eBook - ePub

Dual-Number Methods in Kinematics, Statics and Dynamics

Name: Dual-Number Methods in Kinematics, Statics and Dynamics
Author: Ian Fischer

Ian Fischer,

240 pages
English
ePUB (mobile friendly)
Available on iOS & Android

eBook - ePub

Dual-Number Methods in Kinematics, Statics and Dynamics

Ian Fischer,

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About This Book

This well-organized book uses 3x3 coordinate-transformation matrices and 3-element vectors with dual-number elements to analyze the mechanics of mechanism, robots, and other mechanical systems.
Dual-Number Methods in Kinematics, Statics and Dynamics serves as a text for a course using dual-number methods as well as a manual for the reader to develop his or her abilities for the design of machinery or evaluation of mechanical systems. In addition to the explanatory text and derivations, the author includes numerous examples and exercises to enable the reader to gain insight and perfect skills.

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Information

Publisher

Routledge

Year

2017

ISBN

9781351454483

Edition

Topic

Technology & Engineering

Subtopic

Civil Engineering

Index

Technology & Engineering

Chapter 1 - Definitions

The Dual Number

The concept of the dual number was introduced by Clifford during 1873 and the name was given by Study in 1903.

The dual number ε is defined such that

\begin{matrix} ε \neq 0 & (1) \end{matrix}

\begin{matrix} ε^{2} = 0 & (2) \end{matrix}

A dual number, which perhaps should be called a “duplex” number in analogy with complex numbers, is written as

\begin{matrix} \hat{a} = a + ε a_{0} & (3) \end{matrix}

where symbol a represents the primary (or real) part of duplex (or dual) number â and symbol a₀ represents the dual component of duplex (or dual) number â.

In the mathematics of dual numbers, addition is defined as

\begin{matrix} \hat{a} + \hat{b} = (a + b) + ε (a_{0} + b_{0}) & (4) \end{matrix}

Multiplication is defined as

\begin{matrix} \hat{a} \hat{b} = (a + ε a_{0}) (b + ε b_{0}) & (5) \end{matrix}

\begin{matrix} = a b + ε ({a b}_{0} + a_{0} b) & (6) \end{matrix}

Division is handled in a manner similar to the division of complex numbers

\begin{matrix} \frac{\hat{a}}{\hat{b}} = \frac{a + ε a_{0}}{b + ε b_{0}} & (7) \end{matrix}

\begin{matrix} = \frac{a + ε a_{0}}{b + ε b_{0}} \frac{b - ε b_{0}}{b - ε b_{0}} & (8) \end{matrix}

\begin{matrix} = \frac{a}{b} + ε \frac{a_{0} b - {a b}_{0}}{b^{2}} & (9) \end{matrix}

Division by a pure dual number is not defined.

Raising a dual number to a power is

\begin{matrix} {\hat{a}}^{n} = {(a + ε a_{0})}^{n} & (10) \end{matrix}

\begin{matrix} = a^{n} + ε {n a}_{0} a^{n - 1} & (11) \end{matrix}

Thus the square root of a dual number is

\begin{matrix} \sqrt{\hat{a}} = \sqrt{a} + ε \frac{a_{0}}{2 \sqrt{a}} & (12) \end{matrix}

The Taylor series expansion of a function of a dual number is

\begin{matrix} f (\hat{a}) = f (a + ε a_{0}) & (13) \end{matrix}

\begin{matrix} = f (a) + ε a_{0} \frac{d f (a)}{d a} & (14) \end{matrix}

All formal operations of dual number are the same algebra followed by...

Cover Page
Half title
Title Page
Copyright Page
Preface
Dedication Page
Table of Content
Chapter 1 Definitions
Chapter 2 Coordinate Transformation
Chapter 3 Matrix Modeling of Joints and Links
Chapter 4 Displacement Analysis
Chapter 5 Velocity Analysis
Chapter 6 Static Analysis
Chapter 7 Dynamic Analysis
Chapter 8 Dual-Number Programming
Bibliography
Index