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Über dieses Buch

Our intention in preparing this book was to present in as simple a manner as possible those branches of error analysis which ?nd direct applications in solving various problems in engineering practice. The main reason for writing this text was the lack of such an approach in existing books dealing with the error calculus. Most of books are devoted to mathematical statistics and to probability theory. The range of applications is usually limited to the problems of general statistics and to the analysis of errors in various measuring techniques. Much less attention is paid in these books to two-dimensional and three-dim- sional distributions, and almost no attention is given to problems connected with the two-dimensional and three-dimensional vectorial functions of independent random variables. The theory of such vectorial functions ?nds new applications connected, for example, with analysis of the positioning accuracy of various mechanisms, among them of robot manipulators and automatically controlled earth-moving and loading machines, such as excavators.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Basic Characteristics of Error Distribution; Histograms

Abstract
In Chap. 1 is given introductory basic information concerning the practical aspects of the error analysis. Various examples of histograms show how much information may be deduced from them. Such important parameters deduced from a sample of measurements as a sample average, the sample variance and a sample standard deviation are defined, along with simple examples of practical significance are presented. Using examples of histograms a notion of a cumulative frequency empirical distribution is defined. A number of practical problems illustrate variety of possible empirical distributions. These examples are used to explain the difference between exact values of basic parameters such as for example a mean value or a variance and their estimators when a sample used for measurements contains only a fraction of the total population of a measured quantity.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 2. Random Variables and Probability; Normal Distribution

Abstract
In Chap. 2 basic information concerning probability and random variables as an introduction to error analysis is given. Such important notions as a probability density function and a cumulative distribution function are defined, along with fundamental parameters of a distribution: an average value, an average deviation, a variance, and a standard deviation. Basic parameters of a normal (Gaussian) distribution are defined and illustrated by examples of practical significance. Moreover, simple examples show how this elementary theory of error analysis may be used for analysis of a two-dimensional gravity flow of a granular media in bins and for analysis of terrain subsidence caused by underground exploitation. Experimental simulation with use of an assembly of coins of different diameters located on a glass plate confirms a practical significance of such a stochastic approach for analysis of processes of gravity flow of granular media.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 3. Probability Distributions and Their Characterizations

Abstract
At first in Chap. 3 are described such important notions as characteristic functions of a probability distribution defined as Fourier transform of a probability density function and constants characterizing random variables: an average value, a median value describing the middle of a distribution, a variation ratio closely connected with a standard deviation. Other characteristic constants characterizing random variables are: an asymmetry coefficient and an excess coefficient. Other parameters describing shape of a probability density functions called quantiles are also defined. A number of useful probability distributions are presented, among them are discrete probability distributions: a binomial distribution, a multinomial distribution, a Poisson distribution, and continuous probability distributions: a rectangular distribution, an exponential distribution, a Weibull distribution and a Student distribution are described. Finally are given some introductory remarks concerning multidimensional probability distributions, developed further in Chaps. 5 and 7 for analyzing various practical problems.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 4. Functions of Independent Random Variables

Abstract
At first, in Chap. 4 is given basic information concerning various practical problems in which functions of independent random variables are used in the analysis of the manufacturing or measuring errors. Simple examples illustrate how such functions are used in solving practical problems, among them in non-direct measurements or calculations of tolerance limits in two-dimensional nets of dimensions. For the cases when such nets are complicated is proposed an original method of static analogy. The net is treated as a statically determined system of bars which is loaded by two opposite unit forces acting along a line representing analyzed resulting dimension R(l 1,l 2,…,l n ) being the function of dimensions l i given in the net. This function does not need to be mathematically formulated. Values of partial derivatives of function R are found as a force in a respective bar representing particular dimension l i . Several examples illustrate this procedure.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 5. Two-dimensional Distributions

Abstract
In Chap. 5 is given basic information concerning two-dimensional distributions of random variables. Starting from a classical problem of the accuracy of artillery fire it is shown that, besides a traditional analytical procedure, components of a covariance tensor may be transformed by means of their representation by Mohr circles. It is shown that this representation, used commonly in mechanics of solids, for transformations of stress tensors, may be useful in solving such problems as a linear regression of experimental observations or a linear correlation between experimentally determined quantities. This method is compared with standard procedures. Examples of application illustrate advantages of using the representation of covariance tensors by Mohr circles. Theory of two-dimensional continuous distributions of random variables is described, with special attention given to a two-dimensional normal distribution along with a procedure of determining ellipses of probability concentration including information concerning a chi-squared distribution. It is shown how the normal two-dimensional distribution may be used for analysis of a gravity flow of granular media in bins and for analysis of a terrain subsidence caused by underground exploitation or by tectonic movements of a bedrock.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 6. Two-dimensional Functions of Independent Random Variables

Abstract
Chapter 6 is devoted to procedures in which two-dimensional functions of independent random variables are used for analysis of positioning accuracy of robot manipulators and various mechanisms. It is shown how the tolerance polygons in complex two-dimensional nets of dimensions can be constructed analytically or with the use of the method of static analogy described in Chap. 4. They can be also obtained by constructing the Williot’s diagrams. Respective examples show, how these methods may be used. Procedures of using functions of random variables with normal distributions for determination of a positioning accuracy of robot manipulators are described along with examples of application. Moreover, it is shown how the ellipses of probability concentration in such cases may be determined. Theoretical ellipses are compared with results of a numerical simulation.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 7. Three-dimensional Distributions

Abstract
Chapter 7 presents basic information concerning three-dimensional distributions of random variables. Attention is given to continuous distributions, mainly to a three-dimensional normal distribution of independent random variables. Described is a procedure of determining ellipsoids of probability concentration and their dimensions being determined by a chi-squared distribution with three degrees of freedom.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 8. Three-dimensional Functions of Independent Random Variables

Abstract
Chapter 8 is devoted to three-dimensional functions of independent random variables along with applications to an analysis of a positioning accuracy of robot manipulators. Procedure of determining polyhedrons of the positioning accuracy in such cases when tolerance limits of the positioning accuracy of particular joints of the manipulator are given, is described. When the positioning accuracy of joints is random according to a normal distribution then the procedure of determining of ellipsoids of probability concentration is described. Examples of application illustrate how to proceed.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 9. Problems Described by Implicit Equations

Abstract
Chapter 9 is devoted to more advanced problems of error analysis: problems described by implicit equations and inequalities. After formulation of the problem, some methods of approximate calculation of the probability that inequalities for random variables and functions of random variables are presented. The powerful method helping to solve such problems, the Rosenblatt transform, is presented along with applications in structures’ reliability theory. Finally, examples of simplest tasks easy to exact calculations are presented.
Zbigniew Kotulski, Wojciech Szczepiński

Chapter 10. Useful Definitions and Facts of Probability Theory for Further Reading

Abstract
In Chap. 10 are presented supplementary facts of probability theory which can be useful in studying error analysis problems. Sections of this chapter contain information on statistical linearization, multidimensional regression, limit theorems of probability theory, and elements of mathematical statistics such as: estimators, testing statistical hypotheses or confidence intervals. The final section presents bibliographical notes for future readings.
Zbigniew Kotulski, Wojciech Szczepiński

Backmatter

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