Matrix reduction procedures for test/analysis models are based on transformation methods of the form:
(2.3-1)
where
A = original (large) matrix
B = new (reduced) matrix
G = transformation matrix
A simple example of a matrix transformation is the conversion of a finite element stiffness matrix from elemental to global coordinates. In this example, the transformation matrix G contains the direction cosines between the element and global coordinate systems. This transformation is exact since the number of degrees of freedom is the same for both the original and transformed matrices.
The matrix reduction methods for test-analysis models must satisfy much greater demands than the above simple example. The major challenge is the matrix size difference between the FEM and the TAM. The number of DOF in a finite element model is usually very large (typically thousands to millions of DOF). However, the number of accelerometers in a modal survey test is usually only 10 to 500. The TAM reduction method must be able to accurately estimate or predict the motion of all non-instrumented DOF using a limited set of “known” values at the accelerometer locations. Since the ratio between predicted versus “known” DOF is so large, the prediction procedure must be very precise in order to accurately reduce the FEM matrices to the TAM DOF.
There are two keys to the accuracy of this prediction. The first is the selection of TAM DOF, and the other is the selection of the interpolation functions used to “guess” the motion of the non-instrumented DOF. Five different reduction methods, each corresponding to a different choice of interpolation functions, are presented in this chapter.