Flexbody Theory

Select from the list of How do I’s... to learn about creating various modes for multi-body analysis.

Procedures that involve finite element analysis are highlighted in red. Data that is obtained by the finite element analysis are highlighted in green.

1.	Perform a normal analysis with fixed interface B.C.

2.	Read results.

m	System mass matrix (Lumped mass).
Xn	Fixed-interface modes (displacement of normal analysis).
Dn	Diagonals are the eigenvalues from the normal analysis.

3.	Perform a static analysis with subcases where unit displacement along each DOF of the interface nodes is applied at each subcase.

4.	Read results.

Xc	Constraint modes (displacement of static analysis).
Fc	Nodal forces at B.C. from static analysis.

5.	Form modal mass matrix MHAT:

MHAT=X'*m*X

where X=[Xn, Xc]

Form modal stiffness matrix KHAT:

KHAT = |Dn 0|
| 0 Fc|

6.	Solve the eigen problem to obtain N and D:

KHAT*N=MHAT*N*D

7.	Transform the X to orthoginalized modes Y:

Y=X*N

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.

1.	Perform a normal analysis without constraint (free-free).

Read results.

m	System mass matrix (Lumped mass).
Xr	Rigid body modes (mass orthonormalized (Xr'mXr=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.

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.

This procedure is without the Guyan reduction procedure and without the special static analysis.

1.	Perform a normal analysis without constraint (free-free).

2.	Read results.

m	System mass matrix (Lumped mass).
Xn	Free-Free normal modes including the rigid body modes.
Xe	Extended Free-Free normal modes including Xn and a number of the higher frequency Free-Free normal modes.
Dn	Diagonals are the eigenvalues associated with Xn.
De	Diagonals are the eigenvalues associated with Xe. De is a superset of Dn.

3.	Calculate the frequency response modes:

Xf = Xe * [( De - l * I )^(-1)] * Xe' * Fa

where:

l	A scalar, usually half of the first nonzero frequency of the free-free normal analysis in step 1.
I	Identity matrix of the same size as De.
Fa	Attachment forces at junction nodes, not necessarily unit loads.

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 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 orthogonalized modes Y:

Y=X*N

This is the mode set of rigid body modes, free-free normal modes, and the "frequency response mode" modes from the modified procedure.

The generalized mass and stiffness matrixes are:

M=N'*MHAT*N=I

K=N'*KHAT*N=D

6.	Y, D, and m are used to calculate the flexible MB input file.