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Today’s lecture is oriented to one direction
of modeling of yarn, internal yarn structure
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and their migration models. It is a little
more difficult than the helical model and
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its application. Therefore, I will speak slowly
as possible and please you to imagine as possible
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the geometric relations, which are valid there.
In our starting part to the theme of a structure,
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internal structure of the yarn, we spoke about
a fiber element, about the relations, which
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are valid there; such element was like these
here, is it not?
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And we define two differential equations,
this is the first and this is the second.
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We said that the Z i, which have to sense
as a twist, the local twist of our assembly.
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Sorry, the local twist of fiber element is
given by such equation and the second quantity
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m, is practically tangents alpha. What is
alpha here? Alpha is the angle on our green
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wall between vertical direction and a project
of our element to our green wall on our picture.
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It characterizes the radial migration of element.
These functions must be some deterministic
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function when we want to speak about deterministic
models of yarn structure.
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In such model the real fiber are substituted
by an ideal means representative fiber trajectories
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around which the real fiber path outside it.
Radial Migration was defined by a set of assumptions;
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assumption one is the function z i zeta, is
Z, and it is constant. The same was assumed
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by helical model. nothing new for us. So,
that or. So, the the the. The equation tangents
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beta is two pi r Z and is valid it is known
to you from the last lecture.
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Well the function m i zeta in opposite to
helical model is now not equal 0. It is some
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value which is not equal to 0. What does it
mean? This quantity m is tangents alpha. So,
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the tangents alpha is not equal to 0. Also,
D r, which is the change of radius by the
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zeta incremental increase of vertical coordinate.
So, this ratio is not equal to 0, it means
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the fibers change its radius; the radius is
increasing or decreasing, that is changing
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it radius. Because the fiber changes it radius,
but radius only is constant and as we say
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here Z is constant. Therefore, we speak about
a radial migration.
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We also will assume that a packing density
mu is constant in all places inside the yarn.
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It is analogous idea to an ideal helical model.
Well in years during 1955, I do not know at
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the moment clearly, one professor from U K,
from Manchester, from, his name was Professor
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Treloar, the name is mentioned here, for the
first time constructed such model which is
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not based on the helical concept, such a migrating
model.
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He used assumption three. It was assumption
one and assumption two. Now, it is a general
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assumption three. The absolute value m i is
same for all fiber elements lying at the same
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radius, r. Then at each given radius, also
the tangents alpha absolute value, we speak
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about absolute values; absolute value of tangents
alpha must be constant. What does it means?
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Let us imagine some radius inside the yarn
body, some hypothetical cylinder of some general
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radius r. In our migration model; the fibers
now are going to change its radius. So, they
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are going also through our cylinder of the
radius r; from inside to outside then from
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outside to inside and so on. Do you imagine
it?
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On our radius r, intersected this surface
of our cylinder, with some angle alpha, and
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Treloar’s assumption say that each intersection,
from each fiber of our radius have same angle
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alpha on a given radius. On another radius,
the angle alpha is another, but in given radius,
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all sections have the same angle alpha.
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Earlier, we derived cosines of angle theta
i, angle theta i, this green angle between
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the vertical direction of yarn axis and the
right element of fiber. Is it not? We derived
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it. So, it is defined d r by d l; it was shown
and we derived that it is tangent alpha by
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square root of tangent square alpha plus tangent
square beta plus one.
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The length of fiber element in the differential
layer; what is differential layer? Last time
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we speak about a differential annulus and
differential annulus is a cross section of
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our differential layer. Differential layer
is the space between two cylinders, one have
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the direction r and the second have the direction
r plus d r. So, differentially thin layer
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on the radius r between two cylinders, it
is the differential layer and through this
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differential layer is going the fiber.
The lengths which such fiber have inside of
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our differential layer is dl, and we derived
that or it is possible obtained from here
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that d l is d r times the square root by tangents
alpha. This d l must be constant for each
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fiber in our radius only, because beta is
constant; for each fiber beta is constant,
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for each fiber alpha is constant too, because
of the assumption. So, we have these equation
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and now the symbols which we will use.
Let us imagine the yarn lengths as delta zeta;
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delta zeta it is the length of our yarn, and
n is known value and is the number of fibers
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in yarn cross-section; number of this green
ir here. Capital N, it is number of fiber
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elements intersecting the differential layer
at the radius r in yarn in the length delta
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zeta. This differential layer have the length
delta zeta and throughout lot of times, the
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fiber intersected from inside to outside,
from outside to inside and so on. How many
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times all fibers intersect the differential
layer? Capital N times, inside of our differential
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layer is lying capital N fiber portions, elemental
fiber portions, each have length dl, is it
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imaginable?
Now, how is the packing density in the differential
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layer? So, packing density is the ratio of
fiber volume by total volume and it was the
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general definition. How the fiber volume in
our differential layer is is an elemental
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part of fiber? Each elemental part have the
lengths d l and cross section of area of fiber
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is small s in our set of our lectures, permanently.
So, this is the fiber volume. What is in the
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denominator? Evidently, it must be our volume
of our differential layer; the volume between
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these two cylinders. What is it? 2 pi r d
r, it is the area of elemental annulus as
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you know it from last lecture and this times
the zeta, the length of our yarn and we have
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a volume of differential layer; the total
volume. The absolute value here is because
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based from the track theory of fiber, radius
can increase as well as decrease, but for
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volume, I need a positive value of thickness
of our differential layer. Therefore, symbolically
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we use absolute value here.
After arranging, we obtained here the black
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alphabets here staying on the left hand side
only graphically in other form. We divide
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it by n and multiplied by n too. We can do
it; divide and multiply by the same quantity
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unless the quantity is not equal to 0, isn’t
it? So, we divide it into n and multiply by
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the quantity n and then we are here and now
what we obtained a fourth black here. But
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on the place of d l by d r we used this here,
why because of this one here.
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We can also multiply and divide by capital
Z that is the yarn twist. So, we obtained
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is the two pi r Z, is traditionally we know
it tangents beta. For this ratio capital N
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by small n times delta zeta, we will use one
symbol; nu r, because it based on r on the
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radius it can be changed this radius, so nu
r.
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Using this we can write mu, which is given
by such equation, such expression where the
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ratio nu r is given by this expression. Is
it clear with what we are doing? The question
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is what the sense of the quantity nu r is
or what is the logical sense of the quantity
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nu r? See, what is capital N? It is total
number of intersections inside our differential
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layer of lengths delta zeta, but in our yarn
of n fibers. Let us imagine please because
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it is easier our yarn from filament fibers.
We have n fibers inside in our yarn, so that
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per one fiber we obtained capital N by small
n, it is number of intersections per one fiber
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on to yarn lengths delta zeta and when I divide
this value by delta zeta more, I obtain number
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of intersections of our differential layer
per one fiber on the yarn lengths equal one.
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Is the logical sense is clear?
So, therefore, here it is written that nu
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r is number of elements intersecting the cylinder
with radius r on one fiber per unit length
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of yarn. Generally, this value is a function
of radius, is changed to its on radius on
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another radius, where we seem this quantity
and other, may be it can be in the middle
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or on the middle fiber is often times near
to axis or near to periphery and so on. Therefore,
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this quantity generally say is function of
radius.
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We derived this equation. You know, what is
nu? This equation we can write also in this
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form, that is only rearranging this part is
denominator on left hand side and nothing
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more here, on right hand side, then square
root of both sides of this equation. We obtained
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this here, after a small rearranging, you
want to have tangents alpha or with a tangents
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alpha square explicitly, so that we rearrange
our equation to this form to the final rearranging.
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We are rearranging only because to obtain
tangents alpha in explicit form, in the form
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tangents alpha is equal to or here tangents
square alpha is equal to. Tangents alpha depends
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on radius r, because of two functions, because
nu r is in the moment unknown function of
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r, and tangents beta too, because tangents
beta is 2 pi r Z and its known function, but
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also function of r. Therefore, tangents alpha
must be the function of r.
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Postulate a fourth assumption; he said that
we assume that the quantity nu r is constant;
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that is this nu and this nu is constant; independent
of radius, same value in each radius, so that
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in each radius, number of intersections of
radius, in each radius per one fiber and per
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lengths unit of the yarn, is same. For example,
I do not know six. So, one fiber intersects
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six times this radius, small radius, very
small radius, so that each radius, the fiber
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is intersecting six times, for example. This
is the assumption; I can say ad hoc assumption
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or prior assumption of Treloar’s model.
This assumption corresponds to following idea.
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All fibers have the same path, fiber passes
through the point on the yarn axis, from this
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point here where radius r equal 0; and then
the radius r is continuously increasing along
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the fiber path. You see that the red fiber
increase in its radius to its maximum; the
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maximum is on the yarn surface. Then fiber
trajectory breaks and breaks itself on the
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yarn surface, then the radius r is continuously
decreasing along the fiber, that is symmetrical
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of way to smaller and smaller, smaller, smaller,
to the yarn axis, the second red circle small,
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the point here.
So, the whole described process is repeated
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and so on from here. So, it is a geometrical
interpretation of the trajectory, of deterministic
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trajectory of fiber in Treloar’s migration
model. You can see that the elements of fibers
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are going from inside to outside like this
element and the radius is increasing. In this
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part, the radius is decreasing and they are
going from outside to inside. Is it imaginable?
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The yarn length between two axial points of
fiber, in this idea, the yarn lengths that
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is these lengths, from this point to this
point, we call it anterior chord, called as
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a period of migration. It is in one period
it is repeated.
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So, it is a period of migration, P. On these
lengths the fiber path increases each cylinder;
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sorry intersects each cylinder, each radius
r, just two times. On the yarn lengths P,
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yarn lengths P, for example, the green cylinder
is intersected one time from inside to outside.
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Here, is the intersecting point and from outside
to inside, here is the intersecting point,
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two times.
So, just imagine that the yarn length, delta
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zeta is now equal P, the period of migration.
Then number of intersections per one fiber,
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capital N by small n, is equal to one times
from inside to outside, one times from outside
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to inside. So, nu r, our quantity nu r, which
is now constant, is capital N by n by delta
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zeta, then this is two and this is p. So,
nu is now two by p. The quantity is two by
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p or one half value or reciprocal of one half
of period.
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Our equation, our earlier equation, this equation
can be rearranged using this knowledge, nu
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is two by pi; on the nu, we put two by pi.
So, we obtain this expression, this quantity
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we can also put for shorter writing call K.
Here some parameter K and this parameter is
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independent of radius; only nu can have sometimes
and you can have some problems with mu, but
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by value, by constant value of mu, it is some
parameter. Well using the symbol K some parameters,
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some constant, we obtain our equation in this
form and these equation, it is a fundamental
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equation of ideal migration model according
to Treloar. This is the fundamental equation.
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What is the sign of our parameter K in this
fundamental equation? It is derived here,
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the product and times as is evidently starting
substantial cross sectional area; substantial
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cross sectional area S 0. For a fiber bundle,
cross sectional area of all fibers, its n
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times cross sectional area per one fiber and
in per one fiber bundle it is S. So, that
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S 0 is T 0, starting fineness of per one fiber
starting from which we create the yarn by
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rho; rho is fiber mass density as every times.
For twisted yarn it is valid as cross sectional
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area is T by rho, we cross in first lecture
about the yarn structure and it can be also
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explained as a pi d square by four, the total
area of yarn cross section times packing density
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mu. So, we can write and then we know that
T is T 0 by one minus delta. It was from the
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chapter of yarn retraction, where delta is
yarn retraction. All this equations we know,
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then we can write that ns is S 0 is T 0 by
rho and is equal to one minus delta by T times
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T by rho. So, one minus delta time S and one
minus delta times mu times pi times D square
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by 4. Using this in the formula for K, we
obtain this here, this path it is D times
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tangents beta D, evidently. So, we can also
rearrange K to the form K is 2 pi by D by
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one minus delta and by tangents beta D. You
can see there this quantity is dimensionless.
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Now, to the domain of definition, our equation,
our fundamental equation, this one is written
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here. On left hand side also, tangents square
must be every times either 0 or positive value,
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it must not be negative value. This is positive,
so that this value must be positive and higher
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than 0, of course, because the denominator.
So, we must write K square tangent square
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beta minus one is higher than 0.
So, K square times tangent square beta from
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this equation must be higher one. Using this
earlier expression for K here, we obtained
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this must be, this square must be higher than
one. So, this must be higher than one. So,
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r from this, r must be higher than some positive
value here, one half of yarn diameters or
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yarn radius, the radius of the yarn or yarn
half of yarn diameter by period of migration
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times one minus yarn retraction. It is may
be usually very small value; p, period of
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migration is in relation to yarn diameter
usually very high, but it is some positive
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value. So, you can see that our fundamental
equation is not defined for very small radius;
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r must be higher than some positive value.
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This is our fundamental square root from our
fundamental equation, is shown here. Square
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root from tangent, square is tangent, square
root of this expression is plus minus this
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here, because tangents alpha is d r by d zeta.
We can write this equation. So, we can write
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this differential equation which is valid
for Treloar’s model. This is some function
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of radius d r. So, d r by this square root,
on left hand side is only r, right hand side
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is d zeta. It is possible, principally it
is possible to obtain to solve this differential
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equation, integrate both sides and we obtain
the relation between r and zeta.
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Sorry, this integral d r by this square root
integral do not exist in analytical form.
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May be you heard about some of such integrals,
one group is an elliptical integrals and some
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think about this type of integral that it
exist, it has a graphical interpretation as
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well, but such integral do not have an analytical
form. So, it brings problems evidently. Nevertheless
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it exists and therefore, I can show you the
relation between radiuses of a fiber on abscissa,
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which is scale of radius; on the ordinate
is axial dimension of per point, so zeta coordinates.
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00:34:43,690 --> 00:34:50,690
The fiber points cannot be in this blue region,
because r is too small and we said that radius
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must be higher than some value. So, it started
here, then the radius is zeta and the radius
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is increasing from starting point to the periphery
of the yarn, to the radius capital D by two
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and then symmetrically, because it changed
the sign here back to the minimum radius,
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which is so and so. This is one half of period
and this is lengths of period of migration.
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It is good and it is possible to obtain using
the numerical mathematic, because this elliptical
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integral we must calculate numerically, but
is a question if it is not enough good as
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some approximation.
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00:35:57,520 --> 00:36:04,520
Therefore, Treloar start it with means of
approximation; approximation formula. He mentioned
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period of migration P is a very long period.
So, intensive migration of fiber is a very
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long period of migration. So, called slow
migration and then K is very high value.
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And second it is valid the tangents beta square
is much smaller than quantity one because
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maximum is around 20 to 25 degree in yarn.
Therefore, in a common yarn such quantity
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is very small in relation to value one. Then
we can derive. This is our first and is our
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traditional fundamental equation. We can rearrange
it; this is rearranging using some geometrical
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00:37:09,720 --> 00:37:16,720
functions of these two, so we obtained this
here. Then we obtain this expression because
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K is very high, then K square plus one is
approximately K square. K is high, and square
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of this is very high quantity, million plus
minus one is practically million. So, it is
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00:37:47,010 --> 00:37:54,010
possible to write it. Now, here is one minus
cosine square, it is sine square. We obtained
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00:37:54,370 --> 00:38:01,370
this expression and then we obtained this
expression and because tangent square is a
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very small in relation to one.
Then this is a very small value, and then
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there one by tangent square is very high.
It is much higher than one. This is very high
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in relation to one quantity. So, that one
plus something very high is roughly this.
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This is one by tangent square beta is approximately
only and we can write what tangent square
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alpha as approximation tangent square alpha
is equal to one by k square times tangent
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00:38:45,680 --> 00:38:52,680
square beta, using the set of approximation
equations. Such equation is evident is defined
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00:38:58,630 --> 00:39:05,630
for all radii, from 0 to D by 2. Now, only
this blue area does not exist any more when
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00:39:13,320 --> 00:39:18,790
we use an approximation equation.
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00:39:18,790 --> 00:39:25,790
We can use the approximation equation, but
it brings another problem, because each approximation
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bring some errors which is not totally identically
original equation, and this bring some problem
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all through, which of this version does not
give correct period of migration. Well, tangents
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alpha as it is, we said d r by d zeta. Now,
approximation equation it is this one that
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is tangent beta is two pi r Z. So, we can
write zeta is plus minus K times two pi r
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Z d r. Integral D zeta must be integral of
right hand side over r over radius. After
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00:40:17,190 --> 00:40:23,080
integration we can get zeta is equal plus
minus this and plus C, where C is the constant
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of integration.
Well starting point of first part of fiber
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path is at d equal to 0. So, that r is 0,
zeta is 0. We start in point r 0, zeta 0.
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Then we can derive that the C must be this
integrating constant must be 0 and so, zeta
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is K times pi times Z times r square. Now,
the integration process was possible. It was
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00:41:01,300 --> 00:41:08,300
not elliptical integral and what we obtained
is each of curves, it is parabola, it is here.
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00:41:11,240 --> 00:41:18,240
So, this is the function and on the radius
D by two, it must be or it should be it one
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00:41:23,800 --> 00:41:29,750
half of period, period of migration and of
course, it should be.
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00:41:29,750 --> 00:41:36,750
How is zeta at the radius D by 2? It is K
times pi times Z times D by 2 square, it is
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00:41:39,850 --> 00:41:46,850
this here. So, it is this here using on the
place of K, our known expression from earlier
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00:41:49,510 --> 00:41:56,510
slides we obtained that zeta is P by 2 by
1 minus delta and it is not correct, it is
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00:42:00,120 --> 00:42:07,120
not one half, it should be P by 2 only; not
divided by 1 minus delta. It is because you
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00:42:08,270 --> 00:42:15,270
do not use original function than our easier
approximation. So, it should be P by 2 only.
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00:42:22,940 --> 00:42:29,940
Therefore, Treloar, you have an idea to change
the quantity K to another modified quantity
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00:42:34,920 --> 00:42:41,920
K to the value K dash and tried the tangents
alpha is plus minus one by K dash times tangents
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00:42:47,090 --> 00:42:54,090
beta, where K dash is our original K, but
multiplied by one Y minus yarn retraction.
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00:42:57,280 --> 00:43:04,280
Using this you can check it, you obtain also
for approximation equation one half of period
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00:43:08,580 --> 00:43:15,580
of migration on the radius for zeta D by 2.
The approximate differential equation is now
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00:43:30,040 --> 00:43:37,040
this one, which is tangent beta, is D r by
zeta. So, we obtain this here and then we
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00:43:40,370 --> 00:43:47,370
can integrate it, set it after integrating
starting from point zeta is 0 by r is 0 we
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00:43:49,970 --> 00:43:51,760
obtain this equation.
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00:43:51,760 --> 00:43:58,760
So, we obtain finally this relation. What
is this equation? For a parabolic zeta and
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00:44:03,080 --> 00:44:10,080
radius, on which is laying the trajectory
of fiber. This parabolic, this approximation
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00:44:20,560 --> 00:44:27,560
equation used by Professor Hearle; Hearle
maybe is here and he must be very old man.
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00:44:33,650 --> 00:44:40,650
I was a little boy and he was a top man of
yarn structure in my young years. He was a
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00:44:44,060 --> 00:44:51,060
professor on UMAST, Manchester, on the university
Manchester, and author of lot of special models.
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This equation Treloar equation used here for
definition of characteristics of migration.
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Well, I think this characteristic will be
the starting part for our next lecture. So,
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00:45:22,700 --> 00:45:29,700
at the moment I thank you for your attention
and I will be happy to see you in our next
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00:45:33,420 --> 00:45:40,420
lecture.