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Journal of Sound and Vibration

Journal of Sound and Vibration 330 (2011) 4413–4428

0022-46

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jsvi

Experimental modal substructuring to estimate fixed-base modesfrom tests on a flexible fixture

Matthew S. Allen a,n, Harrison M. Gindlin a, Randall L. Mayes b

a Department of Engineering Physics, University of Wisconsin-Madison, 535 Engineering Research Building, 1500 Engineering Drive, Madison, WI 53706, USAb Sandia National Laboratories, PO Box 5800, Albuquerque, NM 87185, USA

a r t i c l e i n f o

Article history:

Received 13 December 2010

Received in revised form

25 March 2011

Accepted 5 April 2011

Handling Editor: H. Ouyangthe motion at the connection point can be measured, but this approach can be highly

Available online 14 May 2011

0X/$ - see front matter & 2011 Elsevier Ltd.

016/j.jsv.2011.04.010

esponding author. Tel.: þ1 608 890 1619.

ail address: [email protected] (M.S. Alle

a b s t r a c t

Fixed boundary conditions are often difficult if not impossible to simulate experimen-

tally, but they are important to consider in many applications. In principle, modal

substructuring or impedance coupling approaches can be used to predict the fixed base

modes of a system from tests where the system has some other boundary condition if

sensitive to imperfections in the experimental measurements. This work presents two

alternatives that reduce the sensitivity to experimental errors, capitalizing on recent

works where additional degrees of freedom are used to improve the robustness of

substructure uncoupling. The system of interest is tested while mounted on a stiff

fixture, where some modes of the fixture inevitably interact with those of the system of

interest. The modes of the system–fixture assembly are extracted using a modal test

and then a modal substructuring approach is used to apply constraints to eliminate the

motion of the fixture. Two types of constraints are proposed, one based on the modes of

the fixture and the other on a singular value decomposition of the fixture motion that

was observed during the test. Neither approach requires an estimate of the displace-

ments or rotations at the points where the system of interest is connected to the fixture.

The methods are validated by applying them to experimental measurements from a

simple test system meant to mimic a flexible satellite on a stiff shaker table. A finite

element model of the subcomponents was also created and the method is applied to its

modes in order to separate the effects of measurement errors and modal truncation. The

proposed method produces excellent predictions of the first several modes of the fixed-

base structure, so long as modal truncation is minimized. The proposed approach is also

applied to experimental measurements from a wind turbine blade mounted in a stiff

frame and found to produce reasonable results.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

No fixture can reproduce a perfectly rigid boundary condition; at some frequency the interaction between the fixtureand the structure will begin to be important, causing the modes of the assembly to be considerably different from thefixed-base modes that would be predicted by an idealized finite element model. However, it would be very convenient tobe able to estimate the fixed-base modes of a structure experimentally so they could be used to update or validate themodel for the structure.

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M.S. Allen et al. / Journal of Sound and Vibration 330 (2011) 4413–44284414

For example, testing and model validation campaigns for satellite systems often include both a low-vibration levelmodal test, used to extract the modal parameters of the system, and subsequent shaker testing at higher amplitudes toevaluate the durability of the system. Modal parameters extracted from the former are correlated with FEA models whichare then used to predict the life of the structure in a specified environment. The latter tests are meant to verify the FEApredictions either by verifying that the test article survives the vibration environment or that the strains measured atcritical points are the same as those predicted by the model. There are a number of reasons for desiring to combine thosetwo tests. First, the high amplitude shaker tests better approximate the environment of interest, so it would be preferableto extract modal parameters from those tests in case the structure has any nonlinearity that would change its effectivestiffness or damping with excitation level. Second, each test increases the time and cost required to develop the system, sosignificant savings might be realized if one of the tests can be shortened or eliminated. This approach may also helpminimize uncertainty in the boundary conditions that are applied in shaker testing, and it may be preferable to use thefixed-base modal parameters for model updating rather than the free–free modal parameters. Blair discussed some ofthese issues in the context of model validation for space shuttle payloads in [1]. This work presents two methods that canbe used to estimate the fixed-base modes of a structure from tests on a flexible shaker table, addressing many of theseissues.

To illustrate the proposed methodology, consider the system pictured in Fig. 1, which consists of a wind turbine blademounted in a stiff fixture. Suppose one desires to find the modes that the turbine blade would have if it were constrainedperfectly to ground. It is impractical to construct a fixture that is rigid enough to adequately simulate a fixed boundarycondition over the frequency band of interest. However, using the method proposed in this work one could simply find themodes of the blade when it is attached to the fixture shown and then apply constraints to the identified modes to estimatethe fixed-base modes of the blade. This system will be treated in detail in Section 4.

A few previous works have explored methods for predicting fixed-base modal parameters from test measurements.These were discussed in Mayes and Bridgers [2] so only a few key observations will be repeated here. Using classicalfrequency-based substructuring (FBS) [3] or impedance coupling [4,5] methods, one can, in principle, estimate the fixed-base properties of a test article if all of the displacements and rotations, as well as the forces and moments between thetest article and the supporting structure are measured. This is rarely possible and often leads to ill-conditioning in thecoupling equations even if the necessary moments and forces can be measured. Mayes and Bridgers presented a methodthat avoids the need to measure the interface forces and displacements and which seems to minimize this ill conditioning [2].They used frequency based substructuring to couple the modal motions of a shaker table to ground using the authors’ modalconstraint approach [7] (some additional details were presented in their preliminary works [6,8]). The authors’ work and a fewother recent papers [9–11] have all revealed that it is important to consider other degrees of freedom in addition to theconnection point(s) when uncoupling two substructures. Mayes and Bridgers extended these ideas to constrain a substructureto ground (rather than removing it), and showed that their approach was capable of extracting the first fixed-base mode of acantilever beam from measurements on a simplified shake table.

This work expands upon Mayes and Bridgers’ work in two ways. First, this work employs modal substructuring ratherthan frequency based substructuring. The modal substructuring approach is convenient because one only needs tomanipulate the modes of the structure, whereas with frequency based substructuring the entire frequency responsefunction matrix must be manipulated at each frequency line. Second, this work considers a large number of modes of boththe fixture alone and the fixtureþstructure, showing that the approach also works well for higher modes and exploringthe test frequency bandwidth needed to estimate fixed-base modes over a desired frequency range. The proposed

Fig. 1. Photograph of a 4.3 m wind turbine blade mounted in a stiff frame during the modal test described in Section 4.

M.S. Allen et al. / Journal of Sound and Vibration 330 (2011) 4413–4428 4415

methodology is a special case of modal substructuring, specifically, coupling one substructure to another that isinfinitely rigid.

There is a large body of literature concerning modal substructuring, much of which is described in two review articles[3,12]. A few details will be repeated here to explain the basis for the proposed approach. Modal substructuring is knownto provide accurate results so long as the modal bases of the substructures are adequate to describe their motion in theassembled configuration. Several approaches have been proposed to assure that this is the case. The Craig–Bampton [13]approach is the most popular in the analytical realm, but it is poorly suited for experiments since it requires a test on eachof the substructures with rigid boundary conditions, which, as mentioned earlier, can usually not be adequatelyapproximated in the laboratory. The free modes of a structure are much easier to obtain experimentally, but they areknown to constitute a poor basis for substructuring predictions [14], as nicely illustrated in [15]. Better results areobtained when the free modes are augmented with residuals to account for the flexibility of out of band modes, aspioneered in [14,16,17]. However, the residual terms are weakly represented in the measurements and can be difficult toaccurately measure, and one must typically apply many additional inputs to obtain a full set of residual flexibilities.Several researchers have made strides in addressing this issue by, for example, creating and then updating FEA models toestimate the residuals rather than attempting to measure them completely [15,18], but there are limitations to thisapproach and further research is needed. For these reasons, the authors have pursued a different approach in which themass-loaded interface modes of the structure are used in place of the free modes [7,19,20]. Mass-loaded modes naturallyform a much more efficient modal basis than free modes, yet they are relatively easy to measure experimentally, especiallyusing new techniques that do not require the attached masses to be completely rigid [7]. The methodology proposed herebuilds on the mass-loaded modes approach, using it to constrain one subsystem to ground.

The rest of this paper is organized as follows. Section 2 presents the proposed modal substructuring techniques that areused to apply a set of constraints that convert the measured modes to fixed-base modes. In Section 3 the proposedmethodology is evaluated by applying it to a simple plate–beam structure. Experiments are performed to find the modes ofthe fixture (plate) and the fixtureþstructure (plateþbeam), and the substructuring results are presented revealing someimportant considerations. The method is applied to a wind turbine blade in a stiff fixture in Section 4 and Section 5 thensummarizes the conclusions.

2. Theoretical development

Suppose that a structure of interest is attached to a flexible fixture. A modal test [4] could be performed to obtain N

modes of vibration at a certain set of measurement points, resulting in the following modal parameters for thefixtureþstructure:

or , zr ,uf

us

" #, (1)

where or is the rth natural frequency of the structure, zr is the corresponding damping ratio and the matrices uf and us

contain the mass-normalized mode shapes at the measurement points on the fixture, yf, and substructure, ys, respectively.The index r ranges over all of the measured modes, r¼1,y,N. The fixture is a dynamic system itself, although it is meant toapproximate a rigid boundary condition. For example, the shake tables discussed in [2] can be thought of as stiff,translating fixtures.

One can also obtain the modal parameters of the fixture alone (without the structure of interest attached) eitherthrough test or analysis, and its modal parameters are denoted:

ofixtr , zfixt

r , ufixtf : (2)

The natural frequencies and damping ratios of the fixture are not needed in any of the following, only the massnormalized mode shapes of the fixture, ufixt

f .If the fixture were truly rigid and perfectly fixed to ground, then the motion of the fixture, yf, and the corresponding

mode shapes, uf, would be zero. In practice this will not be the case. The most straightforward remedy would be toestimate both the displacement and rotation at the points where the structure joins the fixture and then to applyconstraints to force that motion to be zero. However, that method is not reliable as will be illustrated in Section 3.3. Whenuncoupling one substructure from another, recent research has revealed that one must consider other points in addition tothe connection point in order to avoid ill-conditioning [7,9,11]. Those results seem to be relevant to this problem as well,prompting the alternative approach that is described below.

2.1. Modal constraints

In this work, we shall presume that the measured fixture motion, yfixt, can be approximated as follows in terms of Nfixt

modes of the fixture:

yfixt �ufixtf qfixt, (3)


where qfixt, denotes the modal coordinates of the fixture. This expression can be inverted to find the participation of each

fixture mode (i.e. using a modal filter [21]), as qfixt � ðufixtf Þ

þ yfixt, where ()þ denotes the Moore–Penrose pseudo-inverse of

the matrix. The estimate of the fixture modal amplitudes is only meaningful here if one has at least as many measurement

points as modes of interest and if the measurement locations on the fixture are chosen such that ufixtf has full column rank.

If the modal basis of the fixture is sufficiently rich, then it will be adequate to span the space of the motion of the fixtureeven after the structure is attached (i.e. of the fixtureþstructure). If this is the case, then the fixture modal motions can beestimated from the measured motions, yf, as follows:

qfixt � ðufixtf Þ

þyf ¼ ðufixtf Þ

þuf q, (4)

where q denotes the modal coordinates of the fixtureþstructure . Our desire is to estimate the modal parameters that thestructure would have if it were attached to a truly rigid fixture. One way to do this is to apply the constraints,

qfixt ¼ 0, (5)

to the modes of the fixtureþstructure using the Ritz method [6,7,22]. In terms of the modal coordinates of thefixtureþstructure, the constraint equations are

ðufixtf Þ

þyf ¼ 0, (6)

or

ðufixtf Þ

þuf q¼ 0, (7)

either of which constitutes a set of Nfixt constraint equations. The matrix multiplying q is the matrix [a] in the text byGinsberg [22], or B in the review by de Klerk et al. [3]. The procedure described in either of those works can be used toenforce these constraints and hence to estimate the modes of the fixtureþstructure with the fixture motion nullified. The‘‘ritzscomb’’ Matlabs routine, which is freely available from the first author or on the Matlab Central File Exchange,implements the method in [22] and was used to perform the calculations for this work.

It is important to note that the constraints above only enforce zero motion at the fixture measurement points if thenumber of measurement points equals the number of fixture modes. In practice one should use more measurement pointson the fixture than there are active modes in order to average out noise and measurement errors. However, when this isdone the motions of the physical measurement points may not be exactly zero after applying the constraints. In the bestcase the residual motion would be due only to measurement noise, but there might also be residual motion in the fixturethat is physical, since one is seeking to constrain an infinite dimensional system with a finite number of constraints.Fortunately, one can readily observe the fixture motions after applying the constraints to see whether the constraints wereeffective in enforcing a rigid boundary condition. This is illustrated in Section 3.2 and provides a valuable way to checkwhether enough modes were used in Eq. (7).

2.2. Singular vector constraints

In some cases it may not be feasible to perform a modal test on the fixture alone in order to estimate ufixtf . There also

may be situations in which the free modes of the fixture do not form an efficient basis for the motion of thefixtureþstructure, so some other basis must be considered. In these cases one alternative is to use the dominant singularvectors of the fixture mode shape matrix to form the constraints. The first step is to perform a singular valuedecomposition (SVD) of the fixture motions observed on the fixtureþstructure

uf ¼USVT, (8)

where U is a matrix of singular vectors, S a diagonal matrix of corresponding singular values and V reveals how eachsingular vector participates in each of the modes. Both U and V are orthonormal. If the motion of the fixture is sufficientlysimple, then it is likely to be captured in only a few, NSVD, dominant vectors, so one can approximate the fixture motion asfollows:

yf �UNSVD SNSVD ðVNSVD ÞTqf , (9)

where UNSVD and VNSVD contain only the first NSVD columns of U and V, respectively, and S is an NSVD�NSVD matrixcontaining only the first NSVD singular values along its diagonal. The following constraint will set this approximation to thefixture’s motion to zero:

ðUNSVD ÞTyf ¼ 0: (10)

This is an alternative to the constraints in Eq. (7) and avoids having to measure the modes of the fixture alone. Thisapproach shall be termed ‘‘SVD constraints’’ in the following.


In some cases, one might wish to mix the free-mode approach with this SVD approach. For example, the authors havefound that when a system contains rigid body modes, it is critical to include them before considering any other motionscontained in uf. Specifically, if the mode shape matrix is divided into rigid body and elastic modes as

uf ¼ðuf ÞRB ðuf Þe

h i(11)

then one would perform SVD as in Eq. (8) but only on the partition of elastic modes ðuf Þe. The resulting singular vectorscould be used together with the rigid body modes as a basis for the constraints as follows:

ðuf ÞRB ðUNSVD Þe

h iþyf ¼ 0, (12)

where ðUNSVD Þe are the first NSVD singular vectors of ðuf Þe. This approach is used in Section 3 and compared with the free-modes approach. The standard SVD approach in Eq. (10) is used in Section 4 where the proposed methods are applied to awind turbine blade in a test frame.

3. Evaluation of methodology using plateþbeam system

3.1. Application using experimental measurements from plateþbeam

A simple system, consisting of a 12.35 in. long steel beam attached to a 12�10 in. steel plate was constructed toevaluate the proposed approach. The 1 in.�0.5 in. cross-section beam is the substructure of interest, and the 0.625 in.thick steel plate is stiff and approximates a rigid boundary condition. A schematic of the system is shown in Fig. 2, with theleft picture looking down the beam onto the plate (the beam extends from the origin in the negative x-direction), and theright figure looking parallel to the surface of the plate showing the beam standing up. The measurement grid used inthe experiments is also shown, along with the locations used for three uni-axial accelerometers. The red circles show thelocations of the accelerometers used to test the plateþbeam, the blue show those used when the plate was tested alone tofind ufixt

f (references to color pertain only to the online version of this article). The accelerometers placed on the plate wereactually placed on the underside of the plate so that the structure could be excited from above at each of the points shown.The dimensions of the plate and beam were chosen so that many of the natural frequencies of the plate would be equal tothose of the fixed-base beam. This assured that the modes of the plate and beam would interact creating an interestingcase study.

The assembly was placed on an inflated rubber tube to simulate free-boundary conditions, which were realized quiteeffectively since the first vibration mode of the system was about 10 times higher than the highest rigid body mode of thesystem on the inner tube, conforming to the guidelines in [23]. Fig. 3 shows a photograph of the setup. The beam is

Fig. 2. Schematic of plate–beam system used to estimate the fixed-base modes of a steel beam. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)

Fig. 3. Photograph of the experimental setup showing the plateþbeam on top of an inflated tube. Two of the accelerometers are visible, the 3rd is

located on the back of the beam in this setup so only its cable is visible.


attached to the plate with two hex screws. Some initial measurements were processed with the zeroed-fast Fouriertransform nonlinearity detection method described in [24], which revealed that the natural frequencies of the systemdid decrease significantly with increasing excitation amplitude. To remedy this, the system was disassembled andreassembled with adhesive between the beam and plate and with the bolts very tight. After doing this, the method in [24]no longer revealed significant nonlinearity and testing proceeded.

A roving hammer modal test was used to obtain the modal parameters of the system. Three hits were applied to each ofthe 36 points on the plate, 6 on the y-side, 6 on the z-side, 12 on the y-side of the beam, and 12 also on the z-side of thebeam giving 72 total measurement points. Once a reasonable set of FRFs had been obtained, the modes were identifiedusing the AMI algorithm [26–28] and the mode shapes and natural frequencies were imported into Matlab to perform thesubstructuring analysis. A similar set of tests was also performed on the plate without the beam attached to estimate ufixt

f .Table 1 lists the natural frequencies found for the plate and plateþbeam. The experiments were designed to extract all

modes below 3 kHz and the results show that those and quite a few more were extracted. Comparing the naturalfrequencies before and after adding the beam, we see that the beam causes the first mode of the plate to split, the familiarvibration absorber effect, and similar mode splits for the 3rd and 4th elastic modes of the plate. This mode splitting isillustrated in Figs. 4 and 5, which show the 3rd and 4th mode shapes of the plateþbeam. The surface plot shows thedeflection of the plate and the two line plots show the bending of the beam in both the y- and z-directions. The beambending has an opposite sense in the 4th mode as in the 3rd mode, demonstrating that the beam is acting as a vibrationabsorber for the plate. In either mode shape the motion of the beam resembles the 2nd analytical mode shape for acantilever in the y-direction, so one would expect this pair of modes to merge to a single bending mode of the beam oncethe constraints at the base of the beam are enforced. The beam’s displacement in the z-direction is small and presumablydominated by noise or errors in the alignment of the hammer blows.

The experimentally measured plate and plateþbeam modes were used in the procedure described in Section 2 toestimate the fixed-base modes of the beam. That procedure requires an estimate of the rigid body modes of the system,and rather than measure those a finite element (FEA) model (the same one described later in Section 3.2) was used toestimate them. Hence, CMS was performed by combining six FEA rigid body modes with 18 experimentally measuredplateþbeam modes, and then creating ufixt

f with six FEA rigid body modes for the plate alone and six measured platemodes for a total of 12 constraints in Eq. (7). The density of the FEA model was adjusted to reproduce the measured massof the system. (The FEA model was also validated by comparing it with the experimental results, as described in Section 3.2,but that is not particularly relevant to the results presented here.) Table 2 shows the modal substructuring (CMS) predictions

Table 1Experimentally measured natural frequencies (Hz) of the plate and plateþbeam.

Nat. freqs. (Hz) of elastic modes

Mode Plate alone (fixture) Mode Plateþbeam Plateþbeam modes

1 670.5 1 130.7 Beam 1y

2 893.7 2 224.0 Beam 1z

3 1344.0 3 628.3 Beam 2yþPlate

4 1620.3 4 693.8 Beam 2yþPlate

5 1850.4 5 902.4 Plate

6 2538.0 6 1254.4 Beam 2zþPlate

7 3107.8 7 1350.2 Beam 2zþPlate

8 3143.6 8 1657.8 Plate

9 3591.1 9 1770.7 Beam 3yþPlate

10 4082.0 10 1797.8 Beam 3yþPlate

11 4814.5 11 1906.1 Beam 3yþPlate

12 4837.7 12 2311.9 Plate

13 5220.6 13 2995.7 Plate

14 5561.4 14 3107.7 Plate

15 6765.1 15 3233.0 Plate

16 7093.4 16 3424.5 Beam 3zþPlate

17 7147.8 17 3522.8 Beam 4yþPlate

18 7179.0 18 3845.5 Plate

19 7808.4

20 8288.7

21 8414.8

05

10

15

0

5

10–1

–0.5

0

0.5

1

12" side (z)

Mode 3 at 628.3 Hz

10" side (y)

x di

sp 0 5 10 15–4

–2

0

2

Beam Axial Position (–x)

y di

sp

Beam Deformation

0 5 10 15–3

–2

–1

0

1

2

Beam Axial Position (–x)

z di

sp

Beam

Fig. 4. Mode shape of 3rd elastic mode of the plateþbeam: (a) shows a three-dimensional view of the deformation of the plate, while (b) and (c) show

the deflection of the beam in the y- and z-directions. The beam is fixed to the plate midway between the two square markers, as illustrated

schematically.


as well as the analytical natural frequencies of a cantilever beam with the same properties as the actual beam. The first fiveCMS predicted natural frequencies agree fairly well with the analytical ones, but beyond the 5th the predicted naturalfrequencies are inaccurate. The rule of thumb for CMS predictions is that coupled system predictions are typically valid overa bandwidth, BW Hz, if modes for each substructure are used out to 1.5*BW or 2*BW. Here, modes out to 3000 Hz were usedin the CMS predictions, so one would expect the results to be accurate out to 1500 or 2000 Hz. The errors in the CMSpredicted natural frequencies are below 10 percent for the first five modes, which suggests that this rule of thumb may be

05

1015

0

5

10-1

-0.5

0

0.5

1

12" side (z)

Mode 4 at 693.8 Hz

10" side (y)

x di

sp 0 5 10 15-2

-1

0

1

2

Beam Axial Position (-x)

y di

sp

Beam Deformation

0 5 10 15-2

-1

0

1

2

Beam Axial Position (-x)

z di

spFig. 5. Mode shape of 4th elastic mode of the plateþbeam (see description for Fig. 4).

Table 2Estimated fixed-base natural frequencies of the beam found using the free-mode procedure with 12 modes of the plate and 24 plateþbeam modes. The

natural frequencies of the plateþbeam before substructuring are shown in Table 1.

CMS prediction with12 MCFS constraints

Analytical fixed-basebeam freq. (Hz)

Percent error in CMSprediction

104.1 107.35 �3.0

195.2 214.70 �9.1

651.4 672.8 �3.2

1230.5 1345.5 �8.6

1777.1 1883.4 �5.6

1823.0 3691.4 �50.6

2369.4 3766.8 �37.1

2923.8 6102.2 �52.1

3045.3 7382.9 �58.8

3352.8 9115.6 �63.2

3503.8 12,204.3 �71.3

3620.2 12,731.7 �71.6


valid for this CMS procedure. One also observes that the modes that involve z-direction motion have larger percent errors.The z-direction is the stiffer of the two bending directions of the beam.

As mentioned in Section 2, it is advisable to observe the motion of the fixture after applying the constraints in Eq. (7) tocheck whether enough constraints were used to force the fixture motion to zero. This was done and the norm of themotion over the plate was found to be between 0.6 and 4.3 percent of the maximum mode shape for the first six modesestimated by CMS, suggesting that the constraints were quite effective. The motion of the plate after applying theconstraints had no recognizable pattern, suggesting that it was due to noise in the measured mode shapes, so it is notshown. The mode shapes over the beam are shown in Fig. 6. Each plot overlays the estimated y-bending shape, z-bendingshape and the analytical shape of a cantilever with the same properties as the experimental. The shapes agree very well.There is a scale difference between the first bending modes and the analytical, but otherwise the shapes are quite similar.The second modes also agree closely, but the shapes near the root of the beam suggest that the rotation there may not beexactly zero.

The CMS procedure was repeated using different numbers of plate modes, and hence different numbers of constraints.There were larger errors in the predicted natural frequencies for modes 3 and 4 if fewer than eight constraints were used,while mode 5 was not accurately predicted unless at least 12 constraints were used. However, there was virtually noimprovement in the natural frequencies if the number of constraints was increased from 12 to 16. If more than 16 wereused then one begins to have only a few modes left in the system (i.e. 24–18¼6), causing the results to degrade.

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

3

x-position (in)

Exper yExper zAnalytical

0 2 4 6 8 10 12 14-3

-2

-1

0

1

2

3

x-position (in)

Exper yExper zAnalytical

Fig. 6. Mode shapes of the beam found with the proposed CMS procedure. For comparison, the analytical mode shapes of an Euler–Bernoulli cantilever

are also shown: (a) shows the first mode in both the y- and z-directions while (b) shows the 2nd mode in each direction.


The procedure based on the singular value decomposition was also used on this system. The SVD was computed of theelastic plateþfixture modes that were identified experimentally, and the six rigid body modes were used in Eq. (12)together with the six dominant singular vectors. Quite similar results were obtained with this method. The errors in thefirst three modes were �3.23, �9.78 and �3.16 percent, respectively, which are very similar to the results above inTable 2. The errors were higher for the 4th and 5th modes, �10.1 and �12.0 percent, respectively. The mode shapesobtained were very similar to those shown in Fig. 6. Although the results are not as accurate as was obtained with the free-modes, this method produced reasonable results and has the significant advantage that it does not require an additionaltest to estimate the modes of the fixture alone.

3.2. Finite element simulation of substructuring for plateþbeam

A finite element model was created to estimate the rigid body mode shapes, as mentioned previously, and also todetermine whether the discrepancies between the CMS and analytical modes observed in Section 3.1 were due to noise inthe experimental results or whether modal truncation was the culprit. The FE model was constructed with simple shelland beam elements, with 21 nodes along the length of the beam and 405 eight-node shell elements for the plate. The meshwas such that the elements on the beam were 0.95 in. long while the elements used for the plate were 1.0 in2. The FEMwas validated by comparing the FEM modes and natural frequencies with those obtained experimentally, the results ofwhich are shown in Table 5 in the Appendix. The comparison there shows that the correlation between the experimentand FEA model is excellent, suggesting that the FE model is an accurate representation of the real system. The ModalAssurance Criterion (MAC) [4] between the experimentally measured mode shapes and the FEA mode shapes are all above0.92. This is a simple system and easy to model, so one would expect the FEA model to be highly accurate and to show howwell the proposed CMS procedure would work with near perfect measurements.

The FEA models for the plate and plateþbeam were used in conjunction with the proposed procedure to estimate thefixed-base modes of the beam. Only the mode shapes at the measurement points defined in Fig. 2 were used to facilitatethe comparison with the experimental results that were presented previously. The results were very similar to those foundexperimentally Table 2. The errors in the first five modes were �3.8, �14.4, �3.8, �12.3 and �3.9 percent, respectively,

0 2 4 6 8 10 12 140

0.5

1

1.5

2

2.5

x-position (in)

FEA-CMS yFEA-CMS zAnalytical

0 2 4 6 8 10 12 14-2

-1

0

1

2

3

x-position (in)

FEA-CMS yFEA-CMS zAnalytical

Fig. 7. Mode shapes of fixed-base beam predicted by CMS using FEA-derived mode shapes, compared with the analytical mode shapes (see caption

for Fig. 6).


while the higher modes had much larger errors as in the experimental results. As observed in the experimental results,only the first five modes are well predicted by CMS, and the even natural frequencies (z-direction bending) are lessaccurate than the odd ones. Surprisingly, the 2nd and 4th natural frequencies found here using the FEA modes are lessaccurate than those found using the experimental modes. The mode shapes predicted by CMS, shown in Fig. 7, matchalmost perfectly with the analytical ones, although some small residual rotation is visible at the base of the cantilever.

As mentioned in Section 2, it is advisable to check the motion of the fixture to assure that the constraints were adequateto reduce its motion to a negligible amount. This was investigated by plotting the mode shapes of the plate after applyingthe constraints in Eq. (7). Those plots do show a marked rotation of the plate in the bending direction of the beam for eachof the beam’s bending modes, as illustrated for mode 4 in Fig. 8. The plate motion is less than 0.03, while the maximumplate motion in each of the modes of the plate was about 1.0 before the constraints were applied. The plate motion shows anonzero rotation in the bending direction of the beam (rotation in the z-direction, about the y-axis), which would reducethe effective stiffness of the beam somewhat. However, the rotation is less than a degree, so one would expect it to benegligible.

Another way of determining whether modal truncation is important is to check that the plate’s free modes spanthe space of the observed plateþbeam modes. If this is not the case, then the approximation in Eq. (3) will not be accurate.To check this, the FEA predicted modes for the plateþbeam were projected onto the plate modes usinguproj

f ¼ufixtf ðu

fixtf Þ

þuf . The largest difference between the projected shape and the measured shape was found anddivided by the maximum absolute value of any coefficient in that shape. This gives the maximum percent error in eachexpanded shape, given in Table 3. Modes 1–16 of the plateþbeam are accurately captured with 12 plate modes, but modes 17and 18 show significant errors. Those errors could be reduced to 13 and 12 percent, respectively, by increasing the number ofconstraints from 12 to 16, but, as mentioned previously, the predicted natural frequencies did not change noticeably when 16constraints were used instead of 12.

3.3. Discussion

The fact that the CMS predictions have about the same level of error as the experimental predictions suggests that theerrors in both methods are dominated by modal truncation. This is reassuring, especially considering the large errors thatare sometimes encountered in substructuring predictions due to cross-axis sensitivity and other unavoidable experimental

Table 3Maximum error in expansion of the FEA mode shapes for the plateþbeam onto the 12 plate modes.

Mode Nat. freq. Max. err. (percent)

Zero for modes 1–6

7 126.34 0.6

8 207.2 1.6

9 630.96 1.6

10 688.82 1.3

11 899.33 0.3

12 1207.9 6.2

13 1349.2 1.9

14 1641.3 1.4

15 1786.1 4.3

16 1895.3 5.4

17 2316.9 35.1

18 2996.3 58.0

05

1015

0

5

10-0.03

-0.02

-0.01

0

0.01

0.02

0.03

12" side (z)

Mode 4 at 1181 Hz

10" side (y)

xdis

p 0 5 10 15-4

-2

0

2

Beam Axial Pos (-x)

y di

sp

Beam Deformation

0 5 10 15-4

-2

0

2

Beam Axial Pos (-x)

z di

spFig. 8. Deflection of plate and beam in 4th estimated mode (after constraining plate motion to zero). In the three dimensional plot the beam (not shown)

would be located midway between the square markers on the plate (see the description for Fig.4).


errors [29,30]. A standard, inexpensive modal test was used to obtain the modes that were used in this work, yet thosemodes were still accurate enough to obtain reasonable estimates of the fixed-base natural frequencies of the system.

The alternative to the approach presented in this paper would be to estimate the rotations and displacements at theconnection point and then to apply constraints to force them to be zero. The connection point motions are known to bedifficult to measure, so that method was expected to be more sensitive to noise. To explore this, an alternate analysis wasperformed in which some of the measurements (those points within the rectangle denoted with a dashed line in Fig. 2)were used to estimate the displacement and rotation of the connection point in the out of plane direction. Themeasurements on the perimeter of the plate were used to estimate the in plane motion. The connection point motionin all six degrees of freedom was then constrained to zero to estimate the fixed-base modes of the beam. When thisapproach was used, the estimates for the first two bending modes of the beam were in error by �14 and �21 percent,respectively (as compared to �3 and �9 percent for the proposed method, see Table 2). The next few modes identified bythis approach were spurious with varying degrees of motion of both the plate and beam. Clearly the methods presentedhere produce better qualitative results. It should also be noted that no residual terms were used in the analysis justdescribed. Residuals have been found to significantly improve the modal basis when two free-mode models are joined.Unfortunately, residuals cannot easily be incorporated into the methodology presented here. Residuals could be readilyincorporated into a frequency based substructuring algorithm based on connection point constraints, but here the focuswas on a modal approach that minimizes the ill-conditioning that is present in the connection-point problem.


On the other hand, it is disappointing that the natural frequencies estimated by the proposed methods are not moreaccurate. This was explored in order to identify the cause. The fact that the plate shows residual rotation near theconnection point in Fig. 8 suggests that the modal basis used to describe the plate may not have been adequate tocompletely constrain its motion to ground. The plate seems to be deforming near the point where the beam is connected.In order to understand the effect of this localized deformation on the CMS predictions, another model was created wherethis region was stiffened. In this model, a 4 in.�4 in. square of the plate surrounding the point where the beam connects(outlined with a dashed line in Fig. 2) was made to be three times as thick (1.875 in.) as the rest of the plate (0.625 in.).This significantly increases the stiffness of the plate near the connection point and would be expected to reducethe localized kinking shown in Fig. 8. This model was used in the CMS procedure as described above and the fixed-basenatural frequencies of the beam were estimated. The errors in the CMS predicted frequencies for y-direction bending, the1st, 3rd and 5th natural frequencies, were only �0.3, �0.5 and �0.8 percent, respectively, using this modifiedplate model, as compared to �3.8, �3.8 and �3.9 percent for the regular plate as discussed in Section 3.2. Similarly,the errors in the z-direction bending frequencies reduced from �14.4 and �12.3 percent to �1.3 and �2.1 percent,respectively. This suggests that localized bending of the connection point was responsible for the errors observed in thenatural frequencies. In future works this difficulty can and should be addressed by designing the fixturing to minimizelocalized bending, otherwise a very large number of modes might be needed to obtain accurate predictions of the fixed-base modes.

4. Application to wind turbine blade

The proposed methods were also used to estimate the fixed-base modes of a wind turbine blade, from measurementswhere the blade was mounted in a stiff frame. Fig. 1 shows a photograph of the blade in question, a 4.3 m long blade from a20 kW wind turbine manufactured by Renewegy LLC. The blade is mounted in a frame which was designed to hold theblades during fatigue testing. The frame is stiff, but its flexibility becomes important for many of the modes of the blade, sothe methods presented in this work will be used to estimate its fixed-base modes. The fixed-base modes are desired tovalidate and update the finite element model of the blades.

Fig. 9 shows a schematic of the system indicating all of the measurement points acquired during the test. Only a part ofone day was available for the testing, and the geometry of the frame was not known in detail before arriving, so very littlepre-test planning was performed. Three accelerometers were mounted near the tip of the blade as shown in the schematic,the lower of the two was a high sensitivity triaxial accelerometer, and two of its channels were recorded corresponding toflapwise (z-direction in Fig. 9) and edgewise (y-direction in Fig. 9) motion of the blade. The other accelerometer measuredonly the flapwise motion. The accelerometers can also be seen in the inlaid photograph in Fig. 1. All of the points indicatedin Fig. 9 were excited using an impact hammer and reciprocity was used to form an 89 output by 3 input matrix of FRFs.The accelerometers correspond to points 101z, 2z and 52y. After some investigation, the FRFs corresponding to inputpoints on the fixture were found to be very noisy in the measurements from the low sensitivity accelerometer (at point101z), so the measurements from that accelerometer were discarded. Also note that the frame is anchored to the groundby eight bolts, four of which are seen in Fig. 1. The measurement points 311–363 were chosen away from those anchors,

y

z

211

212

213

201

202

203

221

222

223

Accel.

12x

y

34567891011121314151617181920

101103105107109110111112113114115116117118119120

351 (361) 352 (362) 353 (363)

311 (321) 312 (322) 313 (323)251(261)

252(262)

253(263)

254(264)

TriaxialAccel. ...51y…60y…70y…

Fig. 9. Schematic of wind turbine blade mounted in the fixture showing measurement point locations. Measurement points shown in parenthesis are in

the corresponding positions on the front leg of the frame.

0 20 40 60 80 100 120 140 160 180 200

10-2

10-1

100M

agni

tude

(m s

-2 N

-1)

Data

FitFit Error

0 20 40 60 80 100 120 140 160 180 200-180-90

090

180

Pha

se (

°)

Frequency (Hz)

Fig. 10. Magnitude and phase of frequency response function corresponding to acceleration at 2z and input at 16z. Lines correspond to: (solid gray)

measurement, (dash-dot green) curve fit, and (solid red) difference between measurement and curve fit (references to color are applicable only to the

online version of the article). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


but even then the blade moved little when excited on this lower part of the frame. In all of the measurements, anexponential window was employed to accelerate the decay of the low frequency modes and the FRFs were estimated usingthe H1 method with three averages at each point.

The measurements were then curve fit and the first 18 mass-normalized modes of the system were extracted. Fig. 10shows a sample of the measurements and the curve fit that was obtained using the AMI algorithm [26]. The measurementsappear to be of reasonable quality, although the residual after curve fitting is somewhat larger than might be obtained inan ideal scenario. The first two modes of the beam were fit separately as this was found to produce reconstructed FRFs thatagreed more closely with the measurements.

The natural frequencies of the modes that were identified are shown in Table 4. For brevity the mode shapes are notshown, but they were viewed, and a description is listed for each of the modes, indicating whether the motion of the bladewas primarily bending in the flapwise direction (FW), edgewise direction (EW), or torsion of the blade. Beyond the 18thmode the blade’s modal density became quite high so curve fitting was not attempted. A few of the modes seem to showsimilar motion of the blade occurring at two or more distinct frequencies. For example, the 6th and 7th modes bothresemble the 4th bending mode of a cantilever beam, and their frequencies are quite similar. This is illustrated in Fig. 11,which shows the mode shapes of Modes 6 and 7 before applying the constraint procedure to estimate the fixed basemodes. Two traces are shown for each blade, corresponding to the measurement points on the leading and trailing edges.(The estimate for mode 6 obtained using the SVD method is also shown, as is discussed later.) Inspection of the 6th and 7thmode shapes reveals that the fixture is acting as a vibration absorber for these two modes. Both modes show a similardeformed shape of the blade, but the fixture bends in one direction for the 6th mode and in the opposite for the 7th. The15th and 16th modes showed a similar phenomenon, although torsion of the beam was also apparent in each of thosemodes. The motion of the blades was at least 10 times higher than the motion of the fixture for all of the modes. The lowermodes generally showed less motion of the fixture while the higher frequency modes showed about a factor of 10difference between the two. The motion of the upright part of the frame (points 200–264) was typically larger than themotion on the lower part (points 300–363).

The proposed CMS procedure was applied to the modes in the far left column in Table 4 in order to estimate the fixed-base modes of the blade. Since limited time was available, a separate test could not be performed on the fixture alone,so the SVD method described in Section 2.2 was used. Some initial tests produced some unreasonable results andinvestigation revealed that the curve fits were unreliable at measurements points 300–336 due to noise in themeasurements. As a result, the mode shapes at those points were not used in the CMS procedure. After discarding thosepoints, the first six singular values of the mode shape matrix at the fixture locations, ufixt

f were 0.183, 0.125, 0.079, 0.040,0.026 and 0.017. The first three are at least twice as large as the rest, and the mode shapes suggest that there are two or

Table 4Natural frequencies of wind turbine blade in frame. The mode shape descriptions indicate which modes correspond to edgewise bending (EW), flapwise

(FW) bending and torsion.

Mode num. Mode shape description Natural freq. (Hz) 2 SVD constraints 3 SVD Constraints

fn CMS Percent diff. fn CMS % Diff

1 FW B1 3.36 3.83 12.1 3.84 12.4%

2 EW B1 5.24 5.27 0.5 5.28 0.7%

3 FW B2 11.40 11.44 0.4 11.64 2.1%

4 EW B2 22.42 22.52 0.4 22.77 1.6%

5 FW B3 28.44 28.85 1.4 29.54 3.7%

6 FW B4, Fixtureþ 45.50 48.92 7.0 50.26 9.5%

7 FW B4, Fixture– 52.26 – – – –

8 EWþFW 53.37 – – – –

9 EW B3 58.29 56.52 �3.1 56.96 �2.3%

10 1st Torsion 80.01 79.96 �0.1 79.97 0.0%

11 FW B5 83.54 81.84 �2.1 83.90 0.4%

12 EW B4 107.37 106.85 �0.5 107.01 �0.3%

13 FW B6 118.25 115.77 �2.1 119.75 1.2%

14 2nd Torsion 143.47 143.45 0.0 143.54 0.0%

15 FW B7, Tors. 150.29 150.12 �0.1 154.12 2.5%

16 FW B7, Tors. 156.21 154.18 �1.3 – –

17 EW B5 þFW 169.61 168.30 �0.8 159.09 �6.6%

18 FW B7, EW B8, Torsion 184.11 183.02 �0.6 182.97 �0.6%

00.511.522.533.544.5-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

x-Position

Mod

e Sh

ape

Shapes of 6th & 7th Modes before CMS and 6th Mode after CMS

Leading EdgeTrailing EdgeCMS M6

M6

M7

Leading EdgeTrailing Edge

Fig. 11. Selected mode shapes of wind turbine blade as identified from the measurements and after constraining the base with CMS: (open circles) Mode

7 Identified, (closed circles) Mode 6 Identified, (solid) Mode 7 after CMS. Blue corresponds to the measurements on the leading edge of the blade and

green to the trailing edge. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)


three modes in the measurements due the fixture dynamics, so two and three constraints were applied based on thedominant singular vectors.

The frequencies obtained by the CMS procedure are also shown in Table 4, as well as the percent difference betweenthem and the corresponding natural frequency of the system before applying the constraints. After applying threeconstraints, two of the modes between 45 and 54 Hz disappear, leaving only one mode in that frequency band witha natural frequency of 50.26 Hz. The mode shape of that mode is shown in Fig. 11. The resulting shape is quite similar toboth of the 6th and 7th modes, although the scale is different and there are significant differences in the three shapes nearthe root of the blade. The 6th and 7th identified modes show significant displacement near the root, while thedisplacement is near zero in the mode that was produced by CMS. The 15th and 16th identified modes were also foundto merge when three SVD constraints were applied, and the resulting mode shape seemed to be a plausible 7th flapwisebending mode .


5. Conclusions

This work presented a new method of estimating the fixed-base modes of a structure from measurements with thestructure attached to a flexible fixture. The proposed approach was validated using experimental measurements froma simple plate–beam system. A finite element model of that system was also created to determine how the method wouldperform if the experiments were perfect. The comparison between the finite element and the experimental resultssuggests that the proposed method is quite robust to errors in the measurements, which is important since some level oferror is unavoidable in experimental modal analysis. The method approximates the motion of a fixture as a sum of modalmotions, and then constrains each modal motion to ground. For the system considered here, the results were alwaysqualitatively reasonable even if far too few constraints were used, and the fidelity of the prediction increased as thenumber of constraints increased. For example, the plateþbeam system had pairs of modes where the motion of the beamand plate was highly coupled, those modes merged smoothly to a single mode near the true natural frequency as thenumber of constraints increased. On the other hand, one must remember that each constraint eliminates a mode ofthe fixtureþsubsystem, so physically meaningful modes may be constrained away if too many constraints are used. Whenthe proposed methods were applied to the plateþbeam system, there were still moderate errors in the predictions of thefixed-base natural frequencies near this limit. The analysis revealed that those errors were caused by modal truncation,which was exacerbated by the fact that the plate deforms locally near the point where the beam is connected. This canand should be addressed by designing the experimental fixturing to spread out the load near the connection point.Some metrics were presented that can be used to detect this issue, and so they should be used to check the validity of theCMS predictions when applying this technique to real systems. In any event, the beamþplate problem represents ademanding application in which the fixture was not very stiff relative to the substructure of interest. In most applications,such as the wind turbine blade studied here, this will not be the case. In any event, the results presented here suggestthat this is a simple and effective method for estimating the fixed-base modes of real structures from experimentalmeasurements.

Acknowledgements

This work was supported in part by Sandia National Laboratories. Sandia is a multiprogram laboratory operated bySandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear SecurityAdministration under Contract DE-AC04-94AL85000. The authors also wish to thank Scott Miller for his efforts in helpingto design and perform initial tests on the plateþbeam system.

Appendix A

Table A1 shows the natural frequencies of the experimental plateþbeam system compared with the naturalfrequencies of the FEA model described in Section 3.2. The MAC and modal scale factor between the mode shapes isalso shown for all but the 9th mode, which did not correspond to any of the modes of the FEA model.

Table A1Comparison of experimentally measured natural frequencies with those from the finite element model.

Mode Experiment FEA model MAC MSF

1 130.66 126.34 0.9966 1.03

2 224.0 207.2 0.9960 1.01

3 628.3 631.0 0.9928 0.98

4 693.8 688.8 0.9942 1.04

5 902.4 899.3 0.9655 1.06

6 1254.4 1207.9 0.9676 0.91

7 1350.2 1349.2 0.9266 1.05

8 1657.8 1641.3 0.9675 1.06

9 1770.7 None

10 1797.8 1786.1 0.9586 0.74

11 1906.1 1895.3 0.9893 0.98

12 2311.9 2316.9 0.9765 1.08

13 2995.7 2996.3 0.9713 1.10

14 3107.7 3119.9 0.9800 1.03

15 3233.0 3255.6 0.9617 0.99

16 3424.5 3383.8 0.9681 1.48

17 3522.8 3552.2 0.9752 0.91

18 3845.5 3836.6 0.9704 0.73


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