2.4.           TAM Accuracy and Robustness

The matrix reduction procedures described in Sections 2.3.1 through 2.3.5 reduce the FEM stiffness and mass matrices to the TAM DOF. The intent of the TAM matrix reduction is to retain as much information as possible from the FEM. However, some information is always lost when transforming from a large number of DOF (FEM) to a reduced number of DOF (TAM).

The amount and type of information retained in the TAM is very dependent on the matrix reduction method. Whether or not a TAM matrix adequately represents the FEM depends not only on which DOF are retained in the TAM but also on the TAM reduction method.

There are two important measures of TAM adequacy: accuracy and robustness. A TAM is “accurate” if the TAM mass and stiffness matrices can closely recreate the modal frequencies and mode shapes of the FEM. If the TAM matrices cannot recreate the FEM modes, the TAM matrices are not an equivalent representation of the FEM and cannot be used with confidence to evaluate the results of the modal survey test or to perform test/analysis correlation.

A critical step in creating a TAM is to determine if the TAM is “accurate.” In the context of orthogonality measures, which are standard in the aerospace industry, “accuracy” means that the reduced mass matrix accurately reflects the full model. This is measured by the psuedo-orthogonality and cross-orthogonality matrices. For the purpose of this chapter we assume that both the TAM and FEM modes are mass normalized with respect to their mass matrices. This implies that the self-orthogonality of the FEM and TAM mode shapes with respect to the FEM and TAM mass matrices respectively is identity. The psuedo-orthogonality is the orthogonality of the FEM mode shapes with respect to the TAM mass matrix. This is not necessarily equal to the identity matrix because the FEM mode shapes were not calculated based on the TAM mass matrix. The definition is as follows:

                              (2.4-1)

where

          = FEM mode shapes calculated with full FEM matrices at the TAM A‑set DOF

             = TAM reduced A‑set mass matrix

The accuracy of the TAM can be evaluated based on how close the psuedo-orthogonality matrix is to the identity matrix. Typically values of 95% or higher on the diagonal com­bined with values of 5% or lower on the off-diagonals indicate an acceptable TAM.

An alternative to the psuedo-orthogonality is the cross-orthogonality. This is a measure of how similar the TAM mode shapes are to the FEM mode shapes and is calculated as follows:

                              (2.4-2)

where

          = TAM mode shapes for the TAM A‑set DOF

Typically the psuedo-orthogonality matrix is a more direct measure of the acceptability of a TAM reduction, since it more closely reflects the calculations that will be performed after the extraction of test data. These include a TEST-TEST self-orthogonality calcula­tion and a FEM-TEST cross-orthogonality calculation[4] using the TAM mass matrix. Neither of these calculations involves the TAM modes. However, a good psuedo-orthogonality will typically result in a good cross-orthogonality, so these measures are somewhat equivalent. Furthermore, it may be advantageous to use the TAM modes during the posttest correlation process. An example of this is when mode shapes are tuned using mode shape or cross-orthogonality sensitivities. In this case the sensitivity calculation is much less expensive using the TAM mode shapes, so in this case it is critical that these match the FEM mode shapes as closely as possible.

Another measure of the similarity of two sets of modes is the Modal Assurance Criteria (MAC). A MAC matrix is similar to an orthogonality matrix, except that it does not include the mass matrix.

                                        (2.4-3)

The MAC can be calculated between FEM modes and FEM modes, in which case it is called the self MAC; or it can be calculated between two sets of modes such as the TAM and the FEM modes, in which case it is referred to as a cross MAC. Since the MAC does not require a reduced mass or stiffness matrix, it is much simpler to calcu­late[5]. However, in situations where the goal is to find an accurate reduced model the MAC is not a good measure of accuracy. It is included here for completeness.

The second important measure of TAM adequacy is robustness. A TAM is “robust” if it is not overly sensitive to differences between the pretest FEM and the test article. A robust TAM will facilitate test-analysis correlation, while a sensitive TAM will exag­gerate errors between test and analysis.

The robustness of a TAM is dependent on the number and type of vectors used in developing the TAM transformation matrix. For example, the transformation matrix for a Static TAM is created by a static solution that determines unit displacement vectors at each A-set (retained) DOF. However, the transformation matrix for a Modal TAM is based on the FEM mode shapes. Since the number of TAM A-set DOF is usually much larger than the number of FEM modes, a Static TAM is often more robust than a Modal TAM—i.e., a static TAM is less sensitive to errors between the pretest FEM and the test article. However, the Modal TAM uses mode shapes instead of static vectors and is therefore more accurate than a Static TAM when the modes of the pretest FEM closely match the modes of the test article.

Modal and Hybrid TAMs are usually the most accurate matrix reduction methods. These procedures can produce exact TAMs—i.e., the TAM stiffness and mass matri­ces will exactly recreate the FEM modal frequencies and mode shapes. However, a Modal TAM is also the least robust TAM method and is very sensitive to differences between pretest FEM and test article. A Hybrid TAM is more robust than a Modal TAM because it includes information from the unit displacement static vectors. Static and IRS TAMs are usually more robust than Modal and Hybrid TAMs but are often less accurate. In addition, the IRS TAM is sometimes sensitive to mass errors because of the mass matrix inverse in (2.3.3-2). Reference [10] provides more information on IRS TAM performance and sensitivity and shows that the eigenvalues of the problem  are critical in determining the robustness of the IRS TAM and, by implication, all TAM methods other than static.

Because of the differences in accuracy and robustness, there is no single TAM method which is “best” for all problems. A Static TAM is the easiest and most robust, but least accurate, method. The alternative TAM methods are all more accurate than a static TAM, but can all suffer from poor robustness. The “best” TAM method is therefore problem dependent.

In general, it is not easy to determine the robustness of the TAM a priori. Since the static TAM is the most robust and accuracy can be determined a priori, the goal of the pretest analysis effort usually is to develop an accurate static TAM. In cases where this is not feasible, one of the alternative TAMs can be used. Additional information and example cases evaluating TAM accuracy and robustness are provided in [5].

In summary, five methods are available for creating test-analysis models:

Each method has strengths and weaknesses. Static reduction is the easiest, most robust and most economical method but does not always result in a sufficiently accu­rate reduced model. The alternatives all improve accuracy at the cost of robustness. The best method to use on a particular structure will depend on the dynamic charac­teristics and modeling uncertainty.