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Page 1
,!
i
NASA-CR-1921#O
DAMPING
CItARA CTERIZATION
IN LARGE
STRUCTURES
NASA/JPL
Grant No.: NAGW-1599
)P' t '7
FINAL TECItNICAL
REPORT
Performance
Period:
January
1,
1989
- December
31,
1990
Sulmdtled
by:
School of Engineering
and Architecture
Tuskegee
University
Tuskegee,
Alabama
36088
Principal
Investigator:
Co-Investigator:
Fidelis
O.
Eke,
Dept.
of
Mechanical
Engr.
Estelle
M.
Eke,
Dept.
of
Aerospace
Engr.
January
24,
1991
(NASA-CR-192140)
DAMPING
CHARACTERIZATION
IN
LARGE
STRUCTURES
Final
Technical
Report,
I Jan.
1999
-
31
Dec.
1990
(Tuskegee
Inst.)
17
p
N93-19944
Unc I as
G3/39
0145528
Page 2
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
TABLE
OF
CONTENTS
Executive
Summary
Introduction
Objective
and
Problem
Statement
Analysis
4.1
4.2
Subsystem
Assemblage
Process
Selection
of Component
Damping
Matrices
Minority
Education
Component
Presentations
and
Publications
Conclusion
Recommendation
for Future
Research
References
Page
No.
3
4
5
6
8
12
13
14
15
15
16
Page 3
1.0
EXECUTIVE
SUMMARY
This
research
project
has
as
its
main
goal
the
development
of
method(s)
for
selecting
the
damping
characteristics
of
components
of
a
large
structure
or
multibody
system,
in
such
a way
as
to
produce
some
desired
system
damping
characteristics.
The
main
need
for
such
an
analytical
device
is
in
the
simulation
of
the
dynamics
of
multibody
systems
consisting,
at
least
partially,
of
flexible
components.
The
reason
for
this
need
is
that
all
existing
simulation
codes
for
multibody
systems
require
component-
by-component
characterization
of
complex
systems,
whereas
requirements
(including
damping)
often
appear
at
the
overall
system
level.
The
main
goal
was
met
in
large
part
by
the
development
of
a
method
that
will
in
fact
synthesize
component
damping
matrices
from
a given
system
damping
matrix.
The
restrictions
to
the
method
are
that
the
desired
system
damping
matrix
must
be
diagonal
(which
is
almost
always
the
case)
and
that
interbody
connections
must
be
by
simple
hinges.
In
addition
to
the
technical
outcome,
this
project
contributed
positively
to
the
educational
and
research
infrastructure
of
Tuskegee
University
- a Historically
Black
Institution.
All
the
students
supported
under
this
grant
completed
their
degrees,
and
the
pieces
of
equipment
purchased
via
this
project
are
being
used
to
expand
research
efforts
in
System
and
Structural
Dynamics.
3
Page 4
2.0
INTRODUCTION
Many
engineering
systems
comprise
together,
with
active
control
between
bodies.
Specific
examples
of
such
systems
include
robots
and
manipulators,
space
vehicles,
missiles,
and
precision
pointing
systems.
Because
of
the
increasing
tendency
towards
lightweight
components,
many
such
systems
are
partially
or
totally
composed
of
flexible
bodies.
The
dynamics
of
such
systems
can
be
studied
by
experimentation
or
analysis,
or,
preferably,
both.
When
an
analytical
approach
is
used,
modeling
is
usually
one
of
the
first
issues
to
be
addressed.
In
the
study
of
a complex
structure
or
a system
of
interconnected
flexible
bodies,
most
modeling
strategies
rely
on
a
finite
dimensional
representation
of
each
flexible
component;
and
the
smaller
the
dimension,
the
more
tractable
the
analysis.
Structural
damping
is
one
of
the
most
poorly
understood
parameters
of
a
structure.
Very
often
it
is
simply
ignored.
When
this
is
not
possible,
such
as
when
one
is
interested
in
stability
issues
for
an
actively
controlled
structure,
damping
is
introduced
in
an
ad
hoc
fashion,
usually
in
the
form
of
a system
damping
matrix,
which
is
assumed
to
be
diagonal.
A
rule
of
thumb
is
then
used
to
assign
values
to
the
diagonal
elements,
which
generally
represent
the
damping
ratio
corresponding
to
each
retained
mode
of
the
structure.
several
bodies
connected
There
are
situations
where
one
is
compelled
to
work
with
components
of
a
structure.
Such
a situation
may
arise
in
the
analysis
of
a
large
structure
such
as
an
aircraft
or
a
space
station;
here,
it
is
common
practice
to
assign
different
components
of
the
structure
to
different
analysts.
And,
if
modal
viewpoint
is
adopted,
modal
information,
including
damping
information
is
needed
at
the
component
level.
A
similar
situation
arises
when
it
is
desired
to
simulate
the
motions
of
a system
of
interconnected,
actively
controlled
flexible
bodies,
using
a
simulation
package
such
as
DISCOS[I]
or
TREETOPS[2].
These
programs
require
that
each
body
in
a given
system
be
characterized
Page 5
separately.
That is, mass, stiffness,
and damping matrices
of
each component of the system must be supplied
separately
to the
program.
Generally,
it is desired
to have a diagonal
damping
matrix
for the whole system with each element having a specified
value
(usually
1%).
Knowing what is desired
for the system
damping matrix,
there still
remains a major task of determining
the values that must be assigned to the component damping matrices
such that when they are assembled, they yield
the desired
system
damping matrix.
Experience
with structural
analysis
and
simulation
of the Galileo
spacecraft[3,4]
has shown that
using
component
damping
matrices
that
are
diagonal
leads
to
a system
damping
matrix
that
is
far
from
being
sparse.
3.0
OBJECTIVE
AND
PROBLEM
STATEMENT
The
principal
objective
of
this
research
project
is
to
search
for
a reliable,
systematic,
and
efficient
procedure
for
generating
the
damping
matrices
that
must
be
assigned
to
the
components
of
a given
large
structure
so
that
the
damping
matrix
of
the
structure
as
a whole
(system
damping
matrix)
has
any
desired
form
and
content.
The
secondary
objective
is
to
initiate
a fundamental
re-evaluation
of
current
methods
of
representing
damping
in
structures,
and
indicate
a path
for
future
research
in
this
area.
The
problem
to
be
solved
is
really
an
offshoot
of
a bigger
problem.
The
big
problem
is
that
of
simulating
the
dynamics
of
a
system
of
coupled
rigid/flexible
bodies.
This
simulation
problem
can
be
solved
with
the
aid
of
one
of
the
existing
multibody
simulation
codes
such
as
DISCOS
or
TREETOPS.
In
order
to
use
these
codes,
the
system
is
usually
modeled
in
a NASTRAN-Iike
environment,
so
that
mass,
stiffness,
and
modal
matrices
(among
other
quantities)
are
available
for
the
free-free
vibration
modes
of
each
flexible
body
in
the
system.
Additionally,
these
codes
require
that
a damping
matrix
be
available
for
each
flexible
body
in
the
system.
Since
NASTRAN
does
not
produce
damping
matrices,
5
Page 6
these
component
damping
matrices
must
be
supplied
by
the
analyst.
In general,
it
is desired
that
the
damping
matrix
for
the
whole
system
viewed
as
one,
be
a diagonal
matrix
whose
elements
represent
the
damping
ratios(usually
1%)
for
the
retained
modes.
To
achieve
this
goal,
the
damping
matrices
for
the
flexible
components
in the
system
must
be
selected
judiciously.
These
matrices
cannot
be arbitrary;
they
cannot
even
be
diagonal.
What
is attempted
here,
therefore,
is to
find
a
scientifically
sound
method
or methods
for selecting
the
elements
of
the
component
damping
matrices
so that
the
requirements
on the
system
damping
matrix
are met.
4.0
ANALYSIS
Consider
a system
S
consisting
of n subsystems
S i
(i=l,2,...,n)
connected
together
as
shown
in Fig.
I.
For
each
of
the
subsystems
Si,
it
is possible
to write:
M_i + C_i + Kixl - t_
(1)
And
if body
i has
n i degrees
of
freedom,
then
Mi,
Ci,
and K i have
F_
F2
S2
F3
S/__
dimension
n i by
ni,
and
are
the
mass,
damping
and
stiffness
matrices
respectively.
F i and
x i are
row
vectors
of dimension
n i
by
1 and
represent
the
forcing
function
and
displacement
vector
respectively.
It is also
possible
to view
the
whole
system
as one
structure,
and
write
Page 7
M_ + C:_ + Kx =F
(2)
Suppose
that
modal
analyses
was
performed
for
equation
(2) to
produce
the
system
modal
matrix
u.
This
implies
the
coordinate
transformation
x = uq
(3)
Equation
(3) can
now
be used
to transform
(2) into
l't] + c dl + kq = uTF
(4)
where
I is an
identity
matrix,
k is a diagonal
matrix
with
kj=4
(5)
and
C = uTCu
(6)
Normally,
c is not
diagonal;
but
it is common
practice
to
assume
that
it is, with
q = 2_jcoj
(7)
where
_j is the
damping
ratio
corresponding
to the
jth
mode
of the
system.
If it should
become
necessary
to
reconstruct
the
C matrix
from
c, this
can
be
done
by pre-
and post-multiplying
equation
(6)
by
u and
u T respectively:
C - ucuV
(8)
Similarly,
a modal
matrix
u i can be
found
for
each
subsystem,
so
7
Page 8
that
equation
(i)
can
also
be
transformed
into
llqi + C_i + kiqi = tTiFi
(9)
where,
as
usual,
I i is
an
identity
matrix,
k i
is
a
diagonal
matrix
and
Ci = uTCitl i
(l O)
System
NASTRAN
models
can
be
used
to
generate
k i,
u i,
as
well
as
k
and
u.
To
characterize
the
system,
multibody
simulation
codes
can
only
accept
subsystem
information
as
input.
So
that
a
given
simulation
problem
will
require
that
k i,
u i and
c i
(or
M i,
K i and
C i)
be
available,
k i and
u i are
readily
obtainable
from
NASTRAN
output,
but
c i will
have
to
be
determined
by
the
analyst.
In
general,
the
goal
is
to
pick
the
elements
of
c i
in
such
a way
that
the
system
damping
matrix
c
is
diagonal,
with
the
damping
ratio
for
each
mode
having
the
constant
value
of
about
1%.
In
other
words,
it
is
desired
to
influence
the
elements
of
c through
those
of
the
matrices
ci.
To
do
this
effectively,
it
is
important
to
understand
the
relationship
between
the
ci's
an
c.
That
is,
it
is
necessary
to
examine
the
mathematics
of
the
process
by
which
a
given
multibody
code
assembles
its
subsystems
into
the
full
system.
4.1
5_U_b__y_tem
As_9_blage
Process
The
analytical
basis
of
the
component
assemblage
process
leading
to
the
construction
of
the
system
damping
matrix
from
component
damping
matrices
is
illustrated
below
with
simple
examples.
8
Page 9
Consider
the
two
planar
systems
A and
B shown
in Figure
2.
Each
of the
systems
consists
of two
rigid
rods
connected
together
by
a one
degree
of
freedom
hinge;
and motion
about
the
hinge
is
restricted
by
a torsional
spring
and
damper
system.
All
motions
of
the
systems
are
restricted
to a plane.
Each
such
system
can be
viewed
as
a simple
flexible
body.
Equations
of motion
for
A and
B
can
be written
in matrix
form
as
AI ' IAI
TAI
mBl,
IBI
Fig.
2
Two
Simple
Flexible
Bodies
]At
0
0
IA2
[ ll[bA
-hA
_2
-bh
bA
_lI [ kA
-kA
+
6_2
-kA
kA
iTA'
=
Or2
TA2
(11)
and
0
IB2
32
-bB
bB
132
-kB
kB
_2
TB2
(12)
respectively.
Page 10
Now,
consider
A
and
B
connected
together
at
point
P to
form
one
system
S shown
in
Figure
3 below.
There
are
at
least
three
ways
in
which
the
connection
at
P can
be
implemented:
•
rigid
connection
•
frictionless
hinge
•
hinge
connection
with
spring
and
dashpot.
The
first
option
is
of
no
interest
here.
Assuming
a frictionless
hinge,
the
equations
of
motion
of
S,
when
viewed
as
one
system
reAl,
IAI
I
TA2
mBl,
IBI
i
k
B2'
IB2
TB2
Fig.3
Combined
System
S
becomes
i0
Page 11
IA1
0
I 0
0
I 0
0
0
- ]A2 !
0
0
_ In1
0
0
0
i 0
1B2
..
01..
[
bA
-bA i 0
0
02
+.-bA
bA
0
0
03
0
0
I b.
-b.
..
0
0
a-br_
b.
_04.
Oll
+
-kA
kA I 0
0
02
=|
TA2
"qkn-
03
o
° r-k"
/ T.1
0
0
I-kB
kB
k TB2
_ 04J
01
02
03
.04_
(13)
Comparing
equations
(Ii) and
(12)
with
equation
(13) partitioned
as shown,
it is
immediately
evident
that
the
damping
matrices
in
equations
(ii)
and
(12) are
exactly
the
diagonal
"elements"
of
the
system
damping
matrix
shown
in equation
(13).
There
is,
thus,
a
one-to-one
mapping
between
the
elements
of the
damping
matrices
of
components
A and
B and
the
elements
of the
diagonal
submatrices
of
the
system
damping
matrix.
If the
connection
at P between
A and
B is modified
to
include
a torsional
spring-dashpot
system,
the
equations
of motion
are
modified
somewhat
and
is given
as equation
(14) below.
f
IAI
0
10
0
IA2 I 0
0
0
I IB1
0
0
I 0
kA
+ ,
-kA
0
0
01
[ bA
+ .
-bA
°-
(o-
IB2
0
.04.
-k A
I
0
0
kA+k
I
-k
0
-k
'kB + k-" --_,-
0
I -k.
kB
-bA
I
0
0
-]
Ol.
bA + b
I
-b
0
"1 02
b
03
0
I -bB
bB
.
_04
02=
TA2
03 I
TBI
TB2
_04_1
(14)
Here,
both
the
diagonal
and
the
off-diagonal
submatrices
of the
system
damping
matrix
are
affected.
Note,
however,
that
if b = 0
(no damping
at the
joint),
then
the
one-to-one
mapping
described
Ii
Page 12
earlier
is recovered.
In practice,
damping is rarely
included
at
such hinge connections
of multibody
systems.
Hence, it is
concluded
that
for hinge-connected
systems,
changes
in component
damping
matrices
have
direct
effect
on the
diagonal
submatrices
of
the
system
damping
matrix•
These
effects
are
quantifiable
following
the
relationships
given
in equations
(ii),
(12),
and
(13).
4.2
Selection
of Component
Damping
Matrices
As
stated
earlier,
a multibody
system
containing
flexible
components
can be
viewed
as one
structure;
and
can
therefore
be
represented
by
equation
(2) or equation
(4).
Given
a desired
damping
matrix
for the
system
as a whole,
our
goal
is to determine
the
component
damping
matrices
that
will
produce
the
desired
system
damping
matrix.
The
analyses
presented
in Section
4.1
above
indicate
a clear
path
to the
solution
of the
problem
if the
matrix
C of equation
(2) is the
known
or desired
system
damping
matrix.
However,
this
is generally
not
the
case
in practice.
Normally,
it is the
diagonal
matrix
c of equation
(4) that
is
prescribed.
Each
of
its diagonal
element
is assumed
to be
equal
to
2_i_ i, where
_i
is the
natural
frequency
corresponding
to
the
ith
mode,
and
_i is taken
to be
about
1%.
Thus,
it is assumed
here
that
dt
O]
C =
d2
0
d.
(15)
with
all
the
di's
known.
The matrix
C can
be
found
by using
equation
(8).
The
elements
of C can
thus
be
shown
to be
Cij = _'_ UikUjkdk
k=l
(16)
12
Page 13
where
the
uij's
are
elements
of
the
system
modal
matrix.
The
matrix
C whose
elements
are
given
by
equation
(16),
is
now
partitioned
according
to
the
number
of
degrees
of
freedom
of
each
of
the
components
( see
the
partitioning
scheme
used
in
equation
(13)).
The
ith
diagonal
submatrix
of
C
contains
precisely
the
elements
of
the
undiagonalized
damping
matrix
of
body
i.
In
summary,
the
selection
strategy
consists
of
the
following
steps:
.
,
.
.
Assign
values
to
system
modal
damping
ratios;
this
determines
the
elements
d i of
the
system's
diagonal
damping
matrix
c;
Determine
the
elements
of
the
undiagonalized
system
damping
matrix
C
using
equation
(16);
Partition
the
C matrix
according
to
the
components'
degrees
of
freedom;
The
elements
of
the
undiagonalized
component
damping
matrix
C i for
body
i is
identical
to
the
ith
diagonal
submatrix
of
C.
Note
that
the
component
damping
matrices
that
emerge
from
this
process
are
the
Ci's
and
not
ci's.
This
implies
that
component
information
will
then
have
to
be
supplied
to
the
multibody
simulation
code
in
the
form
of
Mi,
Ci,
and
Ki.
All
the
codes
that
we
know
of
can
accept
component
data
in
this
form.
5.0
MINORITY
EDUCATION
COMPONENT
One
of
the
most
successful
aspects
of
this
project
was
its
education
component.
It
was
particularly
successful
in
exposing
students
and
faculty
at
Tuskegee
University
to
a current
NASA
research
topic.
Three
faculty
members,
two
graduate
students
and
one
undergraduate
student
participated
directly
in
this
project.
The
two
graduate
students
received
their
M.S
degrees
at
Tuskegee
University
with
at
least
partial
funding
from
this
project.
The
undergraduate
student
turned
out
to
become
the
computer
expert
for
13
Page 14
the group; he has also graduated with a B.S. degree in
Mechanical/Aerospace
Engineering
(dual Major).
This project
also contributed
positively
to the research
infrastructure
at Tuskegee University.
The grant made it possible
to purchase some
critical
computer
hardware
and
software
that
were
used
to
start
a
small
Laboratory
in
Systems
and
Structural
Dynamics.
Personnel
Utilized
Senior
Personnel
Dr.
Fidelis
Eke
(Mechan.
Engr.)
- Principal
Investigator
Dr.
Estelle
Eke
(Aerosp.
Engr.)
- Co-Investigator
Dr.
Olusegun
Adeyemi
(Mech.
Engr.)
-
Senior
Investigator
Graduate
Students
Supported
Mr.
Busty
Okundaye
(Mechan.
Engr.)
Mr.
Sheng-Fang
Shen
(Mechan.
Engr.)
Undergraduate
Student
Supported
•
Mr.
Steven
Hill
(Mechan/Aero.
Engr.)
6.0
PRESENTATIONS
AND
PUBLICATIONS
A
presentation
of
some
early
results
of
this
project
was
made
at
the
Sixty-Seventh
Annual
Meeting
of
the
Alabama
Academy
of
Sciences
in
March
1990
at
Mobile,
Alabama.
An
abstract
of
this
work
is
being
submitted
to
the
AAS/AIAA
Conference
Committee
for
presentation
at
the
August
1991
Astrodynamics
Conference
in
Durango,
Colorado.
It
is
planned
to
submit
the
same
material
to
the
AIAA
Journal
of
Dynamic
Systems
and
Control.
14
Page 15
7 .0
CONCLUSION
The main goal of this
research
was achieved.
Some insight
has
been gained into the factors
governing
the selection
of
component
damping
matrices
for
interconnected
multibody
systems.
Specifically,
a workable
selection
strategy
was
developed
for
the
case
where
the
interconnection
between
bodies
is
through
frictionless
hinges
-
this
is
the
normal
assumption
in
most
aerospace
applications.
This
project
was
also
quite
successful
in
exposing
students
and
Faculty
in
a Historically
Black
University
to
a current
NASA
research
effort,
and
contributed
to
the
development
of
research
infrastructure
at
the
University.
8.0
RECOMMENDATION
FOR
FUTURE
RESEARCH
Like
all
engineering
results
and
techniques,
the
component
damping
characterization
method
developed
as
a result
of
this
research
effort
cannot
solve
all
possible
damping
characterization
problems
under
any
circumstances.
The
two
main
limitations
are
that
the
desired
system
damping
matrix
must
be
diagonal,
and
the
inter-component
connections
must
be
by
frictionless
hinges.
These
restrictions
do
not
constitute
major
shortcomings
since
many
engineering
systems
(and
most
aerospace
systems)
actually
satisfy
the
above
conditions.
Nevertheless,
it
is
recommended
that
the
results
obtained
be
extended
to
systems
with
other
than
hinge
connections.
It
is
our
belief
that
this
is
not
only
feasible,
but
it
can
be
done
relatively
easily
from
the
current
results.
At
a more
fundamental
level,
it
is
recommended
that
studies
be
undertaken
to
quantify
the
actual
impact
on
multibody
simulation
results
of
errors
in
component
damping,
with
a possible
view
to
developing
"robust"
simulation
packages.
15
Page 16
9.0
REFERENCES
i.
Bodley,
C. S., Devers, A. D., Park, A. C., and Frisch,
H. P.,
"A Digital
Computer Program for the Dynamic Interaction
Simulation
of Controls
and Structures
(DISCOS)," NASA Technical
Paper 1219,
Vols.
I and II,
May 1978.
2.
Singh, R. P., VanderVoort,
R. J., and Likins,
P. W., "Dynamics
of Flexible
Bodies in Tree-Topology
- A
Computer
Oriented
Approach,"
Paper
Number
AIAA-84-1024,
AIAA/ASME/ASCE
25th
Structures,
Structural
Dynamics,
and
Materials
Conference,
Palm
Springs,
Ca.,
May,
1984.
3.
Eke,
F.
O.,
and
Man,
G.
K.,
"Model
Reduction
in
the
Simulation
of
Interconnected
Flexible
Bodies,"
Paper
No.
AAS
87-455,
AAS/AIAA
Astrodynamics
Specialist
Conference,
Kalispell,
Montana,
August
1987.
4.
Eke,
F.
O.,
Macala,
G.
A.,
Man,
G.
K.,
"Impact
of
Flexibility
on
the
Control
Loops
of
a Flexible
Spacecraft,"
Paper
No.
AAS
85-
363,
AAS/AIAA
Astrodynamics
Specialist
Conference,
Vail,
Colorado,
August
1985.
16