They wrecked out of the center of acoustic
and the reason they stayed Yankees but

you know if I'm right or
not you said that's right.

They might be kind of Ph D.
in our species and here.

I do see a before joining in stay
here position at Sparta Roku and

sidestep and I systems.

I need some gusts or
veins and are going national

research interesting to structure they
now means the first three systems.

You really busted down Being
a few structures energy harvested

dot dot dot dot.

This year.
Service P.R. officer a major part of

programs to get up the structures has
received by society based paper awards.

He's a fellow of a I've made and
served on service on the A couple

a board of directors he was a member
of the mighty You have been here of

the recent National Research Council you
study for the NASA Space Technology hold

on a different side he he did very
interesting things he's not that in there.

Yes I am so he wants.

Bob look and know you from Montreal to the
Gulf of Mexico which you find used to be

amazing and he also a fifty mile automatic

he's going to talk today about dumping
more than four C. of beams without you.

Cation to speak craft.

Spacecraft wiring harness this
well thank you very much.

Thanks for the very nice introduction.

I have to say this is my first time
on the Georgia Tech campus and

I'm truly delighted to be here.

Yes it would be helpful.

I can't see them playing with the fact I
was going to be a little tricky working

with the laser pointer over
the shoulder to go looking.

But simply putting them at risk and
they were delighted to be here this is

one of the premier aerospace engineering
institutions in the US in the world and

it's really an honor for me to us to
be addressed to you today from now.

Tell you about some of the work
we've done in the last couple of.

Years on this particular topic of damping
as a road relates to a particular

aspect of space craft structural dynamics
want to acknowledge the contributions of

Jeff Kaufman who's recently graduated
from Penn State and started his career

as an assistant professor at
the University of Central Florida this

work is not his Ph D. dissertation work
we sort of took this up on the side for

fun because we like doing this stuff.

So here's what I'm going to talk
about fundamentally from a mic

mechanics point of view I assume
that I'm I'm so happy that so

many graduate students seem to be here and
faculty members but

I assume the grad students are here
because they have some interest in

the structural mechanics side of the House
and maybe you'll find this of interest.

I hope so.
So

fundamentally what this will be will
be a new damping model for sure.

Beads and
I'll give you plenty of background for

the faculty members here who don't
know what you're beams are because I

think there might be to talk a little
bit about spacecraft cable dynamics

some background regarding the specifics
of the details of the model and

its development and some of the results
that fall from this and we'll

see if comparison between the predictions
from the mathematical model and

some experimental measurements made by our
colleagues at Sandia and a a for real.

And so forth.

That's kind of the main topics.

One of the unique features fairly unique
features of this damping model is

that it will provide modal damping for
beam like structures that's approximately

constant or uniform with frequency and
I'll show you some background.

Regarding the behavior of more
typical damping models and

you see that these more typical models
don't follow this behavior at all.

So that that's kind of
the unique feature of this.

So by way of background.

Here's a relatively small
spacecraft flown by

the US Air Force and
developed in conjunction.

With some folks and at Sandia all
out in Albuquerque New Mexico.

This is roughly the size
of a washing machine.

It's called the X S S one eleven and
it has the it's

mission is kind of interesting because
I don't know exactly what this means.

Autonomous rendezvous and
proximity maneuvers

the case of the satellite flies around
in visits other satellites and that.

The really interesting thing from
a structures in dynamics point of view is

that thirty percent of the drive
mass of this vehicle is cabling So

you see all this orange
colored stuff on here.

That's kept on this are kept on wrapped
cables thirty percent of the drive masts

your couple reasons for this of
course there must be some sensors and

power electronics and other kinds
of instrumentation on this vehicle.

So there's increasing power communication
systems associated with these

data requirements and structures people
being what they are always try to

build very light efficient structures that
have a pretty low structural mass given

the payloads so I was actually amazed to
learn that this mass fraction was so high.

So if you think about the structural
plane Nam mix of this vehicle.

It's not hard to imagine but
the cables can effect

the dynamics quite significantly
with thirty percent of the mass and

back in the day when I was doing
structural dynamics of spacecraft

as a practicing engineer the number
was nowhere near that high and

usually we just kind of simulate the or
try to capture the first

order effect of the cabling by
putting some extra mass there.

We totally ignored the stiffness even and
it turns out for

a vehicle like that it's not possible.

So for many of these vehicles including
this one an accurate dynamics model

is important to the spacecraft design
invalidation launch loads as many of you.

No a very significant driver
of the structural strength.

And some of the precision
control requirements for

an orbit operation might might require
I don't actually know if it's case for

this particular vehicle high performance
attitude and instrumentation pointing

systems such that if you might have
several or many structural modes of

vibration within the band with the active
control system in a situation like that

it's fairly important to have accurate
models of the structural dynamics and

it's as an aside it also helps to have had
a dip in because it gives you a little bit

of robustness to modeling
errors in your controls were so

based on their activity using current
structural Dynamic Modeling approaches

tended to over predict the response
levels during launch today

and primarily because they
know the cables advantage but

they didn't know how much or
how to model it so.

This is one of those few instances at
least in my experience where the customer

actually comes to you and
says Hey I know you.

I think I heard you know something about
structural damping could you help me

with this problem.

Actually that happens once in a while but
it's nice when it does so

this kind of dropped in our laps and
was very interesting problem to work on.

So I'm not going to spend
a lot of time we came in.

Probably more than halfway
through this multi-year project

with a request to help specifically with
modeling damping of these cable harnesses.

Or wiring harnesses but
researchers at Sandia and

the Air Force Research Lab to
see if the engineering and

chief are all organizations out in
Albuquerque spent several years

just kind of zeroing in on an accurate
stiffness model for those cables.

I'm not going to go through the details
here you can kind of glance said it

in the time I'm going to.

Gifty which is not much but
the fundamental outcome of their Stephanus

modeling effort was the realization
that a Bernoulli Euler beam type model.

The simplest kind of beam model.

You know in which if you imagine
line segments initially straight and

perpendicular to the neutral axis the beam
remain straight in perpendicular to

the neutral axis of the beam upon differ
mation that's a Bernoulli Euler beam that

wasn't quite good enough for these cables
as you might imagine I'll show you some

details of what the cable cross sections
look like but they needed to allow for

the possibility of sheer defamation
in the simplest sheer beam type model

still says that the line segments that
are initially straight remain straight but

not necessarily perpendicular
to the neutral axis and

that's kind of the basic physical.

Kind of assumption about
the beam behavior.

So let's take a little
bit closer look at how

the cables are coupled to the spacecraft
or these orange cable type

titles here are some that aren't
really quite wrap quite soon nicely.

They're attached the cables are attached
to the underlying aluminum structure which

you can see here through these tie downs
there are discrete attachment points

to the base structure see if we can have
a side view maybe there's a side view.

Here's here's a tide out point.

Here's the base structure of
the cable tie down point down point.

At regular intervals.

And you can see what that looks like so

the challenge was to
come up with a model for

the dynamic behavior of this part that
accurately captured stiffness and

damping in our role was the Epping Let's
take a quick look at what those cables

look like they want to I
just call them cables or

wire harness is really easy but
if you take a close look like many things.

A little bit more complicated
than you might expect.

So the basic wire is actually a cop as its
structure itself it's a metallic core with

a polymer sheath these are combined and
twisted pairs for

electronics reasons shielding reasons
either collected in bundles or

stitched together at regular intervals
the wrapped with kept on and

this is kind of what they
look like in cross-section.

They're kind of they're
more complex than you

might think from a structural mechanics
point of view or twisted you can

get extension twist coupling
they're bound together this way you

don't actually know what the normal forces
are from twisted pair to twisted pair.

There's friction involved
when they bend and

I should say at this point we haven't
you can imagine diving into great

detail developing a mechanics model
it's not the approach we took

the approach we took was more like
the one you would take in saying.

OK I'm going to try to
model this mechanically or

stiffness wise as a shear be even though
I understand these various elements

are components might actually be sliding
relative to one another as it bends.

We're going to not have to worry
about that if we don't have to.

Another interesting feature is that
depending on the day of the week and

who assembled the cable you can actually
get different mechanical behavior.

And might even be different if one
graduate student did it versus another.

OK so these folks out in
Albuquerque did construct

many many tests Testaments
segments of these cable harnesses

they measured their mechanical behavior in
the launch a two dollar extensional under

extensional load longitudinal direction
and measure the lateral of the havior of

these to see it as the cable here
it's a tattered being driven

through a shaker through a thing or that
the idea here is to excited dynamically.

And lateral motion transverse to
the long axis of the cable and

try to determine its mechanical
properties or if we're modeling this

is a beam that the Flexeril rigidity E.I.
and some shear stiffness as well.

Kappa G. extensional properties a.

And I've already told you this but they
found that they needed to use the shear

beam model to capture the cable dynamics
accurately and you could see this from

a plot of natural frequency versus mode
number for let's say a clamp clamp be.

OK so they're exciting at laterally if you
know the properties maybe you best fit

the properties for E.I. you know what the
mass density is per unit length you can

actually calculate pretty readily the
natural frequencies versus mode number and

if you assume it's an oil or
a Bernoulli oil or

B to get the dash line and
if you go to the lab and measure it.

You actually get this other line which
indicates the beam is a little bit softer

than you might otherwise expect it's
because it's deforming in shear and

this dark lane actually connects the shear
beam solution so looks pretty good.

They were very happy about that.

Here's what it looks like
from a damping point of view.

So this is modal damping of the scale and.

Typically we call that Zeta for each mode.

If you remember the equations of motion
for a single degree of freedom oscillator

typically Zeta modal
damping versus mode number.

What's collected on here
are the results for

one hundred cables representing twelve
different families that are distinguished

from one another by the number of twisted
pairs the length of the cable and

some other features that we're not going
to get into but all plotted one graph.

So this dark line represents
the mean connecting the dots for

each mode the gray area represents
two standard deviations.

So at a first glance.

So you can see that to first order
you might say that the modal damping

is four percent that's pretty good
to some variation about that.

Now you'd like to model this in your
structural dynamics model there

are different ways for

doing that again we're not going
to get into too much detail.

One way involves restricting the kind
of structural dynamic analysis you're

interested in to frequency domain analysis
looking at frequency response functions

from point A to Point B. under harmonic
loading here are some plots of

the frequency response function magnitude
vs frequency for a couple different test

and analysis cases with the cables
oriented vertically and horizontally.

Maybe you'll have to take my word for
it that the agreement is not dead.

I think you're going to have to
you can you can stare at it for

a while they talk like the kind
of model used here is essentially

a last factor which is something that's
useful for frequency domain analysis.

You can just say sometimes in
commercial find an element

code codes this might be called
a structural damping factor.

Sometimes I think.

McNeil swindler have a Seamaster and
calls it G.

last factor is a common name for material
scientists would call that Ada usually

is a material property in a way
this is modeled it's just.

Basically a complex multiple
of the stiffness matrix.

You just take the step and stiffness
matrix multiply it by the complex last

factor that your damping model.

OK So that's what's done here.

The the agreement is not bad but
one of the issues here is that these folks

in the Air Force really badly needed
a time domain model so they could simulate

the response to transients events
to look at non-linear behavior.

I didn't show you the data but the.

The lateral response of the beams
is actually non-linear with

driving amplitude not surprisingly.

You can't look at that kind of
behavior with a linear model

which a frequency domain
model effectively is so

this is the kind of the point
it which we get involved.

So you might you might say our goal is to
develop a time domain damping model that

that yields.

Maybe approximately
constant damping across

multiple modes of these
cable harness assemblies.

And if anything you know at higher modes

you kind of like the damping
to be even higher.

So when you actually use your time
domain model to do seem to lation.

The higher damping actually kind.

Kinds of takes their contribution to
the structural dynamic response out.

So you have to worry you don't
need to worry about it very much.

Of course it's true that we need a better
understanding of physical mechanisms and

that's a broad statement about structural
damping in general I would say.

OK.

One other bit of background here
is that we've done a little bit

of work in like the two thousand and nine.

Let's call it showing that it was possible
to develop a relatively simple viscous

damping model that yielded uniform
motile damping for a Bernoulli oil or

B. we didn't look at a sure Beamon
has this this kind of structure.

Since I'm guessing many of the grad
students here have looked at an equation

like this in the not distant past.

Now this is the equation of motion for

the transverse vibration of a uniform
planar Bernoulli Euler beam.

Here's the mass term
Here's the stiffness term.

Here's some boundary conditions for
a simply supported

case stiffness the Asari displacement and
moment of both and three zero.

There's the damping term.

And it looks as I'll get to.

In a minute.
It looks a little bit different from the

more conventional damping terms that are
used in this kind of model but let me just

talk it through the analysis because this
is conveniently assumed to have simply

supported Badri conditions we already know
what the mode shapes look like at least

when there's no damping they look like
integer numbers of half sine waves.

So we can stuff that answer which we
know into the differential equation and

go from a partial differential
equation to an ordinary differential

equation we call this the modal E.
Quezon of motion this is Mode him.

The little subscript here indexed.

So you have a mass and
stiffness terms only time derivatives.

This term tells us something about
the natural frequency of the mode this is

Omega squared right.

This this is to zeta and

him from your single degree of
freedom oscillator background.

So there's the natural frequency.

Here's the damping and
it turns out to be constant and

independent of mode number or frequency.

OK only depends on the mass the Flexeril
rigidity in this damping term.

So this is a nice result and and

actually surprising and wonderful to
us when we started working on this but

the question here is is there some way
to carry this a little bit farther and

extend it to the sheer beam problem and
help of these Air Force guys and

their modeling of cables
before we get to that.

Let me just point out ways in which this
particular damping model for the Bernoulli

oiler beam differs from more conventional
damping models for the same problem.

So here is the same equation of motion for
the lateral vibration of the plane or

uniform Bernoulli Euler be mass and
stiffness terms.

Here's a damping term.

For which the damping operator looks very
much like the stiffness operator you can

see there fourth facial derivatives
here it's a uniform beam we might call

this strain by.

Viscous tamping if we were to extend this
to a finite element type analysis look for

discretized versions of this equation
of motion we might call this stiffness

proportional damping It's a special
kind of proportional damping.

We could even make a connection with
the material equation of state or

the constituent of equations
in which the stress would

not only be proportional to the strain but
proportional to the strain rate.

OK so it's a kind of viscous dipping you
can arrive at this through a number of

ways but this is a pretty typical
damping model for this kind of problem.

And one that makes a lot of sense
it's mathematically simple.

It's just that it's a crummy model.

OK So
here's the model equation of motion with

the model frequencies just exactly the
same the damp the here's the equation for

the model damping for
this particular kind of damping model and

I want you to notice when you
squeeze everything out of it.

The modal damping is proportional
to the modal frequency.

So you don't have to go to a very high
mode number before the modal damping is

absurd.

OK and not very useful this is not
a good damping model in general

people insist on using it in a lot of
applications but it's just a bad idea.

It's convenient mathematically.

OK.

Another kind of model that's often used
by way of background as this kind of term

I think controls people might call
this something like skyhook damping.

This is kind of analogous to
a beam on a lastic foundational

only it's a dissipated foundation.

So there's a lateral force that arises
external to the beam somehow that's

proportional to the lateral
velocity in the opposing

OK which it's kind of it
might be the simplest sort of

you might say aerodynamic damping model
is that it resists the motion and

might be associated with the environment
in which the beam is the.

You waited but

you'll see that this is also a bad damping
model in general for a couple of reasons.

If you look at the modal
damping you work it out.

It turns out to be inversely
proportional to the couple.

Well one problem is this with
this is a again you don't have to

go to very very high number before
the damping is quite low and almost zero.

Another problem you have this
again in a finite element context

we would call this mass proportional
damping you get into a damping

matrix that's proportional to the mass
matrix and actually gives you

non-zero damping in the rigid body
modes if you have which is a problem.

OK actually a problem.

OK so nuff said about those more
conventional models it's pretty clear

why we're we'd rather
have something better.

So let's take a quick look
at shear beam modeling

the idea here is I've already said
is is that we've got a plane or

beam model in which line segments
that are initially straight and

perpendicular to the neutral axis are
still straight and perpendicular star or

sorry are still straight but
not necessarily perpendicular and

they vary from being perpendicular to this
neutral axis of the beam wanted to forms

by something called the shearing So
this just allows for the possibility that.

At the bean can shear Now I
know we're not in Texas but

everybody here knows what a case A D.
A is so you know when you heat up a case

the DIA kind of gets soft and
it becomes flexible and shear because that

inner core is shearing when it's
when it's hard at room temperature.

It's more like a Bernoulli oil or
case of DIA and K.

so this is one way to
think about shear beams.

OK Now you'll never forget that.

So I've done my job here already.

OK so we can describe the behavior in
terms of any any two of these three

quantities to transverse displacement
the rotation of the cross-section.

That's associated with bending the
rotation of the cross-section that's due

to the shear angle.

We can pick any two.

OK.

And one of the things
we know from structural

mechanics is that the shear stress and
therefore the shear strain actually

isn't constant from what from
the top to the bottom of the beam.

It's actually zero at the top and

bottom of the surfaces
under normal circumstances.

But nevertheless this model is useful and
we will use it.

And we can relate the sheer force to the
nominal shear strain which is shear angle

beta or Kappa times beta through
this way and this relationship.

The sheer force can also be related to
the rate of change of the rotation angle

due to bending.

This provides a relationship between these
two another relationship between these

three quantities that we can use to
simplify the governing equations.

If we if we care to use.

That I'm not going to go into dig detail
I've already said this we can pick three

the equations of motion for lateral
equilibrium and rotational equilibrium

crossed fiction so there are two equations
emotional like these we can couple them

together as they said by
playing some games and

not looking too closely at the forcing
terms because now this Q. represents

a lateral pressure per unit length acting
on the beam it's kind of the external

distributed force but now we're putting
primes on its spatial derivatives.

I don't want to get into that very much.

We're going to usually set
the right hand side to zero for

what we do could be problematic.

Let's leave it at that for now.

So here's a version we're going to use.

Here's the equation of motion for
this shear beam without any damping or

some familiar looking terms the mass and

stiffness terms associated with
the burning of the or lower beam her.

An additional mass like term.

That arises because we've collapse
these two governing equations into one

we it's useful to study
the behavior of this.

So we can distinguish between bending and

shear dominated type behaviors
from an elastic point of view and

then we'll carry that forward as we look
into the damping response or look at it.

So as it turns out all of these terms have
an even if they have spatial derivatives

on the left hand side they
have an even number that means

this this mode shape will still work for

us because we put a sign sine wave
in take two derivatives we get

a sine wave back out of the model equation
of motion look something like that.

Here's the natural frequency for mode M..

This leading factor is just
the Bernoulli oil or natural frequency

which essentially assumes
infinite year stiffness or

neglecting the the softness
associated with shear and

this other factor at the end is kind
of a correction that takes into account

the fact that it's possible
to deforming shear now.

And this quantity epsilon
is the sheer sheer factor.

It's not dimensional term I'll
show you that here in a second.

It's a sure parameter looks like this.

But the idea here from our point of view
is that if epsilon is small the effect

of shear is small steps align
more properly epsilon M.

squared as small the shear effects
are small or if epsilon M.

squared M. is the most numbers large and
shear relative to one.

Then we say the cheer begins
to be can be significant.

So let's take a closer look at
that not we just divided by

through by the Mode One natural frequency.

So we could define a mode
number below which we might

say the behavior from
a Stephanus point of view is.

Bending dominated and

above which we might say that
the stiffness behavior is sheer dominated.

So let's just plot that out
see what it looks like.

So here's a plot of normalized
model frequency versus mode number.

I've chosen Epsilon to be one percent.

So one over the square root
of one percent is ten so

mode ten I drew this vertical line.

You said below that the modal behavior or

the behavior of these modes tends to be
bending dominated in above it tends to be

more influenced by shear and

should have pointed out in
the preceding plot kind of see it here.

So if you look at asymptotic values for

either cases where M.
is either small like one or

very large compared to one you can
see different asymptotic behavior so

that it's in the spending
dominated region.

The modes frequencies generally
increase with the square of

the modem is a little hard to see
because it's not a log scale so

it looks like it's bending the wrong
way with a linear mode number scale but

the dotted line is a quadratic line.

Above that.

For high values of M. S. and tonic Li
the natural frequencies begin to increase

with linearly with the mode number kind
of see that with this linear plot of him.

So hopefully I'll take my word for
that but that sorry.

That mathematically that's just what I
said but the key thing is that at low

frequencies low mode numbers the baber
can be bending dominated at higher

mode numbers could be more influenced
by the effect of the sheer behavior.

OK I'm going to just see comment that

we can look at we can take our
equations of motion for the sheer.

Beam ad terms said correspond to what I
already described to you for the Bernoulli

earlier being the motion base or mass mass
proportional damping of strain based or

stiffness proportional damping and
add them in and see what happens.

They weren't very satisfactory in the case
of the Bernoulli or lower beam and

they're still not satisfactory.

That's that's the bottom line here.

OK So combine that together.

You can see what that.

So this is modal damping you can kind
of see how these motion based and

stiffness based parameters enter here and
how each of the individual factors.

This is the sum of two terms of the
individual factors dependent mode number.

OK And we can plot it the next we see it.

So this is normalized model damping
versus mode number and model dates so

it's normalized by the mode one damping So
this turns out to be one here and.

With the stiffness or strain based damping
you get model damping that increases

dramatically quickly with modem birds
no good or and in the motion based

model it decreases dramatically and again
it's not acceptable for the same reason.

So we went back to
the equations of motion we

dreamed up lots of possible damping terms
a lot of them are not satisfactory.

For a variety of mathematical reasons.

And for
in the interest of time I'm going to.

Just kind of skip through this.

I don't want to stare at
equations all afternoon.

So the idea we had said if we consider
an internal sense of sheer force.

So this is an internal
force in the beam and

we consist that we have mentioned
that a viscous part of the sheer

force a damping part of it might be a soda
associated at the time rate of change

of the shear angle and
another part with the timer.

Of change of the bending angle.

So these are two separate knobs we might
have is as damping modelers proportional

to these terms we can take one spatial
derivative and add those terms into

the equation of motion for the lateral
equilibrium of beam cross sections.

That's how they look enter.

We can combine the two
equations into the single one.

And follow the same process of
looking at the equations of motion for

the simply see under simply
supported boundary conditions and

we find that the model
damping looks like this.

So it's a product of two
factors one involving

Actually there are two terms because this
term over here is one plus something.

OK but we factor it this way.

So there's a leading term
involving this bending

angle damping term viscous tamping.

And the other term the sheer
angle damping term.

We've put put in here this way.

OK so it's a way to isolate two
parameters we can use this parameter and

then the ratio of the two
damping parameters.

That's

what it's saying from the point
of view of this model.

So there are there's a part that's
associated with the shared defamation or

its rate of change in a part that's
associated with the bending to from Asia.

So you.

I mean even when we model
isotropic materials

we talk about the young as modulus
in the sheer modulus this would

be analogous with their
viscous damping terms instead.

So we can in this way we can kind
of separate the contributions

of the the bending inch and sheer viscous
damping sheer terms that turns out.

In this bending dominated regime or
epsilon him squared is small the damping

turns turns out to be dominated
by this bending angle viscous

damping term because the some other
stuff over here is dominated by this one

in this factor and a higher mode numbers
epsilon M. squared much larger than one.

The important term in terms
term turns out to be this one.

So the modal damping is proportional
to the alphabets of beta.

And it's like anything it's much
easier to see if we plotted out.

So there's a normalized model damping
now on a linear scale not a log scale so

we are already like this for

a linear scale things are generated
like crazy over a few mode numbers.

Epsilon is one percent.

So this line between bending dominated
It's a fuzzy line really but

it's around mode number ten at sea.

So we.

I forget what value I pick for office of
feet but we're looking at the ratio now

of these two parameters the sheer
damping to the bending damping.

No member Our goal is to to arrive at
a viscous damping model that gives

relatively constant modal
damping over some range.

So depending on what range of mode
numbers you're talking about you might

choose this ratio differently.

If you look at a large enough number
of modes even this curve comes back up.

OK So mathematically.

You saw that when I showed you some of
the data of the day a forelady taken in

Sandia had taken that they're
only looking at sixty six or

seven modes in terms of having
experimental data for them.

So if we were to pick this ratio
to be something like one half.

You might be very satisfied with
the result right in terms of

having nearly uniform little
damping it's really quite good.

Remarkably good.

Of course this is for simply supported.

Reconditioned So
there are some other issues we might be.

Looking forward to thinking about and
just for effect.

I want to plot these results against
what I already showed you for

this the stiffness proportional
in mass proportional damping in

the results I just showed you
are these inner ones here.

And in fact this gold line corresponds
to the value of the ratio of

the shear damping to bending
damping of one half.

OK And that's the one

we like to I do have some slides that
are correspond to kind of a closer look.

Mathematically it under what conditions
can we define a damping operator for

this kind of partial differential
equation of motion that will

give us uniform modal damping in the end
I think in the interest of time and

maybe in the hope of keeping you
awake a little while longer.

I'm not going to go through these OK
except to say that it's possible to define

a stiffness operator its implementation
in practice is quite difficult.

I would say in challenging but
we can make a connection between this

damping operator and
the model I've already shown you.

OK this is like a a one
term approximation a two

term approximation of that
damping operator so nuff said.

Another interesting thing
about this is because we

all know that structural
dynamics really earn their keep.

In real life by running fine an element
analysis we have to ask the question

how would one implement this model in
a fight element context and it turns out.

It's not.

I mean from a mathematical point of view
this is just a bunch of differential

equations and we can discretized
them using our favorite method and

structures people tend to like find
it element of approaches to this but.

Turns out if you take these equations
this equation of motion and

using a standard find it own that approach
for a Bernoulli Euler beam using those.

Shape functions thirty cubic shaped
functions and four degrees of freedom

you get matrices here's something
that looks like the math matrix.

It is a bass Matrix or
something that looks like the standard for

two noted for degree of freedom
planar beam stiffness it drifts.

Here to peers again.

Here's a matrix that looks just like what
we call the geometric stiffness matrix

in structural stability analysis buckling
analysis for beams under axial loads.

So it has precisely of the same form
with that shouldn't surprise us

because the damping term has
two spatial derivatives as the.

The term involving the axial load has
been structural stability problems

sort of the stability of both coffee and
columns anyway so

mathematically there are a lot of familiar
looking matrices here it's quite nice.

There are some questions
when you collapses to

a single equation of motion how
you deal with the forcing terms.

We're not going to address this today.

But having this find it I'll find it
elements finite element analysis approach

available to us as we're used to having
and like having we're able to look

at the the behavior of this model under
different kinds of boundary conditions so

we can look both for example and
simply supported boundary conditions from

a numerical point of view we can look
at clap clap boundary conditions.

Whatever whatever we
think is appropriate for

the physical problem at hand and
if I think it's true for

the spacecraft application that clamp
clamp might be more appropriate.

OK so we can look at that have a look
over here so here's model damping again

a familiar plot versus mode number
the set of curves up on tough.

Sorry let me back up here is the kind
of the fuzzy line the divides bending

dominated in sheer dominating behavior.

The sort of curves that emanate from
this point here in the left are for

simply supported by under conditions.

Each of the color lines course
solid colored lines corresponds

to a different value of
this ratio of the shear

to damping to bending damping parameters
and the one we kind of like is this

one half lease for
small values small mode numbers

have to say that the FIND IT element
results appear is dotted lines

that fall right underneath these
solid lengths are right exactly.

On top of the exact solution for
this particular problem clamped on so

at least for this problem.

We haven't shown any exact solution.

So here are the numerical results for
clampdown recorded.

Conditions you can see if our goal is to
get constant modal damping over a range of

mode numbers.

It's going to be a little harder to
achieve because the boundary conditions

actually affect both
the natural frequencies and

the damping much more at
the load numbers and they do it.

The higher move numbers but
these are the kinds of problems

we can address with the numerical tools
afforded by fine an element analysis.

So we can take this just happened.

Power Point but

said All right well we'll get there and
see.

Sorry about that.

I'm not used to that happening on a Mac.
I'm not sure what happened.

So what's sad is so.

All right.
So that's so

let's go back to some experimental data
the plot like this that I showed much

earlier in the talk was a collection
of results for one hundred cables.

This is a collection of just thirteen
cables from a single family of cables

that have a lot of similarities in terms
of cross-section maybe different lengths.

But here's the model.

Sorry not even differently.

So here's the plot of modal
damping versus number again

the dark line corresponds to the mean of
these measurements for thirteen cables.

I think the single gray line is one
standard deviation of the double

grey region is two standard deviations and
the red line

is the fit we got by adjusting some
of the parameters of our model.

I hope you'll agree that's
quite quite satisfactory and

we probably could have done all right.

Even setting the first one to zero.

I told you there were some reasons
we didn't didn't like using that but

it turns out it's actually decreases
with Mode number initially.

So it's helpful to include
it in this particular case.

So the folks at F.R.L. and

Sandia were delighted with this and with
this result and we were happy to answer.

Well let me just say at high end at a high
enough mode number all the solutions grow.

OK you can follow this curve out
far enough in or grow back up.

OK but we don't really care
about what goes on there and

if we're actually using a model like
this is part of a larger model to do

time domain transience simulations
depending on how we do them

if we do them with
the record time simulation.

We just assume those modes
were highly damped anyway so

it's not a bad thing to say and
from a numerical point of view.

OK So to kind of wrap up

the main topic of this talk today was
the time domain damping model for sure.

Beans.

With applications to spacecraft wiring
harnesses some of you might have

learned a little bit about the difference
between shear beams and Bernoulli or

other beams today hopefully for
that interesting and possibly useful but

a key thing it said low frequencies from
a structural dynamic view the behaviors.

Sort of dominated by the bending
behavior at higher frequencies this

year becomes more and more important and

high enough frequencies to become shared
dominate that holds true both for

the stiffness behavior and in the stamping
model for the damping behavior and

from our point of view this
does provide some guidance for

maybe future direction digging in a little
deeper into the physical mechanisms

mechanisms that give rise to damping
in these surprisingly complex

wiring harnesses from a mechanics for
the view that maybe someone will pursue

if our goal as I said was frequent
relatively frequency independent or

uniform overall damping over
a small range of load numbers

was our goal we can get pretty
close to that by controlling these

two parameters self a beta and
Alpha feet and actually

control the damping in the high frequency
or number regime through alpha beta.

This model we hope will be useful to
structural dynamicists in the future

because it can be readily implemented
in a finite element context and

all the major cities that arise are kind
of have a familiar looking structure.

And we're happy with the agreement
with experimental data.

We're going to carry this a little
bit further the Air Force's

assets asked us to look at the application
to it to machine beam the.

Difference between a ten machine beam and

a sheer beam is not much the primary
difference is the inclusion of another

mass term corresponding to the rotational
inertia of cross sections.

It can sometimes be and its effect can
sometimes be as important as the effect

of the shear itself that probably will not
turn out to be the case and this problem.

I think we'll see.

And I'm happy to in the support
of our sponsors it's India and

the Air Force Research Lab and
if you're really interested I can give

you some references to look at from
their work in our more recent work.

So thanks very much for your attention.

One.

I'm in trouble.

Nice piece of yours.

What you did was the word exactly the way

it was being noticed

so I don't think I can.

Say much about the first one.

I'm not sure of the resemblance to a
Kelvin void model hundred not sure I would

have described it that way
hadn't seen that before.

As I heard.

Basically it really is.

Yeah but when you call it that you make it
sound like it's an X. turtle damping force

is a time derivative curvature and and
we don't like talking about it that way.

This is kind of an internal damping.

Mechanism.

The second part was I guess I don't
need to back up the form of this.

Well if you think of the bending angle
fee is being something like dubs say

what do you think of that as being like W.
prime and Fi prime is like W.

double prime in beta is like like Fi so
they they have the same exact nature is W.

double prime they now.

OK You're OK then reset.

Yes thought that way if I were just beta
dot then it would be again a kind of an X.

terminal damping force that resists
sheer defamation somehow and

I guess we can think about
a time you probably tried that.

Probably but
I don't remember what the result was

probably it led to pretty fast stamping
once this year kicked in and I thank

the server so I'm going to be one of those
people who has the stuff like this all

we were suddenly a fraud
nodding your head.

Unless the best against US vs.

It was OK stirred up I
said you wanted it so

that let Little Debbie as your
data suggest are correct.

So I could questions that clearly
the families of families were

glad you were go to movies and Greece.

Two very recently.

How much of that is it's very little
Certainly and the second question is if

you could be it's of a different families
off by a factor of two With little bit.

How does that affect their bottles of
trying to look at your performances

as a backward two hundred workers affected
really that is an excellent question.

In some cases it could be
very important I think I

think if one were to talk about the state
of strong damping modeling in aerospace

structural mechanics you'd say that we're
pretty ignorant would would you argue.

So a lot of what people use in practice
is based on their experience as

a perhaps as a company or organization
as well as is information that's

available in the literature but a typical
approach would either involve this kind of

frequency domain analysis
where you pick some values for

this complex last factor
based on your experience.

You would try to be conservative.

Oftentimes that might be
choosing the damping to be lower

than you really think it is so that
the predicted response would be higher

the stresses would be higher than
you might really expect or hope for

a factor of two in many cases
would probably be pretty good but

it would move to move very easily or
is this a move that would go

to some of the just yeah I think
you're right I think you're right.

Very different.

Yeah I think you're right I was a very

good observation I think you
could as well even That's

because you are trying to measure
what amount to small quantities.

It is these are actually
difficult measurements to make.

As well.

So part of it could be due to experimental
uncertainty at least if not err.

You know.

Yes sir.
Well I

didn't say that exactly but
you could think about that them that way.

Yes

those are great questions.

It turns out the material scientists know
a lot more about material damping than

aerospace structural dynamicists So
there is some knowledge about that and

material scientists can actually use
these kinds of measurements of dynamic

behavior of material specimens to do
to deduce things about the material

structure at a very fine scale but
that kind of

knowledge has not really proven that
useful for designing Aerospace Materials.

Maybe with one if one or two exceptions
that I can think of I mean if if

if if you were a practicing structural
dynamicists in industry and

as often happens you
discover that you have

a dynamic response problem after
you've already built a vehicle.

And your boss is OK we've got to fix this.

Now there are some solutions for
and at what I would call additive

damping treatments you might call them
nonstructural additive damping treatments

that are typically that typically
involve the use of polymers and

have high damping and in their
simplest are incarnation they look

something like scotch tape is kind of
a soft polymer with a stiff backing on it.

These are called constrained
layer damping treatments.

But these polymers are designed to have
high damping in a certain temperature

operating temperature and
frequency range and

they can be fairly effective if you know
what you're doing as far as the design

where to put them on the structure
how much of it to put there.

I've talked I am not quite sure I've
addressed your question but to try to hit.

But but I don't think of these
these damping these properties as

being material properties in the same
way that I think a shear modulus and

the Youngs modulus their
material properties.

You might be able to go to the lab and
measure something and do a fit but I don't

think they're good models of which what's
actually happening from a dissipative

point of view within the material a good
model a good model might a structural

material and they might be good and some
materials reaction have viscous behavior

like a fluid but there are there are more
complicated material models fiscal elastic

models that are time dependent have
built in time dependence that are pretty

good in that we can use instructional
dynamics analysis we just usually don't

use or yes

really in terms of their mechanics.

They're wired.

You know I haven't really
thought about that.

That's very interesting but I've answered
your question with I saying wow.

We never thought about death.

So you have a time to put in
a normal force between the wires

which would affect friction and so on.

Yeah so.

So there might be you know
civil engineers probably.

We haven't dug very far into a more
fundamental mechanics type model but

it wouldn't surprise me to know that civil
engineers who built cables supported

bridges understand those kinds
of well not even those kinds but

the mechanics of cable supported
structures better than I do.

Actually is that one of
the beda beda pride I

might have maybe what you assume I know it

was OK and that you're doing it.

If you don't you feel their forces

because if U.S.
forces indicate there will not be

perceptible Yeah they're right and

if so their intention the normal
force is so high there can't be

any relative motion would have been very
close and you would expect it anyway.

Only you know they cannot actually
move and then push it off.

That is to walk a little slow

actually put close but

it was only that you will

see these are not load bearing to
the best of my knowledge of body

depth at even too low
might be that it's just

the movement but I think that's right.

Very very interesting problem for
such a simple structure.

Yeah.

Yes ma'am.

But once they realized that they tightened
up the process pretty significantly.

Well let me be clear I didn't
do the experiments those for

the folks at the Earth probably see a say

but I think you put your finger on
something very important from their point

of view as they needed better uniformity
in the manufacturing process and

they didn't realize it.

You know at least that they had any
hope of predicting the dynamic response.

More accurately based.