1 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction Techniques MODEL REDUCTION TECHNIQUES Peter Avitabile Mechanical Engineering Department University of Massachusetts Lowell [ K ] n [ M ] n [ M ] a [ K ] a [ E ] a [ ω ] 2 Structural Dynamic Modeling Techniques & Modal Analysis Methods
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1 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
MODEL REDUCTION TECHNIQUES
Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell
Generally, it may be necessary to reduce a finite element model to a smaller size especially when correlation studies are to be performed.Several model reduction techniques are:
5 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
General TransformationFor all model reduction/expansion techniques, there is a relationship between the master dof (adof) and the deleted dof (ddof) which can be written in general terms as
n denotes all FEM dofa denotes master or tested dofd denotes deleted or omitted dof
[ ] ad
an xT
xx
x =
=
[ ] [ ] 2121ad
an xTxorxT
xx
x ==
=
6 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
General TransformationSince the energy of the system needs to be conserved, a balance can be written between the energy at state 1 and state 2 as
Substituting the transformation gives
[ ] [ ] 22T
211T
1 xKx21xKx
21U ==
[ ] [ ] [ ] [ ] 22T
22121T
212 xKx21xTKxT
21U ==
7 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
General TransformationRearranging some terms then yields
Then the reduced stiffness is related to the original stiffness by
The mass is reduced in a similar fashion
[ ] [ ][ ] [ ] 22T
22121T
12T
2 xKx21xTKTx
21U ==
[ ] [ ] [ ][ ] [ ] [ ] [ ][ ]TKTKorTKTK nT
a121T
122 ==
8 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Reduction of System MatricesThe reduced mass and stiffness matrices can be written as
[M] denotes the mass matrix[K] denotes the stiffness matrix
[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]TKTK
TMTM
nT
a
nT
a
=
=
9 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
General TransformationThe transformation T will take on various forms depending on the transformation technique utilized
[ ] ad
an xT
xx
x =
=
XXAA = active set of = active set of dof’sdof’sXXFF = full set of = full set of dof’sdof’s
10 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Eigensolution of Reduced SystemUsing the reduced mass and stiffness matrices, the eigensolution produces frequencies that are higher than those of the original system (for most of the reduction schemes).
The eigensolution of the reduced matrices
yields[ ] [ ][ ] 0xMK aaa =λ−
[ ] [ ]a2a U;ω
11 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Expansion FormulationThe expansion of the adof from the reduced model eigensolution over all the ndof is given by
[ ] ad
an xT
xx
x =
=
XXAA = active set of = active set of dof’sdof’sXXFF = full set of = full set of dof’sdof’s
12 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Guyan CondensationThe stiffness equation
can be partitioned into the ‘a’ active DOF and the ‘d’ deleted or omitted DOF to form two equations given as
[ ] nnn FxK =
[ ] [ ][ ] [ ]
=
d
a
d
a
ddda
adaa
FF
xx
KKKK
13 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Guyan CondensationAssuming that the forces on the deleted DOF are zero, then the second equation can be written as
which can be solved for the displacement at the deleted DOF as
[ ] [ ] 0xKxK dddada =+
[ ] [ ] ada1
ddd xKKx −−=
14 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Guyan CondensationThe first equation can be written as
and upon substituting for the ‘d’ deleted DOF gives the equation becomes
[ ] [ ] adadaaa FxKxK =+
[ ] [ ][ ] [ ] aada1
ddadaaa FxKKKxK =+ −
15 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Guyan CondensationThis can be manipulated to yield the desired transformation to be
[ ] [ ][ ]
−
=
= − ]K[]K[
]I[tI
Tda
1dds
s
16 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Guyan CondensationUsing this transformation, the reduced stiffness can be written as
Guyan proposed that this same transformation be applied to the mass matrix given by
[ ] [ ] [ ][ ]snT
sGa TKTK =
[ ] [ ] [ ][ ]snT
sGa TMTM =
17 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
18 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Guyan Condensation – MATLAB Script
19 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Guyan Condensation• Guyan (static) condensation is only accurate for
stiffness reduction; inertial forces are not preserved
• Eigenvalues of the reduced system are always higher than those of the original system
• The quality of the eigenvalue approximation depends highly on the location of points preserved in the reduced model
• The quality of the eigenvalue approximation decreases as the mode number increases
20 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic CondensationThe equation of motion is cast as a shifted eigenproblem. A shift value, f, is introduced into the set of equations describing the dynamic system, thus
[ ] ( )[ ][ ] 0xMfK nnn =−λ−
21 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic CondensationThe terms are rearranged to group the constant term f times the mass matrix with the stiffness matrix to yield
Then let a new system matrix [D] be used to describe the ‘effective’ stiffness matrix as
[ ] [ ][ ] [ ][ ] 0xMMfK nnnn =λ−+
[ ] [ ] [ ][ ]nnn MfKD +=
22 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques