ARTeMIS Modal Pro  Stochastic Subspace Identification
SSIUPCX / SSIUPC /
SSIPC / SSICVA
ARTeMIS Modal
includes up to five time domain modal analysis techniques. They are all of the
data driven Stochastic Subspace Identification (SSI) type and all implementing
the powerful Crystal Clear SSI feature. This feature result in extremely clear
stabilization diagrams with unseen accuracy of the physical parameters and
nearly no noise modes. The
techniques available are:

Extended Unweighted
Principal Component (SSIUPCX)

Unweighted
Principal Component (SSIUPC)

Principal
Component (SSIPC)

Canonical Variate
Analysis (SSICVA)

Unweighted
Principal Component Merged Test Setups (SSIUPCMerged)
These techniques
estimate the modal parameters directly from the raw measured time
series. The SSI techniques incorporate effective ways of dealing
with noise. As a result, the modal parameter estimations are the
most accurate commercially available today. The SSI techniques can
work with closely space and repeated modes with light or heavy
damping. Since they are working in time domain there are no leakage
bias or lack of frequency resolution, see below. As a result, the
modal parameter estimates are asymptotically unbiased. Further, as
the SSI techniques are low model order estimators, the statistical
errors of the modal parameter estimates are extremely small.
Features
The features of these
methods are:

Modal parameter estimation
based on estimation of state space models in time domain.

State space model estimation can be
done on even large channel counts through the use of Projection
Channels.

Projection Channels can be found
manually, semiautomatic or fully automatic.

Very fast estimation of even hundreds
of models of increasing dimension.

Very limited user interaction required.
Can be fully automated. By default, the techniques always start
with an fully automatic estimation of the global modes.

Unbiased modal parameter estimation
using standard leastsquares or the Crystal Clear SSI®
solver.

Estimation of uncertainties of
extracted modal parameters (SSIUPCX)

Handles single or
multiple Test Setups.

Modal parameters
can be extracted in the presence of deterministic (harmonic)
signals from e.g. rotating machinery.

Extraction of
global mode estimates can
be made manually or automatic.

Estimated state
space models can be validated against data in frequency domain.
Both spectral response and prediction errors can be validated.

Estimated state
space models as well as the entire stabilization diagram can be exported in ASCII file format for use
outside ARTeMIS Modal.

All SSI methods
are available in the ARTeMIS Modal Pro version.
Benefits
Unbiased estimation – No systematic
estimation errors
No leakage – The SSI
techniques work in
time domain and are datadriven methods. Since the model estimation
is not relying on any Fourier transformations to frequency domain no
leakage is introduced. Leakage is always introduced when applying
the Fourier transformation and assuming periodicity. Leakage always
results in an unpredictable overestimation of the damping. No
problems with deterministic signals (harmonics) – Since the modal
parameters are extracted directly by fitting parameters to the raw
measured time histories, the presence of deterministic signals, such
as harmonics introduced by rotating machinery, does not create
problems. Harmonics are just estimated as very lightly damped modes.
This is in contrast to frequency domain methods relying on the estimation of half power spectral densities
that all assume that the excitation is broadbanded (white noise).
Using these methods in presence of deterministic signals introduce bias in the modal
parameter estimation.
Less random errors
Loworder model estimator  SSI algorithms
are born linear leastsquares fitting techniques fitting state space
systems with correct noise modeling. This leads to the use of much
smaller model orders than other commercially available high order
model estimators. These estimators are often used to approximate a
nonlinear least squares problem with a linear leastsquares fitting
problem. This is an often seen approximation when fitting e.g.
polynomial matrix fractions. In order for this approximation to
work, a highorder polynomial order is needed. Since this leads to
the use of many parameters compared to a loworder technique, the
uncertainties of the highorder parameter estimates becomes larger.
More parameters are fitted with the same amount of data available,
meaning less independent information per estimated parameter. All
modal parameters are fitted in one operation. All parameters fitted
are taking advantage of the noise cancellation techniques of the
orthogonal projection of SSI. Other commercially available methods
often fit the poles (frequency and damping) first, and then use the
noisy spectral data and the estimated poles to fit the mode shapes
resulting in poor mode shape estimates.
Automatic Mode Estimation
All SSI techniques
includes automatic mode estimation that searches the stable modes in the SSI
Stabilization Diagram. All stable modes of all estimated models of all test
setups are included in the search, and the result is modal estimates of natural
frequencies, damping ratios and modes shapes of high accuracy. The estimates are
presented in terms of both mean values and standard deviations.
Even in the case of closely spaced or repeated modes and multiple
test setups, the automatic mode estimation for SSI works.
Estimation of Modal Uncertainties
Modal
estimation methods that make use of Stabilization Diagrams only present, in
general, the estimated mean values of the modal parameters. Typically,
Stabilization Diagrams show the mean values of the natural frequencies of
the estimated modes with respect to selected model dimensions. In a diagram like this, the search for stable
modes is made on the basis of the mean values. Even if the stabilization is
clear, it is still difficult to assess the level of confidence that can be
associated with each of the presented modes.
In the SSIUPCX
technique it is possible to visualize the uncertainty of the
individual estimates in terms of confidence bounds around the mean
values. An example of this is shown below.The
confidence bounds represented by the grey
horizontal bars clearly show the
uncertainty for each mode in the Stabilization Diagram.
The confidence bounds shown above in the
stabilization diagram can be further extended. Above only the
uncertainty of the natural frequencies are visualized. However, by
selecting a specific state space model (Cursor Model), and selecting
what modes to visualize in the Modes List, then by showing the new
Frequency vs. Damping Diagram, the mean values and confidence
ellipsoids of the natural frequency / damping ratio pairs can be
visualized. The diagram showing the damping as a function of
frequency can be seen below for some of the modes of the cursor
model. The confidence level has been set to 95%:
In general, the additional covariance
information enhance the assessment of the invidual modes. It is
clear to see what modes can be trusted most.
In addition
to the display of confidence bounds in the Stabilization Diagram, the
uncertainty information can also be used to remove modes that are too
uncertain. For each
estimated modal parameter its Coefficient of Variation (CV)
is calculated as the standard deviation divided with the mean value. Two
screenshots are shown below.
On the picture to the
right, the mouse is pointing at a specific mode in the diagram.
As soon as the mouse pointer
is placed over an estimated value, a tooltip appears. The tooltip presents
the mean value, the standard deviation and the Coefficient of Variation for
the natural frequency and the damping ratio. The picture on the left shows
the Modal Indicator properties. These properties now include the maximum
allowed Coefficients of Variations of the natural frequencies and damping
ratios. These Coefficient of Variations are powerful dimensionless modal
indicators that effectively help filter out the modes that have high
uncertainty from the search for stable modes.
In the
picture below, the maximum allowed Coefficient of Variation of the natural
frequency has been set to 0.01 and to 0.1 for the damping ratio. By only
showing the stable modes that are left, it is much easier now to extract the
most accurate modes from the diagram.
All the stable modes left in the above
picture are like in the case of the other SSI techniques used for
estimation a final set of modal parameters. Usually, the stable
modes of different model orders are simply averaged to find the
final estimate. However, in the SSIUPCX technique
we make use of the additional covariance information, in order to do
an even better estimation of the final set of modal parameters.
These final estimates of the modal parameters can be evaluated e.g.
using the Frequency vs. Damping
Diagram. In the case of the SSIUPCX
technique this diagram not only presents the mean values of the
natural frequencies versus damping ratios, but also the confidence
ellipsoids. The ellipsoids are coming from the estimated combined
covariance matrix of the global natural frequency and damping ratio
estimate.
This diagram
gives a quick graphical overview over the estimation accuracy of the
individual modes of the analysis. The larger confidence bounds a
modal parameter has for a certain significance level, the more
uncertain the estimation of the "true modal parameter" is. The
ellipsoids also reveal the correlation between the natural frequency
and the damping ratio. The first five modes e.g. show a reasonable
uncorrelated nature . This is indicated by the "standing" confidence
bound ellipses. In case of the fifth mode , there is correlation
between the estimates of the natural frequency and the damping
ratios. This is indicated by the "tilting" confidence bounds
ellipses. It is tilting to the left which indicates that a higher
frequency estimate at the same time most probable will result in a
lower damping estimate.
