A Low Order Implementation of the Polyreference Least
Squares Complex Frequency (LSCF) Algorithm
A.W. Phillips, R.J. Allemang
Structural Dynamics Research Laboratory
Mechanical, Industrial and Nuclear Engineering
University of Cincinnati
Cincinnati, OH 45221-0072 USA
Email: [email protected]
ABSTRACT
Recent presentations by several authors concerning a Rational Fraction Polynomial (RFP) algorithm with a
new complex frequency mapping, called the Polyreference Least Squares Complex Frequency (LSCF) and
referred to commercially as the PolyMAX algorithm, have demonstrated an interesting modal parameter
estimation algorithm that has optimal numerical characteristics for a high order frequency domain method.
This paper explores this algorithm, evaluating the performance against both theoretical and experimental data
cases, paying particular attention to the impact of residuals on the final modal parameter estimates.
Comparison to other modal parameter estimation algorithms is also included. In particular, a low order
implementation is evaluated and compared to historical low and high order algorithms in the frequency
domain in order to determine the advantages and disadvantages of using this algorithm.
Nomenclature
N i = Number of inputs.
N o = Number of outputs.
N s = Number of spectral lines (frequencies).
F max = Maximum frequency (Hz).
ω i = Frequency (rad/sec).
ω max = Maximum frequency (rad/sec).
Δ f = Frequency resolution (Hz).
λ r = Complex modal frequency
T = Observation period (Sec.).
si = Generalized frequency variable.
[C] = Companion matrix.
[α ] = Numerator polynomial matrix coefficient.
[ β ] = Denominator polynomial matrix coefficient.
[I ] = Identity matrix.
[H(ω i )] = Frequency response function matrix (N o × N i )).
[T ] = Transformation matrix.
[U] = Left singular vector matrix.
[Σ] = Singular value matrix (diagonal).
[Λ] = Eigenvalue matrix (diagonal).
[V ] = Right singular vector, or eigenvector, matrix.
1. Introduction
An important variation of the historical Rational Fraction Polynomial (RFP) algorithm for estimating modal
parameters, the Polyreference Least Squares Complex Frequency (PLSCF) [21-25] , has been recently
introduced which has optimal numerical characteristics for a high order frequency domain method. This
algorithm uses a complex Z mapping of the generalized frequency variable in order to improve the notably
poor numerical characteristics of the RFP algorithm. In order to understand the characteristics of this PLSCF
method, it is desirable to compare its performance to a traditional, low order frequency domain algorithm
such as the Polyreference Frequency Domain (PFD) method. Additionally, the same complex Z mapping of
the generalized frequency domain variable can be applied to the the low order PFD method, generating
another variation which may provide further insight into the performance of the complex Z mapping of the
generalized frequency variable. This new variation of the PFD method will be referred to as PFD-Z for the
purposes of further discussion.
2. Low Order Frequency Domain Algorithms
Low order, frequency domain algorithms used for estimating modal parameters refer to the class of
algorithms that generate matrix coefficient polynomials of first or second order. These algorithms are
formulated from frequency response function (FRF) matrices that are typically of dimension N o × N i .
Depending upon the experimental method used to generate the FRF matrix, either the number of inputs (N i )
or the number of outputs (N o ) may be larger. Since this matrix is generally considered reciprocal and the
FRF matrix may be transposed if needed, the number of outputs (N o ) will be assumed to the largest
dimension and will be referred to as the long dimension in any further discussion.
In order for low order, frequency domain algorithms to estimate a large number of poles, the long dimension
of the FRF matrix must be at least as large as the number of positive modal frequencies desired. The modal
frequencies result from the solution of the resulting matrix coefficient polynomial.
Second Order Matrix Coefficient Polynomial
⎪ [α ] s2 + [α ] s1 + [α ] s0 ⎪ = 0
1
0
⎪ 2
⎪
(1)
First Order Matrix Coefficient Polynomial
⎪ [α ] s1 + [α ] s0 ⎪ = 0
0
⎪ 1
⎪
(2)
The companion matrix, formed from the coefficients of the matrix polynomial is one approach that can be
used in the following eigenvalue formulation [30] to determine the modal frequencies for the original matrix
coefficient equation:
[C]{Φ } = λ [I ] {Φ }
(3)
Second Order Matrix Coefficient Polynomial
⎡ − [α 1 ]
[C] = ⎢
⎣ [I ]
− [α 0 ] ⎤
⎥
[0] ⎦
(4)
First Order Matrix Coefficient Polynomial
[C] = ⎡ − [α 0 ] ⎤
⎦
⎣
(5)
Note that in the above solutions, the lead coefficient ([α 2 ] or [α 1 ]) is normalized to the identity matrix (high
order coefficient normalization). An alternate solution is to set the trailing coefficient ([α 0 ]) equal to identity
(low order coefficient normalization) which gives different numerical estimates in the presence of noise.
Note also that the numerical characteristics of the eigenvalue solution of the companion matrix will be
different for low order cases compared to high order cases for a given data set.
2.1 First Order Frequency Domain Algorithms
Several algorithms have been developed that fall into the category of first order frequency domain algorithms,
including the Simultaneous Frequency Domain (SFD) algorithm [1] first reported by Coppolino and the
Multiple Reference Simultaneous Frequency Domain algorithm [2] reported by Craig. These algorithms are
essentially frequency domain equivalents to the Ibrahim Time Domain (ITD) and the Eigensystem
Realization Algorithm (ERA) algorithms and effectively involve a state-space formulation when compared to
the second order frequency domain algorithms. The state-space formulation utilizes the derivatives of the
frequency response functions as well as the frequency response function in the solution. These algorithms
have superior numerical characteristics compared to the historical, high order frequency domain algorithms
(Rational Fraction Polynomial). Unlike the low order time domain algorithms, though, sufficient data from
across the complete frequency range of interest must be included in order to obtain a satisfactory solution.
Essentially, these methods are a first order polynomial with square matrix coefficients equal to twice the
dimension of the long dimension of the FRF matrix and it is assumed that the long dimension is greater than
or equal to the number of positive modal frequencies that are desired (2N o ≥ 2N ). In most experimental
cases, the problem is that the long dimension of the FRF matrix is much greater than the number of modal
frequencies desired. The basic linear equations that can be repeated for each measured frequency ω i are
shown in the following for the two common normalizations. Note the nomenclature in the following
equations regarding measured frequency ω i and generalized frequency si . Measured input and response data
are always functions of measured frequency but the generalized frequency variable used in the model may be
altered to improve the numerical conditioning. This will become important in a later discussion of
generalized frequency involving normalized freqeuncy, orthogonal polynomials and complex Z mapping
(Section 2.5)
Basic Equation: [α 1 ] Normalization
0
⎡(si ) [H(ω i )]⎤
⎢(si )1 [H(ω i )]⎥
⎡(s )1 [H(ω i )] ⎤
⎡ [α ] [ β ] ⎤
= −⎢ i 2
⎥
⎢
⎥
0
0
0
⎦2N o ×4N o ⎢ −(si ) [I ] ⎥
⎣
⎣(si ) [H(ω i )] ⎦2N o ×N i
⎣ −(si )1 [I ] ⎦4N o ×N i
(6)
Basic Equation: [α 0 ] Normalization
1
⎡ (si ) [H(ω i )] ⎤
2
⎢ (si ) [H(ω i )] ⎥
⎡ (si )0 [H(ω i )] ⎤
⎡ [α ] [ β ]⎤
= −⎢
⎢
⎥
0
1
⎦2N o ×4N o ⎢ −(si )0 [I ] ⎥⎥
⎣ 1
⎣ (si ) [H(ω i )] ⎦2N o ×N i
⎣ −(si )1 [I ] ⎦4N o ×N i
(7)
2.2 Second Order Frequency Domain Algorithms
The second order frequency domain algorithms include the Polyreference Frequency Domain (PFD) [4-6]
and Frequency domain Direct Parameter Identification (FDPI) [3,6,7] algorithms that were developed at the
Katholieke University of Leuven (KUL), Belgium and the University of Cincinnati, Structural Dynamics
Research Lab (UC-SDRL) in slightly different forms. These algorithms have superior numerical
characteristics compared to the high order frequency domain algorithms. Unlike the low order time domain
algorithms, though, sufficient data from across the complete frequency range of interest must be included in
order to obtain a satisfactory solution. Essentially, these methods are a second order polynomial with matrix
coefficients equal to the long dimension of the FRF matrix and it is assumed that the long dimension is
greater than or equal to the number of positive modal frequencies that are desired(N o ≥ N ). In most cases,
the problem is that the long dimension of the FRF matrix is much greater than the number of modal
frequencies desired. The basic linear equations that can be repeated for each measured frequency ω i are
shown in the following for the two common normalizations.
Basic Equation: [α 2 ] Normalization
0
⎡ (si ) [H(ω i )] ⎤
1
⎢ (si ) [H(ω i )] ⎥
⎡ [α ] [α ] [ β ] [ β ]⎤
= − (si )2 [H(ω i )] N o ×N i
0
1
0
1
⎦ N o ×4N o ⎢⎢ −(si )0 [I ] ⎥⎥
⎣
⎣ −(si )1 [I ] ⎦4N o ×N i
(8)
Basic Equation: [α 0 ] Normalization
1
⎡ (si ) [H(ω i )] ⎤
⎢ (si )2 [H(ω i )] ⎥
⎡ [α ] [α ] [ β ] [ β ]⎤
= − (si )0 [H(ω i )] N o ×N i
2
0
1
⎦ N o ×4N o ⎢⎢ −(si )0 [I ] ⎥
⎣ 1
⎥
⎣ −(si )1 [I ] ⎦4N o ×N i
(9)
2.3 Coefficient Condensation (Virtual DOFs)
For the low order modal identification algorithms, the number of physical coordinates (typically N o ) is often
much larger than the number of desired modal frequencies (2N ). For this situation, the numerical solution
procedure is constrained to solve for N o or 2N o modal frequencies. This can be very time consuming and is
unnecessary. One simple approach to reducing the size of the coefficient matrices is to sieve the physical
DOFs to temprorarily reduce the dimension of N o . Beyond excluding all physical DOFs in a direction, this
is difficult to do in an effective manner that will retain the correct information from the FRF data matrix.
The number of physical coordinates (N o ) can be reduced to a more reasonable size (N e ≈ N o or N e ≈ 2N o )
by using a decomposition transformation from physical coordinates (N o ) to the approximate number of
effective modal frequencies (N e ). These resulting N e transformed coordinates are sometimes referred to as
virtual DOFs. Currently, singular value decompositions (SVD) or eigenvalue decompositions (ED) are used
to preserve the principal modal information prior to formulating the linear equation solution for unknown
matrix coefficients [6,8-10,26-28].
It is important to understand that the ED and SVD transformations yield a mathematical transformation that,
in general, contains complex-valued vectors as part of the transfomation matrix, [T ]. Conceptually, the
transformation will work well when these vectors are estimates of the modal vectors of the system, normally
a situation where the vectors can be scaled to real-valued vectors. Essentially, this means that the target goal
of the transformation is a transformation from physical space to modal space. As the modal density increases
and/or as the level of damping increases, the ED and SVD methods give erroneous results, if the complete
[H] matrix is used. Generally, superior results will be obtained when the imaginary part of the [H] matrix is
used in the ED or SVD transformation, thus forcing a real-valued transformation matrix, [T ]. Another option
is to load both the real and imaginary portions of the complex data into a real matrix which will also force a
real-valued transformation matrix [10] .
In most cases, even when the spatial information must be condensed, it is necessary to use a model order
greater than two to compensate for distortion errors or noise in the data and to compensate for the case where
the location of the transducers is not sufficient to totally define the structure.
[H′] = [T ] [H]
(10)
where:
[H′] is the transformed (condensed) frequency response function matrix .
•
•
[T ] is the transformation matrix.
•
[H] is the original FRF matrix.
The difference between the two techniques lies in the method of finding the transformation matrix, [T ]. In
reality, while the two methods (ED and SVD) appear to be different, the methods yield the same results since
the eigenvalues and singular values are numerically related. Since eigenvalue decomposition (ED) is
numerically more efficient, this method is normally used. Once [H] has been condensed, however, the
parameter estimation procedure is the same as for the full data set. Because the data eliminated from the
parameter estimation process ideally corresponds to the noise in the data, the poles of the condensed data are
the same as the poles of the full data set. However, the participation factors calculated from the condensed
data may need to be expanded back into the full space.
[Ψ] = [T ]T [Ψ′]
(11)
where:
•
[Ψ] is the full-space participation matrix.
•
[Ψ′] is the condensed-space participation matrix.
Eigenvalue Decomposition
In the eigenvalue decomposition method (sometimes referred to as Principal Component Analysis [28]), the
[T ] matrix is composed of the eigenvectors corresponding to the N e largest eigenvalues of the power
spectrum of the FRF matrix as follows:
[H(ω )] N o ×N i N s [H(ω )] HNi N s ×N o = [V ] [Λ] [V ] H
(12)
Note that superior results will generally be obtained when the imaginary part of the [H] matrix, or a real
matrix formed from the real and imaginary parts of the [H] matrix, is used in the above ED. The eigenvalues
and eigenvectors are then found, and the [T ] matrix is constructed from the eigenvectors corresponding to the
N e largest eigenvalues:
T
[T ] N e ×N o
⎡⎧ ⎫⎧ ⎫
⎧ ⎫
⎧ ⎫⎤
= ⎢⎨v 1 ⎬⎨v 2 ⎬ . . . ⎨v k ⎬ . . . ⎨v N e ⎬⎥
⎣⎩ ⎭⎩ ⎭
⎩ ⎭
⎩ ⎭⎦
(13)
where:
•
{v k } is the N o × 1 eigenvector corresponding to the k − th eigenvalue.
This technique may be adapted for condensing on the input space, as well. The power spectrum matrix is
again found, but the FRF matrix must be reshaped (transposed) so that it is an N i × N o matrix for each
spectral line:
[H(ω )] N i ×N o N s [H(ω )] HNo N s ×N i = [V ] [Λ] [V ] H
(14)
The eigenvalues and eigenvectors are again found as before, and the transformation matrix [T ] becomes:
T
[T ] N e ×N i
⎡⎧ ⎫⎧ ⎫
⎧ ⎫
⎧ ⎫⎤
= ⎢⎨v 1 ⎬⎨v 2 ⎬ . . . ⎨v k ⎬ . . . ⎨v N e ⎬⎥
⎣⎩ ⎭⎩ ⎭
⎩ ⎭
⎩ ⎭⎦
(15)
where:
•
{v k } is the N i × 1 eigenvector corresponding to the k − th eigenvalue.
Singular Value Decomposition
The singular value decomposition condensation technique is similar to the eigenvalue-based technique, but
operates on the FRF matrix directly instead of the power spectrum of the FRF matrix. The basis for this
technique is the singular value decomposition [26,28], by which the matrix [H] is broken down into three
component parts, [U], [Σ], and [V ]:
[H] N o ×N i N s = [U] N o ×N o [Σ] N o ×N i N s [V ] HNi N s ×N i N s
(16)
Note that superior results will generally be obtained when the imaginary part of the [H] matrix, or a real
matrix formed from the real and imaginary parts of the [H] matrix, is used in the above SVD. The leftsingular vectors corresponding to the N e largest singular values are the first N e columns of [U]. These
become the transformation matrix [T ]:
T
[T ] N e ×N o
⎡⎧ ⎫⎧ ⎫
⎧ ⎫
⎧ ⎫⎤
= ⎢⎨u1 ⎬⎨u2 ⎬ . . . ⎨u k ⎬ . . . ⎨u N e ⎬⎥
⎣⎩ ⎭⎩ ⎭
⎩ ⎭
⎩ ⎭⎦
(17)
where:
•
{u k } is the k − th column of [U], which corresponds to the k − th singular value.
This technique may also be adapted for condensing the input space, as long as the FRF matrix [H] is
reshaped (transposed) to an N i × N o matrix at each spectral line. The SVD operation then becomes
[H] N i ×N o N s = [U] N i ×N i [Σ] N i ×N o N s [V ] HNo N s ×N o N s
(18)
The transformation matrix [T ] is still composed of the left singular vectors corresponding to the N e largest
singular values,
T
[T ] N e ×N i
⎡⎧ ⎫⎧ ⎫
⎧ ⎫
⎧ ⎫⎤
= ⎢⎨u1 ⎬⎨u2 ⎬ . . . ⎨u k ⎬ . . . ⎨u N e ⎬⎥
⎣⎩ ⎭⎩ ⎭
⎩ ⎭
⎩ ⎭⎦
(19)
where:
•
{u k } is again the k − th column of [U], which corresponds to the k − th singular value.
2.4 Virtual DOF Sieve
One of the drawbacks of low order algorithms is that there will only be one or two solutions for the
algorithm, based upon whether the lead matrix coefficient ([α m ]) or trailing matrix coefficient ([α 0 ]) is
normalized to the identity matrix in the matrix coefficient polynomial. This makes the consistency diagram
hard to utilize if only one or two solutions are displayed. The final step to most implementations of low
order algorithms is to formulate the solution on the basis of a subset of the virtual DOFs that are estimated
from the transformation represented by Equation 10. Since sieving the FRF matrix refers to removing
physical DOFs (Z direction at all locations, for example) prior to using the FRF data in a modal parameter
estimation algorithm, eliminating some of the virtual FRF data is essentially the same process.
For example, once the long dimension of the FRF matrix gets larger than 40, it is common to utilize
coefficient condensation to limit the long dimension to 40. This is done by including only the 40
eigenvectors, or singular vectors, associated with the 40 largest eigenvalues, or singular values. The solutions
can then be formed by taking a subset of these vectors in multiple solutions to generate a pleasing and
meaningful consistency diagram. A common formulation would be to use sucessively larger numbers of
vectors from 20 to 40 in steps of 2, with two normalization solutions at each step. This generates a
consistency diagram with 40 solutions which normally gives a useful result.
2.5 Generalized Frequency Mapping
The important contribution behind the development of the Polyreference LSCF method is the recognition of a
new method of frequency mapping. In the past, the only way to avoid the numerical problems inherent in the
frequency domain methods (Van der Monde matrix), particularly the high order frequency domain methods
that have historically been called the Rational Fraction Polynomial (RFP) methods, is to use a transformation
from power polynomials to orthogonal polynomials [11-18] . This approach is very cumbersome
mathematically and can be replaced by a trigonometric mapping function (complex Z) that is the
fundamental contribution of the Polyreference LSCF method [21-25] . This approach can also be applied to
low order frequency domain methods, as well as high order frequency domain methods, although the
numerical advantage is not as profound.
For the low order frequency domain algorithms, the generalized frequency is normally just the normalized
power polynomial given by the following equation:
si = j * (ω i /ω max )
(20)
This gives a generalized frequency variable that is bounded by (-1,1) with much better numerical
conditioning than utilizing the raw frequency range (- ω max ,ω max ). The graphical plot of this Van der Monde
matrix for orders 0 through 8 is shown in Figure 1.
Order:
8
Condition Number: 5.48e+002
1
0.8
Power Polynomial Functions − Real Part
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Normalized Frequency (Rad/Sec)
Imaginary:
Order:
8
0.4
0.6
0.8
1
0.6
0.8
1
Condition Number: 5.48e+002
1
0.8
Power Polynomial Functions − Imag Part
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Normalized Frequency (Rad/Sec) − Imag Part
Figure 1. Van der Monde matrix - Normalized Frequency - Orders 0-8
The basic complex Z mapping function, in the nomenclature of this presentation, is as follows:
si = z i = e j*π *(ω i /ω max ) = e j*ω *Δt
(21)
sim = z im = e j*π *m*(ω i /ω max )
(22)
This mapping function maps the positive frequency range to the positive unit circle in the complex plane and
the negative frequency range to the negative unit circle in the complex plane. This effectively yields a real
part of the mapping functions which are cosine terms and an imaginary part which are sin functions. Since
sin and cos functions at different frequencies are mathematically orthogonal, the numerical conditioning of
this mapping function is quite good and does not involve a change in the companion (eigenvalue-eigenvector)
solution formulation (required by the Orthogonal Polynomial methods). The graphical plot of this Van der
Monde matrix for orders 0 through 8 is shown in Figure 2.
Order:
8
Condition Number: 1.01e+000
1
0.8
Z Mapping Functions − Real Part
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
Normalized Frequency (Rad/Sec)
Imaginary:
Order:
8
0.4
0.6
0.8
1
0.6
0.8
1
Condition Number: 1.01e+000
1
0.8
Z Mapping Functions − Real Part
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
Normalized Frequency (Rad/Sec) − Imag Part
Figure 2. Van der Monde matrix - Complex Z Mapped Frequency - Orders 0-8
The condition number for this example matrix is 1.01 (Figure 2) compared to a condition number of 548 for
the normalized frequency example (Figure 1).
3. Case Studies
Three case studies were used to evaluate the low order, frequency domain algorithms, with and without
complex Z mapping of the frequency, compared to the Polyreference LSCF algorithm. The first case study
involves a theoretically generated 15 DOF set of FRF with significant random noise added to the data. This
case has the obvious advantage of having a known answer. Two other case studies are presented based upon
historical data sets used in UC-SDRL to test modal parameter estimation algorithms. The first experimental
case is data taken with impact testing from a circular plate. This data contains repeated and very close modes
and is quite easily analyzed with almost any algorithm. The second experimental case is FRF data taken
from a bridge test using impact testing. This data contains many modes and is very difficult for almost any
algorithm. For comparison, all case studies (and solution methods) utilized low order coefficient
normalization. Additionally, the Consistency Diagrams discarded all non-realistic poles and the Pole
Surface Density plots utilized a lower density threshold of three.
3.1 Analytical 15 DOF Case Study
The 15 DOF analytical case, which used heavy gaussian random noise, demonstrates that each algorithm
accurately identifies the system poles. It is also clear that the Consistency Diagrams for each test case
provide clear indications that consistent pole solutions were calculated. Viewing the Pole Surface Density
Plots shows the same consistent pole estimation (Figures 5-7). This is as expected since the algorithms are
formulated to minimize the influence of gaussian random noise.
Frequency Response Function
200
150
100
Phase
50
0
−50
−100
−150
−200
0
10
20
30
Frequency (Hz)
40
50
Frequency Response Function
−4
10
−5
Magnitude
10
−6
10
−7
10
0
10
20
30
Frequency (Hz)
Figure 3. Analytical Case Study: Typical FRF
40
50
Complex Mode Indicator Function
−2
10
−3
Magnitude
10
−4
10
−5
10
−6
10
0
10
20
30
Frequency (Hz)
40
Figure 4. Analytical Case Study: 3 Reference CMIF
50
RFP−Z − Consistency Diagram
12
11
10
Model Iteration
9
8
7
6
5
4
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
3
2
1
0
5
10
Frequency (Hz)
15
20
15
20
RFP−Z − Pole Surface Density
5
4.5
4
3.5
Zeta (%)
3
2.5
2
9
1.5
Density
1
6
0.5
4
2
1
0
0
5
10
Imaginary (Hz)
Figure 5. Analytical Case Study: Polyreference LSCF Method
PFD − Consistency Diagram
9
8
Model Iteration
7
6
5
4
3
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
2
1
0
5
10
Frequency (Hz)
15
20
15
20
PFD − Pole Surface Density
5
4.5
4
3.5
Zeta (%)
3
2.5
2
9
1.5
Density
1
6
0.5
4
2
1
0
0
5
10
Imaginary (Hz)
Figure 6. Analytical Case Study: PFD Method
PFD−Z − Consistency Diagram
9
8
Model Iteration
7
6
5
4
3
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
2
1
0
5
10
Frequency (Hz)
15
20
15
20
PFD−Z − Pole Surface Density
5
4.5
4
3.5
Zeta (%)
3
2.5
2
9
1.5
Density
1
6
0.5
4
2
1
0
0
5
10
Imaginary (Hz)
Figure 7. Analytical Case Study: PFD-Z Method
3.2 Circular Plate Case Study
The circular plate is a solid steel plate with a welded central hub. This test article is very lightly damped and
due to the structural symmetry, there are several repeated roots and most multi-reference modal parameter
estimation algorithms work well on this data set, as well. Comparing the Consistency Diagrams and the Pole
Surface Density plots (Figures 10-12) for the three algorithms (RFP-Z, PFD-Z, and PFD) again show clear
consistent pole solutions. Zooming in on the first repeated pole, Figures 13-15, shows that all algorithms
have reliably identified the repeated pole, but for this case, the complex Z mapped solutions (Polyreference
LSCF and PFD-Z) appear to have a greater variance (as viewed in the s-plane) than the traditional, low order
frequency domain solution (PFD), but as yet, no conclusion as to whether this is a characteristic of the
algorithms is drawn.
Frequency Response Function
200
150
100
Phase
50
0
−50
−100
−150
−200
0
500
1000
1500
Frequency (Hz)
2000
2500
2000
2500
Frequency Response Function
−4
10
−5
10
−6
Magnitude
10
−7
10
−8
10
−9
10
−10
10
0
500
1000
1500
Frequency (Hz)
Figure 8. Circular Plate Case Study: Typical FRF
Complex Mode Indicator Function
−3
10
−4
10
−5
Magnitude
10
−6
10
−7
10
−8
10
−9
10
0
500
1000
1500
Frequency (Hz)
2000
Figure 9. Circular Plate Case Study: 7 Reference CMIF
2500
Model Iteration
RFP−Z − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
500
1000
1500
Frequency (Hz)
RFP−Z − Pole Surface Density
2
1.8
1.6
1.4
Zeta (%)
1.2
1
0.8
38
0.6
Density
0.4
20
0.2
10
0
1
0
500
1000
Imaginary (Hz)
Figure 10. Circular Plate Case Study: Polyreference LSCF Method
1500
Model Iteration
PFD − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
500
1000
1500
Frequency (Hz)
PFD − Pole Surface Density
2.5
2
Zeta (%)
1.5
1
0.5
Density
38
0
20
10
1
−0.5
0
500
1000
Imaginary (Hz)
Figure 11. Circular Plate Case Study: PFD Method
1500
Model Iteration
PFD−Z − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
500
1000
1500
Frequency (Hz)
PFD−Z − Pole Surface Density
2.5
2
Zeta (%)
1.5
1
0.5
Density
31
20
10
1
0
−0.5
0
500
1000
Imaginary (Hz)
Figure 12. Circular Plate Study: PFD-Z Method
1500
Model Iteration
RFP−Z − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
360
361
362
363
364
Frequency (Hz)
365
366
367
RFP−Z − Pole Surface Density
1.5
1.4
1.3
1.2
Zeta (%)
1.1
1
0.9
38
0.8
Density
0.7
20
0.6
10
0.5
360
1
361
362
363
364
Imaginary (Hz)
365
Figure 13. Circular Plate Case Study: Polyreference LSCF Method
366
367
Model Iteration
PFD − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
360
361
362
363
364
Frequency (Hz)
365
366
367
PFD − Pole Surface Density
1.5
1.4
1.3
1.2
Zeta (%)
1.1
1
0.9
38
0.8
Density
0.7
20
0.6
10
0.5
360
1
361
362
363
364
Imaginary (Hz)
365
Figure 14. Circular Plate Case Study: PFD Method
366
367
Model Iteration
PFD−Z − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
360
361
362
363
364
Frequency (Hz)
365
366
367
PFD−Z − Pole Surface Density
1.5
1.4
1.3
1.2
Zeta (%)
1.1
1
0.9
30
Density
25
0.8
0.7
20
15
0.6
10
5
1
0.5
360
361
362
363
364
Imaginary (Hz)
Figure 15. Circular Plate Study: PFD-Z Method
365
366
367
3.3 Bridge Case Study
The final test case involves a 15 reference impact test on a civil engineering bridge structure. This structure is
notoriously difficult to fit using any traditional algorithm and to date the best results have been obtained by
using spatial domain modal parameter estimation methods such as CMIF and EMIF [26,27] . Figure 16
shows a typical FRF for this system. As is evident, the data is not particularly noisy but neither is it an ideal
laboratory measurement. The CMIF plot, Figure 17, shows that there are several modes in the selected band
that are difficult to separate even when using spatial domain parameter estimation methods like CMIF.
Comparing the results, Figures 18-20, shows that only the low order algorithm with complex Z mapping
(PFD-Z) produced a satisfyingly indicative Consistency Diagram. On the other hand, the Pole Surface
Density plots for all three cases were much more revealing. Nonetheless, only the low order, and hence more
spatially sensitive, algorithms identified the very difficult group of modes between 10 Hz and 15 Hz.
Frequency Response Function
200
150
100
Phase
50
0
−50
−100
−150
−200
0
10
20
30
Frequency (Hz)
40
50
40
50
Frequency Response Function
−5
10
−6
10
Magnitude
−7
10
−8
10
−9
10
−10
10
0
10
20
30
Frequency (Hz)
Figure 16. Bridge Case Study: Typical FRF
Complex Mode Indicator Function
−3
10
−4
10
−5
Magnitude
10
−6
10
−7
10
−8
10
−9
10
−10
10
0
10
20
30
Frequency (Hz)
40
Figure 17. Bridge Case Study: 15 Reference CMIF
50
Model Iteration
RFP−Z − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
10
15
Frequency (Hz)
20
25
20
25
RFP−Z − Pole Surface Density
20
15
Zeta (%)
10
5
7
Density
6
5
0
4
3
2
1
−5
5
10
15
Imaginary (Hz)
Figure 18. Bridge Case Study: Polyreference LSCF Method
Model Iteration
PFD − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
10
15
Frequency (Hz)
20
25
20
25
PFD − Pole Surface Density
20
15
Zeta (%)
10
5
Density
21
15
0
10
5
1
−5
5
10
15
Imaginary (Hz)
Figure 19. Bridge Case Study: PFD Method
Model Iteration
PFD−Z − Consistency Diagram
cluster
pole & vector
pole
frequency
conjugate
non realistic
1/condition
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
10
15
Frequency (Hz)
20
25
20
25
PFD−Z − Pole Surface Density
20
15
Zeta (%)
10
5
Density
21
15
0
10
5
1
−5
5
10
15
Imaginary (Hz)
Figure 20. Bridge Case Study: PFD-Z Method
4. Summary and Conclusions
The performance of the PFD and PFD-Z algorithms is very consistent with the performance of the
Polyreference LSCF method. Some difference in the variance of the complex modal frequency estimates is
notable in some data cases but is not a significant difference. In general, the complex Z mapping of the
generalized frequency variable appears to be an important contribution to the implementation of frequency
domain algorithms, both high and low order. With respect to the Bridge Case Study, the combination of the
spatial information in the low order frequency domain method with the improved numerical characteristics
of the complex Z mapping gave quite good results, better than the results obtained by any other method as
viewed from the pole surface density plot. Further studies are necessary to completely understand the subtle
effects of the complex Z mapping of the generalized frequency variable approach but the superior numerical
properties with no apparent notable problems provides another powerful tool that can be used in estimating
modal parameters.
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