Modal correlation for axisymmetric models with repeated roots – wheel rim case study F. Lembregts1 1 Siemens Industry Software, Simulation & Test Solutions Interleuvenlaan 68, B-3001, Leuven, Belgium e-mail:
[email protected]
Abstract This paper describes a number of issues that make correlation of experimental and simulated mode shapes a challenge when applied to industrial examples. It also describes how to cope with them to get reliable and representative correlation measures between the test and FE models. The five issues discussed are the geometric correlation of the models with different orientations, the availability of measured nodes on only a sub-part of the model, the different orientation of local axis systems in the FE and test models, missing measurement directions in the test modes, and –in more detail- the presence of repeated roots as for axisymmetric structures. All topics are applied to a realistic use case of a wheel rim. The paper shows the need for a dedicated approach to each of the challenges to reach a valid MAC-based correlation model in real-life circumstances.
1
Introduction
Modal correlation is a well-established discipline nowadays, with many reliable quantitative tools to assess how well two modal models describe the dynamic behavior of a system in an equivalent way. Today’s software tools make the correlation process quite straightforward in many cases. The challenges appear however when the two models are different in terms of topology or completeness, as is mostly the case when comparing finite element (FE) modal models with experimental modal analysis (EMA) test models. The following sections describe the most common challenges in this domain, and how they can be solved or worked around. The emphasis will be on one of the more difficult problems: correlating modes with repeated (or very closely spaced) modes, as they will occur for axisymmetric structures. All of the discussed topics will be illustrated on a realistic model for a wheel rim.
2
The test model
The tested model is shown in Figure 1. It is an aluminum 5-spokes wheel rim with a diameter of 43cm, about 20cm high. The rim is suspended in free-free conditions, and equipped with 4 uni-axial and 1 tri-axial accelerometer to capture the responses to a roving hammer excitation, normal to the surface. The acceleration over force frequency response functions (FRFs) represent 7 columns of the FRF matrix with 70 rows. Each measurement node only has one degree-of-freedom: normal to the surface.
Figure 1: Wheel rim test setup for roving hammer modal test. A simple geometric model was built with 70 measurement nodes, as shown in Figure 2.
Figure 2: Test wire frame geometry model: 70 measurement nodes.
The displacement over force FRFs were measured with commercial software (LMS Test.Lab), in a range to cover the flexible modes up to 200Hz. As shown in Fig. 3, the FRFs exhibit low damping peaks.
Figure 3: Typical FRF, showing low damping. A modal model with the first 18 flexible modes was extracted from the FRFs using the PolyMAX Plus algorithm [3, 4]. For this modal parameter extraction, only 3 of the 7 FRF columns were used. Because of the double roots, several manual attempts were needed to separate the modes in the 2 nd and 4th pair.
3
The Finite Element model
The simulated FE model contains 20k solid TET10 and 25k shell QUAD4 elements, with in total 36k nodes. It is solved for the first 20 vibration modes with NX Nastran. Except for mode 11, all of these modes occur in pairs, as expected for such an axisymmetric structure. Figure 4 shows a few of them, while the table below lists the frequencies.
Figure 4: Six first flexible FE-modes, in 3 pairs (less than 0.1Hz apart within each pair).
Mode 1 2 3 4 5 6 7 8 9 10
Frequency (Hz) 0.00 0.00 0.00 0.00 0.00 0.00 369.20 369.21 806.83 806.85
Mode 11 12 13 14 15 16 17 18 19 20 21
Frequency (Hz) 1074.65 1301.09 1301.23 1332.45 1332.54 1445.61 1445.64 1576.51 1576.68 1930.19 1931.96
Table 1: NX-Nastran FE-modes up to 200Hz, mostly occurring in pairs.
4
Correlating the geometry models
In many cases, the models from test and FE can be aligned from the start, either by defining the 2 nd model in a consistent way with the 1st one, whichever is created first. But in case the models are created by different teams, or by external providers or suppliers, they may have to be aligned first, before any other correlation step can start. An easy way for this is to define 3 or more pairs of nodes that are supposed to coincide, where each pair has one node on each model. Fig. 2 shows yellow markers at 3 nodes on the test model, which are easy to locate on some border. Pairing them up with their respective counterpart node on the FE model makes it possible to compute the needed translation and rotation values to transform one model to make it coincide with the other one. It is also possible to derive a scaling factor (normally very close to 1.0) between the two models for an even closer match. The applied transformation needs to be included in any other of the pre-processing steps that follow in the correlation process, such as the ones discussed in the next sections.
5
Correlating the test and FE nodes, and mapping the modes
The most popular quantitative measure for correlation of vibration modes is the Modal Assurance Criterion [1], expressing the statistical correlation coefficient for a linear relationship between 2 vectors, in this case 2 mode shapes:
MAC(f1 ,f2 ) =
(f1* · f2 ) 2 (f1* · f1 ) × (f2* · f2 )
(1)
Where f1 and f2 are the two modal vectors, · denotes the vector product operator, and the superscript denotes the complex conjugate.
*
The FE model usually has many more nodes than the test model. To correlate test with FE modes, the modal vectors need to be made equally long, i.e. they need to contain the same number of degrees of freedom (DoFs). The problem of reduction of the FE modal vectors is described elsewhere, e.g. in [2]. The simplest method is to find for each test node the FE node that is closest to it, and compose the reduced FE modal vector by the corresponding FE-modal values. This will however introduce errors because the test and FE nodes do not completely coincide. When the FE element size is small, these errors are negligible. More sophisticated methods minimize these errors by finding for each test node the 2 (or 3 or more) closest FE-nodes and using a weighted average of their FE-modal values to assign it to the corresponding reduced FE-vector element. The weighting factors can be inversely proportional to the respective distance between the FE-node and the target test node. This gives generally very good results for spatially smooth deformation patterns. For complex modes, the averaging of phase values is however less reliable. Therefore, it is advised to first convert the complex test modes to real normal modes, prior to this averaging process. We reduced for our test case the FE modes up to 2000Hz as described above, using the weighted average of close test nodes. The MAC formula applied on the test and these reduced FE modes shows a really poor correlation, see Fig. 5. In this matrix view, the test modes are in columns, on the horizontal axis, and the FE modes (not including the 6 rigid body modes) are in rows, on the vertical axis. The color intensity of each cell is a measure for the MAC value between the corresponding test and FE mode (darker red being a higher MAC value).
Figure 5: Poor first correlation attempt, with only 1 test mode with MAC > 0.6 It is clear that some more pre-processing is needed (discussed in the next sections) to get reliable MAC values for this case.
6
Missing test DoFs
The spatial resolution for a modal test is very often constrained by the number of available transducers, or by the number of measurement channels in the acquisition hardware. On the other hand, to have a good
visual representation or animation of the test modes, the test model must contain enough nodes. A compromise is therefore often needed: single-axis transducers are used in nodes where the motion is known to be dominant in one direction. A typical use case are plates or panels that mainly deform along the surface normal direction. For such uni-directional measurements, the test mode shape will contain only one value for that node, not 3 or 6 as for their FE-counterpart. An approach to complete the test vector by using 0-values for the missing test DoFs will yield wrong (i.e. systematically too low) MAC results. The correct solution is indeed to drop the values for the missing test DoFs also from the FE-mode shapes. But there is an important caveat here. In many cases, the values in the test mode shapes are not exactly zero, but just very small. This can be caused by rounding errors in some processing, like in the normalization of complex modes, or a geometrical transformation step (rotation of the model to align it with the FE model), or conversion to other units, or conversion between axis systems, or after export or import to files to exchange the data between different software packages. These rounding errors make it in reality difficult to distinguish missing values from the very small values where a mode shape has a genuine nodal point with almost no movement. Another point of attention is the fact that a certain part of the structure with single DoF nodes may get relatively under-weighted compared to parts with tri-axial measurements. Depending on the target use of the modal model (e.g. stress or fatigue prediction versus NVH) those panel-like parts may exactly be the more important ones, like to predict the radiated sound.
7
Correlating the DoFs with local axes
Transducers in modal tests are very often positioned onto the structure such that the transducer (or 1 of the tri-axial combination) is aligned with the local surface normal. On a curved surface, this means that the measurement direction will be different for most test nodes, and will certainly not always coincide with the global X, Y, or Z direction. To have valid MAC values, each element in the FE-vector must be the equivalent of the corresponding test value, i.e. in the ‘same’ node, but also in the same direction (DoF). The latter one requires additional processing to be applied. By itself, this is not a difficult axis transformation to perform, when the source (test) and destination (FE) local axis systems are known. The challenge comes from combining this with the previous requirement to keep only the measured DoFs in the modal vectors for the MAC formula. Fig. 6 below shows how a ‘measurement value’ with only non-zero Y and Z components becomes a triplet of 3 non-zero values after transformation to the nodal FE axis system. From these 3 new values alone, it is impossible to conclude that only 2 of the 3 DoFs were measured.
Figure 6: Axis transformations make it difficult to detect missing measured DoFs
A good Test-FE correlation toolbox needs an algorithm to automatically detect such missing test DoFs, also after model rotations, nodal axis transformations, re-scaling or mode shape normalization. It then needs to eliminate the contribution of the missing test DoFs from the FE modal vectors. Fig. 7 shows the improved MAC values after this missing DoFs elimination step.
Figure 7: Improved correlation (compared to Fig 5.) after eliminating missing test DoFs, with now 4 test modes with MAC > 0.8
8
Correlating mode clusters of repeated roots
Fig. 7 shows for example relatively high MAC values of 0.85 and 0.81 between test mode 1 and FE-mode 8 and test mode 2 and FE-mode 7 resp. But FE-mode 4 has MAC values of 0.6 and 0.7 resp. with the test modes 3 and 4, while FE-mode 3 does not have any MAC value above 0.35 at all. The reason is the fact that the structure has repeated roots, because of its symmetry. For a root of multiplicity n, the n modal vectors describe a modal subspace of dimension n, which perfectly can describe the dynamic behavior of the structure for those modes. But any combination of n linearly independent vectors in that subspace would also describe that behavior perfectly. In terms of modal correlation, that means we do not need to correlate the modes one by one, but rather need to correlate the modal subspaces: is the n-dimensional test modal subspace the same as the n-dimensional FE modal subspace? That is easier said than done, because we have ‘sampled’ that modal subspace in two different ways with the modal test and the FE-simulation. In mathematical terms: the ‘basis’ or ‘coordinate axes’ we have chosen to describe that modal subspace is different for the test and FE model. In some cases, the modal test even extracted 1 or 3 ‘axes’ for the 2-dimensional modal subspaces. This problem has been discussed before. Chen proposes two techniques [5] to tackle the correlation of repeated modes: one by geometrically rotating the symmetric model and the mode shapes to align the test and FE modal vectors by removing the spatial phase shifts between them, i.e. based on the position of the node-and-belly deformations. Applying this approach on this wheel rim use case is relatively easy, as we
know the axis of symmetry, and the angle between 2 spokes being 360 / 5 = 72 degrees. Applying this additional “n times 72 degrees” rotation to the original geometry transformation based on the 3 node pair matching approach yields indeed different MAC values as shown in Fig. 8 below.
Figure 8: MAC graphs when rotating the models over 1, 2, 3, or 4 times 72 degrees, with resp. 4, 7, 4, and 3 MAC values above 0.8 > 0.8 The same paper describes also a second approach using a Fourier decomposition to identify the diametral order of each (quasi) axisymmetric mode. This would allow us to avoid the above demonstrated trial-anderror approach to rotate the model, or to apply it in cases where the rotation angle is not that obviously known. De Paula et al. on the other hand take a mathematical approach [6] with a transformation of one set of the modes to express them in the basis of the other set. This latter approach is much more appealing, because it does not require or suppose any a priori knowledge about the geometry of the structure, nor about how exactly the mode shapes are cyclic or symmetric. A practical algorithm can be designed as follows: 1. Identify the modal subspaces in both modal models (test and FE). 2. For each pair of subspaces: express the FE modes in the basis of the test modes, or vice versa. 3. Replace the original FE modes of the subspace with new FE-modes that are aligned (as much as possible) with the test modes. Such a linear re-combination of FE modes will indeed still describe an identical FE modal subspace. 4. Compute the MAC value between each test mode and its aligned FE-mode in that modal subspace. Identifying the clusters of modal subspaces is a fairly simple task. In most cases, this can be done by just looking at modes with (almost) identical natural frequencies. If needed, a side-by-side animation of the modes (as shown in Fig. 3 above) will clearly indicate which modes should go into which cluster on both test and FE sides. When modes of one cluster are interlaced with other modes, this step may be a bit more complicated though. Once the clusters, i.e. the modal subspaces, are defined, the correlation can proceed as follows for each cluster. We try to express the test modes as a linear combination of FE modes within the same cluster:
[T ] = [ F ] ´ [W ] + [ R ]
(2)
Where [T] is the matrix of test modes in the modal subspace, [F] are the FE modes in that same subspace, and [W] are the yet unknown weighting factors. The latter ones are the ‘coordinates’ of the test modes in the basis of the FE modal subspace. If the modal subspaces are completely coinciding, then there will be a zero residual term [R]. If the residual [R] is too big (e.g. based on the matrix norm, expressed relatively to the norm of the test mode matrix [T]) then the two modal subspaces are not matching, and the correlation process stops for this cluster. The weighting factors can be computed from (2), using e.g. a singular value decomposition (SVD) or by solving the equations in a least squares sense:
[W ] = [ F ]+ ´ [T ]
(3)
The new equivalent FE modes of the modal subspace can then be computed as the linear combination of the original FE modes that approximates best (in a least squares sense) the test modes:
[ Feq ] = [ F ] ´ [W ]
(4)
The original FE modes in the cluster are then replaced with these new equivalent FE modes, and the standard MAC-formula (1) is used. Note that the test modes are usually expressed as complex values. In such case, also the [W] matrix is complex, and also the equivalent modes will be complex.
9
Final correlation results for the axisymmetric wheel rim
Finally, all of the discussed pre-processing steps on the modal models of the wheel rim are combined and applied, using the LMS Virtual.Lab Correlation software: 1. The geometric correlation using 3 node pairs 2. Correlating each test node with one or a few FE nodes, and computing the reduced FE model and modes 3. Eliminating the missing test DoFs from the reduced FE modes, taking into account the possibly different local axis systems 4. Clustering the modes in subspaces, and replacing the FE-modes with their best equivalent recombinations The resulting MAC matrix is shown in Fig. 9. Table 2 shows the corresponding correlation table.
Figure 9: Final correlation result, with most shown MAC values > 0.8 (including 4 above 0.92)
Test modes FE modes Frequency Frequency Nr. Nr. (Hz) (Hz) 1 370.4 7 369.2 2 373.3 8 369.2 3 754.4 9 806.8 4 755.3 10 806.8 5 1088.7 11 1074.6 12 1301.1 13 1301.2 6 1235.4 14 1332.5 7 1236.3 15 1332.5 8 1430.7 9 1442.0 17 1445.6 10 1778.4 18 1576.6 11 1785.0 19 1576.6 12 1824.7 20 1931.1 13 1826.4 21 1931.1
0.98 0.99 0.92 0.84 0.86
Frequency Difference (Hz) -1 -4 52 52 -14
0.71 0.87
97 96
0.71 0.70 0.76 0.94 0.82
4 -202 -208 106 105
MAC
Table 2: Correlated mode pair table. The following observations can be made: 1. All of the thirteen test modes (in columns) but mode 8 now have a relatively well correlated FE counterpart, with a MAC value of 0.7 or higher. Overall, the test-FE correlation is good, but not great. 2. FE modes 12, 13, and 16 were not extracted from the modal test. With 2 pairs of double roots between 1300 and 1335Hz, the modal parameter estimator has difficulties is separating the mode shapes, even from a multi-reference FRF data set. Additional attempts to extract modes in that frequency range resulted in quite low quality modes, exhibiting complex deformation, i.e. with large phase delays between the nodal displacements. 3. Despite the clustering and FE-mode recombination, there is still a big cross-correlation between the test modes 3 and 4 versus the FE modes 9 and 10, with cross-MAC values of 0.33 and 0.37. The reason is that the test modes 3 and 4 are quite similar: they have a MAC value of 0.34 between them. This means that the modal parameter estimator was not able to completely separate the two modes. As a result, the test modal subspace for that pair is not completely spanned by the 2 test samples, and the equation (2) has a relatively big residual error of about 35%. Clustering cannot fix that problem. 4. The frequency difference between the correlated pairs Test (10,11) and FE (18,19) is more than 200Hz. This probably means that some part of the FE model is not stiff enough. As the difference in frequency is not consistently positive or negative, the needed correction will most probably not be a global one (like a wrong Young’s modulus or so) but will be much more local and concentrated, e.g. a poor modelling of the connection between the spokes and the outer ring. This would need further investigation, or even a model updating process on the FE-model to match the test results more closely. The above correlation table would be the pre-requisite for such updating step. Visual assessment of the correlated mode pairs is the final validation step. In order to make this effective, the created images on both the test and FE modes must include of course all the applied pre-processing on the modes:
· · · · ·
Geometric translation, rotation, and scaling Reduction of the FE-model, and mapping the nodes Skipping the non-measured DoFs, also from the FE modes, and also taking into account the local measurement directions Re-combination of any clustered FE-modes to align them with the test modes Side-by-side images, with synchronized animation frames to show the maximum deformation at the same moment for both modes
If one of those aspects is missing in the images, the visual comparison becomes more difficult and less conclusive. Depending on the purpose of the correlation exercise, the missed test modes may or may not be important. · ·
If one wants to further use the test modal model to simulate the dynamic response, then it is essential to get also the missing FE-modes into the test model. Without those, the test modal model is indeed not complete. However, if the objective is to validate the FE model (and possibly update it to better align the natural frequencies) then the missing FE-mode #16 would not be too big an issue because at least one of the symmetric modes in that cluster of repeated roots is included in the test model. This is sufficient to confirm that the FE model adequately describes the structural dynamics, and therefore can be used with a high level of accuracy for more advanced simulation studies. The missing FE mode pair #12 and #13 may be a bigger issue in this case.
10 Conclusions Correlating experimental modal test modes with FE-modes can create quite some challenges in practice because of compromises one has to make for a modal test. This paper describes the most important ones, and illustrates on a realistic example how they can be tackled in practice, and how the MAC values can be drastically improved with proper pre-processing of the modal vectors: geometric model correlation, test and FE nodes correlation, transformation of the modal data, compensating for missing test DoFs (even when local test axes are involved). The last section shows how to assess the quality and completeness of the test model when repeated roots are present. The test samples of each modal subspace can be correlated with their FE counterparts by reformulating the test and FE modes using a common vector basis of each modal subspace. The MAC values can only then be used as a reliable correlation measure.
Acknowledgement The author thanks the division PMA of the department of Mechanical Engineering of the K.U. Leuven to provide the test and simulation models that were used in this study to illustrate the discussed topics.
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[4] B. Peeters, M. El-kafafy, P. Guillaume, The new PolyMAX Plus method: confident modal parameter estimation even in very noisy cases, ISMA2012 Conference Proceedings, K.U. Leuven (2012) [5] G. Chen, D. Fotsch, N. Imamovic, D.J. Ewins, Correlation Methods for Axisymmetric Structures, IMAC XVIII Conference Proceedings, Society for Experimental Mechanics (2000), pp. 1007-1012. [6] B. Franca de Paula, G. Rejdych, T. Chancelier, G. Vermot Des Roches, E. Balmes, On the Influence of Geometry Updating on modal correlation of brake components, XVIII symposium Vibrations, Shocks & Noise (VISHNO) (2012), pp.1-13. .
