Joint Identification through Hybrid Models Improved by Correlations
Zeeshan Saeed, Christian Maria Firrone, Teresa Maria Berruti
JJoint Identification through Hybrid Models Improved byCorrelations
Zeeshan Saeed ∗ , Christian M. Firrone, Teresa M. Berruti Department of Mechanical Engineering,Politecnico di TorinoCorso Duca degli Abruzzi 24, 10129Turin, Italy
Abstract
In mechanical systems coupled with joints, accurate prediction of the joint char-acteristics is extremely important. Despite years of research, a lot is yet to belearnt about the joints’ interface dynamics. The problem becomes even moredifficult when the interface Degrees-of-Freedom (DoF) are inaccessible for Fre-quency Response Function (FRF) measurements. This is, for example, the caseof bladed-disk systems with dove-tail or fir-tree type joints. Therefore, an FRFbased expansion method called System Equivalent Model Mixing (SEMM) isused to obtain expanded interface dynamics. The method uses numerical andexperimental sub-models of each component and their assembly to produce therespective expanded or hybrid sub-models. By applying substructure decou-pling to these sub-models, the joint can be identified. However, the joint can benoisy due to expansion and measurement errors which propagate to the hybridsub-models.In this paper, a correlation based approach is proposed in the SEMM methodwherein the quality of the expanded sub-models is improved. In this new ap-proach, several expanded models are generated systematically using differentcombinations of the experimental FRFs and computing a parameter, FrequencyResponse Assurance Criteria (FRAC), to evaluate quality of the contribution ∗ Corresponding author
Email address: [email protected] (Zeeshan Saeed)
Preprint submitted to Journal of Sound and Vibration 23 November, 2020 a r X i v : . [ phy s i c s . a pp - ph ] J a n f the different measurements. The lowest correlated channels or FRFs can befiltered out based on a certain threshold value of FRAC. Using the improved hy-brid sub-models, the joint identification also shows a remarkable improvement.The test object for the method is an assembly of disk and one blade with adove-tail joint. Keywords:
Joint Identification, System Equivalent Model Mixing, CorrelatedSEMM, Blade-root, Turbine disk, System Identification
NomenclatureAbbreviations
DoF Degree(s) of FreedomFBS Frequency Based SubstructuringFRAC Frequency Response Assurance CriteriaFRF Frequency Response FunctionSEMM System Equivalent Model MixingSEREP System Equivalent Reduction Expansion ProcessVP Virtual Point
Latin Symbols B Signed Boolean matrix f External force vectori, j, k Dummy indices m Number of rows of experimental FRF matrix m Vector of virtual forces n Number of columns of experimental FRF matrix2
Index for columns of experimental FRF matrix q Vector of virtual displacements r Index for rows of experimental FRF matrix R Matrix of coordinates’ information about VP T Matrix of interface displacement or force modes u Displacement vector Y FRF matrix, also uncoupled matrix ˆ Y FRF matrix of correlated channels
Greek Symbols (cid:15)
Expansion error φ FRAC ω Frequency in Hz
Superscripts A , B Substructure identifiersavg Average or meanexp Measured or experimental model J JointN Numerical modelov Overaly modelS Hybird or expanded model by SEMM
Subscipts b Set of boundary DoF 3
Set of internal DoF where responses are measured e Set of internal DoF where excitations are applied g Set of global DoF i Set of internal DoF v Set of internal DoF where responses are measured and reserved forvalidation w Set of internal DoF where excitations are applied and reserved forvalidation
1. INTRODUCTION
Joints are found in many mechanical systems and they tend to influence thesystem behaviour significantly. From structural dynamics perspective, knowl-edge of their characteristics is extremely important to accurately predict thestructural response. There have been numerous studies in the past that try toidentify the joint parameters [1–9]. Generally, in spectral methods, the jointsare identified by inverse substructuring [10] or substructure decoupling [11–13].The inverse substructuring is based on a-priori knowledge of the system to beidentified. By using a specific formulation of the dynamic stiffness matrices andmeasuring only the connection dynamics, the joint properties can be identified.This approach is applied to resilient rubber isolators between two substructuresin [5, 7]. On the other hand, substructure decoupling methods do not requirea-priori knowledge of the system and can be used to decouple any (linear) typeof joint. This black-box [14] type identification is more general and allows one tofit or optimize the parameters (stiffness, damping or even mass) later [1–3, 15].The decoupling methods in the class of experimental substructuring have sofar been applied to simpler joint interfaces. This is due to difficulties in exper-imental substructuring [11] which needs the acquisition of Frequency Response4unctions (FRFs) at the interface Degrees-of-Freedom (DoF). Measurement er-rors of the FRFs at the interface can cause spurious peaks [16]. These decouplingmethods also often require square FRF matrices and collocated DoF. One of themain challenge then becomes to accurately acquire the drive-point FRFs [17, 18].When it comes to complex interfaces such as those found in bladed-disk sys-tems, the conventional methods are not applicable. Their dove-tail or fir-treeinterfaces are not reachable for measurements. Therefore, expansion methodsare needed. Using the Frequency based Substructuring (FBS) framework [19],System Equivalent Model Mixing (SEMM) [20] allows expansion of the mea-sured dynamics on the internal DoF to the interface DoF. The peculiarity ofSEMM is that it uses different formulations of the same system, coupling thenumerical model with experimental measurements performed on a limited num-ber of locations. The final result is a hybrid model mimicking and expandingthe dynamics at the measured and unmeasured DoF, respectively. Its modaldomain counterpart System Equivalent Reduction Expansion Process (SEREP)[21] can also be used for expansion. However, SEREP requires mode shapesextraction from the measured FRF which is yet another challenging task forlight-weight and complex geometries. Provided that the interface dynamics areobservable by internal FRF measurements, SEMM allows to avoid:• direct measurements at the interface; neither for response nor for excita-tion [22],• drive point FRF measurements,• modal parameters estimation especially in regards to the damping.This makes SEMM a great candidate for FRF expansion to a complex and in-accessible interface as well as for the identification of the joint properties (linear)by substructure decoupling methods. It should be noted that the decouplingmethods are sensitive to errors in measured FRFs [1, 6, 12, 23]. If SEMM isused for the joint identification by the decoupling method, it is imperative thatany errors in the hybrid FRFs generated by SEMM are reduced. Since SEMM5ixes different models of a structure, discrepancies among them can result inundesirable effects in the hybrid model. The mixed models are, generally, oftwo types:1. experimental, which provides compact FRF content but it contains mea-surement errors including noise.2. numerical, which provides a larger DoF set but its accuracy is influencedby material properties, finite element discretization schemes and boundaryconditions.It is quite common that some discrepancies arise when the sensor/impact po-sitions in the actual test and the corresponding numerical nodes do not coin-cide. As a results, inconsistencies propagate in the component hybrid modelsby SEMM and affect the joint decoupling. In order to reduce the effect of thiserror as well as the noise in experimental FRFs, a correlation based metric ishere introduced in the SEMM method. This new approach will be called cor-related SEMM . In particular, the correlated SEMM generates multiple hybridmodels by systematically using subsets of the experimental model. A statisticalcorrelation parameter called Frequency Response Assurance Criteria (FRAC)is computed between an expanded FRF and an experimental one kept for val-idation. The procedure is repeated for all the response and input channels. Inthis way, the highest and lowest correlated channels (DoF) can be identified andthe lowest ones filtered out, if needed. This approach of computing correlationsof the FRFs is analogous to VIKING (Variability Improvement of Key Inaccu-rate Node Groups) [24, 25] which works in the modal domain by improving themodal basis expanded by SEREP through modal correlations.Moreover, it is good to underline that the constraints or boundary condi-tions are always difficult to implement [26, 27] both in an experimental and ina numerical model. This causes an additional error in the hybrid models. Ina recent work of these authors related to the present study on the joint identi-fication [28], the constraint modelling in one of the substructures – the disk –proved to greatly affect the the joint identification result.6n this paper, the new correlated SEMM method is applied for the identifi-cation of the same blade-root joint of the bladed-disk tested in Ref. [28]. Boththe blade and disk are modelled and tested in free boundary conditions, thereby,minimizing the modelling discrepancies arising from the geometric constraints.The hybrid models’ quality is upgraded by using the correlated SEMM methodby filtering the uncorrelated FRFs (measured and numerical). As a result, theeffect of improved hybrid models is investigated on the joint decoupling.The paper is organized as follows: Section 2 briefly covers the FBS methodwhich allows coupling and decoupling of substructures. Section 3 presents indetail different DoF classification and the models that are mixed together inthe standard SEMM method. Section 4 introduces and elaborates theoreticalbasis of the correlated SEMM method. The standard and correlated SEMM arethen applied to the uncoupled blade and disk test-cases in Section 5 followedby the expanded interface description in Section 6. The SEMM methodology isextended to the coupled system and applied to the blade-disk joint identificationin Section 7 and Section 8, respectively.
2. FREQUENCY BASED SUBSTRUCTURING
This section briefly introduces the key substructuring expressions based onLagrange Multiplier Frequency Based Substructuring (LM-FBS). Some expres-sions used here are described in detail in the next sections. Consider two examplesubstructures A and B of Fig. 1 whose admittances Y A and Y B , respectively,can be computed or measured on the indicated internal i and boundary b DoF.The uncoupled receptance (or accelerance) Y is defined as a block diagonalmatrix. Y (cid:44) diag ( Y A , Y B ) = Y A
00 Y B (1)The two substructures A and B can be coupled by using the LM-FBS form[19] as: u = (cid:0) Y − YB T ( BYB T ) − BY (cid:1) f = ⇒ u = Y AB f (2)7 igure 1: Two dummy substructures A and B with their internal and boundary DoF. Theyare coupled through the boundary DoF u b . where the displacement vector u consists of all the DoF of A and B , as shownin Fig.1. The vector of external forces f is also applied on the same DoF. In theabove equation, equilibrium forces on the interface are already eliminated bythe use of Lagrange multipliers. The displacement compatibility is applied bythe signed Boolean matrix B such that Bu = u Bb − u Ab = . Since Eq. (2) willbe used several times in this paper, it is represented in the function notation: Y AB (cid:44) f bs ( Y , B ) = Y − YB T ( BYB T ) − BY (3)Eq. (3) can also be used:• to decouple A from AB to obtain admittance of B by setting Y = diag ( Y AB , − Y A ) and calculating Y B = f bs ( Y , B ) [12, 13, 29].• to include (linear) effect of the joint flexibility Y J by Y = diag ( Y A , Y J , Y B ) and computing Y AJB = f bs ( Y , B ) .• to couple and decouple different model descriptions of the same substruc-ture like numerical and experimental expansion purposes, as it will bediscussed (although not derived here) in the next section.Of course, B has to be appropriately defined in each case.8 . SYSTEM EQUIVALENT MODEL MIXING The System Equivalent Model Mixing (SEMM) method is an expansion tech-nique that takes different equivalent FRF models of a structure and couples themso that the dynamics of one are overlaid on the other. It relies on three models,namely, an overlay, a parent and a removed model. The result is a hybrid modelthat tends to mimic the dynamic behaviour of the structure. In the followingsubsection, the different DoF classifications and the models that form the basisof SEMM, are described.
Consider a generic FRF model of a component which consists of internal u i and boundary u b displacements. On the same degrees of freedom, a set ofinput forces can also be defined i.e. forces f i acting on the internal and f b onthe boundary DoF. The corresponding FRF matrix Y consists of all the FRFsbetween output and input DoF. Y = Y ii Y ib Y bi Y bb , u = u i u b , and f = f i f b (4)Since Eq. (4) contains the point and transfer functions among all the inputand output DoF, it is called a collocated DoF set. Such DoF set is essentialfor computing the coupled admittance in Eq. (3). This could easily be obtainedfrom an analytical or numerical model. In order to check the reliability of thenumerical model, an experimental validation is always desired. However, thenumber of measurements in the experiment is limited due to inaccessibility ofsome DoF for either response measurement or excitation or even due to limitednumber of measuring equipment. For bladed-disk interfaces such as dove-tail(Fig. 2 and Fig. 9) and fir-tree type joints, the boundary DoF are clearly neithermeasurable nor excitable and hence Y bb , Y ib and Y bi can not be obtainedexperimentally. Only the internal FRF Y ii can be measured since they areaccessible. Moreover, not all FRFs can be measured accurately. Especially, theaccurate measurement of the drive-point FRFs is very challenging in practice9 igure 2: An academic disk on which the different DoF sets are indicated. There are twotri-axial accelerometers labelled c , and one uni-axial accelerometer labelled v for validation.Among five e and w labelled impacts, w is designated as a validation impact. All of themform a set of internal DoF. The boundary DoF b are displayed only for one disk-slot. Thelack of space in the slot inhibits any direct measurement. [18, 26]. Therefore, the set of internal DoF u i is divided into different categoriesbased on whether the DoF is a response measurement, input force or used forvalidation purpose. u i = (cid:110) u Tc u Tv (cid:111) T and f i = (cid:110) f Te f Tw (cid:111) T (5)The different subscripts are explained in the following and in Fig. 2:• c : set of DoF where responses are measured by triaxial accelerometers (orby other sensor types)• e : set of DoF where excitations are applied by a modal impact hammer• v : set of DoF where responses are measured as u c but reserved for vali-dation.• w : set of DoF where excitations are applied as f e but reserved for valida-tion. 10 .2. Experimental FRF Model Since measurements are not possible on the boundary DoF (interface ofFig. 2), an internal DoF based experimental model of FRFs in Eq. (5) can bedefined as: u exp = Y exp f exp = ⇒ u c u e exp = Y ce Y cw Y ve Y vw exp f e f w exp (6)The different subscripts of u exp and f exp indicate that the FRFs are of the non-collocated type, i.e. there is no drive-point FRF. The matrix in Eq. (6) containsall the measurements including the ones used for validation – subscripts ( (cid:63) ) v and ( (cid:63) ) w . In SEMM, an experimental FRF model of the structure is overlaidon its numerical model (to be discussed in the next subsection) to expand themeasured dynamics on the unmeasured DoF [20]. This experimental modelis called the overlay model Y ov and it can be obtained by setting it equal to Y exp or taking its subset. In our case, the overlay model is always a subset ofthe experimental model since some measurements are used only for validationpurposes and they are not included in the overlay model. u ov = Y ov f ov where Y ov ⊆ Y exp (7)If Y ov is a subset of Y exp , different choices for the measurements to include in Y ov would produce different hybrid models. This will be further discussed inSection 4 and 5. Unlike an experimental model, an FE numerical model allows all DoF tobe available, even those not accessible for experimental tests. The definitionsof different DoF in Eq. (5) along with the boundary DoF u b in Eq. (4) canbe expressed in a square collocated DoF set. Note that in an FE model, themass, stiffness and damping matrices are expressed as dynamic stiffness whichis then inverted to compute receptance (or accelerance) form. The numerical11RF model Y N is then expressed as: Y N (cid:44) Y N gg = Y ii Y ib Y bi Y bb N = Y cc Y ce Y cv Y cw Y cb Y ec Y ee Y ev Y ew Y eb Y vc Y ve Y vv Y vw Y vb Y wc Y we Y wv Y ww Y wb Y bc Y be Y bv Y bw Y bb N (8)The subscript g = { i, b } = { c, e, v, w, b } denotes the global DoF set. Using the numerical and experimental FRF models, the hybrid model [20]can be computed from the following single-line expression: Y S (cid:44) Y S gg (cid:44) semm ( Y N , Y ov ) = Y N gg − Y N gg ( Y N cg ) + (cid:0) Y N ce − Y ov (cid:1) ( Y N ge ) + Y N gg (9)where ( (cid:63) ) + represents the Moore-Penrose pseudo inverse. The equation isderived in [20] from the FBS framework [11]. The hybrid model has the followingproperties:1. Within the same set of DoF in the numerical model Y N , the hybrid model Y S has the same experimental features of Y ov .2. Any measurement errors including noise in the experimental FRFs con-tained in Y ov are transmitted to the FRFs in the hybrid model Y S .3. Another source of error in the expansion process is the expansion errorexpressed as norm of the matrix difference (cid:15) = | Y N ce − Y ov | . This, of course,depends on how close the numerical model Y N is to the experimental one.Measurement errors but also approximation in the models, like not fullyrealistic constraint conditions, affect (cid:15) .
4. CORRELATION ANALYSIS OF HYBRID AND EXPERIMEN-TAL MODELS
In the previous section, a structure’s hybrid or expanded model is obtainedby coupling its overlay (experimental) and numerical models. The hybrid model12an be significantly affected by the discrepancies between the measurements andnumerical model. For instance, the location of sensors on the actual structureand the corresponding DoF in its numerical model may not be exactly coinci-dent, thereby, introducing some variations in the respective FRFs. The sameholds for the impact positions and direction. Moreover, the numerical modeldue to its discretization type, material properties and boundary conditions willalways have some differences from its experimental counterpart. In this paper,a statistical metric, Frequency Response Assurance Criteria (FRAC), is used[30, 31] to quantify the discrepancies between the FRFs of the two models in aconvenient way. In particular, the correlation of FRFs is computed between hy-brid model (instead of the numerical model) and the FRFs from measurementskept only for validation and not included in the hybrid model. A strong corre-lation is indicated by 1 whilst a no correlation is indicated by 0. The FRAC isdefined by: φ ij (cid:44) F RAC (cid:0) Y S ij ( ω ) , Y exp ij ( ω ) (cid:1) = | Y S ij ( ω ) Y exp* ij ( ω ) | Y S ij ( ω ) Y S ∗ ij ( ω ) . Y exp ij ( ω ) Y exp ∗ ij ( ω ) (10)where Y S ij ( ω ) and Y exp ij ( ω ) ∈ C n ω × for each i and j . n ω is number of spectralpoints and ( (cid:63) ) ∗ represents the complex conjugate. We introduce the technique of checking the correlation between the FRFs ina systematic way in the SEMM procedure. This new approach, called correlatedSEMM, is described in detail in this subsection. The aim is to improve the qual-ity of the substructure hybrid models as much as possible before a subsequentcoupled structure model is created (see Section 7). Therefore, it is proposedthat different subsets of experimental model Y exp , called overlay models, arecreated to generate the hybrid models by the SEMM method. The process isexplained in detail below:1. Define an overlay model such that one response channel (a row Y exp re ) from Y exp is excluded in the overlay model to be kept for validation, i.e. Y ov ,r ⊂ Y exp : Y exp re / ∈ Y ov ,r (11)13here r = 1 , , ..., m . Since one channel has been excluded, the size of Y ov ,r is ( m − × n . The channel Y exp re is now considered as the movingvalidation channel (MVC) and is graphically shown in the upper left partof Fig. 3a.2. Perform expansion by the SEMM method with Y ov ,r as per Eq. (9) to get Y S ,r , i.e. Y S ,r = semm ( Y N , Y ov ,r ) .3. The corresponding r th expanded channel Y S,rve is correlated with Y exp re (see Fig. 3a) by computing FRAC, as per Eq. (10). FRAC is computedover a fixed frequency band in Eq. (10) for two given FRFs. However,the explicit dependence of the FRFs on frequency ω is not shown for thesake of clarity in the above expressions. The FRAC, thus computed forthe pairs of FRFs in Y S,rve and Y exp re are denoted by φ re and used forcalculating φ avg r as follows: φ avg r = 1 n n (cid:88) j =1 φ rj (12)The parameter in Eq. (12) can be considered an indication of an overallcorrelation level of the response channel r .4. The process is repeated for all the remaining channels up to r = m , i.e.each time one channel r in Y exp is excluded from the r th overlay model.5. The low correlated response channels are identified based on the averagecorrelation in Eq. (12).In a similar way, by successively excluding the columns from the overlay model,the respective correlations can be computed for the input channels. Fig. 3billustrates the procedure by excluding q th column from Y exp . The two schemesof computing correlations are listed side by side in Table 1 for further clarity. The correlation analysis can be interpreted physically from the observabilityand controllability perspective. At the step r , when r th channel is excludedfrom the construction of the hybrid model, there are m − response channels14 a) An example in which the overlay model Y ov ,r is short of the r th response channel (row)in experimental model Y exp . In the respective r th hybrid model Y S ,r , the correlationis calculated between Y S ,rve , Y exp re . For simplicity, the DoF set in Y S ,r consists of only g = { c, e, v } .(b) An example in which the overlay model Y ov ,q is short of the q th input channel (col-umn) in experimental model Y exp . In the respective q th hybrid model Y S ,q , the corre-lation is calculated between Y S ,qcw , Y exp cq . For simplicity, the DoF set in Y S ,r consists ofonly g = { c, e, w } . Figure 3: Illustration of the different models used to find correlated or uncorrelated responsechannels and input channels. The DoF set in the hybrid models Y S are shown only for theinternal DoF. Note the difference in the DoF structure in the top and bottom figure. Thecolour of Y exp is the same in both figures to signify that Y ov ,r and Y ov ,q are its subsets.The same colour appearance of the overlay model in the respective hybrid model shows themimicking behaviour of those DoF. ction Response ChannelCorrelation Input Channel Correlation Define overlay models Y ov ,r ⊂ Y exp : Y exp re / ∈ Y ov ,r Y ov ,q ⊂ Y exp : Y exp cq / ∈ Y ov ,q for r = 1 , , . . . , m for q = 1 , , . . . , nsize ( Y ov ,r ) = ( m − × n size ( Y ov ,q ) = m × ( n − Generate hybrid models Y S ,r = semm ( Y N , Y ov ,r ) Y S ,q = semm ( Y N , Y ov ,q ) Compute correlations φ re = F RAC ( Y S ,rve , Y exp re ) φ cq = F RAC ( Y S ,qcw , Y exp cq ) φ avgr = n (cid:80) nj =1 φ rj φ avgq = m (cid:80) mi =1 φ iq Plot and decide r = { , m } vs φ avgr q = { , n } vs φ avgq Table 1: Summarized action steps to generate overlay and hybrid models in order to findcorrelations among all DoF or channels (both response and input). The dimension of theoverlay matrix is different for response or input channels correlations. Note that size ( Y exp ) = m × n . which try to observe the r th channel through the SEMM expansion. FRAC asa correlation provides a measure or degree of observability. By repeating theprocess until r = m , each response channel has undergone an observability check(performance review) by the rest of the channels in terms of FRAC. Computingthe overall performance, for example, by averaged FRAC values in Eq. (12),the best or least observed channels can be identified. A similar interpretationholds also for the input channels from the controllability perspective. In short,excluding the input channel q , how well n − input channels could control the q th input channel.If one or more channels have low correlation at the end of the process, thefollowing could be the probable reasons:1. the DoF associated with the channel(s) did not have significant dynamiccontribution in the selected frequency band. Thus, they could not be fullyobserved or controlled by the other measured channels.2. the location of sensors or impacts and the corresponding DoF in the nu-merical model were not coincident.16 a) Blade sensor and impact positions.(b) Disk sensor and impact positions. Figure 4: Experimental setup for impact testing for (a) the blade and (b) the disk. Thenumbering sequence on the disk continues from the blade. Sensor 5 and 10 are not visiblefor their mounting on the other side. Free boundary condition is realized by supporting thecomponents by flexible wires.
As a result, the least correlated measurement channel(s) can be filtered out from Y exp . In this case, if needed, the corresponding DoF may be kept in thenumerical model and thus expanded over in the hybrid model. In another case,the DoF can be filtered out from both Y N and Y exp , as it did not contributemuch in the global dynamics of the component.Different criteria can be assumed to select the channels to keep after theFRAC analysis. A minimum threshold value criterion would imply that allthe channels with a correlation level below the threshold will be disregarded.17his could lead to missing the sufficient number of independent measurementchannels necessary for an onward analysis. For this reason, it was chosen todefine a minimum number z required for the joint identification. In this way,only the z channels with the highest correlation levels are kept, all the otherchannels are discarded.
5. APPLICATION OF CORRELATED SEMM TO A BLADE ANDDISK
In order to find the correlations of hybrid models by the above-mentionedmethod, two structural components are considered: i) blade and ii) disk (alreadyshown in Fig. 2). The effect of the improved hybrid models by the correlationanalysis will be studied in Section 7 when the blade and disk are assembled.Here, the details of the experimental setup and their respective numerical modelsin the stand-alone configuration are presented.Two impact testing campaigns were carried out on the blade and disk, asshown in Fig. 4a and Fig. 4b along with sensor and impact positions. The sensorsare triaxial accelerometers and the excitations are made with a modal impacthammer. The details about the sensors and channels are given in Table 2. Boththe disk and blade were tested while they were hanged with flexible wires. Thischoice was adopted for two factors:1. The boundary DoF at the root-joint of the blade and disk have to be leftunconstrained in the substructuring context. FRFs have to be measured(or expanded by SEMM) on these boundary DoF and hence they have toremain unconstrained. The shroud part on the other end is also left freesimply because only one blade is to be coupled to the disk (c.f. Section 7).2. The disk was not constrained at its centre to avoid errors due to a non-optimal constraint model [27]. The impact of the constraint model as apossible source of errors on the final results was shown in a previous workby these authors [28]. 18 ype Description Blade A Disk B ExperimentalSetup Number of accelerometers 5 5Number of available response channels 15 15Number of useful response channels ( m )
14 14Labels for response channels {1-8, 10-15} {16-23, 25-30}Number of input channels ( n )
18 19Labels for input channels {1–18} {19–37}NumericalModelling Young’s Modulus (GPa) 190 178Density (kg/m ) 7800 7800Fixed interface modes 200 200CorrelationAnalysis Poorly correlated response channels ch Table 2: Details of experimental and numerical parameters of the blade and disk. The channelswith the lowest average FRAC levels are also listed after the correlation analysis. Note amissing channel in the response channel labels. This channel had unusual high noise floor andwas not included in the measurements since the beginning of the test campaign.
From these measurements, the accelerance FRFs are collected in Y exp ,A and Y exp ,B for the blade and disk, respectively.Numerical modelling consisted in creating corresponding FE models from thesolid geometries. The discretization was done with Solid elements in ANSYSand with the material properties listed in Table 2. The FE models were thenreduced by Hurty-Craig-Bampton [32] method by retaining only the essentialnodes and fixed interface modal amplitudes. The retained DoF correspondedto the nodes of sensors, impacts and the interface. From the reduced systems,accelerance FRFs are computed and stored in Y N ,A and Y N ,B for the blade anddisk, respectively.From the above experimental and numerical models, different overlay andhybrid models will be generated in the following sections for the correlation19 Response Channel Index r a v g , A r (a) Input Channel Index q a v g , A q (b) Figure 5: Average FRAC of the blade A against response channels and (b) input channels.The channels which are removed from measurements based on the lowest FRAC are indicatedwith arrows. analysis. In the first case, the blade’s overlay models are generated from the mea-sured FRFs Y exp ,A of size × (Table 2). In order to compute correlationsfor the response channels, overlay models are generated where each modelcorresponds to the exclusion of one response channel r . The expansion in eachhybrid model is checked by computing the FRAC. Since FRAC is computed be-tween two FRFs over a range of frequency, this range is set as − Hz. Theaverage FRAC is then plotted against all the response channels in Fig. 5a.It can be seen that overall correlation levels are higher than . except for r = 4 . This means that this channel or DoF could not be well-observed bythe other channels, when this was removed from Y ov ,A . Following the samemethod for the input channels by skipping the q th column in Y exp ,A to generate q th overlay and hybrid models, the average FRAC values are plotted as bars in20
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Frequency (Hz) -2 A cc e l e r an c e ( m / s / N ) [10] Sensor 4 -X / [18] Excitation -X ReferenceStandard SEMMCorrelated SEMM
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Frequency (Hz) -180-90090180 P ha s e ( deg ) Figure 6: FRFs of the blade with standard SEMM and correlation based SEMM when thelowest correlated channels marked with arrows in Fig. 5 are removed from the measurements.The reference FRF is Y exp ,A , with description shown at the top of the FRF. Fig. 5b versus the input channels. The correlations are again good for manyinput channels with the exception of q = 4 and q = 17 marked with two arrows.If the channels or DoF with low correlations are retained in the measurementsand the standard SEMM method is applied, some of the resulting FRFs mayhave some inconsistencies. In the standard SEMM, all the measured FRFs(except the validation) are included in the overlay model such that Y ov ,A = Y exp ,Ace , as per Eq. (7). In Fig. 6, for the sake of validation, an FRF by standardSEMM (thin solid line) is compared with a corresponding experimental FRF(called Reference) and for this reason not included in the construction of Y S ,A .At the first glance, the standard SEMM method Y S ,A expands the dynam-ics really well in most of the frequency band. This is because experimentaland numerical FRF models are quite close. However, comparing the standardSEMM and reference curves, some inconsistencies are visible especially around − Hz. In the same figure, it is plotted as dash-dotted line the FRF (la-belled: ’Correlated SEMM’) obtained from SEMM after filtering out the lowestcorrelated channels r = 4 , q = 4 and q = 17 . These channels have been filtered21 Response Channel Index r a v g , B r (a)
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Input Channel Index q a v g , B q (b) Figure 7: Average FRAC of the disk B against response channels and (b) input channels.The channels which are removed from measurements based on the lowest FRAC are indicatedwith arrows. according to the criterion adopted in Section 4.2. It can be noticed that thisFRF obtained by new correlated hybrid model ˆ Y S ,A agrees extremely well withthe reference FRF both in amplitude and phase (Fig. 6). From an a-posterioricheck on the measurements of each channel, it came out that the channels dis-carded by FRAC were not good for different reasons. In detail:• channel r = 4 had very low response levels in the shown frequency band-width and was, therefore, prone to be easily polluted with noise• input channels q = 4 and q = 17 did not produce good FRF due to humanerrors in the impact direction or location. After the blade’s analysis, the disk’s correlations are calculated. The disk’sFRAC bar graphs similar to that of the blade are shown in Fig. 7 both for the22
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Frequency (Hz) -2 -1 A cc e l e r an c e ( m / s / N ) [22] Sensor 8 +Z / [19] Excitation -X ReferenceStandard SEMMCorrelated SEMM
500 1000 1500 2000 2500 3000
Frequency (Hz) -180-90090180 P ha s e ( deg ) Figure 8: FRFs of the disk with standard SEMM and correlation based SEMM when thelowest correlated channels marked with arrows in Fig. 7 are removed from the measurements.The reference FRF is Y exp ,A , with description shown at the top of the FRF. response channels and input channels. One response channel with label r = 27 and two input channels with label q = 30 and q = 31 are found to be the leastcorrelated. By filtering these channels from the experimental (and the overlay)model and regenerating the hybrid model of the disk, the filtering effect is seenin Fig. 8. From the figure, it is evident that the standard SEMM using allthe measurements produce an FRF that does not overlap with the referenceFRF around the first antiresonance from 1200 to 1800 Hz. On the contrary, aremarkably improved FRF is obtained with the correlated SEMM in both theamplitude and phase.
6. INTERFACE DEFINITION
The expanded hybrid models in the above sections were related only to theinternal DoF in order to be compared and to exclude the low correlated channels.By SEMM, the measured dynamics can be expanded to the inaccessible DoFwhich are boundary DoF u b and f b . The expanded boundary DoF are shownfor the blade in Fig. 9 with blue arrows. In an FE model, these DoF are23sually the translational DoF of some selected nodes. Even in the experiments,translations are also easier to measure compared to rotations. However, theinterface dynamics are not accurately described if only translational DoF areconsidered. There have been numerous studies to approximate rotations [6, 13,33, 34] as well as measure them directly [35–37]. Based on Equivalent MultiPoint Connection (EMPC) [34], a Virtual Point (VP) type interface [38, 39]can be defined to represent the interface motion by virtual translations androtations q A and virtual forces and moments m A (Fig. 9). In this work, a VPtype interface is considered because the blade-disk joint has three sided-interfacei.e. left side, right side and bolted pins pushing from the bottom side, as seenin Fig. 9. This means that some uncertainty is associated with the contact atthe interface [40] and this will be minimized in a least-squared fashion by thevirtual point transformation.Consider a hybrid model of the blade ¯ Y S ,A in which the measured dynamicshave been expanded on the boundary DoF u b , as per the DoF structure ofEq. (8) and the compact form of Eq. (4). The displacements u A relate to thevirtual point displacements q A by: u Ai u Ab = I 00 R Au (cid:124) (cid:123)(cid:122) (cid:125) R A u Ai q A (13)where R Au contains the positions and orientations of the DoF in u Ab withrespect to the virtual point. The vector on the right hand side can then beobtained as u Ai q A = (cid:0) ( R A ) T R A (cid:1) − ( R A ) T (cid:124) (cid:123)(cid:122) (cid:125) T A u Ai u Ab (14)Since the boundary DoF set in Eq. (13) is collocated (also see Fig. 9), thesame transformation T A holds for the virtual forces and moments m A , i.e. f A = ( T A ) T m A . Thus, the hybrid FRF matrix Y S ,A for the virtual pointinterface is computed by: Y S ,A = T A ¯ Y S ,A ( T A ) T (15)24 igure 9: The interface details on the blade. The measured translations in Y exp ,Ace are ex-panded to translational boundary DoF u Ab , f Ab . They are then transformed to the indicatedtwo virtual displacements and rotations q A and the corresponding virtual forces and moments m A . For the sake of clarity, virtual translations and rotations are indicated only for one VP. Similarly, the boundary DoF in the slots of the disk hybrid models can betransformed to the VP interface. In the discussion to follow, it is assumed thatthe hybrid models are described by the VP interface.
7. COUPLED STRUCTURE MODELS
In Section 2, it was discussed how Eq. (3) can be used to decouple thesubstructures. If the joint can be considered a substructure with its accelerance Y J , it can be decoupled from its coupled or assembled model Y AJB . In detail,Eq. (3) is to be used as Y J = f bs ( Y , B ) with Y = diag ( Y AJB , − Y A , − Y B ) .The superscript J emphasizes the explicit presence of the joint or boundary DoFin the coupled model which would not be possible in a directly measured modelof the assembly. This is a dual decoupling method [12]. The methods in [1, 6]related to joint identification use primal formulation. In this work, the SEMMmethod is exploited to generate such a hybrid coupled model Y S ,AJB whichexplicitly contains boundary dynamics. The method was originally proposed in[41] on a numerical test-case and then applied by the authors of the present paper25o the real case in [28, 42]. The method, which this time uses models generatedwith the correlated SEMM, is explained here and applied to the blade-diskassembly in Section 8. As well as the individual components, also the assembly needs a hybridmodel. Since component hybrid models were generated in the preceding sec-tions, they shall be used as a basis for the coupled system’s numerical model. Inthe joint identification context, the boundary dynamics should also be present.Therefore, at the beginning, a guessed joint accelerance Y Jk is introduced. So,the coupled numerical accelerance Y N ,AJB can be written as: Y N ,AJBk = f bs ( Y , B ) with Y = diag ( Y S ,A , Y Jk , Y S ,B ) B = u Ai q A q J,A q J,B u Bi q B − I I 0 0 00 0 0 I 0 − I (16)where q J,A and q J,B represent the DoF of the joint which couple to substructure A and B , respectively. The index k = 0 , , , . . . indicates the iteration number,as the joint is not known a priori. The joint needs to be updated at everyiteration and as a results, Y N ,AJBk would also be updated. The iterative natureof the method has been explained in [28]. The measured FRFs on the coupled system Y exp ,AB contain joint dynamicsimplicitly. From this experimental model, an overlay model Y ov ,AB is takenas a subset and imposed on the numerical model Y N ,AJBk . Some channels in Y exp ,AB are not used for the overlay model and are kept for validation. Notethat this overlay model remains the same at each iteration.26 igure 10: Illustration of the SEMM method applied to an assembled system in order todecouple the joint. The example joint shown is a spring and damper, but the decouplingmethod is not limited to it. The quantities in coloured blocks (2,4 and 6) are updated at eachiteration k . The sign (+) indicates coupling of substructures. With the above coupled numerical and overlay models, the hybrid model isgenerated from Eq. (9) such that Y S ,AJBk = semm ( Y N ,AJBk , Y ov ,AB ) (17)The hybrid model is also updated iteratively. Since Y N ,AJBk has a guessed linearjoint (or no joint) at k = 1 which may be far from the actual one, so there mayexist a high expansion error | Y N ,AJBk − Y ov ,AB | . With the above different coupled models, it is now possible to decouple thejoint dynamics by Eq. (18) Y Jk +1 = f bs ( Y , B ) with Y = diag ( Y S ,AJBk , − Y S ,A , − Y S ,B ) (18) Y Jk +1 is then substituted in Eq. (16) to update the numerical model Y N ,AJBk +1 ,to subsequently generate an updated hybrid model Y S ,AJBk +1 in Eq. (17) and,27hereafter, to decouple the joint Y Jk +2 . The iterative process is graphicallyillustrated in Fig. 10. At each iteration, the updated joint improves because thisis the only part that is updating the numerical model Y N ,AJB . The process isrepeated until the expansion error (cid:15) = | Y N ,AJBk − Y ov ,AB | between the numericaland the experimental model is reduced below a given threshold. It should benoted that:1. The initial guess of Y Jk at k = 1 can be a blank joint i.e. the substructurescan be left uncoupled [28, 41].2. Y Jk +1 obtained by LM-FBS equation has all the rows and columns cor-responding to the DoF of both the coupled and uncoupled models. It isnecessary to retain only the independent entries [23].3. The method converges faster by using weighted pseudo-inverses with higherweights assigned to the boundary DoF in Eq. (17). This aspect has beendeeply discussed in [28].
8. APPLICATION OF SEMM TO THE BLADE-DISK ASSEMBLY
In this section, the SEMM method is applied to the assembly of the bladeand disk of Section 5. Fig. 11 shows the blade-disk assembly in free constraintconditions. The impact test campaign was carried out on the coupled systemwith the same sensor and impact positions as in the case of blade and disk alone.Since there were limited number of sensors and data acquisition channels, theyhad to be mounted once on the blade and then on the disk. To obtain the joint bydecoupling, the blade and disk models (substructures) have to be decoupled fromthe coupled model. The sensor masses are considered part of the substructuremodels. Therefore, by placing dummy masses on one component while thesensors are on the other, the additional mass effect is cancelled. The set of FRFmeasurements on the assembly is denoted by Y exp ,AB whose size is × (seeTable 2). From this, the assembly overlay model Y ov ,AB is taken as a subsetof size × , while one response channel and one input channel are left as28 a) (b) Figure 11: Experimental setup of the blade coupled to the disk. Due to limited number ofsensors and channels in the data acquisition system, the campaign was completed first by(a) mounting the sensors on the blade and the dummy masses on the disk and then by (b)mounting sensors on the disk and the dummy masses on the blade. Each dummy mass valueis equivalent to the sensor’s nominal mass. The sensor and impact positions were preservedexactly as Fig. 4a and 4b, respectively. Positions of sensor 1 and sensor 6 are indicated forreference. the validation channels. In detail, the experimental FRF used for validation is Y exp ,ABvw = Y exp ,AB , . The joint is decoupled using the methodology described in Section 7 withthe substructure hybrid models computed in Section 5. The joint acceleranceis obtained when the expansion error (cid:15) defined in Section 7.4 does not changeanymore [41]. The joint is decoupled (or identified) by finding its acceleranceusing the two methods:• Standard SEMM: Y J = f bs ( Y , B ) with Y = diag ( Y S ,AJB , − Y S ,A , − Y S ,B ) • Correlated SEMM: ˆ Y J = f bs ( ˆ Y , B ) with ˆ Y = diag ( ˆ Y S ,AJB , − ˆ Y S ,A , − ˆ Y S ,B ) Since accelerations are measured on all the structures with their FRFs expressedas accelerance, the identified joint is also represented as accelerance in Fig. 12.As explained in Section 6, the joint is represented by two virtual points (12 DoFon each component) for the whole blade-disk interface i.e. a × system. In29he past, the identified joints were limited to small systems under simplifiedmotion and assumptions. To note a few examples for rigid joints, a × spring-damper system was identified in [6], and a × spring-mass-damper system in[43]. For our realistic case, the complexity of the motion of the interface requiresa larger system description with the size of × – in the form of acceleranceFRFs [40, 44].Amongst those FRFs, only four accelerance (amplitude) plots are discussedas representative of the identified joint for both the standard and correlatedSEMM in Fig. 12. The first two plots are for the translational DoF (Fig. 12a andFig. 12b), and the other two are for the rotational DoF (Fig. 12c and Fig. 12d).It is seen in the figures that despite some noisy behaviour, which is typical after adecoupling procedure [1–3, 6], the joint seems to follow a trend. This behaviouris representative of a system with high stiffness, low damping and low massi.e. a stiffness dominant line on a logarithmic scale [26]. The fluctuations aredue to the measurement and modelling errors, which propagate in the hybridmodels. It can be observed that the accelerances obtained by the correlatedSEMM exhibit slightly less fluctuations than the ones obtained by standardSEMM. Correlated SEMM in fact removes the channels that introduce morevariability in the identification. However, the fluctuations still remain becausethe measurement noise cannot be completely removed.It can be noticed in the translational accelerance (Fig. 12b) and in the rota-tional one (Fig. 12c) that around 100 Hz, the accelerances from standard SEMMhave a kind of a hump. This could be interpreted as an internal resonance ofthe joint. However, the hump disappears in the corresponding accelerance iden-tified by the correlated SEMM, confirming that it was due to some spurious,non-physical effects which were eliminated by the correlated SEMM.In the reconstruction of the response of the assembled system (blade plusdisk with the joint in between), the accelerance curves of Fig. 12 were kept asthey are (with their fluctuations) without any fitting. This is due to the factthat the joint system is large and has high fluctuations (Fig. 12a). It wouldrequire curve-fitting to every FRF (a total of × curves including real and30
200 400 600 800 1000 1200
Frequency (Hz) -1 A cc e l e r an c e ( m / s / N ) (a) Y J , X/X
Frequency (Hz) -1 A cc e l e r an c e ( m / s / N ) (b) Y J , Z/Z
Frequency (Hz) A cc e l e r an c e ( deg / s / N m ) Standard SEMMCorrelated SEMM (c) Y J , RX/RX
Frequency (Hz) A cc e l e r an c e ( deg / s / N m ) (d) Y J , RZ/RZ
Figure 12: The decoupled joint accelerance on two translational DoF (a) and (b) and tworotational DoF (c) and (d) for both standard SEMM and correlated SEMM. imaginary parts) in the joint accelerance with fluctuations while maintaininggood matrix conditioning and symmetry. It is in itself a challenging task and isconsidered beyond the scope of this paper.
In order to check the reliability of the two identified joints, the identifiedjoints are recoupled to their respective substructure models i.e. Y J is recoupledto Y S ,A , Y S ,B for the standard SEMM and ˆ Y J is recoupled to ˆ Y S ,A , ˆ Y S ,B forthe correlated SEMM. Note that the joint accelerance of Fig. 12 is coupledas such without any fitting to the respective substructures due to the above-cited difficulties. A similar approach can be found in other works in literature[45–47]. In fact, the authors in [45] during identification of their known massby substructure decoupling method noted that it was quite straightforward todetect errors in case of the known mass; however, it would not be the case ifthe system to be identified is unknown. They went on to assert that the only31
200 400 600 800 1000 1200
Frequency (Hz) -2 -1 A cc e l e r an c e ( m / s / N ) [2] Sensor 1 +Y / [27] Excitation -X ReferenceStandard SEMMCorrelated SEMM
Frequency (Hz) -180-90090180 P ha s e ( deg ) Figure 13: FRF on the coupled blade and assembly. The reference measured FRF is Y exp ,AB , (see Fig. 4). The FRFs with standard SEMM and correlated SEMM are obtained by recouplingthe identified joint with their respective component hybrid models. The FRFs have beensmoothed for clarity. check that can be performed is to couple the predicted FRFs of the unknownsubsystem with those of the known subsystem (to form the mathematicallycoupled system) and to compare it with the reference measured FRF on theassembly. This type of validation is called on-board validation [46, 47] i.e. thereference FRFs (not included in the identification) in the same measurementcampaign should be predicted by recoupling of the identified joint with therespective substructure models.The recoupled accelerance FRFs obtained with both correlated and standardSEMM are shown in Fig. 13 together with the FRF assumed as reference. Thereference FRF was kept for validation since it was not included in the SEMMexpansion. The values of the amplitude of the FRF peaks obtained for bothstandard and correlated SEMM are also listed in Table 3 for the sake of com-parison with the reference ones. By looking just at the FRF of Fig. 13, it can32 ode Frequency Experiment Standard SEMM Correlated SEMM (Hz) Amplitude Amplitude % Difference Amplitude % Difference1 191.3 59.3 48.6 -18.0% 60.3 1.7%2 530.0 44.7 91.4 104.6% 48.5 8.5%3 691.1 33.6 25.5 -24.2% 41.5 23.3%4 1111.1 298.3 99.0 -66.8% 89.7 -69.9%5 1138.9 14.4 8.3 -42.7% 12.2 -15.5% Table 3: Peak value comparison for the FRFs reconstructed by the standard SEMM andcorrelated SEMM methods. All amplitudes are in m/s /N. be noticed that both the recoupled FRFs (standard and correlated) are almostoverlapped to the reference curve in the regions close to the peak resonances.It can also be observed that, using the correlated SEMM, leads to a generalimprovement in the regions of small amplitudes (ranges 220 - 740 Hz, 300 – 500Hz, 800 - 900 Hz) where the FRF estimated by correlated SEMM is closer (thanthe one obtained by standard SEMM) to the reference FRF.By looking in detail at the values of the peaks’ amplitude in Table 3, thereader may observe that the values predicted by the correlated SEMM at reso-nance are in general better than the standard SEMM (except for mode 4 wherethe difference is negligible). In particular, the amplitude values predicted bycorrelated SEMM are much better for the first two peaks i.e. below 600 Hz.This improvement is given by the correlated SEMM, instead of standard, inthe model of the disk, and this was particularly effective for the disk in 0-600Hz range (see Fig. 8). From Table 3, it can be seen that, using the correlatedSEMM, it is still advantageous in high frequency modes (3 to 5) even if thedifference with standard SEMM is not always as evident as for modes 1 and2. In these high frequency regions, other factors such as interface definitionand singular value filtering, as shown in [40] can also play a key role in betterpredicting the coupled system’s dynamics.33 . CONCLUSIONS In this paper, a procedure to identify a dove-tail joint between a blade anddisk is presented. The identification is based on a substructure decouplingtechnique. The novelty of the paper is that a new correlation based methodis proposed in order to better select the experimental results to include in thejoint identification procedure.Considering that in a typical dove-tail joint, it is not possible to measuredirectly on the interfaces, the dynamics at the interface are then predicted bymeasurements at accessible points far from the joint. The already existing tech-nique of System Equivalent Model Mixing (SEMM) is used here to generate hy-brid numerical-experimental models of the single components (blade and disk)and of the blade-disk assembly. The hybrid models of each component (bladeor disk) allow to predict the dynamics at the interface, that are coupled by thejoint, by measuring in points far from the interface. However, the accuracy ofthese hybrid models is affected by two main error sources: i) expansion error(systematic or bias error) and ii) measurement errors (random as well as bias).The expansion error depends on the difference between the numerical and over-lay (experimental) models. A big source of this error comes from the boundaryconditions, for example, bad modelling of the component constraint.Therefore, in order to reduce at minimum the expansion error, it was herechosen to model both the structural components (blade and disk) in free condi-tions. As a result, a very good agreement between experimental and expandedFRFs from SEMM was obtained for each of the two components (the blade anddisk).The second source of error – the measurement errors – introduce noise and lo-cal inconsistencies in the FRFs. To reduce the effect of these errors, a procedureto check the goodness and usefulness of the measurements is here proposed.The procedure employs the FRAC (Frequency Response Assurance Criteria)correlation based method.When this procedure is introduced in SEMM it allows to systematically34dentify the poorly correlated measurement channels (that is the measurementspolluting the construction of the hybrid model). This new improved approach(called correlated SEMM) here developed has the ability of filtering out the badmeasurements. The correlated SEMM produces multiple hybrid models andeach time computes correlations with a measured FRF taken as reference. Themeasurements with inconsistencies can be identified due to their low correlationslevels and can be filtered out. This improved the overall quality of hybrid modelsof the two substructures (blade and disk).The two SEMM methods (both standard and correlated) were here appliedto the blade-disk assembly to decouple (identify) the joint. The result of theidentification is the accelerance for each joint DoF. To validate this joint iden-tification, the obtained joint accelerances were coupled back to the two hybridmodels of the substructures to obtain the FRF of the assembled structure (bladeplus disk). The obtained recoupled FRFs were then compared with the experi-mental FRFs measured on the assembled structure.This validation procedure was implemented both by using the standardSEMM and the correlated SEMM approaches. The recoupled FRF obtainedby the correlated SEMM proved to be much more overlapped to the measuredreference FRF than the FRF obtained by standard SEMM. In particular, thecorrelated SEMM showed to capture better the FRF plot in the non-resonanceranges.The procedure implemented in the correlated SEMM of filtering out the lowcorrelated measurements proved then to be effective also in the low response(near anti-resonance) regions which are more influenced by the noise level andthe measurement errors.
ACKNOWLEDGMENTS
This work is a part of the project EXPERTISE that received funding fromthe European Union’s H2020 research and innovation program under the MarieSkłodowska-Curie grant agreement No 721865. The authors also acknowledge35he efforts of Meysam Kazeminasab for conducting the experimental campaigns.
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