# Produce S. Dietz's "frequency response mode" for Multibody 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 (Xn=[Xr,X1,X2,....,Xk]).
Dn
Diagonals are the eigenvalues associated with Xn.
2. Perform a "special" static analysis without constraint in FE:
( k - l * m ) * Xf = Fa
where,
k
System stiffness matrix.
l
A scalar, usually half of the first nonzero frequency of the free-free normal analysis in step 1.
m
System mass matrix (Lumped mass).
Fa
Attachment forces at junction nodes, not necessarily unit loads.
Xf
The "frequency respond mode" associated with l and Fa (the displacement from the "special" static analysis).
3. Form modal stiffness matrix KHAT as:
KHAT=	|     Dn  Xn'*Fb |
| Fb'*Xn  Xf'*Fb |
where Fb is the balancing force and is defined 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 following 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 orthoginalized modes Y:
Y=X*N

This is the mode set of rigid body modes, free-free normal modes, and S.Dietz's "frequency response mode" modes.

The generalized mass and stiffness matrix are:
M=N'*MHAT*N=I
K=N'*KHAT*N=D

Y, D, and m are used to calculate the flexible MB input file.