FEM for Test Eng

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Quartus Engineering CopyrightQuartus Engineering Incorporated, 2000.

FEMFOR THE TEST ENGINEER

Christopher C. Flanigan

Quartus Engineering Incorporated

San Diego, California USA

18th International Modal Analysis Conference (IMAC-XVIII)

San Antonio, Texas

February 7-10, 2000


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DOWNLOAD FROM THEQUARTUS ENGINEERING WEB SITE

http://www.quartus.com


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FEM for the Test Engineer

TOPICS

Theres reality, and then theres FEM

FEM in a nutshell

FEM strengths and challenges

Pretest analysis Model reduction

Sensor placement

Posttest analysis

Correlation

Model updating


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Theres Reality, and Then Theres FEM

REALITY IS VERY COMPLICATED!

Many complex subsystems

Unique connections

Advanced materials

Broadband excitation

Nonlinearities

Flight-to-flight variability

Chaos Extremely high order behavior


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Theres Reality, and Then Theres FEM

REMEMBER THAT FEM

ONLY APPROXIMATES REALITY

Reality has lots of hard challenges

Nonlinearity, chaos, etc.

FEM limited by many factors Engineering knowledge and capabilities

Basic understanding of mechanics

Computer and software power

But its the best approach we have

Experience shows that FEM works well when used properly

FEM

Ahead!


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FEM Strengths and Challenges

TEST IS NOT REALITY EITHER!

Test article instead of flight article

Mass simulators, missing items, boundary conditions

Excitation limitations

Load level, spectrum (dont break it!)

Nonlinearities

Testing limitations

Sensor accuracy and calibration

Data processing

But its the best reality check available


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FEMin a Nutshell


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FEM IN A NUTSHELL

Divide and conquer!

Shape functions

Elemental stiffness and mass matrices

Assembly of system matrices Solving

Related topics

Element library

Superelements


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FEM in a Nutshell

CLOSED FORM SOLUTIONS, ANYONE?

Consider a building

Steel girders

Concrete foundation

Can you write an equation tofully describe the building?

I cant!

Even if possible, probably not

the best approach

Very time consuming

One-time solution


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FEM in a Nutshell

DIVIDE AND CONQUER!

Behavior of complete

structure is complex

Example: membrane

Divide the membrane

into small pieces

Buzzword: element

Feasible to calculate

properties of each piece

Collection of pieces

represents structure

1

3

5

7

9

11

13

15

17

19

S1

S3

S5

S7

S9

S11

S13

S15

S17

S19

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00


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FEM in a Nutshell

SHAPE FUNCTIONS ARE THE

FOUNDATION OF FINTE ELEMENTS

Shape function

Assumed shape of element when deflected

Some element types are simple Springs, rods, bar

Other elements are more difficult

Plates, solids

But thats what Ph.D.s are for! Extensive research

Still evolving (MSC.NASTRAN V70.7)

Spring

F = K X

FX

K


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FEM in a Nutshell

ELEMENT STIFFNESS MATRIX

FORMED USING SHAPE FUNCTIONS Element stiffness matrix

Relates deflections of elemental DOF

to applied loads

Forces at element DOF when un it

def lect ion imposed at DOFiand

oth er DOFjare fixed

Example: linear spring (2 DOF)

Spring

F = K X

FX

K

KK

KKKspring


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FEM in a Nutshell

ELEMENT MASS MATRIX

HAS TWO OPTIONS Lumped mass

Apply 1/N of the element mass to each node

Consistent mass

Called coupled mass in NASTRAN

Use shape functions to generate mass matrix

In practice, usually little difference

between the two methods Consistent mass more accurate

Lumped mass faster

M5.00

0M5.0Mspring

1/4 1/4

1/4 1/4


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FEM in a Nutshell

SYSTEM MATRICES FORMED

FROM ELEMENT MATRICES

K = 2

K = 5

K = 1

M = 1

M = 2

M = 3

22

22K1

55

55K2

11 11K3

1100

1650

0572

0022

K

5.1000

05.200

005.10

0005.0

M


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Quartus EngineeringCopyrightQuartus Engineering Incorporated, 2000.

FEM in a Nutshell

CALCULATE SYSTEM STATIC

AND DYNAMIC RESPONSES Static analysis

Normal modes analysis

Transient analysis

PqKqCqM TTTT

0MK ii

XKP


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FEM in a Nutshell

HONORARY DEGREE IN FEM-OLOGY!


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FEM STRENGTHS AND CHALLENGES


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FEM IS VERY POWERFUL FOR

WIDE ARRAY OF STRUCTURES Regular structures

Fine mesh

Sturdy connections Seam welds

Well-defined mass

Smooth distributed

Small lumped masses

Linear response

Small displacements General DynamicsControl-Structure Interaction Testbed


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FEM HAS MANY CHALLENGES

Mesh refinement

How many elements required?

Stress/strain gradients, mode shapes

Material properties A-basis, B-basis, etc.

Composites

Dimensions

Tolerances, as-manufactured

Joints

Fasteners, bonds, spot welds

continued...


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FEM HAS MANY CHALLENGES

Mass modeling

Accuracy of mass prop DB

Difficulty in test/weighing

Secondary structures

Avionics boxes, batteries

Wiring harnesses

Shock mounts

Nonlinearities (large deformation, slop, yield, etc.)

Pilot error!


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FEM ASSISTED BY ADVANCES

IN H/W AND S/W POWER Computers

Moores law for CPU

Disk space, memory

Software Sparse, iterative

Lanczos eigensolver

Domain decomposition

Pre- and post-processing Increasing resolution

Closer to realityMoravec, H., When Will Computer Hardware Match the Human Brain?

Robotics Institute Carnegie Mellon University

http://www.transhumanist.com/volume1/moravec.htm


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FEM CONTINUES TO IMPROVE

ABILITY TO SIMULATE REALITY Model resolution

Local details

Some things still

very difficult Joints

Expertise

Mesh size, etc.

FEM is not exact Big models do not guarantee accurate models

Thats why testing is still required!


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PRETEST ANALYSIS

Develop

FEM

Pretest

AnalysisTest

Posttest

Correlation


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Pretest Analysis

MODAL SURVEY OFTEN PERFORMED

TO VERIFY FINITE ELEMENT MODEL Must be confident that structure will survive

operating environment

Unrealistic to test flight structure to flight loads Alternate procedure

Test structure under controlled conditions

Correlate model to match test results

Use test-correlated model to predict operating responses

Modal survey performed to verify analysis model

Reality check


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Pretest Analysis - TAM

TEST AND ANALYSIS DATA HAVE

DIFFERENT NUMBER OF DOF Model sizes

FEM = 10,000-1,000,000 DOF

Test = 50-500 accelerometers

Compare test results to

analysis predictions

Need a common basis for

comparison

MOrtho T


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Pretest Analysis - TAM

TEST-ANALYSIS MODEL (TAM)

PROVIDES BASIS FOR COMPARISON Test-analysis model (TAM)

Mathematical reduction of finite element model

Master DOF in TAM corresponds to accelerometer

Transformation (condensation)

Many methods to perform reduction transformation

Transfo rmat ion method and senso r select ion c r i t ica l

for accu rate TAM and test-analysis compar isons

gaggT

gaaagaggT

gaaa TMTMTKTK


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Pretest Analysis - TAM Transformation Methods

GUYAN REDUCTION IS THE

INDUSTRY STANDARD METHOD Robert Guyan, Rockwell, 1965

Pronounced Goo-yawn, not Gie-yan

Implemented in many commercial software codes

NASTRAN, I-DEAS, ANSYS, etc.

Start with static equations of motion

Assume forces at omitted DOF are negligible

a

o

a

o

aaao

oaoo

P

P

U

U

KK

KK

0Po


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GUYAN REDUCTION IS A

SIMPLE METHOD TO IMPLEMENT Solve for motion at omitted DOF

Rewrite static equations of motion

Transformation matrix for Guyan reduction

aoa1

ooo UKKU

aaa

oa1

oo

a

oU

I

KK

U

U

aa

oa1

ooGuyan

I

KKT


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TRANSFORMATION VECTORS

ESTIMATE MOTION AT OTHER DOF

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1 2 3 4

Node ID

Displacement


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TRANSFORMATION VECTORS CAN

REDUCE OR EXPAND DATATAM

Display


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DISPLAY MODEL RECOVERED USING

TRANSFORMATION VECTORS

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1 2 3 4

Node ID

Enhance

dDisplay

-1.00

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1 2 3 4

Stan

dardDisplay


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IRS REDUCTION ADDS

FIRST ORDER MASS CORRECTION Guyan neglects mass effects at omitted DOF

IRS adds first order approximation of mass effects

aa

IRSGuyanGuyan

I

GGT

oa1

ooGuyan KKG

aa1aaGuyanoooa1ooIRS KMGMMKG


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DYNAMIC REDUCTION ALSO

ADDS MASS CORRECTION Start with eigenvalue equation

Replace eigenvalue with constant value

Equivalent to Guyan reduction if = 0

i

a

o

aaao

oaooii

a

o

aaao

oaoo

MM

MM

KK

KK

LL

aa

oaoa1

oooodReDyn

I

MKMKT


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MODAL TAM BASED ON

FEM MODE SHAPES Partition FEM mode shapes

Pseudo-inverse to form transformation matrix

ooUaaU

aa

T

a

1

a

T

aoModal

IT

aalmodo UTU


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EACH REDUCTION METHOD HAS

STRENGTHS AND WEAKNESSESADVANTAGES DISADVANTAGES

Easy to use, efficient Limited accuracy

Guyan Works well if good A-set Bad if poor A-set

Widely accepted Unacceptable for high M/K

Better than Guyan Requires DMAP alter

IRS Errors if poor A-set

Better than Guyan Requires DMAP alter

Dynamic Choice of Lamda?

Limited experience

Exact within freq. range Requires DMAP alter

Modal Hybrid TAM option Sensitivity

Limited experience


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STANDARD PRACTICE FAVORS

GUYAN REDUCTION Guyan reduction used most often

Easy to use and commercially available

Computationally efficient

Widely used and accepted

Good accuracy for many/most structures

Use other methods when Guyan is inadequate

Modal TAM very accurate but sensitive to FEM error

IRS has 1st order mass correction but can be unstable

Dynamic reduction seldom used (how to choose )


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Pretest Analysis - Sensor Placement

SENSOR PLACEMENT IMPORTANT

FOR GOOD TAM AND TEST Optimize TAM

Minimize reduction error

Optimize test

Get as much independent data as possible

Focus on uncertainties

High confidence areas need only modest instrumentation

More instrumentation near critical uncertain areas (joints)

Common sense and engineering judgement

General visualization of mode shapes


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MANY ALGORITHMS FOR

SENSOR PLACEMENT Kinetic energy

Retain DOF with large kinetic energy

Mass/stiffness ratio

Retain DOF with high mass/stiffness ratio

Iterated K.E. and M/K

Remove one DOF per iteration

Effective independence

Retain DOF that maximize observability of mode shapes

Genetic algorithm

Survival of the fittest!


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SENSOR PLACEMENT ALGORITHM

CLOSELY LINKED TO TAM METHOD Guyan or IRS reduction

Must retain DOF with large mass

Iterated K.E. or M/K

Mass-weighted effective independence

Modal or Hybrid reduction

Effective independence

Genetic algorithm offers best of all worlds

Examine tons of TAMs! Seed generation from other methods

Cost function based on TAM method


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PRETEST ANALYSIS ASSISTS

PLANNING AND TEST Best estimate of modes

Frequencies, shapes

Accelerometer locations Optimized by sensor placementstudies

TAM mass and stiffness

Real-time ortho and x-ortho

Frequency response functions

Dry runs/shakedown prior to test


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TEST CONSIDERATIONS

Develop

FEM

Pretest

AnalysisTest

Posttest

Correlation


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Test Considerations

PRETEST DATA ALLOWS

REAL-TIME CHECKS OF RESULTS Traditional comparisons

What if test accuracy goals arent met?

Keep testing (different excitement levels, locations, types)

Stop testing (FEM may be incorrect!)

Decide based on test quality checks

Experienced test engin eer extremely valuable!

testTAM

T

test MORTHO testTAM

T

TAM MXORTHO


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POSTTEST CORRELATION

Develop

FEM

Pretest

AnalysisTest

Posttest

Correlation


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Posttest Correlation

CORRELATION MUST BE FAST!

FEM almost always has some differences vs. test

Very limited opportunity to do correlation

After structural testing and data processing complete

Before operational use of model First flight of airplane

Verification load cycle of spacecraft

Need methods that are fast!

Maximum insight Accurate


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NO UNIQUE SOLUTION FOR

POSTTEST CORRELATION More unknowns than knowns

Knowns

Test data (FRF, frequencies, shapes at

test DOF, damping)

Measured global/subsystem weights

Unknowns

FEM stiffness and mass (FEM DOF)

No unique solu t ion

Seek best reasonable solution

When you

have

eliminated

theimpossible,

whatever

remains,

however

improbable,

must be

the truth.


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MANY CORRELATION METHODS

Trial-and-error

Stop doing this! It 's (almost)

the new m il lenium !

Too slow for fast-paced projects Not sufficiently insightful for

complex systems

FEM matrix updating

FEM property updating Error localization

FEM

Test OK?

Done

Updates


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MATRIX UPDATE METHODS

ADJUST FEM K AND M ELEMENTS Objective

Identify changes to FEM K and M so that analysis

matches test

Baruch and Bar-Itzhack (1978, 1982) Berman (1971, 1984)

Kabe (1985)

Kammer (1987)

Smith and Beattie (1991)

and many others

1100

1650

0572

0022

K

5.1000

05.200

005.10

0005.0

M


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MATRIX UPDATE METHODS

HAVE LIMITATIONS Lack of physical insight

What do changes in K, M coefficients mean?

Lack of physical plausibility

Baruch/Berman method doesn't enforce connectivity

Limitations for large problems

Great for small demo models, but ...

Smearing" caused by Guyan reduction/expansion

What if test article different than flight vehicle?

Requires very precise mode shapes (unrealistic)


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PROPERTY UPDATE METHODS

ADJUST MATERIALS AND ELEMENTS Objective

Identify changes to element and material

properties so that FEM matches test

Hasselman (1974) Chen (1980)

Flanigan (1987, 1991)

Blelloch (1992)

Smith (1995)

and many others* Calculate updates using

design sensitivity and optimization

FEM

Test OK?

Done

Updates*


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COMMERCIAL SOFTWARE

FOR CORRELATION SDRC/MTS

I-DEAS Correlation (MAC, ortho, x-ortho, mapping)

LMS

CADA LINK (parameter updating, Bayesian estimation)

MSC

SOL 200 design optimization (modes, FRF)

Dynamic Design Solutions (DDS)

FEMtools (follow-on to Systune)

Others (SSID, ITAP, etc.)


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MODE SHAPE EXPANSION

FOR CORRELATION IMPROVEMENTTAM

Display


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SHAPE EXPANSION IS AN

ALTERNATIVE TO MATRIX REDUCTION Expand test mode shapes to FEM DOF

Expansion and reduction give same results if samematrices used

Dynamic expansion based on eigenvalue equation

Computationally intensive

But computers are getting faster all the time!

agag UTU

i

aoa

i

oaoo

i

oo

i

o MKMK


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SUMMARY

FEM is a simple yet powerful method Complex structures from simple building blocks

FEM must make many assumptions

Joints, tolerances, linearity, mass, etc.

Big models do not guarantee accuracy

Testing provides a valuable reality check

Within limits of test article, excitation levels, etc.

FEM can work closely with test for mutual benefit

Pretest analysis to optimize sensor locations TAM for providing test-analysis comparison basis

Correlation and model updating for validated model


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FEM PEOPLE REALLY ARE SMART!

And maybe test people are smart too !


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Quartus Engineering


RECOMMENDED READING

Finite element method

Concepts and Applications of Finite Element Analysis, 3rd ed.; Cook,

Robert D./Plesha, Michael E./Malkus, David S.; John Wiley & Sons; 1989

Finite Element Procedures, Klaus-Jurgen Bathe; Prentice Hall; 1995

Correlation and model updating Finite Element Model Updating in Structural Dynamics; M. I. Friswell,

J. E. Mottershead; Kluwer Academic Publishers; 1995.

Optimization

Numerical Optimization Techniques for Engineering Design, 3rd edition

(includes software); Garret N. Vanderplaats, Vanderplaats Research &

Development, Inc., 1999

FEM for Test Eng

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