Produce Craig-Chang Modes for MB Analysis
-
Perform a normal analysis without constraint
(free-free).
Read results.
- 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.
-
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.
-
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
).Read results.- Xa
- Inertial relieve attachment modes, or displacement of static analysis.
-
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.
-
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.