Produce S. Dietz's "frequency response mode" for Multibody Analysis

  1. 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.
  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.
    Read results.
    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.