Error analysis with applications in engineering

By Wojciech Szczepiński

Subjects: Error analysis (Mathematics), Engineering mathematics, Machine design

Description: Our intention in preparing this book was to present in a possibly simple manner these branches of the error analysis which find direct applications in solving various problem 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 the 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 threedimensional distributions, and almost no attention is given to the problems connected with the two-dimensional and three-dimensional vectorial functions of independent random variables. The theory of such vectorial functions finds new applications connected, for example, with the analysis of the positioning accuracy of various mechanisms, among them of robot manipulators and automatically controlled earth-moving and loading machines, such as excavators. Besides the basic information concerning the classical simple applications of error calculus, a substantial part of the book is devoted to new aspects of more advanced problems along with numerous examples of practical applications in engineering practice. Among others, the Mohr circles representation of tensors is used for transformation of the components of covariance tensors, for determination of linear regression, for the analysis of the accuracy of artillery fire, and for the analysis of the positioning accuracy of various mechanisms. The methods of determination of the ellipses and ellipsoids of probability concentration have been described in detail, along with examples of practical calculations. Chapters I, II, and III contain presentation of the fundamentals of error calculus: basic characteristics of error distributions, histograms and their various applications, basic continuous distributions of errors and functions of independent random variables. In Chapter IV, two-dimensional distributions of errors are discussed with applications to the analysis of the accuracy of artillery fire, to the determination of linear regression for the sets of experimental points, and to the calculation of correlation coefficients. Fundamentals of the theory of two-dimensional continuous independent and dependent random variables are also discussed in that chapter. Then the methods of determination of the ellipses of probability concentration for the two-dimensional continuous normal distribution are given. Chapter V deals with the two-dimensional vectorial functions of independent random variables along with practical applications to the analysis of the positioning accuracy of mechanisms with two-dimensional movements. The procedure of determination of ellipses of probability concentration is also described. In Chapter VI, the three-dimensional distributions of errors are considered, while Chapter VII deals with the three-dimensional vectorial functions of independent random variables. The theory is illustrated by examples of the analysis of the positioning accuracy of robot manipulators. The examples of determining the ellipsoids of probability concentration are presented. This book has been written for readers whose main interests are applications of error calculus in various problems of engineering.We have indicated that certain important concepts of that calculus such as, for example, the variance and covariance, are notionally analogous to the concepts of inertia moments of plane or solid figures. The standard deviation is analogous to the so-called inertia radius of such figures. The procedure for calculating such values is analogous to that of determination of the centers of gravity of plane or solid figures. In all seven chapters much attention is paid to the practical significance of error analysis. However, some additional information concerning its mathematical foundations has been included in this book. It may be omitted by readers who are mainly interested in applications of error calculus. For convenience, the sections containing such material are marked with an asterisk. Authors want to express their thanks to Professors Marek Sokolowski and Richard Hetnarski for their help and discussions during preparation of this book.