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Developing a Correlation Criterion (SpaceMAC) for Repeated and Pseudo-repeated Modes

One of the most important factors in the validation of a finite element model is whether the modal vectors obtained from the finite element solution match sufficiently with the modal analysis results of the part or a test prototype. Modal assurance criterion(MAC) is usually a very effective way to check this condition. However, in case of repeated modes, MAC can give misleading results. A low MAC value could indicate that the finite element model suffers from some deficiencies or in case of closely spaced modes it could mean that these modes are repeated modes. Hence, there is a need for a method that could indicate how well the finite element method based estimates for the repeated modes correlate with the modal analysis modes. This thesis work is an attempt to develop a correlation criterion between a set of repeated roots from a finite element method solution and a experimental modal analysis solution. Building on a low dimensional modal vector example a vector subspace based approach was identified to help properly define the solution to a characteristic equation with repeated roots. This analogy was extended to higher dimension modal vector cases and vector subspace or hyper planes were identified as a way to model a repeated mode case. Similar to MAC, consistency of the solution was considered an ideal way to establish correlation. But in this case the consistency of the solution subspace was found to be more important than that of normalized modal vectors. The smallest principal angle between the two solution subspace was identified as a way of measuring the consistency. The criterion, called spaceMAC, was developed as a function of this angle such that the range of the criterion is 0 to 1, similar to MAC. SpaceMAC was defined as 1 − sin(θ) where θ is the principal angle between the two solution subspaces. The performance of this criterion was tested with experimental data sets from two different symmetric structures known to exhibit the repeated mode condition.


Introduction 1.1 Motivation
As the modern design process moves further towards computer based modeling and finite element computations, there is a need for higher and higher certainty about the accuracy of these models. In case of dynamic modeling of systems, comparisons of modal parameter estimates from the model and a prototype is a great way to ascertain the validity of the model. The modal frequencies and modal vectors are usually used for this purpose.
Modal Assurance Criterion(MAC) is the most common way of measuring the consistency of estimates of modal vectors when the lengths of the modal vectors are same. It is also a very convenient method because it can be used in a 'black-box' manner where the user can input the modal vectors and get a direct indicator of correlation between various modal vectors. It is often used to check the validity of an analytic model by using the cross-MAC values between analytic and experimental solution. If the diagonal terms in the cross-MAC are close to 1, it means that the model models the dynamic behavior very well. However, there is a drawback in this approach. MAC can give misleading results in case of repeated and psuedo-repeated modes. This can jeopardize the amount of certainty that can be had in a model.
A usual workaround that can be used in these cases is to observe a wireframe animation or to observe the mode shape for the modes in question, that is the modes with low cross MAC values, and check if any shape can be rotated in space to get the shape for some other mode. 1

Existing methods
Apart from using the wireframe animation or mode shape to judge whether a set of modes are a repeated set or not, some methods exist to detect and correlate repeated roots. For example, it has been shown that by taking the singular value decomposition(SVD) of unity scale factor(USF), which is defined similarly to modal scale factor(MSF), the number of repeated roots can be found [1] . A method developed for correlating repeated modes is shown in [2] , where the test mode is expressed as a linear combination of all possible combinations of analytic modes. The cross-MAC values for all these combinations and the test mode shape are calculated. From these cross-MAC values it is found which combination is a repeated mode combination.

New method
A new way of correlating repeated modes is proposed in this thesis. Since repeated modes do not exist in unique modal vectors but form a unique set of modal vectors, the proposed correlation factor evaluates the consistency of the sets obtained from a FEM solution and experimental analysis. Vector subspace theory is used to identify repeated modal vector sets as vector subspace. The concept of principal or canonical angles is then implemented to compare the two vector subspaces.

Summary of contents
The remainder of this thesis consists of four more chapters. The necessary theoretical background can be found in Chapter 2. Some basic information about the repeated root condition is followed by the theoretical background for vector subspaces. This is followed by a discussion on the geometry mapping required to downsample the analytical modal vector to the size of the experimental modal vector so that MAC and spaceMAC can be applied. Then a brief discussion on the existing methods to identify and correlate repeated root conditions is done.
2 Chapter 3 describes the process undertaken to extract and condition the modal vectors for both the analytical and experimental methods so that the resulting modal vectors are accurate and describe the same system. A description of the two case studies and the experimental data and analytical models for the two case studies is also present in this chapter. Then the results from an attempt to implement existing methods and spaceMAC method are discussed in Chapter 4. Chapter 5 conclude this research work with the conclusions from the results and with a discussion on the scope of future work. 3

Chapter 2
Theoretical background 2.1 Repeated root condition A repeated modal frequency condition is said to exist when the modal frequencies of two or more modes occur at exactly same frequency. This condition exhibits equal complex modal frequency, that is both the modal frequency and the modal damping are equal.
This condition can also be referred to as a repeated root condition or a repeated pole condition. Repeated modes are usually found in structures with some symmetry but it is not a necessary for the object to be symmetric to have repeated modes. It is important to correctly estimate repeated modes to get an accurate model of the system. Therefore, it is important to have a method to estimate the certainty in the estimates of repeated In case of repeated roots, the Equation 2.2 still holds true. It can be expressed for a column q of the frequency response function as, where λ s is the repeated modal frequency. It can be seen from the above equation that if there is only one column of frequency response function, the residue estimated at λ s would be the sum of all terms belonging to each modal frequencies. To be able to detect repeated modes experimentally, it is important to have at a minimum the number of independent columns in the FRF matrix equal to the number of repeated modes in the largest repeated mode set that is to be identified. It is usually better to have more references than this number.
A modal vector that belongs to a repeated mode set is orthogonal to other modal vectors but not necessarily to the modal vectors that are from the same repeated mode set.
Although, the condition of orthogonality can be enforced when estimating the analytical modes, it is not necessary. Experimentally determined repeated modal vectors are very often not orthogonal to each other. In the case of a repeated root of multiplicity 2, two independent modes will be required to describe the motion of the plate at that frequency but these two modes are not unique. An infinite number of pairs exist such that their linear combination can describe that motion. Therefore, the exact vector of each mode is not unique. Since the complex valued frequencies may not be perfectly equal in the case of repeated roots in experimental analysis, it can be problematic when regular modes occur at closely spaced frequencies. This condition gives the appearance of being a repeated mode condition. It is called a psuedo-repeated root. Usually checking the wireframe animation can clarify this and confirm which modes belong to a repeated set but with a repeated mode correlation method this can be done in an automatic and more convenient way.

Vector subspace theory
A vector space is a non empty set of vectors, on which the operations of vector addition and scalar multiplication are defined, such that they follow ten basic axioms stated below [3] . Given that u,v,w are vectors in set V and a and b are scalars, the axioms can be stated as When a set satisfies the second condition it is said to be closed in vector addition and the when a set satisfies the third condition it is said to be closed in scalar multiplication.

Modal vectors as vector spaces and subspaces
The set of all solutions to an eigenvalue equation is called an eigenspace. As can be checked with the axioms stated in the previous section, an eigenspace is also a vector space. It is also closed in vector addition and scalar multiplication. So it can be said that the eigenspace is basically a subspace of C n , where n is the length of the eigenvectors.
Because the characteristic equation for a multi degree of freedom system is also a eigenvalue equation, the solution to that equation belongs to an eigenspace. In case of repeated modes, the modes are not unique and any linear combination of the modes of a repeated set can be said to be a mode in the repeated set. Also any mode multiplied by a scalar still describes the same mode. These two considerations mean that a repeated mode set is closed in addition and in scalar multiplication. Since each repeated root set is a set contained completely in the set containing all possible solutions to the characteristic equation, each set of repeated root would also be a vector subspace of the eigenspace corresponding to the characteristic equation. This is because the repeated roots satisfy all three condition laid down in Section 2.2 for a vector space to be called a vector subspace of another vector space.

Principal angles between subspaces
The angle between vector subspaces for a vector dimensionality greater than 3 is a difficult concept to visualize. However, when a R 3 space and the eigenvectors belonging to this space are considered, an analogy can be extended to general concept of angles between subspaces. In a general eigenvector solution belonging to R 3 space, the normalized eigenvectors would be three unit vectors pointing in mutually perpendicular directions.
All vectors in any one of these directions form a subspace of the complete solution set.
In case of an eigenspace belonging to a R 3 space with a repeated eigenvalue of multiplicity two, the subspace described by the repeated root is a plane in the 3 dimensional The angle between the planes is essentially the principal angle between subspaces for this case. The principal angles between subspaces of complex-valued larger dimension modal vectors are just an extension to this idea. This angle can be viewed as the angle between the normals to the planes in a C 3 space. It is also convenient that the eignevectors belonging to the non-repeated set are the normal vectors to the plane or subspace of repeated modes. Hence a dot product of the other normalized eigenvectors should give the cosine of the angle. The principal angles for higher dimensional subspace are defined in a similar way. The minimal angle between subspaces X and Y is given by where x and y are unit vectors in the X and Y subspaces respectively. The computation of the minimal angle using Equation 2.4 is possible but sometimes not very accurate. The angles can be calculated as shown by Wedin [4] . Various methods have been developed over time to improve the calculation. One such method is developed by Bjorck et al. [5] . This method is used in the subspace() function in MATLAB that has been used in this thesis.
The subspace() function first finds the orthonormal bases for both sets of vectors. The orthonormal bases are swapped to maintain an order of rank. It is shown in [5] that small principal angles can be calculated more accurately through the use of sine projections which can be calculated through singular value decomposition(SVD) of a matrix C given by, where, A and B are the orthonormal bases of the two sets of vectors. The 2-norm of C gives the sine of the principal angle.

Analytical model and experimental wireframe
To be able to correlate modal vector results from experimental and analytical solutions, it is necessary to map the geometries that are used to arrive at those respective solutions.
In case of a finite element method modal solution, the modal vector contains modal information from all nodes in the meshed geometry. This modal vector is usually of a very high dimension. In this research work, since the two case studies belong to relatively simple planar objects, it was easy to identify the common plane between the ANSYS model and X-Modal(experimental modal analysis program) wireframe. This is usually a difficult task for more complicated geometries. If the model and wireframe coordinate systems are scaled differently and at an angle other than 90 • , 180 • , 270 • or 0 • it becomes a further complicated task. Horn and Hilden show a way of finding a transformation matrix that can reorient and rescale one of the geometries so that nodes and wireframe points can be matched [6] . Once that is done, the points on the mesh corresponding to the points on the wireframe can be found out by calculating the euclidean distance of the wireframe point from each mesh node. The node with the minimum distance can then be selected as the matching node to the wireframe point. The analytical nodes(ANSYS) can then be truncated to the nodes that best match the experimental nodes.

Existing correlation methods
Cross-MAC is the most common way of correlating modal vectors obtained from two different methods. It can be calculated as [7] , where φ and ψ are the modal vectors obtained from two different methods. The modal vectors in Equation 2.6 are complex-valued in general. Other methods like the shape difference indicator(SDI) are methods that focus on measuring the difference between two modes. The SDI is calculated as [13] , But these criteria do not succeed in correlating repeated modes.   Apart from using the wireframe animation or mode shape to judge whether a set of modes are a repeated set or not, some methods exist to detect and correlate repeated roots. For example, it has been shown [1] that by taking the singular value decomposition(SVD) of unity scale factor(USF), which is defined aŝ where{ψ dr } and{ψ cr } are the modal vectors scaled to unity length, the number of repeated roots can be found [1] . To get to the exact repeated root, this process has to be applied repeatedly. First, it should be applied to a larger set of modes, which would show how many independent modes are present in the set. If this is more than the number of frequencies it can be inferred that repeated mode are present. Then smaller groups of modes can be selected and put through the same process and a group of modes that are a repeated set can be arrived at. Though, this does not help define a way of correlating repeated modes.
A method developed for correlating repeated modes is shown in [2] , where a combined mode is first set up as where {β} is the linear coeffiecients matrix, [ψ k ] is the matrix made up of analytical modal vectors for a set of modes and [ψ Ak ] is the combined modal vector for k th mode.
The combined mode is substituted for k th test mode and β values are calculated. The combined mode is reconstructed from these β values and MAC is calculated between the combined mode and test mode. This is repeated for all combinations for the combined mode. The method has some drawbacks that are discussed in Section 4.1.

13
Chapter 3 Extracting and conditioning modal vectors Before a modal correlation criterion can be implemented, the modal vectors from an analytical and experimental modal analysis have to be extracted. These extracted modes have to be conditioned so that they can be made of same dimension. The processes for extracting modal vectors for experimental and FEM analyses are different. The modes may also be normalized so that the different modal vector sets are similar. Some other changes might also be required depending on the kind of modal vectors obtained.

Description of case studies
To develop and test a method of correlation for modal vectors from repeated modes, two circular plates were chosen. The data for these plates has been obtained from previous experiments at UC-SDRL. Circular plates are very symmetric objects and yield many    values that were significant for this comparison were the ones corresponding to modes that were known to be non repeated. Figure 3.5 shows the cross-MAC values between modal vectors from analysis and experiment. The modal frequencies were compared numerically.  The format of the files obtained can be seen in Table 3   seven contain information about deformation for that particular modeshape.   Plate Dataset, a typical shaker testing procedure was followed in order to obtain extra experimental data to compare different experimental cases for repeated mode correlation.

Experimental modal vectors
The MPE settings were varied to get a set of results that could be compared for different experimental cases for both the plates.     The combinations are [12 and 13], [12,13,14], [12,10] and [12,10,14]. As can be seen from some of the MAC values in the table, there are two problems with this approach The high correlation between the combination [12,13,14] and the 12 th experimental mode suggests that the repeated root set is [12,13,14]. This is clearly not true as can be seen from the Figure 4.1, where 12 th and 13 th can be seen to be similar modes which can be rotated to look the same, whereas the 14 th mode is a very different modal shape as can be seen in Figure 4.2. The rotation test works only for objects with circular symmetry and is not a good general approach. Although this is an issue, it can be addressed by considering the smallest set that has a high correlation as a repeated mode. The high correlation between the combination [12,10,14] and the 12 th experimental mode is more problematic as it suggests that high correlation may be obtained when-  The first issue occurs because once a repeated mode set has been identified, the presence of noise causes some part of the experimental modal vector to be modeled by a linear combination of some other modes. This is why a good correlation between the 12 t h experimental mode with the set [12,13,14] of analytical modes is obtained. The 14th mode models some part of the leftover noise based result, in the experimental modal vector ,that is not modelled by the set [12,13] modes. This second problem is essentially because this method compares a set of analytical modes with one experimental mode, which is a comparison between an analytical vector space and a experimental vector. This is why the sets [12,10,14] and [12,10] give a high MAC with 12 th experimental mode.
Once either of these two problems are observed, a judgment call needs to be made based 25 on observing the mode shape or wireframe animation. This is why this method is not a acceptable solution and it would be advantageous to have a criterion that works towards correlating experimental vector space and analytical vector space.

SpaceMAC
For comparing vector spaces, a criterion was developed. The concept of principal angle between subspaces, θ, was used to estimate the similarity of the vector subspaces. The criterion was calculated as 1 − |sin(θ)|. This is also known as a coversine function. The choice of function of θ that should be selected was based on the the preference of having a 0 to 1 scale such that an output of 0 would mean no spatial correlation while that of 1 would mean full spatial correlation. This is similar and consistent with the original definition of MAC. The first function of choice was cos(θ) but the cosine function shows very little sensitivity to the angle at low angular values so 1−|sin(θ)| was selected instead.
This criterion was used as part of a process which can be seen in the flowchart below.
The matlab script implementation of the SpaceMAC can be seen in Appendix A.5. As can be seen in the flowchart, first both sets of modal vectors(analytical and experimental) have to be extracted. This process is described in detail in Chapter 3. This also includes conditioning the modal vector sets so that both the sets are of same size and at the same degree of freedom(DOFs). Then the user is asked how many rigid body modes are present in both sets respectively. In case there are any rigid body modes, they are removed from the set. This is so that modal vectors that correspond to a particular mode have the same index in both modal vector sets. This makes it easier to find modes that do not have a high cross-MAC value by just comparing the diagonal of the cross-MAC. The next step is to map the geometry of the analytical model with experimental model(wireframe).
The method for this process is detailed in Chapter 2. An important part of mapping the geometry is comparing the error in the mapping process. This can be done by calculating the distance between each point on the experimental wireframe and the corresponding point on the downsampled analytical node geometry.  This procedure was applied for both circular plate data case studies. The procedure was also applied for experimental to experimental results comparison. 28

Discussion of Results
When two different sets of results were compared for both the circular plates, one important observation was that the repeated modal vectors were numerically consistent to begin with. The cross-MAC between two different modal parameter estimation results is shown in Figure 4   and 4.10 show the modeshapes at these six frequencies. As can be seen from these figures, the spaceMAC values around 0.6 seems to be because of spatial aliasing phenomenon.
But clearly the spatial aliasing was not strong enough to affect the cross-MAC values.
SpaceMAC was observed to have higher sensitivity to aliasing effects.       Following an increasing order of selected multiplicity is an important factor in the spaceMAC implementation. SpaceMAC calculations for multiplicity 3 were done. Some sample values can be seen in Table 4.4 below. As is already known this set of modal vectors does not contain any repeated modes of multiplicity 3. But some high values of spaceM AC 3 can be seen. Some of these high values were observed to be around 0.6 and the set of modes corresponding to these values are the same as those that showed high values with multiplicity 2 due to spatial aliasing. These can be put aside as spatial aliasing cases. However, in some cases there were some mode sets that show very high spaceMAC with three multiplicity. It was observed that these sets are made up of one pair that shows high spaceM AC 2 and a normal mode. This means that out of three mode, two made a repeated pair and one was orthogonal to the space described by the repeated pair. Similar effect can be observed in Table 4.5 where the first 20 rows of spaceM AC 4 calculation have been shown. In this, apart from the aliasing effects, high spaceM AC 4 can be observed between quadruples formed by combining two repeated mode pairs. This can be misleading if spaceM AC 2 calculation have not been run beforehand. This is why spaceMAC calculations should be run with an increasing multiplicity.  A high sensitivity to spatial aliasing has been observed in the developed correlation method. However, the spaceMAC values obtained due to spatial aliasing were around 0.6 whereas the spaceMAC values obtained for repeated modes were higher than 0.8.
Also since the modal frequencies are normally quite different, it is easy to eliminate those cases as being possible repeated root cases based upon frequency alone. It has been observed that MAC values between two different experimental results belonging to the same experimental setup were close to unity even for repeated modes. This observation was contrary to expectation. It was the same for different force functions used, signal processing settings and number of reference points.
When compared to another method for correlation of closely spaced modes, the results obtained with spaceMAC were clearer as long as an increasing order of multiplicity of repeated modes was followed. This was largely because of the fact that spaceMAC works with correlation of two modal vector spaces whereas the other method compares a space with a vector. The effect of variation of MAC threshold has been explored. It has been observed that it did not directly affect the spaceMAC calculation for a particular set of modes. It did however affect if a particlar combination of modes is selected or not.

Future Work
One of the main subject of curiosity during this research, apart from the main subject of repeated mode correlation, was that repeated modes for different experiments performed on the same setup were observed to have high correlation even when . Usually it is expected that repeated mode should should vary and not show cross-MAC values close to unity. Although some exploration was done to find out the cause of this, it would be interesting to do a more thorough research work to examine the possible reason for this anomaly. This observation is mostly based upon historical parameter estimation methods and current modal parameter estimation methods may just be that much better.
Another avenue of improvement would be developing a general geometry mapping algorithm and combining it with a repeated mode correlation method like spaceMAC.
This would be extremely helpful in making the whole process applicable to any generic test object and corresponding model. This could also involve a smoother transition from FEM analysis and experimental analysis to the application of spaceMAC.