## Discussion

In the previous subsections four different rotor blades have been examined in order to justify the use of the proposed systematic structural modeling procedure and the resulting models for both model based control design and design optimization purposes. Measured non-rotating eigenfrequencies obtained via either hand-excited displacement measurements or a modal analysis are compared with those from both a finite element and the superelement approximation. The following observations can be made:

• The effect of violating the modeling assumptions is negligible since the detailed finite element models do produce biases of the same order;

• The superelement models are very useful to approximate the first bending modes with only a few superelements. In general, it can be stated that the number of superelements required is equal to the number of modes the model needs to describe accurately plus one;

3rd Flap

el 5

• The difference between the hand-lay-up (APX-45) and resin-infusion moulding (RIM) manufacturing technique (APX-70) could not be detected since the supplied mass distribution was already modified to diminish the difference between the measured and simulated mass and center of gravity location;

• The model accuracy is strongly limited by the quality of the supplied input data.

To stress the importance of the latter observation, we have used measured displacement information of static tests applied to both the APX-70 and the RB-70 rotor blade to modify the torsional spring constants accordingly. The displacement is measured using a LVDT located at a radial position of r = 26.75 m (APX-70) and r = 26.2 m (RB-70). The torsional springs of the APX-70 blade in flap direction are increased with 8.6% and in lead-lag direction reduced with 24.4%, while the the torsional springs of the RB-70 blade are reduced in flap direction with 0.6% and in lead-lag direction reduced with 18.5%. The results are plotted in Fig. 4.12 for Nse = 1, • • • , 6, while for Nse = 6 the numerical values are listed in Table 4.8. The upper figure shows the relative frequency errors of the APX-70 blade, while the lower figure shows the errors of the RB-70 blade.

APX-70 rotor blade (modified)

APX-70 rotor blade (modified)

RB-70 rotor blade (modified)

RB-70 rotor blade (modified)

Figure 4.12: The relative errors for the first three flap and first two lead-lag eigenfrequencies of the APX-70 rotor blade (upper plot) and RB-70 rotor blade (lower plot) as function of the number of superelements Nse. Displacement information of static tests has been used to modify the torsional spring constants accordingly. Dashed-dotted horizontal lines: + 2% and - 2% error bound respectively.

Figure 4.12: The relative errors for the first three flap and first two lead-lag eigenfrequencies of the APX-70 rotor blade (upper plot) and RB-70 rotor blade (lower plot) as function of the number of superelements Nse. Displacement information of static tests has been used to modify the torsional spring constants accordingly. Dashed-dotted horizontal lines: + 2% and - 2% error bound respectively.

It can be concluded by comparing the modified and original frequency error values for Nse = 6 listed in Table 4.8 that adding measured displacement information (even at only one location) significantly reduces the absolute frequency error value. The mean value of the absolute frequency error is reduced from 7.2% to 2.9% in the APX-70 case, while for the RB-70 blade this value is reduced from 7.1 % to 3.3%. It is expected that an even higher accuracy can be obtained if displacement (or acceleration) information is available at more radial positions. After all, this offers the possibility to tune the torsional spring constants individually using the procedure treated in Chapter 5.

Modified APX-70 and RB-70 rotor blade | ||

Mode |
Difference Nse = 6 w.r.t. TNO measurements |
Difference Nse = 6 w.r.t. TNO measurements |

1si flap 1si lead-lag 2 nd flap 2 nd lead-lag 3 rd flap |
+ 0.1% (-3.9%) - 3.5% (+11.0%) + 2.3% (-1.8%) + 2.2% (+17.5%) + 6.3% (+2.0%) |
- 0.4% (-0.1%) -1.3% (+9.3%) + 5.9% (+6.2%) + 1.3% (+12.3%) + 7.5% (+7.8%) |

Table 4.8: Comparison of APX-70 and RB-70 rotor blade non-rotating eigenfre-quencies calculated using the superelement approximation with modified (original) torsional spring constants for Nse = 6 to the ones resulting from the modal analysis performed by TNO.

Table 4.8: Comparison of APX-70 and RB-70 rotor blade non-rotating eigenfre-quencies calculated using the superelement approximation with modified (original) torsional spring constants for Nse = 6 to the ones resulting from the modal analysis performed by TNO.

In the previous six cases the validity of the proposed systematic structural modeling procedure has been examined. We felt that it was important first to verify the proposed systematic structural modeling procedure using a test case, and subsequently to validate the approach using individual modal tests carried out on a number of rotor blades before applying it to a complete flexible wind turbine.

The test case shows that the superelement modeling method represents a consistent approximation to the Euler-Bernouilli beam with an approximation accuracy that increases with an increasing number of superelements. Furthermore, we found a good agreement between the measured non-rotating eigenfrequencies of various wind turbine rotor blades and those calculated using the superelement approximation. This means that the mass and stiffness properties of the rotor blade models are appropriate.

The aforementioned results demonstrate that it is, in principle, possible to derive limited order multibody models suited for time-domain simulation, analysis of dynamic loads, and model based control design directly from physical rotor blade design data using the proposed systematic structural modeling procedure. Depending on the intended model use, tuning of torsional spring constants using either static test data or experimental modal test data is required to reach the desired accuracy. We will now continue the validation using data acquired from a full-scale modal test applied to the Lagerwey LW-50/750 wind turbine.

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