The discussion could get very detailed. I can go into the subject with more if desired. There are a number of applications in structural dynamics that utilize the mass, stiffness, and mode shape matrices along with the resonance frequencies of a vibrating structure. The one I'm focused on at the moment is comparing mode shapes from a simulation model (Finite Element Model) to mode shapes obtained experimentally for a given structure.
When I worked at NASA JSC ('73 - '92) my colleague Walt West and I developed FEM, test and analytical methods that utilized these matrices along with resonance frequencies to compare the FEM mode shape and frequency results to the similar modal test results. We could locate the elements of the FEM that needed adjustment to match the test data. This was used to validate the FEM. We calculated the residual force and displayed the Orbiter component mesh with color coded elements representing the residual force on each element. A red spot covering a small group of elements identified the location that needed adjustment of element properties in order to match the test data.
The residual force is obtained by enforcing the test mode shape (set of displacements) for a selected frequency, then back-calculating the force vectors (for FEM and Test models) required to enforce that mode shape. The residual force is found by simply computing the difference between the two sets of force vector components. I can provide a little more detail on why that works if desired.
We also used the residual force for force vectors calculated for the experimental modal tests performed before launch and after two or three flights (comparing pre-test mode shapes to post-test mode shapes). In this case the red spot showed location of damage to internal structural components. This was needed because much of the internal structure covered by Orbiter tile was unavailable to visual inspection. At that time we were using the SDRC I-deas FEM-test system. That company has sold a couple of times and the software is now owned by Siemens. The product is overly expensive for this limited application, besides Autodesk 360 is too attractive to not pursue if we find potential (we would use the application for our customers internationally).
The problem is actually more involved than outlined here, because we may need to do Guyan reductions and coordinate transformations that further utilize the FEM stiffness matrix and mode shape matrices. But this is the overview. I can do more detail if needed.
Thanks for your interest and response.
Bob Coleman
Fusion
