An overview of experimental strain-based modal analysis methods

Transcription

1 An overview of experimental strain-based modal analysis methods F. L. M. dos Santos 1,2,3, B. Peeters 1, J. Lau 1, W. Desmet 2, L. C. S. Góes 3 1 LMS, A Siemens Business, LMS International N.V. Interleuvenlaan 68, B-3001 Leuven, Belgium fabio.santos@lmsintl.com 2 KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300 B, B-3001, Heverlee, Belgium 3 Instituto Tecnológico de Aeronáutica (ITA), Praça Marechal Eduardo Gomes, 50 - Vila das Acácias CEP São José dos Campos - SP - Brazil Abstract The most established way of performing experimental modal analysis is to use acceleration or velocity based transducers that lead to the calculation of the displacement mode shapes. However, there are applications where the use of strain measurements makes for a more attractive and interesting option. Strain gauges have been commonly used for static load testing of mechanical products in the aeronautic, automotive and mechanical industry. Moreover, fatigue testing, durability analysis and lifetime prediction has also been a common application where strain gauges are used. This sort of testing is a common part of the product development process, and additional information on product durability and dynamic performance can be assessed by obtaining the modal parameters of the system, while still using the same instrumentation. Moreover, since strain measurements are more directly related to stress, fatigue and failure, strain-based measurement methods can be a good option for structural health monitoring methods and monitoring systems. Applications where sensor size and placement might be critical are also good candidates for strain-based methods. Helicopters, wind turbines and gas turbines are a good example where strain gauges are more suited for vibration measurements. Some application cases of dynamic strain measurements and dynamic strain modal analysis are shown in this work, with test subjects such as a composite helicopter blade. 1 Introduction Modal analysis and testing has been for a long time, associated with the use of displacement-based responses and sensors, such as accelerometers and laser vibrometers. The use of strain sensors and measurements for modal testing [1, 16], on the other hand, has been less accentuated, mostly due to the difficulties with respect to the use strain gauges and with the simplicity of the use of accelerometers. On the other hand, there has been increased interest from both industry [8] and academia [4] on assessing the potential benefits of strain modal analysis. Another important use of strain modal testing is on evaluating structural integrity or stress concentration [18] on design prototype stages and also monitoring in real-time (with structural health monitoring systems (SHM) [6]), which has led to an increase in the number of dynamic strain applications and to the development of improved identification and measurement techniques [3, 19], as well as to improved sensor technology [13, 12, 2]. Strain gauges have been commonly used for static load testing of mechanical products in the aeronautic, automotive and mechanical industry. Moreover, fatigue testing [17], durability analysis and lifetime predic- 2453

2 2454 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 tion [15] has also been a common application where strain gauges are used.this sort of testing is a common part of the product development process, and additional information on product durability and dynamic performance can be assessed by obtaining the modal parameters of the system, while still using the same instrumentation. A very important contribution on the field of strain measurement are the fiber optic sensors, or Fiber Bragg Grating (FBG) sensors [5, 7]. Their robustness to magnetic interference, added to the easiness of creating sensor arrays with multiple sensors, plus the possibility of embedding these sensors in composite structures, makes for an attractive solution for use in SHM systems. The availability of such an array of sensors, ready to be used and adequate for modal testing, is another incentive to carrying out a strain modal analysis, saving up on time and instrumentation. Another application of dynamic strain measurements has to do with the strain-displacement relations [11]. In many systems, strain gauges are used as the standard vibration sensor, especially when size or sensor location is an issue. Such is the case in aerospace applications, like gas turbines, wind turbines and helicopters [14], where size and weight are very restricted, and any sensor place on a blade should affect its aerodynamic properties as little as possible. One particular use of the strain measurements and strain to displacement relations is the strain pattern analysis (SPA), where strain measurements are used to predict blade displacements. 2 Theoretical background To obtain the strain modal formulation, one can start with the fundamental theory of modal analysis. Modal theory states that the displacement on a given coordinate can be approximated by the summation of a n number of modes: n u(t) = φ i q i (t) (1) i=1 where u is the displacement response in x direction, φ i is the i th (displacement) vibration mode, and q i is the generalized modal coordinate and t is time. For small displacements, given the theory of elasticity, the strain/displacement relation is: ε x = x u (2) And similarly, the same relationship exists between the strain vibration modes and the displacement modes: ψ i = x φ i (3) This way, by the relations on equations (2) and (3), the expression on (1) can be rewritten as: ε(t) = n ψ i q i (t) (4) Moreover, the relationship between the generalized modal coordinate q and an input forcef is: i=1 q i = Λ 1 i φ i F, withλ i = ( ω 2 m i +jωc i +k i ) (5) where m i, c i and k i are the i th modal mass, modal damping and modal stiffness, and ω is the excitation frequency. Substituting (5) into (4), the relation between a force input and a strain output, in terms of displacement and strain modes is represented as: n ε i = ψ i Λ 1 i φ i F (6) i=1

3 MODAL TESTING: METHODS AND CASE STUDIES 2455 And finally, the strain frequency response function (SFRF) can be obtained, in matrix form: n [H ε ] = Λ 1 i {ψ i }{φ i } = [ψ][λ] 1 [φ] T (7) The expansion of (7) is: H11 ε H12 ε H1N ε i H21 ε H22 ε H2N ε i.... HN ε o1 HN ε o2 HN ε on i i=1 n = Λ 1 i i=1 ψ 1i φ 1i ψ 1i φ 2i ψ 1i φ Ni i ψ 2i φ 1i ψ 2i φ 2i ψ 2i φ Ni i.... ψ Noiφ 1i ψ Noiφ 2i ψ Noiφ Ni i where N o represents the number of strain gauge measurement stations (or the number of output measurements) andn i represents the number of excitation points (or the number of inputs). There are some remarks and considerations that can be obtained from the strain modal analysis theory presented: The columns of the matrix correspond to the strain responses due to the excitation points along the rows of the matrix; the SFRF matrix is not symmetric - for example, H ε AB is different than Hε BA ; from the item above, it can be inferred that there is no reciprocity in strain modal analysis; any column of the SFRF matrix contains all the information regarding the strain modes (ψ); any row of the SFRF matrix contains information about the displacement modes(φ); to obtain the strain mode shapes, one must use a fixed excitation point and measure the strain responses; by using a strain gauge as a fixed reference sensor and moving the excitation point (as with roving hammer impact testing), the displacement mode shapes can be obtained; the similarity of the strain modal formulation and the displacement modal formulation means that the same identification methods can be used fro both. In general, there are drawbacks but also advantages present in strain modal theory. A very important advantage is its versatility - being able to identify both displacement and strain modes with one type of sensor is a property that can be used to reduce the number and types of sensors used in an experimental test campaign, and something not achieved directly with accelerometers. On the other hand, there are some impracticalities involved with the use of strain gauges and strain sensors: Worse signal-to-noise-ratio when compared with accelerometers; harder to instrument (bond sensors to structure surface); sensors are usually not reusable; some data acquisition equipment require the use of additional signal conditioning systems (strain gauge bridges). Of the above mentioned items, instrumentation can be very critical - in general, some skill is required in bonding strain gauges or other strain sensors, and the bonding quality can degrade the quality of the measured signal. In this work, the modal identification methodology used was the PolyMAX identification method [10], which could be used without any modification. Moreover, the additional signal conditioning was not necessary due to the use of an acquisition system which already contained modules with strain gauge bridges (LMS SCADAS mobile with VB8 modules). (8)

4 2456 PROCEEDINGS OF ISMA2014 INCLUDING USD Experimental analysis To illustrate some of the characteristics of strain modal analysis shown in the previous sections, two test cases will be presented. The first test subject is a small composite wind turbine blade, which was tested with strain gauges and an impact hammer and the experimental results were correlated with a finite element (FEM) simulation model. Then, a full size composite helicopter main rotor blade was tested with an electrodynamic shaker and Fiber Bragg Grating (FBG) sensors were used. 3.1 Wind turbine blade A small composite wind turbine blade was used on the first strain modal test [9]. For this purpose, 20 strain gauges were glued to the surface of the blade and an impact hammer with an impedance head was used to excite the structure at several locations. The blade was fixed at its root, to impose a cantilevered condition. Of the 20 strain gauges, one of them consisted of a strain gauge rosette to measure purely shear strain, while the other 19 strain gauges were aligned with the radius of the blade and were measuring normal strain. Figure 1 shows the wind turbine blade, its sensor locations represented in the acquisition software and a finite element model of the blade. (a) Composite small wind turbine blade (b) Sensor locations on wind turbine blade (c) Wind turbine blade FEM model Figure 1: Small wind turbine blade, sensor locations and FEM model The first step of the experimental procedure is to measure the strain frequency response functions (SFRFs), that are used later on the modal analysis procedure. Figure 2(a) shows the SFRF of an arbitrary strain gauge, where the resonance peaks are clearly visible. Moreover, the phase shift due to the resonances is the same for the SFRF, where the phase shifts in 180 degrees whenever there is a resonance peak. Additionally for this experiment, a reciprocity check was carried out to verify if the theory for strain modal analysis was correct - for this purpose, two measurement points were picked and the impact hammer was used to excite those points. A successful reciprocity check should yield identical or almost identical FRFs for a classical displacement modal analysis. In the case of the strain modal analysis, as it can be seen on Figure 2(b), reciprocity is not guaranteed for the strain measurements, since the SFRFs do not match each other.

5 MODAL TESTING: METHODS AND CASE STUDIES Phase ( o ) Amplitude (db) Frequency (Hz) (a) Wind turbine blade - Strain FRF from an arbitrary measurement point Amplitude (db) Frequency (Hz) (b) Wind turbine blade - reciprocity check using strain measurements Figure 2: Wind turbine blade: (a) strain FRF; (b) reciprocity check using a strain gauge - reciprocity is not guaranteed After the reciprocity check, the modal analysis and strain modes identification was carried out. For this purpose, the PolyMAX identification method was used. A bandwidth from 10 to 210 Hz was taken into account as 6 clear modes were identified in that frequency range - these modes consist of bending, in-plane and torsional modes. Table 1 shows the natural frequencies and mode types of the identified modes. Finally, Mode Frequency Mode number [Hz] type bending in-plane bending bending torsional in-plane Table 1: Composite wind turbine blade: natural frequencies and vibration mode types the strain modes obtained with the modal analysis procedure were compared with the strain modes obtained from a FEM analysis of the blade, from the model shown on Figure 1(c). The strain modes of the FEM model are extracted via the strain tensor matrix, where the appropriate strin directions are taken according to the orientation of the strain gauges on the blade, and the strain gauge location and size are approximated to one element of the finite element model. Two methods of comparison were used to compare the simulation and experimental analysis, the percentage difference in natural frequency between experiment and simulation, and the modal assurance criterion (MAC). Table 2 shows the natural frequencies comparison and percentage difference, as well as the diagonal MAC value. Additionally, the natural frequency values obtained using a traditional accelerometer based impact test are also show, for the purpose of comparison. Moreover, Figure 3 shows the full MAC matrix for all the considered modes. An initial analysis of the data from Table 2 shows that there is good agreement between the simulation model and the experiment. Regarding the natural frequencies, most of them are within 10 % of difference, except for modes 2 and 5. Moreover, the MAC values show very good correlation, except for mode 5. The lack of correlation for this mode is understandable - it is a torsional mode, which means that most of its energy

6 2458 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 Mode Natural frequency Natural frequency Natural frequency Difference MAC number strain gauges [Hz] accelerometers Simulation [Hz] % Table 2: Strain modes comparison - experiment and simulation Modal Assurance Criterion (MAC) 1 Experimental Modes FEM Modes Figure 3: Modal Assurance Criterion - comparison of computational strain modes and experimental strain modes for the wind turbine blade comes from shear strain, while only one of the strain gauges (the strain rosette) is measuring shear strain, and the others are measuring normal strain. 3.2 Helicopter main rotor blade This experiment had as the main goal to carry out a strain modal analysis on the helicopter main rotor blade, asd a means to assess how the dynamics of the composite blade are captured by the strain sensors, and a different and newer type of sensor was tested as - the Fiber Bragg Grating (FBG) sensors. These sensors are present in an optical fiber and a laser scans through the fiber, for a whole range of wavelengths. Relative distortions for a given wavelength bandwidth mean that there is strain on the sensor location. Moreover, to better study the blade dynamic behavior, displacement modal analysis sensors, typical piezo ICP accelerometers, were also used - by using both displacement and strain sensors, one can have a better idea of the similarities between both types. In total 20 FBG sensors were used - the sensors are actually gratings marked within the fiber, which means that a single fiber can hold multiple sensors. This can be an advantage in terms of instrumentation time and amount of cables. For this test campaign, two fiber lines were used, each containing 10 sensors. Moreover, 4 regular resistive strain gauges were used, collocated with some FBG sensors. The strain gauges were used to be able to better assess the noise floor of the FBG sensors, and also to synchronize them with the force measurements - since the FBG sensors use their own acquisition unit, while the force and other sensors are measured with the standard LMS SCADAS measurement equipment, it is important that an offline synchronization procedure is carried out to synchronize the FBG sensors with the force sensor. Additionally, 16 accelerometers were collocated with the FBG sensors to investigate the visualize the displacement modes and investigate strain-displacement relations. Figure 4 shows a part of the blade with the above mentioned sensors, while Figure 5 shows the strain gauge collocated with the FBG sensor.

7 MODAL TESTING: METHODS AND CASE STUDIES 2459 Figure 4: Helicopter blade instrumented with accelerometers, strain gauges and FBG sensors) Figure 5: Strain gauge collocated with FBG sensor The full list of the equipment used for the experimental strain modal analysis is: LMS SCADAS mobile with 64 measurement channels and 2 output channels PC with LMS Test.Lab 12A software 16 PCB 333B30 Accelerometers PCB 208C03 impedance head Electrodynamic shaker with stinger and amplifier FiberSensing BraggMETER 2 optical fibers with 10 gratings (sensors) each 4 resistive strain gauges

8 2460 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 The measurement location for all the sensors is shown in Figure 6 - the 16 accelerometers were collocated with the FBG sensors, except on locations 4, 5, 14 and 15, where no accelerometers were placed. The resistive strain gauges were collocated with the FBG sensors on locations 4, 7, 15 and 19. The blade was suspended by elastic cords to obtain an approximate free-free boundary condition. With this sort of boundary condition, strain is close to zero in the two tips of the blade and therefore, the choice was to place the sensors towards the middle of the blade, such that the measured strain would be higher. The blade was excited with an electrodynamic shaker and the driving point is represented in Figure 6 with an FBG sensor Accelerometer Strain gauge Driving point Figure 6: Helicopter blade: location of FBG sensors, strain gauges, accelerometers and driving point The accelerometers were placed in a way to measure acceleration in a direction perpendicular to the one measured by the strain gauge, such that they are both capturing mainly the same types of modes (especially bending). Figure 7 better depicts the sensor orientations and the measurement directions. x y z y Strain gauge and FBG measurement direction Accelerometer measurement direction x z Figure 7: Helicopter blade with strain sensors and accelerometer measurement directions - the accelerometer measures on a direction perpendicular to that of the strain gauge Two types of excitation were used for the experimental campaign - burst random and sine sweep. The data acquisition details used for the SCADAS Mobile system are show in Table 3. For the rest of this work, the data obtained from the sine sweep experiments will be used. Similar results were also achieved using burst random excitation. The frequency range of interest was limited in particular to less than 100 Hz due to the limitations of the FBG acquisition system, that could sample at 200 Hz at the highest. Most importantly, due to the intrinsic nature of the FBG sensor and the way the data is acquired, discretized and digitalized, it is not possible to use an analog anti-aliasing filter, which means that care must be taken not to excite the structure past the maximum allowed bandwidth. A simple check that can be carried out to verify the quality of the measurements is the comparison autopower spectrum between an FBG sensor and its collocated strain gauge. Figure 8 shows this comparison for the sine sweep excitation case. As it can be seen, there are no major deviations between both

9 MODAL TESTING: METHODS AND CASE STUDIES 2461 Excitation signal burst random sine sweep Number of averages Sampling frequency 800 Hz 800 Hz Frequency range of interest 8 to 90 Hz 8 to 90 Hz Frequency resolution Hz Hz FRF estimator H1 H1 Table 3: Data acquisition details spectra. 40 Autopower Spectrum comparison 20 Strain gauge FBG 0 Amplitude (db) Frequency (Hz) Figure 8: Autopower spectrum comparison between strain gauge and FBG sensor The next step for the strain modal analysis using the FBG sensors is to synchronize their signals with the measured force signal. Since the resistive strain gauges, accelerometers and force cell signals were all acquired with the same acquisition unit and are therefore synchronized, the FBG sensors have only to be synchronize them with one of these sensors - hence, the use of the collocated strain gauges. This synchronization procedure is carried out in an offline manner, or practically speaking, after the data acquisition has already been done. The steps for the data synchronization are as follows: Selection of the best suitable strain gauge to be used for the synchronization Division of the data by blocks equal to number of averages Synchronization (alignment) of the data, block-by-block Reassembly of all the blocks in one data signal FRF calculation Data import into LMS Test.Lab for modal analysis After an initial analysis, the strain gauge and FBG sensor on point 4 (from Figure 6) were chosen for the synchronization procedure - overall, all sensor pairs were suitable to be used, but one pair had to be chosen.

10 2462 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 Next, all signals were upsampled to improve the offline synchronization efficiency. This step can help to reduce errors - even if both signals would have the same sampling frequency, it is still possible that the samples are taken at a different time and therefore unsynchronized. Figure 9 shows an example of how this can happen - even though the sampling time T s shown in the Figure is the same for both signals (black and grey), they can still be shifted in time. By reducing the sampling periodt s, one can reduce how big this error will be. The resampling factor for the strain gauge was of 4, and the resampling factor for the FBG sensor was of 16, bringing both sensors sampling frequency up to 3200 Hz. signal T s time T s Figure 9: Sampling time error example Furthermore, the signals were divided by blocks - in total, they were divided in 20 blocks representing the 20 excitation cycles for the sine sweep. As a standard procedure, the first and last blocks are also discarded, so in the end 18 blocks were available. The next step is to align each block individually. This is carried out by using the cross-correlation function. For 2 very similar signals, synchronized in time, the cross correlation function should have its peak value exactly on the 0 lag position in the x-axis. If the signals are misaligned (which is our case), then the peak value will occur outside of the 0 lag position, but will represent how many lags (or sample differences) one of the signals should be shifted to be aligned. The result, after realigning all 18 blocks and putting them back together into one signal, is shown in Figure 10, where one of the realigned blocks is shown. Consequently, the FRF can be computed by using the H1 estimator. To calculate the crosspower and autopower functions to be used in the H1 estimator calculations, a rectangular window was used and no overlap was performed (each block consisted of a full sweep, starting and ending with 0 excitation, so leakage was not a problem). The resulting FRFs for the one of the FBG and strain gauge sensor pairs and their respective coherence functions are shown in Figure 11, where a comparison between the strain gauge and FBG signal quality can be made. Finally, the FRFs can be imported in LMS Test.Lab so that the modal analysis can be carried out. For this purpose, only the FBG sensor FRFs are really needed, since the strain gauge measurements were only used initially for the synchronization. The procedure to identify the modes is the same as in the classic displacement modal analysis. The PolyMAX identification algorithm was used and in total 9 strain mode shapes were identified - higher frequency modes were also seen, but the sensor resolution means that they were hard to be visualized. These mode shapes are shown in Figures 12 through 20. Moreover, an advantage of using both accelerometers and strain gauges in one single acquisition experiment is that both strain and displacement modes can be identified and visualized together. Figures 21, 22 and 23 show the displacement and strain modes together in one animation for the first 3 mode shapes. The deformation of the blade comes from the displacement modes acquired from the accelerometers, while the

11 MODAL TESTING: METHODS AND CASE STUDIES FBG sensor Strain gauge 30 Amplitude (µǫ) Samples x 10 4 Figure 10: FBG and strain gauge time signal alignment: zoomed in one of the blocks FBG sensor Strain gauge FBG sensor Strain gauge Amplitude (db) Coherence Frequency (Hz) (a) FRF comparison for sensor pair on point Frequency (Hz) (b) Coherence comparison for sensor pair on point 4 Figure 11: Comparison of FRFs and coherence for one of the sensor pairs coloring represents the surface strain on the blade. 4 Results analysis and conclusion In this paper, the concepts of strain modal analysis were introduced and verified experimentally. For this purpose, the theoretical formulation for strain modal analysis was presented, as well as its particular features and characteristics. Furthermore, two experimental cases were investigated - a small wind turbine and a helicopter blade. The first experiment was carried out using impact testing and multiple resistive strain gauges.the lack of reciprocity, as described in the theory section of this work, was shown. Moreover, the strain mode shapes obtained from test were correlated with a finite element model, and it was seen that the torsional modes, that

12 2464 PROCEEDINGS OF ISMA2014 INCLUDING USD2014 Figure 12: Helicopter blade strain mode - first bending mode at 3.58 Hz Figure 13: Helicopter blade strain mode - second bending mode at Hz Figure 14: Helicopter blade strain mode - first in-plane mode at 13.9 Hz Figure 15: Helicopter blade strain mode - third bending mode at Hz Figure 16: Helicopter blade strain mode - first torsional mode at 30.4 Hz Figure 17: Helicopter blade strain mode - fourth bending mode at Hz Figure 18: Helicopter blade strain mode - second in-plane mode at Hz Figure 19: Helicopter blade strain mode - fifth bending mode at Hz Figure 20: Helicopter blade strain mode - second torsional mode at Hz

13 MODAL TESTING: METHODS AND CASE STUDIES 2465 Figure 21: Helicopter blade first bending mode: deformation from displacement mode and coloring from strain modes Figure 22: Helicopter blade second bending mode: deformation from displacement mode and coloring from strain modes Figure 23: Helicopter blade third bending mode: deformation from displacement mode and coloring from strain modes induce shear strain, are harder to be correlated when most of the sensors are measuring normal strain. Finally, some other concepts were verified with the helicopter main rotor blade. Fiber Bragg Grating sensors were used to carry out a a strain modal analysis of the blade using shaker excitation. The sensors were able to capture all mode types (bending, in-plane and torsional) but the strain mode shapes were only useful in the visualization of the bending and in-plane modes. Future studies include the investigation of sensor placement for better strain field interpretation, hotspot (high stress and strain) locations, the time and modal relations between strain and displacement and methods of scaling the strain modes. Acknowledgements Fábio Luis Marques dos Santos, first author of this paper, is an Early Stage Researcher at LMS International, under the FP7 Marie Curie ITN project IMESCON (FP7-PEOPLE-2010-ITN, Grant Agreement No ). This research was also carried out in the Framework of FP7 ICT Collaborative project WiBRATE (FP7-ICT , Grant Agreement No ). The authors of this work gratefully acknowledge the European Commission for the support.

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