Advanced Dynamics E-Book

Reza N. Jazar

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch
• Veröffentlichungsjahr: 2011

Seitenzahl: 1265

Beschreibung

A thorough understanding of rigid body dynamics as it relates to modern mechanical and aerospace systems requires engineers to be well versed in a variety of disciplines. This book offers an all-encompassing view by interconnecting a multitude of key areas in the study of rigid body dynamics, including classical mechanics, spacecraft dynamics, and multibody dynamics. In a clear, straightforward style ideal for learners at any level, Advanced Dynamics builds a solid fundamental base by first providing an in-depth review of kinematics and basic dynamics before ultimately moving forward to tackle advanced subject areas such as rigid body and Lagrangian dynamics. In addition, Advanced Dynamics: * Is the only book that bridges the gap between rigid body, multibody, and spacecraft dynamics for graduate students and specialists in mechanical and aerospace engineering * Contains coverage of special applications that highlight the different aspects of dynamics and enhances understanding of advanced systems across all related disciplines * Presents material using the author's own theory of differentiation in different coordinate frames, which allows for better understanding and application by students and professionals Both a refresher and a professional resource, Advanced Dynamics leads readers on a rewarding educational journey that will allow them to expand the scope of their engineering acumen as they apply a wide range of applications across many different engineering disciplines.

Leseprobe

Table of Contents

Title Page

Copyright

Epigram

Dedication

Preface

Level of the Book

Organization of the Book

Method of Presentation

Prerequisites

Unit System

Symbols

Part I: Fundamentals

Chapter 1: Fundamentals of Kinematics

1.1 Coordinate Frame and Position Vector

1.2 Vector Algebra

1.3 Orthogonal Coordinate Frames

1.4 Differential Geometry

1.5 Motion Path Kinematics

1.6 Fields

Key Symbols

Exercises

Chapter 2: Fundamentals of Dynamics

2.1 Laws of Motion

2.2 Equation of Motion

2.3 Special Solutions

2.4 Spatial and Temporal Integrals

2.5 Application of Dynamics

Key Symbols

Exercises

Part II: Geometric Kinematics

Chapter 3: Coordinate Systems

3.1 Cartesian Coordinate System

3.2 Cylindrical Coordinate System

3.3 Spherical Coordinate System

3.4 Nonorthogonal Coordinate Frames

3.5 Curvilinear Coordinate System

Key Symbols

Exercises

Chapter 4: Rotation Kinematics

4.1 Rotation About Global Cartesian Axes

4.2 Successive Rotations About Global Axes

4.3 Global Roll–Pitch–Yaw Angles

4.4 Rotation About Local Cartesian Axes

4.5 Successive Rotations About Local Axes

4.6 Euler Angles

4.7 Local Roll–Pitch–Yaw Angles

4.8 Local Versus Global Rotation

4.9 General Rotation

4.10 Active and Passive Rotations

4.11 Rotation of Rotated Body

Key Symbols

Exercises

Chapter 5: Orientation Kinematics

5.1 Axis–angle Rotation

5.2 Euler Parameters

5.3 Quaternion

5.4 Spinors and Rotators

5.5 Problems in Representing Rotations

5.6 Composition and Decomposition of Rotations

Key Symbols

Exercises

Chapter 6: Motion Kinematics

6.1 Rigid-Body Motion

6.2 Homogeneous Transformation

6.3 Inverse and Reverse Homogeneous Transformation

6.4 Compound Homogeneous Transformation

6.5 Screw Motion

6.6 Inverse Screw

6.7 Compound Screw Transformation

6.8 Plücker Line Coordinate

6.9 Geometry of Plane and Line

6.10 Screw and Plücker Coordinate

Key Symbols

Exercises

Chapter 7: Multibody Kinematics

7.1 Multibody Connection

7.2 Denavit–Hartenberg Rule

7.3 Forward Kinematics

7.4 Assembling Kinematics

7.5 Order-Free Rotation

7.6 Order-Free Transformation

7.7 Forward Kinematics by Screw

7.8 Caster Theory in Vehicles

7.9 Inverse Kinematics

Key Symbols

Exercises

Part III: Derivative Kinematics

Chapter 8: Velocity Kinematics

8.1 Angular Velocity

8.2 Time Derivative and Coordinate Frames

8.3 Multibody Velocity

8.4 Velocity Transformation Matrix

8.5 Derivative of a Homogeneous Transformation Matrix

8.6 Multibody Velocity

8.7 Forward-Velocity Kinematics

8.8 Jacobian-Generating Vector

8.9 Inverse-Velocity Kinematics

Key Symbols

Exercises

Chapter 9: Acceleration Kinematics

9.1 Angular Acceleration

9.2 Second Derivative and Coordinate Frames

9.3 Multibody Acceleration

9.4 Particle Acceleration

9.5 Mixed Double Derivative

9.6 Acceleration Transformation Matrix

9.7 Forward-Acceleration Kinematics

9.8 Inverse-Acceleration Kinematics

Key Symbols

Exercises

Chapter 10: Constraints

10.1 Homogeneity and Isotropy

10.2 Describing Space

10.3 Holonomic Constraint

10.4 Generalized Coordinate

10.5 Constraint Force

10.6 Virtual and Actual Works

10.7 Nonholonomic Constraint

10.8 Differential Constraint

10.9 Generalized Mechanics

10.10 Integral of Motion

10.11 Methods of Dynamics

Key Symbols

Exercises

Part IV: Dynamics

Chapter 11: Rigid Body and Mass Moment

11.1 Rigid Body

11.2 Elements of the Mass Moment Matrix

11.3 Transformation of Mass Moment Matrix

11.4 Principal Mass Moments

Key Symbols

Exercises

Chapter 12: Rigid-Body Dynamics

12.1 Rigid-Body Rotational Cartesian Dynamics

12.2 Rigid-Body Rotational Eulerian Dynamics

12.3 Rigid-Body Translational Dynamics

12.4 Classical Problems of Rigid Bodies

12.5 Multibody Dynamics

12.6 Recursive Multibody Dynamics

Key Symbols

Exercises

Chapter 13: Lagrange Dynamics

13.1 Lagrange Form of Newton Equations

13.2 Lagrange Equation and Potential Force

13.3 Variational Dynamics

13.4 Hamilton Principle

13.5 Lagrange Equation and Constraints

13.6 Conservation Laws

13.7 Generalized Coordinate System

13.8 Multibody Lagrangian Dynamics

Key Symbols

Exercises

References

Appendix A: Global Frame Triple Rotation

Appendix B: Local Frame Triple Rotation

Appendix C: Principal Central Screw Triple Combination

Appendix D: Industrial Link DH Matrices

1,2—Links with R||R or R||P

3, 4—Links with R⊥R or R⊥P

5, 6—Links with RR or RP

7, 8—Links with P||R or P||P

9, 10—Links with P⊥R or P⊥P

11, 12—Links with PR or PP

Appendix E: Trigonometric Formula

Index

Note: A star () indicates a more advanced subject or example that is not designed for undergraduate teaching and can be dropped in the first reading.

This book is printed on acid-free paper.

Copyright © 2011 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Jazar, Reza N.

Advanced dynamics : rigid body, multibody, and aerospace applications / Reza N. Jazar.

p. cm.

Includes index.

ISBN 978-0-470-39835-7 (hardback); ISBN 978-0-470-89211-4 (ebk); ISBN 978-0-470-89212-1 (ebk); ISBN 978-0-470-89213-8 (ebk); ISBN 978-0-470-95002-9 (ebk); ISBN 978-0-470-95159-0 (ebk);

ISBN 978-0-470-95176-7 (ebk)

1. Dynamics. I. Title.

TA352.J39 2011

620.1′04—dc22

2010039778

The answer is waiting for the right question.

To my daughter

Vazan,

my son

Kavosh,

and my wife,

Mojgan

Preface

This book is arranged in such a way, and covers those materials, that I would have liked to have had available as a student: straightforward, right to the point, analyzing a subject from different viewpoints, showing practical aspects and application of every subject, considering physical meaning and sense, with interesting and clear examples. This book was written for graduate students who want to learn every aspect of dynamics and its application. It is based on two decades of research and teaching courses in advanced dynamics, attitude dynamics, vehicle dynamics, classical mechanics, multibody dynamics, and robotics.

I know that the best way to learn dynamics is repeat and practice, repeat and practice. So, you are going to see some repeating and much practicing in this book. I begin with fundamental subjects in dynamics and end with advanced materials. I introduce the fundamental knowledge used in particle and rigid-body dynamics. This knowledge can be used to develop computer programs for analyzing the kinematics, dynamics, and control of dynamic systems.

The subject of rigid body has been at the heart of dynamics since the 1600s and remains alive with modern developments of applications. Classical kinematics and dynamics have their roots in the work of great scientists of the past four centuries who established the methodology and understanding of the behavior of dynamic systems. The development of dynamic science, since the beginning of the twentieth century, has moved toward analysis of controllable man-made autonomous systems.

Level of the Book

More than half of the material is in common with courses in advanced dynamics, classical mechanics, multibody dynamics, and spacecraft dynamics. Graduate students in mechanical and aerospace engineering have the potential to work on projects that are related to either of these engineering disciplines. However, students have not seen enough applications in all areas. Although their textbooks introduce rigid-body dynamics, mechanical engineering students only work on engineering applications while aerospace engineering students only see spacecraft applications and attitude dynamics. The reader of this text will have no problem in analyzing a dynamic system in any of these areas. This book bridges the gap between rigid-body, classical, multibody, and spacecraft dynamics for graduate students and specialists in mechanical and aerospace engineering. Engineers and graduate students who read this book will be able to apply their knowledge to a wide range of engineering disciplines.

This book is aimed primarily at graduate students in engineering, physics, and mathematics. It is especially useful for courses in the dynamics of rigid bodies such as advanced dynamics, classical mechanics, attitude dynamics, spacecraft dynamics, and multibody dynamics. It provides both fundamental and advanced topics on the kinematics and dynamics of rigid bodies. The whole book can be covered in two successive courses; however, it is possible to jump over some sections and cover the book in one course.

The contents of the book have been kept at a fairly theoretical–practical level. Many concepts are deeply explained and their use emphasized, and most of the related theory and formal proofs have been explained. Throughout the book, a strong emphasis is put on the physical meaning of the concepts introduced. Topics that have been selected are of high interest in the field. An attempt has been made to expose the students to a broad range of topics and approaches.

Organization of the Book

The book begins with a review of coordinate systems and particle dynamics. This introduction will teach students the importance of coordinate frames. Transformation and rotation theory along with differentiation theory in different coordinate frames will provide the required background to learn rigid-body dynamics based on Newton–Euler principles. The method will show its applications in rigid-body and multibody dynamics. The Newton equations of motion will be transformed to Lagrangian equations as a bridge to analytical dynamics. The methods of Lagrange will be applied on particles and rigid bodies.

Through its examination of specialist applications highlighting the many different aspects of dynamics, this text provides an excellent insight into advanced systems without restricting itself to a particular discipline. The result is essential reading for all those requiring a general understanding of the more advanced aspects of rigid-body dynamics.

The text is organized such that it can be used for teaching or for self-study. Part I “Fundamentals,” contains general preliminaries and provides a deep review of the kinematics and dynamics. A new classification of vectors is the highlight of Part I.

Part II, “Geometric Kinematics,” presents the mathematics of the displacement of rigid bodies using the matrix method. The order-free transformation theory, classification of industrial links, kinematics of spherical wrists, and mechanical surgery of multibodies are the highlights of Part II.

Part III, “Derivative Kinematics,” presents the mathematics of velocity and acceleration of rigid bodies. The time derivatives of vectors in different coordinate frames, acceleration, integrals of motion, and methods of dynamics are the highlights of Part III.

Part IV, “Dynamics,” presents a detailed discussion of rigid-body and Lagrangian dynamics. Rigid-body dynamics is studied from different viewpoints to provide different classes of solutions. Lagrangian mechanics is reviewed in detail from an applied viewpoint. Multibody dynamics and Lagrangian mechanics in generalized coordinates are the highlights of Part IV.

Method of Presentation

The structure of the presentation is in a fact–reason–application fashion. The “fact” is the main subject we introduce in each section. Then the “reason” is given as a proof. Finally the “application” of the fact is examined in some examples. The examples are a very important part of the book because they show how to implement the knowledge introduced in the facts. They also cover some other material needed to expand the subject.

Prerequisites

The book is written for graduate students, so the assumption is that users are familiar with the fundamentals of kinematics and dynamics as well as basic knowledge of linear algebra, differential equations, and the numerical method.

Unit System

The system of units adopted in this book is, unless otherwise stated, the International System of Units (SI). The units of degree (deg) and radian (rad) are utilized for variables representing angular quantities.

Symbols

Lowercase bold letters indicate a vector. Vectors may be expressed in an n-dimensional Euclidean space: Uppercase bold letters indicate a dynamic vector or a dynamic matrix: Lowercase letters with a hat indicate a unit vector. Unit vectors are not bolded: Lowercase letters with a tilde indicate a 3 × 3 skew symmetric matrix associated to a vector: An arrow above two uppercase letters indicates the start and end points of a position vector: A double arrow above a lowercase letter indicates a 4 × 4 matrix associated to a quaternion: The length of a vector is indicated by a nonbold lowercase letter: Capital letters A, Q, R, and T indicate rotation or transformation matrices: Capital letter B is utilized to denote a body coordinate frame: Capital letter G is utilized to denote a global, inertial, or fixed coordinate frame: Right subscript on a transformation matrix indicates

the departure frames:

Left superscript on a transformation matrix indicates

the destination frame:

Whenever there is no subscript or superscript, the matrices are shown in brackets: Left superscript on a vector denotes the frame in which the vector is expressed. That superscript indicates the frame that the vector belongs to, so the vector is expressed using the unit vectors of that frame: Right subscript on a vector denotes the tip point to which the vector is referred: Right subscript on an angular velocity vector indicates the frame to which the angular vector is referred: Left subscript on an angular velocity vector indicates the frame with respect to which the angular vector is measured: Left superscript on an angular velocity vector denotes the frame in which the angular velocity is expressed:

Whenever the subscript and superscript of an angular velocity are the same, we usually drop the left superscript:

Also for position, velocity, and acceleration vectors, we drop the left subscripts if it is the same as the left superscript:

If the right subscript on a force vector is a number, it indicates the number of coordinate frames in a serial robot. Coordinate frame Bi is set up at joint i + 1:

At joint i there is always an action force Fi that

link (i) applies on link (i + 1) and a reaction force − Fi that link (i + 1) applies on link (i). On link (i) there is always an action force Fi−1 coming from link (i − 1) and a reaction force − Fi coming from link (i + 1). The action force is called the driving force , and the reaction force is called the driven force.

If the right subscript on a moment vector is a number, it indicates the number of coordinate frames in a serial robot. Coordinate frame Bi is set up at joint i + 1:

At joint i there is always an action moment Mi that

link (i) applies on link (i + 1), and a reaction moment − Mi that link (i + 1) applies on link (i). On link (i) there is always an action moment Mi−1 coming from link (i − 1) and a reaction moment − Mi coming from link (i + 1). The action moment is called the driving moment, and the reaction moment is called the driven moment.

Left superscript on derivative operators indicates the frame in which the derivative of a variable is taken:

If the variable is a vector function and the frame in which the vector is defined is the same as the frame in which a time derivative is taken, we may use the short notation

and write equations simpler. For example,

If followed by angles, lowercase c and s denote cos and sin functions in mathematical equations: Capital bold letter I indicates a unit matrix, which, depending on the dimension of the matrix equation, could be a 3 × 3 or a 4 × 4 unit matrix. I3 or I4 are also being used to clarify the dimension of I. For example, Two parallel joint axes are indicated by a parallel sign (||).Two orthogonal joint axes are indicated by an orthogonal sign (). Two orthogonal joint axes are intersecting at a right angle.Two perpendicular joint axes are indicated by a perpendicular sign (⊥). Two perpendicular joint axes are at a right angle with respect to their common normal.

Part I

Fundamentals

The required fundamentals of kinematics and dynamics are reviewed in this part. It should prepare us for the more advanced parts.

Chapter 1

Fundamentals of Kinematics

Vectors and coordinate frames are human-made tools to study the motion of particles and rigid bodies. We introduce them in this chapter to review the fundamentals of kinematics.

1.1 Coordinate Frame and Position Vector

To indicate the position of a point P relative to another point O in a three-dimensional (3D) space, we need to establish a coordinate frame and provide three relative coordinates. The three coordinates are scalar functions and can be used to define a position vector and derive other kinematic characteristics.

1.1.1 Triad

Take four non-coplanar points O, A, B, C and make three lines OA, OB, OC. The triad OABC is defined by taking the lines OA, OB, OC as a rigid body. The position of A is arbitrary provided it stays on the same side of O. The positions of B and C are similarly selected. Now rotate OB about O in the plane OAB so that the angle AOB becomes 90 deg. Next, rotate OC about the line in AOB to which it is perpendicular until it becomes perpendicular to the plane AOB. The new triad OABC is called an orthogonal triad.

Having an orthogonal triad OABC, another triad OA′BC may be derived by moving A to the other side of O to make the opposite triadOA′BC. All orthogonal triads can be superposed either on the triad OABC or on its opposite OA′BC.

One of the two triads OABC and OA′BC can be defined as being a positive triad and used as a standard. The other is then defined as a negative triad. It is immaterial which one is chosen as positive; however, usually the right-handed convention is chosen as positive. The right-handed convention states that the direction of rotation from OA to OB propels a right-handed screw in the direction OC. A right-handed or positive orthogonal triad cannot be superposed to a left-handed or negative triad. Therefore, there are only two essentially distinct types of triad. This is a property of 3D space.

We use an orthogonal triad OABC with scaled lines OA, OB, OC to locate a point in 3D space. When the three lines OA, OB, OC have scales, then such a triad is called a coordinate frame.

Every moving body is carrying a moving or body frame that is attached to the body and moves with the body. A body frame accepts every motion of the body and may also be called a local frame. The position and orientation of a body with respect to other frames is expressed by the position and orientation of its local coordinate frame.

When there are several relatively moving coordinate frames, we choose one of them as a reference frame in which we express motions and measure kinematic information. The motion of a body may be observed and measured in different reference frames; however, we usually compare the motion of different bodies in the global reference frame. A global reference frame is assumed to be motionless and attached to the ground.

Example 1 Cyclic Interchange of Letters

In any orthogonal triad OABC, cyclic interchanging of the letters ABC produce another orthogonal triad superposable on the original triad. Cyclic interchanging means relabeling A as B, B as C, and C as A or picking any three consecutive letters from ABCABCABC ….

Example 2 Independent Orthogonal Coordinate Frames

Having only two types of orthogonal triads in 3D space is associated with the fact that a plane has just two sides. In other words, there are two opposite normal directions to a plane. This may also be interpreted as: we may arrange the letters A, B, and C in just two orders when cyclic interchange is allowed:

In a 4D space, there are six cyclic orders for four letters A, B, C, and D:

So, there are six different tetrads in a 4D space.

In an nD space there are cyclic orders for n letters, so there are different coordinate frames in an nD space.

Example 3 Right-Hand Rule

A right-handed triad can be identified by a right-hand rule that states: When we indicate the OC axis of an orthogonal triad by the thumb of the right hand, the other fingers should turn from OA to OB to close our fist.

The right-hand rule also shows the rotation of Earth when the thumb of the right hand indicates the north pole.