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dynamic reduction methods are needed in

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dynamic reduction methods are needed in

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dynamic reduction methods are needed in
finite element analysis because the

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finite element analysis because the

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finite element analysis because the
model it has been constructed often is

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model it has been constructed often is

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model it has been constructed often is
too large and complex to efficiently

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too large and complex to efficiently

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too large and complex to efficiently
solve for the vibration characteristics

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solve for the vibration characteristics

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solve for the vibration characteristics
for instance an automobile might have a

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for instance an automobile might have a

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for instance an automobile might have a
million or two million degrees of

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million or two million degrees of

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million or two million degrees of
freedom mostly caused by they need to

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freedom mostly caused by they need to

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freedom mostly caused by they need to
model the geometry of the problem not so

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model the geometry of the problem not so

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model the geometry of the problem not so
much stiffness or mass distribution but

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much stiffness or mass distribution but

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much stiffness or mass distribution but
just to get all the angles and holes and

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just to get all the angles and holes and

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just to get all the angles and holes and
corners and brackets and so on therefore

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corners and brackets and so on therefore

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corners and brackets and so on therefore
there's some need to simplify in a

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there's some need to simplify in a

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there's some need to simplify in a
consistent way the scope of the problem

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consistent way the scope of the problem

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consistent way the scope of the problem
so that a person can understand the

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so that a person can understand the

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so that a person can understand the
lower modes of vibration of the system

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lower modes of vibration of the system

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lower modes of vibration of the system
it's been my experience that in stress

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it's been my experience that in stress

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it's been my experience that in stress
analysis and Static problems often the

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analysis and Static problems often the

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analysis and Static problems often the
interest of the designer engineer

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interest of the designer engineer

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interest of the designer engineer
analyst is to find details around local

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analyst is to find details around local

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analyst is to find details around local
portions of the structure where does it

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portions of the structure where does it

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portions of the structure where does it
break where does it fatigue and those

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break where does it fatigue and those

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break where does it fatigue and those
are local ideas in contrast the denied

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are local ideas in contrast the denied

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are local ideas in contrast the denied
the dynamicists wants to get global

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the dynamicists wants to get global

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the dynamicists wants to get global
behavior what are the long wavelength

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behavior what are the long wavelength

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behavior what are the long wavelength
frequencies that are involved so it's

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frequencies that are involved so it's

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frequencies that are involved so it's
interesting it's almost a conflict in

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interesting it's almost a conflict in

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interesting it's almost a conflict in
function between the static stress

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function between the static stress

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function between the static stress
analyst and then the dynamic analyst

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analyst and then the dynamic analyst

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analyst and then the dynamic analyst
well dynamic reduction methods are meant

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well dynamic reduction methods are meant

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well dynamic reduction methods are meant
to bridge that gap one of them is really

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to bridge that gap one of them is really

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to bridge that gap one of them is really
straightforward and easy to discuss and

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straightforward and easy to discuss and

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straightforward and easy to discuss and
that's the so-called static condensation

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that's the so-called static condensation

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that's the so-called static condensation
now this is a normal condensation as

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now this is a normal condensation as

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now this is a normal condensation as
opposed to a mold condensation that we

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opposed to a mold condensation that we

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opposed to a mold condensation that we
studied earlier under the modal solution

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studied earlier under the modal solution

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studied earlier under the modal solution
methods so the concentration is how you

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methods so the concentration is how you

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methods so the concentration is how you
handle the degrees of freedom at nodes

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handle the degrees of freedom at nodes

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handle the degrees of freedom at nodes
in this case unfortunately to move

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in this case unfortunately to move

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in this case unfortunately to move
beyond static condensation gets tough

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beyond static condensation gets tough

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beyond static condensation gets tough
very quickly I find it more difficult to

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very quickly I find it more difficult to

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very quickly I find it more difficult to
teach that material and so I don't

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teach that material and so I don't

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teach that material and so I don't
stress it as much I view it as an

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stress it as much I view it as an

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stress it as much I view it as an
advanced topic when you get in

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advanced topic when you get in

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advanced topic when you get in
component mode synthesis generalized

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component mode synthesis generalized

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component mode synthesis generalized
dynamic reduction some of those methods

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dynamic reduction some of those methods

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dynamic reduction some of those methods
are really fairly theoretically involved

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are really fairly theoretically involved

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are really fairly theoretically involved
so what I'll do is pull my punches a

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so what I'll do is pull my punches a

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so what I'll do is pull my punches a
little here and hit static condensation

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little here and hit static condensation

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little here and hit static condensation
pretty hard and then kind of wave my

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pretty hard and then kind of wave my

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pretty hard and then kind of wave my
arms a little on the more complicated

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arms a little on the more complicated

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arms a little on the more complicated
reduction methods now this is okay I'm

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reduction methods now this is okay I'm

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reduction methods now this is okay I'm
going to argue it's okay because some of

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going to argue it's okay because some of

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going to argue it's okay because some of
the new solvers don't really need this

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the new solvers don't really need this

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the new solvers don't really need this
simplification for instance cilantro's

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simplification for instance cilantro's

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simplification for instance cilantro's
method seems to work for fairly large

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method seems to work for fairly large

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method seems to work for fairly large
problems in a direct way and so I hope

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problems in a direct way and so I hope

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problems in a direct way and so I hope
that the user can figure out in your own

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that the user can figure out in your own

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that the user can figure out in your own
corporate setting of what sort of

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corporate setting of what sort of

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corporate setting of what sort of
dynamic reduction methods you're

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dynamic reduction methods you're

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dynamic reduction methods you're
interested in there are a lot of vehicle

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interested in there are a lot of vehicle

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interested in there are a lot of vehicle
people it's still really like component

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people it's still really like component

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people it's still really like component
mode synthesis and that's a super

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mode synthesis and that's a super

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mode synthesis and that's a super
element type of condensation approach

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element type of condensation approach

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element type of condensation approach
well the procedure will be to talk in

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well the procedure will be to talk in

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well the procedure will be to talk in
general about dynamic reduction hit the

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general about dynamic reduction hit the

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general about dynamic reduction hit the
static condensation rather hard on going

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static condensation rather hard on going

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static condensation rather hard on going
on reduction and do some examples and

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on reduction and do some examples and

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on reduction and do some examples and
then I'll just wave my arms at

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then I'll just wave my arms at

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then I'll just wave my arms at
generalize dynamic reduction in

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generalize dynamic reduction in

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generalize dynamic reduction in
component mode synthesis will follow

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component mode synthesis will follow

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component mode synthesis will follow
with some problem session examples which

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with some problem session examples which

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with some problem session examples which
are primarily the static reduction type

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are primarily the static reduction type

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are primarily the static reduction type
up here I'll start off with a bit of an

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up here I'll start off with a bit of an

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up here I'll start off with a bit of an
overview of dynamic reduction

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overview of dynamic reduction

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overview of dynamic reduction
in dynamic analysis the necessary steps

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in dynamic analysis the necessary steps

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in dynamic analysis the necessary steps
are to assemble the equations solve them

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are to assemble the equations solve them

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are to assemble the equations solve them
and then recover the forces and stresses

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and then recover the forces and stresses

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and then recover the forces and stresses
that are the internal fuel variables in

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that are the internal fuel variables in

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that are the internal fuel variables in
the problem an optional step which is so

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the problem an optional step which is so

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the problem an optional step which is so
important is this one where you might

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important is this one where you might

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important is this one where you might
reduce the number of degrees of freedom

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reduce the number of degrees of freedom

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reduce the number of degrees of freedom
in the system basically you pass from a

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in the system basically you pass from a

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in the system basically you pass from a
set of physical degrees of freedom to a

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set of physical degrees of freedom to a

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set of physical degrees of freedom to a
set of coordinates that are more or less

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set of coordinates that are more or less

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set of coordinates that are more or less
artificial sometimes that reduced set of

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artificial sometimes that reduced set of

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artificial sometimes that reduced set of
coordinates carries within it the more

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coordinates carries within it the more

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coordinates carries within it the more
complicated behavior of the whole system

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complicated behavior of the whole system

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complicated behavior of the whole system
but in some idealized way if you don't

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but in some idealized way if you don't

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but in some idealized way if you don't
do dynamic reduction you'll find that

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do dynamic reduction you'll find that

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do dynamic reduction you'll find that
the solution of the equations here

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the solution of the equations here

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the solution of the equations here
really becomes expensive part by far

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really becomes expensive part by far

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really becomes expensive part by far
more than 90% in most cases so you would

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more than 90% in most cases so you would

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more than 90% in most cases so you would
be willing to do a certain amount of

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be willing to do a certain amount of

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be willing to do a certain amount of
work at this stage in dynamic reduction

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work at this stage in dynamic reduction

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work at this stage in dynamic reduction
in order to reduce the solution cost

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there are several forms for dynamic

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there are several forms for dynamic

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there are several forms for dynamic
reduction we've already spoken about a

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reduction we've already spoken about a

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reduction we've already spoken about a
modal approach using normal malls as our

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modal approach using normal malls as our

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modal approach using normal malls as our
building blocks then the nodal

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building blocks then the nodal

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building blocks then the nodal
condensations are such as the goon

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condensations are such as the goon

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condensations are such as the goon
reduction here generalize dynamic

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reduction here generalize dynamic

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reduction here generalize dynamic
reduction is a bit in-between where you

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reduction is a bit in-between where you

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reduction is a bit in-between where you
have somewhat of a global and somewhat

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have somewhat of a global and somewhat

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have somewhat of a global and somewhat
of a local view on the situation and

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of a local view on the situation and

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of a local view on the situation and
we'll show some of those equations later

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we'll show some of those equations later

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we'll show some of those equations later
component mode synthesis is the dynamic

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component mode synthesis is the dynamic

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component mode synthesis is the dynamic
version of super element Theory where

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version of super element Theory where

217
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version of super element Theory where
you segment your larger structure into

218
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you segment your larger structure into

219
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you segment your larger structure into
pieces then those pieces are joined

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pieces then those pieces are joined

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pieces then those pieces are joined
after they're separately analyzed

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after they're separately analyzed

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after they're separately analyzed
dynamically and

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dynamically and

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dynamically and
one one key question there is how do you

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one one key question there is how do you

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one one key question there is how do you
join them together certainly if use a

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join them together certainly if use a

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join them together certainly if use a
mobile solution alone for really large

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mobile solution alone for really large

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mobile solution alone for really large
problems you may get into troubles with

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problems you may get into troubles with

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problems you may get into troubles with
the cost of eigenvalue extraction and

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the cost of eigenvalue extraction and

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the cost of eigenvalue extraction and
then as you get above ten thousand

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then as you get above ten thousand

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then as you get above ten thousand
degrees of freedom you're really forced

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degrees of freedom you're really forced

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degrees of freedom you're really forced
to go to something pretty powerful

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to go to something pretty powerful

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to go to something pretty powerful
perhaps component mode synthesis or

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perhaps component mode synthesis or

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perhaps component mode synthesis or
these newer Atlantia solvers the

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these newer Atlantia solvers the

245
00:06:07,970 --> 00:06:10,030
these newer Atlantia solvers the
Lentulus solver has pretty well swept

246
00:06:10,030 --> 00:06:10,040
Lentulus solver has pretty well swept

247
00:06:10,040 --> 00:06:12,760
Lentulus solver has pretty well swept
through the industry as I speak and is

248
00:06:12,760 --> 00:06:12,770
through the industry as I speak and is

249
00:06:12,770 --> 00:06:18,870
through the industry as I speak and is
now the dominant large problem solver

250
00:06:18,870 --> 00:06:18,880


251
00:06:18,880 --> 00:06:22,240
now let's discuss this green reduction

252
00:06:22,240 --> 00:06:22,250
now let's discuss this green reduction

253
00:06:22,250 --> 00:06:26,950
now let's discuss this green reduction
or static condensation and see how it

254
00:06:26,950 --> 00:06:26,960
or static condensation and see how it

255
00:06:26,960 --> 00:06:29,620
or static condensation and see how it
works we can follow it rather well in

256
00:06:29,620 --> 00:06:29,630
works we can follow it rather well in

257
00:06:29,630 --> 00:06:32,140
works we can follow it rather well in
fact you should understand this method

258
00:06:32,140 --> 00:06:32,150
fact you should understand this method

259
00:06:32,150 --> 00:06:34,960
fact you should understand this method
cold I think it's useful because it also

260
00:06:34,960 --> 00:06:34,970
cold I think it's useful because it also

261
00:06:34,970 --> 00:06:37,020
cold I think it's useful because it also
gives you the whole spirit of other

262
00:06:37,020 --> 00:06:37,030
gives you the whole spirit of other

263
00:06:37,030 --> 00:06:40,360
gives you the whole spirit of other
condensation methods will illustrate

264
00:06:40,360 --> 00:06:40,370
condensation methods will illustrate

265
00:06:40,370 --> 00:06:44,610
condensation methods will illustrate
this with the MSC Nastran data

266
00:06:44,610 --> 00:06:44,620
this with the MSC Nastran data

267
00:06:44,620 --> 00:06:48,690
this with the MSC Nastran data
definitions and their sets of variables

268
00:06:48,690 --> 00:06:48,700
definitions and their sets of variables

269
00:06:48,700 --> 00:06:53,020
definitions and their sets of variables
the original set of physical variables

270
00:06:53,020 --> 00:06:53,030
the original set of physical variables

271
00:06:53,030 --> 00:06:58,000
the original set of physical variables
would be called use of G and then after

272
00:06:58,000 --> 00:06:58,010
would be called use of G and then after

273
00:06:58,010 --> 00:07:00,220
would be called use of G and then after
reducing out the multi point constraints

274
00:07:00,220 --> 00:07:00,230
reducing out the multi point constraints

275
00:07:00,230 --> 00:07:03,690
reducing out the multi point constraints
you're left with a set u sub Annan

276
00:07:03,690 --> 00:07:03,700
you're left with a set u sub Annan

277
00:07:03,700 --> 00:07:06,700
you're left with a set u sub Annan
at that point you apply the single point

278
00:07:06,700 --> 00:07:06,710
at that point you apply the single point

279
00:07:06,710 --> 00:07:08,860
at that point you apply the single point
constraints and you're left with these

280
00:07:08,860 --> 00:07:08,870
constraints and you're left with these

281
00:07:08,870 --> 00:07:13,290
constraints and you're left with these
degrees of freedom use of F

282
00:07:13,290 --> 00:07:13,300


283
00:07:13,300 --> 00:07:15,900
here's where condensation enters and

284
00:07:15,900 --> 00:07:15,910
here's where condensation enters and

285
00:07:15,910 --> 00:07:17,219
here's where condensation enters and
there are a certain number of degrees of

286
00:07:17,219 --> 00:07:17,229
there are a certain number of degrees of

287
00:07:17,229 --> 00:07:18,990
there are a certain number of degrees of
freedom that are considered to be

288
00:07:18,990 --> 00:07:19,000
freedom that are considered to be

289
00:07:19,000 --> 00:07:22,140
freedom that are considered to be
omitted degrees of freedom when those

290
00:07:22,140 --> 00:07:22,150
omitted degrees of freedom when those

291
00:07:22,150 --> 00:07:24,210
omitted degrees of freedom when those
are taken out you're then left with the

292
00:07:24,210 --> 00:07:24,220
are taken out you're then left with the

293
00:07:24,220 --> 00:07:28,200
are taken out you're then left with the
analysis set of degrees of freedom even

294
00:07:28,200 --> 00:07:28,210
analysis set of degrees of freedom even

295
00:07:28,210 --> 00:07:30,420
analysis set of degrees of freedom even
the analysis set which is what we will

296
00:07:30,420 --> 00:07:30,430
the analysis set which is what we will

297
00:07:30,430 --> 00:07:33,659
the analysis set which is what we will
work with today can be subdivided into

298
00:07:33,659 --> 00:07:33,669
work with today can be subdivided into

299
00:07:33,669 --> 00:07:35,550
work with today can be subdivided into
those degrees of freedom that represent

300
00:07:35,550 --> 00:07:35,560
those degrees of freedom that represent

301
00:07:35,560 --> 00:07:38,219
those degrees of freedom that represent
rigid body modes and those that are or

302
00:07:38,219 --> 00:07:38,229
rigid body modes and those that are or

303
00:07:38,229 --> 00:07:40,559
rigid body modes and those that are or
need to be constrained to if you wish to

304
00:07:40,559 --> 00:07:40,569
need to be constrained to if you wish to

305
00:07:40,569 --> 00:07:44,550
need to be constrained to if you wish to
remove them and those that are the final

306
00:07:44,550 --> 00:07:44,560
remove them and those that are the final

307
00:07:44,560 --> 00:07:46,290
remove them and those that are the final
independent degrees of freedom which are

308
00:07:46,290 --> 00:07:46,300
independent degrees of freedom which are

309
00:07:46,300 --> 00:07:48,360
independent degrees of freedom which are
you can think of is the left your left

310
00:07:48,360 --> 00:07:48,370
you can think of is the left your left

311
00:07:48,370 --> 00:07:51,029
you can think of is the left your left
over degrees of freedom but basically in

312
00:07:51,029 --> 00:07:51,039
over degrees of freedom but basically in

313
00:07:51,039 --> 00:07:53,820
over degrees of freedom but basically in
dynamics there's no need as there is in

314
00:07:53,820 --> 00:07:53,830
dynamics there's no need as there is in

315
00:07:53,830 --> 00:07:56,969
dynamics there's no need as there is in
statics however to remove your javadi

316
00:07:56,969 --> 00:07:56,979
statics however to remove your javadi

317
00:07:56,979 --> 00:07:58,800
statics however to remove your javadi
mode so it's better to work at this

318
00:07:58,800 --> 00:07:58,810
mode so it's better to work at this

319
00:07:58,810 --> 00:08:02,370
mode so it's better to work at this
analysis set of variable level here so

320
00:08:02,370 --> 00:08:02,380
analysis set of variable level here so

321
00:08:02,380 --> 00:08:05,760
analysis set of variable level here so
we will be looking at glan reduction in

322
00:08:05,760 --> 00:08:05,770
we will be looking at glan reduction in

323
00:08:05,770 --> 00:08:09,899
we will be looking at glan reduction in
the way that you progress from this set

324
00:08:09,899 --> 00:08:09,909
the way that you progress from this set

325
00:08:09,909 --> 00:08:12,300
the way that you progress from this set
of degrees of freedom here and how you

326
00:08:12,300 --> 00:08:12,310
of degrees of freedom here and how you

327
00:08:12,310 --> 00:08:17,580
of degrees of freedom here and how you
break that into this and this we won't

328
00:08:17,580 --> 00:08:17,590
break that into this and this we won't

329
00:08:17,590 --> 00:08:20,279
break that into this and this we won't
go to the finest smallest detail were

330
00:08:20,279 --> 00:08:20,289
go to the finest smallest detail were

331
00:08:20,289 --> 00:08:22,260
go to the finest smallest detail were
you worried about degrees of freedom

332
00:08:22,260 --> 00:08:22,270
you worried about degrees of freedom

333
00:08:22,270 --> 00:08:23,760
you worried about degrees of freedom
that characterized rigid body modes

334
00:08:23,760 --> 00:08:23,770
that characterized rigid body modes

335
00:08:23,770 --> 00:08:25,980
that characterized rigid body modes
since in dynamics they can be carried

336
00:08:25,980 --> 00:08:25,990
since in dynamics they can be carried

337
00:08:25,990 --> 00:08:32,790
since in dynamics they can be carried
long anyway let's work out the

338
00:08:32,790 --> 00:08:32,800
long anyway let's work out the

339
00:08:32,800 --> 00:08:35,040
long anyway let's work out the
mathematics for the Guyenne reduction

340
00:08:35,040 --> 00:08:35,050
mathematics for the Guyenne reduction

341
00:08:35,050 --> 00:08:39,420
mathematics for the Guyenne reduction
and we'll do it as I mentioned in terms

342
00:08:39,420 --> 00:08:39,430
and we'll do it as I mentioned in terms

343
00:08:39,430 --> 00:08:41,790
and we'll do it as I mentioned in terms
of those sets of variables sub scripted

344
00:08:41,790 --> 00:08:41,800
of those sets of variables sub scripted

345
00:08:41,800 --> 00:08:43,800
of those sets of variables sub scripted
with F we might call those three degrees

346
00:08:43,800 --> 00:08:43,810
with F we might call those three degrees

347
00:08:43,810 --> 00:08:48,210
with F we might call those three degrees
of freedom just to kind of word to let

348
00:08:48,210 --> 00:08:48,220
of freedom just to kind of word to let

349
00:08:48,220 --> 00:08:50,190
of freedom just to kind of word to let
us discuss what that is and the ACE that

350
00:08:50,190 --> 00:08:50,200
us discuss what that is and the ACE that

351
00:08:50,200 --> 00:08:52,920
us discuss what that is and the ACE that
will be the analysis

352
00:08:52,920 --> 00:08:52,930
will be the analysis

353
00:08:52,930 --> 00:08:55,350
will be the analysis
so if we write the equations of motion

354
00:08:55,350 --> 00:08:55,360
so if we write the equations of motion

355
00:08:55,360 --> 00:08:59,100
so if we write the equations of motion
at this stage then the multi point

356
00:08:59,100 --> 00:08:59,110
at this stage then the multi point

357
00:08:59,110 --> 00:09:01,590
at this stage then the multi point
constraints have been reduced out of the

358
00:09:01,590 --> 00:09:01,600
constraints have been reduced out of the

359
00:09:01,600 --> 00:09:03,570
constraints have been reduced out of the
system and the single point constraints

360
00:09:03,570 --> 00:09:03,580
system and the single point constraints

361
00:09:03,580 --> 00:09:06,510
system and the single point constraints
have been reduced out so these then are

362
00:09:06,510 --> 00:09:06,520
have been reduced out so these then are

363
00:09:06,520 --> 00:09:07,980
have been reduced out so these then are
the three degrees of freedom that

364
00:09:07,980 --> 00:09:07,990
the three degrees of freedom that

365
00:09:07,990 --> 00:09:10,770
the three degrees of freedom that
described the active degrees of freedom

366
00:09:10,770 --> 00:09:10,780
described the active degrees of freedom

367
00:09:10,780 --> 00:09:13,770
described the active degrees of freedom
that remain at that stage there would be

368
00:09:13,770 --> 00:09:13,780
that remain at that stage there would be

369
00:09:13,780 --> 00:09:17,550
that remain at that stage there would be
mass terms and the happening terms and

370
00:09:17,550 --> 00:09:17,560
mass terms and the happening terms and

371
00:09:17,560 --> 00:09:21,150
mass terms and the happening terms and
stiffness terms these are pretty close

372
00:09:21,150 --> 00:09:21,160
stiffness terms these are pretty close

373
00:09:21,160 --> 00:09:23,220
stiffness terms these are pretty close
to physical degrees of freedom at this

374
00:09:23,220 --> 00:09:23,230
to physical degrees of freedom at this

375
00:09:23,230 --> 00:09:26,580
to physical degrees of freedom at this
point with some agglomeration effects

376
00:09:26,580 --> 00:09:26,590
point with some agglomeration effects

377
00:09:26,590 --> 00:09:29,340
point with some agglomeration effects
into the reduced matrices at this level

378
00:09:29,340 --> 00:09:29,350
into the reduced matrices at this level

379
00:09:29,350 --> 00:09:31,650
into the reduced matrices at this level
but closer to a physical set of degrees

380
00:09:31,650 --> 00:09:31,660
but closer to a physical set of degrees

381
00:09:31,660 --> 00:09:33,780
but closer to a physical set of degrees
of freedom but now we want to make a

382
00:09:33,780 --> 00:09:33,790
of freedom but now we want to make a

383
00:09:33,790 --> 00:09:38,460
of freedom but now we want to make a
strong distinction between the original

384
00:09:38,460 --> 00:09:38,470
strong distinction between the original

385
00:09:38,470 --> 00:09:40,470
strong distinction between the original
physical degrees of freedom and then a

386
00:09:40,470 --> 00:09:40,480
physical degrees of freedom and then a

387
00:09:40,480 --> 00:09:44,160
physical degrees of freedom and then a
reduced analysis set by doing this kind

388
00:09:44,160 --> 00:09:44,170
reduced analysis set by doing this kind

389
00:09:44,170 --> 00:09:46,020
reduced analysis set by doing this kind
of partitioning where we partition the

390
00:09:46,020 --> 00:09:46,030
of partitioning where we partition the

391
00:09:46,030 --> 00:09:48,360
of partitioning where we partition the
these three degrees of freedom into the

392
00:09:48,360 --> 00:09:48,370
these three degrees of freedom into the

393
00:09:48,370 --> 00:09:51,150
these three degrees of freedom into the
analysis the omitted set now I'm going

394
00:09:51,150 --> 00:09:51,160
analysis the omitted set now I'm going

395
00:09:51,160 --> 00:09:54,300
analysis the omitted set now I'm going
to do poetic license here or authors

396
00:09:54,300 --> 00:09:54,310
to do poetic license here or authors

397
00:09:54,310 --> 00:09:57,360
to do poetic license here or authors
license to assume that those were

398
00:09:57,360 --> 00:09:57,370
license to assume that those were

399
00:09:57,370 --> 00:09:59,520
license to assume that those were
numbered in a way such that I can

400
00:09:59,520 --> 00:09:59,530
numbered in a way such that I can

401
00:09:59,530 --> 00:10:01,470
numbered in a way such that I can
consecutively partition out these

402
00:10:01,470 --> 00:10:01,480
consecutively partition out these

403
00:10:01,480 --> 00:10:04,500
consecutively partition out these
omitted ones now normally the analysis

404
00:10:04,500 --> 00:10:04,510
omitted ones now normally the analysis

405
00:10:04,510 --> 00:10:08,160
omitted ones now normally the analysis
set ought to be less in ratio of one to

406
00:10:08,160 --> 00:10:08,170
set ought to be less in ratio of one to

407
00:10:08,170 --> 00:10:10,560
set ought to be less in ratio of one to
ten let's say you would like to reduce

408
00:10:10,560 --> 00:10:10,570
ten let's say you would like to reduce

409
00:10:10,570 --> 00:10:13,140
ten let's say you would like to reduce
your degrees of freedom greatly by doing

410
00:10:13,140 --> 00:10:13,150
your degrees of freedom greatly by doing

411
00:10:13,150 --> 00:10:15,960
your degrees of freedom greatly by doing
this method so I don't mean that these

412
00:10:15,960 --> 00:10:15,970
this method so I don't mean that these

413
00:10:15,970 --> 00:10:19,710
this method so I don't mean that these
two vector sub components here are of

414
00:10:19,710 --> 00:10:19,720
two vector sub components here are of

415
00:10:19,720 --> 00:10:23,010
two vector sub components here are of
the same size one of them the omitted

416
00:10:23,010 --> 00:10:23,020
the same size one of them the omitted

417
00:10:23,020 --> 00:10:25,140
the same size one of them the omitted
set is generally way bigger than the

418
00:10:25,140 --> 00:10:25,150
set is generally way bigger than the

419
00:10:25,150 --> 00:10:26,920
set is generally way bigger than the
other

420
00:10:26,920 --> 00:10:26,930
other

421
00:10:26,930 --> 00:10:29,470
other
now what we're going to assume is that

422
00:10:29,470 --> 00:10:29,480
now what we're going to assume is that

423
00:10:29,480 --> 00:10:31,930
now what we're going to assume is that
the relation between the analysis and

424
00:10:31,930 --> 00:10:31,940
the relation between the analysis and

425
00:10:31,940 --> 00:10:35,260
the relation between the analysis and
the omitted variables is based on static

426
00:10:35,260 --> 00:10:35,270
the omitted variables is based on static

427
00:10:35,270 --> 00:10:38,170
the omitted variables is based on static
equilibrium concepts that makes this

428
00:10:38,170 --> 00:10:38,180
equilibrium concepts that makes this

429
00:10:38,180 --> 00:10:41,400
equilibrium concepts that makes this
sort of a first order correction for

430
00:10:41,400 --> 00:10:41,410
sort of a first order correction for

431
00:10:41,410 --> 00:10:45,610
sort of a first order correction for
dynamic condensation and other people

432
00:10:45,610 --> 00:10:45,620
dynamic condensation and other people

433
00:10:45,620 --> 00:10:47,949
dynamic condensation and other people
will come along later and say hey I can

434
00:10:47,949 --> 00:10:47,959
will come along later and say hey I can

435
00:10:47,959 --> 00:10:50,910
will come along later and say hey I can
improve this by five ten percent by

436
00:10:50,910 --> 00:10:50,920
improve this by five ten percent by

437
00:10:50,920 --> 00:10:53,440
improve this by five ten percent by
accounting for those degrees of freedom

438
00:10:53,440 --> 00:10:53,450
accounting for those degrees of freedom

439
00:10:53,450 --> 00:10:56,410
accounting for those degrees of freedom
that are admitted in a dynamic way but

440
00:10:56,410 --> 00:10:56,420
that are admitted in a dynamic way but

441
00:10:56,420 --> 00:10:58,540
that are admitted in a dynamic way but
we don't do this you need to walk before

442
00:10:58,540 --> 00:10:58,550
we don't do this you need to walk before

443
00:10:58,550 --> 00:11:03,040
we don't do this you need to walk before
you can run in this case so you actually

444
00:11:03,040 --> 00:11:03,050
you can run in this case so you actually

445
00:11:03,050 --> 00:11:06,040
you can run in this case so you actually
do a mental experiment really starting

446
00:11:06,040 --> 00:11:06,050
do a mental experiment really starting

447
00:11:06,050 --> 00:11:09,670
do a mental experiment really starting
here and I'll draw an arrow here for a

448
00:11:09,670 --> 00:11:09,680
here and I'll draw an arrow here for a

449
00:11:09,680 --> 00:11:11,440
here and I'll draw an arrow here for a
short time and then I'll tell you when

450
00:11:11,440 --> 00:11:11,450
short time and then I'll tell you when

451
00:11:11,450 --> 00:11:14,019
short time and then I'll tell you when
we stop doing this we drop out the

452
00:11:14,019 --> 00:11:14,029
we stop doing this we drop out the

453
00:11:14,029 --> 00:11:16,900
we stop doing this we drop out the
inertia and damping forces and look at

454
00:11:16,900 --> 00:11:16,910
inertia and damping forces and look at

455
00:11:16,910 --> 00:11:19,930
inertia and damping forces and look at
an a static problem which means that

456
00:11:19,930 --> 00:11:19,940
an a static problem which means that

457
00:11:19,940 --> 00:11:21,670
an a static problem which means that
you're looking at your dynamic problem

458
00:11:21,670 --> 00:11:21,680
you're looking at your dynamic problem

459
00:11:21,680 --> 00:11:24,340
you're looking at your dynamic problem
in a quasi static way as if dynamics

460
00:11:24,340 --> 00:11:24,350
in a quasi static way as if dynamics

461
00:11:24,350 --> 00:11:26,560
in a quasi static way as if dynamics
really didn't matter and all of the

462
00:11:26,560 --> 00:11:26,570
really didn't matter and all of the

463
00:11:26,570 --> 00:11:28,630
really didn't matter and all of the
degrees of freedom are related purely by

464
00:11:28,630 --> 00:11:28,640
degrees of freedom are related purely by

465
00:11:28,640 --> 00:11:30,970
degrees of freedom are related purely by
the springs that connect up the elastic

466
00:11:30,970 --> 00:11:30,980
the springs that connect up the elastic

467
00:11:30,980 --> 00:11:36,250
the springs that connect up the elastic
forces let's carry out a mental

468
00:11:36,250 --> 00:11:36,260
forces let's carry out a mental

469
00:11:36,260 --> 00:11:39,160
forces let's carry out a mental
experiment now to help us eliminate the

470
00:11:39,160 --> 00:11:39,170
experiment now to help us eliminate the

471
00:11:39,170 --> 00:11:41,440
experiment now to help us eliminate the
unwanted degrees of freedom in this

472
00:11:41,440 --> 00:11:41,450
unwanted degrees of freedom in this

473
00:11:41,450 --> 00:11:46,380
unwanted degrees of freedom in this
problem one way to do it is imagine a

474
00:11:46,380 --> 00:11:46,390
problem one way to do it is imagine a

475
00:11:46,390 --> 00:11:48,940
problem one way to do it is imagine a
simplified structure such as this beam

476
00:11:48,940 --> 00:11:48,950
simplified structure such as this beam

477
00:11:48,950 --> 00:11:51,610
simplified structure such as this beam
like structure shown here and identify

478
00:11:51,610 --> 00:11:51,620
like structure shown here and identify

479
00:11:51,620 --> 00:11:53,890
like structure shown here and identify
certain important degrees of freedom

480
00:11:53,890 --> 00:11:53,900
certain important degrees of freedom

481
00:11:53,900 --> 00:11:56,140
certain important degrees of freedom
such as these analysis degrees of

482
00:11:56,140 --> 00:11:56,150
such as these analysis degrees of

483
00:11:56,150 --> 00:11:58,870
such as these analysis degrees of
freedom at the right end the center and

484
00:11:58,870 --> 00:11:58,880
freedom at the right end the center and

485
00:11:58,880 --> 00:12:01,870
freedom at the right end the center and
the left end and then identify less

486
00:12:01,870 --> 00:12:01,880
the left end and then identify less

487
00:12:01,880 --> 00:12:03,730
the left end and then identify less
important degrees of freedom such as

488
00:12:03,730 --> 00:12:03,740
important degrees of freedom such as

489
00:12:03,740 --> 00:12:06,310
important degrees of freedom such as
these at the intermediate spots there

490
00:12:06,310 --> 00:12:06,320
these at the intermediate spots there

491
00:12:06,320 --> 00:12:11,530
these at the intermediate spots there
and there then the analysis nodes are to

492
00:12:11,530 --> 00:12:11,540
and there then the analysis nodes are to

493
00:12:11,540 --> 00:12:13,569
and there then the analysis nodes are to
be treated as master nodes that control

494
00:12:13,569 --> 00:12:13,579
be treated as master nodes that control

495
00:12:13,579 --> 00:12:17,290
be treated as master nodes that control
the whole system this reminds me of the

496
00:12:17,290 --> 00:12:17,300
the whole system this reminds me of the

497
00:12:17,300 --> 00:12:20,140
the whole system this reminds me of the
lofting curves that were used when I

498
00:12:20,140 --> 00:12:20,150
lofting curves that were used when I

499
00:12:20,150 --> 00:12:22,750
lofting curves that were used when I
worked years ago at Vaught aircraft and

500
00:12:22,750 --> 00:12:22,760
worked years ago at Vaught aircraft and

501
00:12:22,760 --> 00:12:24,790
worked years ago at Vaught aircraft and
at Boeing where they had long plastic

502
00:12:24,790 --> 00:12:24,800
at Boeing where they had long plastic

503
00:12:24,800 --> 00:12:27,460
at Boeing where they had long plastic
strips and then these strips were held

504
00:12:27,460 --> 00:12:27,470
strips and then these strips were held

505
00:12:27,470 --> 00:12:30,100
strips and then these strips were held
in place at intervals by what they

506
00:12:30,100 --> 00:12:30,110
in place at intervals by what they

507
00:12:30,110 --> 00:12:33,190
in place at intervals by what they
called ducts and these were heavy lead

508
00:12:33,190 --> 00:12:33,200
called ducts and these were heavy lead

509
00:12:33,200 --> 00:12:36,550
called ducts and these were heavy lead
bodies that had a nose on them and they

510
00:12:36,550 --> 00:12:36,560
bodies that had a nose on them and they

511
00:12:36,560 --> 00:12:39,430
bodies that had a nose on them and they
looked a little bit like a duct that's

512
00:12:39,430 --> 00:12:39,440
looked a little bit like a duct that's

513
00:12:39,440 --> 00:12:42,579
looked a little bit like a duct that's
where the name came from that fit into

514
00:12:42,579 --> 00:12:42,589
where the name came from that fit into

515
00:12:42,589 --> 00:12:45,579
where the name came from that fit into
the slide along the top edge of this

516
00:12:45,579 --> 00:12:45,589
the slide along the top edge of this

517
00:12:45,589 --> 00:12:48,160
the slide along the top edge of this
plastic strip and then by controlling

518
00:12:48,160 --> 00:12:48,170
plastic strip and then by controlling

519
00:12:48,170 --> 00:12:50,019
plastic strip and then by controlling
the shape of that long strip at

520
00:12:50,019 --> 00:12:50,029
the shape of that long strip at

521
00:12:50,029 --> 00:12:52,120
the shape of that long strip at
intervals the draftsman had a nice

522
00:12:52,120 --> 00:12:52,130
intervals the draftsman had a nice

523
00:12:52,130 --> 00:12:55,510
intervals the draftsman had a nice
smooth contour to follow with their pen

524
00:12:55,510 --> 00:12:55,520
smooth contour to follow with their pen

525
00:12:55,520 --> 00:12:58,260
smooth contour to follow with their pen
and this is very much the same

526
00:12:58,260 --> 00:12:58,270
and this is very much the same

527
00:12:58,270 --> 00:13:00,730
and this is very much the same
philosophy that we have here we want the

528
00:13:00,730 --> 00:13:00,740
philosophy that we have here we want the

529
00:13:00,740 --> 00:13:02,829
philosophy that we have here we want the
Anastas degrees of freedom to control

530
00:13:02,829 --> 00:13:02,839
Anastas degrees of freedom to control

531
00:13:02,839 --> 00:13:07,079
Anastas degrees of freedom to control
all of the shape of the body statically

532
00:13:07,079 --> 00:13:07,089
all of the shape of the body statically

533
00:13:07,089 --> 00:13:09,569
all of the shape of the body statically
when we write out the equations

534
00:13:09,569 --> 00:13:09,579
when we write out the equations

535
00:13:09,579 --> 00:13:12,490
when we write out the equations
presuming that the analysis degrees of

536
00:13:12,490 --> 00:13:12,500
presuming that the analysis degrees of

537
00:13:12,500 --> 00:13:15,069
presuming that the analysis degrees of
freedom are above the partitioning marks

538
00:13:15,069 --> 00:13:15,079
freedom are above the partitioning marks

539
00:13:15,079 --> 00:13:16,960
freedom are above the partitioning marks
hair and the omitted degrees of freedom

540
00:13:16,960 --> 00:13:16,970
hair and the omitted degrees of freedom

541
00:13:16,970 --> 00:13:20,050
hair and the omitted degrees of freedom
are below then we have this set of

542
00:13:20,050 --> 00:13:20,060
are below then we have this set of

543
00:13:20,060 --> 00:13:23,110
are below then we have this set of
equations for this mental experiment

544
00:13:23,110 --> 00:13:23,120
equations for this mental experiment

545
00:13:23,120 --> 00:13:25,980
equations for this mental experiment
only we don't put live loads on the

546
00:13:25,980 --> 00:13:25,990
only we don't put live loads on the

547
00:13:25,990 --> 00:13:29,620
only we don't put live loads on the
omitted degrees of freedom many students

548
00:13:29,620 --> 00:13:29,630
omitted degrees of freedom many students

549
00:13:29,630 --> 00:13:32,140
omitted degrees of freedom many students
who study this forget later on and think

550
00:13:32,140 --> 00:13:32,150
who study this forget later on and think

551
00:13:32,150 --> 00:13:33,970
who study this forget later on and think
that you can't have live loads on the

552
00:13:33,970 --> 00:13:33,980
that you can't have live loads on the

553
00:13:33,980 --> 00:13:36,519
that you can't have live loads on the
full structures or meta degrees of

554
00:13:36,519 --> 00:13:36,529
full structures or meta degrees of

555
00:13:36,529 --> 00:13:38,530
full structures or meta degrees of
freedom that's not true now these are

556
00:13:38,530 --> 00:13:38,540
freedom that's not true now these are

557
00:13:38,540 --> 00:13:40,949
freedom that's not true now these are
the physical degrees of freedom

558
00:13:40,949 --> 00:13:40,959
the physical degrees of freedom

559
00:13:40,959 --> 00:13:43,930
the physical degrees of freedom
presently and these are physical forces

560
00:13:43,930 --> 00:13:43,940
presently and these are physical forces

561
00:13:43,940 --> 00:13:46,600
presently and these are physical forces
on the analysis degrees of freedom up

562
00:13:46,600 --> 00:13:46,610
on the analysis degrees of freedom up

563
00:13:46,610 --> 00:13:50,019
on the analysis degrees of freedom up
above you're able to partition out and

564
00:13:50,019 --> 00:13:50,029
above you're able to partition out and

565
00:13:50,029 --> 00:13:51,049
above you're able to partition out and
solve the sight

566
00:13:51,049 --> 00:13:51,059
solve the sight

567
00:13:51,059 --> 00:13:55,459
solve the sight
large set of equations here which

568
00:13:55,459 --> 00:13:55,469
large set of equations here which

569
00:13:55,469 --> 00:13:58,119
large set of equations here which
involve these coefficients shown below

570
00:13:58,119 --> 00:13:58,129
involve these coefficients shown below

571
00:13:58,129 --> 00:14:02,539
involve these coefficients shown below
the KOA matrix which is rectangular the

572
00:14:02,539 --> 00:14:02,549
the KOA matrix which is rectangular the

573
00:14:02,549 --> 00:14:06,109
the KOA matrix which is rectangular the
K all all matrix which is square then by

574
00:14:06,109 --> 00:14:06,119
K all all matrix which is square then by

575
00:14:06,119 --> 00:14:08,929
K all all matrix which is square then by
inverting the hay or matrix you can

576
00:14:08,929 --> 00:14:08,939
inverting the hay or matrix you can

577
00:14:08,939 --> 00:14:12,109
inverting the hay or matrix you can
solve for the omitted degrees of freedom

578
00:14:12,109 --> 00:14:12,119
solve for the omitted degrees of freedom

579
00:14:12,119 --> 00:14:14,809
solve for the omitted degrees of freedom
in terms of the analysis degrees of

580
00:14:14,809 --> 00:14:14,819
in terms of the analysis degrees of

581
00:14:14,819 --> 00:14:17,149
in terms of the analysis degrees of
freedom so this is a mapping that you

582
00:14:17,149 --> 00:14:17,159
freedom so this is a mapping that you

583
00:14:17,159 --> 00:14:19,099
freedom so this is a mapping that you
might want to consider moving in this

584
00:14:19,099 --> 00:14:19,109
might want to consider moving in this

585
00:14:19,109 --> 00:14:22,549
might want to consider moving in this
direction and very effective than that

586
00:14:22,549 --> 00:14:22,559
direction and very effective than that

587
00:14:22,559 --> 00:14:24,499
direction and very effective than that
letting these analysis degrees of

588
00:14:24,499 --> 00:14:24,509
letting these analysis degrees of

589
00:14:24,509 --> 00:14:27,259
letting these analysis degrees of
freedom control the omitted degrees of

590
00:14:27,259 --> 00:14:27,269
freedom control the omitted degrees of

591
00:14:27,269 --> 00:14:34,039
freedom control the omitted degrees of
freedom will give that mapping matrix

592
00:14:34,039 --> 00:14:34,049
freedom will give that mapping matrix

593
00:14:34,049 --> 00:14:36,680
freedom will give that mapping matrix
assembled G

594
00:14:36,680 --> 00:14:36,690
assembled G

595
00:14:36,690 --> 00:14:40,610
assembled G
and here it is

596
00:14:40,610 --> 00:14:40,620


597
00:14:40,620 --> 00:14:42,770
and we will include the minus sign

598
00:14:42,770 --> 00:14:42,780
and we will include the minus sign

599
00:14:42,780 --> 00:14:46,270
and we will include the minus sign
within that definition shown here I

600
00:14:46,270 --> 00:14:46,280
within that definition shown here I

601
00:14:46,280 --> 00:14:51,530
within that definition shown here I
suppose she can as a crutch can be

602
00:14:51,530 --> 00:14:51,540
suppose she can as a crutch can be

603
00:14:51,540 --> 00:14:57,110
suppose she can as a crutch can be
thought of as a transformation here so

604
00:14:57,110 --> 00:14:57,120
thought of as a transformation here so

605
00:14:57,120 --> 00:15:00,530
thought of as a transformation here so
then you have the ability to take the

606
00:15:00,530 --> 00:15:00,540
then you have the ability to take the

607
00:15:00,540 --> 00:15:03,740
then you have the ability to take the
smaller set of variables use of a and

608
00:15:03,740 --> 00:15:03,750
smaller set of variables use of a and

609
00:15:03,750 --> 00:15:07,700
smaller set of variables use of a and
let it drive the larger set use of F

610
00:15:07,700 --> 00:15:07,710
let it drive the larger set use of F

611
00:15:07,710 --> 00:15:12,560
let it drive the larger set use of F
over here interestingly to expand this

612
00:15:12,560 --> 00:15:12,570
over here interestingly to expand this

613
00:15:12,570 --> 00:15:16,400
over here interestingly to expand this
set in the static way we've had just

614
00:15:16,400 --> 00:15:16,410
set in the static way we've had just

615
00:15:16,410 --> 00:15:19,340
set in the static way we've had just
described then you use the identity

616
00:15:19,340 --> 00:15:19,350
described then you use the identity

617
00:15:19,350 --> 00:15:22,520
described then you use the identity
matrix here to map into the smaller set

618
00:15:22,520 --> 00:15:22,530
matrix here to map into the smaller set

619
00:15:22,530 --> 00:15:26,720
matrix here to map into the smaller set
say of relative size one and then you

620
00:15:26,720 --> 00:15:26,730
say of relative size one and then you

621
00:15:26,730 --> 00:15:30,980
say of relative size one and then you
use the G or a matrix to give you the

622
00:15:30,980 --> 00:15:30,990
use the G or a matrix to give you the

623
00:15:30,990 --> 00:15:33,470
use the G or a matrix to give you the
omitted coordinates here so it's as if

624
00:15:33,470 --> 00:15:33,480
omitted coordinates here so it's as if

625
00:15:33,480 --> 00:15:36,050
omitted coordinates here so it's as if
there's two parallel pipe lines here

626
00:15:36,050 --> 00:15:36,060
there's two parallel pipe lines here

627
00:15:36,060 --> 00:15:38,660
there's two parallel pipe lines here
going from right to left the analysis

628
00:15:38,660 --> 00:15:38,670
going from right to left the analysis

629
00:15:38,670 --> 00:15:40,460
going from right to left the analysis
degrees of freedom propagate straight

630
00:15:40,460 --> 00:15:40,470
degrees of freedom propagate straight

631
00:15:40,470 --> 00:15:42,560
degrees of freedom propagate straight
through this system into the three

632
00:15:42,560 --> 00:15:42,570
through this system into the three

633
00:15:42,570 --> 00:15:44,930
through this system into the three
degrees of freedom and the omitted ones

634
00:15:44,930 --> 00:15:44,940
degrees of freedom and the omitted ones

635
00:15:44,940 --> 00:15:48,410
degrees of freedom and the omitted ones
have to be mapped to become their

636
00:15:48,410 --> 00:15:48,420
have to be mapped to become their

637
00:15:48,420 --> 00:15:50,420
have to be mapped to become their
rightful spot in the three degrees of

638
00:15:50,420 --> 00:15:50,430
rightful spot in the three degrees of

639
00:15:50,430 --> 00:15:54,230
rightful spot in the three degrees of
freedom alright so we've got a mapping

640
00:15:54,230 --> 00:15:54,240
freedom alright so we've got a mapping

641
00:15:54,240 --> 00:15:56,060
freedom alright so we've got a mapping
we've got a way to statically condense

642
00:15:56,060 --> 00:15:56,070
we've got a way to statically condense

643
00:15:56,070 --> 00:16:00,050
we've got a way to statically condense
this problem at this point we have ended

644
00:16:00,050 --> 00:16:00,060
this problem at this point we have ended

645
00:16:00,060 --> 00:16:04,180
this problem at this point we have ended
the mental experiment which was up above

646
00:16:04,180 --> 00:16:04,190
the mental experiment which was up above

647
00:16:04,190 --> 00:16:08,240
the mental experiment which was up above
and now we get back to the real world we

648
00:16:08,240 --> 00:16:08,250
and now we get back to the real world we

649
00:16:08,250 --> 00:16:11,240
and now we get back to the real world we
look at the full equation of motion for

650
00:16:11,240 --> 00:16:11,250
look at the full equation of motion for

651
00:16:11,250 --> 00:16:14,470
look at the full equation of motion for
our three degrees of freedom and we

652
00:16:14,470 --> 00:16:14,480
our three degrees of freedom and we

653
00:16:14,480 --> 00:16:17,960
our three degrees of freedom and we
replace the three degrees of freedom

654
00:16:17,960 --> 00:16:17,970
replace the three degrees of freedom

655
00:16:17,970 --> 00:16:21,020
replace the three degrees of freedom
with the equivalent mapping version now

656
00:16:21,020 --> 00:16:21,030
with the equivalent mapping version now

657
00:16:21,030 --> 00:16:23,750
with the equivalent mapping version now
if we wanted to take a single derivative

658
00:16:23,750 --> 00:16:23,760
if we wanted to take a single derivative

659
00:16:23,760 --> 00:16:27,260
if we wanted to take a single derivative
of this that works you can take a second

660
00:16:27,260 --> 00:16:27,270
of this that works you can take a second

661
00:16:27,270 --> 00:16:29,570
of this that works you can take a second
derivative that work so I can map the

662
00:16:29,570 --> 00:16:29,580
derivative that work so I can map the

663
00:16:29,580 --> 00:16:33,500
derivative that work so I can map the
not only the displacement field but the

664
00:16:33,500 --> 00:16:33,510
not only the displacement field but the

665
00:16:33,510 --> 00:16:35,750
not only the displacement field but the
velocity field and the acceleration

666
00:16:35,750 --> 00:16:35,760
velocity field and the acceleration

667
00:16:35,760 --> 00:16:38,240
velocity field and the acceleration
field with the same mapping that's

668
00:16:38,240 --> 00:16:38,250
field with the same mapping that's

669
00:16:38,250 --> 00:16:40,670
field with the same mapping that's
because all of these quantities in here

670
00:16:40,670 --> 00:16:40,680
because all of these quantities in here

671
00:16:40,680 --> 00:16:44,030
because all of these quantities in here
are are not time dependent that was from

672
00:16:44,030 --> 00:16:44,040
are are not time dependent that was from

673
00:16:44,040 --> 00:16:47,840
are are not time dependent that was from
our static experiment a static curve

674
00:16:47,840 --> 00:16:47,850
our static experiment a static curve

675
00:16:47,850 --> 00:16:51,980
our static experiment a static curve
and then on the right we will also

676
00:16:51,980 --> 00:16:51,990
and then on the right we will also

677
00:16:51,990 --> 00:16:55,070
and then on the right we will also
consider that the original forces on the

678
00:16:55,070 --> 00:16:55,080
consider that the original forces on the

679
00:16:55,080 --> 00:16:58,760
consider that the original forces on the
system here were and these are close to

680
00:16:58,760 --> 00:16:58,770
system here were and these are close to

681
00:16:58,770 --> 00:17:02,900
system here were and these are close to
physical forces here are an F bar of a

682
00:17:02,900 --> 00:17:02,910
physical forces here are an F bar of a

683
00:17:02,910 --> 00:17:05,510
physical forces here are an F bar of a
and an F zero I carry the bar

684
00:17:05,510 --> 00:17:05,520
and an F zero I carry the bar

685
00:17:05,520 --> 00:17:10,250
and an F zero I carry the bar
temporarily to to avoid the use of the F

686
00:17:10,250 --> 00:17:10,260
temporarily to to avoid the use of the F

687
00:17:10,260 --> 00:17:13,130
temporarily to to avoid the use of the F
sub a which I'll have later on so I'm

688
00:17:13,130 --> 00:17:13,140
sub a which I'll have later on so I'm

689
00:17:13,140 --> 00:17:16,790
sub a which I'll have later on so I'm
reserving the unbarred symbol F for

690
00:17:16,790 --> 00:17:16,800
reserving the unbarred symbol F for

691
00:17:16,800 --> 00:17:22,880
reserving the unbarred symbol F for
later final use alright we are in the

692
00:17:22,880 --> 00:17:22,890
later final use alright we are in the

693
00:17:22,890 --> 00:17:25,189
later final use alright we are in the
same position as we've been earlier in

694
00:17:25,189 --> 00:17:25,199
same position as we've been earlier in

695
00:17:25,199 --> 00:17:28,340
same position as we've been earlier in
this lecture series where we have say n

696
00:17:28,340 --> 00:17:28,350
this lecture series where we have say n

697
00:17:28,350 --> 00:17:31,850
this lecture series where we have say n
equations in n unknowns and we need to

698
00:17:31,850 --> 00:17:31,860
equations in n unknowns and we need to

699
00:17:31,860 --> 00:17:33,650
equations in n unknowns and we need to
do something to convert this into a

700
00:17:33,650 --> 00:17:33,660
do something to convert this into a

701
00:17:33,660 --> 00:17:36,620
do something to convert this into a
better handle set of equations and again

702
00:17:36,620 --> 00:17:36,630
better handle set of equations and again

703
00:17:36,630 --> 00:17:38,960
better handle set of equations and again
we look toward an energy approach where

704
00:17:38,960 --> 00:17:38,970
we look toward an energy approach where

705
00:17:38,970 --> 00:17:41,230
we look toward an energy approach where
we can pre multiply by something that's

706
00:17:41,230 --> 00:17:41,240
we can pre multiply by something that's

707
00:17:41,240 --> 00:17:44,450
we can pre multiply by something that's
displacement like this particular set of

708
00:17:44,450 --> 00:17:44,460
displacement like this particular set of

709
00:17:44,460 --> 00:17:47,090
displacement like this particular set of
equations here would in a plate

710
00:17:47,090 --> 00:17:47,100
equations here would in a plate

711
00:17:47,100 --> 00:17:49,310
equations here would in a plate
structure have all our net translations

712
00:17:49,310 --> 00:17:49,320
structure have all our net translations

713
00:17:49,320 --> 00:17:52,370
structure have all our net translations
and rotations and it's numerically not

714
00:17:52,370 --> 00:17:52,380
and rotations and it's numerically not

715
00:17:52,380 --> 00:17:56,600
and rotations and it's numerically not
real well posed at this time and so

716
00:17:56,600 --> 00:17:56,610
real well posed at this time and so

717
00:17:56,610 --> 00:17:58,370
real well posed at this time and so
we'll move to the next figure to show

718
00:17:58,370 --> 00:17:58,380
we'll move to the next figure to show

719
00:17:58,380 --> 00:18:05,570
we'll move to the next figure to show
the glucan type approach our set of

720
00:18:05,570 --> 00:18:05,580
the glucan type approach our set of

721
00:18:05,580 --> 00:18:08,300
the glucan type approach our set of
equations then is a set of force and

722
00:18:08,300 --> 00:18:08,310
equations then is a set of force and

723
00:18:08,310 --> 00:18:10,850
equations then is a set of force and
moment balances at this point but if we

724
00:18:10,850 --> 00:18:10,860
moment balances at this point but if we

725
00:18:10,860 --> 00:18:12,760
moment balances at this point but if we
do a pre multiplication with a

726
00:18:12,760 --> 00:18:12,770
do a pre multiplication with a

727
00:18:12,770 --> 00:18:16,580
do a pre multiplication with a
displacement like vector then we're

728
00:18:16,580 --> 00:18:16,590
displacement like vector then we're

729
00:18:16,590 --> 00:18:19,010
displacement like vector then we're
going to end up with an energy balance

730
00:18:19,010 --> 00:18:19,020
going to end up with an energy balance

731
00:18:19,020 --> 00:18:22,640
going to end up with an energy balance
instead so we pre-multiply by our same

732
00:18:22,640 --> 00:18:22,650
instead so we pre-multiply by our same

733
00:18:22,650 --> 00:18:25,700
instead so we pre-multiply by our same
transformation matrix and that gives us

734
00:18:25,700 --> 00:18:25,710
transformation matrix and that gives us

735
00:18:25,710 --> 00:18:29,760
transformation matrix and that gives us
this set of quantities and we get a new

736
00:18:29,760 --> 00:18:29,770
this set of quantities and we get a new

737
00:18:29,770 --> 00:18:33,240
this set of quantities and we get a new
the mass here out of this triple matrix

738
00:18:33,240 --> 00:18:33,250
the mass here out of this triple matrix

739
00:18:33,250 --> 00:18:36,500
the mass here out of this triple matrix
product and we'll call these the

740
00:18:36,500 --> 00:18:36,510
product and we'll call these the

741
00:18:36,510 --> 00:18:41,580
product and we'll call these the
analysis mass terms here we get a new

742
00:18:41,580 --> 00:18:41,590
analysis mass terms here we get a new

743
00:18:41,590 --> 00:18:44,070
analysis mass terms here we get a new
kind of damping matrix that will call

744
00:18:44,070 --> 00:18:44,080
kind of damping matrix that will call

745
00:18:44,080 --> 00:18:48,240
kind of damping matrix that will call
the analysis or reduced damping terms

746
00:18:48,240 --> 00:18:48,250
the analysis or reduced damping terms

747
00:18:48,250 --> 00:18:51,600
the analysis or reduced damping terms
and here's a new kind of stiffness shown

748
00:18:51,600 --> 00:18:51,610
and here's a new kind of stiffness shown

749
00:18:51,610 --> 00:18:56,310
and here's a new kind of stiffness shown
here on the fourth side we get a new set

750
00:18:56,310 --> 00:18:56,320
here on the fourth side we get a new set

751
00:18:56,320 --> 00:18:59,550
here on the fourth side we get a new set
of equivalent loads these are a set

752
00:18:59,550 --> 00:18:59,560
of equivalent loads these are a set

753
00:18:59,560 --> 00:19:02,370
of equivalent loads these are a set
loads these are have become more and

754
00:19:02,370 --> 00:19:02,380
loads these are have become more and

755
00:19:02,380 --> 00:19:04,230
loads these are have become more and
more artificial they're not the original

756
00:19:04,230 --> 00:19:04,240
more artificial they're not the original

757
00:19:04,240 --> 00:19:06,510
more artificial they're not the original
physical set of loads on a but they're

758
00:19:06,510 --> 00:19:06,520
physical set of loads on a but they're

759
00:19:06,520 --> 00:19:09,090
physical set of loads on a but they're
they're the representation of all the

760
00:19:09,090 --> 00:19:09,100
they're the representation of all the

761
00:19:09,100 --> 00:19:11,220
they're the representation of all the
loads in the system and how you have to

762
00:19:11,220 --> 00:19:11,230
loads in the system and how you have to

763
00:19:11,230 --> 00:19:13,950
loads in the system and how you have to
agglomerate those at the remaining few

764
00:19:13,950 --> 00:19:13,960
agglomerate those at the remaining few

765
00:19:13,960 --> 00:19:16,530
agglomerate those at the remaining few
degrees of freedom so these are

766
00:19:16,530 --> 00:19:16,540
degrees of freedom so these are

767
00:19:16,540 --> 00:19:20,850
degrees of freedom so these are
basically equivalent forces over here in

768
00:19:20,850 --> 00:19:20,860
basically equivalent forces over here in

769
00:19:20,860 --> 00:19:23,040
basically equivalent forces over here in
the earlier terminology where we had

770
00:19:23,040 --> 00:19:23,050
the earlier terminology where we had

771
00:19:23,050 --> 00:19:25,830
the earlier terminology where we had
dealt with modal degrees of freedom and

772
00:19:25,830 --> 00:19:25,840
dealt with modal degrees of freedom and

773
00:19:25,840 --> 00:19:28,710
dealt with modal degrees of freedom and
generalize degrees of freedom we got

774
00:19:28,710 --> 00:19:28,720
generalize degrees of freedom we got

775
00:19:28,720 --> 00:19:30,870
generalize degrees of freedom we got
generalized forces and this is

776
00:19:30,870 --> 00:19:30,880
generalized forces and this is

777
00:19:30,880 --> 00:19:34,500
generalized forces and this is
definitely that same philosophy you

778
00:19:34,500 --> 00:19:34,510
definitely that same philosophy you

779
00:19:34,510 --> 00:19:36,450
definitely that same philosophy you
might think of these as reduced degrees

780
00:19:36,450 --> 00:19:36,460
might think of these as reduced degrees

781
00:19:36,460 --> 00:19:44,430
might think of these as reduced degrees
of freedom and reduced forces the theory

782
00:19:44,430 --> 00:19:44,440
of freedom and reduced forces the theory

783
00:19:44,440 --> 00:19:46,530
of freedom and reduced forces the theory
for grand reduction is really rather

784
00:19:46,530 --> 00:19:46,540
for grand reduction is really rather

785
00:19:46,540 --> 00:19:50,250
for grand reduction is really rather
nice and can be incorporated into

786
00:19:50,250 --> 00:19:50,260
nice and can be incorporated into

787
00:19:50,260 --> 00:19:52,410
nice and can be incorporated into
computer codes rather straightforwardly

788
00:19:52,410 --> 00:19:52,420
computer codes rather straightforwardly

789
00:19:52,420 --> 00:19:55,980
computer codes rather straightforwardly
there's a few decisions to be made I'll

790
00:19:55,980 --> 00:19:55,990
there's a few decisions to be made I'll

791
00:19:55,990 --> 00:19:58,320
there's a few decisions to be made I'll
show how those have been resolved in the

792
00:19:58,320 --> 00:19:58,330
show how those have been resolved in the

793
00:19:58,330 --> 00:20:03,400
show how those have been resolved in the
user implementation and MSC Nastran

794
00:20:03,400 --> 00:20:03,410
user implementation and MSC Nastran

795
00:20:03,410 --> 00:20:06,790
user implementation and MSC Nastran
you have a body that is a potato shape

796
00:20:06,790 --> 00:20:06,800
you have a body that is a potato shape

797
00:20:06,800 --> 00:20:10,240
you have a body that is a potato shape
and has live loads in various places on

798
00:20:10,240 --> 00:20:10,250
and has live loads in various places on

799
00:20:10,250 --> 00:20:13,270
and has live loads in various places on
the body some on what are too soon to be

800
00:20:13,270 --> 00:20:13,280
the body some on what are too soon to be

801
00:20:13,280 --> 00:20:15,850
the body some on what are too soon to be
analysis degrees of freedom and some on

802
00:20:15,850 --> 00:20:15,860
analysis degrees of freedom and some on

803
00:20:15,860 --> 00:20:18,330
analysis degrees of freedom and some on
what will be only two degrees of freedom

804
00:20:18,330 --> 00:20:18,340
what will be only two degrees of freedom

805
00:20:18,340 --> 00:20:21,400
what will be only two degrees of freedom
how should you model such a system as

806
00:20:21,400 --> 00:20:21,410
how should you model such a system as

807
00:20:21,410 --> 00:20:26,530
how should you model such a system as
this well for one thing you should take

808
00:20:26,530 --> 00:20:26,540
this well for one thing you should take

809
00:20:26,540 --> 00:20:29,110
this well for one thing you should take
any point or any area here that has a

810
00:20:29,110 --> 00:20:29,120
any point or any area here that has a

811
00:20:29,120 --> 00:20:31,690
any point or any area here that has a
large mass and make sure it has analysis

812
00:20:31,690 --> 00:20:31,700
large mass and make sure it has analysis

813
00:20:31,700 --> 00:20:33,970
large mass and make sure it has analysis
degrees of freedom describing it that

814
00:20:33,970 --> 00:20:33,980
degrees of freedom describing it that

815
00:20:33,980 --> 00:20:36,880
degrees of freedom describing it that
way you're not using the so called

816
00:20:36,880 --> 00:20:36,890
way you're not using the so called

817
00:20:36,890 --> 00:20:40,240
way you're not using the so called
beaming out of the mass effects there to

818
00:20:40,240 --> 00:20:40,250
beaming out of the mass effects there to

819
00:20:40,250 --> 00:20:43,450
beaming out of the mass effects there to
some other node point so it reduces the

820
00:20:43,450 --> 00:20:43,460
some other node point so it reduces the

821
00:20:43,460 --> 00:20:47,650
some other node point so it reduces the
effect of the static reduction you

822
00:20:47,650 --> 00:20:47,660
effect of the static reduction you

823
00:20:47,660 --> 00:20:49,660
effect of the static reduction you
should distribute the eight points over

824
00:20:49,660 --> 00:20:49,670
should distribute the eight points over

825
00:20:49,670 --> 00:20:52,360
should distribute the eight points over
the body somewhat uniformly so that

826
00:20:52,360 --> 00:20:52,370
the body somewhat uniformly so that

827
00:20:52,370 --> 00:20:54,370
the body somewhat uniformly so that
you're doing a fair job of representing

828
00:20:54,370 --> 00:20:54,380
you're doing a fair job of representing

829
00:20:54,380 --> 00:20:57,910
you're doing a fair job of representing
that the total geometry that also means

830
00:20:57,910 --> 00:20:57,920
that the total geometry that also means

831
00:20:57,920 --> 00:21:00,790
that the total geometry that also means
that you don't have to beam out the

832
00:21:00,790 --> 00:21:00,800
that you don't have to beam out the

833
00:21:00,800 --> 00:21:04,060
that you don't have to beam out the
inertia effects so far and the damping

834
00:21:04,060 --> 00:21:04,070
inertia effects so far and the damping

835
00:21:04,070 --> 00:21:07,990
inertia effects so far and the damping
effects now these two are rather common

836
00:21:07,990 --> 00:21:08,000
effects now these two are rather common

837
00:21:08,000 --> 00:21:10,060
effects now these two are rather common
sense the third one is not so intuitive

838
00:21:10,060 --> 00:21:10,070
sense the third one is not so intuitive

839
00:21:10,070 --> 00:21:12,820
sense the third one is not so intuitive
and that's the feeling that it's better

840
00:21:12,820 --> 00:21:12,830
and that's the feeling that it's better

841
00:21:12,830 --> 00:21:15,100
and that's the feeling that it's better
for the a set to have both the

842
00:21:15,100 --> 00:21:15,110
for the a set to have both the

843
00:21:15,110 --> 00:21:17,410
for the a set to have both the
translational and rotational set of

844
00:21:17,410 --> 00:21:17,420
translational and rotational set of

845
00:21:17,420 --> 00:21:19,900
translational and rotational set of
degrees of freedom and that means you're

846
00:21:19,900 --> 00:21:19,910
degrees of freedom and that means you're

847
00:21:19,910 --> 00:21:23,290
degrees of freedom and that means you're
going to do this breaking down of

848
00:21:23,290 --> 00:21:23,300
going to do this breaking down of

849
00:21:23,300 --> 00:21:25,480
going to do this breaking down of
degrees of freedom into analysis and all

850
00:21:25,480 --> 00:21:25,490
degrees of freedom into analysis and all

851
00:21:25,490 --> 00:21:29,100
degrees of freedom into analysis and all
meta degrees of freedom node by node

852
00:21:29,100 --> 00:21:29,110
meta degrees of freedom node by node

853
00:21:29,110 --> 00:21:33,610
meta degrees of freedom node by node
that's not obvious I did a problem years

854
00:21:33,610 --> 00:21:33,620
that's not obvious I did a problem years

855
00:21:33,620 --> 00:21:36,340
that's not obvious I did a problem years
ago with a woman professor from China

856
00:21:36,340 --> 00:21:36,350
ago with a woman professor from China

857
00:21:36,350 --> 00:21:40,660
ago with a woman professor from China
mrs. Zhu and we decided at that time to

858
00:21:40,660 --> 00:21:40,670
mrs. Zhu and we decided at that time to

859
00:21:40,670 --> 00:21:43,750
mrs. Zhu and we decided at that time to
include only translational degrees of

860
00:21:43,750 --> 00:21:43,760
include only translational degrees of

861
00:21:43,760 --> 00:21:46,450
include only translational degrees of
freedom in the analysis set the

862
00:21:46,450 --> 00:21:46,460
freedom in the analysis set the

863
00:21:46,460 --> 00:21:49,450
freedom in the analysis set the
rationale was okay there because it was

864
00:21:49,450 --> 00:21:49,460
rationale was okay there because it was

865
00:21:49,460 --> 00:21:52,000
rationale was okay there because it was
a civil engineering structure and it was

866
00:21:52,000 --> 00:21:52,010
a civil engineering structure and it was

867
00:21:52,010 --> 00:21:53,250
a civil engineering structure and it was
going to be easier to measure

868
00:21:53,250 --> 00:21:53,260
going to be easier to measure

869
00:21:53,260 --> 00:21:56,740
going to be easier to measure
translational accelerations at points

870
00:21:56,740 --> 00:21:56,750
translational accelerations at points

871
00:21:56,750 --> 00:21:57,670
translational accelerations at points
and

872
00:21:57,670 --> 00:21:57,680
and

873
00:21:57,680 --> 00:21:59,170
and
have transducers that measured

874
00:21:59,170 --> 00:21:59,180
have transducers that measured

875
00:21:59,180 --> 00:22:01,360
have transducers that measured
rotational effects I think might now

876
00:22:01,360 --> 00:22:01,370
rotational effects I think might now

877
00:22:01,370 --> 00:22:02,980
rotational effects I think might now
they're probably good transducers for

878
00:22:02,980 --> 00:22:02,990
they're probably good transducers for

879
00:22:02,990 --> 00:22:05,140
they're probably good transducers for
both but it was somewhat experimentally

880
00:22:05,140 --> 00:22:05,150
both but it was somewhat experimentally

881
00:22:05,150 --> 00:22:08,110
both but it was somewhat experimentally
motivated to not do this

882
00:22:08,110 --> 00:22:08,120
motivated to not do this

883
00:22:08,120 --> 00:22:13,270
motivated to not do this
third user-friendly action as a result

884
00:22:13,270 --> 00:22:13,280
third user-friendly action as a result

885
00:22:13,280 --> 00:22:15,070
third user-friendly action as a result
of this third one though it's relatively

886
00:22:15,070 --> 00:22:15,080
of this third one though it's relatively

887
00:22:15,080 --> 00:22:18,340
of this third one though it's relatively
easy to set up runs in MSE nastran

888
00:22:18,340 --> 00:22:18,350
easy to set up runs in MSE nastran

889
00:22:18,350 --> 00:22:20,410
easy to set up runs in MSE nastran
because you only need to mention the

890
00:22:20,410 --> 00:22:20,420
because you only need to mention the

891
00:22:20,420 --> 00:22:23,740
because you only need to mention the
analysis nodes and then proceed notice

892
00:22:23,740 --> 00:22:23,750
analysis nodes and then proceed notice

893
00:22:23,750 --> 00:22:26,560
analysis nodes and then proceed notice
that you'd rather call out the analysis

894
00:22:26,560 --> 00:22:26,570
that you'd rather call out the analysis

895
00:22:26,570 --> 00:22:29,740
that you'd rather call out the analysis
nodes than the omitted nodes because

896
00:22:29,740 --> 00:22:29,750
nodes than the omitted nodes because

897
00:22:29,750 --> 00:22:31,840
nodes than the omitted nodes because
they are perhaps ten or a hundred times

898
00:22:31,840 --> 00:22:31,850
they are perhaps ten or a hundred times

899
00:22:31,850 --> 00:22:35,080
they are perhaps ten or a hundred times
as numerous so that's the convention in

900
00:22:35,080 --> 00:22:35,090
as numerous so that's the convention in

901
00:22:35,090 --> 00:22:38,500
as numerous so that's the convention in
typical problems run on ms seen a

902
00:22:38,500 --> 00:22:38,510
typical problems run on ms seen a

903
00:22:38,510 --> 00:22:45,040
typical problems run on ms seen a
stripper my teaching style in the past

904
00:22:45,040 --> 00:22:45,050
stripper my teaching style in the past

905
00:22:45,050 --> 00:22:48,160
stripper my teaching style in the past
has typically been to try to break down

906
00:22:48,160 --> 00:22:48,170
has typically been to try to break down

907
00:22:48,170 --> 00:22:51,130
has typically been to try to break down
a larger finite element process into a

908
00:22:51,130 --> 00:22:51,140
a larger finite element process into a

909
00:22:51,140 --> 00:22:54,700
a larger finite element process into a
simple hand calculation aside from the

910
00:22:54,700 --> 00:22:54,710
simple hand calculation aside from the

911
00:22:54,710 --> 00:22:57,130
simple hand calculation aside from the
use of the computer this helps develop

912
00:22:57,130 --> 00:22:57,140
use of the computer this helps develop

913
00:22:57,140 --> 00:23:00,340
use of the computer this helps develop
intuition and it can be done luckily for

914
00:23:00,340 --> 00:23:00,350
intuition and it can be done luckily for

915
00:23:00,350 --> 00:23:04,109
intuition and it can be done luckily for
Guyenne reduction here

916
00:23:04,109 --> 00:23:04,119


917
00:23:04,119 --> 00:23:06,419
a physical problem that I will take is

918
00:23:06,419 --> 00:23:06,429
a physical problem that I will take is

919
00:23:06,429 --> 00:23:08,519
a physical problem that I will take is
going to be trivial then because it's a

920
00:23:08,519 --> 00:23:08,529
going to be trivial then because it's a

921
00:23:08,529 --> 00:23:11,609
going to be trivial then because it's a
simple beam element and a single one

922
00:23:11,609 --> 00:23:11,619
simple beam element and a single one

923
00:23:11,619 --> 00:23:14,700
simple beam element and a single one
undergoing small amplitude vibration in

924
00:23:14,700 --> 00:23:14,710
undergoing small amplitude vibration in

925
00:23:14,710 --> 00:23:17,849
undergoing small amplitude vibration in
the XY plane the idea is to find the

926
00:23:17,849 --> 00:23:17,859
the XY plane the idea is to find the

927
00:23:17,859 --> 00:23:21,509
the XY plane the idea is to find the
fundamental flexural frequency in the XY

928
00:23:21,509 --> 00:23:21,519
fundamental flexural frequency in the XY

929
00:23:21,519 --> 00:23:25,019
fundamental flexural frequency in the XY
plane ignoring the axial torsional and

930
00:23:25,019 --> 00:23:25,029
plane ignoring the axial torsional and

931
00:23:25,029 --> 00:23:28,080
plane ignoring the axial torsional and
out of plane vibrations again we're

932
00:23:28,080 --> 00:23:28,090
out of plane vibrations again we're

933
00:23:28,090 --> 00:23:30,119
out of plane vibrations again we're
going to have a compact cross-section

934
00:23:30,119 --> 00:23:30,129
going to have a compact cross-section

935
00:23:30,129 --> 00:23:33,719
going to have a compact cross-section
say a rectangular cross-section so if

936
00:23:33,719 --> 00:23:33,729
say a rectangular cross-section so if

937
00:23:33,729 --> 00:23:36,810
say a rectangular cross-section so if
these are principal coordinate

938
00:23:36,810 --> 00:23:36,820
these are principal coordinate

939
00:23:36,820 --> 00:23:39,479
these are principal coordinate
directions for principal axes of inertia

940
00:23:39,479 --> 00:23:39,489
directions for principal axes of inertia

941
00:23:39,489 --> 00:23:42,209
directions for principal axes of inertia
then the problem uncouples nicely and we

942
00:23:42,209 --> 00:23:42,219
then the problem uncouples nicely and we

943
00:23:42,219 --> 00:23:44,369
then the problem uncouples nicely and we
can just study flexural vibration in the

944
00:23:44,369 --> 00:23:44,379
can just study flexural vibration in the

945
00:23:44,379 --> 00:23:48,869
can just study flexural vibration in the
XY plane physical modeling is to use the

946
00:23:48,869 --> 00:23:48,879
XY plane physical modeling is to use the

947
00:23:48,879 --> 00:23:50,999
XY plane physical modeling is to use the
conventional Euler Bernoulli beam theory

948
00:23:50,999 --> 00:23:51,009
conventional Euler Bernoulli beam theory

949
00:23:51,009 --> 00:23:54,139
conventional Euler Bernoulli beam theory
so that plane sections remain plane and

950
00:23:54,139 --> 00:23:54,149
so that plane sections remain plane and

951
00:23:54,149 --> 00:23:58,289
so that plane sections remain plane and
we constrain those out of plane motions

952
00:23:58,289 --> 00:23:58,299
we constrain those out of plane motions

953
00:23:58,299 --> 00:24:01,219
we constrain those out of plane motions
and the axial and torsional motions

954
00:24:01,219 --> 00:24:01,229
and the axial and torsional motions

955
00:24:01,229 --> 00:24:04,049
and the axial and torsional motions
boundary conditions that the root are

956
00:24:04,049 --> 00:24:04,059
boundary conditions that the root are

957
00:24:04,059 --> 00:24:06,889
boundary conditions that the root are
that there are no rotations nor

958
00:24:06,889 --> 00:24:06,899
that there are no rotations nor

959
00:24:06,899 --> 00:24:10,289
that there are no rotations nor
translations there at the free tip you

960
00:24:10,289 --> 00:24:10,299
translations there at the free tip you

961
00:24:10,299 --> 00:24:13,619
translations there at the free tip you
find that you have no moment and no

962
00:24:13,619 --> 00:24:13,629
find that you have no moment and no

963
00:24:13,629 --> 00:24:21,329
find that you have no moment and no
shear in the finite element modeling I'm

964
00:24:21,329 --> 00:24:21,339
shear in the finite element modeling I'm

965
00:24:21,339 --> 00:24:22,999
shear in the finite element modeling I'm
going to first take the most odious

966
00:24:22,999 --> 00:24:23,009
going to first take the most odious

967
00:24:23,009 --> 00:24:27,089
going to first take the most odious
comparison that I can by constraining

968
00:24:27,089 --> 00:24:27,099
comparison that I can by constraining

969
00:24:27,099 --> 00:24:30,659
comparison that I can by constraining
the tip translation to be the omitted

970
00:24:30,659 --> 00:24:30,669
the tip translation to be the omitted

971
00:24:30,669 --> 00:24:32,729
the tip translation to be the omitted
degree of freedom and then using the tip

972
00:24:32,729 --> 00:24:32,739
degree of freedom and then using the tip

973
00:24:32,739 --> 00:24:36,419
degree of freedom and then using the tip
of rotation to be the analysis degree of

974
00:24:36,419 --> 00:24:36,429
of rotation to be the analysis degree of

975
00:24:36,429 --> 00:24:40,200
of rotation to be the analysis degree of
freedom now that means that we are

976
00:24:40,200 --> 00:24:40,210
freedom now that means that we are

977
00:24:40,210 --> 00:24:42,089
freedom now that means that we are
trying to make the rotation at the tip

978
00:24:42,089 --> 00:24:42,099
trying to make the rotation at the tip

979
00:24:42,099 --> 00:24:44,249
trying to make the rotation at the tip
control the entire dynamics of the

980
00:24:44,249 --> 00:24:44,259
control the entire dynamics of the

981
00:24:44,259 --> 00:24:44,909
control the entire dynamics of the
problem

982
00:24:44,909 --> 00:24:44,919
problem

983
00:24:44,919 --> 00:24:47,219
problem
furthermore if I use the lumped mass

984
00:24:47,219 --> 00:24:47,229
furthermore if I use the lumped mass

985
00:24:47,229 --> 00:24:49,739
furthermore if I use the lumped mass
approach that means I'm concentrating

986
00:24:49,739 --> 00:24:49,749
approach that means I'm concentrating

987
00:24:49,749 --> 00:24:52,889
approach that means I'm concentrating
all the mass at the tip as a point mass

988
00:24:52,889 --> 00:24:52,899
all the mass at the tip as a point mass

989
00:24:52,899 --> 00:24:55,649
all the mass at the tip as a point mass
and there's no rotatory inertia involved

990
00:24:55,649 --> 00:24:55,659
and there's no rotatory inertia involved

991
00:24:55,659 --> 00:24:58,529
and there's no rotatory inertia involved
somehow this condensation method has got

992
00:24:58,529 --> 00:24:58,539
somehow this condensation method has got

993
00:24:58,539 --> 00:25:00,959
somehow this condensation method has got
to account for that and convert that

994
00:25:00,959 --> 00:25:00,969
to account for that and convert that

995
00:25:00,969 --> 00:25:03,119
to account for that and convert that
translational inertia effect into

996
00:25:03,119 --> 00:25:03,129
translational inertia effect into

997
00:25:03,129 --> 00:25:04,919
translational inertia effect into
something governed by the rotational

998
00:25:04,919 --> 00:25:04,929
something governed by the rotational

999
00:25:04,929 --> 00:25:07,260
something governed by the rotational
degree of freedom so that's really not

1000
00:25:07,260 --> 00:25:07,270
degree of freedom so that's really not

1001
00:25:07,270 --> 00:25:09,120
degree of freedom so that's really not
easy now it will turn out that this

1002
00:25:09,120 --> 00:25:09,130
easy now it will turn out that this

1003
00:25:09,130 --> 00:25:11,730
easy now it will turn out that this
works although we have some 20% error in

1004
00:25:11,730 --> 00:25:11,740
works although we have some 20% error in

1005
00:25:11,740 --> 00:25:14,670
works although we have some 20% error in
this case so when I push the problem

1006
00:25:14,670 --> 00:25:14,680
this case so when I push the problem

1007
00:25:14,680 --> 00:25:16,620
this case so when I push the problem
this hard I thought maybe I won't get

1008
00:25:16,620 --> 00:25:16,630
this hard I thought maybe I won't get

1009
00:25:16,630 --> 00:25:18,710
this hard I thought maybe I won't get
answers that make any sense at all

1010
00:25:18,710 --> 00:25:18,720
answers that make any sense at all

1011
00:25:18,720 --> 00:25:21,570
answers that make any sense at all
here's the degrees of freedom that are

1012
00:25:21,570 --> 00:25:21,580
here's the degrees of freedom that are

1013
00:25:21,580 --> 00:25:22,200
here's the degrees of freedom that are
shown

1014
00:25:22,200 --> 00:25:22,210
shown

1015
00:25:22,210 --> 00:25:24,180
shown
of course we're constraining the tip

1016
00:25:24,180 --> 00:25:24,190
of course we're constraining the tip

1017
00:25:24,190 --> 00:25:27,030
of course we're constraining the tip
with these degrees of freedom so the

1018
00:25:27,030 --> 00:25:27,040
with these degrees of freedom so the

1019
00:25:27,040 --> 00:25:29,520
with these degrees of freedom so the
only three degrees of freedom are those

1020
00:25:29,520 --> 00:25:29,530
only three degrees of freedom are those

1021
00:25:29,530 --> 00:25:36,180
only three degrees of freedom are those
two here and I'm going to make the u3e

1022
00:25:36,180 --> 00:25:36,190
two here and I'm going to make the u3e

1023
00:25:36,190 --> 00:25:39,720
two here and I'm going to make the u3e
be the omitted degree of freedom and in

1024
00:25:39,720 --> 00:25:39,730
be the omitted degree of freedom and in

1025
00:25:39,730 --> 00:25:42,150
be the omitted degree of freedom and in
other words a slave to the u4 degree of

1026
00:25:42,150 --> 00:25:42,160
other words a slave to the u4 degree of

1027
00:25:42,160 --> 00:25:45,540
other words a slave to the u4 degree of
freedom so this is a toughest case

1028
00:25:45,540 --> 00:25:45,550
freedom so this is a toughest case

1029
00:25:45,550 --> 00:25:48,330
freedom so this is a toughest case
possible particularly when shown below

1030
00:25:48,330 --> 00:25:48,340
possible particularly when shown below

1031
00:25:48,340 --> 00:25:51,210
possible particularly when shown below
that I have a point mass out at the tip

1032
00:25:51,210 --> 00:25:51,220
that I have a point mass out at the tip

1033
00:25:51,220 --> 00:25:54,060
that I have a point mass out at the tip
when I use the lumped mass matrix

1034
00:25:54,060 --> 00:25:54,070
when I use the lumped mass matrix

1035
00:25:54,070 --> 00:25:56,130
when I use the lumped mass matrix
remember that it moves half the main

1036
00:25:56,130 --> 00:25:56,140
remember that it moves half the main

1037
00:25:56,140 --> 00:25:58,620
remember that it moves half the main
mass to the tip and half to the root

1038
00:25:58,620 --> 00:25:58,630
mass to the tip and half to the root

1039
00:25:58,630 --> 00:26:01,440
mass to the tip and half to the root
well the root mass is disabled and

1040
00:26:01,440 --> 00:26:01,450
well the root mass is disabled and

1041
00:26:01,450 --> 00:26:04,260
well the root mass is disabled and
doesn't even move then by moving half

1042
00:26:04,260 --> 00:26:04,270
doesn't even move then by moving half

1043
00:26:04,270 --> 00:26:06,470
doesn't even move then by moving half
the mass to the tip we're somewhat over

1044
00:26:06,470 --> 00:26:06,480
the mass to the tip we're somewhat over

1045
00:26:06,480 --> 00:26:09,480
the mass to the tip we're somewhat over
emphasizing the inertia effect by moving

1046
00:26:09,480 --> 00:26:09,490
emphasizing the inertia effect by moving

1047
00:26:09,490 --> 00:26:11,760
emphasizing the inertia effect by moving
it outboard but then we're requiring

1048
00:26:11,760 --> 00:26:11,770
it outboard but then we're requiring

1049
00:26:11,770 --> 00:26:13,890
it outboard but then we're requiring
that that be governed by the static

1050
00:26:13,890 --> 00:26:13,900
that that be governed by the static

1051
00:26:13,900 --> 00:26:17,850
that that be governed by the static
motion of the rotation of the tip so

1052
00:26:17,850 --> 00:26:17,860
motion of the rotation of the tip so

1053
00:26:17,860 --> 00:26:19,410
motion of the rotation of the tip so
right off the bat you might say you've

1054
00:26:19,410 --> 00:26:19,420
right off the bat you might say you've

1055
00:26:19,420 --> 00:26:21,900
right off the bat you might say you've
lost the entire inertial effect that you

1056
00:26:21,900 --> 00:26:21,910
lost the entire inertial effect that you

1057
00:26:21,910 --> 00:26:25,080
lost the entire inertial effect that you
haven't because in the static relation

1058
00:26:25,080 --> 00:26:25,090
haven't because in the static relation

1059
00:26:25,090 --> 00:26:28,470
haven't because in the static relation
that has developed if there is a tip

1060
00:26:28,470 --> 00:26:28,480
that has developed if there is a tip

1061
00:26:28,480 --> 00:26:30,500
that has developed if there is a tip
rotation there will have to be

1062
00:26:30,500 --> 00:26:30,510
rotation there will have to be

1063
00:26:30,510 --> 00:26:33,300
rotation there will have to be
correspondingly a static tip deflection

1064
00:26:33,300 --> 00:26:33,310
correspondingly a static tip deflection

1065
00:26:33,310 --> 00:26:36,420
correspondingly a static tip deflection
and that was the whole idea of that

1066
00:26:36,420 --> 00:26:36,430
and that was the whole idea of that

1067
00:26:36,430 --> 00:26:37,970
and that was the whole idea of that
approach

1068
00:26:37,970 --> 00:26:37,980
approach

1069
00:26:37,980 --> 00:26:41,040
approach
now dynamically they may not be related

1070
00:26:41,040 --> 00:26:41,050
now dynamically they may not be related

1071
00:26:41,050 --> 00:26:43,050
now dynamically they may not be related
but statically as soon as I give a

1072
00:26:43,050 --> 00:26:43,060
but statically as soon as I give a

1073
00:26:43,060 --> 00:26:46,080
but statically as soon as I give a
positive angle here this beam will

1074
00:26:46,080 --> 00:26:46,090
positive angle here this beam will

1075
00:26:46,090 --> 00:26:50,250
positive angle here this beam will
naturally go upward so the inertial

1076
00:26:50,250 --> 00:26:50,260
naturally go upward so the inertial

1077
00:26:50,260 --> 00:26:57,260
naturally go upward so the inertial
effect is properly accounted for

1078
00:26:57,260 --> 00:26:57,270


1079
00:26:57,270 --> 00:27:00,780
now let's go to the solution will gather

1080
00:27:00,780 --> 00:27:00,790
now let's go to the solution will gather

1081
00:27:00,790 --> 00:27:03,750
now let's go to the solution will gather
the equations of motion for all of those

1082
00:27:03,750 --> 00:27:03,760
the equations of motion for all of those

1083
00:27:03,760 --> 00:27:07,020
the equations of motion for all of those
original degrees of freedom this would

1084
00:27:07,020 --> 00:27:07,030
original degrees of freedom this would

1085
00:27:07,030 --> 00:27:09,270
original degrees of freedom this would
be called a G set of degrees of freedom

1086
00:27:09,270 --> 00:27:09,280
be called a G set of degrees of freedom

1087
00:27:09,280 --> 00:27:11,430
be called a G set of degrees of freedom
because we are just now applying the

1088
00:27:11,430 --> 00:27:11,440
because we are just now applying the

1089
00:27:11,440 --> 00:27:14,030
because we are just now applying the
single point constraints at the wall

1090
00:27:14,030 --> 00:27:14,040
single point constraints at the wall

1091
00:27:14,040 --> 00:27:16,890
single point constraints at the wall
those cause these first two degrees of

1092
00:27:16,890 --> 00:27:16,900
those cause these first two degrees of

1093
00:27:16,900 --> 00:27:21,240
those cause these first two degrees of
freedom to be zero when we partition

1094
00:27:21,240 --> 00:27:21,250
freedom to be zero when we partition

1095
00:27:21,250 --> 00:27:23,820
freedom to be zero when we partition
those out then we really are going from

1096
00:27:23,820 --> 00:27:23,830
those out then we really are going from

1097
00:27:23,830 --> 00:27:26,370
those out then we really are going from
the G set now down to what we've called

1098
00:27:26,370 --> 00:27:26,380
the G set now down to what we've called

1099
00:27:26,380 --> 00:27:29,880
the G set now down to what we've called
the F set of degrees of freedom we

1100
00:27:29,880 --> 00:27:29,890
the F set of degrees of freedom we

1101
00:27:29,890 --> 00:27:32,970
the F set of degrees of freedom we
haven't yet done the condensation we're

1102
00:27:32,970 --> 00:27:32,980
haven't yet done the condensation we're

1103
00:27:32,980 --> 00:27:36,480
haven't yet done the condensation we're
left with these reduced matrices here

1104
00:27:36,480 --> 00:27:36,490
left with these reduced matrices here

1105
00:27:36,490 --> 00:27:39,660
left with these reduced matrices here
which are still physical terms as they

1106
00:27:39,660 --> 00:27:39,670
which are still physical terms as they

1107
00:27:39,670 --> 00:27:43,050
which are still physical terms as they
stand now we want to do the condensation

1108
00:27:43,050 --> 00:27:43,060
stand now we want to do the condensation

1109
00:27:43,060 --> 00:27:52,050
stand now we want to do the condensation
approach our lumped mass matrix

1110
00:27:52,050 --> 00:27:52,060
approach our lumped mass matrix

1111
00:27:52,060 --> 00:27:54,060
approach our lumped mass matrix
effectively moves half of the

1112
00:27:54,060 --> 00:27:54,070
effectively moves half of the

1113
00:27:54,070 --> 00:27:56,310
effectively moves half of the
translational mass out to the free tip

1114
00:27:56,310 --> 00:27:56,320
translational mass out to the free tip

1115
00:27:56,320 --> 00:27:58,350
translational mass out to the free tip
and half of it to the route where it's

1116
00:27:58,350 --> 00:27:58,360
and half of it to the route where it's

1117
00:27:58,360 --> 00:28:02,190
and half of it to the route where it's
constrained anyway here's the mass

1118
00:28:02,190 --> 00:28:02,200
constrained anyway here's the mass

1119
00:28:02,200 --> 00:28:05,340
constrained anyway here's the mass
matrix reduced to the free coordinates

1120
00:28:05,340 --> 00:28:05,350
matrix reduced to the free coordinates

1121
00:28:05,350 --> 00:28:09,150
matrix reduced to the free coordinates
and here you see the tip mass of one

1122
00:28:09,150 --> 00:28:09,160
and here you see the tip mass of one

1123
00:28:09,160 --> 00:28:12,210
and here you see the tip mass of one
half the total being mass stiffness

1124
00:28:12,210 --> 00:28:12,220
half the total being mass stiffness

1125
00:28:12,220 --> 00:28:14,570
half the total being mass stiffness
comes from the euler-bernoulli Theory

1126
00:28:14,570 --> 00:28:14,580
comes from the euler-bernoulli Theory

1127
00:28:14,580 --> 00:28:19,440
comes from the euler-bernoulli Theory
simple beam theory we now do our mental

1128
00:28:19,440 --> 00:28:19,450
simple beam theory we now do our mental

1129
00:28:19,450 --> 00:28:22,530
simple beam theory we now do our mental
experiment which is a static problem so

1130
00:28:22,530 --> 00:28:22,540
experiment which is a static problem so

1131
00:28:22,540 --> 00:28:24,750
experiment which is a static problem so
right here we draw a line and say we're

1132
00:28:24,750 --> 00:28:24,760
right here we draw a line and say we're

1133
00:28:24,760 --> 00:28:28,440
right here we draw a line and say we're
going to do this mental experiment we

1134
00:28:28,440 --> 00:28:28,450
going to do this mental experiment we

1135
00:28:28,450 --> 00:28:31,890
going to do this mental experiment we
don't allow forces on the omitted degree

1136
00:28:31,890 --> 00:28:31,900
don't allow forces on the omitted degree

1137
00:28:31,900 --> 00:28:33,690
don't allow forces on the omitted degree
of freedom which is translational but we

1138
00:28:33,690 --> 00:28:33,700
of freedom which is translational but we

1139
00:28:33,700 --> 00:28:37,830
of freedom which is translational but we
do allow a moment here on the on the

1140
00:28:37,830 --> 00:28:37,840
do allow a moment here on the on the

1141
00:28:37,840 --> 00:28:40,350
do allow a moment here on the on the
analysis degree of freedom then we use

1142
00:28:40,350 --> 00:28:40,360
analysis degree of freedom then we use

1143
00:28:40,360 --> 00:28:42,720
analysis degree of freedom then we use
the first equation in this case to solve

1144
00:28:42,720 --> 00:28:42,730
the first equation in this case to solve

1145
00:28:42,730 --> 00:28:46,470
the first equation in this case to solve
for the translation in terms of the tip

1146
00:28:46,470 --> 00:28:46,480
for the translation in terms of the tip

1147
00:28:46,480 --> 00:28:51,110
for the translation in terms of the tip
rotation because maybe

1148
00:28:51,110 --> 00:28:51,120
rotation because maybe

1149
00:28:51,120 --> 00:28:54,340
rotation because maybe
moves from zero slope at the root to a

1150
00:28:54,340 --> 00:28:54,350
moves from zero slope at the root to a

1151
00:28:54,350 --> 00:28:59,299
moves from zero slope at the root to a
tip rotation out here you might say that

1152
00:28:59,299 --> 00:28:59,309
tip rotation out here you might say that

1153
00:28:59,309 --> 00:29:02,720
tip rotation out here you might say that
this makes sense because the tip

1154
00:29:02,720 --> 00:29:02,730
this makes sense because the tip

1155
00:29:02,730 --> 00:29:07,670
this makes sense because the tip
deflection is taken to be the L over 2

1156
00:29:07,670 --> 00:29:07,680
deflection is taken to be the L over 2

1157
00:29:07,680 --> 00:29:11,720
deflection is taken to be the L over 2
times the tip rotation and if the

1158
00:29:11,720 --> 00:29:11,730
times the tip rotation and if the

1159
00:29:11,730 --> 00:29:14,360
times the tip rotation and if the
average rotation over the length of the

1160
00:29:14,360 --> 00:29:14,370
average rotation over the length of the

1161
00:29:14,370 --> 00:29:17,030
average rotation over the length of the
beam is half of the tip rotation which

1162
00:29:17,030 --> 00:29:17,040
beam is half of the tip rotation which

1163
00:29:17,040 --> 00:29:19,160
beam is half of the tip rotation which
would be approximate then you're getting

1164
00:29:19,160 --> 00:29:19,170
would be approximate then you're getting

1165
00:29:19,170 --> 00:29:21,560
would be approximate then you're getting
a displacement by measuring an angle

1166
00:29:21,560 --> 00:29:21,570
a displacement by measuring an angle

1167
00:29:21,570 --> 00:29:25,070
a displacement by measuring an angle
through a distance so it's pretty close

1168
00:29:25,070 --> 00:29:25,080
through a distance so it's pretty close

1169
00:29:25,080 --> 00:29:28,760
through a distance so it's pretty close
to the normal definition of arc lengths

1170
00:29:28,760 --> 00:29:28,770
to the normal definition of arc lengths

1171
00:29:28,770 --> 00:29:30,919
to the normal definition of arc lengths
and saw so you can get some physical

1172
00:29:30,919 --> 00:29:30,929
and saw so you can get some physical

1173
00:29:30,929 --> 00:29:32,630
and saw so you can get some physical
feeling for this expression so it's not

1174
00:29:32,630 --> 00:29:32,640
feeling for this expression so it's not

1175
00:29:32,640 --> 00:29:36,190
feeling for this expression so it's not
completely stupid

1176
00:29:36,190 --> 00:29:36,200


1177
00:29:36,200 --> 00:29:38,450
now let's form our coordinate

1178
00:29:38,450 --> 00:29:38,460
now let's form our coordinate

1179
00:29:38,460 --> 00:29:43,220
now let's form our coordinate
transformation and the analisys degree

1180
00:29:43,220 --> 00:29:43,230
transformation and the analisys degree

1181
00:29:43,230 --> 00:29:46,190
transformation and the analisys degree
of freedom u 4 which is a rotation maps

1182
00:29:46,190 --> 00:29:46,200
of freedom u 4 which is a rotation maps

1183
00:29:46,200 --> 00:29:49,669
of freedom u 4 which is a rotation maps
directly into u 4 but to the surprises

1184
00:29:49,669 --> 00:29:49,679
directly into u 4 but to the surprises

1185
00:29:49,679 --> 00:29:54,110
directly into u 4 but to the surprises
and then it also creates the translation

1186
00:29:54,110 --> 00:29:54,120
and then it also creates the translation

1187
00:29:54,120 --> 00:29:58,000
and then it also creates the translation
here by this factor L over 2

1188
00:29:58,000 --> 00:29:58,010
here by this factor L over 2

1189
00:29:58,010 --> 00:30:01,659
here by this factor L over 2
now that's the end of our mental

1190
00:30:01,659 --> 00:30:01,669
now that's the end of our mental

1191
00:30:01,669 --> 00:30:05,380
now that's the end of our mental
experiment and now we progress to the

1192
00:30:05,380 --> 00:30:05,390
experiment and now we progress to the

1193
00:30:05,390 --> 00:30:08,380
experiment and now we progress to the
real problem again the real problem that

1194
00:30:08,380 --> 00:30:08,390
real problem again the real problem that

1195
00:30:08,390 --> 00:30:10,690
real problem again the real problem that
we're interested in is not forced motion

1196
00:30:10,690 --> 00:30:10,700
we're interested in is not forced motion

1197
00:30:10,700 --> 00:30:13,780
we're interested in is not forced motion
but finding the natural frequency of

1198
00:30:13,780 --> 00:30:13,790
but finding the natural frequency of

1199
00:30:13,790 --> 00:30:16,570
but finding the natural frequency of
this system the lowest natural frequency

1200
00:30:16,570 --> 00:30:16,580
this system the lowest natural frequency

1201
00:30:16,580 --> 00:30:18,909
this system the lowest natural frequency
so we put zeros on the right hand side

1202
00:30:18,909 --> 00:30:18,919
so we put zeros on the right hand side

1203
00:30:18,919 --> 00:30:22,150
so we put zeros on the right hand side
there are no live loads then we've got

1204
00:30:22,150 --> 00:30:22,160
there are no live loads then we've got

1205
00:30:22,160 --> 00:30:25,840
there are no live loads then we've got
our various matrices here and we've got

1206
00:30:25,840 --> 00:30:25,850
our various matrices here and we've got

1207
00:30:25,850 --> 00:30:28,600
our various matrices here and we've got
our transformed variable here or so down

1208
00:30:28,600 --> 00:30:28,610
our transformed variable here or so down

1209
00:30:28,610 --> 00:30:33,340
our transformed variable here or so down
to the analysis set we pre-multiply by

1210
00:30:33,340 --> 00:30:33,350
to the analysis set we pre-multiply by

1211
00:30:33,350 --> 00:30:36,880
to the analysis set we pre-multiply by
the transpose of that coordinate

1212
00:30:36,880 --> 00:30:36,890
the transpose of that coordinate

1213
00:30:36,890 --> 00:30:39,280
the transpose of that coordinate
transformation here to put this on more

1214
00:30:39,280 --> 00:30:39,290
transformation here to put this on more

1215
00:30:39,290 --> 00:30:45,970
transformation here to put this on more
of an energy basis so we collect terms

1216
00:30:45,970 --> 00:30:45,980
of an energy basis so we collect terms

1217
00:30:45,980 --> 00:30:48,880
of an energy basis so we collect terms
and simplify them and get the resulting

1218
00:30:48,880 --> 00:30:48,890
and simplify them and get the resulting

1219
00:30:48,890 --> 00:30:51,480
and simplify them and get the resulting
differential equation for free vibration

1220
00:30:51,480 --> 00:30:51,490
differential equation for free vibration

1221
00:30:51,490 --> 00:30:54,180
differential equation for free vibration
here's the equation and you have a

1222
00:30:54,180 --> 00:30:54,190
here's the equation and you have a

1223
00:30:54,190 --> 00:30:57,039
here's the equation and you have a
generalized mass term and a generalized

1224
00:30:57,039 --> 00:30:57,049
generalized mass term and a generalized

1225
00:30:57,049 --> 00:31:01,060
generalized mass term and a generalized
stiffness term now from here down to the

1226
00:31:01,060 --> 00:31:01,070
stiffness term now from here down to the

1227
00:31:01,070 --> 00:31:02,890
stiffness term now from here down to the
actual answers just a matter of putting

1228
00:31:02,890 --> 00:31:02,900
actual answers just a matter of putting

1229
00:31:02,900 --> 00:31:05,280
actual answers just a matter of putting
in the harmonic motion

1230
00:31:05,280 --> 00:31:05,290
in the harmonic motion

1231
00:31:05,290 --> 00:31:08,110
in the harmonic motion
realizing that derivatives bring out -

1232
00:31:08,110 --> 00:31:08,120
realizing that derivatives bring out -

1233
00:31:08,120 --> 00:31:13,120
realizing that derivatives bring out -
Omega squared collect terms you have all

1234
00:31:13,120 --> 00:31:13,130
Omega squared collect terms you have all

1235
00:31:13,130 --> 00:31:16,930
Omega squared collect terms you have all
of these terms that multiply a term here

1236
00:31:16,930 --> 00:31:16,940
of these terms that multiply a term here

1237
00:31:16,940 --> 00:31:19,960
of these terms that multiply a term here
which is an amplitude term

1238
00:31:19,960 --> 00:31:19,970
which is an amplitude term

1239
00:31:19,970 --> 00:31:24,110
which is an amplitude term
solve for the unusual situation where

1240
00:31:24,110 --> 00:31:24,120
solve for the unusual situation where

1241
00:31:24,120 --> 00:31:27,400
solve for the unusual situation where
you can have an arbitrary amplitude and

1242
00:31:27,400 --> 00:31:27,410
you can have an arbitrary amplitude and

1243
00:31:27,410 --> 00:31:32,000
you can have an arbitrary amplitude and
it occurs when this coefficient here is

1244
00:31:32,000 --> 00:31:32,010
it occurs when this coefficient here is

1245
00:31:32,010 --> 00:31:36,260
it occurs when this coefficient here is
zero so you can have equilibrium with

1246
00:31:36,260 --> 00:31:36,270
zero so you can have equilibrium with

1247
00:31:36,270 --> 00:31:39,310
zero so you can have equilibrium with
any amplitude at this specific frequency

1248
00:31:39,310 --> 00:31:39,320
any amplitude at this specific frequency

1249
00:31:39,320 --> 00:31:42,370
any amplitude at this specific frequency
sometimes you might plot a curve of

1250
00:31:42,370 --> 00:31:42,380
sometimes you might plot a curve of

1251
00:31:42,380 --> 00:31:45,920
sometimes you might plot a curve of
frequency over here versus this

1252
00:31:45,920 --> 00:31:45,930
frequency over here versus this

1253
00:31:45,930 --> 00:31:49,550
frequency over here versus this
amplitude you for here and you find that

1254
00:31:49,550 --> 00:31:49,560
amplitude you for here and you find that

1255
00:31:49,560 --> 00:31:52,040
amplitude you for here and you find that
there is a point in along here at a

1256
00:31:52,040 --> 00:31:52,050
there is a point in along here at a

1257
00:31:52,050 --> 00:31:54,170
there is a point in along here at a
certain frequency and that's what we've

1258
00:31:54,170 --> 00:31:54,180
certain frequency and that's what we've

1259
00:31:54,180 --> 00:31:56,450
certain frequency and that's what we've
just found which is a so-called natural

1260
00:31:56,450 --> 00:31:56,460
just found which is a so-called natural

1261
00:31:56,460 --> 00:31:59,540
just found which is a so-called natural
frequency here is the resulting number

1262
00:31:59,540 --> 00:31:59,550
frequency here is the resulting number

1263
00:31:59,550 --> 00:32:02,270
frequency here is the resulting number
and when you compare this with the

1264
00:32:02,270 --> 00:32:02,280
and when you compare this with the

1265
00:32:02,280 --> 00:32:05,930
and when you compare this with the
classical solution which has a constant

1266
00:32:05,930 --> 00:32:05,940
classical solution which has a constant

1267
00:32:05,940 --> 00:32:10,430
classical solution which has a constant
out front of 3,5 - rather than this 2,8

1268
00:32:10,430 --> 00:32:10,440
out front of 3,5 - rather than this 2,8

1269
00:32:10,440 --> 00:32:16,310
out front of 3,5 - rather than this 2,8
- you find our solution is low by 20% at

1270
00:32:16,310 --> 00:32:16,320
- you find our solution is low by 20% at

1271
00:32:16,320 --> 00:32:18,080
- you find our solution is low by 20% at
first you say wait a minute you're never

1272
00:32:18,080 --> 00:32:18,090
first you say wait a minute you're never

1273
00:32:18,090 --> 00:32:21,110
first you say wait a minute you're never
supposed to be low on these approximate

1274
00:32:21,110 --> 00:32:21,120
supposed to be low on these approximate

1275
00:32:21,120 --> 00:32:23,740
supposed to be low on these approximate
solutions if you're truly within the

1276
00:32:23,740 --> 00:32:23,750
solutions if you're truly within the

1277
00:32:23,750 --> 00:32:26,030
solutions if you're truly within the
really rich or the potential energy

1278
00:32:26,030 --> 00:32:26,040
really rich or the potential energy

1279
00:32:26,040 --> 00:32:29,210
really rich or the potential energy
format but in this case our lumped mass

1280
00:32:29,210 --> 00:32:29,220
format but in this case our lumped mass

1281
00:32:29,220 --> 00:32:31,010
format but in this case our lumped mass
approximation is a non-physical

1282
00:32:31,010 --> 00:32:31,020
approximation is a non-physical

1283
00:32:31,020 --> 00:32:33,440
approximation is a non-physical
approximation and it has moved so much

1284
00:32:33,440 --> 00:32:33,450
approximation and it has moved so much

1285
00:32:33,450 --> 00:32:35,660
approximation and it has moved so much
mass out to the tip that it is lowered

1286
00:32:35,660 --> 00:32:35,670
mass out to the tip that it is lowered

1287
00:32:35,670 --> 00:32:38,420
mass out to the tip that it is lowered
the frequency in this way and so it is

1288
00:32:38,420 --> 00:32:38,430
the frequency in this way and so it is

1289
00:32:38,430 --> 00:32:45,230
the frequency in this way and so it is
explainable generalize dynamic reduction

1290
00:32:45,230 --> 00:32:45,240
explainable generalize dynamic reduction

1291
00:32:45,240 --> 00:32:47,120
explainable generalize dynamic reduction
was developed by Richard McNeil some

1292
00:32:47,120 --> 00:32:47,130
was developed by Richard McNeil some

1293
00:32:47,130 --> 00:32:48,920
was developed by Richard McNeil some
years ago and it's been very powerful

1294
00:32:48,920 --> 00:32:48,930
years ago and it's been very powerful

1295
00:32:48,930 --> 00:32:52,250
years ago and it's been very powerful
used by a lot of vehicle companies for

1296
00:32:52,250 --> 00:32:52,260
used by a lot of vehicle companies for

1297
00:32:52,260 --> 00:32:55,430
used by a lot of vehicle companies for
studying large problems it's a

1298
00:32:55,430 --> 00:32:55,440
studying large problems it's a

1299
00:32:55,440 --> 00:32:59,390
studying large problems it's a
combination of two methods of a guy anti

1300
00:32:59,390 --> 00:32:59,400
combination of two methods of a guy anti

1301
00:32:59,400 --> 00:33:02,090
combination of two methods of a guy anti
production with the mobile analysis so

1302
00:33:02,090 --> 00:33:02,100
production with the mobile analysis so

1303
00:33:02,100 --> 00:33:04,810
production with the mobile analysis so
it's both a modal and a nodal reduction

1304
00:33:04,810 --> 00:33:04,820
it's both a modal and a nodal reduction

1305
00:33:04,820 --> 00:33:09,260
it's both a modal and a nodal reduction
it brings in some coordinates here that

1306
00:33:09,260 --> 00:33:09,270
it brings in some coordinates here that

1307
00:33:09,270 --> 00:33:13,520
it brings in some coordinates here that
are generalized coordinates UQ

1308
00:33:13,520 --> 00:33:13,530
are generalized coordinates UQ

1309
00:33:13,530 --> 00:33:16,280
are generalized coordinates UQ
and the analysis set is broken into some

1310
00:33:16,280 --> 00:33:16,290
and the analysis set is broken into some

1311
00:33:16,290 --> 00:33:17,840
and the analysis set is broken into some
generalized and some physical

1312
00:33:17,840 --> 00:33:17,850
generalized and some physical

1313
00:33:17,850 --> 00:33:20,120
generalized and some physical
coordinates at the very end the

1314
00:33:20,120 --> 00:33:20,130
coordinates at the very end the

1315
00:33:20,130 --> 00:33:22,040
coordinates at the very end the
generalized coordinates become the

1316
00:33:22,040 --> 00:33:22,050
generalized coordinates become the

1317
00:33:22,050 --> 00:33:29,420
generalized coordinates become the
master one let's break down those F set

1318
00:33:29,420 --> 00:33:29,430
master one let's break down those F set

1319
00:33:29,430 --> 00:33:31,400
master one let's break down those F set
of variables which i've been calling the

1320
00:33:31,400 --> 00:33:31,410
of variables which i've been calling the

1321
00:33:31,410 --> 00:33:34,340
of variables which i've been calling the
free set and assuming that prior to that

1322
00:33:34,340 --> 00:33:34,350
free set and assuming that prior to that

1323
00:33:34,350 --> 00:33:36,470
free set and assuming that prior to that
you've already taken out multi point

1324
00:33:36,470 --> 00:33:36,480
you've already taken out multi point

1325
00:33:36,480 --> 00:33:38,170
you've already taken out multi point
constraints a single point constraints

1326
00:33:38,170 --> 00:33:38,180
constraints a single point constraints

1327
00:33:38,180 --> 00:33:41,750
constraints a single point constraints
so here's your total F set on the left

1328
00:33:41,750 --> 00:33:41,760
so here's your total F set on the left

1329
00:33:41,760 --> 00:33:44,530
so here's your total F set on the left
and on the right here we have the

1330
00:33:44,530 --> 00:33:44,540
and on the right here we have the

1331
00:33:44,540 --> 00:33:47,090
and on the right here we have the
omitted degrees of freedom as before

1332
00:33:47,090 --> 00:33:47,100
omitted degrees of freedom as before

1333
00:33:47,100 --> 00:33:50,540
omitted degrees of freedom as before
with a clue an approach and then we have

1334
00:33:50,540 --> 00:33:50,550
with a clue an approach and then we have

1335
00:33:50,550 --> 00:33:53,510
with a clue an approach and then we have
the generalized coordinates those that

1336
00:33:53,510 --> 00:33:53,520
the generalized coordinates those that

1337
00:33:53,520 --> 00:33:55,690
the generalized coordinates those that
are used for rigid body mode description

1338
00:33:55,690 --> 00:33:55,700
are used for rigid body mode description

1339
00:33:55,700 --> 00:33:59,060
are used for rigid body mode description
and then some free coordinates here and

1340
00:33:59,060 --> 00:33:59,070
and then some free coordinates here and

1341
00:33:59,070 --> 00:34:01,400
and then some free coordinates here and
some rigidly restrained coordinates some

1342
00:34:01,400 --> 00:34:01,410
some rigidly restrained coordinates some

1343
00:34:01,410 --> 00:34:06,110
some rigidly restrained coordinates some
of these are new concept now I won't go

1344
00:34:06,110 --> 00:34:06,120
of these are new concept now I won't go

1345
00:34:06,120 --> 00:34:08,360
of these are new concept now I won't go
into details on that it's a generalized

1346
00:34:08,360 --> 00:34:08,370
into details on that it's a generalized

1347
00:34:08,370 --> 00:34:14,350
into details on that it's a generalized
coordinate this is surviving set though

1348
00:34:14,350 --> 00:34:14,360


1349
00:34:14,360 --> 00:34:16,940
so those generalized coordinates are the

1350
00:34:16,940 --> 00:34:16,950
so those generalized coordinates are the

1351
00:34:16,950 --> 00:34:19,460
so those generalized coordinates are the
ones that drive the whole problem here's

1352
00:34:19,460 --> 00:34:19,470
ones that drive the whole problem here's

1353
00:34:19,470 --> 00:34:21,879
ones that drive the whole problem here's
the kind of mapping that's developed and

1354
00:34:21,879 --> 00:34:21,889
the kind of mapping that's developed and

1355
00:34:21,889 --> 00:34:24,680
the kind of mapping that's developed and
progresses from these coordinates on

1356
00:34:24,680 --> 00:34:24,690
progresses from these coordinates on

1357
00:34:24,690 --> 00:34:28,159
progresses from these coordinates on
back to these use of F coordinates over

1358
00:34:28,159 --> 00:34:28,169
back to these use of F coordinates over

1359
00:34:28,169 --> 00:34:31,750
back to these use of F coordinates over
here

1360
00:34:31,750 --> 00:34:31,760


1361
00:34:31,760 --> 00:34:33,790
there are some approximate modes

1362
00:34:33,790 --> 00:34:33,800
there are some approximate modes

1363
00:34:33,800 --> 00:34:39,220
there are some approximate modes
involved here and this is related in a

1364
00:34:39,220 --> 00:34:39,230
involved here and this is related in a

1365
00:34:39,230 --> 00:34:41,080
involved here and this is related in a
sense to the super element approach

1366
00:34:41,080 --> 00:34:41,090
sense to the super element approach

1367
00:34:41,090 --> 00:34:42,990
sense to the super element approach
where you divide a structure into these

1368
00:34:42,990 --> 00:34:43,000
where you divide a structure into these

1369
00:34:43,000 --> 00:34:46,060
where you divide a structure into these
segments this is really an advanced

1370
00:34:46,060 --> 00:34:46,070
segments this is really an advanced

1371
00:34:46,070 --> 00:34:49,300
segments this is really an advanced
topic and the mappings are intricate and

1372
00:34:49,300 --> 00:34:49,310
topic and the mappings are intricate and

1373
00:34:49,310 --> 00:34:52,450
topic and the mappings are intricate and
I have to admit failure I wanted to show

1374
00:34:52,450 --> 00:34:52,460
I have to admit failure I wanted to show

1375
00:34:52,460 --> 00:34:54,730
I have to admit failure I wanted to show
a simple hand calculation for this

1376
00:34:54,730 --> 00:34:54,740
a simple hand calculation for this

1377
00:34:54,740 --> 00:34:57,040
a simple hand calculation for this
method and I could not do it there's

1378
00:34:57,040 --> 00:34:57,050
method and I could not do it there's

1379
00:34:57,050 --> 00:34:59,140
method and I could not do it there's
just too many variables involved and

1380
00:34:59,140 --> 00:34:59,150
just too many variables involved and

1381
00:34:59,150 --> 00:35:01,240
just too many variables involved and
that's if you have to start out with six

1382
00:35:01,240 --> 00:35:01,250
that's if you have to start out with six

1383
00:35:01,250 --> 00:35:03,490
that's if you have to start out with six
five or six degrees of freedom by the

1384
00:35:03,490 --> 00:35:03,500
five or six degrees of freedom by the

1385
00:35:03,500 --> 00:35:05,140
five or six degrees of freedom by the
time you do the mappings to bring it

1386
00:35:05,140 --> 00:35:05,150
time you do the mappings to bring it

1387
00:35:05,150 --> 00:35:07,930
time you do the mappings to bring it
down to the final solution it's just too

1388
00:35:07,930 --> 00:35:07,940
down to the final solution it's just too

1389
00:35:07,940 --> 00:35:09,490
down to the final solution it's just too
complicated for a simple hand

1390
00:35:09,490 --> 00:35:09,500
complicated for a simple hand

1391
00:35:09,500 --> 00:35:12,250
complicated for a simple hand
calculation but many people like this

1392
00:35:12,250 --> 00:35:12,260
calculation but many people like this

1393
00:35:12,260 --> 00:35:14,260
calculation but many people like this
implementation in a strand and have

1394
00:35:14,260 --> 00:35:14,270
implementation in a strand and have

1395
00:35:14,270 --> 00:35:22,290
implementation in a strand and have
gotten used to it for vehicle dynamics

1396
00:35:22,290 --> 00:35:22,300


1397
00:35:22,300 --> 00:35:25,300
another reduction method is popular with

1398
00:35:25,300 --> 00:35:25,310
another reduction method is popular with

1399
00:35:25,310 --> 00:35:27,370
another reduction method is popular with
some people is the component mode

1400
00:35:27,370 --> 00:35:27,380
some people is the component mode

1401
00:35:27,380 --> 00:35:31,330
some people is the component mode
synthesis this competes nowadays with

1402
00:35:31,330 --> 00:35:31,340
synthesis this competes nowadays with

1403
00:35:31,340 --> 00:35:33,820
synthesis this competes nowadays with
the Lantos method and it's useful for

1404
00:35:33,820 --> 00:35:33,830
the Lantos method and it's useful for

1405
00:35:33,830 --> 00:35:37,060
the Lantos method and it's useful for
very large problems it's a super element

1406
00:35:37,060 --> 00:35:37,070
very large problems it's a super element

1407
00:35:37,070 --> 00:35:39,610
very large problems it's a super element
concept it's the dynamic version of

1408
00:35:39,610 --> 00:35:39,620
concept it's the dynamic version of

1409
00:35:39,620 --> 00:35:43,180
concept it's the dynamic version of
super elements or substructuring and you

1410
00:35:43,180 --> 00:35:43,190
super elements or substructuring and you

1411
00:35:43,190 --> 00:35:45,370
super elements or substructuring and you
use those vibration modes of the super

1412
00:35:45,370 --> 00:35:45,380
use those vibration modes of the super

1413
00:35:45,380 --> 00:35:47,620
use those vibration modes of the super
elements to infer what the properties of

1414
00:35:47,620 --> 00:35:47,630
elements to infer what the properties of

1415
00:35:47,630 --> 00:35:51,970
elements to infer what the properties of
the entire system are there are at least

1416
00:35:51,970 --> 00:35:51,980
the entire system are there are at least

1417
00:35:51,980 --> 00:35:54,190
the entire system are there are at least
two versions of this kind of approach

1418
00:35:54,190 --> 00:35:54,200
two versions of this kind of approach

1419
00:35:54,200 --> 00:35:56,410
two versions of this kind of approach
and one is where at the joints between

1420
00:35:56,410 --> 00:35:56,420
and one is where at the joints between

1421
00:35:56,420 --> 00:36:01,260
and one is where at the joints between
the super elements you develop the

1422
00:36:01,260 --> 00:36:01,270
the super elements you develop the

1423
00:36:01,270 --> 00:36:04,240
the super elements you develop the
individual element or the super element

1424
00:36:04,240 --> 00:36:04,250
individual element or the super element

1425
00:36:04,250 --> 00:36:07,000
individual element or the super element
properties with fixed boundaries and the

1426
00:36:07,000 --> 00:36:07,010
properties with fixed boundaries and the

1427
00:36:07,010 --> 00:36:08,920
properties with fixed boundaries and the
other method is where you have free

1428
00:36:08,920 --> 00:36:08,930
other method is where you have free

1429
00:36:08,930 --> 00:36:12,870
other method is where you have free
boundaries and this is an advanced topic

1430
00:36:12,870 --> 00:36:12,880
boundaries and this is an advanced topic

1431
00:36:12,880 --> 00:36:15,730
boundaries and this is an advanced topic
some people prefer the fixed boundaries

1432
00:36:15,730 --> 00:36:15,740
some people prefer the fixed boundaries

1433
00:36:15,740 --> 00:36:18,550
some people prefer the fixed boundaries
even though experimentalist prefer the

1434
00:36:18,550 --> 00:36:18,560
even though experimentalist prefer the

1435
00:36:18,560 --> 00:36:20,110
even though experimentalist prefer the
free boundaries this may be another one

1436
00:36:20,110 --> 00:36:20,120
free boundaries this may be another one

1437
00:36:20,120 --> 00:36:22,120
free boundaries this may be another one
of these areas where depending on the

1438
00:36:22,120 --> 00:36:22,130
of these areas where depending on the

1439
00:36:22,130 --> 00:36:23,800
of these areas where depending on the
problem you're doing you would like to

1440
00:36:23,800 --> 00:36:23,810
problem you're doing you would like to

1441
00:36:23,810 --> 00:36:28,930
problem you're doing you would like to
use one approach or the other our

1442
00:36:28,930 --> 00:36:28,940
use one approach or the other our

1443
00:36:28,940 --> 00:36:31,060
use one approach or the other our
problem session will concentrate on the

1444
00:36:31,060 --> 00:36:31,070
problem session will concentrate on the

1445
00:36:31,070 --> 00:36:34,150
problem session will concentrate on the
Guyenne reduction it's one that we can

1446
00:36:34,150 --> 00:36:34,160
Guyenne reduction it's one that we can

1447
00:36:34,160 --> 00:36:37,640
Guyenne reduction it's one that we can
actually do some hand calculations with

1448
00:36:37,640 --> 00:36:37,650
actually do some hand calculations with

1449
00:36:37,650 --> 00:36:39,850
actually do some hand calculations with
the first problem I take is a

1450
00:36:39,850 --> 00:36:39,860
the first problem I take is a

1451
00:36:39,860 --> 00:36:42,590
the first problem I take is a
two-element line assembly which is

1452
00:36:42,590 --> 00:36:42,600
two-element line assembly which is

1453
00:36:42,600 --> 00:36:47,510
two-element line assembly which is
vibrating along its axis and the idea is

1454
00:36:47,510 --> 00:36:47,520
vibrating along its axis and the idea is

1455
00:36:47,520 --> 00:36:50,270
vibrating along its axis and the idea is
to determine the frequency response of

1456
00:36:50,270 --> 00:36:50,280
to determine the frequency response of

1457
00:36:50,280 --> 00:36:53,270
to determine the frequency response of
this system using this static

1458
00:36:53,270 --> 00:36:53,280
this system using this static

1459
00:36:53,280 --> 00:36:57,170
this system using this static
condensation I'm going to ask two things

1460
00:36:57,170 --> 00:36:57,180
condensation I'm going to ask two things

1461
00:36:57,180 --> 00:36:59,150
condensation I'm going to ask two things
I don't want the whole problem solved

1462
00:36:59,150 --> 00:36:59,160
I don't want the whole problem solved

1463
00:36:59,160 --> 00:37:01,850
I don't want the whole problem solved
but just to find the mapping matrix for

1464
00:37:01,850 --> 00:37:01,860
but just to find the mapping matrix for

1465
00:37:01,860 --> 00:37:04,400
but just to find the mapping matrix for
the coordinates for either choice of

1466
00:37:04,400 --> 00:37:04,410
the coordinates for either choice of

1467
00:37:04,410 --> 00:37:08,090
the coordinates for either choice of
analysis degrees of freedom namely the

1468
00:37:08,090 --> 00:37:08,100
analysis degrees of freedom namely the

1469
00:37:08,100 --> 00:37:10,550
analysis degrees of freedom namely the
motion at the midpoint here or at the

1470
00:37:10,550 --> 00:37:10,560
motion at the midpoint here or at the

1471
00:37:10,560 --> 00:37:14,330
motion at the midpoint here or at the
right end and then to explain which of

1472
00:37:14,330 --> 00:37:14,340
right end and then to explain which of

1473
00:37:14,340 --> 00:37:16,550
right end and then to explain which of
those