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We start as set of information, set of lectures
about the yarns about the yarns. Yarns are
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very important objects in textile and very
old objects in textile practices. Around 25000
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years, the people of society know to produce
some yarns. So, it is very old and very very
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original type of fibers assembly. In my speech,
I want first to introduce some general vision
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about the possibility, how to model the structure
of the yarn, then we will speak about the
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special type of models, so called helical
models of the yarn, then something about the
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alternative to helical helical model, which
is called as a migration model of the yarn.
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Let us start let us start this general introduction
part, on your picture, you is a scheme of
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some some some general yarn is one that fiber
inside. The fiber in the yarn is very, have
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the very complicated shape and we must use
some coordinates for description of this of
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this shape, usually it is a cylindrical coordinates
r, may be this is some general point on the
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fiber, r is a reduced and phi and the lengths
zeta, the shape of the fiber in the yarn have
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the same moment random character as well as
the general deterministic trend. You know
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intuitively that this, this the general trend
is your to helical trend, is it not?
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Let us think character about this one fiber
element inside the yarn, this is yarn axis
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and this is a red part is some general fiber
element, the distance is a here from r to
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r plus d r this angle is d phi so that these
lines is r times d phi and they have this
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d zeta. Using this this analysis, we can define
free angles which are here; angle alpha on
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the green wall angle beta on the yellow wall
and angle gamma on this violet wall, it is
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evident that these three equations are valid
and from this three, it is evident that also
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we saw that tangents alpha is tangents beta
times tangents gamma yes sums more no it is
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evident that this angle beta relate it, relates
to the yarn twist intensity, we will speak
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about it later.
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It takes it also our second possibility, second
triplet of angles which can characterize our
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element there are the angles theta a to axial
direction theta r to radial direction and
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theta t to some tangent direction. From this
picture, you can see what is theta, theta
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r theta t and theta a and evidently based
on the Pythagorean theorem in three dimensional
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space, this equation must be valid after dividing
by d l we obtain this, so that we obtain this
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equation which is very known rule of direction
cosines.
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From definition of all angles, it is evident
that following expressions are right, we can
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also refine lines of our red fiber element
which is this here, d zeta times this here.
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After you arrange, you obtain this equation
derivation between, so that derivation between
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angles theta, theta t and theta a are given
by this triplet of functions. Now, too difficult
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it is only the describing of geometry of fibers
element. It is evident that two independent
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values they find the direction of fiber element,
we use one value d phi by d zeta and the second
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d r by d zeta to, let us define d phi d phi
by d zeta for one fiber element, let us call
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as a value two times pi times z where z is,
we can call as a twist of element. We will
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explain it more precisely, why it is this
twist of element can that need not be identical
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this yarn list which we know from technological
terminology.
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So, from so defined equation, z is d phi by
d zeta by two times pi, we can multiply and
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divide by here blue value r, and this was
tangents beta. So, z is tangents beta by 2
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pi r. The second characteristic is the characteristic
of the first one, is characteristic of twist
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for our fiber element, the second characteristic
is the d r by d zeta which is m is a characteristic
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of radial migration.
Evidently going back to our equations about
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angles, we can also write about this m is
sin than tangents alpha, this couple of two
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here, I must say that we conventionally use
d zeta. So, increasing of vertical coordinates
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as a positive as a positive value, elemental
but positive, we we assume that that our zeta
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coordinate is increasing when we go, when
we have some like a microbe and when we go
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through this fiber path, inside of the yarn
then our zeta coordinate is higher and higher
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and higher and higher. So, using these two
parameters z and n for possible characters
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of fibers element can be can be defined, it
is shown here.
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I say that we assume that all that that zeta
coordinates must every times increase in our,
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this is the graph reduce of fiber for the
points and zeta direction.
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It is possible that the, generally that the
fiber is can go also back some loop inside
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of the yarn. So, how to explain it when we
say that we want zeta, where you have zeta
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increasing coordinate, it is easy, let us
imagine that we divide our fiber so that we
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obtain this part, this blue part, this red
part and that this blue part to some segments
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and on each such segments, we can imagine
that the zeta coordinate is increasing so
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that that our assumption about the increasing
value of coordinate zeta is possible to accept
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also for generally for each fibers.
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Free dimensional path of a general i th fiber,
some fiber; one fiber in yarns rapture, let
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us call it i th fiber, general i th fiber
we determine by two differential equations,
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we because d phi by d zeta is 2 pi z by d
z for i th fiber is changed from point to
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point, from the fiber from element to element
of our i th fiber. So, this z i is some function
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of zeta from this, of this we obtain d phi
is 2 pi z i zeta times d zeta.
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Similarly, also the quantity m i is changed
on the path of our i th fiber therefore, it
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must be a function of zeta. So, from that
d r is the function m i zeta times d zeta,
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this couple of two formally very easy, but
both are a differential equations, is it not?
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This couple of two differential equations,
they find three dimensional path of a general
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i th fiber and now we have two possibilities;
how to create a model of the yarn, the first
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is to interpret a fiber path as a stochastic
process.
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Let us imagine to set a fibers inside of yarn,
exist a set of functions z i zeta over o i
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for first fiber, for second fiber, for third
fiber and so on. Each such function is order,
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it is individual to fiber. So, with a set
of such functions can be usually interpreted
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as a set of realization of a stochastic process.
And similarly, also the set of functions and
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i zeta over o i can be interpreted as a stochastic
process and the path of fibers in yarn can
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be interpreted as a stochastic process, so
that each fiber is a not is a bit larger random
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oriented, now then now sin 1 as the order.
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If it is so, then we need to use for evaluation
of the yarns structure, the tools relate it
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to stochastic processes; it is for example,
correlograms or maybe you heard about the
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parameters from professor Hero, fiber mean
position deviation and mean migration in terms
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so on and so on. These all are tools how to
evaluate the random process, also of of migration
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and so on.
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It is one way, one way now too easy in our
lectures we were not to more discuss this
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direction; second, the second concept is to
interpret a fiber fiber paths as a deterministic
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process. Individual fiber paths have usually
a random character, but a dominant trend is
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commonly deterministic, for example, helical.
Therefore, we often substitute the real fibers
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by an ideal, represent the fiber trajectory
around which the real fiber paths are, we
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can say intuitively ask to write it, then
this from on the, on our function z i zeta
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m i zeta and this this couple of this fibers
must be deterministic, now probabilistic.
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If m i zeta is not equals 0 for all fibers,
for all i, what is it m was d r 2 d zeta,
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if it is not equal 0, then the radials of
fibers path is changed, is it not?
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We speak about a radial migration of fibers,
if zeta is different from constant z was d
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phi by d z. And now, increase of angle by
zeta coordinates, if this value is not constant,
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then learn to twist, the twist of fibers elements
is from point to point another so that we
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must speak about our twisted migration of
fiber. And if both both, then we must speak
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about a general migration of fibers in yarn.
In opposite, if each m i zeta is equal 0,
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if each zeta zeta is the same constant, we
speak about non migrate non migrating or helical
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model.
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It brings us to this table which can in short
to classify the lot of different types of
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models of yarn structure through some, in
some system. For its group of this stochastic
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models, we do not want more to divide this
possibility.
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The second group is the deterministic models,
and it base on the character of functions
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z i zeta and m i zeta. If both are equal,
what it means reduce reduces stable for fiber,
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for each fiber d r is now different from 0
and d phi by d zeta is also 0. So, it is bundle
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of parallel fibers, parallel fiber bundle.
If z function is equal 0, but m is different
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from 0 that fiber have not twist, but the
fiber is on different rally then the.
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If z is constant and m is 0, it is our non
helical model; if z is constant and m is not
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0, then it is the traditional radial migration
model. For example, like if z if z is not
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constant and m is constant, then the fibers
are lying on some imaginary cylinder every
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times. But on the cylinder, they are, do not
create helices, then some other curve. We
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can speak about a twisted migration and the
last position is if all is possible and this
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is the general migration.
We will speak, we will in more details are
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the speak about this free, about this is not
necessary more speak bundle of parallel fibers
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was in our earlier lecture so that to important
for helical model utmost, and when also the
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radial migration.
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So, it was an slow introduction to how to
general, in general to divide different types
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of models of yarns structure. Now, let us
speak about the helical model of the yarn,
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we said the helical, each model is given by
two differential equations; this is the first
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and this is the second. It is the same as
first, in helical model, this functions z
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i zeta is constant for each fibers and each
point, each segments, each elements on each
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fiber. This constant have named Z, it is constant
and I have to some one or two slides to show
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that it is the twist of the yarn and the second
and the second equation, this function m m
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zeta m i zeta is equal 0.
What we obtain, what we obtain from this first
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equation, when this function is equal Z. Let
us integrate this equation so that integral
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over phi from some starting point coordinates
phi 0 zeta 0 r 0 from phi 0 to phi, is equal
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to the integral from zeta 0 to zeta on right
hand side that is function z is constant.
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Therefore, we obtain this here and then these
here, what it there, what is it zeta minus
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zeta 0 is an increase of zeta coordinates,
actual coordinated in the yarn, 5 5 minus
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5 is 0, is increase of angle phi divided by
2 pi, 2 pi is one times round. So, it is number
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of coils, is it not? Number of coils number
of coils by corresponding coils is definition
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of twist, is it so? So, the such constant
Z have the sense as a yarn twist so that also
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the height height per one coil must be 1 by
Z evidently.
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From the same equation, we can write also
if this is what it my Z is constant, is not
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the function as a general it is constant after
lot application if dividing by r, we obtain
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this here now this is tangents beta, so tangents
beta is 2 pi r Z.
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From the second equation, from the second
differential equation after integrating from
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r 0 to r here from zeta 0 to zeta from starting
point to some general point, because this
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is 0 then r minus r 0 must be equal 0. So,
r is equal r 0, what it means, the fiber is
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lying on the same cylinder by constant, by
the given fiber is lying on the cylinder is
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constant reduced of course, r is it each r
is r 0.
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There are this two this two equations are
dizic equations for helical model that we
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can also to interpret this helical model very
easy, we can say this is some scheme of the
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yarn according helical model on the general
cylinder diameter r is lying one red fiber.
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Once this length is 1 by z, I know of between
the tangent to our fiber and vertical direction
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is along a beta, is it not? And after unwiring
of this surface of such imaginer cylinder,
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we obtain this, from this is also possible
to calculate the tangents beta is 2 pi r by
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1 by Z, so 2 pi r times Z.
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The, so we called differential of differential
equation every times, some some starting point
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or something. So, our starting points from
our fibers based on packing density, if some
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fiber is starting from some radius then this
fiber is lying on this radius and have path
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to the packing density on this radius, how
is the packing density, so much fibrous can
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be on this or that radius r.
Generally, the packing density is a function
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of radius, in a based on experiment to experience
is, this is radius and this is packing density,
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then the real curve is something like this
black curve on our small picture.
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But it is difficult to obtain such function,
we can obtain our experimentally based on
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now to easy experimental process or make some
mechanical model of the yarn that is one of
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the easy. Therefore, we often use some, use
a fourth assumption for simplification of
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our problems, we assume in that fourth packing
density in all places inside of the yarn is
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constant like this red line.
Also, this assumption is valid, and then we
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can speak about the ideal helical model. Helical
model is reserved this assumption ideal, in
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ideal helical model we accept also this assumption.
The helical models and ideal helical models
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are in lot of publications, some of known
authors, I mention here, from Kochlin I think
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1828 to Treloar which which completed this
helical model in a relatively complete concept
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well.
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There are this is the, I can say definition
of the, or description what is it a helical
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model. In our helical model, let us find the
number of fibers and then also the shortening
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of the yarn due to twist. The first theme
will be number of fibers in cross section
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of yarn, ideal yarn, yarn which corresponds
to our helical model.
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So, let us imagine some cross section of the
yarn, the cross section of the yarn is over
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here, the grey islands here represents the
section islands, islands of individual fibers.
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We create two circles, here two green circles
here; the small one have the radius r and
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the higher one radius r plus d r plus differentially
elemental increase of radius, so that we obtain
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a differential annulus, is it not?
Differential annulus is a how is the thought
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to area of this differential annulus its evident
because its differential annulus its 2 pi
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r times thickness, when you cut it this you
obtain something some long and its intuitively
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clear 2 pi r times d r is the r is inside
of all annulus the area of fibers it is here
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as a red area
Now, how area of such annulus with a area
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of fiber section there are also some parts
here isn’t it well, but you know the how
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what is it a packing density so that the area
of fibers d S must be 2 pi r d r times mu
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times packing density well and see here it
also an order and for us known relation that
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the section of area there one fiber which
is lying with z axis on the radius r of course,
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is a cross section area by cos inverse of
this angle beta.
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Now, is the question how is the number of
fibers in differential annulus to this to
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solve this problem lets imagine one abstract
situation lets imagine that I am nobody of
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us is here in this room and I am standing
here one foot is inside of room the second
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my foot is outside of this room how many people
is here may be one half because I am here
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only one half you understand it well.
You can see that number of fibers need not
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be only 1 2 3 4 natural number then it can
be a real number for example, 1 half or something.
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So, second how many people is here we can
say calculate 1 2 3 4 5, but I have another
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idea, let us go all together to some writing
machine for tracks and. So, on and you will
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find a, our common right and then idea why
this, is it possible this way logically is
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00:30:37,039 --> 00:30:38,119
it possible?
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00:30:38,119 --> 00:30:45,119
We may use this style of thinking fiber area
in differential annulus is d S and area pair
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00:30:51,369 --> 00:30:58,369
one fiber is a star a star so that d S by
S star must be number of fibrous in all differential
196
00:31:03,090 --> 00:31:10,090
annulus, using equations derive we obtain
for d and this, formula this expression. And
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00:31:11,440 --> 00:31:18,440
now, how is the substantial cross sectional
area of yarn substantial cross sectional area
198
00:31:19,070 --> 00:31:26,070
of yarn is this area in earlier, but integral
of this, sum of this over all over all possible
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00:31:33,850 --> 00:31:40,850
annulus, they are from r equal 0 to on the
periphery r equal D by 2, about half of an
200
00:31:44,609 --> 00:31:51,609
yarn diameter so that it is this here; using
s we create it, we obtain this equation. Mean
201
00:31:56,269 --> 00:32:03,269
packing density of the yarn, it is total substance
cross section of the yarn by total area of
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00:32:05,669 --> 00:32:10,799
cross section of the yarn pi D squared by
4, after we arranging we obtain this here,
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00:32:10,799 --> 00:32:17,619
number of fibers in yarn cross section. Now
it is integral from this the yarn of number
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00:32:17,619 --> 00:32:24,619
pair one annulus from r equals 0 to r equal
D by 2. After using, we use here some geometrical
205
00:32:28,210 --> 00:32:35,210
formula which we now 1 by cosine square equal
to 1 plus tangents square and after slowly
206
00:32:35,519 --> 00:32:42,519
arranging we obtained this here, because tangents
is 2 pi r Z squared, we said tangents beta.
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00:32:42,559 --> 00:32:49,559
Well, but it was also derived in our lecture
one, starting lecture to this lecture that
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00:32:50,029 --> 00:32:57,029
number of fibers is tau times k n where tau
relative to 1 is the ratio yarn yarn count
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00:33:02,129 --> 00:33:09,129
more precisely for example, in tax when your
density a by fiber linear density fiber.
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00:33:16,549 --> 00:33:23,549
Coefficient k n, which is in this in this
expression can be derive from this s n by
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00:33:25,119 --> 00:33:32,119
tau, tau was 1 also cut it as by s times n,
we now so that we obtain this expression for
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00:33:37,309 --> 00:33:44,309
k n, this expression are valid for a helical
model. It means therefore, I have mu insight
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00:33:54,859 --> 00:34:00,580
of integrals because mu can be a function
of radius.
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00:34:00,580 --> 00:34:07,580
I spoke about a difficult thing is done, so
that let me now to make this very rearrange
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00:34:07,809 --> 00:34:14,809
of our equations for the case of ideal helical
model, it is what it is the model on which
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00:34:15,429 --> 00:34:22,270
the packing density in all places inside out
yarn is constant. Therefore, mu is possible
217
00:34:22,270 --> 00:34:28,250
to give B for integral as a constant.
How this, how is substantial cross section
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00:34:28,250 --> 00:34:35,250
of area mu is going before, so we obtain this
area formalize this here, now pi D square
219
00:34:39,090 --> 00:34:44,970
by 4 times mu, corresponds to our knowledge.
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00:34:44,970 --> 00:34:51,970
Mean packing density, we derive this equation
times mu sorry, mu is constant. So, before
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00:34:53,650 --> 00:35:00,650
integral for r d r its trivial so that you
obtain final in final position, mu bar equal
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00:35:03,810 --> 00:35:10,810
mu, evident if mu is constant then each all
mu must be there for all yarn. For future,
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00:35:13,560 --> 00:35:20,560
we are arranging we need to solve one integral,
this integral is shown here, is shown also
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00:35:24,530 --> 00:35:31,530
to why how to obtain it is not too difficult
to use in such such substitution as we obtain
225
00:35:31,600 --> 00:35:38,600
this result. So, I think do not want to comment
integration.
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00:35:38,640 --> 00:35:45,640
A number of fibers in yarn cross section,
for this we had this expression. Now, mu is
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00:35:48,530 --> 00:35:55,480
constant can go before integral we obtain
this expression, but this is our early integral
228
00:35:55,480 --> 00:36:02,480
this one on the place of this, we can give
this expression in a in a brackets and after.
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00:36:04,430 --> 00:36:11,430
So, rearranging, where we use here is we multiply
and divide by D square, then we understand
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00:36:13,900 --> 00:36:20,900
that this is s capital S so that this is tau,
you know it is earlier equations to the r
231
00:36:22,000 --> 00:36:29,000
equations, we obtain number of fibers in this
form.
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00:36:31,590 --> 00:36:38,590
And k n k n because it is n by tau, k n is
given by this equation. We can also we can
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00:36:43,770 --> 00:36:50,740
also use another rearranging this is, this
expression is identical is this expression,
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00:36:50,740 --> 00:36:57,740
but we know for us along the pi D Z is tangents
beta D of tangents of peripheral angle beta
235
00:37:01,010 --> 00:37:08,010
on the yarn.
So, tangents beta D, after rearranging multiply
236
00:37:10,920 --> 00:37:17,920
and divide by 1 plus cos beta D. Here, we
obtain finally k n in this form, all rearranging
237
00:37:19,400 --> 00:37:26,400
pure trivial mathematical, rearranging which
you know know from university, from high school,
238
00:37:27,470 --> 00:37:34,470
no difficultly. How the, how is k n how is
k n graphically? How is k n when we use this
239
00:37:38,980 --> 00:37:41,940
last formula for k n?
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00:37:41,940 --> 00:37:48,940
In or express it in graphical form, we obtain
such graph here, this is axis of peripheral
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00:37:49,620 --> 00:37:56,620
angle of beta D, and this is axis of value
of k n, we obtain this thick curve, in the
242
00:38:01,080 --> 00:38:08,080
textile is usually usually we twist the yarn
so that the peripheral angle is something
243
00:38:12,590 --> 00:38:19,590
between 20 and 30 degrees.
So, let us imagine average average value 25
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00:38:22,780 --> 00:38:29,780
degree, this is this red dotted line to this
angle corresponds the coefficient 0.95. When
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00:38:35,270 --> 00:38:42,270
you experimentally measured coefficient k
n, are evaluated based on cross sectional
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00:38:45,240 --> 00:38:52,240
microscopic triplets, cross sections of yarns,
then really we obtain the value 0 point, around
247
00:38:56,130 --> 00:39:03,130
0.95 for yarns, ring spun yarns.
Now for rotor yarns, for rotor yarns we obtain
248
00:39:07,400 --> 00:39:14,400
much more smaller value, because the angles
of fibers are not in dominant effect, create
249
00:39:18,060 --> 00:39:25,060
it to twist that important is also the intensive
unparallelity of ribbon in rotor, and so called
250
00:39:31,170 --> 00:39:37,770
birch fibers on the periphery of rotor yarn.
You know this term, you know this problem
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00:39:37,770 --> 00:39:44,660
so that in rotor yarn the coefficient k n
is smaller.
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00:39:44,660 --> 00:39:51,660
Now this is, this is all for coefficient k
n, the theoretical value of this coefficient
253
00:39:57,130 --> 00:40:04,130
based on ideal helical model. In reality,
it can be a bit larger why, because the structure
254
00:40:05,070 --> 00:40:12,070
is not perfectly ideal helical model. Well,
to the problem of number of fibers and coefficient
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00:40:20,300 --> 00:40:27,300
k n, rewrite it also the theme about the yarn
retraction. You have found an individual experience,
256
00:40:31,540 --> 00:40:38,540
institutive experience, when you twist it
something, some bundle of something may be
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00:40:39,810 --> 00:40:46,810
fibers, may be also a bundle of yarns or something.
So, then when you twist it this bundle, the
258
00:40:47,530 --> 00:40:53,770
bundle is shorter and shorter and shorter
and shorter and shorter, is it not? It is
259
00:40:53,770 --> 00:41:00,770
not possible more twist inside; it means by
twisting the fiber bundle is shorter and shorter.
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00:41:06,560 --> 00:41:13,560
Let us imagine some bundle of parallel fibers
which is here, runs at zeta 0 after twisting
261
00:41:18,840 --> 00:41:25,840
the lengths of resulting yarn is zeta smaller
than zeta 0, the zeta 0 minus zeta, it is
262
00:41:32,150 --> 00:41:39,150
the difference of ones between non twisted
and twisted form of our bundle. This column
263
00:41:41,420 --> 00:41:48,420
here represents non twisted structure; this
structure the second column here represents
264
00:41:48,420 --> 00:41:55,420
this structure. So, once a bundle non twisted
bundle is zeta 0 twisted is zeta.
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00:41:58,750 --> 00:42:05,750
Yarn retraction, we define as ratio zeta 0
minus zeta, these lengths, the starting lengths
266
00:42:09,000 --> 00:42:16,000
zeta 0. So, we can write it 1 minus zeta by
zeta 0, number of fibers by twisting is not
267
00:42:20,240 --> 00:42:27,240
changed, so here is n and here is also n,
volume of fibers generally, we can say that
268
00:42:31,280 --> 00:42:38,280
it can be different volume of fibers can be
different in this bundle, and in this bundle.
269
00:42:40,030 --> 00:42:47,030
Therefore, starting value is 0, final value
is V, mass of fibers must be same, non twisted
270
00:42:50,570 --> 00:42:57,570
as well as in twisted structure, starting
count starting count count of parallel fiber
271
00:42:59,370 --> 00:43:06,370
bundle is mass by lengths; mass is m, lengths
is zeta 0. So, starting yarn count T 0 is
272
00:43:10,300 --> 00:43:17,300
m by zeta 0, is it not? After twisting the
yarn count, I mean linear density that is
273
00:43:21,630 --> 00:43:28,630
a, the m count is now m by another lengths,
lengths zeta.
274
00:43:30,230 --> 00:43:37,230
The ratio between T 0 and T, there is under
definition of yarn retraction is this one,
275
00:43:41,120 --> 00:43:48,120
number of coils in parallel fiber bundle is
0, here number of coils is N c. And we can
276
00:43:51,790 --> 00:43:58,790
construct two, or we were construct two quantities
for twist; the first, which is here will be
277
00:44:02,060 --> 00:44:09,060
latent yarn twist, it is number of coils per
lengths per lengths of starting non twisted
278
00:44:14,400 --> 00:44:21,400
structure clear.
How many coils I were give to 1 meter of non
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00:44:23,080 --> 00:44:30,080
twisted parallel fiber bundle, do you understand
this term? In opposite to start yarn twist
280
00:44:34,620 --> 00:44:41,620
which is which is the same number of coils,
but by length of yarn how many coils is in
281
00:44:45,060 --> 00:44:52,060
on 1 meter of yarn, final yarn, so that it
is N c by zeta between latent yarn twist,
282
00:44:55,620 --> 00:45:02,620
zeta 0 z 0 and yarn twist is Z, related to
this expression is valid.
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00:45:07,160 --> 00:45:14,160
Latent twist coefficient, we can also construct
the latent twist confident which is Z 0 times
284
00:45:17,540 --> 00:45:24,540
square root of T 0, latent twist and starting
yarn count in opposite to this coefficient
285
00:45:29,800 --> 00:45:36,800
real which is Z times square root of T clear.
So, this latent quantity related to starting
286
00:45:40,880 --> 00:45:47,880
lengths, now to final lengths, there is a
difference here. It is starting quantities,
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00:45:57,240 --> 00:46:04,240
now in a set of our of our slides are free
variations of model for yarn retraction. We
288
00:46:09,590 --> 00:46:16,590
will comment only the second one, we will
jump the first one and the third one, it is
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00:46:18,870 --> 00:46:25,870
not too necessary to sign it here, when somebody
will study deeper the problem of yarn retraction,
290
00:46:29,670 --> 00:46:36,670
he can use my my slides and immediately for
this, from this slides to understand also
291
00:46:38,350 --> 00:46:41,100
the first and the second variation.
292
00:46:41,100 --> 00:46:48,100
So, the first variation it is idea of neutral
radius according a book, thus at we will not
293
00:46:49,830 --> 00:46:53,280
to to to do it.
294
00:46:53,280 --> 00:47:00,280
We will start this variation 2, the second
variant of model idea of total fiber volume
295
00:47:03,010 --> 00:47:10,010
which is, which was created from Brasher around
1935. It was some special set of textile on
296
00:47:15,830 --> 00:47:22,830
the university in Switzerland, but how is
this theoretical concept; I want to show you
297
00:47:27,500 --> 00:47:34,500
in the next lecture. So, in the moment, thank
you for your attention.