Expansion

For many vibration measurements, the low spatial resolution (limited number of sensors, hidden or inaccessible parts, …) limits uses of test results. Expansion, a name from the vibration community for methods also known as state-estimation in controls, seeks

  • To estimate states (motion at all model DOF)
  • Find an estimation trade-off between experimental and modeling errors
  • Provide a hybrid model combining test and analysis that can be used for further predictions

Developments in SDT 7.5 have notably improved the accessibility of these methods by better GUI integration, handling of superelements generated from very large FEM, ability to deploy the method in industrial contexts. The following gives a few recent published illustrations.

A first objective of expansion methods is the estimation of experimental shape motions at all model DOFs.

Experimental shape [2]
Expanded shape [2]

The Minimum Dynamics Residual Expansion method [1] used in SDT introduces a cost function composed of two errors

  • a test error evaluating the gap between the expanded shape and the experimental shape,
  • a model error evaluating the amount of force required for the model to deform according to the expanded shape.

The two errors are typically balanced to have the expanded shape follow test enough (low enough test error containing mainly noise) but not too much (high model error and nonphysical shape).

Balance between test and model error [2]
Test error [2]
Model error [2]

At this stage, a second purpose of expansion is to perform model updating [3,4]. The spatial distribution of the model error gives insights on which parameters should be tuned. Optimal parameter values are found by minimizing the model error after expansion.

Parameter optimization minimizing the model error after expansion

A third illustration [2] deals with Experimental Modal Analysis from which a modal model defined at model DOFs can be built through expansion. The modal model obtained is thus spatially continuous, which is very helpful to perform Structural Dynamic Modification or to couple the modal model with other dynamic systems.

SDM after expansion procedure [2]

SDTools thanks Hitachi Astemo for the illustrations used in this page

References

[1] Expansion in structural dynamics: a perspective gained from success and errors in test/FEM twin building 

[2] A Structural Dynamics Modification Strategy based on Expanded Squeal Operational Deflection Shapes 

[3] MDRE: An efficient expansion tool to perform model updating from squeal measurements 

[4] Model Quality Quantification And Improvement Based On Operational Squeal Deflection Shapes ExpansionÂ