Produce S. Dietz's "frequency response mode" for MB Analysis
-
Perform a normal analysis without constraint
(free-free).
Read results.
- 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.
-
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.
Read results.- Xf
- The "frequency respond mode" associated with l and Fa (the displacement from the "special" static analysis).
-
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]
-
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.
-
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.