Model Reduction 061904 - Faculty Server...

57
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

Transcript of Model Reduction 061904 - Faculty Server...

Page 1: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 2: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 3: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 4: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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:

Page 5: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 ==

=

Page 6: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 ==

Page 7: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 ==

Page 8: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

=

=

Page 9: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 10: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 11: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 12: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 13: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 −−=

Page 14: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 =+ −

Page 15: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 16: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 =

Page 17: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 18: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

18 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

Guyan Condensation – MATLAB Script

Page 19: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 20: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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−

Page 21: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 +=

Page 22: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 23: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 −−=

Page 24: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 =+ −

Page 25: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 26: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 =

Page 27: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 28: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

28 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

Dynamic Condensation – MATLAB Script

Page 29: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 30: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

−−

=

−=

Page 31: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

31 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

IRS Reduction– MATLAB Script

Page 32: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 33: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 =

Page 34: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

==

=

=

=

−−

Page 35: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

=

=−

Page 36: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

=

Page 37: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Ua =Un =

Page 38: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

=

=

Page 39: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 40: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 41: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

41 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

SEREP Reduction – MATLAB Script

Page 42: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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.

Page 43: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 =

Page 44: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 −+=

Page 45: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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 −+=

Page 46: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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:

Page 47: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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

Page 48: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

48 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

SEREP - Exact System Reduction

Page 49: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

49 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

SEREP - Effect of Mode Selection

Page 50: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

50 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

SEREP - Effect of Point Selection

Page 51: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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:

Page 52: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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:

Page 53: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

53 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

Case 3 - Poor Selection of 6 DOF

Page 54: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

54 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

Case 4 - Better Selection of 6 DOF

Page 55: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

55 Dr. Peter AvitabileModal Analysis & Controls LaboratoryModel Reduction Techniques

Case 5 - Larger Selection of DOF

Page 56: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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:

Page 57: Model Reduction 061904 - Faculty Server Contactfaculty.uml.edu/pavitabile/22.515/Model_Reduction_061904.pdf · 37 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Model Reduction

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