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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Quartus Engineering CopyrightQuartus Engineering Incorporated, 2000.
DOWNLOAD FROM THEQUARTUS ENGINEERING WEB SITE
http://www.quartus.com
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Quartus Engineering CopyrightQuartus Engineering Incorporated, 2000.
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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Quartus Engineering CopyrightQuartus Engineering Incorporated, 2000.
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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Quartus Engineering CopyrightQuartus 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
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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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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 for the Test Engineer
FEM STRENGTHS AND CHALLENGES
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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 DynamicsControl-Structure Interaction Testbed
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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...
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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!
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FEM Strengths and Challenges
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 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
Thats why testing is still required!
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FEM for the Test Engineer
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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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
aoa1
ooo UKKU
aaa
oa1
oo
a
oU
I
KK
U
U
aa
oa1
ooGuyan
I
KKT
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Pretest Analysis - TAM Transformation Methods
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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Pretest Analysis - TAM Transformation Methods
TRANSFORMATION VECTORS CAN
REDUCE OR EXPAND DATATAM
Display
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Pretest Analysis - TAM Transformation Methods
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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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
IRSGuyanGuyan
I
GGT
oa1
ooGuyan KKG
aa1aaGuyanoooa1ooIRS KMGMMKG
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Pretest Analysis - TAM Transformation Methods
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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Pretest Analysis - TAM Transformation Methods
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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Pretest Analysis - TAM Transformation Methods
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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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 )
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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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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!
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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
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Pretest Analysis - Sensor Placement
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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FEM for the Test Engineer
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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FEM for the Test Engineer
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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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 solu t ion
Seek best reasonable solution
When you
have
eliminated
theimpossible,
whatever
remains,
however
improbable,
must be
the truth.
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Posttest Correlation
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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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
1100
1650
0572
0022
K
5.1000
05.200
005.10
0005.0
M
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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)
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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*
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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.)
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Posttest Correlation
MODE SHAPE EXPANSION
FOR CORRELATION IMPROVEMENTTAM
Display
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Posttest Correlation
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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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
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FEM PEOPLE REALLY ARE SMART!
And maybe test people are smart too !
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Quartus Engineering
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
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