S5-1 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation SECTION 5 REDUCTION IN DYNAMIC ANALYSIS

S5-2 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation

S5-3 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation INTRODUCTION TO DYNAMIC REDUCTION n Definition u Dynamic reduction means reducing (decreasing) the number of degrees-of-freedom of a dynamic model to create a model with fewer degrees-of-freedom. n Why reduction for dynamics? u The un-reduced model may be too large to be solved without reduction. u The un-reduced model may have much more detail than required. u Dynamic reduction will result in a model whos solution can be obtained more quickly than that for the corresponding un-reduced (large) model. Reduction is approximately quadratic in cost (time-to-solve), whereas eigen-analysis is approximately cubic in cost. u Dynamic reduction is more accurate, and probably cheaper, than constructing a separate, smaller dynamic model.

S5-4 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation REDUCTION METHODS FOR DYNAMICS AVAILABLE WITH MSC.NASTRAN n Guyan reduction u Static condensation n Modal reduction

S5-5 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation STATIC CONDENSATION FOR STATICS n Let {u f } be the set of the unconstrained (free) structural coordinates, f-set coordinates. n Partition the f-set coordinates u where u {u o } = omitted set, o-set, coordinates l The omitted set coordinates are not constrained, they are partitioned-out u {u a } = analysis set, a-set, coordinates l The values of the analysis set coordinates are solved for DOF omitted, removed DOF, analysis set

S5-6 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation n Static condensation for a static analysis n Form a static equation for {u f } n Partition the stiffness matrix and vectors into the o-set and a-set n Set {P o } to zero (no loads applied to omitted DOFs) and solve for {u o } in terms of {u a } STATIC CONDENSATION FOR STATICS (Cont.) K oo K oa T K aa

S5-7 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation n Solution for {u o } in terms of {u a } n Transformation from the a-set to the f-set using static condensation n The o-set coordinates are dependent upon the a-set coordinates. The motion of the o-set is a linear combination of the a-set motions. The columns of [G oa ] are static shape vectors. STATIC CONDENSATION FOR STATICS (Cont.)

S5-8 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation STATIC CONDENSATION FOR DYNAMICS n Static condensation for a dynamic analysis n Form a dynamic equation for {u f } n Ignore the damping [B ff ] for this discussion n Partition the matrices and vectors into the o-set and a-set. Solve for {u o } in terms of n Set {P o } to zero, and choose the members of the a-set carefully so can eliminate [M oo ] and [M oa ]. Then, {u o } is given by

S5-9 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation STATIC CONDENSATION FOR DYN (Cont.) n The following equation is arrived at n The response for the f-set coordinates is found from

S5-10 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation n Dynamics problems are solved in terms of the reduced coordinates (a-set). The o-set coordinates are recovered from the a-set coordinates. n The o-set mass, damping, and stiffness is distributed to the a-set. n The largest computational effort is associated with the formulation of [M aa ] and [B aa ], particularly for nondiagonal (coupled mass) [M ff ]. n The resulting [K aa ], [B aa ], and [M aa ] are small and dense (i.e., the matrices may be fully populated). STATIC CONDENSATION FOR DYN (Cont.)

S5-11 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation n Comments/summary n Separate free degrees-of-freedom {u f } into the omitted set {u o } and the analysis set {u a } by means of the OMIT or ASET entries. Retain only a small fraction of the DOFs (typically 10% or less) in the analysis set (a-set) because the matrices may be fully populated, because of the static condensation, and the computational effort will greatly increase because it will not be possible to use banded solvers. Otherwise, retain all of the DOFs. Retain DOFs with large concentrated masses in the analysis set. STATIC CONDENSATION FOR DYN (Cont.)

S5-12 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation n Comments/summary (continued). Retain DOFs (assign to a-set) that are loaded (in transient and frequency response analysis). Retain the appropriate DOFs to be able to adequately represent normal modes of interest, e.g., accurately display mode shapes. There is no loss of accuracy when omitting massless DOFs. STATIC CONDENSATION FOR DYN (Cont.)

S5-13 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation SOLUTION CONTROL FOR GUYAN REDUCTION n Executive Control Section u Any SOL n Case Control Section u No special commands required n Bulk Data Section u ASET (optional, specifies a-set) u OMIT (optional, specifies o-set)

S5-14 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation ASETID1C1ID2C2ID3C3ID4C4 ASET SOLUTION CONTROL FOR GUYAN REDUCTION (Cont.)

S5-15 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation SOLUTION CONTROL FOR GUYAN REDUCTION (Cont.) n From the menu Loads/BCs: Create/Displacement/Nodal

S5-16 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation DIFFICULTIES WITH GUYAN REDUCTION n User effort in selecting a-set DOFs. n Accuracy depends on the users skill in selecting a-set DOFs. n Regardless of a users skill, high accuracy requires a relatively large number of a-set DOFs. Approximately 2 to 5 times the number of normal modes, that are desired to be accurately described, a-set DOFs are needed. n Stiffness reduction is exact, where as mass and damping reductions are approximate.

S5-17 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation DIFFICULTIES WITH GUYAN REDUCTION (Cont.) n Errors are most pronounced in higher modes. n Local modes may be missed altogether. n Note generally recommended, except when performing test-analysis correlation. n The static condensation approximation may miss dynamic effects that are important. Using the equation for {u o }, the following is found

S5-18 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation DIFFICULTIES WITH GUYAN REDUCTION (Cont.) n Any dynamic effects that are represented by {u o 1 } will be missed using only the static condensation transformation

S5-19 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation MODAL REDUCTION n ALL MSC.Nastran linear dynamic solutions have two methods. u Direct – the solution is solved for in terms of a-set coordinates. u Modal – the solution is solved for in terms of modal coordinates (i-set). n In the modal solution sequences (e.g., SOL 112, modal transient response) a transformation is made from the a-set coordinates to the modal coordinates. where [ ai ] is the matrix of normal modes, and { i } is the vector of modal coordinates n Modal vectors (normal modes) are solutions to the undamped eigenvalue problem (a-set coordinates)

S5-20 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation MODAL REDUCTION (Cont.) n The equations of motion are transformed from the a-set to the set of modal coordinates (i-set). The modal coordinates are handled internally in MSC.Nastran. (Note: e-set DOFs are not shown here for clarity.) If [ ai ] is mass normalized and there are no K2PP, M2PP, B2PP, or TF, then

S5-21 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation MODAL REDUCTION (Cont.) n Note: a-set matrices may be reduced matrices from Guyan reduction or GDR. Transformation from modal coordinates (i-set) to free coordinates (f-set) may require two transformations.

S5-22 NAS122, Section 5, August 2005 Copyright 2005 MSC.Software Corporation