Estimation of Unmeasured DOF s on a Scaled Model of a Blade Structure Anders Skafte 1, Rune Brincker 2 ABSTRACT This paper presents a new expansion technique which enables to predict mode shape coordinates and responses in unknown degrees of freedom, using only a limited amount of sensors. The method relies on the Local Correspondence Principle which assumes that two sets of mode shapes which a alike, but not the same, can be described as a linear combination of one another. The principle is used by combining experimentally obtained mode shapes, with mode shapes from an finite element model. The limited but true information from the experimentally obtained mode shapes is used to alter the extensive but less accurate mode shapes from the finite model, so these fit the structure. To validate the technique tests have been performed on a 2.4 m model resembling a bearing beam from a wind turbine blade. Both mode shapes and responses have been expanded and errors between measured and expanded responses are determined. The results show that the technique has a very high accuracy, an enables to expand displacements with an average error less than 6 %. Keywords: Expansion, Mode shapes, Local Correspondence 1. INTRODUCTION The work resulting in this paper was performed as a part of a feasibility study for the Danish wind turbine manufacture Vestas. A number of elements concerning monitoring of wind turbine blades where investigated, and the ability to expand the measured response on a blade played a central part of this study. When failures occur on a wind turbine blade one of the reasons can be due to fatigue which is highly depending on the stress history of the structure. A precise expansion of the response in the blade using only a limited amount of sensors, would be the first step to estimate the stress history of the entire blade. It would therefore serve as an important part of the monitoring of a blade. The Local Correspondence [LC] principle was chosen as expansion technique due to its simplicity and its robustness. Since it all depends on a linear transformation between two sets of mode shapes, the expansion of mode shapes is easy to compute and gives fast results. The LC principle is valid for any set of eigenvectors which are alike, but in this article the focus is on expanding experimentally obtained mode shapes by using mode shapes from an Finite Element [FE] model of the same structure. The experimentally obtained mode shapes have the quality of being the true mode shapes of the structure. Set aside noise from measurements and identification algorithms these mode shapes represents how the structure vibrates when subjected to dynamic loads. 1 Ph.D. Student, Engineering College of Aarhus, Aarhus, Denmark, ask@iha.dk 2 Professor, Engineering College of Aarhus, Aarhus, Denmark, rub@iha.dk
The technique presented here has a wide variety of applications. As mentioned it could serve as part of a fatigue monitoring algorithm. It is also likely to be used as an advanced version of Operation Deflection Shape (ODS). In traditional ODS [1] the vibrations of a structure are animated in either a certain time period, or at a certain frequency. By using the LC principle it is possible to make a much more precise animation due to the expansion of the measured data. Furthermore it makes it possible to predict displacement in a point of a structure where it is impossible to place a sensor. There are several monitoring techniques [2,3] which use changes in mode shapes to determine damage on a structure. By a precise expansion of the mode shapes the amount of information is increased, which could be beneficial when using these techniques. 2. THEORY The theoretical background for this expansion technique comes from two different assumptions. The first is the standard assumption in modal dynamics that a response of a structure can be expressed as a linear combination of the structures mode shapes and modal coordinates. The second is the linearity between two sets of mode shapes, which are alike [4]. The first assumption is often described as: (1) Where indicates the modes shapes of the structure, the response and the modal coordinates. If the response of a structure has been measured using a limited amount of sensors, an estimate of the modal coordinates can be found by: (2) Where indicates an estimate, indicates the pseudo inverse, and subscript a indicates the active DOF s as in the DOF s where sensors have been placed. Once an estimate of the modal coordinates is found, the response can be reconstructed again using Eq.1. But this time the mode shape matrix can contains as many DOF s as wanted resulting in an expanded response. The theory behind the LC principle is used to expand the mode shapes. The LC Principle is deducted from the sensitivity equations [5]. These state that a change in a mode shape can be described analytically by all the other mode shapes and eigenfrequencies in the system. The LC principal uses this expression to state that if the difference between two eigenvectors is small, the changes of an eigenvector can be expressed as a combination of the eigenvectors nearby, in terms of frequency: (3) Where indicates a set of FE mode shapes, and indicates the transformation matrix. When is known from experimental data, and is extracted from the FE model, an estimate of the transformation matrix can be found. (4) Here are the FE mode shapes, but only with the DOF s coinciding with where the sensors are placed on the structure.
An estimate of the expanded modes shapes can then easily be found using Eq. 3. (5) And then again an estimate of the expanded response can be found using Eq.1. (6) To get the best result for the expanded response, Eq. 4 and 5 are performed using one experimental mode at a time. The trick is here to find the right amount of mode shapes in the FE mode shape matrix, and the right order of the mode shapes. There is no analytical expression to determine the optimal size of the FE mode shape matrix when expanding a given experimental mode so this must be done by an iterative process. 3. TEST SETUP AND EXPERIMENTALLY OBTAINED MODE SHAPES The focus of this study was to predict and expand measured responses on a model of a wind turbine blade. In order to validate the technique a test object was performed. Due to limited funding the model was kept relatively simple, but included some of the characteristics from the bearing a blade the cantilever beam, different materials, varying cross-section and glued joints. 3.1. The Test Object The test object was performed as a 2.4 m cantilever beam, with a varying hollow cross section. The edges where made with 22 mm timber and the flap sides were made with 4 mm plywood. At the bottom the cross section had the dimensions 109 x 205 mm and at the top 47 x 102 mm see fig. 2. Figure 1 Picture of test setup. Figure 2 Top- and bottom cross-section of the structure. The Youngs Modulus for both timber and plywood was found due to static test. Their values were as followed: E timber 18000 MPa E plywood 16500 MPa A rectangular steel plate was mounted to the bottom of the blade, so the blade could be bolted to a fixed plane.
3.2. Equipment The response was measured using 15 Brüel & Kjaer 4508-B-002 accelerometers. The blade was divided into 5 cross-sections, and 3 accelerometers were placed in each section to ensure that both translations and rotation could be measured. By a rule of thumb the recording time was set to 1000 times the lowest period in this case being ~100 s. - and the sampling ratio was set to 2048 hz. Both sampling time and sampling frequency was chosen due to initial tests. Load was induced to the structure simply by scratching the blade with a sharp objects. This was done to ensure a stochastic response which are essential when the modal parameters are found using Operational Modal Analysis [OMA]. 3.3. Experimentally Obtained Mode Shapes To extract the modal parameters from the measured data, the Frequency Domain Decomposition [FDD] [6] was used as the identification technique. 5 mode shapes and frequencies were determined and used in the further post processing. Figure 3 Plot of singular values. The 5 experimentally obtained mode shapes came out as followed: Table 1 Experimentally obtained mode shapes Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Shape: Frequency: 10.3 Hz 25.7 Hz 59.6 Hz 92.7 Hz 114.7 Hz The OMA enabled identification of higher modes than shown in table 1. But in this paper it is chosen only to focus on the first 5. 4. FINITE ELEMENT MODEL The finite element model was created using Robot Millennium [7]. All elements were created using rectangular panels, and a total of 1460 nodes were included in the model. Thickness of the panels was design according to the test structure, as well as Young modulus of elasticity. Mode shapes from two different versions of the FE model were used in the expansion. The first model was designed with a charnier at the bottom resulting in 2 rigid body modes. These 2 modes were
combined with the 7 first mode shapes from a second model which had fixed supports at the bottom. The reason for creating two different FE models was that the test structure couldn t be expected to be fully fixed to the floor. This results in experimental mode shapes that are linear combinations of rigid body modes and bending/torsional modes. Table 2 Finite element mode shapes RBM RBM Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Mode 7 Shape: Frequency: - - 18.9 Hz 49.4 Hz 83.8 Hz 199 Hz 208 Hz 209 Hz 303 Hz RBM indicates rigid body modes. The frequencies are not taken into account, since they have no physical meaning. A total of 9 FE modes were used to estimate the response in unknown DOF s. 5. RESULTS When expanding the mode shapes using Eq. 4 and Eq. 5, the expansion is done mode by mode and the equations are changed to: (7) (8) The DOF s in the experimentally obtained mode shapes are then divided into 2 sets an estimation set and a prediction set. The estimation set is used in Eq. 7 and Eq. 8, to create the expanded mode shapes. The prediction set is used to measure the quality of the expanded mode shape. If all active DOF s were used in the expansion, the fit of the active DOF s would be exact when holds as many modes as DOF s in Eq. 8. But the fit on all the unknown DOF s would be all wrong. By dividing the active DOF s into two set it enables the possibility to optimize each expanded mode shape with respect to which and how many FE modes that should be taken into account when using Eq. 7 and Eq. 8. When the expanded mode shapes are found, these can be multiplied with the estimate of the modal coordinates to create the expanded response. In this example the response has been bandpass filtered so accelerations only exist between 5 Hz and 120 Hz. When the expanded response is compared with the measured response for a given sensor, the error is calculated without taken the signal of the present sensor into account. For example when calculating the error between the measured- and expanded signal in DOF1, the measured signal from the sensor placed in DOF1 will not be a part of the expansion. The error is calculated by: () 100% (9)
The errors on the different channels are presented in fig.4. x 10-3 Blue Measured, Red Expanded 6 4 2 Acceleration [g] 0-2 -4-6 -8 Figure 4 Error for each sensor. 4.54 4.55 4.56 4.57 4.58 4.59 4.6 4.61 4.62 Timestep x 10 4 Figure 5 Measured- and expanded response. Fig. 4 shows that the error of the sensors placed at the bottom of the structure are higher than the rest. This is explained by the higher signal-to-noise ratio. The average error for all 15 sensors is 5.19 %. 6. CONCLUSION This paper presents a method for which a measured response in a limited amount of points of a structure can be expanded to cover the entire structure. The method relies on a linear transformation between experimentally obtained mode shapes, and mode shapes from a finite element model. When an estimate of the transformation is found, the mode shapes can be expanded by multiplying the finite elements matrix with the transformation matrix. The method is tested on a scaled blade structure, within a frequency band that holds the first 5 modes. Set aside the 3 sensors near the support, the tests show good results with errors within 1.5% 5.5 %. 7. REFERENCES [1] Døssing, Ole. Structural Stroboscopy - Measurement of Operational Deflection Shapes. s.l. : Journal of Sound and Vibration, August 1988. [2] Parloo, E., Guillaume, P., Van Overmeire, M. Damage Assessment using Mode Shape Sentivities, Mechanical Systems and Signal Processing, 17 (3), 499-518, 2003 [3] Stubbs, N., J.-T. Kim, Topole, K., An Efficient and Robust Algorithm for Damage Localization in Offshore Platforms, Proc. ASCE Tenth Structure Congress, 543-546. [4] Brincker, R., Skafte A., López-Aenlle, M., Sestieri, A., D Ambrogio, W., Canteli, A.: A Local Correspondence Principle for Mode Shapes in Structural Dynamics. To be published. [5] Heylen, W., Lammens, S. and Sas, P.: Modal analysis theory and testing. Katholieke Universiteit Leuven, Faculty of Eng., Dept. of Mech. Eng., 2007. [6] Brincker, R; Zhang, LM; Andersen, P.: Modal identification of output-only systems using frequency domain decomposition, SMART MATERIALS & STRUCTURES Volume: 10 Issue: 3 Pages: 441-445 [7] Autodesk Robot Structural Analysis Professional 2011, Autodesk.