# Perform a Modified Procedure of Producing S.Dietz's "frequency response mode" for MB Analysis

This procedure is without the Guyan reduction procedure and without the special static analysis.
1. Perform a normal analysis without constraint (free-free).
m
System mass matrix (Lumped mass).
Xn
Free-Free normal modes including the rigid body modes.
Xe
Extended Free-Free normal modes including Xn and a number of the higher frequency Free-Free normal modes.
Dn
Diagonals are the eigenvalues associated with Xn.
De
Diagonals are the eigenvalues associated with Xe. De is a superset of Dn.
3. Calculate the frequency response modes:
Xf = Xe * [( De - l * I )^(-1)] * Xe' * Fa
where,
l
A scalar, usually half of the first nonzero frequency of the free-free normal analysis in step 1.
I
Identity matrix of the same size as De.
Fa
Attachment forces at junction nodes, not necessarily unit loads.
Form modal stiffness matrix KHAT as:
Fb = Fa + l*m*Xf
Form modal mass matrix MHAT as:
MHAT=X'*m*X
where X is the combined mode:
X=[Xn Xf]
4. Orthogonalize X by solving the eigen problem:
KHAT*N=MHAT*N*D

If X is not independent, then one of the following occurs:

The eigenvalues/vectors are complex

Some highest eigenvalues are infinite

Extra zero eigenvalue rigid body modes

In either case, the corresponding modes can be filtered out so this step removes dependent modes as well.

5. Transform X to orthogonalized modes Y:
Y=X*N

This is the mode set of rigid body modes, free-free normal modes, and the "frequency response mode" modes from the modified procedure.

The generalized mass and stiffness matrixes are:
M=N'*MHAT*N=I
K=N'*KHAT*N=D
6. Y, D, and m are used to calculate the flexible MB input file.