dynamic reduction methods are needed in
finite element analysis because the
model it has been constructed often is
too large and complex to efficiently
solve for the vibration characteristics
for instance an automobile might have a
million or two million degrees of
freedom mostly caused by they need to
model the geometry of the problem not so
much stiffness or mass distribution but
just to get all the angles and holes and
corners and brackets and so on therefore
there's some need to simplify in a
consistent way the scope of the problem
so that a person can understand the
lower modes of vibration of the system
it's been my experience that in stress
analysis and Static problems often the
interest of the designer engineer
analyst is to find details around local
portions of the structure where does it
break where does it fatigue and those
are local ideas in contrast the denied
the dynamicists wants to get global
behavior what are the long wavelength
frequencies that are involved so it's
interesting it's almost a conflict in
function between the static stress
analyst and then the dynamic analyst
well dynamic reduction methods are meant
to bridge that gap one of them is really
straightforward and easy to discuss and
that's the so-called static condensation
now this is a normal condensation as
opposed to a mold condensation that we
studied earlier under the modal solution
methods so the concentration is how you
handle the degrees of freedom at nodes
in this case unfortunately to move
beyond static condensation gets tough
very quickly I find it more difficult to
teach that material and so I don't
stress it as much I view it as an
advanced topic when you get in
component mode synthesis generalized
dynamic reduction some of those methods
are really fairly theoretically involved
so what I'll do is pull my punches a
little here and hit static condensation
pretty hard and then kind of wave my
arms a little on the more complicated
reduction methods now this is okay I'm
going to argue it's okay because some of
the new solvers don't really need this
simplification for instance cilantro's
method seems to work for fairly large
problems in a direct way and so I hope
that the user can figure out in your own
corporate setting of what sort of
dynamic reduction methods you're
interested in there are a lot of vehicle
people it's still really like component
mode synthesis and that's a super
element type of condensation approach
well the procedure will be to talk in
general about dynamic reduction hit the
static condensation rather hard on going
on reduction and do some examples and
then I'll just wave my arms at
generalize dynamic reduction in
component mode synthesis will follow
with some problem session examples which
are primarily the static reduction type
up here I'll start off with a bit of an
overview of dynamic reduction
in dynamic analysis the necessary steps
are to assemble the equations solve them
and then recover the forces and stresses
that are the internal fuel variables in
the problem an optional step which is so
important is this one where you might
reduce the number of degrees of freedom
in the system basically you pass from a
set of physical degrees of freedom to a
set of coordinates that are more or less
artificial sometimes that reduced set of
coordinates carries within it the more
complicated behavior of the whole system
but in some idealized way if you don't
do dynamic reduction you'll find that
the solution of the equations here
really becomes expensive part by far
more than 90% in most cases so you would
be willing to do a certain amount of
work at this stage in dynamic reduction
in order to reduce the solution cost
there are several forms for dynamic
reduction we've already spoken about a
modal approach using normal malls as our
building blocks then the nodal
condensations are such as the goon
reduction here generalize dynamic
reduction is a bit in-between where you
have somewhat of a global and somewhat
of a local view on the situation and
we'll show some of those equations later
component mode synthesis is the dynamic
version of super element Theory where
you segment your larger structure into
pieces then those pieces are joined
after they're separately analyzed
dynamically and
one one key question there is how do you
join them together certainly if use a
mobile solution alone for really large
problems you may get into troubles with
the cost of eigenvalue extraction and
then as you get above ten thousand
degrees of freedom you're really forced
to go to something pretty powerful
perhaps component mode synthesis or
these newer Atlantia solvers the
Lentulus solver has pretty well swept
through the industry as I speak and is
now the dominant large problem solver
now let's discuss this green reduction
or static condensation and see how it
works we can follow it rather well in
fact you should understand this method
cold I think it's useful because it also
gives you the whole spirit of other
condensation methods will illustrate
this with the MSC Nastran data
definitions and their sets of variables
the original set of physical variables
would be called use of G and then after
reducing out the multi point constraints
you're left with a set u sub Annan
at that point you apply the single point
constraints and you're left with these
degrees of freedom use of F
here's where condensation enters and
there are a certain number of degrees of
freedom that are considered to be
omitted degrees of freedom when those
are taken out you're then left with the
analysis set of degrees of freedom even
the analysis set which is what we will
work with today can be subdivided into
those degrees of freedom that represent
rigid body modes and those that are or
need to be constrained to if you wish to
remove them and those that are the final
independent degrees of freedom which are
you can think of is the left your left
over degrees of freedom but basically in
dynamics there's no need as there is in
statics however to remove your javadi
mode so it's better to work at this
analysis set of variable level here so
we will be looking at glan reduction in
the way that you progress from this set
of degrees of freedom here and how you
break that into this and this we won't
go to the finest smallest detail were
you worried about degrees of freedom
that characterized rigid body modes
since in dynamics they can be carried
long anyway let's work out the
mathematics for the Guyenne reduction
and we'll do it as I mentioned in terms
of those sets of variables sub scripted
with F we might call those three degrees
of freedom just to kind of word to let
us discuss what that is and the ACE that
will be the analysis
so if we write the equations of motion
at this stage then the multi point
constraints have been reduced out of the
system and the single point constraints
have been reduced out so these then are
the three degrees of freedom that
described the active degrees of freedom
that remain at that stage there would be
mass terms and the happening terms and
stiffness terms these are pretty close
to physical degrees of freedom at this
point with some agglomeration effects
into the reduced matrices at this level
but closer to a physical set of degrees
of freedom but now we want to make a
strong distinction between the original
physical degrees of freedom and then a
reduced analysis set by doing this kind
of partitioning where we partition the
these three degrees of freedom into the
analysis the omitted set now I'm going
to do poetic license here or authors
license to assume that those were
numbered in a way such that I can
consecutively partition out these
omitted ones now normally the analysis
set ought to be less in ratio of one to
ten let's say you would like to reduce
your degrees of freedom greatly by doing
this method so I don't mean that these
two vector sub components here are of
the same size one of them the omitted
set is generally way bigger than the
other
now what we're going to assume is that
the relation between the analysis and
the omitted variables is based on static
equilibrium concepts that makes this
sort of a first order correction for
dynamic condensation and other people
will come along later and say hey I can
improve this by five ten percent by
accounting for those degrees of freedom
that are admitted in a dynamic way but
we don't do this you need to walk before
you can run in this case so you actually
do a mental experiment really starting
here and I'll draw an arrow here for a
short time and then I'll tell you when
we stop doing this we drop out the
inertia and damping forces and look at
an a static problem which means that
you're looking at your dynamic problem
in a quasi static way as if dynamics
really didn't matter and all of the
degrees of freedom are related purely by
the springs that connect up the elastic
forces let's carry out a mental
experiment now to help us eliminate the
unwanted degrees of freedom in this
problem one way to do it is imagine a
simplified structure such as this beam
like structure shown here and identify
certain important degrees of freedom
such as these analysis degrees of
freedom at the right end the center and
the left end and then identify less
important degrees of freedom such as
these at the intermediate spots there
and there then the analysis nodes are to
be treated as master nodes that control
the whole system this reminds me of the
lofting curves that were used when I
worked years ago at Vaught aircraft and
at Boeing where they had long plastic
strips and then these strips were held
in place at intervals by what they
called ducts and these were heavy lead
bodies that had a nose on them and they
looked a little bit like a duct that's
where the name came from that fit into
the slide along the top edge of this
plastic strip and then by controlling
the shape of that long strip at
intervals the draftsman had a nice
smooth contour to follow with their pen
and this is very much the same
philosophy that we have here we want the
Anastas degrees of freedom to control
all of the shape of the body statically
when we write out the equations
presuming that the analysis degrees of
freedom are above the partitioning marks
hair and the omitted degrees of freedom
are below then we have this set of
equations for this mental experiment
only we don't put live loads on the
omitted degrees of freedom many students
who study this forget later on and think
that you can't have live loads on the
full structures or meta degrees of
freedom that's not true now these are
the physical degrees of freedom
presently and these are physical forces
on the analysis degrees of freedom up
above you're able to partition out and
solve the sight
large set of equations here which
involve these coefficients shown below
the KOA matrix which is rectangular the
K all all matrix which is square then by
inverting the hay or matrix you can
solve for the omitted degrees of freedom
in terms of the analysis degrees of
freedom so this is a mapping that you
might want to consider moving in this
direction and very effective than that
letting these analysis degrees of
freedom control the omitted degrees of
freedom will give that mapping matrix
assembled G
and here it is
and we will include the minus sign
within that definition shown here I
suppose she can as a crutch can be
thought of as a transformation here so
then you have the ability to take the
smaller set of variables use of a and
let it drive the larger set use of F
over here interestingly to expand this
set in the static way we've had just
described then you use the identity
matrix here to map into the smaller set
say of relative size one and then you
use the G or a matrix to give you the
omitted coordinates here so it's as if
there's two parallel pipe lines here
going from right to left the analysis
degrees of freedom propagate straight
through this system into the three
degrees of freedom and the omitted ones
have to be mapped to become their
rightful spot in the three degrees of
freedom alright so we've got a mapping
we've got a way to statically condense
this problem at this point we have ended
the mental experiment which was up above
and now we get back to the real world we
look at the full equation of motion for
our three degrees of freedom and we
replace the three degrees of freedom
with the equivalent mapping version now
if we wanted to take a single derivative
of this that works you can take a second
derivative that work so I can map the
not only the displacement field but the
velocity field and the acceleration
field with the same mapping that's
because all of these quantities in here
are are not time dependent that was from
our static experiment a static curve
and then on the right we will also
consider that the original forces on the
system here were and these are close to
physical forces here are an F bar of a
and an F zero I carry the bar
temporarily to to avoid the use of the F
sub a which I'll have later on so I'm
reserving the unbarred symbol F for
later final use alright we are in the
same position as we've been earlier in
this lecture series where we have say n
equations in n unknowns and we need to
do something to convert this into a
better handle set of equations and again
we look toward an energy approach where
we can pre multiply by something that's
displacement like this particular set of
equations here would in a plate
structure have all our net translations
and rotations and it's numerically not
real well posed at this time and so
we'll move to the next figure to show
the glucan type approach our set of
equations then is a set of force and
moment balances at this point but if we
do a pre multiplication with a
displacement like vector then we're
going to end up with an energy balance
instead so we pre-multiply by our same
transformation matrix and that gives us
this set of quantities and we get a new
the mass here out of this triple matrix
product and we'll call these the
analysis mass terms here we get a new
kind of damping matrix that will call
the analysis or reduced damping terms
and here's a new kind of stiffness shown
here on the fourth side we get a new set
of equivalent loads these are a set
loads these are have become more and
more artificial they're not the original
physical set of loads on a but they're
they're the representation of all the
loads in the system and how you have to
agglomerate those at the remaining few
degrees of freedom so these are
basically equivalent forces over here in
the earlier terminology where we had
dealt with modal degrees of freedom and
generalize degrees of freedom we got
generalized forces and this is
definitely that same philosophy you
might think of these as reduced degrees
of freedom and reduced forces the theory
for grand reduction is really rather
nice and can be incorporated into
computer codes rather straightforwardly
there's a few decisions to be made I'll
show how those have been resolved in the
user implementation and MSC Nastran
you have a body that is a potato shape
and has live loads in various places on
the body some on what are too soon to be
analysis degrees of freedom and some on
what will be only two degrees of freedom
how should you model such a system as
this well for one thing you should take
any point or any area here that has a
large mass and make sure it has analysis
degrees of freedom describing it that
way you're not using the so called
beaming out of the mass effects there to
some other node point so it reduces the
effect of the static reduction you
should distribute the eight points over
the body somewhat uniformly so that
you're doing a fair job of representing
that the total geometry that also means
that you don't have to beam out the
inertia effects so far and the damping
effects now these two are rather common
sense the third one is not so intuitive
and that's the feeling that it's better
for the a set to have both the
translational and rotational set of
degrees of freedom and that means you're
going to do this breaking down of
degrees of freedom into analysis and all
meta degrees of freedom node by node
that's not obvious I did a problem years
ago with a woman professor from China
mrs. Zhu and we decided at that time to
include only translational degrees of
freedom in the analysis set the
rationale was okay there because it was
a civil engineering structure and it was
going to be easier to measure
translational accelerations at points
and
have transducers that measured
rotational effects I think might now
they're probably good transducers for
both but it was somewhat experimentally
motivated to not do this
third user-friendly action as a result
of this third one though it's relatively
easy to set up runs in MSE nastran
because you only need to mention the
analysis nodes and then proceed notice
that you'd rather call out the analysis
nodes than the omitted nodes because
they are perhaps ten or a hundred times
as numerous so that's the convention in
typical problems run on ms seen a
stripper my teaching style in the past
has typically been to try to break down
a larger finite element process into a
simple hand calculation aside from the
use of the computer this helps develop
intuition and it can be done luckily for
Guyenne reduction here
a physical problem that I will take is
going to be trivial then because it's a
simple beam element and a single one
undergoing small amplitude vibration in
the XY plane the idea is to find the
fundamental flexural frequency in the XY
plane ignoring the axial torsional and
out of plane vibrations again we're
going to have a compact cross-section
say a rectangular cross-section so if
these are principal coordinate
directions for principal axes of inertia
then the problem uncouples nicely and we
can just study flexural vibration in the
XY plane physical modeling is to use the
conventional Euler Bernoulli beam theory
so that plane sections remain plane and
we constrain those out of plane motions
and the axial and torsional motions
boundary conditions that the root are
that there are no rotations nor
translations there at the free tip you
find that you have no moment and no
shear in the finite element modeling I'm
going to first take the most odious
comparison that I can by constraining
the tip translation to be the omitted
degree of freedom and then using the tip
of rotation to be the analysis degree of
freedom now that means that we are
trying to make the rotation at the tip
control the entire dynamics of the
problem
furthermore if I use the lumped mass
approach that means I'm concentrating
all the mass at the tip as a point mass
and there's no rotatory inertia involved
somehow this condensation method has got
to account for that and convert that
translational inertia effect into
something governed by the rotational
degree of freedom so that's really not
easy now it will turn out that this
works although we have some 20% error in
this case so when I push the problem
this hard I thought maybe I won't get
answers that make any sense at all
here's the degrees of freedom that are
shown
of course we're constraining the tip
with these degrees of freedom so the
only three degrees of freedom are those
two here and I'm going to make the u3e
be the omitted degree of freedom and in
other words a slave to the u4 degree of
freedom so this is a toughest case
possible particularly when shown below
that I have a point mass out at the tip
when I use the lumped mass matrix
remember that it moves half the main
mass to the tip and half to the root
well the root mass is disabled and
doesn't even move then by moving half
the mass to the tip we're somewhat over
emphasizing the inertia effect by moving
it outboard but then we're requiring
that that be governed by the static
motion of the rotation of the tip so
right off the bat you might say you've
lost the entire inertial effect that you
haven't because in the static relation
that has developed if there is a tip
rotation there will have to be
correspondingly a static tip deflection
and that was the whole idea of that
approach
now dynamically they may not be related
but statically as soon as I give a
positive angle here this beam will
naturally go upward so the inertial
effect is properly accounted for
now let's go to the solution will gather
the equations of motion for all of those
original degrees of freedom this would
be called a G set of degrees of freedom
because we are just now applying the
single point constraints at the wall
those cause these first two degrees of
freedom to be zero when we partition
those out then we really are going from
the G set now down to what we've called
the F set of degrees of freedom we
haven't yet done the condensation we're
left with these reduced matrices here
which are still physical terms as they
stand now we want to do the condensation
approach our lumped mass matrix
effectively moves half of the
translational mass out to the free tip
and half of it to the route where it's
constrained anyway here's the mass
matrix reduced to the free coordinates
and here you see the tip mass of one
half the total being mass stiffness
comes from the euler-bernoulli Theory
simple beam theory we now do our mental
experiment which is a static problem so
right here we draw a line and say we're
going to do this mental experiment we
don't allow forces on the omitted degree
of freedom which is translational but we
do allow a moment here on the on the
analysis degree of freedom then we use
the first equation in this case to solve
for the translation in terms of the tip
rotation because maybe
moves from zero slope at the root to a
tip rotation out here you might say that
this makes sense because the tip
deflection is taken to be the L over 2
times the tip rotation and if the
average rotation over the length of the
beam is half of the tip rotation which
would be approximate then you're getting
a displacement by measuring an angle
through a distance so it's pretty close
to the normal definition of arc lengths
and saw so you can get some physical
feeling for this expression so it's not
completely stupid
now let's form our coordinate
transformation and the analisys degree
of freedom u 4 which is a rotation maps
directly into u 4 but to the surprises
and then it also creates the translation
here by this factor L over 2
now that's the end of our mental
experiment and now we progress to the
real problem again the real problem that
we're interested in is not forced motion
but finding the natural frequency of
this system the lowest natural frequency
so we put zeros on the right hand side
there are no live loads then we've got
our various matrices here and we've got
our transformed variable here or so down
to the analysis set we pre-multiply by
the transpose of that coordinate
transformation here to put this on more
of an energy basis so we collect terms
and simplify them and get the resulting
differential equation for free vibration
here's the equation and you have a
generalized mass term and a generalized
stiffness term now from here down to the
actual answers just a matter of putting
in the harmonic motion
realizing that derivatives bring out -
Omega squared collect terms you have all
of these terms that multiply a term here
which is an amplitude term
solve for the unusual situation where
you can have an arbitrary amplitude and
it occurs when this coefficient here is
zero so you can have equilibrium with
any amplitude at this specific frequency
sometimes you might plot a curve of
frequency over here versus this
amplitude you for here and you find that
there is a point in along here at a
certain frequency and that's what we've
just found which is a so-called natural
frequency here is the resulting number
and when you compare this with the
classical solution which has a constant
out front of 3,5 - rather than this 2,8
- you find our solution is low by 20% at
first you say wait a minute you're never
supposed to be low on these approximate
solutions if you're truly within the
really rich or the potential energy
format but in this case our lumped mass
approximation is a non-physical
approximation and it has moved so much
mass out to the tip that it is lowered
the frequency in this way and so it is
explainable generalize dynamic reduction
was developed by Richard McNeil some
years ago and it's been very powerful
used by a lot of vehicle companies for
studying large problems it's a
combination of two methods of a guy anti
production with the mobile analysis so
it's both a modal and a nodal reduction
it brings in some coordinates here that
are generalized coordinates UQ
and the analysis set is broken into some
generalized and some physical
coordinates at the very end the
generalized coordinates become the
master one let's break down those F set
of variables which i've been calling the
free set and assuming that prior to that
you've already taken out multi point
constraints a single point constraints
so here's your total F set on the left
and on the right here we have the
omitted degrees of freedom as before
with a clue an approach and then we have
the generalized coordinates those that
are used for rigid body mode description
and then some free coordinates here and
some rigidly restrained coordinates some
of these are new concept now I won't go
into details on that it's a generalized
coordinate this is surviving set though
so those generalized coordinates are the
ones that drive the whole problem here's
the kind of mapping that's developed and
progresses from these coordinates on
back to these use of F coordinates over
here
there are some approximate modes
involved here and this is related in a
sense to the super element approach
where you divide a structure into these
segments this is really an advanced
topic and the mappings are intricate and
I have to admit failure I wanted to show
a simple hand calculation for this
method and I could not do it there's
just too many variables involved and
that's if you have to start out with six
five or six degrees of freedom by the
time you do the mappings to bring it
down to the final solution it's just too
complicated for a simple hand
calculation but many people like this
implementation in a strand and have
gotten used to it for vehicle dynamics
another reduction method is popular with
some people is the component mode
synthesis this competes nowadays with
the Lantos method and it's useful for
very large problems it's a super element
concept it's the dynamic version of
super elements or substructuring and you
use those vibration modes of the super
elements to infer what the properties of
the entire system are there are at least
two versions of this kind of approach
and one is where at the joints between
the super elements you develop the
individual element or the super element
properties with fixed boundaries and the
other method is where you have free
boundaries and this is an advanced topic
some people prefer the fixed boundaries
even though experimentalist prefer the
free boundaries this may be another one
of these areas where depending on the
problem you're doing you would like to
use one approach or the other our
problem session will concentrate on the
Guyenne reduction it's one that we can
actually do some hand calculations with
the first problem I take is a
two-element line assembly which is
vibrating along its axis and the idea is
to determine the frequency response of
this system using this static
condensation I'm going to ask two things
I don't want the whole problem solved
but just to find the mapping matrix for
the coordinates for either choice of
analysis degrees of freedom namely the
motion at the midpoint here or at the
right end and then to explain which of
those