Rigid Body Kinematics

Book Preface

Origin and Scope

This textbook, together with Rigid Body Dynamics (Cambridge University Press, forthcoming), is the result of a whole professional life devoted to Newtonian Mechan-ics, both from an educational and a research point of view. Through more than 40 years, our way of teaching this subject to second-year undergraduate students (UG) in Indus-trial Engineering at the Universitat Politècnica de Catalunya (UPC) has evolved pro-gressively. To find the right words and examples to make every concept clear, to discover the misconceptions that lead to the most frequent errors, and to kindle the students’ curiosity for this discipline have been our major concerns throughout all these years.

We have found our personal point of view to present this classical subject, and have generated hundreds of exercises, questionnaires, and interesting case studies. Sharing this material with others involved in the higher education field seems nearly an obligation!

Our regular course on Newtonian Mechanics at UPC spans over 15 weeks and contains three main parts: Kinematics, Dynamics, and Energy. Students enrolling in this course already have a background in fundamental physics (mainly focused on 2D cases) and mathematics, and they will not be studying any other course on general mechanics (though they may follow some particular applications as “Machine and mechanism theory” and “Vibrations”). This context conditions the syllabus of our course: it has to be complete and provide the main tools to face any problem that may be encountered in Mechanical Engineering.

A percentage of our regular students are really enthusiastic about this discipline. For them we propose an additional and optional 15-week course where some concepts presented in the regular course are revisited and enlarged, and some well-known methods to perform dynamical analysis of mechanical systems are introduced. The course includes a final section devoted to an interesting problem: percussive dynamics.

Our goal when writing the book was not just to cover the syllabus of both courses but also to provide answers to questions raised by the best students (often out of scope of the course). The result has been a long text that includes more material than is likely to be covered in formal lectures to second-year undergraduates (including both the regular and the optional courses), and thus may serve as a reference also for graduate students, professional engineers working in the industry, and even researchers in robotics and biomechanics.

Though the main target are the undergraduate students, enlarging the potential audience was the main reason for splitting it into two volumes (Rigid Body Kinematics and Rigid Body Dynamics). We are aware that some people working in the fields of robotics and biomechanics and game developers are mainly interested in the kinematical description of mechanical systems. Thus, devoting a first volume dealing exclusively with kinematics seemed appropriate.

The contents of the UG standard course on Mechanics is not the same in the United States and in Europe. In the United States, this text may be better suited for intermediate (and even master) courses rather than for UG courses. Conversely, classical US textbooks may correspond to first-year UG students in Europe but may fall below the level we seek in second-year students.

Main Features

This textbook presents the general problem of rigid body kinematics in a very rigorous and precise way, and with a particularly suitable approach for engineering students.
Rigor is not at odds with the very visual and geometric approach we pursue throughout the book. We have included hundreds of illustrations which emphasize the geometrical properties of rigid body kinematics and help understand the concepts visually.
The mathematical notation throughout the book is highly explicit not only to avoid any ambiguity but also to allow an automatic and straightforward interpretation.

Every chapter starts with a short introduction that provides an overview of its content. Then, the different concepts and methods are introduced. The proofs are clearly isolated from the main text so that the reader may skip them. Every single concept is illustrated through figures (expressly designed for this work) and many fully worked-out examples (nothing is left to the reader as a further exercise!). Some of them are strictly peda-gogical exercises, but many others correspond to simplified models of usual mechanical systems.

Advanced topics are presented in appendices, and they may be overlooked without impairing the understanding of the following chapters.
The book includes an extensive collection of multiple-choice quiz questions (with answers) and exercises (with final results). The quiz questions do not require long calculations, but a clear understanding of the fundamental concepts. Each of them is illustrated through a figure showing the mechanical system under study plus the precisions an engineer would add to complete the description (as dimensions and motion).

Thus, it is not strictly necessary to read the text as the figures are self-explanatory.
The exercises are pencil-and-paper problems to be solved symbolically: symbolic results allow to detect possible errors (undetectable in numerical solutions) and give the whole picture of the system behavior (showing which parameters may be responsible for a bifurcation or transition from one regime to another). The student may implement the results in a computer, but that is not the main purpose of the exercises. Computer simulations may be time-consuming, and the required skills (programming, numerical calculation, etc.) do not fall within the objectives of our textbook.
The results for the exercises are given at the end of the chapters, but we do not provide the detailed resolution. Thus, the students are obliged to work on their approach and find by themselves the mistakes leading to wrong results: you have to do kinematics to learn kinematics!

A few puzzles complete the application material given at the end of the chapters. They confront the student with formally simple systems and situations found in everyday life. Justifying their design and/or understanding their kinematic behavior may be far from intuitive and calls for a rigorous application of the concepts introduced in the chapter. A complete explanation is given for each puzzle.