An alternate reduction procedure was developed by Kammer [3]. This procedure uses the FEM mode shapes as the interpolation functions to estimate the motion at the omitted DOF. This method is based on partitions of the FEM mode shapes instead of stiffness and mass matrices as used by the Static and IRS reduction methods. Writing the displacements as functions of the FEM mode shapes and a modal scale factor:
(2.3.4-1)
The upper and lower partitions of (2.3.3-1) can be written as:
(2.3.4-2a)
(2.3.4-2b)
Premultiply (2.3.4-2b) by faT:
(2.3.4-3)
Premultiply (2.3.4-3) by :
(2.3.4-4)
Substituting (2.3.4-4) into (2.3.4-2a) provides the relationship between the A-set and O-set DOF:
(2.3.4-5)
The transformation matrix for the modal reduction is:
(2.3.4-6)
Note that (2.3.4-6) requires that be invertible. This is true whenever enough
DOF are chosen so that the modes are linearly independent. In practice
the number of instrumented DOF is typically much larger than the number
of mode shapes, so this criterion is nearly always met. However, if not
enough ASET DOF or the ASET DOF that result in linearly dependent modes
are selected, the modal reduction can fail.