Experimental Uncertainty Quantification of a Class of ...

Experimental Uncertainty Quantification of a Class of Wind Turbine Blades

D. Todd Griffith, Miguel Casias, Gregory Smith, Josh Paquette, and Todd W. Simmermacher Sandia National Laboratories P.O. Box 5800 Albuquerque, New Mexico 87185-0557 Abstract This paper covers modal test results and quantification of uncertainty for two series of wind turbine blades tested at Sandia National Laboratories. The CX-100 and TX-100 designs have unique features which are being evaluated for emerging wind power applications. A specific aim of these tests is the characterization of the blade structural dynamics properties for model validation purposes. Further, in this paper, we evaluate the testing regime by quantifying the variations in the measured frequencies and damping values found from test to test, for example, by changing the location of the forcing input, and for varied environmental factors such the amount of mass loading or the support conditions. We compare these effects with unit to unit variability to assess the test regime. Introduction and Motivation The size of utility-grade wind turbine rotors continues to increase, especially for offshore applications. Designs of wind turbines with ratings of 3-5 MW are now being considered [1]. This growth in size puts pressure on blade designers to constrain the associated weight increases while maintaining acceptable tip deflections. To address this challenge, the Sandia National Laboratories (SNL) Wind Energy Technology Department has initiated a Carbon-Hybrid Blade Developments: Standard and Twist-Coupled Prototypes project and a Blade System Design Studies project that seeks to design, model and fabricate 100-KW-sized, carbon-hybrid, 9-meter blades to study potential weight reductions and investigate ways to incorporate both epoxy-glass and carbon fiber in the blades in an efficient structural manner. The projects have the goal of designing, modeling, manufacturing and testing a standard blade design, referred to as the carbon experimental or CX-100 blade; a twist-coupled experimental or TX-100 blade design; and an integration of advanced blade concepts into the BSDS blade design. Laboratory and field test results will be used to improve the blade design tools, the manufacturing process, and help validate the blade structural models.

This paper covers the SNL modal test results for two nominally identical CX-100 and two nominally identical TX100 wind turbine blades. The CX-100 blade incorporates a standard carbon reinforcement scheme while the TX100 blade is unique in that it features a passive braking, force-shedding mechanism where bending and torsion are coupled to produce desirable aerodynamic characteristics. A specific aim of these tests is to characterize the unique properties of each blade so that numerical models can be developed and validated. Here we report the values for and uncertainties in the natural frequencies and damping values. Further information on the tests or the mode shapes can be found in the test report [2]. To date, no modal tests have been performed on the BSDS blade. In summary, this paper focuses on as assessment of test to test variability in modal parameters due to force input location and environmental variability. We also assess unit to unit variability. There are limitations in quantifying each of these effects. For example, only two blades for each design are available for testing which

limits quantifying unit variation. Environmental variation due to mass loading, support conditions, and temperature are considered in this paper. With the exception of temperature, these can be closely controlled because the mass loading and support conditions do not change once the blade is instrumented and suspended for a particular set of tests. Furthermore, it is difficult to isolate the effect of temperature and its effect on mass loading and support effects since it was not possible to control temperature in these tests. Thus, the number of available samples for evaluating individual environmental effects is somewhat limited for these tests since no additional testing time was allotted for evaluating the individual effects contributing to uncertainty in the measured frequencies and damping values. Test Setup and Instrumentation A number of key factors were considered in the test setup. These include the support conditions, protection of the blade surface, safety, environmental factors, instrumentation and mass loading effects. Each of these is described in this section. A photo of a fully instrumented blade sitting in the storage cradles is shown in Figure 1.

Figure 1. Fully Instrumented Wind Turbine Blade Sitting in Storage Cradles The blade was tested in a free-free condition. The free boundary condition is typically easy to realize in practice and is thus a preferable support condition to perform a modal test for the purpose of model validation. Any other idealized constraint condition is, in practice, affected by the compliance of the system applying the constraint. For example, if the blade was tested while being mounted at the hub, the compliance of the mount would affect the results. This compliance, of course, would need to be characterized for the purpose of any model validation. Therefore, special care must be taken so that the suspension system will not have a noticeable influence on the test results. For these tests, the blade was suspended by bungee cords to approximate the free-free boundary condition. The blades were tested in two configurations. In one configuration, the free-free boundary condition was approximated by suspending the blade at two locations using two independent tripods positioned fore and aft the blade’s center of gravity (CG). At each location, the blade was supported by a single nylon strap which was linked to the tripod with a coil of bungee rope. No load was applied to the relatively fragile blade trailing edge. A photo of the double strap suspension configuration is shown in Figure 2. In order to evaluate the suspension setup, some tests were repeated for a different suspension configuration using only a single nylon strap and tripod positioned at the CG as shown in Figure 3. Additionally, there was adequate frequency separation for the modes related to the suspension. All rigid body modes occur below 2.7 Hz with the rigid body bounce mode occurring at 2.0 Hz for the single strap test with the first flexible mode inline with the bungees at about 25 Hz. This provided greater than the normally accepted factor of 10 in frequency separation from the suspension dynamics [3]. If inadequate frequency separation between the suspension and blade modes were to occur, it would be possible to reduce the stiffness of the coiled bungee rope by reducing the number of turns of the rope used to support the blade. This was not necessary for these tests.

Figure 2. Double Strap Suspension Configuration

Z Y X

Figure 3. Single Strap Suspension Configuration The instrumentation for each fully instrumented test consisted of 68 accelerometers rated to 50 g’s. The accelerometers were carefully placed on the blade so as not to damage the blade’s finish. Aluminum tape was used to protect the blade with the instrumentation adhered using a type of dental cement. All of the mounting locations utilized a biaxial setup as seen in Figure 4. Accelerations were measured in both the xand z-direction while acceleration in the y-direction (axial direction) was neglected. See Figure 3 for a triad defining these reference directions, and Figure 5 for the measurement locations. The final instrumentation consisted of two perpendicular accelerometers positioned at all 34 mounting locations. All the accelerometers were oriented with no more than 3 degree error for all three axes as determined by a digital inclinometer. Additional tests were performed after removing all but 4 accelerometers to assess mass loading.

Figure 4. Biaxial Accelerometer Mounting

Nine data sets were acquired for each blade corresponding to nine different force excitation locations. The structure was excited using a 3 lb impact hammer. Six excitation locations were in the x-direction while the remaining three were in the z-direction. In this way, we expect that both flapwise and edgewise bending modes are excited. Flapwise bending refers to bending about the z-axis while edgewise bending refers to bending about the x-axis (See Figure 3 for definition of directions). Excitation in the x-direction was performed at points low and high on the blade (i.e. near the leading edge and near the trailing edge) and near and distant from the root end and suspension in an attempt to better excite the structure.

Figure 5. Accelerometer Mounting Locations on Both Sides of the Blade

These tests were conducted at the FAA Airworthiness Assurance NDI Validation Center in Albuquerque, NM in a large aircraft hangar in an uncontrolled temperature environment. The temperature increase can be more than 10 degrees over the course of a typical work day from morning to afternoon. For each test, temperature was recorded. Modal Parameter Estimation The modes for each test were extracted from each data set using the Synthesis Modes and Correlate (SMAC) algorithm developed at SNL [4,5]. The SMAC algorithm is based upon modal filtering, and was designed to extract modal parameters from frequency response functions (FRFs). The SMAC algorithm is highly automated – over 90% of the modes of interest are identified automatically. Any remaining modes (frequency and damping) which are not identified automatically must be fit manually. Basically, a frequency and damping range is chosen, and an improved guess for frequency and damping can be found by maximizing the correlation between the experimental data and the SMAC single degree of freedom curve fit. When converged, the new mode is added to the mode set. The final product is a synthesis of the FRFs using the analytically fit modes. This synthesized FRF is compared to the test FRF. The analytically fit modes of each impact test were then compared using the MultiSort algorithm that was also developed at SNL [6]. This program is used to compare modal parameters from multiple modal tests. Modal parameters identified from each of the tests performed at different locations of force input are compared in an effort to extract the most satisfactory modes. This algorithm provides insight through numerical metrics in choosing the “best” mode. The Multisort algorithm does not provide modal parameter uncertainty estimates; however it was used to evaluate the quality of the reported test modes. For each blade, we only report in this paper the first seven and eight modes, respectively for the CX-100 and TX100 blades, although several more modes can be identified, which are reported in Reference 2. FE models have not yet been validated above this frequency range [7] thus we go no further in this analysis. The description of each mode is determined by viewing the experimental mode shapes and by comparing them with the theoretical mode shapes for a free-free beam [8]. The modes reported are from lightly instrumented tests with single strap suspension, and were considered to be the most accurate modal data from all the tests as determined using the Multisort algorithm. These modes include the first four (or five for the TX-100 blade) flapwise bending modes, the first two edgewise bending modes, and the first torsional mode. Flapwise bending refers to bending about the zaxis while edgewise bending refers to bending about the x-axis (See Figure 3). Test to Test Variability Assessment The objective of this paper is to quantify the uncertainty in the measured natural frequencies and damping values for two designs of wind turbine blades. More specifically, we are interested in assessing the uncertainty in these parameters from any given test, that is, the test to test variability. In the following sections, we evaluate the individual uncertainties that contribute to the test to test variability. These include force input location, and environmental factors such as to temperature, mass loading effects, and support conditions. We consider the test to test variability as a measure of the total uncertainty in the measured frequency and damping values for a particular test. In an effort to assess the worst case scenario, we do not distinguish between bias and random sources of error until the end of the paper. In the final section, we assess unit to unit variability and compare to the test to test variability. There are a number of numerical methods to consider for quantifying the uncertainty in the modal parameters. We desire information on the expected value (the mean) and spread or uncertainty in the values (the standard deviation). Unfortunately, in a typical test regime, such as the one reported here, we do not have enough samples to rigorously compute the data uncertainty. Therefore, here we compute our data dispersion metric by looking at the largest difference in the value relative to the mean value. That is we divide the difference by the mean when comparing only two samples, and when considering three or more samples we divide the largest difference by the mean. This calculation is made throughout the paper when comparing the modal parameters of the two blades tested in each series. All differences reported here are computed in this way, unless otherwise noted. Furthermore, the signs of the differences reported are arbitrary depending on how the difference is computed. One method of assessing test to test variability which we do not consider in this paper is test repeatability. No tests were conducted to assess test to test repeatability, which could be determined by conducting a series of

tests in succession. Furthermore, we do not consider analysis to analysis variability. That is, the variability associated with the modal parameter estimation algorithm. We do not characterize the accuracy of the estimation algorithm or compare available algorithms; although, we assume that the frequency estimates are highly accurate, while the damping estimates are accurate but subject to larger uncertainty. Furthermore, model validation does not place great importance on the damping values for these tests, so we do not consider this to be a limitation of the results presented. Force Input Location Variability Assessment In this section we consider the variability of the modal parameters due to changing the location of the forcing input. Here we do not consider small variations due to individual impacts, that is, the small variations in the precise location and in the force amplitude of the individual impacts which are averaged to complete one set of test data. We assume these effects are negligible. We consider the variation from one test location to another. For the CX-100-006 blade, we look at tests conducted with forcing input at 33 and 61 in the x-direction and 31 in the z-direction, or simply 33x, 61x, and 31z. Since we only have three samples, we compute the largest difference observed in the parameters and divide by the mean. These results are given in Table 1. Table 1. Force Input Location Variability for CX-100-006 for Tests at 33x, 61x and 33z CX100-006 Test to Test Mode Variability (%) Damping Frequency st 1 flapwise 0.0 -6.0 nd 2 flapwise 0.0 58.7 st 1 edgewise 0.1 70.2 rd 3 flapwise 0.3 52.9 nd 2 edgewise 0.1 33.8 th 4 flapwise 0.0 40.5 st 1 torsion 0.1 24.8

As can be seen, the variations in frequency are small from test to test for this blade. In fact, several tests were identical up to 0.001 Hz in frequency. Damping values show much larger variation relative to the frequency values. Typically, modal parameter estimation algorithms estimate frequency more accurately than damping estimates. This partially explains the relatively large damping variations in Table 1, and those presented throughout this paper. Now we look at force input location variability for the TX-100 blades. In Figures 6 and 7 we show the variation for the TX100-001 and TX-100-003 respectively. As opposed to percent difference, here we normalize each parameter estimate to the mean value. We do this because there are a larger number of samples to analyze, and from the plots we get a good visual representation of the uncertainty in each mode. The six tests considered are those with forcing at locations 131x, 133x, 131z, 141x, 143x, 141z. We plot each figure on the same scale. Note that the frequency and damping plot in each figure have different scales. For both blades, test to test variability (the largest difference divided by the mean) in frequency is 0.6% with damping variation of 50% for the first mode, and can be seen from the figures to be less for the other modes.

Figure 6. TX-100-001 Test to Test Variability

Figure 7. TX-100-003 Test to Test Variability Environmental Variability Assessment In this section, we look at the test to test variability due to environmental factors. The testing procedure and test site presented several uncontrollable variables most notably ambient temperature. The tests for the CX100 o blades were performed in the winter with an average temperature of about 40 F while the tests for the TX100 o blades were at an average temperature of 75 F in the summer. Although there was no control over temperature, it was recorded for each test.

We had more control over the amount of mass loading and the support conditions. The amount of mass loading can be controlled by the choice of the number of sensors used in the test; however, some amount of mass loading cannot be avoided. For this testing regime, separate tests were performed with a fully instrumented blade using 68 accelerometers and a lightly instrumented blade using 4 accelerometers. From these tests, we can evaluate the mass loading effect. Of course, included in the mass loading effect are the mounting blocks and the accelerometer cables. Mass loading due to the accelerometer cables was reduced as much as possible, yet was not quantified and thus is considered a small uncertainty in the total mass loading. The accelerometers and mounting blocks account for the majority of the approximately 3 pounds of accelerometers loaded on the nominally 375 pound blades. The support conditions result from a choice of either two supporting straps located about the blade CG or one supporting strap at the CG. Each support -- comprised of one strap and a coil of bungee rope connected to a tripod -- is virtually identical. Due to practical constraints on test time, only a small number of tests were performed to evaluate these environmental effects. Therefore, these results are not the end result of a complete test matrix, but are an attempt to quantify uncertainty for a practical limited test regime. In the following plots, we show the effect of these environmental factors on the FRF plots. Subsequently, we summarize these uncertainties numerically in tabular form. Mass loading effect. We can perform some calculations to estimate the mass loading effect. Consider Eq. (1) which gives the ratio of measured natural frequency ( ωm ) for an instrumented blade to the true natural frequency ( ωt ). We assume a simple single degree of freedom model with no change in stiffness.

ωm k / mm mt mt = = = k / mt mm mt + ∆m ωt

(1)

For our 375 pound blade with 3 pounds of mass loading, we compute a frequency change due to mass loading of 0.4%. The plot in Figure 8 is an overlay of two FRFs (Frequency Response Functions), one represents a fully instrumented blade (red) and the other represents a lightly instrumented blade (blue), for the same blade and excitation location. Both FRFs in Figure 8 are from TX-100-001 and represent a driving point FRF for location 131 in the x-direction. For the fully instrumented blade, we can see a small but noticeable shift in the FRF to a lower frequency.

Figure 8. FRF Comparison of Fully and Lightly Instrumented TX-100-001 Blade

Support Conditions. Reference 3 provides some relations for assessing the effects of the support stiffness and damping on the measured frequencies and damping values. These relations are based on simplified massspring-damper models for the structure and the support. The true frequency is given by Eq. (2)

ª § ω ·2 º ωt = ωm «1 − ¨ s ¸ » «¬ © ωm ¹ »¼ Thus one can understand why the

ωs 1 < ωm 10

1/ 2

(2)

rule is desired to be satisfied in practice based on Eq. (2), and we

can compute an estimate of the effect of support stiffness on the measured frequencies. For our test, the bungee mode was found to have frequency of 2 Hz and the first mode inline with the support was at 20 Hz for the CX-100 blades and 25 Hz for the TX-100 blades. The frequency change here is computed to be 0.5% and 0.3% for the CX-100 and TX-100 blades, respectively. Equation (3) gives a relation for the effect of the support damping on the measured damping values.

ζt = ζm

ωm ωt

ª ωs ζ s º «1 − » ¬ ωm ζ m ¼

(3)

For our tests, the damping ratio for the bungee support mode was estimated at about 4% thus making the value of final term in braces in Eq. (3) nearly equal to unity, and the term in braces nearly equal to zero (first inline mode at 25 Hz and 0.3% damping for the TX-100 blades). This shows that for certain ratios of support and measured frequency and damping that the support can have a very large effect on the uncertainty of the damping measurement, according to Eq. (3). We now look at a plot in Figure 9 which is an overlay of two FRFs, one is for a double strap support condition (red) and the other for a single strap support condition (blue). Both FRFs are from TX-100-003 and represent driving point FRFs at location 163 in the x-direction. The FRFs are very similar throughout the frequency range.

Figure 9. FRF Comparison of Single and Double Strap Support Conditions

Temperature. We could estimate the effect of temperature on frequency if data were available on the dependence of the material elastic modulus on temperature. Yet, we rely solely on the estimated modal parameters in this case. Figure 10 shows an overlay of two FRFs, one was taken at 34°F (blue) and the other was taken at 46°F (green). Both FRFs are from CX-100-002 for a driving point at location 61 in the x-direction. Small but noticeable differences are found due to temperature change.

Figure 10. Effect of Temperature Change of 12ºF We now tabulate the variations in frequency and damping from the tests. For the CX-100 blades tested, we look at the mass loading, support conditions and temperature effects in Tables 2, 3, and 4 respectively. The percent difference values are reported on the right side of each table. Table 2. Mass Loading Effect: CX100-006: 31z – Fully and lightly instrumented one strap data sets % Difference Reduced Full Mode Frequency Damping Frequency Damping Damping Frequency (%) (Hz) (%) (Hz) st 1 flapwise 7.992 1.176 7.978 0.600 -0.2 -64.9 nd 2 flapwise 17.181 0.906 17.360 0.438 1.0 -69.6 st 1 edgewise 20.351 0.785 20.541 0.604 0.9 -26.1 rd 3 flapwise 34.216 0.643 34.362 0.484 0.4 -28.2 nd 2 edgewise 44.367 0.725 44.750 0.682 0.9 -6.1 th 4 flapwise 53.750 0.571 54.259 0.357 0.9 -46.1 st 1 torsion 73.073 0.642 73.824 1.874 1.0 97.9

Table 3. Support Conditions: CX100-002: 61x – One strap and two strap fully instrumented data sets % Difference Two Straps One Strap Mode Frequency Damping Frequency Damping Damping Frequency (%) (Hz) (%) (Hz) st 1 flapwise 8.043 2.063 7.992 1.641 -0.6 -22.8 nd 2 flapwise 17.153 1.943 17.257 2.073 0.6 6.5 st 1 edgewise 20.261 1.728 20.404 1.752 0.7 1.4 rd 3 flapwise 33.796 1.199 33.932 1.176 0.4 -1.9 nd 2 edgewise 44.685 1.248 44.865 1.224 0.4 -1.9 th 4 flapwise 53.159 0.888 53.318 0.963 0.3 8.1 st 1 torsion 72.635 0.870 72.781 0.963 0.2 10.1

Mode st

1 flapwise nd 2 flapwise st 1 edgewise rd 3 flapwise nd 2 edgewise th 4 flapwise st 1 torsion

Table 4. Temperature Effect: CX100-002: 61x – 52F and 34F data sets % Difference 34F 52F Frequency Damping Frequency Damping Damping Frequency (%) (Hz) (%) (Hz) 7.968 3.038 7.992 1.641 0.3 -59.7 17.239 1.943 17.257 2.073 0.1 6.5 20.363 1.693 20.404 1.752 0.2 3.4 33.898 1.248 33.932 1.176 0.1 -5.9 44.686 1.199 44.865 1.224 0.4 2.1 53.212 0.943 53.318 0.963 0.2 2.1 72.635 0.888 72.781 0.963 0.2 8.1

We see that the mass loading has the largest effect as much as 1%, while the support conditions are as much as 0.7% for the frequency variation. Based on Eqs. (1) and (2), we estimated these variations to be 0.4% and 0.5% respectively. Thus, the values from Tables 2 and 3 seem reasonable. Damping value variations are found to be small due to varied support conditions relative to the mass loading effect. However, it cannot be expected that damping values are this consistent from test to test based on an analysis of Eq. (3) and the uncertainty attributed to the estimation of the damping values. With regard to temperature and Table 4, the frequency and damping values are found to be fairly insensitive to temperature changes as large as 18F. Now, we look at the environmental variability in the TX-100 blades. More tests were performed for the TX-100 blades although not enough samples are available for rigorous uncertainty calculations. In Tables 5 and 6 we look at the effect of mass loading and support conditions for two tests in each case conducted with forcing input at locations 133x and 161x. Table 5. Mass Loading Effect for TX100-003 133x and 161x % Difference: 161x % Difference: 133x Mode Damping Frequency Damping Frequency st 1 flapwise 0.7 -6.0 1.2 -133.8 nd 2 flapwise 0.9 -27.8 1.1 -31.4 st 1 edgewise 0.7 -65.8 No Data No Data rd 3 flapwise 1.7 -4.1 1.9 25.6 th 4 flapwise 0.7 19.2 0.9 -11.0 nd 2 edgewise 0.6 -54.4 0.6 -24.4 st 1 torsion -0.3 -2.2 0.4 -26.1 th 5 flapwise 0.4 -22.9 0.9 10.3

Table 6. Support Condition Effect for TX100-003 133x and 161x % Difference: 161x % Difference: 133x Mode Damping Frequency Damping Frequency st 1 flapwise 0.2 79.7 0.6 35.7 nd 2 flapwise -0.6 22.5 -0.2 10.8 st -0.3 22.1 0.3 72.5 1 edgewise rd 3 flapwise -0.7 -19.5 -0.3 6.0 th 4 flapwise -0.1 41.1 0.0 46.8 nd 2 edgewise -0.5 21.9 -0.1 37.6 st 1 torsion -0.3 11.6 0.0 15.5 th 5 flapwise -0.5 37.7 -0.3 31.9

From Table 5, mass loading effects results in as much as 1.9% variation in frequency, and much larger variation in damping. As was the case with the CX-100 blades, the support conditions have a smaller effect on frequency of 0.6%. From Eqs. (1) and (2) we expect mass loading and support effects to change the natural frequency by 0.4% and 0.3% which matches well with Tables 5 and 6. In summary of the environmental effects, we see that worst case the environmental factors of mass loading, temperature, and number of straps have 1.0%, 0.4%, and 0.7% variation for the CX-100 blades; and 1.9%, 0.4%(assumed), and 0.6% variation for the TX-100 blades on natural frequency. These can be assumed to act independently in their contribution to test to test variability for a conservative estimate of test to test variability. For the present study, we make no further attempt to quantify the uncertainty in damping because of the large variations found, and the uncertainty in their estimation. Unit to Unit Variability Assessment The results for the reported natural frequencies and damping values for the CX-100 and TX-100 blades are given in Tables 7 and 8. These values have been determined to be the best modes in each of the set using the Multisort algorithm. Table 7. Frequency and Damping Comparison for CX-100 Blades Difference (%) Blade CX 100-006 Blade CX 100-002 Damping Frequency Frequency Damping Mode Mode Frequency Damping (%) (Hz) (%) (Hz) Description Number st 1 flapwise 1 0.3 -74.9 7.99 1.11 7.97 2.44 nd 2 flapwise 2 0.2 -20.5 17.18 1.58 17.15 1.94 st 1 edgewise 3 0.5 -20.0 20.35 1.53 20.24 1.87 rd 3 flapwise 4 0.9 -36.5 34.10 0.83 33.80 1.20 nd 2 edgewise 5 -1.1 -52.8 44.38 0.71 44.87 1.22 th 4 flapwise 6 1.1 140.0 53.75 0.51 53.16 0.09 st 1 torsion 7 0.5 -5.9 73.00 0.82 72.64 0.87

As can be seen from Table 7, CX-100-002 is slightly stiffer than CX-100-006 with generally higher natural frequencies, and damping generally follows the opposite trend. In a relative sense, no mode varies by more than 1.1% in frequency relative to the mean parameter value. Damping values show much larger variation. Table 8 shows that TX-100-003 is slightly stiffer than TX-100-001 with generally higher natural frequencies. No mode varies by more than 1.7% in frequency. Again, damping values, although following a strong trend from blade to blade, have large variation. This is not surprising because damping values are typically poorly determined in parameter estimation routines relative to the accuracy of the frequency estimate.

Mode Number 1 2 3 4 5 6 7 8

Table 8. Frequency and Damping Comparison for TX-100 Blades Difference (%) Blade TX 100-006 Blade TX 100-002 Damping Frequency Frequency Damping Mode Frequency Damping (%) (Hz) (%) (Hz) Description st 1 flapwise 0.7 -26.0 6.44 0.27 6.49 0.20 nd 2 flapwise -0.1 -11.8 15.16 0.26 15.14 0.23 st 1 edgewise 1.0 6.0 25.00 0.32 25.25 0.34 rd 3 flapwise 1.7 63.5 28.44 0.31 28.94 0.60 th 4 flapwise 0.1 -6.8 43.89 0.37 43.92 0.34 nd 2 edgewise 1.4 -4.0 54.55 0.41 55.31 0.39 st 1 torsion 1.4 -37.9 58.00 0.71 58.82 0.48 th 5 flapwise 0.8 -44.4 63.69 0.49 64.20 0.31

Although we only have two sets of data for each blade design, we can estimate the unit to unit variability to be 12% or less. In the following two sections, we look at environmental variability due to mass loading, temperature and the support conditions and test to test variability. Quantification of Test to Test Uncertainty in Modal Parameters It is clear form the preceding sections that estimates of the uncertainty in the modal parameters from a limited number of comparable test data sets show uncertainty in frequency is small and in damping is large. The lack of a large number of data sets is a limitation to more rigorous uncertainty quantification. This situation is common in practice, yet some analysis can still be performed. In summary, we consider quantifying the total uncertainty in the modal parameters due to environmental and force input location variability. Without rigorous uncertainty estimation, we look to worst case uncertainty of each source of uncertainty and consider the following measure of total uncertainty [9]: 2 2 2 U test = U mass + U spcond + U temp + U 2force

(4)

where the first three terms on the right hand side represent the environmental uncertainty, and the final term represents the force input location uncertainty. We apply this metric to the worst case values in Table 9, with the test to test or total uncertainty given on the last line of the table. Table 9. List of Worst Case Individual Uncertainties on Frequency TX-100 Uncertainty CX-100 Uncertainty Uncertainty (%) (%) Parameter 1.0 1.9 U-mass loading U-support 0.7 0.6 conditions 0.4 0.4 U-temperature (assumed) 0.3 0.6 U-input location U-test to test 1.3% 2.1% (worst case) U-test to test 0.5% 0.7%

The worst case unit to unit variability is found to be 1.1% and 1.7% for the CX-100 and TX-100 blades respectively. This shows that the test to test variability, computed as conservatively as possible using the worst case scenario in which mass loading and support conditions are considered as random error, is slightly larger than the unit to unit variations. Thus, we cannot determine if the trends in the unit variations are valid by this method. On the other hand, we can consider the mass loading and support conditions as bias error or simply a shift in the data and ignore them in Eq. (4). In this case we find test to test variability of 0.5% and 0.7% for the CX-100 and

TX-100 blades, respectively. In this case, the unit variations are valid considering only test to test variations due to temperature and force input location. This is the more appropriate case as mass loading and support conditions are nominally constant for a test. Conclusion This paper details work on the quantification of uncertainty for a class of wind turbine blades. An important result is the quantification of uncertainty from limited number of modal test data sets. However, enough test data was available to quantify uncertainty due to environmental factors such as mass loading, support conditions, and temperature, as well as unit to unit variation and test to test variation. Mass loading shows the largest effect of the environmental factors on the modal parameters if not accounted for in the test setup, although all individual effects have been shown to be less than the unit to unit variations for each blade design allowing some conclusions to be drawn about the validity of the unit to unit variations. References [1] Joosse et al., “Cost Effective Large Blade Components By Using Carbon Fibres,” Proc. of the 2002 ASME th Wind Energy Symposium Technical Papers, 40 AIAA Aerospace Sciences Meeting and Exhibit. Reno, Nevada. January 14-17, 2002, 47-55. [2] Griffith, D. Todd, Smith, Gregory, Casias, Miguel, Reese, Sarah, and Simmermacher, Todd; “Modal Testing of the TX-100 Wind Turbine Blade,” Sandia National Laboratories, Albuquerque, New Mexico, SAND2005-6454. [3] 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. [4] Mayes, Randall L. and Klenke, Scott E., “The SMAC Modal Parameter Extraction Package,” Proceedings of the th 17 International Modal Analysis Conference, pp. 812-818, February 1999. [5] Mayes, Randall L. and Klenke, Scott E., “Automation and Other Extensions of the SMAC Modal Parameter th Extraction Package,” Proceedings of the 18 International Modal Analysis Conference, pp. 779-785, February 2000. [6] Mayes, Randall L. and Klenke, Scott E., “Consolidation of Modal Parameters from Several Extraction Sets,” th Proceedings of the 19 International Modal Analysis Conference, pp. 1023-1028, February 2001. [7] Paquette, J., Laird, D., Griffith, D. Todd, "Modeling and Testing of 9m Research Blades," ASME/AIAA Wind Energy Symposium, Reno, NV, 2006. [8] Blevins, Robert D., Formulas for Natural Frequency and Mode Shape, Kreiger, Malabar, Florida, 1995. [9] Coleman, Hugh W., and Steele, W. Glenn, Experimentation and Uncertainty Analysis for Engineers, WileyInterScience, New York, 1999.