# Produce Craig-Chang Modes for MB Analysis

1. Perform a normal analysis without constraint (free-free).
m
System mass matrix (Lumped mass).
Xr
Rigid body modes (mass orthonormalized (Xr'*m*Xr=I))
Xn
Free-Free normal modes including the rigid body modes.
(Xn=[Xr,X1,X2,....,Xk]).
Dn
Diagonals are the eigenvalues associated with Xn.
2. Form the equilibrated load matrix Fe:
Fe = P*Fa
Where,
P=I-m*Xr*Xr'

and Fa has unit force along each DOF of the interface nodes.

3. Perform a static analysis without constraint and with (1) restraint to remove the rigid DOF. Allow all elastic deformation subcases (2) where columns of Fe are applied at each subcase (i.e. k*Xa=Fe).
Xa
Inertial relieve attachment modes, or displacement of static analysis.
4. Form modal stiffness matrix KHAT as:
KHAT	|     Dn  Xn'*Fe |
| Fe'*Xn  Xa'*Fe |
and modal mass matrix MHAT as:
MHAT=X'*m*X
where X is the combined mode:
X=[Xn Xa]
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 orthoginalized modes Y:
Y=X*N
This is the mode set of rigid body modes, free-free normal modes, and the residual inertial relieve attachment 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.