10.2.2.3. CMR of RS
The set-up of a flex body modal solution in NX Nastran for export to ADAMS MNF or Recurdyn RFI files is possible with the MBDEXPORT case control command. See the NX Nastran Quick Reference Guide for input details.
The use of MBDEXPORT requires special considerations for the modal solution. This is because flex bodies are attached to other components in the multi-body dynamic (MBD) simulation and local flexibility effects at the connection locations are thus important.
In a standard normal mode solution, the solved modes give good representation of the global dynamics of the component, but the local stiffness effects at the connections are typically not captured because of modal truncation. A modal solution method called Component Mode Reduction of Residual Structure (CMR of RS) is recommended for flex body solutions because it includes both global and local effects. The normal modes method could also capture the residual flexibility effects if residual vectors at the connection DOF are computed.
The CMR of RS method is a variation to the standard SOL 103 modal solution in NX Nastran. It has long been employed by dynamic analysts to include the local stiffness effects in critical areas of a model. The CMR of RS method is a two-step modal solution approach and is equivalent to a Craig-Bampton superelement reduction on the residual structure.
10.2.2.3.1. CMR of RS Theory
The connection degrees-of-freedom (DOF) for the flex body are equivalent to the exterior DOF of a superelement. In Nastran set terminology, this means the connection DOF are in the t-set. The interior DOF of the flex body component are part of the o-set. The union of the t-set and o-set is the f-set.
The first solution step of the CMR or RS method is to compute normal modes and static constraint modes. The normal modes are computed with the specified t-set DOF restrained. The normal mode shape matrix, \(\Phi\), can be used to transform the physical displacement \(u_f\) (and its partitions \(u_t\) and \(u_o\)) into the generalized modal DOF, \(q\). The transformation relation is
The static constraint modes are static deflection shapes computed by applying unit deflections at the t-set DOFs. This is the same thing as performing a Guyan reduction on the o-set DOF. The static constraint modes transform full physical displacement \(u_f\) into just the physical DOF, \(u_t\). The transformation relation is
The matrices \(K_{oo}\) and \(K_{ot}\) are partitions of the \(K_{ff}\) stiffness matrix.
The normal mode transformation is a good reduction for the global dynamics and the constraint mode transformation is a good reduction for the local stiffness at the connection DOF. These transformations can be combined into a single transformation as
The combined transformation captures both the global dynamics and the local stiffness of the component in the reduction.
In the second solution step, the transformation in equation (3) is used to mathematically reduce the model from the full physical DOF set to a reduced set of generalized modal DOFs and the t-set physical DOFs. The reduced DOF set size is equal to the sum of the number of normal modes and number of connection DOFs. Since the normal modes and static modes are not orthogonal to each other, the reduced mass and stiffness matrices are not diagonal as they would be with a pure normal mode reduction. However, a second modal solution is performed on the reduced system resulting in a new set of modes.
The final modes are orthogonal to each other and importantly, capture both global dynamics and local stiffness characterizations of the flex body. It is this second set of modes shapes that are exported to the flex body file. It should be noted that it is critical that all the modes of the reduced system should be solved in the second modal solution so that full global and local effects embodied in the reduction are retained.
10.2.2.3.2. CMR of RS Specification
The CMR of RS method can be set-up in an NX Nastran solution by adding a few additional solution control cards to a standard SOL 103 solution. In the case control, an RSMETHOD card is needed in addition to the standard METHOD card. The purpose of each is:
RSMETHOD : identifies modal solution controls cards (EIGR or EIGRL) to be used for the first modal solution. Typically the control cards define a frequency range to compute the modes. Modes outside that range are neglected.
METHOD : identifies modal solution controls cards (EIGR or EIGRL) to be used for the second modal solution. This is the modal solution of the reduced system. It is important to solve for all the modes of the reduced system. This can be done by specifying a wide frequency range on the EIGR or EIGRL cards.
The bulk data section of the input file needs definition of the t-set and q-set DOF. These are defined by these cards:
ASET : define the connection DOFs (these are in the t-set)
SPOINT : creates additional DOFs
QSET : identifies SPOINT DOFs as modal reduction DOFs and places them in the q-set
For the first modal solution (RSMETHOD), it is recommended that the solved frequency range be at least twice the frequency range of interest in the MBD solution. For example, if the MBD solution is to be accurate to 1500 Hz, the first modal solution of the components should capture modes to at least 3000 Hz. The user needs to be sure that the number of q-set DOF created by QSET and SPOINT cards is larger than the actual number of modes computed from RSMETHOD.
As mentioned already, all the modes of the reduced system need to be computed for the second modal solution (METHOD). The user can verify that all the modes have been computed by confirming that the number of modes is equal to the reduced DOF set size (the sum of the number of normal modes and number of connection DOFs).
10.2.2.3.3. Mixed Boundary Reduction
Another variation of the CMR of RS method is to use a mixed boundary reduction. In the standard solution, all the t-set DOF are restrained in the first eigenvalue solution. With mixed boundary conditions, the user can designate that some of the t-set DOF is unrestrained in this solution. All t-set DOF are treated the same in the constraint mode solution.
The reason for using the mixed boundary approach is to add exterior DOF to the solution that are more appropriately considered as a free DOF rather than a connection DOF. For example if a component has four nodes used as connection points in the MBD analysis, those points should be restrained exterior points. If there are other points on the component that are not connections but may be a marker location to track response, or to apply a force, those nodes may best be treated as an unrestrained exterior point.
Additional bulk data cards for the mixed boundary approach are:
CSET : define the exterior DOFs that is unrestrained in the first eigenvalue solution.
BSET : define the exterior DOFs that is restrained in the first eigenvalue solution. BSET and ASET are treated the same in this case.
10.2.2.3.4. Example
Here is a sample deck set-up for doing a CMR of RS solution for a RecurDyn flexible body.
$
SOL 103
CEND
$
$ CASE CONTROL
$
TITLE = Solution 1
ECHO = NONE
MBDEXPORT(RECURDYNRFI,FLEXBODY=yes, FLEXONLY=no) $ Creates RFI file
RSMETH=100 $ Points to 1st modal extraction method
METHOD = 200 $ Points to 2nd modal extraction method
$
DISPLACEMENT(PLOT) = ALL
STRESS(PLOT,CORNER) = ALL
$
$ BULK DATA
$
BEGIN BULK
DTI,UNITS,1,KG,N,MM,S $ Units used in MBD solve
$
$ SOLUTION CARDS
$
EIGRL, 100, , 400.00 $ Eigenvalue extraction method for 1st modes
EIGR, 200, AHOU, , , , 150 $ Eigenvalue extraction method for 2nd modes
SPOINT,100001,thru,100101 $ SPOINTS > 1st Modes
QSET1,,100001,thru,100101 $ QSET points same as SPOINTS
PARAM AUTOSPC YES
PARAM GRDPNT 0
PARAM POST -2
$
$ Flex body connection ASET points
USET1, U6, ,123456,26
USET1, U6, ,123456,35
USET1, U6, ,123456,44
USET1, U6, ,123456,53
USET1, U6, ,123456,62
USET1, U6, ,123456,71
USET1, U6, ,123456,80
USET1, U6, ,123456,89
$
$ Rest of model …
$
Case Control Section
MBDEXPORT(RECURDYNRFI,FLEXBODY=yes,FLEXONLY=no)
Used to request calculation of the flex body in RecurDyn RFI file format
RSMETH = 100
This card points to the EIGRL card used for the first modal solution
METHOD = 200
This card points to the EIGR card used for the second modal solution
Bulk Data Section
DTI,UNITS,1,KG,N,MM,S
This defines the set of units used and is required by the MBD solver
EIGRL, 100, , 400.00
This is the extraction method for the step-1 modal solve. Recall that you want the upper frequency limit to be about twice the maximum frequency of interest in the structure
EIGR, 200, AHOU, , , , 150
This is the extraction method for the step-2 modes
The Automatic Householder option is recommended since it assures that you get all the modes of the reduced component
ND (or 150 in this example), is the desired number of eigenvectors, this number has to be greater that the number of ASET plus the number of step-1 modes
In this example, there are 8 ASET points, each with 6 DOF, and about 74 step-1modes, so we need 122 step-2 modes (8 x 6 + 74, it’s OK to request more)
SPOINT,100001,thru,100101
SPOINTs than expected step-1 modes, it doesn’t hurt to have too many
The numbering range has to be outside the range of grid points
QSET1,,100001,thru,100101
The QSET card puts the SPOINTs in the q-set
Use the same range as the SPOINTs
USET1, U6, ,123456,grid number
The ASET identifies the connection grids and the DOF at those grids. In this case all 6 DOF are used as connection DOF, which assumes that all is connected by fixed joints in the MBD model