an overview of experimental strain-based modal analysis...

An overview of experimental strain-basedmodal analysis methods

F. L. M. dos Santos1,2,3, B. Peeters1, J. Lau1, W. Desmet2, L. C. S. Goes31 LMS, A Siemens Business, LMS International N.V.Interleuvenlaan 68, B-3001 Leuven, Belgiume-mail: [email protected]

2 KU Leuven, Department of Mechanical Engineering,Celestijnenlaan 300 B, B-3001, Heverlee, Belgium

3 Instituto Tecnologico de Aeronautica (ITA), Praca Marechal Eduardo Gomes, 50 - Vila das Acacias CEP12.228-900 Sao Jose dos Campos - SP - Brazil

AbstractThe most established way of performing experimental modal analysis is to useacceleration or velocity basedtransducers that lead to the calculation of the displacement mode shapes. However, there are applicationswhere the use of strain measurements makes for a more attractive and interesting option. Strain gaugeshave been commonly used for static load testing of mechanical products in the aeronautic, automotive andmechanical industry. Moreover, fatigue testing, durability analysis and lifetime prediction has also been acommon application where strain gauges are used. This sort of testing is a common part of the product devel-opment process, and additional information on product durability and dynamic performance can be assessedby obtaining the modal parameters of the system, while still using the same instrumentation. Moreover, sincestrain measurements are more directly related to stress, fatigue and failure, strain-based measurement meth-ods can be a good option for structural health monitoring methods and monitoring systems. Applicationswhere sensor size and placement might be critical are also good candidates for strain-based methods. Heli-copters, wind turbines and gas turbines are a good example where strain gauges are more suited for vibrationmeasurements. Some application cases of dynamic strain measurements and dynamic strain modal analysisare 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 ofdisplacement-based responsesand sensors, such as accelerometers and laser vibrometers. The use of strain sensors and measurements formodal testing [1, 16], on the other hand, has been less accentuated, mostlydue to the difficulties with respectto the use strain gauges and with the simplicity of the use of accelerometers. On the other hand, there hasbeen increased interest from both industry [8] and academia [4] on assessing the potential benefits of strainmodal analysis. Another important use of strain modal testing is on evaluating structural integrity or stressconcentration [18] on design prototype stages and also monitoring in real-time(with structural health mon-itoring systems (SHM) [6]) , which has led to an increase in the number of dynamic strain applications andto the development of improved identification and measurement techniques [3,19], as well as to improvedsensor 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-

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tion [15] has also been a common application where strain gauges are used.This sort of testing is a commonpart of the product development process, and additional information onproduct durability and dynamicperformance can be assessed by obtaining the modal parameters of the system, while still using the sameinstrumentation.

A very important contribution on the field of strain measurement are the fiber optic sensors, or Fiber BraggGrating (FBG) sensors [5, 7]. Their robustness to magnetic interference, added to the easiness of creatingsensor 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, readyto be used and adequate for modal testing, is another incentive to carryingout a strain modal analysis, savingup 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 sensorlocation is an issue. Such is the case in aerospace applications, like gas turbines, wind turbines and heli-copters [14], where size and weight are very restricted, and any sensor place on a blade should affect itsaerodynamic properties as little as possible. One particular use of the strain measurements and strain to dis-placement relations is the strain pattern analysis (SPA), where strain measurements are used to predict bladedisplacements.

2 Theoretical background

To obtain the strain modal formulation, one can start with the fundamental theory of modal analysis. Modaltheory states that the displacement on a given coordinate can be approximated by the summation of annumber of modes:

u(t) =n

∑

i=1

φiqi(t) (1)

whereu is the displacement response inx direction,φi is theith (displacement) vibration mode, andqi isthe generalized modal coordinate andt is time. For small displacements, given the theory of elasticity, thestrain/displacement relation is:

εx =∂

∂xu (2)

And similarly, the same relationship exists between the strain vibration modes and the displacement modes:

ψi =∂

∂xφi (3)

This way, by the relations on equations (2) and (3), the expression on (1) can be rewritten as:

ε(t) =

n∑

i=1

ψiqi(t) (4)

Moreover, the relationship between the generalized modal coordinateq and an input forceF is:

qi = Λ−1

i φiF , with Λi = (−ω2mi + jωci + ki) (5)

wheremi, ci andki are theith modal mass, modal damping and modal stiffness, andω is the excitationfrequency.

Substituting (5) into (4), the relation between a force input and a strain output, in terms of displacement andstrain modes is represented as:

εi =n

∑

i=1

ψiΛ−1

i φiF (6)

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And finally, the strain frequency response function (SFRF) can be obtained, in matrix form:

[Hε] =n

∑

i=1

Λ−1

i {ψi} {φi} = [ψ] [Λ]−1 [φ]T (7)

The expansion of (7) is:

Hε11

Hε12

· · · Hε1Ni

Hε21

Hε22

· · · Hε2Ni

......

......

HεNo1

HεNo2

· · · HεNoNi

=n

∑

i=1

Λ−1

i ·

ψ1iφ1i ψ1iφ2i · · · ψ1iφNii

ψ2iφ1i ψ2iφ2i · · · ψ2iφNii

......

......

ψNoiφ1i ψNoiφ2i · · · ψNoiφNii

(8)

whereNo represents the number of strain gauge measurement stations (or the numberof output measure-ments) andNi represents the number of excitation points (or the number of inputs).

There are some remarks and considerations that can be obtained from thestrain modal analysis theory pre-sented:

• The columns of the matrix correspond to the strain responses due to the excitation points along therows of the matrix;

• the SFRF matrix is not symmetric - for example,HεAB is different thanHε

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 rovinghammer impact testing), the displacement mode shapes can be obtained;

• the similarity of the strain modal formulation and the displacement modal formulation means that thesame identification methods can be used fro both.

In general, there are drawbacks but also advantages present in strain modal theory. A very important advan-tage is its versatility - being able to identify both displacement and strain modes with one type of sensor is aproperty 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 gaugebridges).

Of the above mentioned items, instrumentation can be very critical - in general, some skill is required inbonding strain gauges or other strain sensors, and the bonding quality can degrade the quality of the measuredsignal.

In this work, the modal identification methodology used was the PolyMAX identification method [10], whichcould be used without any modification. Moreover, the additional signal conditioning was not necessarydue to the use of an acquisition system which already contained modules with strain gauge bridges (LMSSCADAS mobile with VB8 modules).

MODAL TESTING: METHODS AND CASE STUDIES 2455

3 Experimental analysis

To illustrate some of the characteristics of strain modal analysis shown in the previous sections, two test caseswill be presented. The first test subject is a small composite wind turbine blade, which was tested with straingauges 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 testedwith an electrodynamicshaker 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 straingauges were glued to the surface of the blade and an impact hammer with an impedance head was used toexcite the structure at several locations. The blade was fixed at its root, toimpose a cantilevered condition. Ofthe 20 strain gauges, one of them consisted of a strain gauge rosette to measure purely shear strain, while theother 19 strain gauges were aligned with the radius of the blade and were measuring normal strain. Figure 1shows the wind turbine blade, its sensor locations represented in the acquisition software and a finite elementmodel 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) showsthe SFRF of an arbitrary strain gauge,where the resonance peaks are clearly visible. Moreover, the phase shift due to the resonances is the samefor the SFRF, where the phase shifts in 180 degrees whenever there is aresonance peak.

Additionally for this experiment, a reciprocity check was carried out to verify if the theory for strain modalanalysis was correct - for this purpose, two measurement points were picked and the impact hammer wasused to excite those points. A successful reciprocity check should yield identical or almost identical FRFsfor a classical displacement modal analysis. In the case of the strain modalanalysis, as it can be seen onFigure 2(b), reciprocity is not guaranteed for the strain measurements, since the SFRFs do not match eachother.


20 40 60 80 100 120 140 160 180 200−160

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20 40 60 80 100 120 140 160 180 200−200

0

200

Frequency (Hz)

Pha

se(o )

Am

plitu

de(d

B)

(a) Wind turbine blade - Strain FRF from an arbitrary measure-ment point

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Am

plitu

de(d

B)

(b) Wind turbine blade - reciprocity check using strain measure-ments

Figure 2: Wind turbine blade: (a) strain FRF; (b) reciprocity check usinga strain gauge - reciprocity is notguaranteed

After the reciprocity check, the modal analysis and strain modes identificationwas carried out. For thispurpose, the PolyMAX identification method was used. A bandwidth from 10 to210 Hz was taken intoaccount as 6 clear modes were identified in that frequency range - thesemodes consist of bending, in-planeand torsional modes. Table 1 shows the natural frequencies and mode types of the identified modes. Finally,

Mode Frequency Modenumber [Hz] type

1 17.25 bending2 46.29 in-plane3 63.99 bending4 148.97 bending5 178.38 torsional6 199.79 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 obtainedfrom a FEM analysis of the blade, from the model shown on Figure 1(c). The strain modes of the FEMmodel are extracted via the strain tensor matrix, where the appropriate strin directions are taken according tothe orientation of the strain gauges on the blade, and the strain gauge locationand size are approximated toone element of the finite element model.

Two methods of comparison were used to compare the simulation and experimental analysis, the percent-age 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 diago-nal MAC value. Additionally, the natural frequency values obtained usinga traditional accelerometer basedimpact test are also show, for the purpose of comparison. Moreover, Figure 3 shows the full MAC matrix forall the considered modes.

An initial analysis of the data from Table 2 shows that there is good agreement between the simulation modeland the experiment. Regarding the natural frequencies, most of them arewithin 10 % of difference, exceptfor modes 2 and 5. Moreover, the MAC values show very good correlation, except for mode 5. The lackof correlation for this mode is understandable - it is a torsional mode, which means that most of its energy


Mode Natural frequency Natural frequency Natural frequency Difference MACnumber strain gauges [Hz] accelerometers Simulation [Hz] %

1 17.25 17.22 15.6 9.56 0.8342 46.29 46.30 38.6 16.61 0.83 63.99 63.95 65.5 2.35 0.754 148.97 147.52 148.4 0.38 0.8335 178.38 178.33 158.6 11.08 0.336 199.79 200.0 183.9 7.95 0.863

Table 2: Strain modes comparison - experiment and simulation

1

2

3

4

5

6

1 2 3 4 5 6

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Modal Assurance Criterion (MAC)

Exp

erim

enta

lMod

es

FEM Modes

Figure 3: Modal Assurance Criterion - comparison of computational strainmodes and experimental strainmodes 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 adifferent and newer type of sensor was tested as - the Fiber Bragg Grating (FBG) sensors. These sensors arepresent in an optical fiber and a laser scans through the fiber, for a whole range of wavelengths. Relativedistortions 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 piezoICP accelerometers, were also used - by using both displacement and strain sensors, one can have a betteridea of the similarities between both types.

In total 20 FBG sensors were used - the sensors are actually gratings marked within the fiber, which meansthat a single fiber can hold multiple sensors. This can be an advantage in termsof instrumentation time andamount 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 usedto be able to better assess the noise floor of the FBG sensors, and also to synchronize them with the forcemeasurements - since the FBG sensors use their own acquisition unit, while the force and other sensors aremeasured with the standard LMS SCADAS measurement equipment, it is importantthat an offline synchro-nization procedure is carried out to synchronize the FBG sensors with theforce sensor. Additionally, 16accelerometers were collocated with the FBG sensors to investigate the visualize the displacement modesand investigate strain-displacement relations. Figure 4 shows a part of theblade with the above mentionedsensors, while Figure 5 shows the strain gauge collocated with the FBG sensor.


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 analysisis:

• 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


The measurement location for all the sensors is shown in Figure 6 - the 16 accelerometers were collocatedwith the FBG sensors, except on locations 4, 5, 14 and 15, where no accelerometers were placed. Theresistive strain gauges were collocated with the FBG sensors on locations 4, 7, 15 and 19. The blade wassuspended by elastic cords to obtain an approximate free-free boundary condition. With this sort of boundarycondition, strain is close to zero in the two tips of the blade and therefore, the choice was to place the sensorstowards the middle of the blade, such that the measured strain would be higher. The blade was excited withan electrodynamic shaker and the driving point is represented in Figure 6with an×.

12345678910

11121314151617181920

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 onemeasured by the strain gauge, such that they are both capturing mainly the same types of modes (especiallybending). Figure 7 better depicts the sensor orientations and the measurement directions.

Accelerometer

measurement direction

Strain gauge and FBG

measurement direction

x

x

y

y

z

z

Figure 7: Helicopter blade with strain sensors and accelerometer measurement directions - the accelerometermeasures 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 dataacquisition details used for the SCADAS Mobile system are show in Table 3. For the rest of this work, thedata obtained from the sine sweep experiments will be used. Similar results were also achieved using burstrandom excitation.

The frequency range of interest was limited in particular to less than 100 Hz due to the limitations of the FBGacquisition system, that could sample at 200 Hz at the highest. Most importantly,due to the intrinsic natureof the FBG sensor and the way the data is acquired, discretized and digitalized, it is not possible to use ananalog anti-aliasing filter, which means that care must be taken not to excite thestructure past the maximumallowed bandwidth. A simple check that can be carried out to verify the qualityof the measurements is thecomparison autopower spectrum between an FBG sensor and its collocatedstrain gauge. Figure 8 shows thiscomparison for the sine sweep excitation case. As it can be seen, there are no major deviations between both


Excitation signal burst random sine sweepNumber of averages 20 20Sampling frequency 800 Hz 800 Hz

Frequency range of interest 8 to 90 Hz 8 to 90 HzFrequency resolution 0.048 Hz 0.048 Hz

FRF estimator H1 H1

Table 3: Data acquisition details

spectra.

0 20 40 60 80 100 120 140 160 180 200−100

−80

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0

20

40

Strain gaugeFBG

Frequency (Hz)

Am

plitu

de(d

B)

Autopower Spectrum comparison

Figure 8: Autopower spectrum comparison between strain gauge and FBGsensor

The next step for the strain modal analysis using the FBG sensors is to synchronize their signals with the mea-sured force signal. Since the resistive strain gauges, accelerometers and force cell signals were all acquiredwith the same acquisition unit and are therefore synchronized, the FBG sensors have only to be synchronizethem with one of these sensors - hence, the use of the collocated strain gauges. This synchronization proce-dure is carried out in an offline manner, or practically speaking, after thedata acquisition has already beendone. 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 (fromFigure 6) were chosen for thesynchronization procedure - overall, all sensor pairs were suitable to be used, but one pair had to be chosen.


Next, all signals were upsampled to improve the offline synchronization efficiency. This step can help toreduce errors - even if both signals would have the same sampling frequency, it is still possible that thesamples are taken at a different time and therefore unsynchronized. Figure 9 shows an example of how thiscan happen - even though the sampling timeTs shown in the Figure is the same for both signals (black andgrey), they can still be shifted in time. By reducing the sampling periodTs, one can reduce how big this errorwill be. The resampling factor for the strain gauge was of 4, and the resampling factor for the FBG sensorwas of 16, bringing both sensors’ sampling frequency up to 3200 Hz.

time

sign

al

Ts

Ts

Figure 9: Sampling time error example

Furthermore, the signals were divided by blocks - in total, they were divided in 20 blocks representing the 20excitation cycles for the sine sweep. As a standard procedure, the firstand last blocks are also discarded, soin the end 18 blocks were available. The next step is to align each block individually. This is carried out byusing the cross-correlation function. For 2 very similar signals, synchronized in time, the cross correlationfunction should have its peak value exactly on the 0 lag position in thex-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 manylags (or sample differences) one of the signals should be shifted to be aligned. The result, after realigning all18 blocks and putting them back together into one signal, is shown in Figure 10, where one of the realignedblocks is shown.

Consequently, the FRF can be computed by using the H1 estimator. To calculatethe crosspower and au-topower functions to be used in the H1 estimator calculations, a rectangular window was used and no overlapwas performed (each block consisted of a full sweep, starting and ending with 0 excitation, so leakage wasnot a problem). The resulting FRFs for the one of the FBG and strain gaugesensor pairs and their respectivecoherence functions are shown in Figure 11, where a comparison between the strain gauge and FBG signalquality can be made.

Finally, the FRFs can be imported in LMS Test.Lab so that the modal analysis canbe carried out. Forthis purpose, only the FBG sensor FRFs are really needed, since the strain gauge measurements were onlyused initially for the synchronization. The procedure to identify the modes is the same as in the classicdisplacement modal analysis. The PolyMAX identification algorithm was used and in total 9 strain modeshapes were identified - higher frequency modes were also seen, but the sensor resolution means that theywere 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 experimentis that both strain and displacement modes can be identified and visualized together. Figures 21, 22 and23 show the displacement and strain modes together in one animation for the first 3 mode shapes. Thedeformation of the blade comes from the displacement modes acquired from the accelerometers, while the


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30

40

50

FBG sensorStrain gauge

Samples

Am

plitu

de(µǫ)

Figure 10: FBG and strain gauge time signal alignment: zoomed in one of the blocks

10 20 30 40 50 60 70 80 90−60

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Frequency (Hz)

Am

plitu

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(a) FRF comparison for sensor pair on point 4

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1


Frequency (Hz)

Coh

eren

ce

(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 thispurpose, the theoretical formulation for strain modal analysis was presented, as well as its particular featuresand characteristics. Furthermore, two experimental cases were investigated - a small wind turbine and ahelicopter blade.

The first experiment was carried out using impact testing and multiple resistive strain gauges.The lack ofreciprocity, as described in the theory section of this work, was shown. Moreover, the strain mode shapesobtained from test were correlated with a finite element model, and it was seenthat the torsional modes, that


Figure 12: Helicopter blade strain mode - first bending mode at 3.58 Hz

Figure 13: Helicopter blade strain mode - second bending mode at 10.27 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 20.30 Hz

Figure 16: Helicopter blade strain mode - first torsional mode at 30.4 Hz

Figure 17: Helicopter blade strain mode - fourth bending mode at 33.79 Hz

Figure 18: Helicopter blade strain mode - second in-plane mode at 37.61 Hz

Figure 19: Helicopter blade strain mode - fifth bending mode at 49.48 Hz

Figure 20: Helicopter blade strain mode - second torsional mode at 61.15 Hz


Figure 21: Helicopter blade first bending mode: deformation from displacement mode and coloring fromstrain modes

Figure 22: Helicopter blade second bending mode: deformation from displacement mode and coloring fromstrain modes

Figure 23: Helicopter blade third bending mode: deformation from displacement mode and coloring fromstrain 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 sensorswere used to carry out a a strain modal analysis of the blade using shakerexcitation. The sensors were ableto capture all mode types (bending, in-plane and torsional) but the strain mode shapes were only useful inthe visualization of the bending and in-plane modes.

Future studies include the investigation of sensor placement for better strainfield interpretation, hotspot (highstress and strain) locations, the time and modal relations between strain and displacement and methods ofscaling the strain modes.

Acknowledgements

Fabio Luis Marques dos Santos, first author of this paper, is an Early Stage Researcher at LMS Interna-tional, under the FP7 Marie Curie ITN project “IMESCON” (FP7-PEOPLE-2010-ITN, Grant Agreement No.264672). This research was also carried out in the Framework of FP7 ICT Collaborative project “WiBRATE”(FP7-ICT-2011-7, Grant Agreement No. 289041). The authors ofthis work gratefully acknowledge the Eu-ropean Commission for the support.


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