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Let us start today’s lecture, today’s
lecture is have the team fiber orientation.
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You all know that the fiber orientation is
very very important phenomenon in the textile
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practice.
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So that, we want to to to analyze this program
more deeper. When we say orientation we need
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to say, what we mean under this term? We must
say something about the fiber segment, which
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we use for orientation and then about the
definition of direction; how to define the
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orientation vector? We can speak about very
short fiber fiber segments extremely short
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like here, then the direction is our problem
it is is evidently tangent of this small fiber
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segment. In other case, when we use this green
green segment having a longer plans, we usually
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use the line from end points located to end
points A and B. Yes.
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We were to speak about the easiest case; it
means we will speak about planar orientation
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of fibers. It is often a case; for example,
of web and yarder typical extract structures.
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Let we have some coordinate x and y and then
we want to measure the directions from minus
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pi by 2 to 0, 2 plus pi by 2 understandable.
So, from minus 90 degree to 0 to plus 90 90
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degree, the easiest case is if our probability
density function represents pure pure random
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it means isotropic structure. So, that in
each case the direction is in the same probability,
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how must be this probability density function
evidently must be constant is not it.
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If it is constant, we can write we can write
that such function, this function we call
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f 0 psi 0 is this our staring probability
the density function, this integral from this
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PDF f 0 psi 0 d psi 0 from minus pi by t 2
plus by 2 must be equal 1. As every time integral
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from probability, each probability density
functions, but because this function is constant
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we carry this one. So, this one and from this
we obtained that the such probability density
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function is constant and equal 1 pi by 1 pi
1 by pi. Normal way, normal way this isotropic
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structure is not real why.
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Usually, no every thing, but usually one direction
is preferred. In the example of it is usually
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to longitudinal direction. Why is why the
one direction is preferred because first sorry
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first other segments connected our segment.
How to say it? Kick to this segment through
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the process because oriented it more to to
preferential direction. So, it can be other
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segments or adjacent from the same fiber or
from the other fibers and also of course,
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the textile machine elements different pins
cylinders and so on and so on.
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In the reality in the reality, we obtained
some structure, which preferential direction.
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The real physical mechanism of this process
is very vey complicated it is very difficult
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to to describe it.
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So that, we need to use some some thing, which
is more easier easier. Therefore, we start
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listen idea, let us imagine like an imaginary
flexible belt equipped with this perpendicular
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spikes, such spikes substitute the influence
of surrounding of our our fiber segment. This
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is shown here, let us imagine such, imaginary
flexible belt having some spikes some spikes
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like like flicker that here. And this substitute
influence of surrounding of our green fiber.
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On the end, our short fiber segment sometimes
I says, fiber what I mean a very short fiber
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segment, which is possible intemperate as
straight on the end of our fiber segment,
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let us take some some choke some orange choke
and make on our imaginary belt and orange
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point. Our fiber segment is lying on the is
lying on this straight line P, evidently end
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point of fiber is is coordinates x 0 y 0.
So, z tangent psi 0 is x 0 by y 0 evidently.
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And now let us elongate let us elongate our
imaginary flexible belt. Please our belt is
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imaginary and not real. And using this moving
the x x coordinate. I must say the the fiber
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fiber segment is a slip between the slipping
between the amount of the the spikes, but
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it stay on our straight line P.
So, that after our elongation, after our moving
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the situation is alike on our picture here,
x coordinates still be same y coordinate it
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is higher in the traditional a spinning technology.
We know the quantity and drafting value, which
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is ratio in our case it means y by y 0. Let
us use this quantity as a degree of elongation
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of our imaginary belt, well if C is equals
to zero then if C is equal to zero sorry sorry,
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if C is equal to 1, then it is without without
elongation, higher it is, so high is high
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is the elongation. Now, from the from the
picture, we see that the tangent psi, hang
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up psi is here, must be x 0 by y, but y from
this equation is C times y 0. So, that is
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this one, x 0 by y 0 is tangent, psi 0 so
that this one. Now we can make very differentiate,
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we can differentiate this equation and after
few steps you can own to to derive it, we
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obtain we obtain the such equation is here.
This this of three arranging in final, we
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obtain this equation to be important for us.
Well, it was the discussion about one fiber,
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now about all fibers. The starting probability
density function is the starting situation
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was an isotropic structure probability density
function f 0 psi 0.
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You know, that the f 0 psi 0 times d psi 0
probability density function times differential
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quantity have some logical sense. It represents
the reality frequency, but a little abnormal
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relative frequency, the relative frequency
related to a elemental class interval; class
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interval, which is very very short, differentially
short.
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So, we can imagine such elemental class, the
relative frequency of such class is every
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time probability density function times differential
quantity. We will use this logic logical moment
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more times in our lecture. So, it is f 0 psi
0 time d psi 0, relative frequency of fiber
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segments in the elementary class before drafting.
After drafting a new probability density function
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is coming. It is a probability density function
f psi, this also a subscribe. So, the f psi
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d psi is relative frequency of fiber segments
in the elementary after our drafting of our
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imaginary belt, but both such relative frequencies
must be same. Because the fibers inside of
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inside of a starting of a staring angular
elemental class must be same then after our
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elongation of our belt, imaginary belt is
not it.
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So, therefore, it must vary, f 0 psi 0 d psi
0 relative frequency before drafting must
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be same that f psi d psi means the same after
drafting. Using this equation, after substitution
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of our known expressions, do you think the
equation for probability density function
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of fiber short segments after after using
some drafting, it means after preferential
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of one may be longitudinal direction. I will
see, how it is graphically, how is this graphically
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on.
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So, more are the fiber concentrated round
our in our imagine the imagine longitudinal
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longitudinal direction.
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Well, now only for your information, I do
not want to in detail to explain it. When
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you need to the to have some distribution
functions is here is derivation of this. We
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can say in the moment that, we are that we
can be very proud to obtain quite new quite
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new probability density function. We we can
published it in the special journal of applied
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the theory of probability sorry.
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Now, this this pictures show that, how as
a tangents of our angle f psi have so called
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Gaussian distribution. So that, it is this
is a one of a known distribution from theory
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of probability.
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Even even you want, you can study it in more
details. Sometimes it is coming or often it
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is coming the the following program to this
moment, our preferential direction was same
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than the y axis vertical axis.
But you sometimes preferential direction is
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an order, we have y axis here, x axis here
and let us imagine the preferential direction
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is the the direction of our blue arrow, which
have 2 y axis some angle alpha. So, we can
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write the angle psi to y axis is now are the
then the angle psi to preferential distribution.
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Our earlier equation is now valid to the angle
to preferential distribution to angle psi
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and our angle psi is another and its valid
from this easier from this very trivial picture
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that psi is alpha plus psi.
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Yes. So, I said our earlier equation is valid
Also, but angle psi and towards valid for
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now for probability density function. I call
it now, as a new symbol, under the new symbol
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g psi, it is angle to to y axis now obliged
that it through preferential direction. It
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must be it must be our own function, but this
angle psi using it we obtained final probability
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density function g psi, which is given by
such expression having 2 parameters, our known
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parameters C. If parameters is represents
the generally say the intensity of the preference
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of this are the things direction and parameter
alpha, which is angle between 2 directions
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y, y axis direction and preferential direction.
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Well it was our theory, now is the question
and how it is in the in practice? We analyzed
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using thrice the fiber technique for you may
be known known experimental method. We analyze
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the curves, the of fibers in a fiber vamp,
it was research for non woven textiles and
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we we take lot of points from this fiber to
computed and we reconstruct whole curve of
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fiber in computer then we divide our fiber
to very small, very short parts. It in this
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picture it was 0.1 millimeter and measured
or calculate angles of orientation and then
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we construct the histogram of this this this
distribution.
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On this two pictures are in first it is a
fiber viscose fibers finite 3.5 and the second
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one 0.7 decitex and there are this such such
histograms obtain experimentally. When we
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use our curve and there are if we using some
sophistical regression or something. So, two
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parameter C and alpha, we obtained parameter
C rough 1.84 alpha may be 2.3 degree minus
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2.3 degree and the second case 1.97 and the
alpha is minus 4.5 degree, may be that is
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an alpha was the result of the experiment
are or our our mistakes definitions of opposite
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longitudinal direction is very very small,
but you can see that the comparison bring
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a good results, good correlation of this experiments
and theory, theoretical result.
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We applied our equation for long longer fiber
segments through in this case it is 12.8 millimeter,
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in this moment it is it is empirical application
because we derive it for very very short fiber
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segments. So, is it this distribution, you
can see that the result is also acceptable
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is also acceptable, but what is interesting
the C is quite harder, C is then for the question
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is why? Why it is, how it is possible? It
was the same structure, we evaluated the for
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a longer fiber segments, how it is possible?
It is diagrammatically show this picture.
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Let us imagine a fiber green fiber on our
picture having given given given shape, in
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macro in macro trend we can say it is due
to vertical direction. In a shorts segments
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it has lot of loops and waves and so on and
so on. Long fiber segments are the direction
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of long fiber segments are shown through yellow
arrows here.
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You can see that they are near to the longitudinal
position, but the short fiber segments, the
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set of short fiber segments is straight through
a short erect arrows and you can see that
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it is not so. Therefore, C for in web C value
for long fiber segments is much more is much
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more, something is wrong? The the distribution
of red of red arrows is quite other than the
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distribution of the yellow arrows. Well, and
it is answer why the C is increasing from
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refer to the value over for.
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Sometimes, it is necessary to define probability
density function for a non-oriented angle
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what I mean? Let us imagine some fiber segment,
our angle psi is or to your, from your side
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plus 30 degree or minus 30 degree. There are
2 different angle side plus 30 degree minus
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30 degree is not it? But sometimes I want
to know, how is the distribution of angles
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between fiber and our y axis independent of
them if it is on the right hand side or on
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your left hand side? So, I need to understand,
how it is probability density function of
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an angle of an angle theta, which is absolute
value of our angle psi.
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I think you can home more think about his
picture, it is also intuitively evident that
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this probability density function must be
some of two our g functions. One is in angle
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minus theta because theta is every time positive,
theta is 30. So, for example, 30 degree then
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then minus theta is minus 30 degree and the
angle of psi minus plus probability density
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in plus here. So, that u theta is g minus
theta plus g plus theta, which is evident
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of intuitively using our equation we obtained
the final equation in our model as shown.
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Well, so, we that is all for probability density
function of orientation of fibers same direct
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segment, segments in fiber assembly, but we
often we often cut our our fiber assembly
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or by preparation, by practice in microscopic
microscopic of textile structure, it is real
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resection or some imaginary sections by by,
when when we use a breaking machine then the
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jaw have some some line, which is something
like imaginary sections is not it? So, section
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is very often, the question is; how is the
orientation of fiber the direction of distribution
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of fibers? But, only this these fiber sequents,
which are cut it, which was cut it, only this
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one.
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Let us think about his programmed, let us
imagine some books like this here full of
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fibers, full of fibers. These books have the
dimension a, b, c and sectional plan is direct
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plan sigma, this one. The fiber portion fiber
segment number 1, which is shown here is a
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special segment, which have its its end points
a, immediately in the cutting plan alpha then
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the second thing of such fiber segment is
lying in an height h over our our sectional
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plan sectional plan sigma, such segment have
the lines delta l, the angle theta is here
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and then h is delta times casino theta, it
is evident for own picture got a same picture
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is here.
How is it? Now, let us think about a fibers
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having angle theta only, having angle theta
no other. There are lot of fiber in side of
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our books, lot of fibers no no all fiber source
cut it, is not it? Some are the fiber having
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angle theta like fiber number 2 or fiber number
3, we do not know, they are all over or under
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our sectional plan. The question is, which
of fibers, how is the probability that fiber
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having angle theta will be cut it? Well, let
us imagine the parallel plan to our sectional
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red sectional plan sigma, this blue which
is lying on the distant h from sectional plan.
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So, that point b of our fiber number 1, fibers
fiber fiber segment I shall say, this is laying
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on our blue blue plan sigma dash. And now,
how must be the position of fiber segment,
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when such segment shall be cut it, evidently
the upper point b of such fiber segment must
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lay between these two plan, sectional plan,
sigma and our blue plan sigma dash. I think
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this evident. So, how is the probability of
this situation? Using so called geometrical
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definition of probability, we can say probability
of such section is given by a ratio of 2 volumes,
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volume between our red and blue plans by total
volume of our box of course, if fiber, if
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the fibers are hologram distributed in our
in our books. Now in the case in which all
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the fiber having angle theta are laying near
to this this corner only, by hologram distribution.
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So, the probability of section of our section
of our our our fiber, segment is a b h by
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a b c after the arranging, we obtain this
this expression.
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Well, now let us think about fibers having
angle theta, all fibers with all angle theta
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together, all fibers in our books are capital
n and its number is very high. Just imagine
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it; u theta is probability density function
of angle theta. So, that u theta, t theta,
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this u theta t theta have some logical sense,
what is it? Is a really yes, is relative frequency
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of segment in the elementary class, interval
of the given angle theta is not it? What is
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it, n times u theta d theta?
A relative frequency time’s total number,
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it is number of cases in class knows it from
also from laboratory and so on. So, that n
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times u theta d theta is numbered of all segments
elementary class interval on the given angle
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theta and now no all this fibers was sectioned
through our sectional of plan, but also some
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of them; how is it the number of a cut it
fibers. It is total numbers times probability,
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is not it? Times probability, this time this,
using our equation is here. It is number of
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sectioned fiber segments from the group of
fibers having angle theta.
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00:30:51,049 --> 00:30:58,049
But a total number of a cut a fiber segments
is higher because not only our angle theta
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00:30:58,990 --> 00:31:05,990
exist in our box also are there angles theta
are there well. So, what is total number of
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00:31:08,950 --> 00:31:15,950
cut it fiber segments, it must be integral.
May be you know that, the symbol of integral
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00:31:17,519 --> 00:31:24,519
is from known mathematician, which start it
with symbol s, s like latent some sumac and
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00:31:31,960 --> 00:31:38,960
because the s was similar to other a x y and
so on. Then he use longer and longer s through
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00:31:42,059 --> 00:31:47,809
today’s symbol of integral.
So, integral is sum, that is some type of
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00:31:47,809 --> 00:31:54,750
summation is not it? So, in three must sum
our result over all directions. So, that you
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00:31:54,750 --> 00:32:01,260
have obtained this here than this here, it
is number of all intersected fibers valiantly.
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00:32:01,260 --> 00:32:08,260
And now probability density fiber, the probability
density function of of such fibers, let us
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00:32:08,440 --> 00:32:15,440
call this probability density function u star
theta, u star theta then u star theta d theta
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00:32:15,570 --> 00:32:19,970
is relative frequency for the intersected
segments in the elementary class interval
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00:32:19,970 --> 00:32:26,970
of the given angle theta. Is not it? Well,
u theta u star theta d theta, relative frequency
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00:32:29,840 --> 00:32:36,840
of such fibers, but in other way we can say
the relative frequency is in class. It is
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00:32:40,570 --> 00:32:47,570
a number of fibers in class or elemental class
d n to all fibers to all sectioned fibers
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00:32:48,559 --> 00:32:55,559
n, using expressions after small arising obtain
these equation or calling this then I mean
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00:32:57,649 --> 00:33:04,649
this then under symbol k n, we obtained this
formula.
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00:33:09,039 --> 00:33:15,789
In the moment k and its something is some
symbol no more, but I want to show you that
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00:33:15,789 --> 00:33:22,789
this is k and f have some logical sense. Which
one, let us imagine a very short fiber segment
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00:33:26,919 --> 00:33:33,919
lines d l, having a section area s star section
in a positive cross section perpendicular
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00:33:37,380 --> 00:33:44,380
to fiber axis, which is here. And it is s
from our earlier lectures here, lines of this
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00:33:44,700 --> 00:33:51,700
segment is delta l, the height of such segments
perpendicular to cutting line is the delta
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00:33:53,510 --> 00:34:00,510
y, this fiber segment have angle theta. Well,
we can write, evidently we can write the volume
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00:34:05,590 --> 00:34:12,590
of such fiber segment it it can be might,
it can be used to equivalent method, I can
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00:34:16,220 --> 00:34:23,220
say you know it from high school. One is one
is cross section area s times lengths delta
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00:34:25,230 --> 00:34:31,470
l.
The second is sectional area red shaded times
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00:34:31,470 --> 00:34:38,470
a height delta y, both gave, both give the
same result, both here both the expression
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00:34:42,530 --> 00:34:49,530
at from the from this equivalency, we obtained
that s s star this area, s star is this so
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00:34:55,290 --> 00:35:02,290
s by casino theta.
Well, evidently section area is larger than
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00:35:02,610 --> 00:35:09,610
the cross section area in this case. Well,
and now to our symbol, what it the mean sectional
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00:35:12,140 --> 00:35:19,140
area star bar as a symbol for a mean, what
is it now? As each means, the u star times
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00:35:24,660 --> 00:35:31,660
relative frequency means probability density
function time, differential quantity and integral.
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00:35:33,790 --> 00:35:40,790
Using this, you obtain, we obtained this formula
here. After rearranging this one and forth
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00:35:41,920 --> 00:35:46,900
is here this is integral from probability
density function must be equal to 1. So that,
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00:35:46,900 --> 00:35:53,900
it is s by k n resulting equation is that
k n is s by s star bar, it means cross sectional
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00:35:55,260 --> 00:36:02,260
area of fiber by the mean divided by the mean
value of sectional area in our section from
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00:36:03,900 --> 00:36:05,490
all fibers.
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00:36:05,490 --> 00:36:12,490
So, this is the logical sense of earlier defined
k n coefficient. It to be very important for
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00:36:15,290 --> 00:36:22,290
yarns, now about the applications of our model,
it is the equations which we which we need
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00:36:29,400 --> 00:36:36,400
it. Let us solve the problem, how is the distribution
of direction, we have some, let us imagine,
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00:36:39,270 --> 00:36:46,270
we have some for example, warp black black
tin fibers like in our picture here. We make
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00:36:48,190 --> 00:36:55,190
some section in planar case, the section is
ready to section line, this thick red this
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00:36:58,830 --> 00:37:05,830
thick red line is our section. We defined
a y axis perpendicular to to sections. So,
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00:37:08,440 --> 00:37:15,440
this is this is y axis and purple was x axis
of course, exist some preferential directions
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00:37:18,190 --> 00:37:25,190
by giving by blue arrow. And it is interesting
for us to, there are if the probability density
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00:37:26,800 --> 00:37:33,800
function of the short fiber segments, which
are which which are cut it to the sectional
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00:37:33,930 --> 00:37:39,300
line.
So, the distribution of this green arrows,
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00:37:39,300 --> 00:37:46,300
is not it? But, this position of such green
arrows, which characterized the the directions
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00:37:47,180 --> 00:37:53,970
in sections of short fiber segments, which
was set. Well, how is the strategy of our
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00:37:53,970 --> 00:38:00,970
work? I said sectional plan is reduce now
to sectional line, u theta, we derived it
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00:38:02,580 --> 00:38:09,580
in our model is this here is from earlier,
slight u theta u star theta be there are in
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00:38:14,310 --> 00:38:17,970
this equation where k n is this here.
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00:38:17,970 --> 00:38:24,780
So, on the on the position of u theta here
and here, we need to use this here and make
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00:38:24,780 --> 00:38:31,780
this integration, which is not to not to short,
formally now too much easy. And when you want
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00:38:36,640 --> 00:38:43,640
to check all the way by integration and you
can use this this and the result you such
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00:38:46,700 --> 00:38:53,700
value of k, it is possible to do analytically
and to obtain this equation for k n and then
251
00:38:53,820 --> 00:39:00,820
this this equation for probability density
function of orientation of fiber sequence
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00:39:04,440 --> 00:39:11,440
in cutting line. Now, it is that if that k
n is by 2 2 by pi, if for for isotropic structure,
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00:39:18,380 --> 00:39:21,880
these two are valid.
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00:39:21,880 --> 00:39:28,880
This is the graphical interpretation for our
last equations, you can see that there are
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00:39:34,380 --> 00:39:41,310
two different, that there are differences
between probability density function of orientation
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00:39:41,310 --> 00:39:48,310
of fiber segments in whole fiber assembly
u theta and u star theta probability density
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00:39:49,940 --> 00:39:56,940
function of distributional of fibers in cutting
line. You can see that the in this four pictures,
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00:39:58,020 --> 00:40:05,020
it is for alpha equals zero. So, longitudinal
we cut our web perpendicular to longitudinal
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00:40:05,250 --> 00:40:12,250
direction, this is for this is for pi by 6,
30 degree, 60 degree, 90 degree. In this example,
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00:40:17,890 --> 00:40:24,890
C equal 1.9 was use typical value for for
web card, we can see that a cutting process
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00:40:28,160 --> 00:40:35,160
preference directions need to normal of sectional
plan perpendicular to sectional plan. You
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00:40:38,140 --> 00:40:44,980
can see every time, here it is, increasing
u star is increasing, in the ratio to u and
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00:40:44,980 --> 00:40:51,980
the opposite side by 90 degree is decreasing
is going to 0 also here by alpha equal 30
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00:40:53,090 --> 00:41:00,090
degree also here as well as here.
Well, now now let us let us speak about a
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00:41:03,970 --> 00:41:10,970
number of fibers in a section, numbers of
fibers. Let us imagine an something like metal
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00:41:11,630 --> 00:41:18,630
plate on which you have a very thin slot,
very long length C and very thin delta x delta
267
00:41:24,150 --> 00:41:29,420
delta h that I slot like here.
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00:41:29,420 --> 00:41:36,420
So, that you are not see another fiber, you
will see only a fiber segment lying in our
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00:41:36,700 --> 00:41:43,700
very short slot; see, the length of the slot
delta a thickness of slot and its number of
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00:41:44,130 --> 00:41:51,130
all intersected fibers, intersected because
I can take use my knife and make this this
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00:41:55,370 --> 00:42:02,370
section rarely is not it here? And number
of all intersection fibers, i is subscribe
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00:42:04,210 --> 00:42:11,210
for fibers, from 1 this green fibers segments
from 1 to end, g is mass per unit area known
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00:42:13,070 --> 00:42:20,070
as a area rate in industry raw fiber density
s, s i star is the sectional area after one
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00:42:25,210 --> 00:42:32,210
fiber for example, if i th fiber is this one
or this. Here and it is as by casino of corresponding
275
00:42:34,920 --> 00:42:41,920
annual theta i. How is the mass of i th fiber
segment i th fiber segment have it a mass
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00:42:48,840 --> 00:42:55,840
cross section are a times perpendicular height,
its volume times rho mass density, this is
277
00:43:00,840 --> 00:43:07,840
the mass per 1 fiber. Mass per fibers in our
slot is some of them.
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00:43:08,670 --> 00:43:15,670
So, it is it is delta m which is some of this
because delta h as well as rho constant, it
279
00:43:18,580 --> 00:43:25,580
is possible to write it in such form, is not
it? The total area of slim slot is C times
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00:43:29,490 --> 00:43:36,490
delta h and then the mass per unit area of
a planar fiber assembly is g, which is mass
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00:43:40,140 --> 00:43:47,140
by area by, for delta and here this is the
expression C times delta h same here, delta
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00:43:52,500 --> 00:43:58,370
h is here as well as in the denominator.
So, that I can write its black, this black
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00:43:58,370 --> 00:44:05,370
relation, but nevertheless I can multiply
and divide by n total number of fibers in
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00:44:09,120 --> 00:44:16,120
our slot. It is possible, how is the sense
of our the mathematical structures now? What
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00:44:16,260 --> 00:44:23,260
is it n by C, number of our earlier green
fibers by c, it is number of fibers per unit
286
00:44:26,570 --> 00:44:33,570
per unit length of our section and our slot,
is not it? And total number by C rho, what
287
00:44:38,350 --> 00:44:45,350
is this here? Sum of all fiber sections, divided
by number of fibers.
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00:44:46,440 --> 00:44:53,440
Now, it is mean value, it is s star bar is
not it? Mean value of sectional of sectional
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00:44:55,400 --> 00:45:02,400
plan. So, that nu number of section fibers
per unit plan and s bar star means sectional
290
00:45:04,250 --> 00:45:11,250
area of fiber they are symbols. The nu the
nu is g by rho times s star bar nu times rho
291
00:45:17,110 --> 00:45:24,110
times s star bar. So, this from this nu to
this here. So, we can divide multiply and
292
00:45:24,940 --> 00:45:31,940
divide by s, we obtained this structures,
but t is s times rho, it is from lecture from
293
00:45:34,860 --> 00:45:38,600
first lecture.
You know that from our first lecture that
294
00:45:38,600 --> 00:45:45,600
t is s time rho; fiber fineness is fiber cross
section specific mass, this is this 1 and
295
00:45:48,360 --> 00:45:54,770
s by s star bar it was k n. So, that we can
write the nu number of sectioned fiber per
296
00:45:54,770 --> 00:46:01,770
unit lengths is fiber mass per unit area by
fiber fineness times the coefficient k. Then
297
00:46:08,430 --> 00:46:15,280
all depends of course, of alpha it means to
to the angle in which we cut in relation to
298
00:46:15,280 --> 00:46:21,340
preferential angle, why it is it is commented
area?
299
00:46:21,340 --> 00:46:28,340
The intersection method is one method, which
is possible to use for experimental evaluation
300
00:46:30,850 --> 00:46:37,850
of distribution of fiber directions. It it
is special method, which based on our equations
301
00:46:43,230 --> 00:46:49,600
and it is possible to do it in laboratory
also it is not too easier I must say.
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00:46:49,600 --> 00:46:56,600
When you want, you can study this way in my
lectures. We were not comment it today, I
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00:46:57,730 --> 00:47:04,730
only want to say, how is the experimental
background of this such method, you need to
304
00:47:06,250 --> 00:47:13,250
measure number of of cross sections fibers
may be optically or something so. In different
305
00:47:14,690 --> 00:47:21,690
directions in our web, when you know, when
you have the set of experimental obtained,
306
00:47:22,770 --> 00:47:29,770
where use for different alpha. You can from
this set of experimental, from experimental
307
00:47:33,680 --> 00:47:40,680
data to evaluate u theta probability density
function of oriental of directions of our
308
00:47:44,110 --> 00:47:46,790
fiber segments.
309
00:47:46,790 --> 00:47:53,790
It is this here and then it is also an example,
which documented correspondence our model
310
00:47:55,190 --> 00:48:01,790
and experimental research from another author
is are well.
311
00:48:01,790 --> 00:48:08,790
Now, stop of this lecture and in the following
lecture, we will continue and where is how
312
00:48:15,110 --> 00:48:22,110
to apply our knowledge about fiber orientation
to a mechanical behavior. So, thank you for
313
00:48:24,530 --> 00:48:31,530
your attention.