the use of strain sensors for modal identification of

The use of strain sensors for modal identification of aeroelastic structures

Y. Govers1, G. Jelicic1, T. Akbay1 1 German Aerospace Center (DLR), Institute of Aeroelasticity, Bunsenstr. 10, 37073 Göttingen, Germany e-mail: [email protected]

Abstract Acceleration measurements are most commonly employed to determine mode shapes by modal identification. There are two simple reasons for this: an acceleration sensor is easy to apply and deflection mode shapes can directly be estimated from acceleration signals by experimental or operational modal analysis techniques. In the past strain gauges have played a minor role in modal identification because they were of single use and more difficult to apply. Another drawback is that strain modes need to be converted into deflection modes. Nevertheless there are applications where strain measurements show better performance than acceleration measurements if rotating structures are considered. Recent progress in strain measurement technology with fiber optical strain sensors offering spatially nearly continuous measurements up to more than 100Hz of sampling rate raise the interest for aeroelastic applications. This study will analyze the use of strain sensors for aeroelastic applications.

1 Introduction

The following study has been made to analyze the applicability of strain measurements for aeroelastic applications. It is most common to use accelerometers for the analysis of the structural dynamics behavior. Acceleration sensors are easy to apply and can directly be used for modal identification and visualization of mode shapes. Strain sensors are usually much more difficult to apply but have benefits for special applications like blade testing of wind turbines [1, 2], helicopters [3] or turbomachinery [4] where high centrifugal accelerations influence the measurement accuracy of accelerometers.

(a) (b)

Figure 1: long exposure of wing in wind tunnel (a) and its measured strain in the span-wise direction (b)

Strain can be measured from strain gauges, fiber Bragg grating, continuous optical fibers and also 3D laservibrometry and photogrammetry. For non-laboratory applications the first three methods are most interesting since they can also be used in harsh conditions.

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Strain measurements can be used for load identification or fatigue assessment as discussed in [2] or to determine modal parameters like eigenfrequencies and damping [3, 5]. A procedure about how to obtain displacements from strain measurements is shown in [6] which describes the principle of local correspondence (LC) to estimate a mode shape as a linear combination of a given set of mode shapes. Recent developments in strain measurement technologies offer a variety of interesting features for aeroelastic applications. Figure 1 shows strain measurements of the static deformation of a very flexible aeroelastically tailored and forward swept wing that was analyzed in the crosswind tunnel at DLR Göttingen. The photo on the left shows a long exposure of an angle of attack sweep. The diagram on the right shows strain measurements in span-wise direction for discrete angles of attack in the range of –10° to +10°. The strain measurements have been obtained using an optical fiber that offers finely resolved strain information along the fiber. This technique is called optical frequency domain reflectometry (OFDR) [7]. The in strain discontinuities observed from the optical fiber measurements result from a variable stiffness composite design along the span of the wing. Stiffness variations along the span can be realized with composite material through a variation of the fiber angles along the span to achieve bending-torsion coupling. This method allows checking the validity of the numerical model in detail; in particular, the design of the composite’s stacking sequence can be compared with the manufactured wing. Another advantage of OFDR is the ability to measure with a sampling rate of 250Hz and a spatial resolution of 2.5mm in a fiber optical cable of 2m length. The only drawback is that the data needs to be post-processed before it can be evaluated.

(a) (b)

Figure 2: DLR research aircraft HALO (a) with underwing carriers for atmospheric research (b)

During a flight test with one of DLR’s research aircraft the dynamical behavior of the structure was monitored with accelerometers and additional strain gauges at the attachment points of underwing carriers (shown in Figure 2b). Here, it is useful to apply analog sensors like strain gauge bridges that can be measured simultaneously with accelerometers in a single data acquisition system. The current study is made to analyze the suitability strain modal testing (SMT) in contrast to displacement modal testing (DMT). Strain sensors shall be evaluated in terms of accuracy and handling capability within aeroelastic applications. A simple, easy to simulate, aluminum beam laboratory structure was built to investigate sensor placement optimization procedures and the aptitude of strain sensors for the validation of numerical models.

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2 Test setup

2.1 General

The setup consists of a simple aluminum beam with a rectangular hollow section (150×40mm) of 2000mm length and 4mm thickness, welded onto a 400×400mm aluminum plate. The plate is fastened with screws using a symmetrical pattern onto a seismic block which is isolated by air cushions from the structural dynamics laboratory’s building at DLR Göttingen. A National Instruments measurement system with a cDAQ-9188XT-Chassis equipped with 24 channels for strain gauges, 4 channels for IEPE sensors (force and acceleration) and a signal generator card has been used for structural dynamics testing. The deformation of the beam was measured by a laser Doppler vibrometer (PSV-400) from Polytec to obtain spatially highly resolved modes from the structure.

Figure 3: aluminum beam structure with installed strain gauges and shaker attachment

The structure was excited by an electrodynamic shaker with a maximum force of 20N and a maximum coil stroke of 12mm. The excitation force was measured with an ICP force cell; the driving point acceleration was measured with an ICP accelerometer.

2.2 Sensor placement

An MSC.NASTRAN finite element model of the structure has been built. The aluminum beam with a rectangular hollow section is modelled with ~9000 CQUAD4 shell elements to also represent local modes. The plate the beam is fastened to is also modelled with CQUAD4 elements. The nodal points of the plate are fixed at the screws. The results from NASTRAN’s SOL103 analysis are imported into MATLAB using an OP2 reader. Results of the numerical modal analysis are shown in Figure 4. This figure visualizes the deflection of the mode shape and at the same time the element strain εxx resulting from the modal deflection in beam direction. The strain information is color coded. Figure 4 shows different types of modes: mode 01 is a 1st flap-wise bending in the z-direction with maximum strain εxx at the root, mode 05 is the 2nd edge-wise bending, mode 06 is the 1st torsion mode, mode 24 is a higher flap-wise bending and mode 26 is a local mode where the surfaces of the beam vibrate 180° phase shifted. There are also longitudinal modes which are not of interest for our study.

MODAL TESTING: METHODS AND CASE STUDIES 2209

#01FEA 11.6Hz #05FEA 194.9Hz #06FEA 207.7Hz #24FEA 844.5Hz #26FEA 896.3Hz

Figure 4: different types of modes from finite element analysis

For strain sensor placement the numerical model was used. It was decided to only apply sensors in the x-direction. Even a single direction for strain measurements should be sufficient to distinguish between the first twelve modes of the structure. Figure 4 already gives the impression that strain gauges for εxx give a good observability for bending and torsion modes. As described in section 2.1 the strain gauges are applied as half bridges. Half-bridges make sense on a beam-like structure and can be configured to measure bending or normal strain. In this study all strain gauges have been applied to measure bending strains. This implies that mode 26 from Figure 4 cannot be detected since the half-bridge gives zero values for such a deformation. In order to use automatic sensor placement methods like the QR decomposition described in [8] or the effective independence (EI) method [9] it is necessary to generate strain modal matrices according to bending strains, which means that top and bottom surface element strains have to be subtracted. For ease of installation and better visual interpretation, only two rows of elements in the x-direction close to left and right edge of the top and bottom surface have been selected (see Figure 5a) for the strain modal matrix subject to sensor placement. Another row of elements lying on the edge has been included to measure edge-wise bending modes.

(a) (b) (c)

Figure 5: results of sensor optimization for a subset of selected elements (a) of QR (b) and EI (c)

Twelve modes have been selected for sensor placement; the results of both methods presented in Figure 5(b) and (c) show no difference. In the following discussion the sensor placement results from the QR-decomposition have been taken for installation. The selected 12 sensor positions (9 for flap-wise and 3 for edge-wise bending) have been extended to 24 positions for better visualization, which also improves the off-diagonal MAC-value of modes 6 and 8. This effect is visible in the final comparison between calculated and measured strain modes which is shown in the MAC matrix of Figure 14.

2.3 Excitation of the structure

The structure is excited with an electrodynamic shaker using a logarithmic swept sine signal of 2oct/min from 6 to 1000Hz with constant force amplitude of 9N. The swept sine signal has been chosen for better

x y

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Elements selected for strain sensor placement optimization for flap-wise bending


signal-to-noise ratio. The start and the end of the excitation force signal is extended by ramp up and ramp down of the force amplitude which is shown in Figure 6. This extension guarantees that only the desired frequencies are excited during start and end of the signal. Furthermore, the attachment of the push-pull rod, which is typically realized with superglue (see Figure 3), lasts longer.

Figure 6: ramp-up of excitation signal

3 Results

3.1 Time data

Figure 7 shows two excitation runs with 24 time records of the measured strain gauges for the flap and edge-wise excitation. Each dataset shows several resonance peaks at different stages of the excitation signal.

(a) (b)

Figure 7: strain gauges time signals from flap-wise (a) and edge-wise (b) excitation

While the strain gauges have been acquired simultaneously, the scanning LDV measurements need to be acquired point-by-point like a rowing accelerometer test, therefore the acquisition time is multiplied by the number of response points and the accuracy of the measurement is dependent on the repeatability of the excitation and the linearity of the structure. 350 measurement points for flap-wise bending have been acquired altogether. Even here a swept-sine signal was used for excitation.

3.2 Signal processing

The recorded time data signals have been processed into strain frequency response functions (SFRF) by referencing the input force in order to estimate strain mode shapes within the framework of experimental modal analysis.


(a) (b)

Figure 8: strain frequency response functions from flap-wise (a) and edge-wise (b) excitation

Transfer functions have been obtained using the H1 estimator by applying Welch’s modified periodogram with an overlapping Hann window with DLR’s object-oriented modal analysis MATLAB toolbox. The SFRFs obtained from flap-wise and edge-wise excitations are presented in Figure 8. The transfer functions in log-scale show very clear resonance peaks up to the maximum excitation frequency of ~1150Hz. The transfer functions from LDV measurements are shown in Figure 9. Here only the flap-wise excitation was applied and analyzed. Clear resonance peaks are again visible up to the maximum excitation frequency.

Figure 9: frequency response functions from z-velocities and flap-wise excitation

3.3 Modal analysis

Strain gauge modal analysis can be applied on strain frequency response functions in the same way as on transfer functions calculated from acceleration, velocity or displacement signals.

Figure 10: stabilization diagram from flap excitation


Figure 10 shows the stabilization diagram of the strain modal analysis including a summation of experimental (red) and fitted (blue) FRF amplitudes. The eigenvalues and eigenvectors are estimated using the well-known least-squares complex frequency method (LSCF), from which the stabilization diagram is constructed using the DLR’s Modal Analysis MATLAB toolbox. A selection of identified modes from DMT and SMT is presented in Figure 11. The arrows in case of strain modes (STR) represent the strain from flap (arrows in z-dir.) and edge-wise (arrows in y-dir.) bending. The visual inspection makes clear that it is hard to distinguish between some modes from SMT, e.g. 1st and 2nd torsion mode at 210 and 482Hz.

#01VIB 11.0Hz #01STR 11.0Hz #02VIB 68.3Hz #03STR 68.3Hz



Figure 11: comparison of modes from laservibrometry (VIB) and strain gauge (STR) modes

Nevertheless, a number of 23 different modes could be identified from SMT up to 1100Hz. From LDV measurements 20 modes up to 1000Hz could be identified. The difference stems from the missing edge-wise excitation for LDV measurements as well as the missing vibrometer points on the edge of the beam. Figure 12 shows pure edge-wise bending modes identified from SMT that cannot be identified from LDV.

#02STR 30.9Hz #05STR 190.6.0Hz #10STR 518.9Hz #20STR 962.7Hz

Figure 12: pure edge-wise bending modes from strain gauges


The application of the modal assurance criterion (MAC) [10] on the identified strain modal matrix shows a nice auto-MAC matrix (see Figure 13a). The MAC matrix shows that all 23 identified strain modes can clearly be separated from each other. The MAC matrix from DMT (see Figure 13b) shows off-diagonal that are closer to zero but is built from 350 response points and not from 24.

(a) (b)

Figure 13: auto-MAC matrices from strain modes obtained by excitation in the y and z directions

The values on the y-axis of the auto-MAC matrices show the mode indicator function introduced in [11] normalized to 1000 for real modes. Modes from SMT show higher values than the ones from DMT which is due to the “rowing accelerometer” principle testing from LDV measurement. This effect still has to be analyzed in more detail in a further study. Nevertheless, the strain gauge sensors reveal a very good phase behavior even in the high frequency range.

Figure 14: comparison of numerical and experimental strain modes

The MAC matrix between the identified strain modes and the numerical strain modes shown in Figure 14 shows a very good comparison even up to 1000Hz and above. This MAC matrix excludes all numerical modes that cannot be identified with the strain gauge half bridges like mode 26 shown in Figure 4 where εxx on top and bottom surface is equal.


4 Conclusions

The paper shows results from strain modal testing in contrast to displacement modal testing on a simple beam structure. For sensor placement of the strain gauges, two optimization methods have been analyzed that are typically chosen for accelerometer sensor placement; both methods show similar results. The results of the QR-decomposition method have been chosen and 24 locations have been selected on the structure. 48 strain gauge sensors have been configured as 24 half bridges measuring bending strain. The excitation of the structure was performed with an electrodynamic shaker using a logarithmic swept-sine signal with constant force amplitude which yields strain time data signals with a good signal to noise ratio. Modal identification on strain frequency response functions can be performed with known estimators and yield strain mode shapes which can even be interpreted visually. The strain mode shape vectors show nice phase purity even up in the very high frequency range which underlines the accuracy of the used strain gauge sensors. The comparison with a 350-point measurement LDV scan shows that there are no significant drawbacks. SMT results can be used for numerical model calibration as also the modal assurance criterion can be used for automatic mode pairing with a numerical model. Therefore it can be stated that strain sensors that are well positioned yield a significant contribution for vibration measurements and can even replace accelerometers when necessary. Next, strain modes will be included in a model updating process. Furthermore, the strain-to-displacement relation will be analyzed in more detail to investigate if strain measurements can instantaneously be transformed into displacements.

References

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[2] Maes, K., et al., Dynamic Strain Estimation for Fatigue Assessment of an Offshore Monopile Wind Turbine Using Filtering and Modal Expansion Algorithms. Mechanical Systems and Signal Processing, (2016). 76–77: pp. 592-611.

[3] Santos, F.L.M.d., et al., An Overview of Experimental Strain-Based Modal Analysis Methods, Proceedings of the International Conference on Noise and Vibration Engineering. Leuven, Belgium, (2014).

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[7] Soller, B.J., et al., High Resolution Optical Frequency Domain Reflectometry for Characterization of Components and Assemblies. Optics Express, (2005). 13(2): pp. 9.

[8] Schedlinski, C., Link, M., An Approach to Optimal Pick-up and Exciter Placement, Proceedings of the International Modal Analysis Conference. Dearborn, Michigan, USA, (1996).

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