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本帖最后由 VibInfo 于 2016-4-21 14:22 编辑
The LMS PolyMAX method is a further evolution of
the least-squares complex frequency-domain (LSCF)
estimation method. That method was fi rst introduced to
fi nd initial values for the iterative maximum likelihood
method [7]. The method estimates a so-called commondenominator
transfer function model [8]. Quickly it
was found that these “initial values” yielded already
very accurate modal parameters with a very small
computational effort [7, 9, 10]. The most important
advantage of the LSCF estimator over the available and
widely applied parameter estimation techniques [2] is the
fact that very clear stabilization diagrams are obtained.
A thorough analysis of different variants of the commondenominator
LSCF method can be found in [10]. A
complete background on frequency-domain system
identifi cation can be found in [11].
It was found that the identifi ed common-denominator
model closely fi tted the measured frequency response
function (FRF) data. However, when converting this
model to a modal model by reducing the residues to a
rank-one matrix using the singular value decomposition
(SVD), the quality of the fi t decreased [9].
Another feature of the common-denominator
implementation is that the stabilization diagram can only
be constructed using pole information (eigenfrequencies
and damping ratios). Neither participation factors nor
mode shapes are available at fi rst instance [12]. The
theoretically associated drawback is that closely spaced
poles will erroneously show up as a single pole.
These two reasons provided the motivation for a
polyreference version of the LSCF method, using a socalled
right matrix-fraction model. In this approach, also
the participation factors are available when constructing
the stabilization diagram. The main benefi ts of the
polyreference method are the facts that the SVD step
to decompose the residues can be avoided and that
closely spaced poles can be separated. The method was
introduced in [12, 13]. Here we briefl y review the theory.
Figure 7: Comparison of the measured FRFs (green/grey)
with FRFs synthesized from the identifi ed modal model
(black/red). (Top) Sensor at the wing tip; (Bottom) Sensor at
the back of the plane.
LMS PolyMAX: Theoretical
Foundation
Data model
Just like the FDPI (Frequency-domain direct parameter
identifi cation) method1 [4,5], the LMS PolyMAX method
uses measured FRFs as primary data. Time-domain
methods, such as the polyreference LSCE method2 [6],
typically require impulse responses (obtained as the
inverse Fourier transforms of the FRFs) as primary
data. In the LMS PolyMAX method, following so-called
right matrix-fraction model is assumed to represent the
measured FRFs:
p p
[H(jw)]=sum (z ^r[beta_r])*(sum (z ^r[alpha_r])^-1 (1)
r=0 r=0
where is the matrix containing the FRFs
between all m inputs and all l outputs; are the
numerator matrix polynomial coeffi cients; are
the denominator matrix polynomial coeffi cients and is
the model order. Please note that a so-called -domain
model (i.e. a frequency-domain model that is derived from
a discrete-time model) is used in (1), with:
z=exp(-j.w.deltt) (2)
where is the sampling time.
Equation (1) can be written down for all values of the
frequency axis of the FRF data. Basically, the unknown
model coeffi cients are then found as the Least-
Squares solution of these equations (after linearization).
More details about this procedure can be found in[12,13].
Poles and modal participation factors
Once the denominator coeffi cients
Poles and modal participation factors
are determined,
the poles and modal participation factors are retrieved
as the eigenvalues and eigenvectors of their companion
matrix:
(3)
The modal participation factors are the last m rows
of ; the matrix contains the
(discrete-time) poles on its diagonal. They are
related to the eigenfrequencies [rad/s] and damping
ratios [-] as follows ( •* denotes complex conjugate):
(4)
This procedure is similar to what happens in the
time-domain LSCE method and allows constructing a
stabilization diagram for increasing model orders
and
using stability criteria for eigenfrequencies, damping
ratios and modal participation factors.
Mode shapes
Although theoretically, the mode shapes could be derived
from the model coeffi cients , we proceed in a
different way.
The mode shapes can be found by considering the socalled
pole-residue model:
(5)
where n is the number of modes; denotes complex
conjugate transpose of a matrix; are the mode
shapes; are the modal participation factors
and are the poles (4). are respectively the
lower and upper residuals modeling the infl uence of the
out-of-band modes in the considered frequency band.
The interpretation of the stabilization diagram yields a
set of poles and corresponding participation factors
Since the mode shapes and the lower and upper
residuals are the only unknowns, they are readily obtained
by solving (5) in a linear least-squares sense. This second
step is commonly called least-squares frequency-domain
(LSFD) method [2,3]. The same mode-shape estimation
method is normally also used in conjunction with the
time-domain LSCE method.
References
[1] VAN DER AUWERAER H., C. LIEFOOGHE, K.
WYCKAERT AND J. DEBILLE. Comparative study of excitation
and parameter estimation techniques on a fully equipped car.
In Proceedings of IMAC 11, the International Modal Analysis
Conference, 627–633, Kissimmee (FL), USA, 1–4 February 1993
[2] HEYLEN W., S. LAMMENS AND P. SAS. Modal Analysis
Theory and Testing. Department of Mechanical Engineering,
Katholieke Universiteit Leuven, Leuven, Belgium, 1995.
[3] LMS INTERNATIONAL. The LMS Theory and Background
Book, Leuven, Belgium, 2000.
[4] LEMBREGTS F., J. LEURIDAN, L. ZHANG AND H.
KANDA. Multiple input modal analysis of frequency response
functions based direct parameter identifi cation. In Proceedings of
IMAC 4, the International Modal Analysis Conference, 589–598,
Los Angeles (CA), USA, 1986.
[5] LEMBREGTS F., R. SNOEYS AND J. LEURIDAN.
Application and evaluation of multiple input modal parameter
estimation. International Journal of Analytical and Experimental
Modal Analysis, 2(1), 19–31, 1987.
[6] BROWN D.L., R.J. ALLEMANG, R. ZIMMERMAN AND
M. MERGEAY. Parameter estimation techniques for modal
analysis. Society of Automotive Engineers, Paper No. 790221,
1979.
[7] GUILLAUME P., P. VERBOVEN AND S. VANLANDUIT.
Frequency-domain maximum likelihood identifi cation of modal
parameters with confi dence intervals. In Proceedings of ISMA
23, the International Conference on Noise and Vibration
Engineering, Leuven, Belgium, 16–18 September 1998.
[8] GUILLAUME P., R. PINTELON AND J. SCHOUKENS.
Parametric identifi cation of multivariable systems in the
frequency domain - a survey. In Proceedings of ISMA 21, the
International Conference on Noise and Vibration Engineering,
1069–1082, Leuven, Belgium, 18–20 September 1996.
[9] VAN DER AUWERAER H., P. GUILLAUME, P.
VERBOVEN AND S. VANLANDUIT. Application of a faststabilizing
frequency domain parameter estimation method.
ASME Journal of Dynamic Systems, Measurement, and Control,
123(4), 651–658, 2001.
[10] VERBOVEN, P. Frequency domain system identifi cation for
modal analysis. PhD Thesis, Vrije Universiteit Brussel, Belgium,
2002.
[11] PINTELON R. AND J. SCHOUKENS. System
Identifi cation: a Frequency Domain Approach. IEEE Press, New
York, 2001.
[12] GUILLAUME P., P. VERBOVEN, S. VANLANDUIT, H.
VAN DER AUWERAER AND B. PEETERS. A poly-reference
implementation of the least-squares complex frequency-domain
estimator. In Proceedings of IMAC 21, the International Modal
Analysis Conference, Kissimmee (FL), USA, February 2003.
[13] PEETERS B., P. GUILLAUME, H. VAN DER AUWERAER,
B. CAUBERGHE, P. VERBOVEN AND J. LEURIDAN.
Automotive and aerospace applications of the LMS PolyMAX
modal parameter estimation method. In Proceedings of IMAC
22, Dearborn (MI), USA, January 2004.
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