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In earlier lectures, we spoke about the terms
like yarn count, yarn twist, packing density;
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also, we mentioned yarn diameter.
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How is the relation among these often used
quantities? It will be theme of today’s
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lecture.
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It is nothing new. 200 years ago, mister Koechlin
in one French town Mews, presented his first
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model about the relation among these quantities.
So, we start a very alternatical concept,
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which is roughly 200 years old. This model
is usually quoted as a model-like Koechlin.
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Let us accept initial assumptions, which limited
our program. Let us think that our yarns are
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produced from same fibrous material, from
same type of technology and from same kind
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of use or similar kind of use. I will not
repeat these assumptions, but we automatically
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will think about this limit of yarns.
The Koechlin’s first assumption substitutes
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our knowledge of mechanics. We discussed earlier
about the possibility how to calculate the
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relation between pressure and packing density.
The fibrous material is compressed due to
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twist; is not it, in the yarn. But, in the
time of mister Koechlin, this relation was
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not known, so that he must use some assumption.
This assumption is here. Let us assume that
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packing density is a function of twist intensity
only. Is in reality packing density the function
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of twist intensity? Evidently, yes. When I
have higher twist, the twist intensity is
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increasing and the packing density is increasing
too; it is evident. But, the assumption is
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that the packing density is the function of
intensity of twist only. Later, we will show
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that packing density more precisely is the
function of intensity of this, but also, it
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is a function of other quantities.
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In Koechlin, mu is a function of kappa; where
kappa was pi D Z – twist intensity. You
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know it from our lesson 1. How are the consequences
of Koechlin’s first assumption? We will
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use equations known from our lesson 1 based
now in the form, which accepts first assumption
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of Koechlin. We derived areal Koechlin’s
type of twist coefficient; we called it as
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alpha s. It was Z times square root of S;
yarn twist times substance cross sectional
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area of the yarn. And, after arranging in
lesson 1, it was also this expression, where
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kappa is twist intensity and mu is packing
density; is not it? So, generally, alpha is
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a function of two variables: kappa and mu.
But, first assumption of Koechlin says that
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the mu is function of kappa, so that now,
alpha is a function of kappa only. Similarly,
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this areal Koechlin’s type of twist factor
is using the theory.
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In textile practice, use some common Koechlin
type, which is this here. This expression
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was derived in lesson 1 too; more easier rho
– specific mass of fibrous material. And
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similarly, the alpha is given by such equation.
You can see that alpha as well as alpha s
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as well as alpha are functions of kappa of
intensity of this only;only one variable on
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the right-hand side of these two equations.
And, how it is these diameter multipliers?
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Areal diameter multiplier for D was K s; and
K s was 2 by square root of pi mu; back to
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our lesson 1. Now, because mu is function
of intensity of twist only, K s is a function
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of kappa only. And similarly, the common multiplier
K, because this is K times square root of
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T, is after such arranging function of intensity
of twist only. These four quantities are now
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based on the first assumption; the function
of only one variable – it is kappa; it is
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intensity of twist.
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The second Koechlin’s assumption is directed
to suitable twist, said that the twist intensity
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of yarns of different finenesses, different
counts, shall be same; kappa shall be constant.
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What is the logical root of this assumption?
This logical root based on geometrical similarity
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– you know that when we have different geometrical
objects, which are similar means geometrically
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similar, then corresponding angles are same;
is not it? And, Koechlin thought that the
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yarn – some course yarn, some fine yarn;
both will have same possibility; for application,
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we will have some similar properties; especially
mechanical properties, geometrical properties
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when they are same from point of view of geometrical
similarity are similar. Therefore, if these
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ideas we accept as logical root, therefore,
also, the angle beta d – the angle of peripheral
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fiber in our idealized yarn, must be same
in each yarn for the same use and so on.
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And, what is the same angle? Tangent of peripheral
angle, tangent of beta d is intensity of twist
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kappa, so that kappa shall be constant. What
is now with these four equations and the first
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assumption when we accept that kappa is constant?
In these four expressions, you do not know
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this function f, but we know that it is a
common function for each equation; by the
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way, monotone increasing function it must
be. So, when we use kappa is constant, then
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evidently, alpha is alpha K s as well as K,
must be constant; and, mu is constant too;
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is not it? Yes.
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We said that good idea based on Koechlin’s
model, Koechlin’s concept, is to have the
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same angle – peripheral angle of fiber,
because geometrical similarity on each yarn.
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Now, this idea we can say to the people in
spinning mill, you must measure the angle
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of peripheral fiber in your yarns. You can
imagine what they can answer to you when you
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give this idea to your spinning mill. Nevertheless,
it is possible so much for a region, because
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we said that the result alpha s, alpha, K
s and K must be constant. So, we do not need
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to measure the angle; we can say, for example,
for practice, in spinning mill, alpha must
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be constant. What is alpha? From definition
of alpha of twist factor, twist coefficient,
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we can say that twist is alpha by square root
of yarn count, means finenesses.
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Then, we can say to working people in spinning
mill, yes, you must twist each yarns. So,
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then, alpha is… I do not know what based
on your experience in such spinning mill;
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for example, 120 meters to the power minus
1 kilo tex to the power one half. This is
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dimension – physical dimension of alpha;
this is metrical alpha. And, when the people
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will produce another yarn count, then they
use the same alpha. And, using this equation,
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they can very easily calculate in spinning
mill, which of twist is necessary for this
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or that yarn count. This is the first result
of Koechlin’s theory.
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The second result is, write it to our specialist
in weaving technology. In weaving technology,
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you know, is necessary to know the yarn diameter,
because covering cover factors, maybe covering
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can similar quantities, which defined the
density of woven fabrics and so on. How to
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obtain the yarn? How to obtain the yarn diameter?
Koechlin’s theory said it is easy. Yarn
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diameter D is parameter K times square root
of T and K must be constant. Diameter multiplier
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must be constant for given material, given
technology and given type of use of our yarns.
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Maybe based on our experiences, we can say,
it must be 0.0395 for example. And, people
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can calculate it; very easy, very elegant
theoretical model, 200 years old, but not
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too enough precise. It is very often used,
because roughly, it is possible to apply it,
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but when you want to work more precisely,
then these results are not enough. Why? The
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practical experiences say that such alpha
must be a little different for different groups
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of yarn count, so that in textile hand books,
you can read that from count, these two counts,
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usual use of this alpha; then, from count
these two counts, this interval of counts
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and a little larger alpha and so on and so
on. It is typical for hand books for spinning
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practice. This is one way.
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Second empirical way is to empirically change
the Koechlin’s equations. On the place square
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root, it means power to one half. It is possible
for yarn twist to use the ratio alpha by T
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power to some exponent q; where, q exponent
of twist is an empirical value; a little different
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from 0.5. Ruther fortress studied a problem
in relation to this equation, which of exponent
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is the best; Koechlin’s at 0.5; is not it
square root? Then, lots of others have different
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ideas based on this or that experimental experiences.
The yarn diameter can be empirically generalized
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to such form, D is some parameter Q alpha
times yarn count power to some exponent times
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alpha power to another exponent.
An example for these values may be good for
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carbon coated yarns is here. This exponent
w is usually something around 0.56 in this
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equation and the exponent by alpha is usually
minus 0.22. But, my experiences may be another
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can see another world based on pipe of cotton
beds or pipe of technology; based of lot local
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influences.
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Why the model of Koechlin is not enough precise?
His second assumption is very good; and, it
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is the earlier after Koechlin’s experiences
show that the geometrical similarity is very
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good idea. What is not too good is the first
assumption that the packing density is a function
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of twist intensity only. I mentioned it; Koechlin
in 1828, had a chance to use some models,
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which respect the physical relation between
pressure and compression of fibers inside
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of the twisted yarn. It exists some second
way how to solve it. This way is in my check
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book, which professor Ishteyak gave; he can
show you, it will be especially interesting
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for you.
The second way is go out from some differential
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equation of equilibrium of radial forces inside
of the yarn body. And, based on tools of continue
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mechanics solve this problem for you as a
problem of continue mechanic. I proved it
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earlier, lot years ago. But, there is a problem
here. To this time, we do not know the relation
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between the stress tensor and the strain tensor.
Stress tensor and strain tensor are some structures
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very popular to say; to these days, we do
not understand enough general the relation
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between stress and strain in multidimensional,
three dimensional case especially for fibrous
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assembly. Therefore, we can calculate, we
can derive the differential equation, but,
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we have not enough well input to this equation.
We must make some assumptions, some simplifications
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and so on. All these are very difficult from
point of view of mathematical tools; you must
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solve some differential equation and so on.
Nice theme – it was lot years ago; the theme
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of my PhD thesis. But, to these days, this
way is not… and the position to be practical
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tool for application. Therefore, we derived
something in between, which is very easy,
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but not too precise theory of Koechlin. And
physically, the best version differential
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equation of radial equilibrium solving of
this one; something in between, which is easier,
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not so precise from point of view of Physics,
but better than Koechlin’s type. Let us
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assume the non-assumptions from ideal helical
model. All fibers have the helical shape;
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all helixes have common axis, which is yarn
axis; all helixes have the same sense of rotation;
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each fiber coil have same height. We mentioned
these assumptions when we analyzed the helical
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model. And, fifth – packing density is same
in each place in the yarn.
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Then, it is more repetition for a helical
model. Let us imagine some general fiber inside
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the yarn body; yarn body is this. Here schematically,
the cylinder having diameter D. And, inside
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on some general radius r. This thick black
cylinder is like one fiber – red fiber on
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the general radius r helix shape. After unwind
of this cylinder, we obtain such triangle,
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which is possible to the tangent beta, which
is 2 pi r Z; tangent beta is 2 pi r Z is known
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for you from our earlier lecture. When you
open some hand book about the mathematic and
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when you find some properties of different
curves, also for 3D curves, space curves,
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you can read also what is so-called first
curvature – also, means flection is used.
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This first curvature of three dimensional
curves in the space in the case of helix is
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constant, independent to body to points on
which you measure it. And, it is k 1 – sine
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of beta by r; you can read it in each hand
book.
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Reciprocal value of first curvature – it
is radius of curvature; it is the radius of
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some rings, which can approximate our curve
in a very theoretically infinitesimal path.
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You know what this radius of curvature is,
so that the radius of curvature of such helix
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is r by sine square beta; is not it?
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Let us think now about a fiber lying in yarn
body on a hypothetical cylinder of harass.
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It is not from metal; it is only imagination.
On a radius r, this is the red fiber. Let
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us think about the elemental part of this
fiber – part UV. This is the radius of curvature,
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r k. On the angle here is differentially small;
it is angle D phi. In the fiber, let us be
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some axial force – capital F – axial fiber
force. So, when we make the picture of this
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part especially here, that you can see UV
– our elemental part of infinitesimal small
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part of fiber on which is tangential force
F – axial force in fiber. This is the angle
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phi, one half; it is no wider the straight
line is not to see here. This is so-called
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angle d phi; and, this is one half and this
is second half of angle d phi; so, d phi by
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2 and d phi by 2.
The projection of force F to the vertical
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direction is F times
sine of d phi by 2. And, we have two forces;
then, the resulting radial force is two times
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F sine of d phi by 2. If angle is very small,
we can write that value of sine is same that
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the angle in radial in each theoretical work,
we shall calculate or think about the angles
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with radials. So, the sine of d phi by 2 is
d phi by 2, because it is elementally small.
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Then, we obtain F times d phi. Volume of such
differentially small part of the fiber is
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which of fiber cross section is s. So, the
volume of this d V is length of the fiber
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times cross section; length of the fiber is
a part of ring on the radius r; it is radius
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r k; is not it? And, angle is d phi. So, r
k times d phi is length of fiber times fiber
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cross section; it is this. This is on the
r k, we use r by sine square. Once more – length
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of part UV of fiber is which? Radius times
angle. Radius is r k; angle is d phi. So,
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r k times d phi is the length UV. Now, it
is clear. This is the length; length times
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r k times D phi; length times fiber cross
section is volume of the red fiber segment
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or elemental segment. It is easy; is not it?
And, use now on the place of r k; our earlier
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expression we obtained is here; I think now
it is clear.
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Let us calculate centripetal force per unit
volume of fiber. is dP here by the volume
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of fiber. So, dP by dv. Using our equations
after rearranging, we obtain this force; P
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1 is given by F times sine square beta by
r s; beta is given by equation tangent beta
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is 2 pi r Z. Back to this picture – Theoretically,
each fiber each fiber compressed the material
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or can bring some compression, but really,
it cannot be too real. They are fibers around
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yarn surface. In reality, the packing density
in vicinity of yarn surface is in reality
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small. The fiber to fiber contacts are not
so intensive; different silly pitch fiber
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is possible, so that the friction is not fully
used; so that the radial force from such fibers
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is not too high; you can imagine. Second – the
fibers around yarn axis – there is lot of
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fibers. They have good normal forces for friction,
but they are need to straight line; the angle
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beta is very small; the radius of curvature
is very high. Therefore, the radial component
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from such fiber is extremely small. Result
– these fibers also do not influence significantly
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for compression of fibrous material inside
the yarn. So, this is the simplification.
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Significance centripetal force is present
only in the green compressing zone in-between
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the layers mentioned before. So, now, whole
material, but only the material in some schematically
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green layer has significant influence to compression
of fibrous material.
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The thickness of this green zone go under
the symbol a. This a – middle radius of
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this green layer is called r, so that the
radii of this green layer are going from r
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minus a by 2 to r plus a by 2. The area of
this green annulus is shown here is 2 pi r
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a evidently. Total volume of compressing zone
is green zone. It is V c, a – 2 pi r a times
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l; volume area times r. Fiber volume in compressing
zone is volume times mu from definition of
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packing density. So, 2 pi r a l times mu.
And, the total centripetal force in compressing
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00:33:49,019 --> 00:33:56,019
zone, P is P 1 times… P 1 we know; we know
P 1 times V a. Using this equation after more
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00:34:04,109 --> 00:34:11,109
rearranging, we obtain such equation.
Our assumption for simplification is the centripetal
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force P acts on the cylinder at radius r.
They are fibrous like on the smaller radii
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in our green zone as well as some other fibers,
which are lying on higher radius than r – the
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00:34:32,659 --> 00:34:39,659
average layer of our zone. But, we make it
easier and we all affects concentrate to some
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00:34:41,210 --> 00:34:48,210
average radius, our radius r. Do understand
this assumption. Then, how is the surface
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00:34:53,679 --> 00:35:00,679
area of the cylinder on the radius r, that
is, 2 pi r times… And, how is the pressure,
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00:35:12,730 --> 00:35:19,730
which creates our yarn, which compressed our
yarn. The pressure is force by A. We calculate
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00:35:24,000 --> 00:35:31,000
this A on the average radius of our green
zone; yes, using our equations for P and for
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A – this is for P; this is for A – we
obtain this after small rearranging. The pressure
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is given by such equation; this yellow equation.
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00:35:55,000 --> 00:36:02,000
You see here is some part of some fiber. In
such fiber, the axial for F exist. The axial
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component of this data – the force F – the
direction of fiber axis; the component of
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00:36:29,520 --> 00:36:36,520
this force in direction to yarn axis is a
component F a; is not it? Axial means the
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direction to yarn axis. It is evident that
F a is F times cosine of beta. By the way,
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it exists also such tangential force, which
from all fibers together give some thousand
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00:37:01,650 --> 00:37:08,650
moments in yarn, but, it is other. The green
section area of the fiber s star is – we
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00:37:14,849 --> 00:37:21,849
mentioned it lot of times earlier – it is
s by cosine of beta, so that the normal stress
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00:37:26,390 --> 00:37:33,390
on the green area – it is normal force to
green area F a by area s star; F a by s star.
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00:37:41,359 --> 00:37:48,359
F times cosine of beta by s by cosine of beta
when we use expressions derived earlier. So,
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00:37:53,799 --> 00:38:00,799
it is F by s times cosine square beta. So,
F therefore, is sigma s by cosine square beta.
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That is rearrangement.
Now, our formula for pressure – using this
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00:38:17,779 --> 00:38:24,779
expression, p is possible to write also the
black symbols here; identical is our earlier
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equation. When I multiply and divide by s,
I obtain this expression. And, in brackets,
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00:38:38,200 --> 00:38:45,200
– what is it in brackets? Now, this is the
normal stress sigma, so that from this equation,
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00:38:48,529 --> 00:38:55,529
we can say that this is sigma s by cosine
square beta; and, this is force F. We can
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00:39:05,890 --> 00:39:12,890
write this equation – tangents beta – it
is 2 pi r Z; use 2 pi r Z on the place of
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00:39:16,210 --> 00:39:23,210
tangents beta. Now, we black symbols are the
same expressions as here. Here we multiply
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00:39:31,440 --> 00:39:38,440
and divide by blue D, yarn diameter; we also
multiply and divide by green D, yarn diameter;
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00:39:42,700 --> 00:39:49,700
then, pi D z is kappa intensity of twist.
So, we can write this expression. And, D yarn
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00:39:54,140 --> 00:40:01,140
diameter here is D s by square root of mu.
D s was in our lecture 1 – substance diameter
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– diameter of hypothetic yarn, which has
not added inside of body. Yes. So, for pressure,
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00:40:17,760 --> 00:40:24,190
we obtain now this expression after such rearrangement.
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Let us continue our work with rearranging.
This is repetition from last slide. We can
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00:40:36,750 --> 00:40:43,750
graphically calculate; write it also – these
black symbols. And, we multiply and divide
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by 4 times pi. And, we do it. Then, this expression
when you compare it with equation for alpha
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00:40:56,529 --> 00:41:03,529
s in our lecture 1, you can see if alpha square…
So, we can write it in such form. Last, multiply
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00:41:07,809 --> 00:41:14,809
and divide blue diameter of fiber; we divide
by fiber diameter here; and, divide it in
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00:41:18,599 --> 00:41:25,599
denominator; it mean multiply; so, we can,
but here it is square root of Tau of relative
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00:41:27,510 --> 00:41:34,510
count, relative finenesses of the yarn – also,
from lecture 1. So, we obtain for pressure,
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00:41:38,579 --> 00:41:45,579
this equation – this expression for pressure.
We can call on the symbol C – this part.
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00:41:56,049 --> 00:42:02,710
And so, we obtain what is here; we obtain
the formula P is C times square root of mu
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00:42:02,710 --> 00:42:09,510
times alpha s square by square root of Tau.
What is this symbol? Here sigma is the normal
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00:42:09,510 --> 00:42:16,510
pressure and on fiber area in yarn cross section;
a is the thickness of the green zone or compressing
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00:42:24,890 --> 00:42:31,890
zone; d is fiber diameter; r is average radius
of the green zone; and, D is yarn diameter.
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00:42:35,170 --> 00:42:39,039
So, we obtain this equation.
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00:42:39,039 --> 00:42:46,039
Let us discuss the quantity C; 2 r by D – it
is said the position of the green zone – more
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00:42:55,740 --> 00:43:02,740
precisely, average radius in the ring of yarn
cross section. We can assume that this position
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00:43:08,410 --> 00:43:15,410
is ratio, because the geometric similarity
is a constant for yarn of given technological
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00:43:19,559 --> 00:43:26,559
material and so on. Axial stress sigma in
yarn cross-section – we assume is constant
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00:43:30,000 --> 00:43:37,000
too, for example, centrifugal force due to
spinning is perhaps the same.
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00:43:37,480 --> 00:43:44,480
Now, the centrifugal force – this stress
from the centrifugal effect, so that we can
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00:43:54,180 --> 00:44:01,180
imagine that also the sigma is constant. Relative
thickness of compressing zone is the ratio
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00:44:08,619 --> 00:44:15,619
a by d. It is difficult to explain. All experiments
say that the least ratio shall be also constant
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00:44:22,250 --> 00:44:29,250
value in the yarn. Why? We have some semi
hypothesis for this, but often say the assumption
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00:44:35,579 --> 00:44:42,579
that this ratio is constant, is not fully
theoretically analyzed and based at most on
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00:44:46,420 --> 00:44:53,420
the experimental results. Nevertheless, we
will use it. We will assume that a by d is
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00:44:57,000 --> 00:45:04,000
constant too. Then, hold this parameter C,
is some characteristic constant in the yarn.
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00:45:06,930 --> 00:45:13,930
And, we can write p is C times square root
of mu times alpha s square by square root
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00:45:16,160 --> 00:45:23,160
of Tau, where C is some constant.
Our hour is out. Thank you for your attention.
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00:45:31,980 --> 00:45:38,980
In next lecture, we will show how to apply
derived pressure. This pressure is derived
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00:45:43,690 --> 00:45:50,690
from geometrical relations inside of the yarn
structure for the yarn structure. In next
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00:45:51,240 --> 00:45:58,240
lecture, we will use also our known equation
for pressure from our lecture about the compressibility
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of fibrous material. Thank you for your attention.