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In todayís lecture, we want to finish with
the theoretical concept of radial migration
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00:00:35,860 --> 00:00:42,860
according to professor Treloar. And then,
I want to introduce an alternative model ñ
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model of equidistant migration to you.
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Last time, we obtained the equation zeta is
plus minus 2 p by D square times r square
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minus C. So, it is an equation of paraboloid
on which is lying Treloarís idealized fiber.
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So, based on this equation, we can make a
small rearranging, so that first, say that
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we start from point r 0 zeta 0; then, the
C must be equal 0; and, this ratio 2 p by
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D square; explain in this for r square by
D by 2 square; and, this is divided by 2.
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Now, it is divided and this divided denominator;
it means multiply. So, that it is well.
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Hearle used this r square by D by 2 square
as a quantity capital Y; r square by D by
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2 square ñ what is it? r by D by 2 ñ what
is it? It is a relative position of fiber
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point between 0 and yarn surface on the yarn
axis. This quantity is zero on the yarn surface;
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this quantity is 1; is not it? But, this is
in square. So, Y is square of this ratio.
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So, defined Hearle is Y. Then, our equation
is possible to write in the form zeta is P
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by 2 times capital Y, because this is capital
Y; why is this? Then, this equation is a linear
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equation between zeta and quantity capital
Y.
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And, Hearle used the traditional statistical
quantities; for example, he saw the interesting,
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can be so-called fiber mean position. What
is its mean value of this quantity capital
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Y? It is called under the term fiber mean
position. When we use the equations derived
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for Treloarís ideal model, we can use it
to obtain that fiber mean position; here is
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1 by 2 on the place of Y; zeta is this one;
is not it. And, the responsible result we
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can obtain on the lengths from 0 to pi by
2 one half of period, because it is periodically
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same. Fiber mean position for Treloarís model
is one half. The second quantity which recommend
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Hearle for calculation in the yarn is so-called
R.M.S. deviation ñ deviation. What is it?
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This is the deviation of quantity Y; the definition
is this here. Using this, after rearranging
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which is shown here, we obtain from Treloarís
ideal model 1 by 2 times square root of 3,
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so that it is 0 .289.
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And, the third characteristic of Hearle ñ
it is the intensity of migration. It is d
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Y by d zeta square times d zeta; integral
of them by d zeta; and, square root then from
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this one. Because this is linear, it is slightly
something like linear; linear is the relation
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between zeta and the Hearleís quantity Y;
it is linear. So, fiber path in this diagram
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is this red ñ linear from 0 to yarn periphery
and then back and so on here. And, this angle
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ñ this is the sense of the intensity of migration
or better say, tangents of this angle. So,
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there are three quantities, which are very
often used in the works, which study the internal
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structure of different yarns.
Lot of others do not think about these quantities;
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only take it as final equations from wealth
of Hearle; and apply it on some experimental
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research work, which they are doing. It is
in also generalized; now, only the Treloarís
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model also exists something, some terms in
literature like the incomplete migration,
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which is rather fiber is not going from 0
to yarn periphery; then, these terms of is
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onlyÖ and so on and so on; more big than
you can obtain. These relations ñ all these
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you can in more details to read; for example,
in the very known book of Hearle and structural
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mechanical fibers ñ yarns and fabrics. This
book is very known; professor Ishteyakor;
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can you show withÖ
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Let us compare the result of Treloar ñ is
an experimental experiences. This is the fundamental
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equation; from this fundamental equation,
square root of this one and multiply by tangents
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beta. And, we obtain such equations. Its relation
between some function of beta on the right-hand
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side and tangent alfa times tangent beta on
left-hand side. When we calculate this, we
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obtain the curves, the thick black lines;
this is substantially tangents alpha times
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tangents beta; here is tangents beta ñ 2
pi r Z. We obtain the thick lines when we
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when use an approximated equation. So, we
obtain this equation, so that the absolute
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theory of tangent alpha tangent beta is constant.
Such constant is the dotted line here to this
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here. The dotted line ñ this is an approximation
of original function of Treloarís model.
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If K is high, you can see here that in most
part, it is roughly very good. How is now
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our experimental experience? We measured three
dimensional curves of fibers inside of the
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yarn. How? Principally only. We used so-called
Morton tracer fiber technique. When you use
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the good immerse liquid to yarn or to each
textile object, then all fibers stay be transparent
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like from glass. When you produce yarn having
small value of black fibers ñ fibers which
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was before processed to blithe black; then,
in immerse liquid, you see transparent yarn
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and the path, the projection of black fiber.
You see it; other fibers do not disturb in
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immersion liquid. And, when you have two projections
of same fiber ñ perpendicular projection
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ñ one time from this light; one time from
this light to this yarn.
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When you have two, then from the known relations
from descriptive geometry, you can reconstruct
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three-dimensional curve of fiber; is not it?
Principally, it is clear. So, we reconstruct
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lot of paths ñ three-dimensional paths of
fibers in the yarn; then, in each radius or
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class interval of radii, we obtain the corresponding
angles, the mean angle alpha, mean angle beta,
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mean angle gamma and so on here. And, through
this way, using calculator, using computer,
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we were able to evaluate this relation experimentally
too. The experimental curve based on different
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yarns ñ it was relatively large set of yarns;
this work was produced, is mentioned by Doctor
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Mohan Kumar Soni. These curves have the trends
as shown in the green area ñ like this or
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this or this area. So, set of curves each
for another yarn. So, the strength of such
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curve is quite other than the trends, which
we obtained from Treloarís ideal migration
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model.
We have totally different trends; what we
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need to say, Treloarís model is not valid;
sorry, but it is so. In the time of Treloarís
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creation of this theoretical model, it was
not enough experimental possibilities to experimentally
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verify this model. If is second time that
Treloarís model was very good; from other
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side for other scientists, which can use a
lot of ideas, lot of imaginations, which started
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professor Treloar his work stay be from this
point of view useful to these days.
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Because this model was not good, was not corresponded
to this yarn, we said we need to start another
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model ñ model which we called model of equidistant
migration. In Treloarís model, we introduce
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number nu of elements intersecting the cylinder
with radius r on one fiber per unit length
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of yarn. Generally, it is a function of radius
r, but the model of ideal migration assumed
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constant value of nu based on the idea of
representative fiber, intersecting all radii.
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This idea is not well imaginable, because
since the differential layer near yarn axis,
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for example, this blue layer has a very small
volume, rather very small surface. Hence,
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the number of fiber intersections can be also
very small there. B ñ since the differential
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layer near yarn surface ñ the green one here
ñ has already small packing density. Hence,
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the number of fiber intersections will be
also small there. And, the yarn periphery
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packing density is every time smaller in reality,
so that the fibers cannot chance to be here
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so often than in the middle radii; is not
it? Therefore, maximum number of intersections
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is in the red layer between blue and green.
Is it imaginable?
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So, it is evident that the model path of fiber
can be regular, but random; count to go every
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time from axis to periphery; then, back to
axis; then, back to periphery and so on. Regular
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can be only directions ñ means derivatives
of elemental increments, so that then, the
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created model is not pure deterministic as
earlier. The diagram here r zeta illustrates
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the given idea. On the place of assumption
4 nuÖ Nevertheless for our model, we use
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on the place of earlier assumption 4; what
was earlier assumption 4? Earlier assumption
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4 say that nu is a constant number. What is
nu? Sense of nu ñ number of intersections
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in given radii in yarn lengths and per one
fiber; we say no; it is not constant; it will
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be constant. This number of elements is proportional
to the fiber volume into the vicinity round
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radius r. When in the vicinity round radius
r is lot of fiber material, then probably
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lot of intersections will be there when you
know that only a few intersections will be
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through our radius r; is not it? This is our
modification of assumption and other versions.
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Then, the assumption used by Treloar. Highest
fiber volume in the differential layer at
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the radius r ñ so, 2 pi r d r because it
must be positive for calculation of volume
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and so on; 2 pi r d r is a total area per
one differential annulus ñ our known differential
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annulus times yarn lengths delta zeta. It
is volume of our differential annulus, so
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that this is volume of our differential annulus
times packing density; it is volume of fibers
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in our differential annulus; is not it? After
small rearranging, this volume of fibers in
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our differential annulus ñ we can write it
is only graphical rearranging; r is changed
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ñ this r evidently. And, nu can be changed
too. We need to think about the constant value
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of nu. So, they are changed. All other numbers
in this expression are stable, are constant.
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So, this expression in brackets ñ it is some
differentially small constant; same for each
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differential layer; is not it?
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And, this value say volume of fibers in differential
layer on the radius r. So, we can say this
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is constant ñ variably r and nu. So, we can
say that our nu must be proportional to the
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quantity r times mu, because we said the higher
is volume of fibers in vicinity round the
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radius r. So, higher is the number of intersections
of [FT] So, this equation must be right in
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our model, where C is some constant of proportionality.
Number of intersecting points on one fiber
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per unit length of yarn, related to the cylinder
with radius r and it is nu r. Just two intersecting
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points must on the length of period of migration
outside-inside, from center of the yarn to
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periphery and back. Then, the number of periods
on the fiber per unit length of yarn is one
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half of nu. I think it must be evident here;
number of intersections per one fiber and
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two intersections ñ the lengths corresponding
to two intersections represent the period
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lengths. So, this equation is valid, so that
the period is 1 by nu r by 2, because nu by
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2 is number of periods per lengths unit. So,
we can derive the period of migration. Now,
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after rearranging, we obtain thisÖ And, using
this small c as a constant, which is 2 by
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constant ñ capital C. We obtained the peri
of p is C by r mu. Then also, nu r is 2 by
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p. So, it is 2 r mu by c.
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And, how is now the fundamental equation for
our equidistant migration? Before using assumption
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4, we derived such equation. So, it is valid
also now. On the place of nu, we use we use
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this expression. So, this is here; we obtain
this. We multiply and divide by 2 pi; and,
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it must be equal to this. It is only rearranging
of this equation based on nu, which have a
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new definition, so that we obtain this here
for tangent square alpha. This is constant;
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inside of brackets, it is a constant. This
constant ñ I want to call capital Q. So,
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we can write the tangents alpha square is
1 plus tangents square beta by Q square minus
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1, where Q is given by this. Have this structure;
constant Q has this structure. This equation
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is a fundamental equation of our equidistant
migration.
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On
the page 61, it was derived n times as this
and this too. The derivation is there. We
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can use this relation; then also, Q, which
is pi c by n s. Using this on the place of
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n s, we obtain this here. So, Q is C by D
by 2 square times mu times 1 minus C. It is
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some constant, which is related to our earlier
constant of proportionality; then, one half
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of yarn diameter packing density and 1 minus
delta.
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Our couple of differential equations for equidistant
migration is following. First, is Treloar
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is same for helicon model, same for Treloarís
model. Second is interesting. Now, from fundamental
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equation, square root of this is here; tangents
alpha is d r by d zeta; tangents beta is 2
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pi r Z. So, the d r by d zeta is plus minus
square root of this quantity. This is differential
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equation between increment of radius and increment
of increment of zeta. This equation is possible
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to solve analytically; it is not integral.
So, after integration, we obtain this equation
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plus minus r is r two parts from inside to
outside from outside to inside every time.
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Now, I think we need not to comment it step
by step; we must do this integral; r is constant,
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but r is integrating variable here. It is
possible how you areÖ See here include the
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substitution, which is here. So, you can quietlyÖ
How to study, how to integrate? Its resulting
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integral value is here. The second is trivial.
So, integral for this zeta; zeta is evident.
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And, using this, we obtain this here, where
some K in the moment is integrating constant.
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So, the trick to the relation between r and
zeta is given by such equation in equidistant
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migration model. Graphically, it is shown
on our picture; the yellow colour is too light.
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This is a set of directions, which are valid,
so that our fiber must follow the trends,
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which are in this yellow net; it is relation
between r and zeta. You can see it is not
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linear; there are some curves. It was really
calculated by program; it is not linear lines;
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it is some curves. But, practically, you can
see that these curves are very near to the
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straight line. We will then make some Why
I call this model as an equidistant model?
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We know that cosine theta r is d r by d l
and is given by this expression. By using
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the fundamental equation in this expression,
we get d r by d l, is this one; evidently,
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this is the same ñ tangents square beta plus
one is also in our fundamental equation; where
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is this here? It is here. So, it is Q square
minus 1 times tangents square alpha, where
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I use it here. So, I obtain this here; tangents
square alpha is going before square root.
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So, that says here. And, what is this? Tangents
alpha by absolute value plus or minus; or,
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plus 1 or minus 1; plus minus.
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So, we obtain dr by dL is plus minus 1 by
this; but, this is constant. So, it is plus
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minus if
q
is this one; it is a constant. In some approximation,
we can say it we can obtain it in given expression.
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So, here the fiber length increases ñ fiber
lengths dl ñ this is important; fiber length
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increases equidistantly with steps of radius;
increasing of radius. Increasing of lengths
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is proportional fiber length is proportional
to increasing of radius. Therefore, increases
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equidistantly the step of radius; therefore,
this model is called as equidistant migration.
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So, is not that equation? Why the term equidistant?
Also, in this model, it is possible to construct
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some approximation.
You saw that our curve was very similar to
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the straight lines; so, let us assume as earlier
as by Treloarís model. That tangents square
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beta is much more smaller than 1; then, tangents
square alpha isÖ This is very small value,
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so that 1 plus something very small is roughly
1; this is one. So, that is 1 by Q square
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minus 1. So, it is constant. And, tangents
square is constant square alpha, so that tangents
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alpha is constant too; plus minus. So, that
dr is constant times d zeta; or, d zeta is
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constant times is how you want; and then,
we obtain that zeta is some constant square
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root of Q square minus 1 times r; it is linear
function. So, linear function plus k So, starting
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00:33:59,700 --> 00:34:06,700
point is in the point 0, 0; then is k is 0,
2; it is integrating constant; which of equation
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00:34:07,200 --> 00:34:14,200
it is linear relation between radius and axial
coordinate zeta; what it have cones. So, it
186
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is an equation, which characterize cone. In
this model, the idealized fiber is lying on
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00:34:27,659 --> 00:34:34,659
a cone surface.
This picture can help to how to imagine our
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00:34:42,559 --> 00:34:49,559
fiber. At first, fiber is not going every
time from center to periphery from periphery
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00:34:50,179 --> 00:34:57,179
to center; it is going maybe from center to
some radius; then, back. The point is not
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00:34:59,650 --> 00:35:05,130
on the periphery; it is on a radius; then,
back to small radii; then, back to higher
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00:35:05,130 --> 00:35:12,130
radii and so on and so o, so that the track
here of the fiber can be like this red curve
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00:35:15,960 --> 00:35:22,960
in the diagram r zeta here. You can imagine
it as aÖ The fiber is going, is on the surface
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00:35:26,900 --> 00:35:33,900
of some cone; then, back its curve; and go
to smaller radii or to another cone; then,
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00:35:38,730 --> 00:35:45,730
break it and go back to the surface of the
third cone and so on and so on like this here.
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00:35:51,750 --> 00:35:58,750
So, is the picture of fiber tractor e, which
is random from point of view points in which
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00:36:02,010 --> 00:36:09,010
break; and, change the direction from inside
to outside or from outside to inside.
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00:36:09,289 --> 00:36:16,289
And, we obtain such result. Now, the comparison
is experiment. We change only one assumption
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00:36:21,930 --> 00:36:28,930
ñ assumption called number 4. On the place
of idea of constant nu, which represents it;
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00:36:32,200 --> 00:36:39,200
each fiber must go from periphery to center
and from center to periphery. We assume that
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00:36:39,240 --> 00:36:45,630
there are quantities for intersections of
radius, is proportional to the vicinity of
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00:36:45,630 --> 00:36:52,630
fibrous material around this radius. We measured
it experimentally how I explained when I discussed
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00:36:57,420 --> 00:37:04,420
the results by Treloar. And, the same diagram
here is tangents beta; it means 2 pi r Z.
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00:37:09,940 --> 00:37:16,940
And, here is absolute value of alpha times
tangents beta. our curves in equidistant model;
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00:37:17,130 --> 00:37:24,130
other thick curve is here; the thick curves
here. And, the approximation, which we used
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00:37:26,339 --> 00:37:33,339
are the dotted lines here. And, the same trend
of experimental curves ñ it is this and the
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00:37:37,950 --> 00:37:44,950
experimental curve was so; or, so on and so
on. It is not ideal, but I think it is much
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00:37:47,779 --> 00:37:54,779
more better correspondent, the theoretic model
of the experimental experiences. And, only
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00:37:59,240 --> 00:38:06,240
one idea is changed from assumption 4 to newer
assumption 4 star.
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00:38:10,960 --> 00:38:17,960
An experimental graphs fromÖ You can see
this is a mean value of cosine absolute value
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00:38:19,760 --> 00:38:26,760
of angle theta r on different radii inside
of yarn from 0 to yarn surface. Here is radius
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00:38:30,440 --> 00:38:37,440
of theta ñ have this k of radius from yarn
center from 0 toÖ; one is because here is
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00:38:38,609 --> 00:38:45,609
r by D by 2. Cosine of theta experimentally
measured is this here like this. Two examples
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00:38:52,650 --> 00:38:59,650
are here ñ carded rotor yarn and carded ring
yarn; both form viscose fibers. We analyzed
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00:39:01,230 --> 00:39:08,230
viscose yarns, because viscose fibers were
the best for experimental work based on the
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00:39:09,010 --> 00:39:16,010
transparency of such fibers in immerse liquid
using Mortonís tracer fiber technique. You
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00:39:16,500 --> 00:39:23,500
can see that roughly the values can follow
our ideal model. This way it is constant ñ
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00:39:28,680 --> 00:39:35,680
equidistant migration. Here now, but this
is for outside of yarn radius.
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00:39:40,940 --> 00:39:47,940
Periodic migration ñ pure experimental results;
also, ring yarn and rotor yarn; length of
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00:39:52,200 --> 00:39:59,200
mean periodic migration and also r by D by
2. You can see that periodic migration is
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00:40:00,309 --> 00:40:07,309
not constant based on radius. As we assume,
it is higher value indulged in the round axis,
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00:40:15,019 --> 00:40:22,019
because small surface of cylinder in small
radius. Therefore, no too often the fibers
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00:40:22,150 --> 00:40:29,150
have the chance to go inside, then outside.
Therefore, number of intersections is small.
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00:40:29,849 --> 00:40:36,849
Therefore, the mean length of period is relatively
high. The same is on the yarn. The surface
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00:40:37,460 --> 00:40:44,460
ñ it is increasing, because packing density,
the border between yarn body and higher yarn
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00:40:46,220 --> 00:40:53,220
is small. Therefore, fiber is here only sometimes.
Therefore, the is higher; and, minimum periods
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00:40:59,190 --> 00:41:06,190
ñ maximum of frequency of intersection is
in this central part as we assume.
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00:41:10,930 --> 00:41:17,930
Some notes to the source of migration ñ the
people, the colleagues of professor Hearle
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00:41:24,240 --> 00:41:31,240
studied the migration in filament yarn. They
take filament yarn having a small twist from
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00:41:36,490 --> 00:41:43,490
production; you know that we often give small
twist to filament yarn, because be enough
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00:41:45,200 --> 00:41:52,200
compact for manipulation technological process;
is not it? This filament yarn ñ re-twist
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00:41:54,569 --> 00:42:01,569
it on a twisting machine; and then, they studied
the migration and he obtained some migration.
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00:42:09,990 --> 00:42:16,990
Some period ñ it is very periodical effect.
Why it is shown here? From couple of cylinders,
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00:42:20,240 --> 00:42:27,240
last couple of cylinders, the filament yarn
is going out flat as a ribbon; and then, the
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00:42:32,279 --> 00:42:39,279
twisting followed the mechanisms of ribbon
twisting. What is it? I have a microphone
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00:42:39,720 --> 00:42:46,720
be carefully, but I prove it. This is a ribbon
ñ my tie. When I twist it so, this is the
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00:42:49,319 --> 00:42:56,319
character of ribbon twisting. And, the fiber
which is on this side; then, on this side;
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00:42:59,430 --> 00:43:06,430
for example, yellow stripes on my tie.
Now, the yellow strip is here; but also, inside;
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00:43:08,420 --> 00:43:15,420
outside as well as inside. Therefore, they
obtained periodical migration, which is bring-in
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00:43:16,670 --> 00:43:23,670
migration from the character of starting product,
a little twisted filament yarn. In more details,
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00:43:27,450 --> 00:43:34,450
it is in the book of Hearle and The second
mechanism of bring-in migration is random
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00:43:37,000 --> 00:43:44,000
bring-in migration. When you will see the
thin sliver, very small sliver, which is going
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00:43:45,859 --> 00:43:52,859
out from last couple of cylinders in ring
spin machine, you can see that no all fibers
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00:43:54,730 --> 00:44:01,730
are perfectly parallel. They have some distribution
of directions; is not it? And, this distribution
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00:44:06,019 --> 00:44:13,019
is going inside to the yarn structure. Therefore,
we can see the change of radius as far as
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00:44:14,779 --> 00:44:21,779
the change of local twist to fiber element,
so that migration, the second influence is
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00:44:28,730 --> 00:44:35,730
random orientation of fibers in the ribbon,
which is coming to creation of the yarn.
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00:44:41,000 --> 00:44:48,000
This may be the most frequent mechanism of
migration ñ Morton ñ his name was here mentioned
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00:44:50,880 --> 00:44:57,880
fiber technique was also on professor on Younis
in Manchester; or, then, Treloar and Hearle.
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00:45:00,480 --> 00:45:07,480
And, he mentioned that exist some mechanism,
which is known as a mechanism of fiber length
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00:45:11,339 --> 00:45:18,339
compensation. When we twist yarn, the peripheral
fibers have some force, axial force; thus,
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00:45:27,240 --> 00:45:34,240
elongated there wasÖ Therefore, they want
on this red fiber, inside exists some axial
252
00:45:38,200 --> 00:45:45,200
force. So, is it a fiber? Wants to go in because
resulting force is this one; is not it? You
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00:45:50,029 --> 00:45:57,029
imagine. If it is possible, this fiber wants
to have this tractor E as this straight line.
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00:45:59,710 --> 00:46:06,710
And, the fibers, which are inside of the yarn,
have the lengths enough, because it is a little
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00:46:09,789 --> 00:46:16,789
compressed; it can go outside. Therefore,
the fibers can change mutually. This is the
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00:46:18,039 --> 00:46:25,039
principle of Mortonís idea.
But, based on my personal meaning, this effect
257
00:46:28,220 --> 00:46:35,220
is not too high. Why? It exists very significant
friction between fibers and changed the fibers;
258
00:46:41,089 --> 00:46:48,089
need very high forces, because the frictions
do not want to do it. Therefore, based on
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00:46:52,519 --> 00:46:59,519
my personal meaning, I think that this mechanism
is not too important. Then, let us see the
260
00:47:03,000 --> 00:47:10,000
third mechanism. When you have watched what
is doing; and, in the spinning triangle, you
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00:47:12,730 --> 00:47:19,730
will not see only the fibers, which are going
immediately to this top point of triangle
262
00:47:21,190 --> 00:47:28,029
to d r; you will see its scheme, but you will
see something like here is in this triangle
263
00:47:28,029 --> 00:47:35,029
ñ some twisting of the yarn together. This
mechanism then the structure, because friction
264
00:47:38,490 --> 00:47:45,490
and so on is partly coming to the yarn, so
that the fiber changed radius, because the
265
00:47:46,390 --> 00:47:53,390
starting structure can be this one. Also,
this is the mechanism for fiber migration
266
00:47:55,880 --> 00:48:01,609
and lot of others.
Although some of them is spoken in my book,
267
00:48:01,609 --> 00:48:08,609
but sorry for you it is in check language.
Good for you is that the check language is
268
00:48:12,259 --> 00:48:19,259
possible to read your professor Ishteyak in
those pictures. This is all for today. This
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00:48:25,549 --> 00:48:32,549
theme was a little more theoretical, but I
hope through these models, minimum you can
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00:48:37,740 --> 00:48:44,740
better imagine how the structure of real yarn
is and then you can imagine what you need
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00:48:48,720 --> 00:48:55,720
to do in a process of spinning. In spinning
process, I think also different modifications
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00:48:59,460 --> 00:49:06,460
of spinning technologies; try to make better
structure of starting, maybe a ribbon or starting
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00:49:14,150 --> 00:49:21,150
product at this small sliver, because to obtain
the yarn, which is now to intensive migration.
274
00:49:24,839 --> 00:49:31,839
This is more. It has the chance to have higher
value of packing density, be more compact
275
00:49:33,740 --> 00:49:40,740
and so on. But, it will be question of our
next lesson ñ packing density, compactness
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00:49:51,170 --> 00:49:58,170
diameter and so on .