Study On The Influence Of Fiber Length Distribution Based On Probabilistic Density Function

The length of fibers is one of the important factors affecting the spinning process and the properties of yarn.It is significant to study the influence of fiber length distribution on the spinning process and properties of yarn.Understanding the mechanism of the influence is helpful in improving spinning process and producing the yarns of good quality.The purpose of this research is to reveal the mechanism of the influence through the fiber length distribution denoted by its density function,and to obtain general results which can be theoretical reference and can be used in spinning process.This research is one of the studies of fundamental textiles.Starting from the density function of fiber length,this thesis studies the relation between commonly-used fiber length indicators and fiber length density function,and discusses the application methods.Then the density function method is applied to the study of yarn irregularity,to the distribution of accelerated points of fibers in drafting zone,and to the yarn tenacity research.In the research,the geometric-probabilistic method is applied using density function of fiber lengths,and models concerning yarn unevenness,yarn tenacity,and accelerated points of floating fibers in drafting are given and discussed.The path of the research is to denote the fiber length distribution by density function,and construct a mathematical model,and then study the general roles.In the research,mathematical language is used to describe the quantitative impact of the fiber lengthy distribution on the spinning process and properties of yarns.The results are demonstrated by the experimental analysis.In the modeling process,strict mathematical deduction is conducted and the results have shown the influence of fiber length on the spinning process and properties of yarns.Chapter 1 introduces the background of the study.International and domestic research in current years on the influence of fiber length is described.The geometric-probabilistic method and classical research in theory are depicted.The existing problems in the research are introduced.Some problems are discussed concerning the denotation of fiber lengths,the impact of fiber length on the theoretical unevenness of yarn,the influence of fiber length distribution on the drafting process, and the influence of fiber length on the yarn tenacity.Chapter 2 mainly discusses the problems in the denotation of fiber length distribution using probabilistic density function.The relation between fiber length density functions by weight and that by number is developed and proved theoretically. The analysis gives a general relation between the tow distributions.A non-parameter kernel estimate of density function of fiber length distribution is introduced.The calculation and the fitness of kernel function to fiber length distribution show a good result and can be used in the calculation of theoretical models.Experiment data is fitted between the fiber length indicators and the mathematical expectation and variance calculated from the kernel density function.A principle component analysis is conducted and shows that two components exist dominating the fiber length distribution.The study of this chapter shows that the relation of the denotation of fiber length by density function and the length measures in common use is calculable and the theoretical research by density function can be totally reflected in the commonly used length measures.Chapter 3 discusses the decomposition properties of fiber length density function, and the mathematical proof of those properties is derived.Those properties can be a good calculation method in investigating fiber length change in textile engineering. This chapter uses the method in assessing cotton combing machine with respect to its effects.The transition function of the combing machine is estimated.The fiber breakage during combing is studied and a breakage index in introduced.The theory and calculation example demonstrates that the density function method is powerful in analyzing fiber length change in engineering and therefore can be used in optimizing textile engineering so as to achieve a best result.The Chapter 2 and 3 show that the density function of fiber length has a good advantage in the study of fiber length distribution and its compact on textile engineering,and the geometric-probabilistic method not only still has a value in theoretical research but also in textile applications.And this application can be realized by using some calculation of density function such as kernel estimate.Chapter 4 touches on the issue of the theoretical irregularity caused by fiber length distribution.The problems in the traditional Martindale's equation for yarn unevenness are analyzed.A novel model about fiber alignment in yarn is given using a binomial distribution.The model shows that the fiber length has a dramatic influence on the fiber alignment and the unevenness of number of fibers in yarn.The model is derived to a semi-empirical forecasting of the yarn.A fiber alignment index is put forward and proved to be an indicator measuring the random unevenness of fibers in yarn.An experiment is carried out to prove the normal distribution of the number of fibers in the yarn cross-section.Chapter 5 studies the influence of fiber length distribution on the drafted yarns. The influence of fiber length distribution on the accelerated points of floating fibers in drafting zone is discussed.With the geometric-probabilistic method,the theoretical variance of accelerated points is presented,with the fiber length distribution function incorporated into the expression.For the sake of calculation,a parabolic density function family model is assumed.On this basis,the variance of accelerated points can be calculated for a given fiber length distribution.The experiment data analysis shows that the parabolic function model reveals the statistics principle in the accelerated points.The calculation implies that 73%of the unevenness of the yarn is caused by accelerated points of the fiber being drafted.The model can also be used in the optimization of the drafting and can play a role in the design of the drafting device.Chapter 6 is the analysis of the influence of fiber length distribution on the tenacity of span yarn.Firstly,the twisted rover experiment is conducted and discussed with the critical slipping length equation given by Gregory.Secondly,the strength distribution model of staple yarn is presented.The experiment shows that the increase of the yarn strength with changing twist multiplier largely depends on the fiber length distribution.The longer the fiber length is,the velocity of strength increase will be. The new model of strength distribution of yarns uses the basic principle that the strength at break for yarn is due to the critical slipping length,and uses the random variables to construct the expression of the yarn strength.The expression quantitatively gives the influence of fiber length distribution on the yarn strength which can be used to forecast and simulate yarn strength with the given properties of fibers and the variance of strength.The analysis of the model shows that in many aspects the stochastic strength model is consistent with the practical roles of yarn strength,and can be used to study the strength properties of yarns.Through the established yarn strength model,the critical slipping length of fibers in yarn is calculated.The calculation solves the problem which has been remained for many years.Through the critical slipping length,the mean slipping forces of fibers in yarn is then calculated.Chapter 7 discusses the model of strength efficiency given by Zeidman in 2002. A joint influence of fiber length and fineness on the yarn strength efficiency is given base on the principle result from Zeidman.The results show that when the fiber length uniformity and fiber fineness are better,the strength efficiency of fibers in yarn will be larger.Through the analysis of the model,an equation of critical slipping length is obtained given that the fiber cross section areas are identical.Calculations of the critical slipping length for some yarns are given and the mean slipping forces are calculated as well.The thesis uses probabilistic density function method to analyze the classical theoretical problems in spinning.Some theoretical results with quantitative methods are achieved for yarn properties and spinning process.The theoretical research in this thesis enriches the foundation of textiles,offering a theoretical and quantitative method for the research of textiles.The methods in this study have generalization significance and can be used in other fields in textiles.All these work established a basis for the study of fiber length distribution and its influence on fiber engineering and yarn properties,and can be researched in the future.