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Today’s lecture is oriented to the yarn strength as a stochastic process. It is known
from practical experiences from textile laboratories and so on, that the mean value of yarn strength
as well as yarn breaking strength and coefficient of variation of yarn strength are changed
in relation to gauge strength which we use for making machines; why? It is a theme of
our today’s lecture.
We will speak about a Peirce’s model which is relatively old and the easiest model and
the second part of this theme I wrote in short comment our model which is much more difficult
and need to have understood the tools from most special tools from theory of probability;
especially from stochastic processes.
Well, let us start the model according to Peirce. You know that yarn strength is a random
variable; it is evident. The probability of breakage of one section of yarn of length
l at a force s we can call under the symbol f s l; s is strength of yarn portion, yarn
part, which is studied l is gauge length which we use ok. So, that s is random variable:
l, it is some parameter which, we know in which lengths we have our yarn which is the
gauge lengths by our experiment. So, that f s l has a sense as a distribution function
of strength s, clear? At the lengths l and the probability of survival is 1 minus f s
l probability of break is f s l. So, that the probability of the portion that does not
break is 1 minus. Well, let us imagine some yarn lengths l like
this line is here f n lengths l between couple of this lengths l lets imagine divide it to
n small lengths is l 0; then evidently l 1 is n times 1 0 as is shown. The probability
of breakage of l section of lengths 1 0 is independent of the probability of breakage
of any other section. This assumption is very important and it create a basis of Peirce’s
model. We assume that the short the strength of short parts of the yarn lengths 1 0 are
mutually independent. You can imagine abstractly when you cut it
your yarn to set of short segments lengths 1 0. Then, you mix the segments together and
then randomly you take one beside the other and glue it together you obtain new yarn.
But, from point of view of break of strength your result will be the same because the strengths
are not mutually dependent they are independent. Based on this assumption, the model is created
The second assumption is with the longer section must not break until any of n shorter sections
breaks. This is the principle of the weakest link theory. Weakest link theory started from
the idea of chain. Let us imagine a long chain; create it from links; one link beside the
other. So, how is the strength of this long chain? This is equal as the strength of this
link which have the minimum links; which is minimum quality, mechanical quality like inside.
Well, so, this is the principle of the weakest link theory. Under these two assumptions,
the following expression is valid; it is shown here. What the sense of this equation F s
l is distribution function of strength on the yarn portion lengths – l 1, relatively
long. Imagine this are long what is 1 minus F s l is it a 1 minus F s l.
Now, let us see the probability that these long lengths will not be distrait will not
be break to our force s is it. So, is the distribution function and what is it lets
imagine that this long lengths of the yarn is not break; it means that the strength of
the first link must be higher than our force s. How is the probability of this? 1 minus
F s l 0; this is distribution function of the force F breaking force F by lengths of
yarn 1 0 short lengths of the yarn l 0 is not it. So, first link the probability that
will not break is 1 minus s F s l 0 ok. How it is with the second link? Same; the
links are independent in our model now. So, the probability of the second link will not
be destroyed, is also 1 minus F s l 0 clear? How is a probability that both will not be
destroyed? You must multiply probabilities and so, we can go from first to the last link.
How many links are there? 1 sorry n. So, we must multiply this quantity and times
is from first link from second link third and then to the end. So, that it is 1 minus
F s l 0 power to n, but n it was l by 1 0 is not it? So, it must be in the place of
n we can write l by 1 0; we can obtain this equation as well as this equation F s l distribution
function of strength of our yarn by lengths l it is 1 minus bracket is 1 minus F s l 0
distribution function of same force s by short fiber segment l 0 powered to l by l 0
So, now we have summarized it how from distribution of the strength from short segments to obtain
the distribution of strengths by longer portions of the yarn. Probability density function
is principally clear? Question is now because probability density function it is a derivative
of distribution function every times, is it not? So, we can say that the probability density
function F s l, it is derivative of the function F s l by s. So, its partial because of the
quantity is a parameter; it is not random quantity.
Well, analogically for length l 0 f s l 0 is the derivative function f s l 0 by l 0
by the s. So, that the probability density function F s l is the you made the derivative
length left as well as right hand side and we obtain F s l having this here, this here,
then this by this was p d f of short for short derivative for short analysis.
The probability density function of the distribution of yarn strengths using long gauge length
l is given by this equation clear. So, it is good to know because in short lengths I
do not k now for example, for 5 centimeter gauge lengths we can in laboratory do it;
it is very difficult to say how it will be made. Some prediction how it will be for example,
the distribution of yarn strengths using the yarn lengths I do not know for example, 5
meter or 3 meter or 500 meter very long so. So, constructed instrument we have not in
our laboratory we have not breaking machines where its possible to have gauge lengths I
do not know, 5 meter or 500 meter it is sometimes very important because for example, in modern
weaving rooms can be 2 – 3 meters which are programs; now 50 centimeter its standard
gauge lengths based on standards in laboratory. So, we can recalculate the probability density
function or distribution function from short lengths right to short gauge lengths to the
distribution on the long run the Peirce’s. Assumed also a third assumption he said that
the in a short gauge lengths in our short gauge lengths 1 0 the distribution of strength
of yarn strength is Gaussian normal Gaussian distribution and you know what is more frequent
in India or Gaussian distribution normal distribution, but I hope you know both these parts, yes.
Then, the probability density function F S l must be given by known expression of probability
density function in a Gaussian distribution.
You can say in the moment that right hand side we have not 1 0; yes, explicitly we have
not, but we have because on the 1 0 this is the 1 0 determent sigma 0 standard deviation
and mean value in our short length 1 0 therefore, implicitly it is there and the distribution
function the distribution function F S l 0 is an integral of this. It is renamed 1 and
the quantity because integrating variable must be added in the border. So, that using
integral from our probability density function we obtain this here. We know that this integral
this in normal distribution, Gaussian distribution, this it is an integral have not the primitive
function in analytic form. That it is not possible to write it in an analytic form it
is. So, it is called integral; under this theorem you can find it in some mathematical
hand books and it takes it a lot of methods - how to numerically obtain the best approximation
of this in general integral. As you know, repetitive possible that we often
write that so called standardized Gaussian distribution is not it; if our random quantity
s f Gaussian distribution by the quantity s minus s bar 0, it means, mean value mean
value of our yarn strength in a short fiber lengths by standard deviation that is standard
deviation it is a quantity u having standardized Gaussian distribution. You know from the theory
of probability I think and this quantity u have probability density function is phi u
which is given by this expression. Also, very known from lighter age and this distribution
function capital phi u it is integral from this to this here, the values exist some numerical
method because probability integral. I said hence, we can write F S l 0 is this here or
this here a very phi is a probability density function of standardized Gaussian distribution
and for f capital F S l distribution function we obtain using integral of there is such
substitution we obtain this formula, but this is distribution function of standardized normal
distribution which we called here as a phi if whole function is very known is in each
book.
This is distribution function of standardized Gaussian distribution. Well, now how it is
with our distribution of yarn strength on the long lengths. We will derive 1 minus F
S l is 1 minus F S l 0 power to l by l 0. Using relations derived on earlier slide,
we obtain that it is 1 minus phi s minus s 0 bar by sigma 0 power to l by l 0 or F S
l is 1 minus this here. If it is right that the distribution of strength
in short lengths is ah giving according the Gaussian distribution then on the long gauge
length l the distribution function is given by this expression. So, that we must the distribution
function of standardized Gaussian distribution and we calculate it in the point s minus s
0 bar by sigma 0.
What we need to know? We need to know the mean value s bar 0 and standard deviation
sigma 0 which have the short analysis for example, when we now we make some experiment
there from this experiment we know that mean value on 5 centimeter lets imagine that the
short lengths is 5 centimeter. So, we will use the gauge length short gauge length 5
centimeter 1 0 and from a laboratory we obtain mean value of strength and standard deviation
very standard method for evaluation of set of data.
We have s as 0 bar and sigma 0 clear and now how is the value of distribution function
and I do not know half meter gauge lengths conductively, we use half meter gauge lengths
in yarn half meter gauge lengths l is 1 half meter how is this value. How is this distribution
function is easy you must calculate this functions phi standardized distribution function in
all points or points numerical points s minus s 0 bar by sigma 0.
So, is it clear how to apply it and we obtain distribution function for long lines. When
we show through this way we when through this way we obtain the distribution function for
one half meter gauge lengths we can compare it with experimental results take it on one
half meter standard gauge lengths. But, we can through this we also obtain the distribution
of strengths which probably we write for the gauge lengths 5 meter for example, and we
have not chanced experimentally obtained from laboratory. So, special breaking machine we
have not, from this we need may be dance room for such breaking machine ok.
The distribution function you can see this over here that this function this distribution
function and the gauge lengths l; another gauge lengths follows this expressions. So,
that it is not Gaussian distribution; Gaussian is in this case only the distribution on the
starting short lengths l is also 1 0. We can also derive the distribution function
which is derivative from one probability density function which is derivative from distribution
function and so, we obtain such using this easy way yes. Because, with understanding
in the moment I do not know what it is your value as 0 bar and sigma 0 what is your value
mean value of strength in short in short gauge lengths and what is your standard deviation.
Nevertheless we want to have some better picture how is also graphically proved explain how
are the relations how is the change this of quantities of functions therefore, lets introduce
some linear transformation which is here u is s minus s 0 bar sigma 0 when on the place
of s breaking forces give the values from gauge lengths 1 0 then it is standardized
random quantity clear, but when on the place of s you use random quantities from general
from another gauge lengths gauge lengths l, then I cannot call it as a standardizing then
it is linear transformation ok. For example, I have values from one half meter
breaking that the strengths from one half meter and now I construct the quantities u
random quantities u every times the strength from the yarn by lengths one half meter minus
mean value mean value of mean strengths value by gauge lengths 5 centimeter by standard
deviation by gauge lengths 5 centimeter by 1 0 clear it is a linear transformation.
It is possible to rearrange our equation to anew a distribution function all distribution
of function of s breaking force then distribution function of. So, defined linearly transformed
quantity name u and I obtain the distribution function g u 1 it is evident this from this
derivation very short and very easy, but this g u r is 1 minus 1 5 u power to 1 by 1; 1
by 1 0 and the probability density function we obtain.
Similarly, its derivative from this here it is shown here after derivative it was f this
is under place F S l and this is from the s. So, that we obtain a probability density
function n such from or especially because from the… because Gaussian distribution
then capital phi is distribution function of standardized Gaussian distribution and
we obtain we obtain these equation for practical application graph and we can show how is the
probability density function for different lengths l.
If the lengths l is on the graph, a quantity u, linear transformed breaking force strength
of the yarn and on the ordinate is the probability p d f probability density function g u for
different gauge length is l if l is equal 1 0. So, that our experiment is also on our
minimum gauge lengths 1 0 then we must obtain because it was our assumption; we must obtain
Gaussian distribution. In this case, it is standardized Gaussian distribution this stick
cover here this stick cover here. When we have for example, length 50 centimeter
and starting 1 0 was 5 centimeter then fifty centimeter is ten times longer gauge lengths.
So, then one by one, 0 l by 1 0 where is the curve for by 1 0 is 10 it is this curve here,
it is this curve, clear? It is another distribution it is another distribution the position of
this hat of probably curve of the probability density function is going to the right hand
side to a smaller values and a I can say thickness of this hat is a little smaller standard deviation
is smaller as well as coefficient variation. You can see that we can calculate it for different
values 1 by 1 0 here and it corresponds principally good to our laboratory experiences means when
we use longer gauge length then the distribution have the mean value is small and standard
deviation is smaller too by the same products same yarn yes. So, you can see how it is you
can also see that the curves this is right Gaussian distribution.
When we use very high value 1 by 1 0 it is also by our eyes to see that this curve is
not Gaussian for example, this curve is not fully symmetric see it clear. So, its another
distribution; its another type of distribution. Each gauge length is have some other distribution
only the lines gauge lengths l equal 1 0 have our starting Gaussian distribution.
It is possible to evaluate also some statistical characteristics u bar which is even here or
ultimately s bar mean values which can be then evaluate using this expression we can
evaluate s square of standard deviation of this dispersion variants of u and using the
this 1 also the variants of s strength variants which is here then of course, standard deviation
as a square root from dispersion variant variants then we can evaluate.
Another important statistic as for example, coefficient of variation v u as well as s
we can obtain also skewers using this expression here or this one as well as Kurtosis curve
for probability density function this graph show different statistical quantities as a
function of gauge lengths important line is two thick curves the mean value oh sorry the
mean value is this curve it is shown that the mean value of strength is increasing of
the gauge lengths is ratio 1 by 1 0 is decreasing and the standard deviation is decreasing too
there is also the path for shorter analysis then 1 0 that is possible to go to another
side, but for us is enough to understand how it from 1 0 to higher gauge lengths.
Well this all is nice, but it need to use a relatively know to nice numerical method
by calculation today its principle possible nevertheless for people which are able to
prepare some computer programmed some software for such evaluation Peirce’s create it around
nineteen thirty in that tine do not exist computers and I must say that based on my
meaning mister Peirce’s was very junior and very high educated because this is was
too complicated for to calculation this final values by hand only.
He proved to construct some approximation equations and when you studied this material
you can see how deep was education in theory of probability and properties of probability
density functions and. So, on resulting equations which we which we obtain as a approximation
of our way because.
We need for we need for mean value this integral we need for standard deviation over yarns
than this here. So, this very difficult integral is not analytical I said it is Gauss integral.
So, it was very difficult, but he found something which is relatively good approximation in
my book as well as in other books for example, cosigns structure of the yarn that polish
book is compact how is the difference between the approximation original results and I can
say that not practically too important. So, we can go; we can very well use following
equations. He said that sigma e which is sigma s by sigma
0 sigma s is standard deviation of the strength by lengths l our lengths longer lengths l
sigma 0 is standard deviation by of yarn strength by gauge length 1 0 and that is ratio is l
by 1 0 times minus 1 by 5 the second equation is the u bar which is s star minus bar minus
s bar 0 mean value of yarn strength and the cosigns 1 minus mean value of yarn strength
by gauge lengths at 0 by sigma 0 standard deviation of yarn strength of short lengths
1 0 therefore, 0 is four point two times 1 by 1 0 power to 1 minus by minus 1.
We can also this result of mister Peirce’s rearrange a flowing from some parameter a
is s 0 bar minus four point two times sigma 0 the other parameter b is c by 0 times 1
0 power to one by 5 and then s bar is a plus four time four by two b times one power two
minus one by 5 how to practically use this equations for example, you are you have studied
some weaving process lets imagine in our modern weaving loom weft yarn where the I do not
know free meter for example, long . We need to have some information how is distribution
of yarn strength by gauge lengths gauge lengths is this the lengths on which is on which we
exponent our left yarn. So, symbolically gauge length three meter
its much possible is standard breaking machine, but you can same yarn study in your laboratory
and you can study it for example, using one half meter standard gauge lengths. So, you
can think my standard my short gauge lengths is one half meter 1 0 my long lengths 3 meters
I observe the work in laboratory and I evaluate standard the values of yarn strength and what
I obtain I obtain as 0 bar mean value of yarn strength by 1 half meter gauge length and
standard deviation sigma 0 from our data know. So, this two are known then can obtain a parameter
a I can calculate the parameter a as well as the parameter b where on the place of 1
0 is now 0 point 5 meter our short lengths I able to calculate both and how is the mean
how is the mean as strength by three meter length long gauge lengths you have these equations
here. And so on. So, that through this way is relatively
easy possible to obtain the mean value and standard deviation and.
So, on to the gauge length we should do not measure based on the results which you measured
this graph is one of graphs from work of our doctor dipayan das why it in the from here
which he was university is not it. The points here are experimental points it
is an example from some yarn which of combed yarn seven point four from cotton it is the
yellow point are experimental points based how we how we measured in our method we used
5 centimeter 5 centimeter gauge lengths as a shortest 5 centimeter was for us was for
us the shortest 1 0 gauge length why use 5 centimeter because the small is gauge length
must be longer than each fiber. In other case, you do not measure the strength
of the yarn then partly you measure strength of fibers when we use for example, 1 centimeter
gauge lengths and lot of fibers you will have in both the jaws. So, it is not the strength
of the yarn then the from fifty percent each strength of the fibers .
So, that we choose 5 centimeter as a short gauge lengths and ah then we then we measure
one beside other one portion 5 sentence second that and so on. So, we obtain lot of 5 centimeter
lengths and. So, on and from this we evaluate also the strength on the different are the
different are the lengths the experimental points from different gauge lengths. Yellow
based on the shortest it means 5 centimeter gauge lengths we derive the parameters which
is necessary and we derive the red curves which are here these red curves show the Peirce’s
mean value in the relation to mean values measure experimental values and the second
graph show the standard deviation standard deviation which is measured the yellow points
and the values constructed from the shortest gauge lengths.
Therefore, in this point 5 centimeter, it is 5 centimeter, 15 millimeter the experimental
point as well as the red curve is in same point. Well, you can see principally this
theoretical concept corresponds to the reality. It corresponds to the reality; nevertheless,
some distances you can see the red curves are little under our experimental is not it
its typical for this equations why I want to say in our next lecture. So, in the moment
thank you for your attention and the next lecture, we will continue it is notes to the
possible modification of the Peirce’s concept; thank you for your attention.