CDI 1 1 1 1 1

— ANALYTICAL PREDICTIONS

1 1 1 1 I 1

100 200 300 400 500 600 700these variatio~s are similar to those ROTOR RPMassociated with the whirling shaft

Fig. 7problem discussed in the previous Predicted Fan Plot for the 2-msection. VAWT

11

In addition to complicatedfrequency variations, the mode shapesalso change in character witfi rprn.Specifically, mode shapes which arecompletely uncoupled when the turbineis parked become coupled duringrotation. In qeneral, the couplinqoc$urs between pairs of parked modes90 out of phase with each other. InFigure 8 this coupling is shown forthe 2nd rotor out-of-plane mode. AS

the turbine rotates, a mode which issimilar to the 1st rotor in–plane modeis drawn into and coupled ~ith theoriginal mode, but at a 90 phaseshift. In mathematical terms, themodes chanqe from real to complex whenthe turbine rotates. Furthermore, asindicated in Figure 8, these complexmode shapes change slightly as afunction of r~m.

0° PHASE 90” PHASE

““pMO}o300RpM0’) o

Fig. 8 Predicted Modal Coupling withTurbine Rotation

Experimental Procedure

There are two objectives indeveloping a modal testing capabilityfor a rotating turbine. First,experimental data on representativehardware is required to verify theaccuracy of the finite elementanalysis technique described above,and secondly, a need exists for anestablished experimental technique to

measure rotating modal frequencies so

that, in the absence of otherinformation, design modifications ofan existing turbine can be made purelyon the basis of the test data.

In this section, the experimentalprocedures that were developed formodal testing of both the parked androtatinq turbine will be described.The general technique of modal testinQwith a mini-computer based FastFourier Transform (FFT) will bediscussed briefly, followed by itsapplication to the parked Sandia 2-mresearch turbine. This particularturbine was selected for the testbecause of its manageable size and itsvariable speed capability. Finally,for the rotating modal test, a fulldescription of the instrumentation,the excitation of the structure, andthe extraction of the modal parametersfrom the data will be qiven.

The FFT technique, which involvesexciting the structure with a forcehaving a linear spectrum containingthe frequency band of interest, isgenerally faster and more versatilethan the older analog swept-sinetechnique. The applied force andresponse are measured in the timedomain and transformed to thefrequency domain using the FFT. Thefrequency response function is thencomputed using the cross–spectral andauto-spectral densities of the appliedforce and the response. Typically anumber of measurements are averaged soas to reduce the effects ofuncorrelated noise. A more completedescription of FFT modal testing iscontained in Ref. 16. The greaterversatility of this technique is aresult of more relaxed requirements onthe excitation. For example, thetechnique is equally applicable toshaker driven structures or structuresexcited by an instrumented hammer.

As the first step in the rotatingmodal test, a detailed parked modaltest was conducted. The turbine wasextensively instrumented withpiezoelectric accelerometers on theblades, tower, and turbine base.Figure 9 shows a diagram of theturbine with the measurement pointsindicated by numbers. The excitationwas provided by an instrumented hammerfor impacting either the tower or theblades. The accelerometer and forcetime histories were filtered, recordedon FM tape, and, using mini-computerbased software, digitized and combinedto form a set of frequency responsefunctions.

12

Fig. 9 Measurement Locations for theParked 2-m VAWT Modal Test

Prior to the test, a two channelanalyzer was used to determine thefrequency range of interest, theresolution required, and the bestlocations for the driving points.From this procedure it was determinedthat all the modes of interest werebetween ten and sixty Hz. The inputforce spectrum was tailored for thisrange, and the time histories werezoomed between its extremes to give adesirable frequency resolution(approximately 0.1 Hz).

Figure 10 shows a typicalfrequency response function usingthree samples of data with themagnitude plotted versus frequency.This response function is the responseof a blade in the flatwise direction(perpendicular to the blade chord) dueto a force input on the tower. Thesharp resonant peaks clearly indicatemodal frequencies and, because oftheir sharpness, the corresponding lowmodal damping.

‘sing additi!?a~jthmini–computer based softwarea complete set of frequency responsefunctions (one for each measurementpoint and direction) as input, themodal frequencies, damping, and modeshapes are computed. Figure 11 showsone view of four of the fourteen modesthat were extracted from the data,with the deformed shapes indicated bythe dashed lines.

io.o 60.0

Frequency (Hz)

Fig. 10 Typical Frequency ResponseFunction From the ParkedModal Test

1st AI?tisymmetricDumbbell

1

,/ \,\

/’ j\\ ,’\t,/’

/./,/,

<’?

‘, /’

Bl!!lFreq = 12.3 Hz Freq = 24.4 Hz

2nd Antisymmetric 3rd RotorFlatwise Out-o -Plane

@l

-.,.-

< :.,

/ “, j ‘j; ‘. t‘.

i > $;/b,‘,/ ,:, :/’

L ; ,’ ‘i1’I /’-.-.-, f’

i!!h!!iFreq = 31.5 Hz Freq = 36.5 Hz

Fig. 11 Four Measured Mode Shapes forthe Parked 2-m VAWT

13

Having obtained the modalcharacteristics of the parked turbine,the rotating modal test wasconducted. Modal data was collectedat six different rotational speeds in100 rpm increments from 100 to 600rpm. In this way, the chanqinqfrequencies could be tracked fromtheir parked values up to the nomoperating speed of 600 rpm.

For the rotating test seven sgages and two accelerometers wereto measure the response of the

nal

rainused

turbine. The two accelerometers wereplaced on tower in–the-plane andout-of–the–plane of the rotor. Thesteady–state centrifugal accelerationwas sufficiently low on the tower sothat accelerometers could be usedconveniently. Two of the seven straingages also were used on the tower withthe other five on the blade roots.The strain gages, which have twoactive elements, were placed on theblades and tower so that they measuredeither purely in–plane or out-of–planedeformations. On the blades in-planeis generally referred to as fl atwise(perpendicular to the blade chord),and out–of–plane as edgewise (in thedirection of the chord).

Slip rings were used to transmitthe transducer signals from therotating turbine. However, beforepassing the signals through the sliprings, they were amplified by DCamplifiers mounted on the tower. Inspite of the complications involved,transducers attached to the rotatingstructure were used so that the datataken would be relative to therotating frame of the analyticalpredictions. Any measurements takenon fixed hardware would have to betransformed to the rotating frame forpurposes of comparison. This processadds complexity to the data processincland cannot adequately determine blatbehavior.

All testing in the rotatingconfiguration was done in very lowwinds (less than 5 mph) to avoidaerodynamic excitation which couldoverwhelm the forces applied to exc-modal resr)onse. In addition. the

e“

te

aerodynamic forces, which ex~ite theturbine at integral multiples of therotational frequency, might obscureany modal information at thosefrequencies. In the absence of wind,the motor-generator was used to propelthe turbine and maintain its speed ata constant, preset value.

In selectinq a device for excitingthe rotating turbine, several conceptswere considered. Any type of impactexcitation is virtually eliminated

because of the hiqh blade tip speeds(over 190 ftlsec). Also, since thistestinq technique is to be applicableto turbines of all sizes, the devicemust be capable of excitinq largeturbines as well as the relativelysmall ?-m turbine used in this test.

The scheme finally chosen consistsprimarily of a pretensioned cableattached between one blade and thetower. The cable is suddenly releasedafter the turbine is rotatinq at apreset speed. Fiqure 4 is aphotograph of the turbine, showinq thesnap release device and the otherinstrumentation mentioned above. Thesnap release device consists of aforce transducer which measures thetension in the cable, a burn wire, andan easily reulaceahle nylon cord.After preloadinq the cable, theturbine is rotated at the desiredspeed. A current is passed throuqhthe sliprinqs to the burn wire whichin turn cuts the nylon cord, releasinqthe tension in the cable. Fiqure 12shows a detailed diaqram of thesnap-release device. The cable isrestrained at both ends so that itbecomes slack after release and ‘not fly out from the turbine.

aoes

\

/’NY1ON CORD

“\\

/AIRCRAFTCABLE

TO TURBINE

\

BLADE

L

rl “ 11Fig. 12 The Snap-Release Device

14

It would appear that thissnap–release device would only producemotion in the plane of the turbine,and this is the case for the parkedturbine. However, when it isrotatinq, the in-plane andout–of-plane motions are cou~ledthrough Coriolis forces.Consequently, all modes of interestare excited by this mechanism.

One difficulty which is related tothe FFT algorithm, arises in usinqthis excitation device. The snaorelease basically applies a forcewhich is a Heaviside function. As theFourier transform for the Heavisidefunction does not exist, the FFTcannot be used on this force.However, if the Heaviside function ispassed through a high-pass filter, itcan be converted to a well behavedfunction that is easily transformablewith the FFT. Figure 13 shows a traceof a typical force history afterpassing it through the hiqh-passfilter. As shown, the force signalhas dropped to essentially zero afterabout 0.4 sec., which is numericallycompatible wit$ the FFT. Also on thisforce trace, evidence exists (thethree/rev response prior to release)that the turbine is beinqaerodynamically driven at some smalllevel.

c1.

c1

\\\\\

-0.4 0.0 0.4 0.8

Time (See)

Fig. 13 A Typical Filtered Force HistoryProduced by the Snap-ReleaseDevice

The high pass filter used in thistest is the AC coupling circuit of thesame set of amplifiers which amplifythe signals before recordinq them onFM tape. Of course, the responsesignals must also pass through matchedfilters to avoid relative phase shiftsbetween the response and theexcitation. Figure 14 shows themagnitudes of the frequency responsefunction for the low and high pass

filters used here, with the three dBpoints at approximately 2.0 and 100.0Hz.

J—d———J1.0 1(1.O 100.0

~requency (Hz)

Fig. 14 The Magnitude of the FrequencyResponse Function for the FilterUsed in the Rotating Modal Test

Figures 15 and 16 show typicalfrequency response functions from therotating test at 600 rpm. Thesefunctions were obtained from datataken by in-plane and out-of-planeaccelerometers located near the top ofthe tower. Taking into account thescale factors on the two plots, themagnitudes of the two responsefunctions are approximately equal.This indicates that the Corioliscoupling has indeed producedout-of-plane response, even though theinitial excitation was strictlyin-plane.

s,&_JJ0.0 50.0

Frequency (Hz)

Fig. 15 Measured In-Plane FrequencyResponse Function for aRotational Speed of 600 rpm

15

I %/rev

0.0 5U. LFrequency (Hz)

Fig. 16 Measured Out-of-Plane FrequencyResponse Function for aRotational Speed of 600 rpm

Although the general data qualityis represented by that in Figures 15and 16, at higher rotational speedsthe aerodynamic excitation of theturbine became more prevalent in spiteof the low wind environment. This isindicated in the frequency response byhigh levels at integral multiples ofthe rotational frequency.

At each rotational speed,frequency response functions werecomputed from the data taken with theseven strain gages and the twoaccelerometers. Ignoring those peaksassociated with the aerodynamicexcitation, the frequencies of theremaining peaks on each of theresponse functions were extracted andtabulated. Small variations in thefrequency of some of the modes didoccur, and in those cases an averagevalue was used. Since the mode shapesare complex, magnitude data were usedrather than the imaginary component toobtain these modal frequencies.

In view of the small number oftransducers on the turbine, a fulldescription of the mode shape isimpossible. However, ei?envectors canstill be computed with nine componentsin each vector using the methodoutlined above. One typicaleigenvector corresponding to thesecond rotor out–of-plane mode and arotational speed of 300 rpm, is shownbelow. The magnitude and phase areseparated by commas in the vector.The first two components are theaccelerations with units of mini-q’s,and the last seven are stresses withunits of psi.

tower out–of–olane

tower in-planetower out–of-Diane

tower in-planeblade 1, lower edqew

blade 1, lupper eflqewblade 7. lower edqew

sesese

blade 7; upper edgewiseblade 2, lower fl atwise

1.00,1.28,7.57,4.75,

1.06,0.83,1.17,0.80,0.00,

90166779178749?75256255——-

;

This eiqenvector has been normalizedso that the first comDonent has aunit magnitude and a uhase of 90deqrees. Note the larqe ohasevariations which are normally notseen when non-rotating structuresare analyzed for their complexmodes. The out-of-plane accelerationhas a phase of 76 deqrees (nearly 90degrees) qreater than the in-planeacceleration. Likewise theout-of-plane tower strain qaqe isadvanced by 101 degrees over t+ein-Diane qage. The four edqewisestrain gages are basically in phasewith the out-of-plane deformation of

the tower. These phase relationshipsare caused urimarilv by rotatinacoordinate system effe~ts.Structural damping, errors fin the initial data andapproximations in the extracalgorithms also contribute.from the uhase components invector, it is clear that theand out-of-plane motion isapproximately ninety degreesohase.

.,

om noise

ionHowever,thein–Diane

out of

Comparison and Discussion of Results7TiT_Tan pTot of Figure 7-titiT6~ti~;--

-——. .—.. -——--—

generated for the 2-m turbine usinqthe finite element predictivecapability is reproduced in Fiqure17. The experimental data aresuperimposed as shown. In qeneralthe aqreement between the predictedand measured values is excellent. Infact, the averaqe absolute deviationof the experimental points from thepredicted is 0.52 Hz., or 2.2percent. Althouqh the deviation doesnot appear to be dependent on therpm, the aqreement is not as qood atthe hiqber frequencies. Thisdeqrarlation for the higherfrequencies occurs in qeneral whenfinite element methods are used andis primarily caused by aninsufficient number of deqrees offreedom in the model to accuratelyrepresent the associated modeshapes. The coarse representation ofthe base of the turbine may also be acontributor to this mild lack ofaqreement, especially since basemotion was experimentally observed tobe more prevalent in the hiqher modes.

16

Acknowledgments————- ---

. EXPERIMENTAL DATA’11_ ANALYTICAL PREDICTIONS

n 1 1

-o 100 200 300 400 500 600 700

ROTOR RPM

Fig. 17 A Comparison of Predicted andMeasured Modal Frequencies forthe 2-m VAWT as a function ofrpm

Due to the small number oftransducers on the turbine, theexperimental mode shapes are coarseand quantitative comparisons to thosepredicted are difficult. However, asindicated in the previous section,the measured complex modes aredefinitely composed of in-plane andout-of-plane motion approximatelyninety degrees out of phase,consistent with the predicted results.

Conclusions-——— -——

In view of the excellent agreementbetween predicted and measured modalfrequencies, the accuracy of thefinite element technique forcomputing the modal characteristicsof rotating VAWTS has been verifiedfor lightly damped turbines. Asshown, these predictions are of thesame level of accuracy as is obtainedin the finite element modal analysisof stationary structures.Additionally, a rotating modaltesting technique for VAWTS has beendemonstrated which is applicable tolarge as well as small turbines.Thus, if a predictive capability isunavailable, the rotating modalcharacteristics of establishedhardware can be measured by thetechnique described herein.

The authors wish to acknowledgeW. N. Sullivan for providing theimpetus to do the rotatinq modaltest. C. M. Grassham made invaluablecontributions in the desiqn of thesnap release device and in theplanning and maintenance of thesignal processing system. Others whomade significant contributions in thetesting and data reduction areP. H. Adams and his strain gageinstallation team, ,1. D. Burkhardt,B. K. Cloer, M. R. Weber, andL. H. Wilhelmi, Also, the work ofJ. R. Koteras in establishing sPecialNASTRAN analysis procedures for thefinite element predictive capabilityis gratefully acknowledged.

References

1. Hoa, S. V., “Vibration of aRotatinq Beam with Tip Mass,”Journal of Sound and Vibration,~-~No. 7;-T57~:-~~:-T69=381.

----

2. Friedmann, p. p. and Strauh, F.,“Application of the Finite ElementMethod to Rotary WingAeroelasticity,” Journal of theAmerican Helicopter Society, Vol.

——— _—_

nsli:-T!T80~m6:TT:--

3. Hodges, (). H. and Rutkowski, M.J “Free-Vibration Analysis ofR;;atinq Beams by a Variable-OrderFinite-Element Method. ” AIAAJournal, Vol. 19, Nov~ 1~ pp.~=T~66 .

4. Nigh, G. L. and Olson, M. D.,“Finite Element Analysis ofRotating Disks,” Journal of Sound———-—and Vibration, Vo~77, No. ~–––~l_;-LiLi:-6~=78.

5. Patel, J. S. and Seltzer, S. M.,“Complex Eiqenvalue Solution to aSpinning Skylab Problem, ” NASATMX-2378, Vol. II, Sept. 1971.

6. Klahs, J. W., Jr., and Ginsberg,J. H., “Resonant Excitation of aSDinnina. Nutatina Plate.” Journal

7. Warmbrodt, W. and Friedmann, p.,“Coupled Rotor/Tower Aeroel asticAnalysis of Larqe Horizontal AxisWind Turbines, ” AIAA Journal, Vol.18, No. 9, 1980,~TTT~T24.

17

8. Voll an, A. J. and Zwaan, R. J.,“Contribution of NLR to the FirstPhase of the N?tional Researchprogram on !dind Energy, Part II:El astomechanical and Aeroel asticStability, ” National AerospaceLaboratory NLR, The Netherlands,NLR TR 77030 U Part II, March1977.

9. ottens, H. H. and Zwaan, F/. ,].,

“Description of a Method toCalculate the Aeroel astic

Stability of a Vertical Axis WindTurbine,” National AerospaceLaboratory NLR, The Netherlands,NLR TR 78072 L, June 1978.

10. Ottens, H. H. and Zwaan, R. J.,“Investigations on theAeroelastic Stability of LarqeWind Turbines, ” Proceedings ofthe Second InternationalSymposium on Wind Enercjy Systems,Paper C3, Amsterdam, TheNetherlands, Oct. 1978.

11. Hunter, W. F.,“Integrating-Matrix Method forDetermining the Natural VibrationCharacteristics of PropellerBlades, ” NASA TN D-6064, Dec.1970.

12. Carpenter, J. E. and Sullivan, E.M “Strl~ctural and Vibrationalc~;racteristics of WAOC S_5 ModelPropeller Blades,” WADC Tech.Rep. 56-298, DDC AD 130 787, U.S. Air Force, June 1957.

13. Popelka, D., “AeroelasticStability of a Darrieus WindTurbine, ” Sandia NationalLaboratories, SAND82-0677, March1982.

14. Dettman, J. W., MathematicalMethods in Physi~

———— _

~ri~iri66FTn~- ~;~;niition ,———McGraw-!%Tf Book Co., New York1969, p. 43.

15. MSC/NASTRAN User’s Manual Vols.—.—1 and II,

—— —— ——,‘~k{ MacNeaT-$chwendler

Corporation, [Los Angeles, 19/31.

16. Klosterman, A. L. and Zimlnerman,R “Modal Survey Activity ViaF~~quency Response Functions, ”SAE Paper No. 751068, NationalAerospace Engineering andManufacturing Meeting, CulverCity, Nov. 1975.

17. Modal-Plus Reference Manual—.. —,Version 6,

—-—___ 7The SDRC Corporation,

Milford, Ohio, Aug. 1981.

18