

The CraigBampton Method
Scott Gordon
NASA Goddard Space Flight Center
May 6, 1999 (updated January 2008)
I. Background
Who is Craig Bampton?
Coupling of Substructures for Dynamic Analysis
by Roy R. Craig Jr. and Mervyn C. C. Bampton
AIAA Journal, Vol. 6, No. 7, July 1968
(Link is to a Word document directly from Mr. Craig's website)
What is the CraigBampton Method?
 Method for reducing the size of a finite element model.
 Combines motion of boundary points with modes of the structure assuming the boundary points are held fixed.
 Similar to other reduction schemes
Guyan Reduction
Modal Decoupling
CraigBampton
Why is the CB Method Used?
 Allows problem size to be reduced
 Accounts for both mass and stiffness (unlike Guyan reduction)
 Problem size defined by frequency range
 Allows for different boundary conditions at interface (unlike modal decoupling)
 Example Problem Model (see below)

Spacecraft Model: 
270 DOF’s
K,M = 270 x 270
10 Modes up to 50 Hz
Single Boundary grid at interface 

CB Reduction:
to 50 Hz 
16 DOF (6 i/f + 10 Modes)
K,M = 16 x 16 
II. CraigBampton Theory
 Equation of motion (ignoring damping)

(1) 
 The CraigBampton transform is defined as:

(2) 
Where:
u_{b} = boundary DOFs
u_{L} = internal (leftover) DOFs
= rigid body vector
= fixed base mode shapes
q = modal DOFs
The 2 X 2 matrix in Equation 2 is the CraigBampton transformation matrix.
 Combining equations (1) and (2) and premultiplying by

(3) 
 Define the CB mass and stiffness matrices as

(4) 

(5) 
 Write equation (3) using equations (4) and (5)

(6) 
where input forces are applied at the boundary only (F = 0).
 Important properties of the CB mass and stiffness matrices
 M_{bb} = Bounday mass matrix  total mass properties translated to the boundary points

(7) 
 K_{bb} = Interface stiffness matrix  stiffness associated with displacing one boundary DOF while other are held fixed
 If the boundary point is a single grid (i.e. nonredundant) then
K_{bb} = 0
 If the mode shapes have been mass normalized (typically they are) then

(8) 
 We can finally write the dynamic equation of motion (including damping) using the CB transform as

(9) 
where = modal damping ( = % critical)
 Summary of CB Theory
 CB Mass and Stiffness Matrices fully define system
 Dynamics problem solved using CB DOFs
 CB boundary DOFs provide location to apply BC’s and Forces or to couple with another structure
 CB transform is used to calculate physical responses from CB responses
III. How to Create a CB Model
$ CB Output File
ASSIGN,OUTPUT4='cb_ex1.kmnp',STATUS=unknown,UNIT=31
ID GORDON,CB_EX1
$ Normal Modes Solution
SOL SEMODES
TIME 5
DIAG 8,14
$ CB DMAP
INCLUDE 'path/cb103_2005r2b.dmp'
CEND
$
TITLE = CraigBampton Example 1
ECHO = NONE
METHOD = 1
$ SPC = 1
$ Request modal effective mass output
MEFFMASS(ALL) = YES
$
$ Request Output Grid Point Forces, Element Forces, and Element Stresses
$ This is recovered in the .F06 file
SET 100 = 209,210
SET 101 = 110,114
GPFORCE = 100
ELFORCE = 101
ELSTRESS = 101
$
BEGIN BULK
$
PARAM,AUTOSPC,YES
PARAM,GRDPNT,0
PARAM,WTMASS,0.00259
$
$ Print Gset (U1 in the DMAP) & Rset internal order
PARAM,USETSEL,0
PARAM,USETSTR1,U1
PARAM,USETSTR2,R
PARAM,USETPRT,0
$
$ Acceleration Output
$ Recover accleration response at GRID 11
$ This is output in the .kmnp file and processed in FLAME
USET1 U1 123456 11
$
$ Define number of modes to recover (or frequency range)
EIGRL 1 10 0 MASS
$
$ Boundary (Rset) Defined on SUPORT cards
$ 1 boundary point x 6 DOFs
$ = 6 physical boundary points
SUPORT 999 123456
SPC 1 999 123456 0.
$
$ 7) Don't forget the rest of your bulk data
INCLUDE 'bulkdata'
$
ENDDATA
What is created?
 file cb_ex1.kmnp which contains CB stiffness and mass matrices (k,m), net CG ltm (n), and the CB transformation matrix (phi_{g})
 .kmnp file is in NASTRAN binary output4 format
 K and M size is [CB DOFs (boundary + modal) x CB DOFs]
 phi_{g} size is [Gset rows x CB DOFs]
 Net CG LTM recovers CG accelerations and I/F Forces, Size is [6+boundary DOFs x CB DOFs]
How do you use this?
 Solve dynamics problem for CB DOF response using the K and M matrices
 Transform CB responses using phi_{g} to get physical responses
IV. Load Transformation Matrices (LTMs)
 LTM is a generic term referring to the matrix used to transform from CB DOFs to physical DOFs (also referred to at OTMs, ATMs, DTMs …)
 In its simplest form, the LTM is simply the phi_{g} matrix

(10) 
(Only the rows corresponding to the physical DOFs of interest are needed)
 There are other useful LTMs that can be created
 Interface (I/F) forces
 Net CG accelerations
 Stress and force LTMs
 I/F Force LTM (created by CB DMAP)

(11) 
(If boundary is nonredundant, then K_{bb} = 0)
 Net CG LTM (created by CB dmap)

(12) 
where = mass matrix about CG (6 X 6)
and = rigid body transform from I/F to CG (BDOF X 6)
 PHIZ LTM
 Allows physical displacements to be calculated from CB accelerations:

(13) 
 Same as modal acceleration approach in NASTRAN
 Useful in calculating relative displacements between DOFs
 Also used to calculate stresses and forces which are a function of displacements
 Calculated from CB dmap using
PARAM,PHZOUT,1
 LTM’s can be created using FLAME, MATLAB or using DMAP
 LTM’s can (and usually do) contain multiple types of responses

(14) 
 LTM’s can be used to recover responses for nested CB models

(15) 
where = row partition of CB_{1} DOFs from the CB_{0} PHI_{G} matrix
 Creating LTMs  See example file below for a sample FLAME script for creating an LTM
V. Checking CB Models AND LTMs
 CB Models and LTMs should be verified to make sure that they have been created correctly (especially for complicated LTM’s or nested CB models)
 CB Mass and stiffness matrices can be checked by computing freefree and fixedbase modes
 CB boundary Mass matrix can be transformed to CG and compared with NASTRAN GPWG
 LTMS can be checked by applying unit acceleration at the boundary:

(16) 
where = boundary rigid body vector (b X 6)
 Each response column represents acceleration in a single direction
 Accelerations should be in correct directions
 Forces should recover weight or correct moments
 Unit acceleration applied to PHI_{Z} can be checked by gravity run with physical model and comparing displacements
VI. Example and Files
 Following are links to the CraigBampton DMAP as well as a CB example:
 cb103_2005r2b.dmp — DMAP for creating a CB model in MSC Nastran v.2005r2b (REMOVED FOR ITAR REASONS)
 cb_pres2.ppt — Scott's Powerpoint presentation on the CB method (160KB)
 Primer_on_the_CraigBampton_Method.pdf — John Young's Primer on all things CraigBampton (2MB)
 CB_example.zip — Example files for creating a CB model and the load transformation matrix (LTM) in MSC Nastran v.2005r2b
(16MB  includes the above files and the CraigBampton paper) (REMOVED FOR ITAR REASONS)
Scott Gordon
