Sem.org IMAC XV 15th Int 15-5-3 the MAC Revisited Updated

7/27/2019 Sem.org IMAC XV 15th Int 15-5-3 the MAC Revisited Updated

http://slidepdf.com/reader/full/semorg-imac-xv-15th-int-15-5-3-the-mac-revisited-updated 1/8



of the best straight line through the data points of two cor-related modes is described es the model scale factor MSFand is defined in [4] and the result is a real number even if

4 is complex..

2.2 The Orthogonality MethodsAnother method of comparison based on the property ofmodal orthogonality is the cross orthogonality check:

and the mixed orthogonality check

(2 )

(3)

techniques which are referred to in 15.61

For perfect correlation, the leading diagonal elements of theorthogonality matrices must all be equal to 1 while the off-diagonal terms remain at 0.

2.3 The Modal Assurance Criterion (MAC)The Modal Assurance Criterion was introduced at IMAC 1

as ” A Correlation Coefficient For Modal Vector Analysis ,,by R.J. Allemang end D. L. Brown [7l The modal assur-ance criterion was build in this paper in e analogy of thefrequency response function and the coherence functionwith respect to noise contamination end error evaluation.The MAC is now used as one of the primary tools for corre-lating two sets of mode shapes in modern modal analysissofhvere. The modal assurance criterion is a correlation ofeach single mode shape pair end is defined in [;1 for testmode j end enalysed mode, i, as:

A MAC value close to 1 suggests that the two modes arewell correlated and a value close to 0 indicates un-corre-

lated modes. State-of-the-art is to compare only trensla-

tionel DOFs. By combining straight forward data sets withrotational end translational DOFs the MAC gives invalidresults due to the in-coherent units and normalization.

2.4 The Co-ordinate Modal Assurance Gi-

terion (COMAC)The Co-ordinate Modal Assurence Criterion (COMAC)seeks to identify those WFs of the structure which demon-strate low degrees of correlation. The COMAC correlates totwo sets of mode shapes, either from a test or finite elementmodel, and then identifies those DOFs at which the modepairs consistently disagree.

Before using the COMAC, other correlation tools have to beused to identify the “pre-correlated” mode pairs.After constructing a set of L correlated mode pairs, a corre-

lation value is calculated for each co-ordinate over allcorrelated mode pairs es follows [S]:

This correlation method weights all degrees of freedomequally regardless of whether or not they have large orsmall mode shape coefficients.

A slightly different formulation has been invented by SDRC(Structural Dynamic Research Corporation) which weightsthe relative difference between degrees of freedom 191.

&4),r -fwxA,I

COMC(i) = I - r=’2L

(6 )

A difference at a degree of freedom which has relativelylarge amplitudes will have more a significant contributionto the final COMAC value then a difference at a degree offreedom with relatively small coefficients, Hence, discrepancies that might occur between mode shapes at points onthe structure that exhibit very little motion will not heavilyaffect the calculation.

3.0 A New Iterative Correlation MethodThe starting point for the new correlation method is thedefinition of the Modal Assurance Criterion (MAC). TheMAC value as described in [4] gives the correlation factor

for each mode pair and can also written as:

148



By an iterative cancelling of a single DOF(k) in each calcula-tion, the DOF which produce a poor correlation in eachmode pair can be identified in each iteration I.

under the condition of

U4C,(A,X)-~C,.,(A,X)=maw (9)

The optimised result of the MAC, (A,X) gives a differentcancelled DOF for each mode pair. Therefore, the outputafter I iterations is a better MAC value but the list of can-celled DOFs differs from mode pair to mode pair. Moredetailed information is needed to investigate a systematic

connection, why and where are the DOFs which leads tothe best MAC value over all modes?

The accumulated improvement by leaving out a single WF

over all mode pairs and iterations gives a value which indi-cates the increase of the MAC value by cancelling one spe-cific DOF, d. The higher this value, the higher is the i&u-

ence of the specific DOF. To focus only the MAC valuesthat give reliably high MAC values (along the diagonal),the result is weighted by the starting MAC value. Theproduct of this is called the High-Sensitivity-Point-Value(HPV) and is defined with following equations.

The first formula is the iteration without the specific DOF,

d .

The idea is to identify the specific DOFor set of WFs withthe highest improvement by comparison of two models(e.g. analytical and experimental). The Eq. 10 gives a resultof the MAC value without the specific DOF, d. If, bycancelling an arbitrary COF, the improvement is no betterthan the value without the DOF, d, a discrimination value(DV) can be set. Where in MAG(A,X) the specific DOF, d, isexcluded and in MACu(A,X), it is included.

If there is no better improvement by cancelling the DOF, d,the discrimination factor DV will be equal to 0.

Further, the high sensitive point value (HPV) is then de-fined as:

w h e r e

b

total number of iterationsnumber of experimental modes

a number of analytical modes

The advantages of the new method are:

(i) normalisation of the mode shapes is not necessary,(ii) there is no mode pairing required,(iii) n o wr o ng i nt e gr a ti o n o f fa l se m od e pa i ri n g.

As a result this teclmique is more robust than the COMACand the enhanced COMAC. In section 6, results are shownin a case study, where the efficiency of the new technique isdemonstrated.

4.0 Case Study: A fully equipped Automo-

tive Alternator for a Dynamic Test and

Simulation in High Frequency RangeAn experimental modal analysis and f inite element analysiswere made of a fully equipped automotive alternator underoperating boundary conditions. The dynamic test andanalysis were undertaken on the non rotating alternator.

For simulation software for the automotive alternator, the

pre- and post-processor from I-DEAS [9] was chosen, andfor a FE solver, the software PERMAS was employed [lo].After validation of each single part the major task is tocouple the individual parts to simulate the complete ma-chine for the purpose of predicting its sound.

4.1 The Finite Element Model of an Auto-motive AlternatorThe alternator FE model consists of six substructures andindividual parts which are outlined in Fig. 1. The substruc-tures consists of A-housing, &housing, stator, coolrng plateand mounting arm and brackets are fixed together byscrews in the experimental structure. In the FE model allscrews are idealised with volume elements. The nodes onthe circumference are only coupled in the three transla-

149



tional degrees of freedom. The ball bearings are idealised

with spring elements in the radial direction and, at the A-housing. additionally in axial direction.

Also the poly-v-belt is idealised with simple spring ele-ments. Finally the whole F&model has the size of morethan 17,ooO Elements, 27,ooO nodes and 81,MM DOFs

4.2 Experimental Modal Analysis of an

Automotive AlternatorOne aim of an experimental modal analysis is to validatethe finite element model over a frequency range up to 2KM

Hz. Hence, there is a fine mesh distribution required toavoid aliasing. At the alternator the mesh has 137 nodeseach with three degrees of freedom. The distributions of the

nodes are as follows:(i.) nodes on the rotor

(ii.) nodes on the A-housing(iii.) nodes on the &housing(iv.) nodes on the cooling plate(v.) nodes on the mounting arm(vi.) nodes on the stator

Dynamic tests were carried out on the alternator using atest rig specially developed for non-linear structures underoperating condition [ll].

The extraction of the mode shapes from the measured FRFswas made using the software package I-DEAS (IntegratedDesign Engineering Analysis Software) from SDRC(Structural Dynamics Research Corporation) on an HPWorkstation 7000 and for more complicated close modes adetailed analysis with ICATS Software (Imperial CollegeAnalysis and Test Sofhuare) was performed.

Previous studies 1121 have shown that the natural frequen-cies of the alternator are very closely spaced and the damping was greater their spacing. To identify each single mode,the MDOF method is the adequate method,

5.0 Application of the New Correlation

Method on a Large Modal Model

A primary difficulty in noise and vibration simulation ofcomplex structures is that the frequency range of interestencompasses “dozens” of system modes in the noise-rele-

“ant frequency range.

In addition, the effect of the connection between each as

sembled part and variations in construction makes it quiteimpossible to develop a finite element model that is accu-rate up to Zoo0 Hz. Furthermore, it is impossible to researchthe structure without a set of independent test methods.

The first task to validate an assembled structure is to vali-date each single part of the structure under free-free condi-tions. Despite the good correlation results of single designparts (diagonal MAC values >O.B), the correlation of theassembled structure gives a poor result with MAC valuesbelow than 0.3 (Fig. 2) .

Fig. 2: The MAC-matrix between analytical and experimen-tal modes of the alternator

Also after several improvements in the FE model the MACvalues did not increase significantly. In spite of the smallMAC value (below 0.3), a correlation could be observed byeye particularly for the modes at the frequencies which aremost interesting from an acoustic point of view. Here, thecorrelation by eye is much clearer than the MAC valuesuggest.

For example, the gapping effect (Fig. 3) could be correlatedby eye, but the original MAC value of 0.35 was quite low.

150



r

Fig. 5: The ranked 15 WFs (indicated hy arrows) with tht

perimental m o d e . highest HPV value on the structure

This led to the idea that a subset of DOFs might pollute the The un-correlated nodes in the Zdirection are caused by an

MAC value. To improve the model, the developed method idealisation error at the brackets on the bottom, where thewas employed. After 15 iterations and extraction of 15 structure was fixed. So the whole body could move moreLXX’s, the MAC-matrix showed a better quantitative result flexibly only in the Z-direction, with the highest influence

(Fig. 4). on the amplitudes located on the top.

6.0 A Comparison between the COMACand the New Iteration Procedure in Situ

The large FE and experimental models of the alternatorwere used to test the correlation methods in practice on areal model. After this test, the differences are discussed.

The enhanced COMAC need the same normalisation of the

mode shapes for a calculation. Usually the experimentaland the analytical result came from different packages andit can happen that the normalisation of the modes is differ-ent (e.g. I-DEAS Test and Permas). Therefore, it has to becarefully checked, if there is a unique normalistion of themode shapes, otherwise the result is wrong!

’Fig. 4: MAC-matrix after 15 iterations between analyticaland experimental modes of the alternator

The highest MAC value in the MAC in the Previous matrix(Fig. 3) was only 0.58. This means there is a poor correle-

tion between the two models. After cancelling less than 4 96

of the DOFs the diagonal MAC values increased to 0.8. The

further calculation by ranking the sensitive DOFs showsthat the most sensitive DOFs were mostly in the Z-directionwhich is shown in Fig. 5.

After normalization of the mode shapes, the relevant vet-

tars have to be paired. For models with a prominent diago-nal line in the MAC-matrix this will he an easv task, hut ifthere is no such prominent line the user must choose themode pairing. In this example, the following mode orders

were paired (Fig. 6).

151



EMA42c

Mode-Pair 3

FEM 1 MAC0 1 MAC203 I 0.54 I 0 .855 0.7? I 0.89e I ,._)_I I ^ “r

Fig. 6: Mode Pairing for the COMAC calculation: FE versu:

experimental modal analysis and the MAC values after I

and 20 iterations

ig. 8: COMAC/enhanced COMAC of mode pair 2

The previous figures show clearly the influence betweenmode pairing and the COMAC results. In Fig. 8 completely

different regions of the alternator are indicated thm in theFig. 7.

To give a more detailed look at the correlation values, thedifferent methods are plotted separately against each de-gree of freedom (Figs. 9 - 11).

The different results inlluenced by the COMAC, enhancedCOMAC and different Mode pairing are shown in Figs. 7 -9 .

I r-

ig. 7: COMAC/enhanced COMAC of mode pair 1

:ig. 9: Normal COMAC

ig. 10: Enhanced COMAC

152



It cm be seen in Figs. 9 - 11 that the values obtained fromthe iteration procedure are less sensitive to the mode orderthan are both the COMAC calculations. With the new

ranking technique it was possible to identify the least un-correlatedCOFs in one run without mode pairing.

7.0 Discussion of the Results

The comparison of the correlation tools shows the powerand the weakness of each. The most robust method is theHPV criterion. The main advantage to this method is thatthere is no mode pairing or normalisation necessary and,therefore, it is easier to use.The enhanced COMAC comes very close to the result of theHPV-value when the correct mode pairs are used.

The normal COMAC does not weight the amplitudes of the

DOFs. It can happen that in fixed structures DOFs withsmall amplitudes in the fixation have a great effect on thebody motion. This effect can also be used as an indicator. Agood example for this is also seen on the alternator. Onlythe COMAC indicates that in one case the bracket region isa prominent part (Fig. 8). Small effects in the bracket regioncan indeed influence the vibration in the upper region ofthe alternator.

The differences between and how to use the methods are

outlined in Fig. 13 for the COMAC and enhanced COMACand the more straight forward calculation route for theiteration procedure is outlined in Fig. 14.

Izig. 12 The COMAC /enhanced COMAC

ig. 13: The developed ranking method,

153



Documents

Sem.org IMAC XV 15th Int 15-5-3 the MAC Revisited Updated