Analytical Model Improvement Localization Approachesfaculty.uml.edu/pavitabile/22.515/Hiro_AMI_SSO_MSSO_thesis.pdf · 3 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Analytical

1 Dr. Peter AvitabileModal Analysis & Controls LaboratoryAnalytical Model Improvement & Localization

Analytical Model Improvement&

Localization Approaches

Peter AvitabileModal Analysis and Controls Laboratory

University of Massachusetts Lowell


Model Improvement Considerations

• Describe the Analytical Model Improvement approach (AMI)

• Describe the Skyline Stiffness Optimization technique and the Mass and Stiffness Optimization technique (SSO/MSSO)

• A significant amount of effort is required to completely describe all aspects of these techniques

Objectives of this lecture:


Several approaches have been customized for this application

System Optimization with Phantom Elements

With a specified performance level identified, modification or adjustment of the system matrices is required to meet requirements

CAB IMPEDANCES

FRAME IMPEDANCES

COMBINEDSTRUCTURE

RESPONSE

AMI

SSO/MSSO

PHANTOM

These modified systems are then used for system disassembly to determine the cascaded component required characteristics


Phantom elements can be used with FEM skyline topology

Optimization with Phantom Elements

Alternate scenarios are needed when the component is obtained from reduced model or superelementformulation since no topology exists

ComponentSkyline

Physical Component Matrix Physical Component Matrixwith Phantom Connectivities

ComponentSkyline

Matrix Values within Skyline Matrix Values within Skylineand connectivities specified

ApplyPhantom

ConnectivityTechnique

PhantomConnectivity

Reduced Matrix Component Reduced Matrix Componentwith Phantom Connectivities

Matrix Values exist

ApplyPhantom

ConnectivityTechnique

No Skyline Exists

over all possible DOF

PhantomConnectivities

Matrix Values exist

over all possible DOF


Any optimization approach can be used

Use of phantom elements easily accomplished with FEM skyline topology

Tsuji, Avitabile Modal Analysis & Controls Laboratory University of Massachusetts Lowell

Optimization Scenarios with Phantom Elements

Alternate scenarios are needed when the component is obtained from reduced model or superelementformulation since no topology exists

Superelement or reduced

Superelement or reduced

Change allowedanywhere in matrix

Change allowedanywhere in matrix

Change not allowedin white space

Connector allowedto change

MAP WITH

MAP WITH

PHANTOMS

PHANTOMS


Analytical Model Improvement

Equation of Motion

with no damping assumed becomes

The eigen solution of this is

with resulting eigenvalues and eigenvectors as

[ ] [ ] [ ] nnnnnnn tFtxKtxCtxM )(=)(+)(+)( &&&

[ ] [ ] nnnnn tFtxKtxM )(=)(+)(&&

[ ] [ ][ ] 0=- nnn xMλK

[ ]nn Uω ,



The modal transformation is given by

which is substituted into the equation of motion

The equations are then put into normal form

Modal MassModal Stiffness

[ ] mnmn pUx =

[ ] [ ] [ ] [ ] nmnmnnmnmn tFpUKpUM )(=+&&

[ ] [ ] [ ] [ ] [ ] [ ] [ ] n

Tnmmnmn

Tnmmnmn

Tnm tFUpUKUpUMU )(=+&&

[ ] [ ] [ ] [ ] [ ]mmnmnTnm IMUMU ==

[ ] [ ] [ ] [ ] [ ]mmnmnTnm KUKU 2Ω==



The redcution of large models follows the typical model reduction methodology addressed elsewhere

[ ] [ ] [ ][ ]nanTnaa TMTM =

[ ] [ ] [ ][ ]nanTnaa TKTK =



The Modal Assurance Criteria (MAC) and the Pseudo-Orthogonality Check (POC) are useful tools for assessment of updaed models

[ ] [ ] [ ]jT

jiT

i

jT

iji

eeuu

euMAC

2

, =

[ ] [ ] [ ]nettnI

TnIndof UMUPOC arg=



The mass and stiffness matrices can be estimated from the direct inversion of the measured or target data of modal mass and modal stiffness

which results in

but the resulting matrices are questionable.

[ ] [ ] [ ] [ ] [ ]mmnmnTnm IMUMU ==

[ ] [ ] [ ] [ ] [ ]mmnmnTnm KUKU 2Ω==

[ ] [ ] [ ] [ ]ngnmm

gT MUMU nm =

[ ] [ ] [ ] [ ]ngnmm

gTnm KUKU =


Analytical Model Improvement (Berman & Nagy)

The discrepancy of modal mass in modal space can be written as

and can be projected back to physical space through a mass weighted generalize inverse

where

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]SIST

IIIT

I MIUMUUMUM -=-=∆

[ ] [ ] [ ] [ ][ ] [ ][ ] [ ] [ ][ ]TSTIS

TTS

TIS MUMMMUMM 1-1- ∆=∆

[ ] [ ] [ ][ ]VMVM T ∆=∆ [ ] [ ] [ ][ ]VMVM T ∆=∆


Analytical Model Improvement – Mass Alone

The improved mass can then be written as

This mass is not necessarily related to the stiffness of the system.

[ ] [ ] [ ] [ ] [ ][ ][ ]VMIVMM ST

SI -+=


Analytical Model Improvement – Summary

The improved mass is given by

[ ] [ ] [ ] [ ] [ ][ ][ ]VM-IVMM ST

SI +=

[ ] [ ] [ ][ ]EMEM ST

S = Projection of analytical FEM seed mass from physical space to modal space

Discrepancy of madal mass in modal space

Projection of the discrepancy of madalmass in modal space to physical space

[ ] [ ] [ ][ ]SM-IM =∆

[ ] [ ] [ ] [ ][ ][ ]VM-IVM ST=∆

[ ] [ ] [ ] [ ]ST1

S MEMV −=

Note: E and UIare often used

interchangeably as “target” shapes


Analytical Model Improvement (Berman & Nagy)

The discrepancy of modal stiffness in modal space can be written as

and can be projected back to physical space through a mass weighted generalize inverse

where

[ ] [ ] [ ][ ] [ ] [ ][ ] [ ] [ ]SIST

IIIT

I KUKUUKUK -Ω=-=∆ 2

[ ] [ ] [ ] [ ][ ] [ ][ ] [ ] [ ][ ]TSTIS

TTS

TIS MUMKMUMK 1-1- ∆=∆

[ ] [ ] [ ][ ]VKVK T ∆=∆ [ ] [ ] [ ] [ ][ ]ST

IS MUMV -1=


Analytical Model Improvement – Stiffness Alone

The improved stiffness can then be written as

This stiffness is not necessarily related to the mass of the system.

[ ] [ ] [ ] [ ] [ ][ ][ ]VKVKK ST

SI -Ω+= 2



The mass and stiffness will be somewhat representative of the actual mass and stiffness since the process is “seeded” with approximate estimates of the actual system.However, the eigensolution using these mass and stiffness matrices may not necessarily produce the exact same frequencies and mode shapes used as the target (measured) information.

[ ] [ ] [ ] [ ] [ ][ ][ ]VMIVMM ST

SI -+=

[ ] [ ] [ ] [ ] [ ][ ][ ]VKVKK ST

SI -Ω+= 2


Analytical Model Improvement (Leung/O’Callahan)

The force balance associated with the eigenvalueproblem is given as

and is projected to modal space as

[ ] [ ][ ] 0=- nnn xMλK [ ]

mnmn pUx =

[ ] [ ] mmmnmnn pUxλx 2Ω-=-=&&

[ ][ ] [ ][ ][ ]2Ω= IIII UMUK

[ ][ ][ ] [ ][ ][ ][ ]2Ω= IITIII

TI UMUUKU


Analytical Model Improvement (Leung/O’Callahan)

Using the weighted generalized inverse for the common form of the solution gives

This uses the equations of motion as part of the process with the improved mass.

This has very special features when updating models

[ ] [ ] [ ] [ ] [ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ]TISISST

SI VUKVUKVKVKK --+Ω+= 2



Finite Test UpdateElement Results Results

42.2 42.5 42.545.7 48.1 48.171.4 75.9 75.9

100.3 100.3130.2 130.2157.5 157.5188.8 188.8210.2 210.2

[ ] [ ] [ ] [ ] [ ][ ][ ]VMIVMM ST

SI -+= [ ] [ ] [ ] [ ] [ ][ ][ ] [ ][ ][ ][ ] [ ][ ][ ][ ]TISISST

SI VUKVUKVKVKK --+Ω+= 2


AMI – Matrix Smearing

System Model

System Matrix


Stiffness Skyline Optimization (SSO) - Kammer

Using the original finite element matrices as seeding matrices, the updated system characteristics can be developed from the motion of equation as

A series of substitutions are needed to put the equations into a packed form with non-zero terms separated from the zero terms of the matrix

[ ][ ][ ][ ] [ ][ ][ ][ ][ ]ITIIII

TIII MUUMMUUK 2Ω=


Stiffness Skyline Optimization (SSO) - Kammer

These packed matrices are

sK

nI Y]K[ ⇒ sn

ITnrnr B]M[]U[]U[ ⇒

snIT

nr2

nrnI D]M[]U[][]U[]M[ ⇒Ω

ssK

s DY]B[ =

s

qK

pK

sqsp DY

Y]]B[]B[[ =

LM


15

5

4

3

2

1

1100012100

012100012100011

D

yyyyy

=

−−−

−−−−

−

Matrix Multiplication Example

Zero terms exist outside of FEM connectivity in [M] & [K]

Matrix multiplication efficiency

Partitioning out zero terms

000000

11

5

4

3

22

11

=×=×=×

−=×−=×

yyy

yyyy

Example of Matrix Multiplication


Mass and Stiffness Skyline Optimization (MSSO) - Wu / O’Callahan

Using the original finite element matrices as seeding matrices, the updated system characteristics can be developed from the motion of equation as

A series of substitutions are needed to put the equations into a packed form with non-zero terms separated from the zero terms of the matrix

[ ][ ][ ] [ ][ ][ ][ ] 0=Ω- 2I

TIII

TII MUUKUU


Mass and Stiffness Skyline Optimization (MSSO)

These packed matrices are

fgTnrnr ]P[]U[]U[ ⇒ fg

Tnrr

2nr ]P[]U[][]U[ ⇒Ω

Kgn

I Y][K ⇒ Mgn

I Y][M ⇒

Mghgnrn

ITnr Y]R[]U[]M[]U[ ⇒

hr V[I] ⇒


SSO and MSSO – Skyline Containment

System Model

System Matrix


SSO and MSSO – Phantom Connectivity Technique

System Model

System Matrix

Phantom Connectivity Technique

Documents

Analytical Model Improvement Localization Approachesfaculty.uml.edu/pavitabile/22.515/Hiro_AMI_SSO_MSSO_thesis.pdf · 3 Dr. Peter Avitabile Modal Analysis & Controls Laboratory Analytical