To my wife, Yi-Ling, and my family PrefaceMultibody dynamics analysis was originally developed as a tool for modeling rigid multibody systems with simple tree-like topologies, but has considerably evolved to the point where it can handle linearly and nonlinearly elastic multibody systems with arbitrary topologies. It is now used widely as a fundamental design tool in many areas of engineering.This textbook has emerged over the past two decades from efforts to teach graduate courses in advanced dynamics and exible multibody dynamics to engineering students. Although this book reviews the basic principles of dynamics, it is assumed that students enrolling in these graduate courses have completed a comprehensive set of undergraduate courses in statics, dynamics, deformable bodies, energy methods, and numerical analysis. The advanced dynamics course is, of course, a prerequisite for the exible multibody dynamics course.The book is divided into six parts. The rst part presents the basic tools and concepts that form the foundation for the other parts. It begins with a review of basic operations on vectors and tensors. The second chapter deals with coordinate systems. The differential geometry of both curves and surfaces is presented and leads to path and surface coordinates. Chapter 3 reviews the basic principles of dynamics, starting with Newton's laws. The important concept of conservative forces is discussed. Systems of particles are then treated, leading to Euler's rst and second laws.Chapter 4 concludes the rst part of the book with a detailed description of threedimensional rotation. For most graduate students, this chapter is not really a review. Indeed, many undergraduate dynamics courses focus primarily on two-dimensional systems. Problems involving three-dimensional rotation, if treated at all, are often rushed in the last few weeks of the semester, leaving most students with insuf cient time to absorb this dif cult material.Part 2 develops rigid body dynamics, the foundation of multibody dynamics. The analysis of the kinematics of rigid bodies is the focus of chapter 5. It starts with the analysis of the general displacement and velocity elds of a rigid body. The classical topics of relative velocities and accelerations are also addressed. The motion tensor and its properties are given an in-depth treatment.

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A hallmark of multibody dynamics is that most formulations involve a number of constraints. Typically, when redundant generalized coordinates are used, equations of motion are simpler to derive but constraint equations are present. While the dynamic behavior of constrained systems is well understood, the numerical solution of the resulting equations, potentially of differential-algebraic nature, remains problematic. Many different approaches have been proposed over the years, all presenting advantages and drawbacks: The sheer number and variety of methods that have been proposed indicate the difficulty of the problem. A cursory survey of the literature reveals that the various methods fall within broad categories sharing common theoretical foundations. This paper summarizes the theoretical foundations to the enforcement in constraints in multibody dynamics problems. Next, methods based on the use of Lagrange’s equation of the first kind, which are index-3 differential-algebraic equations in the presence of holonomic constraints, are reviewed. Methods leading to a minimum set of equations are discussed; in view of the numerical difficulties associated with index-3 approaches, reduction to a minimum set is often performed, leading to a number of practical algorithms using methods developed for ordinary differential equations. The goal of this paper is to review the features of these methods, assess their accuracy and efficiency, underline the relationship among the methods, and recommend approaches that seem to perform better than others.

This paper presents a vectorial parameterization of motion that generalizes the vectorial parameterization of rotation. The Pl¨ucker coordinates of an arbitrary material line of a rigid body subjected to a screw motion are shown to transform by the action of a motion tensor. The proposed vectorial parameterization completely describes an arbitrary motion by means of two vectors that constitute an eigenvector of the motion tensor associated with its positive unit eigenvalue. The first vector is conveniently selected as the vectorial parameterization of rotation, and the second is related to the displacement of a point of the rigid body. A complete description of motion is presented in terms of a generic vectorial parameterization. Relevant formulæ for specific parameterizations of this class can then be easily obtained. More details are given for three parameterizations that present desirable properties: the Euler, Cayley-Gibbs-Rodrigues, and Wiener-Milenkovic motion parameters.

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