Recent work with autonomous modal parameter estimation has shown great promise in the quality of the modal parameter estimation results when compared to results from traditional methods by experienced users. While autonomous modal parameter estimation means slightly different things to different researchers and practitioners, for the purpose of this discussion, autonomous will require an automated procedure which sorts and processes a large number of possible modal parameter solutions to yield one consistent estimate with no user interaction after initial thresholds are chosen. In the work discussed, this final, consistent set of modal parameters is identifiable due to the combination of temporal and spatial information in a
The desire to estimate modal parameters automatically, once a set or multiple sets of test data are acquired, has been a subject of great interest for more than 40 years. In the 1960s, when modal testing was limited to analog test methods, several researchers were exploring the idea of an automated test procedure for determining modal parameters [
Before proceeding with a discussion of the use and impact of spatial information in autonomous modal parameter estimation, some philosophy and definitions regarding what is considered autonomous is required. In general, autonomous modal parameter estimation refers to an automated procedure that is applied to a modal parameter estimation algorithm so that no user interaction is required once the process is initiated. This typically involves setting a number of parameters or thresholds that are used to guide the process in order to exclude solutions that are not acceptable to the user. When the procedure finishes, a set of modal parameters is identified that can then be reduced or expanded if necessary. The goal is that no further reduction, expansion, or interaction with the process will be required.
For the purposes of further discussion, the autonomous modal parameter estimation procedure is simply an efficient mechanism for sorting a very large number of solutions into a final set of solutions that satisfies a set of criteria and thresholds that are acceptable to the user. This user is assumed to be very experienced and uses autonomous modal parameter estimation as a sophisticated tool to highlight the most likely solutions based upon statistics. The experienced user will realize that the final solutions may include unrealistic solutions or nonoptimal solutions and further evaluation will be required.
In order to discuss the impact and use of spatial information in autonomous modal parameter estimation, some background is needed to clarify terminology and methodology. This background has been provided in previous papers and will only be highlighted here in terms of spatial information, modal parameter estimation, and autonomous modal parameter estimation.
Spatial information, with respect to experimental modal parameter estimation, refers to the vector information and dimension associated with the inputs and outputs of the experimental test. Essentially, this represents the locations of the sensors in the experimental test. It is important to recognize that an experimental test should always include multiple inputs and outputs in order to clearly estimate different modal vectors and to resolve modal vectors when the complex natural frequencies are close, what is called repeated or pseudorepeated roots.
Since the data matrix, normally involving frequency response functions (FRF) or impulse response functions (IRF), is considered to be symmetric or reciprocal, the data matrix can be transposed, switching the effective meaning of the row and column index with respect to the physical inputs and outputs. Consider
Therefore, the nomenclature of the number of outputs
All modern, commercial algorithms for estimating modal parameters from experimental inputoutput data utilize matrix coefficient, polynomial models. This general matrix coefficient, polynomial formulation yields essentially the same polynomial form for both time and frequency domain data. Note, however, that this notation does not mean that, for an equivalent model order, the associated matrix coefficients are numerically equal.
For the frequency domain data case, this is
When the modal frequencies are estimated from this eigenvalue problem, a unique estimate of the unscaled modal vector is identified from the associated eigenvector at the same time. The length or dimension of this unscaled modal vector is equal to the dimension of the square, alpha coefficients which, in general, is equal to
In all cases, the size of the square, alpha coefficient matrices
The most commonly used modal identification methods can be summarized as shown in Table
Summary of modal parameter estimation algorithms.
Algorithm  Domain  Matrix polynomial order  Coefficients  

Time  Freq.  Zero  Low  High  Scalar  Matrix  
Complex Exponential Algorithm (CEA)  •  •  •  
Least Squares Complex Exponential (LSCE)  •  •  •  

•  • 


Ibrahim Time Domain (ITD)  •  • 


Multireference Ibrahim Time Domain (MRITD)  •  • 



•  • 



•  • 


Simultaneous Frequency Domain (SFD)  •  • 


Multireference Frequency Domain (MRFD)  •  • 


Rational Fraction Polynomial (RFP)  •  •  • 


Orthogonal Polynomial (OP)  •  •  • 


Polyreference Least Squares Complex Frequency (PLSCF)  •  •  • 



•  •  • 


Complex Mode Indication Function (CMIF)  •  • 

All of the methods summarized in Table
Table
Four corners of modal parameter estimation.
Time domain  Frequency domain  

Low order models  ITD 



High order models  CEA 
RFP 
As a matter of further clarification, two different Polyreference Frequency Domain (PFD) algorithms were developed independently in the 1980s. Both methods are low order, frequency domain methods. PFD1 is the designation for the first order form and PFD2 is the designation for the second order form. A recent paper explains that the two methods are theoretically equivalent [
The interest in automatic modal parameter estimation methods has been documented in the literature since at least the mid1960s when the primary modal method was the analog, force appropriation method [
Each of these past procedures has shown some promise but has not yet been widely adopted. In many cases, the procedure focussed on a single modal parameter estimation algorithm and did not develop a general procedure. Most of the past procedural methods focussed on modal frequency (pole) density but depended on limited modal vector data to identify correlated solutions. Currently, due to increased computational speed and availability of memory, procedural methods can be developed that were beyond the computational scope of available hardware only a few years ago. These methods do not require any initial thresholding of the solution sets and rely upon correlation of the vector space of hundreds or thousands of potential solutions as the primary identification tool.
The discussion in the following sections of the use and impact of spatial information in autonomous modal parameter estimation is based upon recent implementation and experience with an autonomous modal parameter estimation procedure referred to as the common statistical subspace autonomous mode identification (CSSAMI) method [
Note that much of the background of the CSSAMI method is based upon the Unified Matrix Polynomial Algorithm (UMPA) described in the previous section. This means that this method can be applied to both low and high order methods with short or long dimension modal (base) vectors. This also means that most commercial algorithms could take advantage of this procedure. Note that high order matrix coefficient polynomials normally have coefficient matrices of a dimension that is based upon the short dimension of the data matrix
When comparing modal (base) vectors, at either the short or the long dimension, a poleweighted vector can be constructed independent of the original algorithm used to estimate the poles and modal (base) vectors. For a given order
Figures
Eighth order, poleweighted vector (state vector) example.
Eighth order, poleweighted vector (state vector) example, top view.
The order
While the order
Since the magnitude of the
Consistency diagrams have historically been developed for a specific modal parameter estimation algorithm. As such the numerical implementation can be different as a function of basis dimension (
The data used for this, and all following examples in this paper, is FRF data taken from an impact test of a steel disc supported in a pseudofreefree boundary condition. The steel disc is approximately 2 cm. thick and 86 cm. in diameter with several small holes through the disc. The center area of the disc (diameter of approximately 25 cm.) has a thickness of approximately 6 cm. There are seven reference accelerometers and measured force inputs from an impact hammer are applied to thirtysix locations, including next to the seven reference accelerometers. The frequency resolution of the data is 5 Hertz. While the disc is not as challenging as some industrial data situations that contain more noise or other complicating factors like small nonlinearities, the disc has a number of pseudorepeated roots spaced well within the 5 Hertz frequency resolution and a mix of close modes involving repeated and nonrepeated roots which are very challenging. Based upon the construction of the disc, realvalued, normal modes can be expected and the inability to resolve these modes can be instructive relative to both modal parameter estimation algorithm and autonomous procedure performance. For the interested reader, a number of realistic examples are shown in other past papers including FRF data from an automotive structure and a bridge structure [
Figure
Extended consistency diagram, conventional version.
The figure includes a complex mode indicator function (CMIF) for the FRF dataset in the background. The blue diamond symbols indicate that modes have been identified with consistent frequency, damping and modal vector estimates as both model order and algorithm are changed. From the CMIF plot in the background, it is clear that most of the identified modes are closely spaced, repeated roots. For these cases, there are two blue diamonds at each solution on top of one another. The multialgorithm approach is sensitive to the transitions between algorithms as the modal parameter estimates and symbol designations are estimated sequentially across these transitions.
Figure
Extended consistency diagram, poleweighted MAC version.
Both methods work very well but the implementation of Figure
The CSSAMI autonomous procedure utilizes all solutions indicated by a symbol in the consistency diagram. If some symbols are not present, it means that the user has decided not to view solutions identified by those symbols. This provides a way to remove solutions from the autonomous procedure that are clearly not reasonable. However, experience with the CSSAMI autonomous procedure has shown that some solutions that are often eliminated by users in an attempt to have a clear consistency diagram are often statistically consistent and useful.
Figure
Poleweighted MAC of all consistency diagram solutions, before and after threshold applied.
For the examples used in this paper, a minimum cluster size is chosen to be four. This choice is up to the user and generally should not be chosen too large. The philosophy is to allow the autonomous procedure to identify all possible clusters and then have the experienced user make the final decision if some of the clusters yield nonphysical results. These decisions can be made based upon statistics associated with the cluster or physical characteristics of the modal parameters. If the cluster size is too small, some modal vectors might be excluded. This would require the user to reprocess the data looking for a missing modal vector. It is far easier for the user to remove unwanted or nonphysical modes than to try to find modes that have been marginally excited in the data. The first column of Table
Summary of autonomous modal parameter estimation statistics.

Freq. (SV)  Damp (SV)  % Zeta  MPC  Freq. (Mean)  Damp (Mean)  Std. Dev.  NMVR_{1}%  NMVR_{2}%  NSVR_{1}%  NSVR_{2}% 

69  362.356  −3.128  0.8633  0.9199  362.356  −3.128  0.1049  0.7686  6.3402  0.7686  6.3402 
69  363.696  −3.403  0.9358  0.8915  363.696  −3.403  0.1112  1.2308  10.2036  1.2312  10.2062 
71  557.008  −2.888  0.5184  0.9926  557.008  −2.888  0.1144  0.0810  0.6281  0.0812  0.6281 
69  761.172  −5.091  0.6688  0.9701  761.172  −5.091  0.1345  0.8888  7.1128  0.8889  7.1129 
70  764.175  −2.581  0.3377  0.9862  764.175  −2.581  0.0865  0.3335  2.2550  0.3337  2.2566 
64  1222.940  −4.074  0.3332  0.8286  1222.940  −4.074  0.0855  0.9708  6.4450  0.9712  6.4492 
67  1224.080  −3.962  0.3236  0.8316  1224.079  −3.962  0.1645  1.0343  8.3785  1.0338  8.3748 
68  1327.955  −6.704  0.5048  0.8264  1327.953  −6.705  0.3577  1.2250  9.9197  1.2248  9.9176 
57  1328.699  −5.421  0.4080  0.8169  1328.698  −5.421  0.1400  0.8649  6.3990  0.8645  6.3960 
69  2019.161  −8.153  0.4038  0.9655  2019.163  −8.159  1.0262  1.0610  7.3389  1.0623  7.3411 
56  2023.589  −7.525  0.3719  0.9481  2023.590  −7.525  0.5596  1.8525  11.7555  1.8518  11.7490 
























47  2337.946  −4.194  0.1794  0.9654  2337.945  −4.201  0.9058  0.9132  4.5838  0.9129  4.5811 
Once the final set of modal parameters, along with their associated statistics, is obtained, quality can be assessed by many methods that have been used in the past. The most common example is to perform comparisons between the original measurements and measurements synthesized from the modal parameters. Another common example is to look at physical characteristics of the identified parameters such as reasonableness of frequency and damping values, normal mode characteristics in the modal vectors, and appropriate magnitude and phasing in the modal scaling. Other evaluations that may be helpful are modal assurance criterion (MAC) evaluation of the independence of the complete modal vector set, mean phase correlation (MPC) of each vector, or any other method available. Naturally, since a significant number of poleweighted vectors are used in a cluster to identify the final modal parameters, traditional statistics involving mean and standard deviation are now available. Some of these statistical methods are discussed in the next section.
Statistical evaluation parameters can be estimated for each common cluster of poleweighted vectors on the basis of the complex modal frequency, the modal vector, and the modal scaling. The number of poleweighted vectors will in general be different in each cluster so the statistics will be based upon the number of estimates available (sample size
The weighted modal frequency for the cluster is found by constructing the poleweighted vector (typically 10th order) for each pole retained in a cluster, then taking the SVD of the group of poleweighted vectors and selecting the singular vector associated with the largest singular value. This chosen singular vector contains both the shape and the modal frequency information. The modal frequency is identified by dividing the first order portion by the zeroth order portion of the vector in a least squares sense. (Note that it is also possible to solve the frequency polynomial which would result from using the complete vector.) Also, for numerical reasons, the poleweighted vector is actually computed in the
For comparison purposes, the actual mean value and standard deviation of the poles (as well as the separate frequency and damping means and standard deviations) which were used in the computation of the weighted solution are computed. Since these results are unweighted by the vector characteristics, they may be somewhat different from the vector weighted solution and provide comparative feedback about the pole.
In order to evaluate the quality of the resulting modal vectors, several different parameters (representing noise to signal ratios) are calculated. These ratios are evaluated for both the original normalized vectors and the poleweighted (state extended) vectors and are computed using the singular value decomposition of each of the set of vectors.
The first modal vector parameter is evaluated by taking the total residual magnitude (the Frobenius norm of the residuals) divided by the magnitude of the principal vector magnitude. In other words, the square root of the sum of the squares of the residual singular values divides by the first (largest) singular value. This provides an indication of the consistency of the original contributing vectors. Small values tend to indicate greater consistency. Large values indicate greater variance or the possibility that more than one mode has been included in a cluster. Consider
The associated state vector parameters are calculated analogous to the above except that the complete poleweighted vector (state vector) is used:
If modal scaling (residue) information is available in the original vectors, it can be applied to the principal poleweighted vector such that the consistency of modal scaling (Modal A) can also be evaluated. By scaling the largest driving point response to unity, a uniform, comparable value for Modal A is chosen. Since the modal scaling parameter (Modal A) contains the physical units and hence the relative contribution of each pole/vector combination to the original data set, a means of evaluating the significance and confidence in each estimate is possible. Because the original vectors represent multiple scaled solutions and since Modal A is a single scalar complex value for each pole/vector combination, the statistical distribution of Modal A (mean and standard deviation) of the original cluster can be compared with the scaling estimated from the poleweighted (state extended) vector. The magnitude of Modal A can be used to evaluate the contribution of this vector to the total set of identified vectors. When the user believes that the identified vectors should be normal modes, the phase of the Modal A can be used to evaluate the closeness to a normal mode.
Table
Figure
Pole surface consistency, 2320 Hertz region.
The two modes in this region, within one
Much of the CSSAMI autonomous procedure, as well as most commonly used modal parameter estimation methods, utilize the traditional modal assurance criterion (MAC) computation, restated in (
Note that the CSSAMI autonomous procedure treats each cluster of estimates independently and does not constrain the final set of modal vector estimates in any way. At this point in the autonomous procedure, the MAC computation is again utilized to evaluate the quality of the solutions. This begins with an evaluation of the MAC between all of the modal vectors in the final set to ascertain whether the modal set is an independent set of vectors. This often involves including the estimates of the modal vectors associated with the conjugate poles. Since the conjugate poles and vectors are estimated separately, if nonconjugate relationships exist between the associated modal vector estimates (between the modal vector for pole and the modal vector of the conjugate pole), the MAC between these two associated vectors will not be unity as expected. A number of users have noted that this often correlates with modal vectors that are exhibiting some unexpected, nonphysical characteristics.
Since much of the autonomous procedure is based upon numerical processing methods like singular value decomposition (SVD), solutions that are identified, based upon the data associated with a cluster of estimates, have no physical or causal constraint. An example of a physical or causal constraint would be the expectation of realvalued, normal modes for systems where no expectation of nonproportional damping is likely. SVD methods will identify the most dominant unitary (orthogonal and unit length) vectors in a cluster, yielding a complexvalued vector. Experience has shown that when modes are very close in frequency with minimal spatial resolution, the complexvalued vectors will still show significant dependence or correlation.
However, when these complexvalued vectors are examined closely, the nondominant portion of the complexvalued vector often correlates very highly with one or more nearby modal vectors. This can be examined by the MAC calculation or several variants of the MAC calculation (like the weighted MAC calculation defined in (
For this evaluation of the modal vector contamination, it will be easiest to first rotate each complexvalued modal vector to a real (or imaginary) dominant vector. This is done by using a least squares method to identify the rotation of the modal vector away from the real or imaginary axis and then using the associated complex phasor to rotate each original complexvalued modal vector to a new complexvalued modal vector that aligns with the real or imaginary axis. For all following discussions, the original complexvalued modal vectors are rotated to be dominantly realvalued. It is convenient, for display reasons, to also normalize the new complexvalued modal vector to a unity maximum or unity vector length. Naturally, this rotation and rescaling must be considered in any final estimates of modal scaling.
To understand the nature of the possible modal vector contamination in a complexvalued modal vector, three conventional MAC calculations can be performed
The above MAC evaluations identify whether, and how, the contamination of a complexvalued modal vector is related to another of the identified modal vectors. However, the MAC computation is normalized to vector length, vector by vector, for the vectors used in the calculation. A weighted MAC can be used to determine the scale of the contamination. The following three definitions of the weighting for each of the above MAC calculations limit the associated MAC value to a fraction of the zero to one scale. If near normal modes are expected,
With the advent of more computationally powerful computers and sufficient memory, it has become practical to evaluate sets of solutions involving hundreds or thousands of modal parameter estimates and to extract the common information from those sets. In many cases, autonomous procedures give very acceptable results, in some cases superior results, in a fraction of the time required for an experienced user to get the same result. However, it is important to reiterate that the use of these autonomous procedures or
Future work will involve better numerical methods for combining algorithms into single consistency diagrams and autonomous methods for identifying the best causal (physical) solution are still needed. The application of MAC and related weighted MAC estimates is helpful for detecting contaminated modal vector solutions but numerical solution methods that identify both realvalued modal vectors (normal modes) and complexvalued modal vectors, when appropriate, are still needed. Initial positive results have been obtained, with respect to removing this close mode contamination, by constraining the modal participation factors via real normalization [
Number of inputs
Number of outputs
Short dimension size
Long dimension size
Number of vectors in cluster
Complex modal frequency (rad/sec)
Modal damping
Damped natural frequency
Base vector (modal vector)
Poleweighted vector (state vector)
Mode number
Discrete frequency (rad/sec)
FRF matrix (
Numerator polynomial matrix coefficient
Denominator polynomial matrix coefficient
Model order for denominator polynomial
Model order for numerator polynomial
Model order for base vector
Standard deviation
Normalized modal vector residual 1
Normalized modal vector residual 2
Normalized state vector residual 1
Normalized state vector residual 2
Singular value
Modal assurance criterion
Weighted modal assurance criterion.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to acknowledge the collaboration and assistance from the graduate students and faculty of the Structural Dynamics Research Lab at the University of Cincinnati. In particular, the discussions and collaborations with Dr. David L. Brown have been instrumental in the progress made to this point.