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Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM

FOR 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

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

DOWNLOAD FROM THE

QUARTUS ENGINEERING WEB SITE

http://www.quartus.com

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM PEOPLE ARE REALLY SMART


Or so they would have you believe!

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


TOPICS


There’s reality, and then there’s FEM


FEM in a nutshell


FEM strengths and challenges


Pretest analysis


Model reduction


Sensor placement


Posttest analysis


Correlation


Model updating

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

There’s Reality, and Then There’s FEM


REALITY IS VERY COMPLICATED!


Many complex subsystems


Unique connections


Advanced materials


Broadband excitation


Nonlinearities


Flight
-
to
-
flight variability


Chaos


Extremely high order behavior

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

There’s Reality, and Then There’s FEM


FEM ATTEMPTS TO

SIMULATE REALITY


Fortunately, reality is
surprisingly linear


Material properties (


v献s

)


Tension vs. compression


Small deflections (sin




)


Load versus deflection


Allows reasonable
opportunity simulate reality
using FEM

-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
-1
-0.5
0
0.5
1
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

There’s Reality, and Then There’s 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 it’s the best approach we have


Experience shows that FEM works well when used properly

FEM

Ahead!

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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 (don’t break it!)


Nonlinearities


Testing limitations


Sensor accuracy and calibration


Data processing


But it’s the best “reality check” available

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM

in a Nutshell

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


FEM IN A NUTSHELL


Divide and conquer!


Shape functions


Elemental stiffness and mass matrices


Assembly of system matrices


Solving


Related topics


Element library


Superelements

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM in a Nutshell


CLOSED FORM SOLUTIONS, ANYONE?


Consider a building


Steel girders


Concrete foundation


Can you write an equation to
fully describe the building?


I can’t!


Even if possible, probably not
the best approach


Very time consuming


One
-
time solution

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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

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S1
S3
S5
S7
S9
S11
S13
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S19
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
0.80-1.00
0.60-0.80
0.40-0.60
0.20-0.40
0.00-0.20
-0.20-0.00
-0.40--0.20
-0.60--0.40
-0.80--0.60
-1.00--0.80
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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 that’s what Ph.D.’s are for!


Extensive research


Still evolving (MSC.NASTRAN V70.7)

Spring

F = K X

F

X

K

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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 unit
deflection imposed at DOF
i

and
other DOF
j

are fixed


Example: linear spring (2 DOF)

Spring

F = K X

F

X

K










K
K
K
K
K
spring
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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








M
5
.
0
0
0
M
5
.
0
M
spring
1/4

1/4

1/4

1/4

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM in a Nutshell



SYSTEM MATRICES FORMED

FROM ELEMENT MATRICES

K = 2

K = 5

K = 1

M = 1

M = 2

M = 3










2
2
2
2
K
1









5
5
5
5
K
2









1
1
1
1
K
3



















1
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0
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5
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2
2
K













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M
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM in a Nutshell



CALCULATE SYSTEM STATIC

AND DYNAMIC RESPONSES


Static analysis




Normal modes analysis




Transient analysis

P
q
K
q
C
q
M
T
T
T
T















0
M
K
i
i




X
K
P

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM in a Nutshell


COMMERCIAL FEM ISSUES


Element libraries


Springs, rods, beams, shells, solids, rigids, special


Linear and parabolic (shape functions, vertex nodes)


Commercial codes


NASTRAN popular for linear dynamics (aero, auto)


ABAQUS and ANSYS popular for nonlinear


Superelements (substructures)


Simply a collection of finite elements


Special capabilities to reduce to boundary nodes


Assemble system by addition I/F nodes

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM in a Nutshell


HONORARY DEGREE IN FEM
-
OLOGY!

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


FEM STRENGTHS AND CHALLENGES

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM Strengths and Challenges


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 Dynamics

Control
-
Structure Interaction Testbed

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM Strengths and Challenges


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...

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM Strengths and Challenges


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!

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM Strengths and Challenges


FEM ASSISTED BY ADVANCES

IN H/W AND S/W POWER


Computers


Moore’s law for CPU


Disk space, memory


Software


Sparse, iterative


Lanczos eigensolver


Domain decomposition


Pre
-

and post
-
processing


Increasing resolution


Closer to reality

Moravec, H., “When Will Computer Hardware Match the Human Brain?”

Robotics Institute Carnegie Mellon University

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

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM Strengths and Challenges


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


That’s why testing is still required!

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


PRETEST ANALYSIS

Develop

FEM

Pretest

Analysis

Test

Posttest

Correlation

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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”

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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




M
Ortho
T
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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


Transformation method and sensor selection critical
for accurate TAM and test
-
analysis comparisons

ga
gg
T
ga
aa
ga
gg
T
ga
aa
T
M
T
M
T
K
T
K


Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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
aa
ao
oa
oo
P
P
U
U
K
K
K
K
0
P
o

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


GUYAN REDUCTION IS A

SIMPLE METHOD TO IMPLEMENT


Solve for motion at omitted DOF




Rewrite static equations of motion





Transformation matrix for Guyan reduction

a
oa
1
oo
o
U
K
K
U



a
aa
oa
1
oo
a
o
U
I
K
K
U
U
























aa
oa
1
oo
Guyan
I
K
K
T
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


TRANSFORMATION VECTORS

ESTIMATE MOTION AT “OTHER” DOF

-0.2
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1.0
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Node ID
Displacement
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


TRANSFORMATION VECTORS CAN

REDUCE OR EXPAND DATA

TAM

Display

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


DISPLAY MODEL RECOVERED USING
TRANSFORMATION VECTORS

-1.00
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-0.25
0.00
0.25
0.50
0.75
1
2
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Node ID
Enhanced Display
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1
2
3
4
Standard Display
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


IRS REDUCTION ADDS

FIRST ORDER MASS CORRECTION


Guyan neglects mass effects at omitted DOF


IRS adds first order approximation of mass effects









aa
IRS
Guyan
Guyan
I
G
G
T
oa
1
oo
Guyan
K
K
G





aa
1
aa
Guyan
oo
oa
1
oo
IRS
K
M
G
M
M
K
G




Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


DYNAMIC REDUCTION ALSO

ADDS MASS CORRECTION


Start with eigenvalue equation




Replace eigenvalue with constant value
L





Equivalent to Guyan reduction if
L

㴠=

i
a
o
aa
ao
oa
oo
i
i
a
o
aa
ao
oa
oo
M
M
M
M
K
K
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K








































L

L




aa
oa
oa
1
oo
oo
d
Re
Dyn
I
M
K
M
K
T
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


MODAL TAM BASED ON

FEM MODE SHAPES


Partition FEM mode shapes





Pseudo
-
inverse to form transformation matrix




o
o
U



a
a
U
















aa
T
a
1
a
T
a
o
Modal
I
T
a
al
mod
o
U
T
U

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


EACH REDUCTION METHOD HAS

STRENGTHS AND WEAKNESSES

ADVANTAGES
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
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods


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
L
)

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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

Quartus Engineering

Copyright
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Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

Sensor Placement


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!

Quartus Engineering

Copyright
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Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

Sensor Placement


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

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

Sensor Placement


PRETEST ANALYSIS ASSISTS

PLANNING AND TEST


Best estimate of modes


Frequencies, shapes


Accelerometer locations


Optimized by sensor placement

studies


TAM mass and stiffness


Real
-
time ortho and x
-
ortho


Frequency response functions


Dry runs/shakedown prior to test

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


TEST CONSIDERATIONS

Develop

FEM

Pretest

Analysis

Test

Posttest

Correlation

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Test Considerations


PRETEST DATA ALLOWS

REAL
-
TIME CHECKS OF RESULTS


Traditional comparisons






What if test accuracy goals aren’t met?


Keep testing (different excitement levels, locations, types)


Stop testing (FEM may be incorrect!)


Decide based on test quality checks


Experienced test engineer extremely valuable!

test
TAM
T
test
M
ORTHO



test
TAM
T
TAM
M
XORTHO



Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


POSTTEST CORRELATION

Develop

FEM

Pretest

Analysis

Test

Posttest

Correlation

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

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

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


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 solution


Seek “best” reasonable solution


When you
have
eliminated
the
impossible,
whatever
remains,
however
improbable,
must be

the truth
.”

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


MANY CORRELATION METHODS


Trial
-
and
-
error


Stop doing this! It's (almost)

the new millenium!


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

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


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




















1
1
0
0
1
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5
0
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5
7
2
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0
2
2
K













5
.
1
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5
.
2
0
0
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1
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0
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.
0
M
Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


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)

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


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*

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


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.)

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


MODE SHAPE EXPANSION

FOR CORRELATION IMPROVEMENT

TAM

Display

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

Posttest Correlation


SHAPE EXPANSION IS AN

ALTERNATIVE TO MATRIX REDUCTION


Expand test mode shapes to FEM DOF




Expansion and reduction give same results if same
matrices used


Dynamic expansion based on eigenvalue equation



Computationally intensive


But computers are getting faster all the time!

a
ga
g
U
T
U





i
a
oa
i
oa
oo
i
oo
i
o
M
K
M
K








Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


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

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM PEOPLE REALLY ARE SMART!


And maybe test people are smart too!

Quartus Engineering

Copyright

Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer


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