Experimental Uncertainty Quantification of Modal Test Data

Experimental Uncertainty Quantification of Modal Test Data

D. Todd Griffith* and Thomas. G. Carne† Sandia National Laboratories‡ Albuquerque, New Mexico 87185-0557 ABSTRACT In this paper we present the results of a study to quantify uncertainty in experimental modal parameters due to test set-up uncertainty, measurement uncertainty, and data analysis uncertainty. Uncertainty quantification is required to accomplish a number of tasks including model updating, model validation, and assessment of unit-tounit variation. We consider uncertainty in the modal parameters due to a number of sources including force input location/direction, force amplitude, instrumentation bias, support conditions, and the analysis method (algorithmic variation). We compute the total uncertainty due to all of these sources, and discuss the importance of proper characterization of bias errors on the total uncertainty. This uncertainty quantification was applied to modal tests designed to assess modeling capabilities for emerging designs of wind turbine blades. In an example, we show that unit-to-unit variation of the modal parameters of two nominally identical wind turbine blades is successfully assessed by performing uncertainty quantification. This study aims to demonstrate the importance of the proper pre-test design and analysis for understanding the uncertainty in modal parameters, in particular uncertainty due to bias error. INTRODUCTION When one performs an experiment, it is very important that uncertainty in the information derived from the experimental data be quantified. For experiments designed to validate analytical models uncertainty quantification is not merely important, it is required. In this paper we focus on quantifying the uncertainty for one type of test, a modal test. The uncertainty in the modal parameters estimated from the modal test data are due to a number of sources that are inherent to the process involved in a modal test: test set-up, measurement acquisition, and data analysis. Each of these categories of uncertainty is described in more detail in the next section. There are many reasons why the uncertainty in modal data should be quantified. As was already mentioned, one reason is for updating of analytical models. There are formal strategies for model updating based on the uncertainty in the comparison objects between the experimental and analytical data. For example, if one wants to update a structural dynamics model (or a damping model) by comparing measured and predicted natural frequencies (damping ratios), it would be necessary to have an understanding of the uncertainty in the natural frequency (damping ratio) of each measured mode. The analyst would want to update the model such that the natural frequency (damping ratio) of each predicted mode is within the quantified uncertainty bounds. Detailing one formal strategy, model updating would include a weighted least squares solution [1] in which the objective is to determine a set of model parameters which minimize the following objective function:

J = eT We

*

Limited Term Member of Technical Staff, [email protected] Distinguished Member of Technical Staff, [email protected] ‡ Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. †

(1)

where e is a vector whose components are the difference between the measured and predicted natural frequencies (damping ratios). Here, the optimal choice of weighting W is a diagonal matrix whose diagonal elements are one divided by the variance associated with the uncertainty of the corresponding modal data:

W = diag (1/ σ i2 ) . In this way, modes with small uncertainty are given a greater weight (a larger confidence) in the optimization process. There are other reasons why uncertainty quantification is useful. For example, suppose that one wants to evaluate the variation in modal properties from unit-to-unit of some product that is being mass produced. In order to accomplish this, the uncertainty associated with the modal properties extracted by the test evaluation method must be known. Later in the paper, we present an example of assessing unit-to-unit variation for two nominally identical wind turbine blades. One very important technical challenge related to our example application is fatigue modeling and prediction in wind turbine blades. Reliability of wind turbine blades is strongly dependent upon understanding and design of fatigue life into wind turbine blades, which is dependent upon the damping characteristics of the blade. Thus, when the damping data includes the uncertainty in its measurement one will have a better understanding of fatigue life and reliability. Thus, we quantify the uncertainty in damping from the wind turbine blade modal tests to improve our understanding of damping information. For example, uncertainty quantification can be used to determine bounding values for damping that can used for validation of blade damping models or fatigue models. These types of models may be developed in the future. In this work, we have attempted to do a complete evaluation of the many sources of uncertainty that can be present in a single test. In Reference [2], some ideas were presented on uncertainty quantification of modal test data, although in this study the uncertainty quantification was not considered in the design of the test, and as a result it was only applied to a limited number of data sets. In contrast, for the current study, uncertainty quantification was considered in the pre-test analysis and design in order to assess the total uncertainty. Some work has been done on assessing algorithmic variation [3]. A large amount of work has been published in the literature regarding bias errors in modal test data. A few papers we note include works on assessing support condition effects [4] and correcting for mass-loading effects [5]. Certainly many more works have been undertaken to understand algorithmic variation, support conditions, and mass loading than those listed here. The outline of the paper is as follows. First, we take a detailed look at the sources and types of uncertainty which are inherent to modal test data. Then, we discuss how we arrive at a total uncertainty estimate that results from each of the individual sources. Next, we describe the modal tests performed to quantify each source of uncertainty experimentally. We look at an illustrative example for one source of uncertainty, the support conditions of the modal test. Finally, we apply the uncertainty quantification approach to a series of tests conducted on wind turbine blades to illustrate the experimental uncertainty quantification approach, and its application to assessing unit-to-unit variations. SOURCES AND TYPES OF UNCERTAINTY Sources of uncertainty in parameters estimated from modal data are present throughout the entire test process. Uncertainty sources exist in the test set-up; they arise due to measurement technique and measurement choice; and they occur in the data analysis process. Sources of test set-up uncertainty include support conditions, mass loading and damping due to instrumentation, and environmental factors. The uncertainty associated with boundary or support conditions must be well characterized. Fixed or cantilever type boundary conditions must be characterized because it is practically impossible to realize a perfectly rigid boundary condition. The same holds for realizing the free boundary condition for a freely suspended test article, even though it is usually assumed that soft suspension systems can be neglected. When validating a model with experimental data, the boundary conditions of the experiment must be characterized well enough that they can be accurately represented in the model. Measurement uncertainty can result from the choices made in exciting a structure in measuring the structural response. For example, variation of force level and variation of the location of the force input can contribute to the measurement uncertainty. Different sources of measurement uncertainty exist for different types of excitation

(impact hammer versus shaker) and different types of response measurement (acceleration versus strain). Certainly, the sources of uncertainty vary depending on the type of test being performed. A complete list of uncertainty sources for all test situations is not attempted in this paper although the basic categories of uncertainty are applicable to most tests. Analysis uncertainty arises due to choices made in the data reduction, or parameter estimation approach. The application of different algorithms results in a variation of the modal parameters, and this variation is considered as a source of uncertainty. Another source of analysis uncertainty can be associated with the user of the algorithm because choices must be made in applying an algorithm. Different users may choose different frequency ranges, model orders, or initial guesses for parameter values. Before proceeding much further, we must define a few terms that are used in this paper. Variability and uncertainty are concepts that many times are used interchangeably; however, their meanings are quite different in a technical sense. Uncertainty occurs because of a lack of knowledge. Variability refers to a known diversity in some quantity. If a larger number of observations or better quality observations can be made, uncertainty can be reduced. Variability cannot be reduced or simplified; however, it can be better characterized with more observations. These basic definitions, taken from the Environmental Protection Agency (EPA) online glossary [6], are simple enough to express the fundamental difference between uncertainty and variability, and seem appropriate for a modal test. Errors in experimental data are composed of two components: random and bias error. Bias error (systematic error) is a constant in the total error measure, whereas the random component is governed by its statistical properties. For a modal test, we consider the following sources of uncertainty and categorize them as random and bias as given here: Random sources:

Force Level Input Location Algorithmic

Bias sources:

Support Conditions Mass Loading Instrumentation Cable Effects Ambient Environment

Throughout, we assume that the uncertainties from each of these sources are not correlated because we make an effort to isolate each individually. Furthermore, we assume that we know the sign associated with bias errors because a sufficient amount of analysis and testing is performed to understand them. It is very important to note that this allows for the bias errors to be combined. As a result, one can reduce the total uncertainty in the test data by identifying and quantifying the bias errors because the data can be adjusted or corrected accordingly. If we do not understand the magnitude and sign associated with the bias errors, then we must assume they are random errors in the total uncertainty. An important consideration in evaluating the size of the uncertainty due to each source is to understand the inherent uncertainty in a single measurement. In this case, the single measurement we consider can be the result of a number of averaged measurements. This inherent uncertainty can be assessed by repeating a measurement with all sources of uncertainty held constant. Here one can simply acquire two or more measurements successively. A comparison of the information derived from these measurements provides a measure of this inherent uncertainty. One would then consider that any random or bias uncertainty, that is determined to be larger than the inherent measurement uncertainty, is significant enough to be included in the total uncertainty estimate.

COMPUTATION OF TOTAL UNCERTAINTY IN THE MEASUREMENT In the previous section, the individual sources and types of uncertainty were discussed. In this section, we show how we arrive at the total uncertainty of each modal parameter due to the individual sources. The method we adopt is based on Reference [7]. The two basic assumptions here are 1) the random uncertainty sources, characterized by an associated standard deviation, are uncorrelated, and 2) the bias uncertainty is also characterized by a standard deviation which equals ½ of the total bias error (essentially the bias error is conservatively assumed to equal two standard deviations). Together, these assumptions permit the combination of the random and bias errors into a single uncertainty measure. The bias error must be included in the total uncertainty along with random error when the sign associated with the bias error is not known. In this case we must treat the bias as a random error, which increases the total uncertainty.

U

f total

 n 2  B 2  = k  ∑σ i +     i  2   

As Equation 2 shows, the total uncertainty in the parameter

(2)

f , for example, is given by a factor of k times the

square root of the sum of the variances of the random sources,

σ i2 , and the variance of the bias source,

( B2 )

2

,

[7]. The factor k is called a coverage factor. It is intended that this coverage factor expand the coverage of the uncertainty estimate depending on the percent confidence desired in the estimate and the quantity of observations available for computing the random and bias variances. For a 95% confidence interval and sufficiently large number of observations, an appropriate value for the coverage factor is 2 [7]. This is the value we use in this report. Of course, the coverage factor is larger for higher desired confidence and/or a small number of observations. Ultimately, we apply Equation 2 to natural frequency and damping data from a series of modal tests designed to quantify uncertainty. We mention here that the total uncertainty can be reported in a variety of formats, which depends upon primarily the quality of the data. For example, the bias errors can be evaluated in a gross fashion for all the modes in a particular subset (e.g. bending modes, torsional modes) or they can be evaluated on a mode by mode basis. The choice depends on the quality of the measured data, and the ability to draw conclusions about the sign and magnitude of the bias errors on a mode by mode basis. As will be detailed later, some sources of bias error can be assessed on a mode by mode basis. We should also make a note here regarding how we determine random uncertainty. Equation 2 requires a single standard deviation measure ( 1σ ) for each random source of uncertainty. The approach we adopt is to determine the largest or worst case error in a parameter from a set of experiments which isolate a single source of uncertainty and call this the single standard deviation measure associated with that particular source. This is a conservative approach because the largest or worst case error measure is typically considered to be three standard deviations. One has a choice as to how much conservatism is built into the uncertainty quantification. DESCRIPTION OF MODAL TESTS PERFORMED TO QUANTIFY UNCERTAINTY We consider quantifying the uncertainty in the modal data for a wind turbine blade. For this test series, two nominally identical blades were modal tested using impact excitation. Each blade is nominally 8.325 meters (27.3 feet) and 127 kilograms (290 lbs). A photo of one of the blades is shown in Figure 1. Additionally we note that the root end of the blade is 0.5 meters (20 inches) in diameter, and the blade CG location is nominally at 2.1 meters (84 inches) from the root end.

Flatback region

Z Y

X Figure 1. Photo of BSDS Wind Turbine Blade

The blade was instrumented with 48 biaxial accelerometers plus 10 strain gauges in the flatback trailing edge region for a total of 106 measurement channels. No measurements were made in the axial direction (Z-direction). The instrumentation layout in the axial direction was determined by optimizing over the first three flat-wise bending modes (bending about the X-axis) as determined by the pre-test finite element analysis. The instrumentation was placed in the X-direction (vertical direction of Figure 2) such that no sensors were placed on the internal shear web of the blade, which is essentially an I-beam used to stiffen the blade in flat-wise bending. We avoided mounting on it in order to measure localized panel (plate) modes. Also, the accelerometers were placed on the blade extremity in the X-direction to better resolve the torsional modes. The blade is relatively flexible at the tip end. As a result, a large sensor density was deemed necessary near the tip as shown in the instrumentation layout of Figure 2. Accelerometers were placed on both sides of the blade for the first 6 stations starting from the root end. The blue markers in Figure 2 indicate corresponding sensors on the opposite side of the blade. The total mass of the instrumentation including the accelerometers, mounting blocks, and adhesive was measured after the test and found to equal 0.88 kilograms (1.9 lbs).

Z X Figure 2. Instrumentation Layout for BSDS Wind Turbine Blade Excitation was provided with a 3 lb impact hammer with a soft tip. The frequency range of interest for this test was 0-160 Hz, and this hammer configuration provided good input characteristics. This structure has

approximately 20 modes in this frequency range; however, only the first 6-8 modes up to about 60 Hz are of interest for the purpose of model validation. The blade was supported softly at two locations. Straps were used to hold the blade using a choker style loop. The suspension was softened using bungee cords as can be seen in Figure 1. A variety of bungee cord configurations were used for these tests, including a set of bungee cords deemed to be optimal. The two support locations were placed on the nodes of the principally effected mode, the first edge-wise bending mode (about the Y-axis). This was done so as to minimize the effect of the support condition on the first edge-wise bending mode. Furthermore, the supports were chosen to be as soft as possible. More discussion on support conditions is provided in the following section. Minor modifications to the standard test plan were needed to perform the uncertainty quantification. In summary, the uncertain variables include input force location and input force level; ambient conditions; analysis algorithms uncertainty; and support conditions and instrumentation. Uncertainty due to input location and force level was evaluated simply by performing a few additional tests at different locations and repeats at a single location with varying level. For each individual test, input level for all averages was maintained at +/- 10% of the nominal value; anything outside this range was not accepted. Because these tests were performed in a controlled environment indoors, there was no significant variation in the ambient condition to record. Therefore, ambient conditions uncertainty is assumed insignificant for these tests. No additional tests were required to evaluate the algorithmic uncertainty. This uncertainty was evaluated by using different algorithms on the same data sets. The two algorithms used were the Synthesis Modes and Correlate Algorithm (SMAC) [8] developed at Sandia and the PolyReference Frequency Domain algorithm implemented in the IDEAS software. Throughout the paper, SMAC is the nominal algorithm used unless otherwise noted. Support conditions, mass loading, and damping due to the instrumentation are bias sources which contribute to the uncertainty. Support condition uncertainty was evaluated both experimentally and analytically. A detailed look at the support condition analysis is given in the next section. Mass loading and damping due to the instrumentation were evaluated experimentally in a final set of tests. A series of three tests were conducted for a group of reference locations including 1) a fully instrumented baseline test with measurements for all 96 accelerometers, 2) a reduced cable test in which cables were removed from all but 12 accelerometers while removing no accelerometers, and 3) a reduced cable and accelerometer mass test in which the remaining unused accelerometers from the second test in the series were then removed. The objective of these tests was to determine the individual effect of the mass and damping of the cables from the mass effect of the accelerometers by comparing each of the lightly instrumented data sets with the baseline set. This set of 12 accelerometers included biaxial measurements at 6 locations, two biaxials at the root end, two at the tip end, and two near the mid-span of the blade. AN ILLUSTRATIVE EXAMPLE OF ONE SOURCE OF UNCERTAINTY: SUPPORT CONDITIONS In this section, we illustrate the uncertainty due to the support conditions. In particular, these results show that a large bias error is introduced if the support conditions are not well designed. In this case we are concerned with the design of free-free support conditions for the BSDS wind turbine blade (as shown in Figure 1). We aim to make the stiffness of the supports as low as possible in a dynamic sense by reducing the frequency of the rigid body modes. Usually one designs the support with only the stiffness effect in mind; however, the supports can also add a significant amount of damping. Here, we compute the effect of the support on natural frequency and damping by including the support stiffness and damping into an analytical model of the structure. For the BSDS wind turbine blade, we created a FEM model which approximates the tapered blade as a uniform beam. The mass, length, and natural frequency of the first two principal bending modes (first flat-wise -- bending about the X-axis; and first edge-wise -- bending about the Y-axis) were correlated to those of the pre-test high fidelity finite element model. The mass and stiffness matrices of this model were loaded into MATLAB so that calculations could be performed by adding stiffness and damping associated with the bungee suspension. Some additional details regarding support condition analysis can be found in Reference 9. The stiffness and damping coefficients can be computed in two ways: (1) by measuring them from dedicated tests on bungee cords, and (2) deducing them from rigid body bounce modes determined during the modal test. For any pre-test analysis one must rely on data from dedicated tests to get an estimate of the support conditions

effect. We are mainly concerned with the effect of the support on the natural frequency and damping of the first edge-wise bending mode, which is a bending mode in the direction of the bungees. The suspension has an effect on the flat-wise bending modes as well; however, the stiffness of the support associated with the flat-wise direction is small compared to the stiffness in the edge-wise direction. Furthermore, the nodes of the first flat-wise mode are very close to the support location (within 10 cm for the root end support and 33 cm for the tip end support). Dedicated bungee cord testing revealed that a bungee stiffness coefficient of 7004 N/m (40 lb/in) and a bungee damping coefficient of 56 Ns/m (assuming a damping coefficient of 3% consistent with test data) are appropriate at a pre-load equal to ½ of the blade weight for 6 loops of bungee cord. For this pre-test analysis the weight distribution on each bungee is equal because the supports are located symmetrically about the CG. Figure 3 shows the predicted error in the natural frequency of the first edge-wise mode as the bungee suspension are moved symmetrically from the blade ends to the CG (The CG is located at L = 4.15, where L is defined in the schematic shown in Figure 3). This error is computed by differencing the natural frequency computed using the modified stiffness matrix and the nominal free-free value. One can see that this error varies significantly as a function of the location of the supports. The error is zero when the supports are placed exactly at the nodes for this uniform free-free beam model. The worst location to place the supports are outside the node locations, especially near the beam ends (L=0). The largest percent error predicted by this analysis for this 16.5 Hz mode is about 1.9% with supports at the beam tips. Figure 3 illustrates the importance of the location of the supports. Of course, this error strongly depends on the stiffness of the bungee cords as well as their location. The actual magnitude of any errors introduced when the supports are not placed at the node locations must be computed using stiffness values determined at the appropriate deformation levels. A comparison of the predicted error with experimental results are given later in this section.

L

L

Figure 3. Error in First Edge-wise Natural Frequency due to Support Stiffness Figure 4 shows the amount of modal damping predicted by the model to be added to the first edge-wise bending mode as a result of the damping of the bungees. These damping values were determined by forming a damping matrix by placing the bungee damping coefficients on the appropriate degrees of freedom of the damping matrix, and then projecting the mode shape vector matrix from the undamped eigenanalysis (which included the stiffness

of both supports) onto this damping matrix. This assumes that the mode shapes are not affected very much by the addition of these damping elements. The trends in the damping added by the bungees to the first edge-wise mode as a function of location are the same as that on the natural frequency. As much as 1.6% (of critical) damping can be added to this mode if supported at the tips, and 0.7% damping can be added if supported near the CG. This is a significant amount of damping added to a lightly damped structure such as a wind turbine blade. Again, we compare these estimates of damping with those determined experimentally later in the section.

Figure 4. Modal Damping Added as a Result of Support Damping Some preliminary modal tests were conducted to experimentally assess the effect of the support conditions. These tests were performed using very massive high sensitivity accelerometers which tended to significantly mass load the structure; therefore, we see that these values tend to be inconsistent with those reported later in the paper. Table 1 shows the effect of the support conditions on the natural frequency and damping of the first edge-wise mode as measured from four different support conditions (various locations and bungee cord configurations).

Table 1. Experimental Assessment of Support Condition Effect First Edgewise Support Characteristics Mode Location Configuration Number (inches Number freq Description Damp (%) from of (Hz) root Loops end) +/- 75 cm (30 1 8 , 8 54 , 114 16.38 1.00 in) from CG +/- 75 cm (30 2 6 , 6 54 , 114 16.18 0.80 in) from CG 3 On the nodes 6 , 6 38 , 232 16.09 0.73 4

On the nodes

4,2

38 , 232 16.07

0.63

Configuration 1 is a stiff configuration with 8 loops located 30 inches to each side of the blade CG. If we reduce the number of loops to 6 for each support, the frequency drops by 1.2% and the damping drops by 20%. Moving to Configuration 3 results in a 1.8% reduction in frequency and a 27% reduction in damping. Configuration 4 is an optimal configuration in which the supports are soft and placed on the nodes. We see that the support may add more than 35% damping to this mode and can shift the frequency by 2% if the support is poorly designed. We now compare the analytical predictions with the experimental results. The results of Figures 3 and 4 for a value of L equal to 3.4 meters (+/- 0.75 meter from the CG) correspond to Configuration 2. in Table 1. We find that the analysis predicts a frequency error of 0.55% and added modal damping of 0.45% of critical. Experiments show that the frequency error is 0.68% and the added modal damping is 0.17% of critical. The agreement is good for estimation of the frequency effect, but not for the damping effect for our approximate uniform blade model. This type of model can be quite useful; however, the values for the stiffness and damping coefficients must be accurate for it to be useful for frequency and damping. We would like to point out that in our dedicated tests of bungee cords we found that bungee cords are quite complicated as they have a strong dependence on amplitude of motion. This is confirmed in modal tests which reveal that the bungees are stiffer at modal levels of deformation as determined from the natural frequency of the rigid body bounce mode than at the amplitudes of the dedicated tests. EXPERIMENTAL EXAMPLE: CALCULATION OF UNCERTAINTY IN MODAL DATA Ideally, we prefer to quantify the total uncertainty in the natural frequency and damping on a mode by mode basis. For some individual sources of uncertainty we can do this; however, there are other individual sources for which this cannot be accomplished. When we cannot, a reasonable compromise is to assess the individual sources for subsets of modes. For the BSDS wind turbine blade, the types of modes are: 1) bending modes, 2) torsional modes, and 3) localized panel (plate) modes. The modes which are analyzed in this uncertainty quantification for the BSDS 001 blade and BSDS 004 are listed in Table 2 along with the modal parameters extracted from fully instrumented data sets.

Table 2. Modes for BSDS 001 and BSDS 004 Blades

Mode Number Mode Description

BSDS 001 Fully Instrumented

BSDS 004 Fully Instrumented

Freq (Hz)

Damp (%)

Freq (Hz)

Damp (%)

1

1st Flap-wise

5.45

0.45

5.25

0.46

2

2nd Flap-wise

13.5

0.43

13.5

0.55

3

1st Edge-wise

16.5

0.97

17.2

1.2

4

Panel Mode

21.4

1.1

22.1

1.1

5

3rd Flap-wise

25.4

0.43

24.5

0.56

6

4th Flap-wise

38.6

0.60

37.9

0.62

7

2nd Edge-wise

40.1

0.94

40.7

1.0

8

Torsional

52.5

1.2

53.2

0.72

9

Torsional

56.1

0.82

55.8

0.77

10

3rd Edge-wise

72.1

0.72

74.0

0.70

In any test, practical limits are placed on the amount of measurements that can be taken. Of course, if the resources were available we could record and analyze a larger number of data sets so that statistically meaningful results can be attained. Furthermore, we would prefer to have a large number of test articles so that unit-to-unit variation can be better assessed; however, in this case we only have two units available. In this case, we perform the analysis on a limited number of data sets and as a result choose to be conservative. When determining the uncertainty due to a single source, we specify the uncertainty to be a one sigma error. A less conservative approach would be to determine the largest error and assign to this value a two sigma or three sigma error. Conservatively, we choose to assign one sigma to the largest error. The first step in this uncertainty quantification process is to evaluate the inherent uncertainty in a measurement by performing test repeats as described earlier. Repeat measurements at a few different excitation locations were needed to evaluate this inherent uncertainty. Upon analyzing the chosen frequency range of 5-80 Hz, we find that the inherent uncertainty in frequency is 0.05% and for damping is 2%. For the majority of the modes, the inherent frequency uncertainty is less than 0.001%; however, we choose to be conservative. This analysis has been applied to data from each of the two blades tested. We note a few special cases, namely 1) for panel modes, the inherent frequency uncertainty is 0.4% and inherent damping uncertainty is 15%, and 2) the inherent damping error for the first flap-wise bending mode is 5%. Usually, these special cases can be linked to our ability to well excite these modes. We consider any random or bias errors found to be larger than these criteria to be significant enough to contribute to the total uncertainty. First we look at the random sources of uncertainty. Force Level. Tests were performed at ½ times and 2 times the nominal force level. No variation was found because this structure is quite linear. Therefore, we conclude that the uncertainty due to force level is insignificant compared to the inherent uncertainty. Nonetheless, we still follow our requirements that for any set of measurements the input force level for the set must be within +/- 10% of the chosen target value. Force location. In determining the uncertainty due to force location, we compare inputs which are in the same direction, but are located at opposite ends of the beam. One issue that must be considered throughout the entire analysis is how well excited is a particular mode. It would not be correct to determine uncertainty by comparing two modes which are not excited equally well; therefore, for the case of evaluating force location we compare data sets for which the direction of input is the same.

As an example showing how we estimate the uncertainty, consider Table 3 which is a comparison of excitation at the root end and tip end of the blade both in the Y-direction. The panel mode was not excited for these tests. After analyzing four data sets including that shown in Table 3, we conclude that for bending modes the uncertainty due to force location is 0.2% and for damping is 15%. For torsional modes, we find 0.2% for frequency and 20% for damping. Each of these is conservatively assumed to be a single standard deviation measure as will be done for all random sources. For the panel mode we assume that the uncertainty is 0.4% for frequency and 20% for damping. We note that for modes 5,6, and 7 the damping uncertainty is typically lower than 15%. The frequency data in Table 3 indicates that the force location may be a bias error; however, in general this is not the case when looking at the other three references. Table 3. Force Location Comparison: BSDS 004 Root End versus Tip End Mode Number

Mode Description

Force Location Percent Difference Freq (Hz) Damp (%)

1

1st Flap-wise

-0.13%

-13%

2

2nd Flap-wise

-0.15%

-0.6%

3

1st Edge-wise

-0.10%

11%

5

3rd Flap-wise

-0.15%

0.0%

6

4th Flap-wise

-0.16%

2.0%

7

2nd Edge-wise

-0.29%

9.6%

8

Torsional

-0.17%

3.9%

9

Torsional

-0.26%

-11%

10

3rd Edge-wise

-0.29%

0.0%

Algorithmic Uncertainty. The algorithmic uncertainty was estimated by applying two different algorithms to the same data. The two algorithms used were the SMAC algorithm and the PolyReference Frequency Domain algorithm. SMAC is a single reference algorithm; therefore, we utilized the PolyReference algorithm on a single reference data set in order to be consistent. PolyReference, as the name implies, is capable of processing multiple references simultaneously. We determine from the analysis of five data sets that the frequency uncertainty for bending and torsional modes to be 0.05% and for damping 6%. For panel modes, we find a 1% uncertainty in frequency and 30% in damping. The damping uncertainty for the bending and torsional modes is moderately conservative; however, several data sets indicate that for modes 1 and 2 the uncertainty in frequency is 0.1% and 10% for damping. Again, each of these is conservatively assumed to be a single standard deviation measure. Overall, the algorithmic uncertainty is found to be somewhat smaller than force location uncertainty. As an example, consider the comparison of the two algorithms shown in Table 4.

Table 4. Algorithmic Uncertainty (BSDS 001 Root End Flat-wise) Mode Number

SMAC Fit (1) Mode Description

IDEAS PolyRef Freq (2)

Percent Difference (1-2)

Freq (Hz) Damp (%) Freq (Hz) Damp (%) Freq (Hz) Damp (%) 1

1st Flap-wise

5.432

0.669

5.438

0.679

-0.11%

-1.49%

2

2nd Flap-wise

13.473

0.527

13.470

0.452

0.02%

14.2%

3

1st Edge-wise

16.525

1.063

16.524

1.129

0.01%

-6.21%

5

3rd Flap-wise

25.313

0.538

25.331

0.561

-0.07%

-4.28%

6

4th Flap-wise

38.558

0.569

38.553

0.580

0.01%

-1.93%

7

2nd Edge-wise

40.000

0.800

39.998

0.812

0.01%

-1.50%

8

Torsional

52.926

0.754

52.927

0.806

0.00%

-6.90%

9

Torsional

56.472

0.739

56.483

0.766

-0.02%

-3.65%

10

3rd Edge-wise

72.155

0.739

72.244

0.646

-0.12%

12.6%

Now we look at the sources of uncertainty due to bias errors. Support Conditions. As was shown in the previous section, the support conditions introduce a bias principally in the frequency and damping of the first edge-wise bending mode. Because the supports were placed on the nodes of the first edge-wise mode, we conclude that the uncertainty due to support conditions is zero for this mode. Furthermore, the effect on the first flat-wise mode is small because the stiffness of the support acting on the flat-wise mode is much smaller than that acting on the first edge-wise mode and the supports were also located closely to the nodes of the first flat-wise mode as well. We assume that the effect on the other modes is insignificant, although experimental characterization at different support locations would have been beneficial to confirm this assertion. Proper evaluation and design of the support conditions result in a practical elimination of uncertainty in the support conditions. Instrumentation. We now consider the effect of the instrumentation mass loading and cable damping. We use the IDEAS PolyReference algorithm to evaluate the instrumentation effect because some data sets only include 12 responses. The SMAC algorithm requires as many responses as modes. As was mentioned earlier, these effects were evaluated by performing a series of three tests. First, a baseline test was conducted with the full instrumentation set. Second, cables were removed from all but 12 accelerometers leaving all other instrumentation mass on the structure. Third, the accelerometers not used to record response were removed. The second test is used to determine the effect of the cables, and the third test is used to determine the combined mass loading effect. We found that the mass-loading effect can be assessed for each mode because of the consistent bias determined using 2 references on each blade. Damping data is consistent with very few exceptions. The damping added by the instrumentation is solely due to the cables. Modes 1 and 3 and the torsional modes experience the largest damping bias with 30-50% added to modes 1 and 3 and 20-30% added to the torsional modes. In broad terms, we find that the bending modes are uncertain by 0.5-1.5% in frequency due to mass loading. Torsional modes are 2.5-4.0% uncertain in frequency due to mass loading. Because no instrumentation was placed such that panel modes could be measured for the lightly instrumented data sets, we assume that panel modes are similar to bending modes with 1.0% frequency uncertainty, and 20% damping uncertainty. For the BSDS 004 blade, one of these series of three tests was performed with input at the root end. These results are tabulated in Table 5. These results indicate that the cable effect on mass loading is about ¼ the total mass loading effect. Interestingly, the torsional modes are 3-4 times as mass loaded as the bending modes.

Another observation is that the damping effect of instrumentation is solely due to the cables because not much change in damping is found from test 2 to test 3. Series of tests conducted at other reference locations produced results that were consistent in frequency and damping changes for each test in the series. Therefore, we can apply the uncertainty quantification on a mode by mode basis for the uncertainty due to mass loading and cable damping. Table 5. Instrumentation Effects for BSDS 004 Root End Flat-wise

Mode Number

Test (1) Fully Instrumented Baseline Freq (Hz)

Test (2) Minus Cables

Percent Difference (2-1)

Test (3) Lightly Instrumented

Percent Difference (3-1)

Damp Damp Damp Freq (Hz) Freq (Hz) Damp (%) Freq (Hz) Freq (Hz) (%) (%) (%)

Damp (%)

1

5.25

0.46

5.27

0.31

0.4%

-33.4%

5.32

0.32

1.4%

-31.2%

2

13.5

0.55

13.5

0.49

0.2%

-11.2%

13.6

0.48

1.0%

-13.2%

3

17.2

1.2

17.3

0.79

0.6%

-35.7%

17.5

0.76

1.7%

-37.8%

5

24.5

0.57

24.5

0.48

0.2%

-16.5%

24.7

0.46

1.0%

-19.1%

6

37.9

0.60

38.0

0.53

0.2%

-11.5%

38.4

0.52

1.1%

-13.7%

7

40.7

0.96

40.8

0.86

0.2%

-10.2%

41.1

0.85

0.9%

-11.1%

8

53.2

0.81

53.4

0.63

0.4%

-22.1%

54.2

0.58

1.8%

-28.8%

9

55.8

0.83

56.3

0.66

0.8%

-20.7%

58.1

0.66

4.1%

-20.8%

10

74.0

0.71

74.1

0.68

0.2%

-5.2%

74.6

0.61

0.8%

-15.1%

In the following section, we compute the total uncertainty in the measurements using Equation 2 based on the evaluations performed in this section. APPLICATION: ASSESSMENT OF UNIT VARIABILITY In summary, we find uncertainty in the natural frequency and damping for the modal tests conducted on the BSDS wind turbine blade to be due to four sources: 1) force location, 2) algorithm choice, 3) mass loading due to cables and accelerometers, and 4) damping due to instrumentation cables. We found that force level and ambient conditions have zero uncertainty for these tests. Additionally, we assume no uncertainty in support conditions because of a thorough pre-test analysis and the resulting placement of the supports on nodes of the principally affected mode. To summarize the results of the previous section, we list the uncertainty associated with each source in Table 6.

Table 6. Summary of Uncertainty Sources Source of Uncertainty Class of Mode

Force Location

σf

General force Case σ ζ force

Bending Modes Specific Cases

= 5%

σ

σf

General force Cases σ ζ force

For modes 1 and 2

= 0.2% σ

General Torsional Cases σ ζ force = 20% Modes Specific None Cases

Specific Cases

σ aζlg orithmic = 6%

f σ ζforce = 15% σ a lg orithmic = 0.1% ζ For modes σ a lg orithmic = 10%

f force

Instrumentation Cable Bias

Accelerometer Bias

f f = 0.2% σ aflg orithmic = 0.05% Bcables = 0.2% − 0.6% Baccel = 0.9% − 1.1%

1,2, and 3

Panel Modes

Algorithmic

f a lg orithmic

ζ Bcables = 15 − 20%

ζ Baccel = 0%

f Bcables = 0%

None

ζ Bcables = 30 − 50%

For modes 1 and 2

f f = 0.05% Bcables = 0.4% − 0.8% Baccel = 1.4% − 3.3%

σ aζlg orithmic = 6%

ζ Baccel = 0%

None

None

= 0.4% σ aflg orithmic = 1%

f Bcables = 0.5%

f Baccel = 1.0%

= 20% σ aζlg orithmic = 30%

ζ Bcables = 20%

ζ Baccel = 0%

None

None

ζ Bcables = 20% − 30%

None

None

None

We now compute the total uncertainty measure for the BSDS wind turbine blade modal tests using Equation 2. We use the values for the sources listed in Table 6 in this calculation for each class of modes; however, there is one exception in that for the effect of mass loading on frequency due to the accelerometers and cables (total mass loading) we use the uncertainty which was determined on a mode by mode basis as given in Table 5. The results are tabulated in Table 7. First we list the total uncertainty not including the bias errors. The bias errors dominant the uncertainty for this test; however, they are characterized well enough that bias can be removed from the data.

Mode Number

Table 7. Total Uncertainty for Each Mode Total without Bias Total with Bias Unit Variation Mode Description Freq (Hz) Damp (%) Freq (Hz) Damp (%) Freq (Hz) Damp (%)

1

1st Flap-wise

0.4%

36.1%

1.5%

53.9%

3.6%

-3.1%

2

2nd Flap-wise

0.4%

36.1%

1.1%

40.1%

0.0%

-28%

3

1st Edge-wise

0.4%

32.3%

1.7%

51.4%

-4.0%

-24%

4

Panel Mode

2.2%

72.1%

2.6%

74.2%

-3.6%

1.9%

5

3rd Flap-wise

0.4%

15.6%

1.1%

23.5%

3.5%

-28%

6

4th Flap-wise

0.4%

15.6%

1.2%

23.5%

1.7%

-4.4%

7

2nd Edge-wise

0.4%

15.6%

1.0%

23.5%

-1.5%

-7.8%

8

Torsional

0.4%

41.8%

1.8%

48.7%

-1.3%

40%

Torsional

0.4%

41.8%

4.1%

48.7%

0.5%

6.2%

0.4%

15.6%

0.9%

23.5%

-2.5%

1.9%

9 10

rd

3 Edge-wise

Thus we can now say with confidence that the unit variation (Tabulated in Table 7) in natural frequency for BSDS 001 to BSDS 004 is valid for all modes except mode 2 and mode 9 as determined by comparing the magnitude of the unit variation and the uncertainty excluding the bias terms. In fact, in this example even if we did not know the

signs associated with the bias terms we would reach the same conclusion because of the size of the unit variations. The uncertainty associated with damping is too large to conclude that the variation in damping from one blade to another is statistically significant for any of the modes. CONCLUSIONS In this paper, we have presented an experimental study for quantifying the uncertainty in modal test data. A number of factors were considered which create uncertainty in the natural frequency and damping derived from experimental data. These factors come from test-setup uncertainty, measurement uncertainty, and data analysis uncertainty. Typically, modal test engineers assume that frequency estimates contain small uncertainty, and tend to discount damping data as being fairly uncertain. In this report, we quantify the uncertainty in both natural frequency and damping so that both are useful for formal efforts such as model validation and the evaluation of unit variation. For example, structural dynamics models can be validated using the uncertainty in natural frequency. Also, damping models and fatigue models can be validated taking into account the uncertainty in damping. However, in this paper we applied this uncertainty quantification to evaluation of unit variation of two nominally identical wind turbine blades. We found that the observed unit variations are indeed statistically significant for natural frequency – a conclusion that cannot be made if one does not know the uncertainty in the measured frequencies. This study emphasizes the importance of carefully designing a modal test by performing pre-test analysis to reduce bias errors resulting from the test design. We make a few additional comments on the major findings in this work as they relate to modal test design. First, we find that random errors tend to have a smaller effect on the uncertainty than bias errors although tests must be performed to assess how much uncertainty the random errors introduce. Secondly, bias errors tend to be larger sources of uncertainty than random errors. Thus, one must do a careful study of these bias sources because when they are well characterized the data can be adjusted and the total uncertainty is greatly reduced. Pre-test analysis of the test design is useful; however, one must conduct additional tests in order to experimentally assess the bias errors. For example, tests should be conducted to identify the frequency and damping bias of instrumentation cables independent of the effect of the accelerometer mass loading. It was found that the instrumentation cables added more than 30% damping to some modes, and accounted for about 25% of the total mass loading. This is a large amount of damping for this moderately damped structure. Lastly, we mention that rules of thumb for calculating the mass loading effect cannot always be trusted. For approximately 0.9 kg of mass loading on this 127 kg structure, we estimate 0.4% frequency effect. One would then conduct the test assuming mass loading to be a small effect. This is an underestimation by a factor of 3 or more for the bending modes, and a factor of 10 for torsional modes. One must be careful in choosing a method to estimate the mass loading effect, as well as the other sources of bias error. Otherwise, additional tests should be performed to experimentally quantify the bias errors. ACKNOWLEDGEMENTS Miguel Casias and Scott Broome are acknowledged for setting up the instrumentation for these tests. Miguel also performed a number of tests for characterization of the stiffness and damping of various bungee cord configurations. REFERENCES [1] Crassidis, John L., and Junkins, John L., Optimal Estimation of Dynamic Systems, Chapman & Hall/CRC Press, Boca Raton, FL, 2004. [2] Griffith, D. Todd, Casias, Miguel, Smith, Gregory, Paquette, Josh and Simmermacher, Todd W., “Experimental th Uncertainty Quantification of a Class of Wind Turbine Blades,” 24 International Modal Analysis Conference, January 30-February 2, 2006, St. Louis, MO. [3] Pritchard, Jocelyn, Buehrle, Ralph, Pappa, Richard, and Grosveld, Ferdinand, “Comparison of Modal Analysis th Methods Applied to a Vibro-acoustic Test Article,” Proceedings of the 20 International Modal Analysis Conference, pp. 1144-1152, February 2002. [4] Carne, Thomas G., and Dohrmann, Clark R., “Support Conditions, Their Effect on Measured Modal th Parameters,” Proceedings of the 16 International Modal Analysis Conference, pp. 477-483, February 1998. [5] Ashory, M.R. “Correction of Mass-loading Effects of Transducers and Suspension Effects in Modal Testing,” th Proceedings of the 16 International Modal Analysis Conference, pp. 815-828, February 1998.

[6] Website: http://www.epa.gov/iris/gloss8.htm [7] Coleman, Hugh W., and Steele, W. Glenn, Experimentation and Uncertainty Analysis for Engineers, WileyInter-Science, New York, 1999, pg. 40. [8] Mayes, Randall L. and Klenke, Scott E., “The SMAC Modal Parameter Extraction Package,” Proceedings of th the 17 International Modal Analysis Conference, pp. 812-818, February 1999. [9] Carne, Thomas G., Griffith, D. Todd, and Casias, Miguel, “The Effect of Support Conditions on Free-Boundary th Condition Modal Tests,” 25 International Modal Analysis Conference, February 19-February 22, 2007, Orlando, FL.