Urban and Civil

Nov 25, 2013 (4 years and 7 months ago)

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Quartus Engineering


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


Quartus Engineering Incorporated, 2000.

QUARTUS ENGINEERING WEB SITE

http://www.quartus.com

Quartus Engineering


Quartus Engineering Incorporated, 2000.

FEM PEOPLE ARE REALLY SMART

Or so they would have you believe!

Quartus Engineering


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


Quartus Engineering Incorporated, 2000.

Quartus Engineering


Quartus Engineering Incorporated, 2000.

There’s Reality, and Then There’s FEM

REALITY IS VERY COMPLICATED!

Many complex subsystems

Unique connections

Nonlinearities

Flight
-
to
-
flight variability

Chaos

Extremely high order behavior

Quartus Engineering


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

)

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


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

Quartus Engineering


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


Quartus Engineering Incorporated, 2000.

FEM

in a Nutshell

Quartus Engineering


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


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


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

1
3
5
7
9
11
13
15
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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
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


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


Quartus Engineering Incorporated, 2000.

FEM in a Nutshell

ELEMENT STIFFNESS MATRIX

FORMED USING SHAPE FUNCTIONS

Element stiffness matrix

Relates deflections of elemental DOF

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


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


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
1
0
0
1
6
5
0
0
5
7
2
0
0
2
2
K

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


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


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


Quartus Engineering Incorporated, 2000.

FEM in a Nutshell

HONORARY DEGREE IN FEM
-
OLOGY!

Quartus Engineering


Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer

FEM STRENGTHS AND CHALLENGES

Quartus Engineering


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


Quartus Engineering Incorporated, 2000.

FEM Strengths and Challenges

FEM HAS MANY CHALLENGES

Mesh refinement

How many elements required?

Material properties

A
-
basis, B
-
basis, etc.

Composites

Dimensions

Tolerances, as
-
manufactured

Joints

Fasteners, bonds, spot welds

continued...

Quartus Engineering


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


Quartus Engineering Incorporated, 2000.

FEM Strengths and Challenges

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


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


Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer

PRETEST ANALYSIS

Develop

FEM

Pretest

Analysis

Test

Posttest

Correlation

Quartus Engineering


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


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


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


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.

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


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


Quartus Engineering Incorporated, 2000.

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
Quartus Engineering


Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods

TRANSFORMATION VECTORS CAN

REDUCE OR EXPAND DATA

TAM

Display

Quartus Engineering


Quartus Engineering Incorporated, 2000.

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
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


Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods

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


Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods

DYNAMIC REDUCTION ALSO

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
K
K

L

L

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


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


Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

TAM Transformation Methods

EACH REDUCTION METHOD HAS

STRENGTHS AND WEAKNESSES

Easy to use, efficient
Limited accuracy
Guyan
Works well if good 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


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


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


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


Quartus Engineering Incorporated, 2000.

Pretest Analysis
-

Sensor Placement

SENSOR PLACEMENT ALGORITHM

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


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


Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer

TEST CONSIDERATIONS

Develop

FEM

Pretest

Analysis

Test

Posttest

Correlation

Quartus Engineering


Quartus Engineering Incorporated, 2000.

Test Considerations

PRETEST DATA ALLOWS

REAL
-
TIME CHECKS OF RESULTS

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


Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer

POSTTEST CORRELATION

Develop

FEM

Pretest

Analysis

Test

Posttest

Correlation

Quartus Engineering


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

Need methods that are fast!

Maximum insight

Accurate

Quartus Engineering


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


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

Quartus Engineering


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
6
5
0
0
5
7
2
0
0
2
2
K

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


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


Quartus Engineering Incorporated, 2000.

Posttest Correlation

PROPERTY UPDATE METHODS

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

design sensitivity and optimization

FEM

Test

OK?

Done

Quartus Engineering


Quartus Engineering Incorporated, 2000.

Posttest Correlation

COMMERCIAL SOFTWARE

FOR CORRELATION

SDRC/MTS

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

LMS

MSC

SOL 200 design optimization (modes, FRF)

Dynamic Design Solutions (DDS)

FEMtools (follow
-
on to Systune)

Others (SSID, ITAP, etc.)

Quartus Engineering


Quartus Engineering Incorporated, 2000.

Posttest Correlation

MODE SHAPE EXPANSION

FOR CORRELATION IMPROVEMENT

TAM

Display

Quartus Engineering


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


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


Quartus Engineering Incorporated, 2000.

FEM PEOPLE REALLY ARE SMART!

And maybe test people are smart too!

Quartus Engineering


Quartus Engineering Incorporated, 2000.

FEM for the Test Engineer

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,