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1 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
MODEL REDUCTION TECHNIQUES
Peter AvitabileMechanical Engineering DepartmentUniversity of Massachusetts Lowell
[ K ]n
[ M ]n [ M ]a[ K ]a [ E ]a
[ ω ]2
Structural Dynamic Modeling Techniques & Modal Analysis Methods
2 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Model Reduction TechniquesDynamic reduction means :
reducing a given dynamic finite element model
to one with fewer degrees of freedom
while maintaining the dynamic characteristics of the system
XA = active set of dof’sXF = full set of dof’s
3 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
• Guyan condensation
• Dynamic Condensation
• Improved Reduced System
• System Equivalent Reduction Expansion Process
• Hybrid Reduction
Model Reduction Techniques
4 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Model Reduction
• Guyan/Irons condensation• Dynamic condensation• Improved Reduced System• System Equivalent Reduction Expansion Process• Hybrid Reduction (Kammer)
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
Summary - Guyan CondensationThe stiffness equation
can be written as
The transformation matrix can be written as
[ ] nnn FxK =
[ ] [ ][ ] [ ]
=
d
a
d
a
ddda
adaa
FF
xx
KKKK
[ ] [ ][ ]
−
=
= − ]K[]K[
]I[tI
Tda
1dds
s
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
Dynamic CondensationThis ‘effective’ stiffness equation
can be partitioned into the ‘a’ active DOF and the ‘d’ deleted or omitted DOF to form two equations given as
[ ] nnn FxD =
[ ] [ ][ ] [ ]
=
d
a
d
a
ddda
adaa
FF
xx
DDDD
23 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic 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
[ ] [ ] 0xDxD dddada =+
[ ] [ ] ada1
ddd xDDx −−=
24 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic CondensationThe first equation can be written as
and upon substituting for the ‘d’ deleted DOF this equation becomes
[ ] [ ] adadaaa FxDxD =+
[ ] [ ][ ] [ ] aada1
ddadaaa FxDDDxD =+ −
25 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic CondensationThis can be manipulated to yield the desired transformation to be
[ ] [ ][ ]
−
=
= − ]D[]D[
]I[tI
Tda
1ddf
f
26 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic CondensationUsing this transformation, the reduced stiffness can be written as
This same transformation can be applied to the mass matrix given by
[ ] [ ] [ ][ ]fnT
ffa TKTK =
[ ] [ ] [ ][ ]fnT
ffa TMTM =
27 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Summary - Dynamic CondensationThe equation of motion is cast as a shifted eigenproblem
and can be written as
The transformation matrix can be written as
[ ] [ ] [ ][ ] 0xMfKxD nnn =−=
[ ] [ ][ ] [ ]
=
00
xx
DDDD
d
a
ddda
adaa
[ ] [ ][ ]
−
=
= − ]D[]D[
]I[tI
Tda
1ddf
f
28 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic Condensation – MATLAB Script
29 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Dynamic Condensation• If the shift frequency is zero, then this
reduces to Guyan reduction• The reduced model will at most contain an
eigenvalue equal one from the full model• If the shift equals as eigenvalue of the original
system, then the reduced system will also contain this eigenvalue
• All other eigenvalues of the reduced system may not be good representations of the system
• Dynamic condensation is useful when only one mode of the system is to be retained in the model
30 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Improved Reduced SystemExtensions to the Guyan reduction process are used to account for the effects of mass inertia associated with the deleted dof
[ ] [ ][ ] [ ]is
i ttI
T +
=
[ ]
[ ] [ ] [ ][ ] [ ] [ ][ ][ ] [ ]a1
asn1dd
i
da1
dds
KMTMK000
t
]K[]K[t
−−
−
=
−=
31 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
IRS Reduction– MATLAB Script
32 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Improved Reduced System• Adjustment terms to the Guyan reduction
process allow for the better representation of the mass associated with the deleted dof
• Improves on the accuracy of the reduced model when compared to Guyan especially for the higher modes of the system
33 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
System Equivalent Reduction Expansion Process
The modal transformation equations can be written as
and for the active set of dof
[ ] [ ][ ] pUU
pUxxx
d
ann
d
a
===
[ ] pUx aa =
34 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
System Equivalent Reduction Expansion Process
Least Squares Solution - ma ≥
[ ] [ ] [ ] [ ]
[ ] [ ]( ) [ ] [ ] [ ]( ) [ ] [ ]
[ ] [ ]( ) [ ] [ ] agaa
Ta
1a
Ta
aT
a1
aT
aaT
a1
aT
a
aT
aaT
a
aa
xUxUUUp
pUUUUxUUU
pUUxU
pUx
==
=
=
=
−
−−
35 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
System Equivalent Reduction Expansion Process
Using a generalized inverse, this is
[ ] [ ]( ) [ ]
[ ] aga
aT
a1
aT
a
xUp
xUUUp
=
=−
36 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
System Equivalent Reduction Expansion Process
Substituting into the modal transformation equation gives
[ ][ ] agann xUUx =
[ ][ ] [ ] ag
ad
a
d
a xUUU
xx
=
37 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
System Equivalent Reduction Expansion Process
The SEREP transformation matrix can be written as
[ ] [ ][ ] [ ] [ ] [ ]( ) [ ][ ][ ] [ ] [ ]( ) [ ][ ]
== −
−
Ta
1a
Tad
Ta
1a
Taag
anuUUUUUUUUUUT
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Ua =Un =
38 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP Computational Advantages
The SEREP transformation is given by
The reduced mass and stiffness are computed as
[ ] [ ][ ]ganu UUT =
[ ] [ ] [ ] [ ][ ] [ ] [ ] [ ]Un
TU
Sa
UnT
USa
TKTK
TMTM
=
=
39 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP Computational Advantages
Substituting Tu into the reduced mass computation gives
But recall from mass orthogonality that
[ ] [ ] [ ] [ ][ ][ ]gannT
nTg
aSa UUMUUM =
[ ] [ ][ ] [ ]IUMU nnT
n =
The reduced mass is efficiently computed as
[ ] [ ] [ ][ ] [ ] [ ]gaTgaUn
TU
Sa UUTMTM ==
NOTE: Unit modal mass scaling
40 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP Computational Advantages
Substituting Tu into the reduced stiffness computation gives
But recall from stiffness orthogonality that
[ ] [ ] [ ] [ ][ ][ ]gannT
nTg
aSa UUKUUK =
[ ] [ ][ ] [ ]2nnT
n UKU Ω=
The reduced stiffness is efficiently computed as
[ ] [ ] [ ][ ] [ ] [ ][ ]ga2TgaUn
TU
Sa UUTKTK Ω==
NOTE: Unit modal mass scaling
41 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP Reduction – MATLAB Script
42 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
System Equivalent Reduction Expansion Process• The eigenvalues of the reduced system always
equals the eigenvalues of the full system for the modes of interest retained in the model
• The modes that are preserved in the reduced model may be arbitrarily selected from those modes of interest in the original model
• The eigensolution of the reduced system is exact and does not depend on the location or number of points preserved in the reduced model.
43 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Hybrid Reduction
Hybrid reduction combines the advantages of the full rank nature of Guyan Reduction along with the accuracy of the SEREP process
[ ] [ ] [ ] [ ][ ][ ]PTTTT SUSH −+=
[ ] [ ][ ] [ ]SaTaa MUUP =
44 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Hybrid Reduction
The projection operator can be rewritten as
This transformation equation can be manipulated to give
[ ] [ ][ ] [ ] [ ][ ]gaaSa
Taa UUMUUP ==
[ ] [ ][ ] [ ] [ ] [ ][ ]PITPTT SUH −+=
45 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Hybrid Reduction
Substituting the projection operator gives
And recalling the Moore-Penrose conditions this can be rewritten as
or
[ ] [ ][ ] [ ][ ] [ ] [ ] [ ][ ][ ]gaaSg
aag
anH UUITUUUUT −+=
[ ] [ ][ ] [ ] [ ] [ ][ ][ ]gaaSg
anH UUITUUT −+=
[ ] [ ] [ ] [ ] [ ][ ][ ]gaaSUH UUITTT −+=
46 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP Applications
• Exactness of the technique• arbitrary selection of modes included in the
reduced model• arbitrary selection of dof included in the
reduced model
Several simple examples are investigated in order to demonstrate the unique features of the SEREP process:
47 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
FRAME Modal DataN11
*****
*
*
*
*
*
*N1
N2
N3
* * * * * *
*
*
**N13
N14
N15
N16
*
*
*N19
N18
N17
N24N23N22N21N20N4
N5
N6
N7 N8 N9 N10 N12
Aluminum frame1-1/2 x 3-1/2 x3/1624 nodes24 planar bean elements
48 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP - Exact System Reduction
49 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP - Effect of Mode Selection
50 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP - Effect of Point Selection
51 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
SEREP Applications Summary
• Arbitrary selection of modes preserved in the reduced model
• Reduced model accuracy is not dependent on the selection of master dof
• Reduced model frequencies are identical to those of the full system model
• Expanded reduced mode shapes are identical to those of the full system model
A new modeling/mapping techniques referred to as the System Equivalent Reduction Expansion Process (SEREP) reveals the following salient features:
52 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Comparison of Reduced Models
• Guyan• IRS• Dynamic• SEREP
Several simple examples are investigated in order to compare the different model reduction techniques - those investigated were:
53 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Case 3 - Poor Selection of 6 DOF
54 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Case 4 - Better Selection of 6 DOF
55 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Case 5 - Larger Selection of DOF
56 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Model Reduction Application Summary
• Guyan condensation always produces frequencies that are greater than those of the full model; dof selection is critical to its success
• IRS improves on Guyan by making adjustments to the inertial effects associated with the ddof
• dynamic condensation will preserve at most one of the eigenvalues of the original system
• SEREP always produces the same frequencies and mode shapes as the full system
Comparison of several different model reduction methods were presented to show distortion that results from various schemes. Main points are:
57 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques
Model Reduction SummaryThe different reduction forms are:
Guyan
Dynamic
IRS
SEREP
[ ] [ ] [ ] [ ][ ] [ ] [ ][ ][ ] [ ]a1
asn1ddda
1dd
i KMTMK000
]K[]K[I
T −−−
+
−
=
[ ] [ ][ ]
−
=
= − ]B[]B[
]I[tI
Tda
1ddf
f
[ ] [ ][ ]
−
=
= − ]K[]K[
]I[tI
Tda
1dds
s
[ ] [ ][ ] [ ] [ ] [ ]( ) [ ][ ][ ] [ ] [ ]( ) [ ][ ]
== −
−
Ta
1a
Tad
Ta
1a
Taag
anuUUUUUUUUUUT