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ARTeMIS Modal Pro - Stochastic Subspace Identification
SSI-UPCX / SSI-UPC / SSI-PC / SSI-CVA

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 un-seen accuracy of the physical parameters and nearly no noise modes.  The techniques available are:    

  • Extended Unweighted Principal Component (SSI-UPCX)

  • Unweighted Principal Component (SSI-UPC)

  • Principal Component (SSI-PC)

  • Canonical Variate Analysis (SSI-CVA)

  • Unweighted Principal Component Merged Test Setups (SSI-UPC-Merged)

 

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, semi-automatic 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 least-squares or the Crystal Clear SSI® solver.

  • Estimation of uncertainties of extracted modal parameters (SSI-UPCX)

  • 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 data-driven 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 broad-banded (white noise). Using these methods in presence of deterministic signals introduce bias in the modal parameter estimation.

Less random errors

Low-order model estimator - SSI algorithms are born linear least-squares 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 non-linear least squares problem with a linear least-squares fitting problem. This is an often seen approximation when fitting e.g. polynomial matrix fractions. In order for this approximation to work, a high-order polynomial order is needed. Since this leads to the use of many parameters compared to a low-order technique, the uncertainties of the high-order 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 SSI-UPCX 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 screen-shots 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 SSI-UPCX 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 SSI-UPCX 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.  

 

 


Structural Vibration Solutions A/S • Tel: +45 9635 4422 • Fax: +45 9635 4575 • svibs@svibs.com




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