Produce Craig-Chang Modes for MB Analysis

  1. 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.
  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).
    Read results.
    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.